## Chamok hasan

Chamok Hasan is a Bangladeshi writer and mathematician. He and Hasibul Ahsan were in same department in buet. They are known to each other for long time.

Chamok Hasan has a good skill of teaching mathematics.

## Chamok Hasan calculus

Limit, after all, is such a concept which was developed using logic. The defintion of limit was rooted in the idea of converging quantities:
If a quantity approach another quantity making the difference between them less than any arbitrary small number in some interval of time then they become ultimately equal at the end.
This was Newton's idea of converging numbers. But he was never able to base calculus on perfect definition of limit. He knew calculus worked but never questioned about its validity. Later a mathematician named Cauchy defined limit mathematically. Limit is the value which some function approaches but never reaches it. For more explanation follow this page.
Mathematics follow logic. It took a long time to establish connection between mathematics and logic. The most notable endeavour to find such connection was undertaken by Bertrand Russell and Dr. Alfred North Whitehead. Unlike we arevery mistaken, all mathematics are logical deduction from a very few logical premises. Pure mathematics has shown that ten logical premises and almost twenty principles of deductions are enough to define whole of mathematics from algebra to geometry. This is how it goes.
Pure mathematics contains no constant except logical contant and consequently no premises, or indemonstrable propositions, but such as are concerned exclusively with logical constants and variables . It is precisely this that distinguishes pure mathematics from applied mathematics. In applied mathematics, results which have been shown by pure mathematics to follow from some constant satisfying the hypothesis in question. Thus terms which were variables become constants, and a new presmise is always required, namely: this particular entity satisfies the hypothesis in question. Thus for example Euclidean Geometry , as a branch of pure mathematics, consists wholly of propositions having hypothesis "S is a Euclidean space" . If we go on to : "The space that exists is Euclidean" this enables us to assert of the space that exists the consequents of all the hypothetical constituting Euclidean Geometry, where now the variables S is replaced by the constant "actual space". But by this step we pass from pure to applied mathematics.
Implication of pure mathematics can be put in a truth table like this :

The gemoetrical interpretation of calculus can be put like this:

The derivative of a function at a point x is the slope of the tangent drawn on that point. So tangent is actually defined through differentiation. It always finds the rate of change of some function f(x) at specific value of independent variable x. For more explanation vist this page.

## Integration

Integration is the inverse process of differentiation. It is a process of summing infinite number of quantities to find the area under a curve. We can define integration in the following way:

If we diffentiate the integral then we get the original function back. This is called the fundamental theorem of calculus.
Some more integration formulas are given :

We sometimes integrate an argument by substitution.

## Functional derivative

Functional is a function of a function . We can define a rule for differentiating functional like the ordinary differentiation of function.

A more detailed explanation involves variational method

A formal representation would be like as follows:

An example of a functional derivative can be given here.

We treat the function as if it were an independent variable.

## Arithmetic

Arithmetic is a mathematical system which is concerned with addition , subtraction , multiplication and division of numbers. Artithmetic must satisfy these properties concerning numbers

## Fuzzy Logic

In usual logic like boolean logic the truth value of logical statemenet can only be either zero or one. It is equivalent to switching on or switching off. In fuzzy logic the truth value can take any value ranging from 0 to 1. That is to say it can be .4 or .5. Google search algorith uses fuzzy logic everytime to return the most relevant results of the searched keywords. The set of truth values can be anything in between 1 and 0 .

## Fibonacci Sequence and Golden ration

Fibonacci sequence is the sequence where any number in it is the sum of previous two consecutive numbers just before it.

Two quantities are in a golden ratio if it is the same as the ratio of their sum to the larger of the two quantities.

## Munchausen Number

It is one of some weird numbers , which has special property:

## Series and sequence

There are two kinds of series and sequences : one is arithmetic series and the other is geometric series. In arithmetic series there is a constant difference between two consecutive terms and in geometric series there is a constant ratio between two consecutive terms. There are special rules for summing series. These are as follows :

There is another type of series named harmonic series. If you take reciprocal of every term of an arithmetic progression you get a harmonic series. It is a divergent infinite series. Every term after the fast term is the harmonic mean of the neighboring terms.
The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations.
An example of harmonic series is :

There is a simple proof that harmonic series diverges. The proof goes like this :

Musical interval and notes are related to the terms of harmonic series:

The first note leveled at one(1) is the double of second note 1/2, which is called an octave. An octave increases the frequency of a note to the double of that. The difference of the note 3 and 4 is called perfect fourth. It is a difference of frequency (1/3 - 1/4) = 1/12 of the frequency of first note ( it may be 350 hz). Similary various intervals comprise of various musical intervals. All have specific names. Music is nothing but precision of harmony.

## Continued fraction

Continued fraction is a special type of fraction where the denominator repeats over itself. It looks something like this:

## Logarithm

Logarithm was invented by Napier. We all know about Napier's scale. It is a scale which turns multiplication into summation. How is that possible? well it is based on rules of logarithm. Lets first define logarithm. If a to the power x is equal to y than we define log (a) y = x . Usually we use 10 base logarithm. That is to say if 10 to power x = y then log(10) y = x .
If we muliply to logarithm log(y) and log (z) we get log(y) X log (z) = a + b where 10^z = b and 10^y = a . New scale will read a+b . This is easy to verify.
Now suppose there is two scale which reads logarithmic value of 10^a and 10^b respectively corresponding to values of a and b. Now if we multiply the two scales we get a+b as a new scale. Thus logarithm can help us convert multiplication into addition. This was John Napier's invention which is still very useful. What is scale? scale is the ratio of one quanity to another. The time scale according to relativity is different at different location in space around the whole universe. This is although a topic to be discussed in Theory of relativity.
Logarithm of some variable can be expanded as power series. Here are some examples:

## Algebra

Algebra is perhaps the most important tool in mathematics. We can not solve any equation without the rules of algebra. But what is algebra?
Algebra deals with variables which obeys the rules of arithmetic. There is a basic difference between algebra and arithmetic. Algebra is exclusively concerned with variable which can take many numberic value while arithmetic is concerned with constants and numbers. And of course there are rules of subtraction, multiplication, addition and division which are defined in a certain way.
A very basic equation of algebra can be mentioned :

So an equation involves one or more variables and for some value or values of the variable satisfy the equation. Other meaning of equation can be that it expresses an equality. Now let us go back to set theory.
Solving exponential equation is somewhat different. Exponential equation contains exponent of some variables. This is the way exponential equation is solved:

You can watch this video of Chamok Hasan on algebra for elementary ideas:

My video on pure mathematics

Algebra , by dint of equations, helps us solve many real world problems. We can turn many quantitative problems into algebra and solve the equations for understanding. For example, if you are given that the interest rate of your bank account is compounded interest and you deposited certain amount of money. How much money will you get after certain amount of time? Compound interest means that your amount of interest will change over time based on your current holding, which includes all the previous accumulated interest based on your deposit.

This is the kind of problem algebra deals with exclusively. There are many harder problems. All problems are just converting the variables into numbers and solve for unknown. The first successful marriage between algebra and geometry was done by Rene Descartes. Descartes named it coordinate geometry.
Now we come to the notion of constant. Variable is some term in mathematical statement, that does not change it value. But what does it mean to not change its value? we can consider a class of straight lines given by the equation y = ax + b . As long as a and b are fixed we only get only one staight line. But for different values of a and b we have different straight lines on a plane. Are a and b necessary a constant? It is as long as we consider one straight line but they become varaibles when the above equation represents a class of straight lines. Similarly value of π is constant as long as we consider all the circles in Euclidean space. It will be different in different curved spaces.

## Vector algebra

Vector algebra is a very useful branch of mathematics which deals with the vectors and its properties. It is useful in physics also. Vector is a quantity which has both a direction and a magnitude. Some elementary vector algebra equations are given below :

Cross product of two vectors are defined in the following way:

It is a vector which also has a magnitude and a direction.
Linear algebra is the branch of mathematics , which deals with vectors and matrices. It is very useful in both mathematics and theoretical physics.
A set of vectors is said to be linearly independent if the following conditions hold.

## Eigen Value

Eigenvalue is the which , when muyltiplied by some quantity equals some operator acting on the quantity. For example , some operator T acting on a vector v produces a scaled version of v. Mathematically it is
T (v) = λv
In this case , v is called eigenvector. Similar condition can be apllied to functions as well.

Where I is the identity matrix.

A block matrix is a matrix which has other matrices as its elements.

## Matrix function

Matrix function is defined using Taylor series. Here the independent variable is matrix. It is a function which takes a matrix and outputs another matrix.
Tylor polynomial is can be written as :

If we replace matrix A for x then addition becomes matrix sums and multiplication becomes scaling operations. This way we can find exponential of any function.
If the input matrix is diagonalizable : That is to say, A = PD(1/P) here 1/P is actually inverse matrix of P, then f(A) is defined by

## Cayley Hamiltonian theorem

Cayley Hamiltonian theorem is a proposition of matrix algebra which says the a square matrix satisfies its own characterstics equation. The statement can be stated as below:

Statements of algebra have precise interpretations as they conveys truth about the consequences of some hypotheses of pure mathematics. For example we all know 1+1 = 2 bears a true fact. But what does this mean? It is a mathematical statement which is always true. Bertrand Russell and Whitehead proved this statement using a long argument.

Let us interpret this statement in context of pure mathematics. In context of pure mathematics it states that " if both x is one and y is one and x is not y then x and y are two". It also means " if a is not b then whatever x may be , x is an a is always equivalent to x is b or x is γ ". Now we have no numbers but only variables and propositions.
Some basic algebra equations are related to bionomial theorem :

As I mentioned bionomial theorem it is better to give its formula explicitly

Equations or formulas form the backbone of algebra and consequently of all the mathematics. Without formula we could not solve anything and compare one thing with other things. Bionomial theorem needs the concept of permutation and combination.
Various components or parts of an equation can be identified like this :

Permutation is the different arrangement of specific number of things, that we can make out of some given number of things. Suppose we have three letters A, B and C. If we are to ask how many ways we can arrange all those letters or some of those then we need the formula of permutation as follows:

In permutating three letters A, B and C we come up with six different arrangements. That is the result obtained from 3! = 6 but if we want to make a combination of A, B and C we get only one such collection of A, B and C. So to get the result of combination we divide the result by the of permutation by the factorial of the number of things we want to combine. The order of permutation in the combination does not matter. That is the basic difference between permutation and combination and that is the reason why r! is included in the denominator of the combination formula.

## Laws of thought

Goerge Boole first developed a system for logical treatment of any number of propositions having truth values. These truth values are normally zero or one. Modern computers have been developed using this system that George Boole invented. It is best known as boolean algebra. Boole named it laws of thought. Laws of thought obeys these three rules of logic :

## Linear Algebra

Linear algebra is the branch of mathematics , which deals with the linear equations. We can use matrices to solve a system of linear equations. This is called Cramer's rule :

## Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied. A linear Diophantine equation equates the sum of two or more monomials, each of degree 1 in one of the variables, to a constant.

## PEDMAS

The rule of simplifying algebra problems is a special method named PEDMAS. It means P = parenthesis, E = exponential, D = division, M = multiplication, A = addition, S = summation. That is say, we always maintain this order in simplifying algebraic expression. Take a look at this example:

## Number theory

Mathematics is the queen of science whereas number theory is the queen of mathematics. Number theory is concerned with properties and relationship of numbers , expecially the positive integers. Euclid first developed an algorithm to compute greatest common divisor of two integers. This was the first application of modern number theory. The algorithm can be stated as follows:
suppose we want to calculate GCD of two integer 64, 12 . We fist check the remainder of 64 and 12 when dividing the fist number by the second number.
64/12 => 4 as 5X12 = 60
Now divide 12 by 4 to check whether there is any remainder. We get a remainder of 0 when doing that. So the greatest common divisor of 64 and 12 is 4. When we get 0 as a remainder we finish our division. We can check that by finding the gcd of 64 and 12 in usual way.
Factors of 64 are 4X8X2 = 2X2X2X2X2X2 and factors of 12 are 2X2X3 . So the factors that are in common in 64 and 12 are 2, 2 which means 2X2 =4 is the greatest common divisor of 64 and 12.

## Chineese Remainder theorem

The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime.
If you are given certain number of things and you are told them that when you count them by seven six things left over, if you count them by five four things left over and if you count them by three two things are left over. What is the actual number things? This kind of problem needs to be solved using the chineese remainder theorem.
Statement algorithm of chinese remainder theorem:

## Linear equation

A linear equation is the equation where the dependent variable changes with the independent variable linearly. That is to say, if we plot the graph of such equation, we get a straight line. General forms of linear equation are as follows:

## Absolute Value

Absolute value of a number x is defined as |x| = x if x is positive and |x| = -x if x is negative . If x is a real number then |x| <= x

## Complex number

Complex number is written in the form z = a + ib where a, b are real numbers and i = square root(-1 ). There are some identities and inequalitites related to modulus of complex number. These are as follows :

Where modulus of the complex number is defined as :

Complex analysis is perhaps one of the most intriguing topics in mathematics. It has a large class of applications in subjects like electricity , quantum electrodynamics and quantum field theory.

## Euler Identity

Euler Identity can be proved in the following way:

## Set theory

Famous mathematician Goerge Cantor first gave the concept of set. Set is any collection of individuals. It is a class in some sense. For example class of people of Bangladesh can be regarded as set. Class of addresses in Dhaka city can be considered a set. It is not necessary that set must contain real objects. Set can contain abstract elements. I remember now Russell's paradox. Russell found that some class contains itself. For example class of all abstact concepts is also a concept. But most of the classes are not members of themelves. Like the class of all bicycles is not bicycle itself. This also applies to sets. Then Russell formed a class of all the classes which does not contains themselves. He thought also if this class contains itself or not. If this class contains itself then by definition it follows that it does not contain itself. Again if it does not contain itself then it contains itself. So we come to a contradictory situation. Russell spend a long time to resolve the paradox and at last he proposed the theory of type which he later abandoned as a failure.
Set is too useful concept in mathematics. It has turned out to be eventually. We can expresses numbers in terms of sets , particularly empty sets. Empty set is a set which has no element. Now consider an set which has an empty set as its element. Like this : φ = {} , A = {φ , φ }
Continue this up to sets with three and four elements and five elements..
B = {φ, φ, φ) , C = {φ, φ, φ, φ} and D = {φ, φ, φ, φ , φ} and so on .. We can thus express numbers without pointing anything that is physical. This is a simple example of how much powerful the concept of set can be.
Another example of empty set can be found in the definition of ordinal numbers. Ordinal numbers can be defined as class of all well-ordered sets. For example ordinal 6 will be class of all sets which contains 1,2,3,4,5,6 . But instead of defining ordinal as class of well ordered sets , it can be defined as a particular well-ordered-set which cannonically represents the class. For example, we can set:
0 = {} = φ
1 = {0} = { φ}
2 = {0,1} = {φ, {φ} }
3 = {0,1,2} = {φ, {φ}, {φ {φ}} }
and so on. Each set will represent a specific ordinal number. Ordinal arithematic is much harder and at the same time interesting. Cardinal arithematic is easier. These two kind of numbers namely ordinal and cardinal were defined by Cantor for the first time.

## Geometry

Geometry is perhaps the most important branch of mathematics. It is also the highest exercise of human brain. We all know Euclid's book named "elements". In elements Euclid has proved lots of propositions and theorems of his geometrical system known as Euclidean geometry. But what is this geometry actually. Euclid defined some terms and gave few axioms based on which he developed his geometrical system. His axioms were self-evident truth. One of his axioms is like "from one point to another only one straight line can be drawn". Bertrand Russell was once being taught a course on Euclidean geometry by his tutor. His tutor told him to accept the axioms as true without any objection. But Russell refused to accept those at first. His tutor retorted if you cannot accept those we cannot go on. So Russell had no other way than to accept those as true and go on. In logic we do not accept such method. When mathematics is blended with logic , no such self -evidence truth is aceptable. What we can only assert is implication. P implies Q means either P is true or Q is false. P and Q are here propositions. So in mathematics we do not actually know what we are talking about nor that it is actually true. It only assures us that if such and such thing are true of something then such and such other thing are also true of the same thing. Yet mathematics can be very powerful scientific branch to search for the secrets of the nature. By the way we now get back to Euclid's elements.
Euclid's "elements" has thirteen books. Book 1 contain five postulates and 5 notions. Book 2 contains number of lemmas (a subsidiary or intermediate theorem in an argument or proof) concerning rectangles and squares. Euclid proved a whole bunch of propositions and theorems which are now parts of elementary geometry books.
What is the definition of parallel line according to Euclid's ideas?
Given a straight line , there are two classes of lines. All the lines corresponding to one class cut the straight line and lines belonging to the other do not. There is a limiting class between these two classes. This is called the class of parallel lines with respect to that given line. In Euclid's geometry given a point and a line lying not on the point there is only one straight line parallel to it. And parallel lines never meet with each other even at infinity. But in non-Euclidean system there are many lines through a given point , which are all parallel to a given line. Non-Euclidean geometry is counter-intuitive although it only asserts implications like in Euclidean geometry. We can not visualize curved space but it exists. Einstein showed us that.
There are other geometries like affine geometry, metrical geometry, descriptive geometry , projective geometry and others. In metrical geometry only spaces equipped with a metric is dealt with. Metric allows us to make distance measurement. What do we mean by distance in mathematics? Distance is a relation between points. In two dimensions distance is a function of four coordinates (x1, x2, x3, x4) which we can deduce by applying Pythagoras Law. Distance only asserts quantitative property. It is not necessary that distance is what we mean by 5m(five metres). It can be 5s (give seconds) too.

## Affine space

Affine space is a geometric structure that generalizes some of the properties of Euclidean space in such a way that is independent of the distance and angle measurement, keeping only the properties related to parallelism and ration of lengths of parallel line segments. In affine space there is no distinguished point that serves as the origin. It a vector space with origin removed. This let us give the concept of affine function. It is a function the form f(x) = ax + b where a and b are constants ;

## Polynomial equation

A polynomial equation is an equation containing linear combination of various powers of an unknown variable x as follows:

Polynomian equation relates fundamental theorem of algebra. It says that a polynomial equation of nth degree having a complex coefficients has at least one solutions. This includes polynomials with real coefficients too since real coffiecients are the complex number having imaginary part equal to zero.

## Probability

Probability is the ratio of two numbers. It is the ratio of the number of way a specific event can happen and the total number of events that can happen. The exact expression would be :

## Conditional Probability

So the general mathematical formula would turn out to be :

Byesian model
In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person’s age.
Mathematically it can be forulated with the following expression:

This is almost similar to conditional probability theorem but can be used in more complicated cases.

## "Know the formula and solve the problem"

Any quadratic equation can be solved using a closed form formula. This is a well known formula which any school student know more or less. This is as follows:

## Projective geometry

Projective geometry comes from , what the name suggests , projection. It deals with all the properties of geometrical figures that are unaltered by continuous projections. It is not concerned with the quantitative aspects but rather qualitative aspects of lines and points. Projective coordinates are assigned in projective space.

In the figure above, an example of projection from a point is illustrated. Point A, B, C, D are projected into points A`, B`, C` and D`. In this projection certain quantity named anharmonic ratio remains constant. It is the ratio of different line segments formed by the points A, B, C and D.

There is a concept in projective geometry , called duality. The principle of duality says that two distinct points determine a unique line (i.e. the line through them) and two distinct lines determine a unique point (i.e. their point of intersection). This is a proposition of projective geometry. This is how it is so :

## Trigonometry

Trigonometry is a branch of mathematics which deals with triangles and its angles. First it is better to define angle. What is angle? Angle is the relation between two straight lines. That is to say, when two straight lines meet they create an angle between them. The common point they share is called the vertex. It is better to replace straight lines with rays. Rays are any line extending from a point to infinity. The plane where the angle is created does not need not to be Euclidean plane. When two rays are perpendicular they create a right angle between them.
Solid angle is defined using a sphere in 3D space. A total of 4π steredian is subtended from the center of the sphere. A sphere has a total of 4πr^2 area. So there is 1/r(square) steredian per unit area of a sphere.
These are the basics of trigonometry. In addition we divide the plane with four quadrants at the origin(0,0). The main ingredient of trigonometry is the triangle.

Cosine is the ratio of adjacent to hypotenuse of a tright triangle. The side which makes the angle with the hypotenuse is the angle in consideration. The opposite side of the right angle is called hypotenuse.
Sine is the ratio of hypotenuse to the opposite or perpendicular.

## Asymptote

Asymtote of a curve is the line such that the distance between the curve and the line approaches zero as one or more of the coordinates x, y tend to infinity.

As an example an equation of asymtote can be given :

We can see that the distance between the line y= x+1 and the curve become zero as the corrdinate x tends to zero in both side.

## Functions

Function is the most important concept in Mathematics. A function is a relation or mapping between two sets. In mathematics , we are always concerned with numeric functions. But in principle, functions other than numeric are also possible. A function is defined from one set which is called domain to another set which is called co-domain. So a function maps elements from doamin to co-domain. Range is the value b = f(a) which is an element of co-domain. Not every element of co-domain is the element of range.
Suppose a function f : inhabitants -> R+ finds the age of inhabitants of a country. The age is a positive real number. If there are 30 million people in america we can find the age of all those inhabitants according to this function. Here all the ages of the inhabitants of america form the image of the function. Image is sometimes called the range of the function too. One-one function is a function which map exactly one element of domain to exactly one element of co-domain. That is , if x1, and x2 is two elements of domain then f(x1) != f(x2). One-one function is also called injective function too.
f(x) is called surjective if the co-domain equals the image. That is to say, f(x) equals Y (co-domain). In other words, pre-image 1/f(y) is not empty. Every element of the co-domain is mapped to some element of the domain.
A function is called bijective of it is both injective and surjective. Function f(x) is bijective if the pre-image 1/f(y) contains exactly one element. It is also called a one-to-one correspondence.

## Generating function

Generating function is a way of encoding infinite sequence of numbers( a[n]) by treating them as coefficients of a power series. This formal power series is the generating function.
A generating function is like a bag. Instead of carrying many objects detachedly we put them into one bag. Then we would have only one object to carry. Some example of generating functions are :

## Math formula cheat sheet

All the major equations can be put into a package for better correlation between various parts.

## Statistics

Statistic is the method of collecting a vast number of data and do various kinds of calculations on them. Most useful terms of statistics are median, mode and mean. Here are some useful equations of statistics.

## Histogram

Histogram is an accurate distribution of data in numerical form. It provides us with the probability distribution of a continuous variable. Suppose you have some data about the height of a population. Histogram will tell you how much height is possesed by how many people. To construct a histogram, the first step is to "bin" (or "bucket") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval.

## Prime polynomial

Some polynomials can not be factored. This kind of polynomials are called prime polynomials. An example is given below:

## Linear Transformation

Linear transformation T(A) = B where A and B are vectors. T is some opertor or matrix if these two conditions hold:
1. T(a + b) = T(a) + T(b).
2. T(c.A) = cT(A)
Here is an example of a linear transformation :

## Big O notation

We always encounter this big O notation in mathematics. This notation was introduced by the brilliant mathematician Landau, and has been found to be very useful in analysis and in number theory. Let g(n) be a given function of n defined for all n=> a where a is some positive integer. If there exist a simple monotonic positive function h(n) defined for all n =>a and a constant k independent of nn such that
|g(n)| <= kh(n) for all n => a then we write
g(n) = O(h(n)).
Note that k may be any real positive number but in what follows we shall always assume k to be a positive integer. h(n) is usually a simple function such as
n, logn, nlogn, 1 , ..

## The logistic model of population growth

Estimates the change in a population of creatures across generations with limited resources. The equation of logistic model is

## Chaos Theory

“A butterfly flaps its wings, and it starts to rain,” the narrator of one episode of How I Met Your Mother begins begins, “It’s a scary thought but it’s also kind of wonderful.” He continues:
“All these little parts of the machine constantly working, making sure that you end up exactly where you’re supposed to be, exactly when you’re supposed to be there. The right place at the right time.”
It’s depicted differently in each one, but you see a popularized version of the Chaos Theory everywhere in movies and TV shows – a butterfly flaps its wing, and the course of history is changed forever. The theory isn’t as crazy as you would think.
Traditionally, scientists believed that all natural processes were either deterministic or nondeterministic – implyingg we can either predict their behavior, or not at all. Throwing a ball is deterministic, because if you throw a ball at exactly the same angle and speed, you can predict how far it goes. Uranium decay is nondeterministic, because it’s impossible to predict which exact atom will shrivel at a given time.
Mathematicians later discovered that those two categories aren’t enough (are they ever satisfied?) Some processes seem predictable, but small changes would bring catastrophic consequences. For example, consider a double pendulum swinging wildly. The locations of the two joints seem easy to calculate, but can vary wildly with the initial acceleration and position.

Another example of chaos is the Lorentz attractor. It is a solution of three body problem. The solution tends to converge to two sets as depicted below:

This is better known as the butterfly effect. There is a famous movie named "butterfly effect" which may or may not be related to this concept of chaos. But the movie is awesome and thrilling. The chaotic system is very sensitive to initial conditions.

## Cantor's Theorem

2^|S| > |s| where |S| is the cardinality of the set S. If S is a finite set then its cardinality is the number of elements in it, and things are not very interesting. But the concept of cardinality makes sense also for infinite sets. The power set of a set is the set of its subsets. It is easy to see that for finite sets S the cardinality of the power set equals 2^|S|. Thus we denote by 2^|S| the cardinality of the power set even for infinite sets S. Cantor's Theorem states that the cardinality of the power set of a set S always exceeds the cardinality of S itself. That's obvious for finite sets but far from trivial for infinite sets.

## Boltzman Entropy

A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by identifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate designates collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.

scientific method, and technological tools like computers and telescopes. With the help of these tools, scientists have pieced together a lot of knowledge about space. But what do we really know about the universe, and how do we know it? Where did the universe come from? Where is it going? Did the universe have a beginning, and if so, what happened before then? What is the nature of time? Will it ever come to an end? Can we go backward in time? Recent breakthroughs in physics, made possible in part by new technology, suggest answers to some of these long-standing questions. Someday these answers may seem as obvious to us as the earth orbiting the sun—or perhaps as ridiculous as a tower of turtles. Only time (whatever that may be) will tell. 2

## Perturbation Theory

Relatively few problems in quantum mechanics have exact solutions, and thus most problems require approximations. Perturbation theory is a useful method of approximation when a problem is very similar to one that has exact solutions. The approximate results differ from the exact ones by a small correction term. Perturbation theory fails when the correction terms are not small. Consider a set of eigenfunctions and eigenvalues of a given Hamiltonian operator: ! H ˆ (0) "n (0) = En (0) "n (0) (1) Here the label n identifies a specific solution in a set and the superscript (0) denotes the “order” of approximation. The set of functions ψ form an orthonormal basis. Zeroth-order approximation means exact. Now, if we consider a first-order correction such that the true Hamiltonian is ! H ˆ = H ˆ (0) + H ˆ (1) (2). We can expect the corrected wavefunction for a state n to be of the form ! "n = "n (0) + i#n \$ ci "i (0) (3), where the summation runs over all other states i in the basis set and ci are real coefficients in the linear expansion. This implies that the corrected wavefunctions are not normalized. It can be shown that (to first order) the mixing coefficients have the values ! ci (1) = "i (0) H ˆ (1) "n (0) En (0) # Ei (0) , i \$

### Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
perihelion of mercury by Feynman
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