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Topics - Masuma Parvin

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46
Interesting Maths / Fun Application of pigeonhole principle
« on: November 13, 2012, 03:48:15 PM »
In a group of six people, there will always be three people that are mutual friends or mutual strangers. (Assume that “friend” is symmetric-if x is a friend of y, then y is a friend of x.)

The problem can be thought of geometrically. Imagine the six people as points and let an edge between points indicate friendship. No matter how the graph is drawn, we want to show there is a set of three points that are all connected or a set of three points that has no connecting edges.

Consider any single point. There are five other points it could possibly connect to. By the pigeonhole principle, the point is either connected to at least three other points or not connected to at least three other points.

Case 1: the point is connected to (at least) three other points
If any of these points are connected to each other, then we have found a triangle of three mutual friends. (These two points are connected, plus they are both connected to the original point).

Otherwise, that means none of these three points are connected and hence they are mutual strangers. This would be a set of three points without any edges.

Case 2: the point is not connected to (at least) three other points
If any of these points are not connected to each other, then we have found a triangle of three mutual strangers. (These two points are not connected, plus they are both not connected to the original point).

Otherwise, that means all of these three points are connected and hence they are mutual friends. This would be a set of three points with all connecting edges.

47
Interesting Maths / Pigeonhole principle
« on: November 13, 2012, 03:46:20 PM »
In mathematics, the pigeonhole principle states that if n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. This theorem is exemplified in real-life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves". It is an example of a counting argument, and despite seeming intuitive it can be used to demonstrate possibly unexpected results; for example, that two people in London have the same number of hairs on their heads.
The pigeonhole principle is a powerful tool used in combinatorial math. But the idea is simple and can be explained by the following peculiar problem.

Imagine that 3 pigeons need to be placed into 2 pigeonholes. Can it be done? The answer is yes, but there is one catch. The catch is that no matter how the pigeons are placed, one of the pigeonholes must contain more than one pigeon.

The logic can be generalized for larger numbers. The pigeonhole principle states that if more than n pigeons are placed into n pigeonholes, some pigeonhole must contain more than one pigeon. While the principle is evident, its implications are astounding. The reason is that the principle proves the existence (or impossibility) of a particular phenomenon.

One Example Of this principle:
Two or more people reading this blog will have the same birthday.

There are 366 possible birthdays (including February 29 in a leap year) and this blog has many more than 367 readers. Therefore two of you must share the same birthday.

48
Interesting Maths / Tower of Hanoi
« on: November 13, 2012, 02:51:40 PM »
The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower,[1] and sometimes pluralised) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape.

The objective of the puzzle is to move the entire stack to another rod, obeying the following rules:

   1. Only one disk may be moved at a time.
   2.Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that   may already be present on that rod.
   3. No disk may be placed on top of a smaller disk.

Origins:
The puzzle was first publicized in the West by the French mathematician Édouard Lucas in 1883. There is a history about an Indian temple in Kashi Vihswanath which contains a large room with three time-worn posts in it surrounded by 64 golden disks. Brahmin priests, acting out the command of an ancient prophecy, have been moving these disks, in accordance with the immutable rules of the Brahma, since that time. The puzzle is therefore also known as the Tower of Brahma puzzle. According to the legend, when the last move of the puzzle will be completed, the world will end.[2] It is not clear whether Lucas invented this legend or was inspired by it.


Solution:

The following solution is a simple solution for the toy puzzle.
Alternate moves between the smallest piece and a non-smallest piece. When moving the smallest piece, always move it to the next position in the same direction (to the right if the starting number of pieces is even, to the left if the starting number of pieces is odd). If there is no tower position in the chosen direction, move the piece to the opposite end, but then continue to move in the correct direction. For example, if you started with three pieces, you would move the smallest piece to the opposite end, then continue in the left direction after that. When the turn is to move the non-smallest piece, there is only one legal move. Doing this will complete the puzzle using the fewest number of moves to do so.

Recursive solution

A key to solving this puzzle is to recognize that it can be solved by breaking the problem down into a collection of smaller problems and further breaking those problems down into even smaller problems until a solution is reached. The following procedure demonstrates this approach.

    label the pegs A, B, C—these labels may move at different steps
    let n be the total number of discs
    number the discs from 1 (smallest, topmost) to n (largest, bottommost)

To move n discs from peg A to peg C:

  1.  move n−1 discs from A to B. This leaves disc n alone on peg A
  2.move disc n from A to C
 3.   move n−1 discs from B to C so they sit on disc n

The above is a recursive algorithm: to carry out steps 1 and 3, apply the same algorithm again for n−1. The entire procedure is a finite number of steps, since at some point the algorithm will be required for n = 1. This step, moving a single disc from peg A to peg B, is trivial. This approach can be given a rigorous mathematical formalism with the theory of dynamic programming, and is often used as an example of recursion when teaching programming.


49
Applied Maths / Population Dynamics
« on: November 13, 2012, 02:41:20 PM »
Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. Population dynamics deals with the way populations are affected by birth and death rates, and by immigration and emigration, and studies topics such as ageing populations or population decline.

One common mathematical model for population dynamics is the exponential growth model. With the exponential model, the rate of change of any given population is proportional to the already existing population.
Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. The first principle of population dynamics is widely regarded as the exponential law of Malthus, as modelled by the Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic model.

A more general model formulation was proposed by F.J. Richards in 1959, further expanded by Simon Hopkins, in which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general formulation. The Lotka–Volterra predator-prey equations are another famous example. The computer game SimCity and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics.

In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been proposed and analysed, and provide important results that may be applied to health policy decisions.

Source: Online Resource.

50
Applied Maths / Influence of age structure on population growth
« on: November 13, 2012, 02:32:03 PM »
 Influence of age structure on population growth:
Even if the replacement-level fertility rate of 2.1 were magically achieved globally tomorrow, the world population would keep growing for at least another 50 years.

The reason for this is a population's age structure: the proportion of the population (or of each sex) at each age level.

Population Geographers typically construct  a population age structure diagram by plotting the percentages of numbers of males and females in the total population in each of three age categories:

        prereproductive (ages 0-14)

        reproductive (ages 15-44)

        postreproductive (ages 45 and up)


*** How Does Age Structure Affect Population Growth? ***

Any country with many people below 15 years old (represented by a wide based population structure diagram) has a powerful built-in momentum to increase its population size unless death rates rise sharply.

 

The number of births rises even if women have only one or two children because of the large number of girls who will be soon moving into their reproductive years.

 

In 2002, 30% of the world's people were under 15 years old.

 

These 1.9 billion people are poised to move into their reproductive years.

This powerful force for continued population growth, mostly in developing countries, could be slowed by:

        *an effective program to reduce birth rates

       * a sharp rise in death rates

     Population size has stabilized or is declining in Japan and most European countries.

 

However, population size is expected to double or even triple before stabilizing for many developing countries with fairly large populations and large numbers of people under age 15.

 

Example countries are:

        Nigeria

        Ethiopia

        Mexico

        Brazil

        Bangladesh

        Pakistan

For example, 44% of Nigeria's population is under the age of 15 and its TFR is 5.8 children per woman.


As a result, Nigeria's current population of about 130 million is projected to reach about 305 million by 2050.

 


51
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics. Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave–particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.

The term classical mechanics was coined in the early 20th century to describe the system of physics begun by Isaac Newton and many contemporary 17th century natural philosophers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, some sources exclude Einstein's theory of relativity from this category. However, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and most accurate form.

The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz, and others. This is further described in the following sections. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newton's work, particularly through their use of analytical mechanics. Ultimately, the mathematics developed for these were central to the creation of quantum mechanics.

52
Applied Maths / Fluid dynamics
« on: November 13, 2012, 02:06:30 PM »
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion. It has several sub disciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and reportedly modeling fission weapon detonation. Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.

Historically, hydrodynamics meant something different than it does today. Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability—both also applicable in, as well as being applied to, gases.
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum (also known as Newton's Second Law of Motion), and conservation of energy (also known as First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds Transport Theorem.

In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption considers fluids to be continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and velocity are taken to be well-defined at infinitesimally small points, and are assumed to vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.

For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier-Stokes equations, which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form.

53
Science Discussion Forum / Fibonacci Numbers and The Golden Number
« on: October 29, 2011, 06:17:22 PM »
If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13...) and we divide each by the number before it, we will find the following series of numbers:
1/1 = 1,   2/1 = 2,   3/2 = 1•5,   5/3 = 1•666...,   8/5 = 1•6,   13/8 = 1•625,   21/13 = 1•61538...

  The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1•618034
The golden ratio 1•618034 is also called the golden section or the golden mean or just the golden number. It is often represented by a Greek letter Phi . The closely related value which we write as phi with a small "p" is just the decimal part of Phi, namely 0•618034.

54
It is a common question which I have to face everyday,Why is mathematics taught at university not related to the engineering course we study?
Mathematics seems to be an abstract idea which does not overlap with our engineering course.
Shouldn't it be taught in an engineering context?
No, it shouldn't be taught in any particular context. Mathematics is a common derivative of all other forms of language, in that it precisely describes relationships between objects, whether they're abstract concepts or features of physical objects. Often the abstract and physical overlap and so you just never know what part of mathematics you'll need and so you have to study as much of it as you can. You can't get it all, of course, but at least you need to be able to find what you're going to need later in your professional life. The broader the base you can achieve now the better off you'll be later.

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