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# Quantum field theory

Path integral   |   S Matrix   |   Quantum electrodynamics

# Theory of relativity

Special theory of relativity   |   General theory of relativity

# Algebra for dummies free

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.
Algebra is the language that the field of mathematics uses to talk about abstract world numbers.

A typical algebraic equation contains variables and terms

one or more v values of the variable are usually solutions to algebraic equation. Algebra was first combined with geometry by Rene Descates to study coordinate geometry. This was a revolution in mathematics and science. Coordinates are levels for points in two or more dimensional space.

## Ramanujan

Ramanujan was an Indian born mathematician and algebraist. He developed many theorems of algebra. One of his discoveries is the formula of number three(3).

He was invited to Oxford from great mathematician Tomas Hardy. Ramanujan had no idea how he invented the equations. He claimed that he saw those eqations in his dreams. Hardy was very much reluctant to accept all his formulas because those had no proof. Ramanujan was a man of great mystery.

## Generators

Generator can have different meaning in different field of study. In mathematics it has a unique definition. Generators are used in group theory. It arises in Lie algebra or Lie group. First let us see what Lie group is.
A Lie group is a group which is also a differentiable manifold. The group operations are smooth.
The rotation matrices form a subgroup of GL(2, R). It is a Lie group under it own right. It is a one-dimensional compact connected Lie group which is diffeomorphic to a circle.

A set of generators is the set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements of the same group.

## Riemann Sphere

Riemman sphere is simply the complex one dimensional manifold with infinity ∞ added to it. Simply adjoining an extra point called '1' to the complex plane does not make it completely clear that the required seamless structure holds in the neighbourhood of 1, the same as everywhere else. The way that we can address this issue is to regard the sphere to be constructed from two 'coordinate patches', one of which is the z-plane and the other the w-plane. All but two points of the sphere are assigned both a z-coordinate and a w-coordinate (related by the Mo¨bius transformation above). But one point has only a z-coordinate (where w would be 'infinity') and another has only a w-coordinate (where z would be '∞'). We use either z or w or both in order to deWne the needed conformal structure and, where we use both, we get the same conformal structure using either, because the relation between the two coordinates is holomorphic.
In fact, for this, we do not need such a complicated transformation between z and w as the general Mo¨bius transformation. It suffices to consider the particularly simple Mo¨bius transformation given by
w = 1/z, z = 1/w ,
where z = 0 and w = 0, would each give 1 in the opposite patch. I have indicated in Figure below how this transformation maps the real and imaginary coordinate lines of z.

Patching the Riemann sphere from the complex z- and w-planes, via w = 1/z, z = 1/w. (Here, the z grid lines are shown also in the w-plane.) The overlap regions exclude only the origins, z = 0 and w = 0 each giving '1' in the opposite patch.
All this defines the Riemann sphere in a rather abstract way. We can see more clearly the reason that the Riemann sphere is called a ‘sphere’ by employing the geometry illustrated in Figure below. I have taken the z-plane to represent the equatorial plane of this geometrical sphere. The points of the sphere are mapped to the points of the plane by what is called stereographic projection from the south pole. This just means that I draw a straight line in the Euclidean 3-space (within which we imagine everything to be taking place) from the south pole through the point z in the plane. Where this line meets the sphere again is the point on the sphere that the complex number z represents. There is one additional point on the sphere, namely the south pole itself, and this represents z =1. To see how w fits into this picture, we imagine its complex plane to be inserted upside down (with w = 1, i, -1, -i matching z = 1, -i, -1, i, respectively), and we now project stereographically from the north pole (Figure b). An important and beautiful property of stereographic projection is that it maps circles on the sphere to circles (or straight lines) on the plane.

(a) Riemann sphere as unit sphere whose equator coincides with the unit circle in z's (horizontal) complex plane. The sphere is projected (stereographically) to the z-plane along straight lines through its south pole, which itself gives z=1. (b) Re-interpreting the equatorial plane as the w-plane, depicted upside down but with the same real axis, the stereographic projection is now from the north pole (w=1), where w = 1/z. (c) The real axis is a great circle on this Riemann sphere, like the unit circle but drawn vertically rather than horizontally.

## Plato's mathematical world

This was an extraordinary idea for its time, and it has turned out to be a very powerful one. But does the Platonic mathematical world actually exist, in any meaningful sense? Many people, including philosophers, might regard such a 'world' as a complete fiction—a product merely of our unrestrained imaginations. Yet the Platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the world of physical things. Moreover, it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the world—or, rather, of certain aspects of the world—and these models may be tested against previous observation and against the results of carefully designed experiment. The models are deemed to be appropriate if they survive such rigorous examination and if, in addition, they are internally consistent structures. The important point about these models, for our present discussion, is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers.

If the model itself is to be assigned any kind of 'existence', then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato's world to be in any sense absolute and 'real'. Yet, there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise, unreliable, and inconsistent in their judgements. The precision, reliability, and consistency that are required by our scientific theories demand something beyond any one of our individual (untrustworthy) minds.
It may be helpful if I put the case for the actual existence of the Platonic world in a different form. What I mean by this ‘existence’ is really just the objectivity of mathematical truth. Platonic existence, as I see it, refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture. Such ‘existence’ could also refer to things other than mathematics, such as to morality or aesthetics , but I am here concerned just with mathematical objectivity, which seems to be a much clearer issue. Let me illustrate this issue by considering one famous example of a mathematical truth, and relate it to the question of 'objectivity'. In 1637, Pierre de Fermat made his famous assertion now known as ‘Fermat’s Last Theorem’ (that no positive nth power3 of an integer, i.e. of a whole number, can be the sum of two other positive nth powers if n is an integer greater than 2), which he wrote down in the margin of his copy of the Arithmetica, a book written by the 3rd-century Greek mathematician Diophantos. In this margin, Fermat also noted: ‘I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.’ Fermat’s mathematical assertion remained unconfirmed for over 350 years, despite concerted efforts by numerous superb mathematicians. A proof was Wnally published in 1995 by Andrew files (depending on the earlier work of various other mathematicians), and this proof has now been accepted as a valid argument by the mathematical community.
Now, do we take the view that Fermat's assertion was always true, long before Fermat actually made it, or is its validity a purely cultural matter, dependent upon whatever might be the subjective standards of the community of human mathematicians? Let us try to suppose that the validity of the Fermat assertion is in fact a subjective matter. Then it would not be an absurdity for some other mathematician X to have come up with an actual and specific counter-example to the Fermat assertion, so long as X had done this before the date of 1995.4 In such a circumstance, the mathematical community would have to accept the correctness of X’s counter-example. From then on, any effort on the part of Wiles to prove the Fermat assertion would have to be futile, for the reason that X had got his argument in first and, as a result, the Fermat assertion would now be false! Moreover, we could ask the further question as to whether, consequent upon the correctness of X's forthcoming counter-example, Fermat himself would necessarily have been mistaken in believing in the soundness of his 'truly marvellous proof', at the time that he wrote his marginal note. On the subjective view of mathematical truth, it could possibly have been the case that Fermat had a valid proof (which would have been accepted as such by his peers at the time, had he revealed it) and that it was Fermat's secretiveness that allowed the possibility of X later obtaining a counter-example! I think that virtually all mathematicians, irrespective of their professed attitudes to 'Platonism', would regard such possibilities as patently absurd
Of course, it might still be the case that Wiles's argument in fact contains an error and that the Fermat assertion is indeed false. Or there could be a fundamental error in Wiles's argument but the Fermat assertion is true nevertheless. Or it might be that Wiles's argument is correct in its essentials while containing 'non-rigorous steps' that would not be up to the standard of some future rules of mathematical acceptability. But these issues do not address the point that I am getting at here. The issue is the objectivity of the Fermat assertion itself, not whether anyone's particular demonstration of it (or of its negation) might happen to be convincing to the mathematical community of any particular time.

Three 'worlds'— the Platonic mathematical, the physical, and the mental—and the three profound mysteries in the connections between them.

## Gauss-Bonnet theorem

Gauss-Bonnet theorem relates curvature of a topological space with the Euler characterstic. In any given curved space the angles of a figure do not sum up to that of Euclidean space. In that case there is an excess of those angles. This is usually expressed by an indentity as follows:

If M is a compact surface with a Riemannian metric, then Where K =Gauss curvature , X = the Euler characteristic of M and dA=the area measure on determined by the Riemannian metric.
For an triangle the value of X(M) = V -E + F = 2 . This is different for different Euelerian shapes and figures.

## Linear Programming

Linear programming a method to solve linear inequalities and euations to find the best optimal condition. It is a mathematical method to finf minimum or maximum of a linear function of several variables , such as output or cost.
Suppose a company produces two commodities of x and y. The total cost must come as x commodity and twice the total cost of y. So the total cost comes as C = x + 2y in a given time. This may serve as our objective function which extremum value we seek. There are other constraints as to how many commodities of x and y can be produced in the mean time simultaneously. So our mathematical argument is as follows:

The shadded area contains all the solutions of the objective function for which the cost is minimum or maximum.

## Binomial theorem

Bionomial theorem was developed by Issac Newton. It is a theorem used to describe the expansion of powers of bionomial. As an example some powers of binomial are :

This can be generalized as :

The coefficient of each term can be computed using pascal triangle :

First row corresponds to n=0; Second row corresponds to n=1 expansion and so on to for all finite value of n. In pascal triangle all the values of a row other than 1's at the end are computed by two numbers just above it in the upper row. Pascal first discovered such a triangle and that is why it is called Pascal's triangle.
Where we assumed first term a = 1 for simplicity. Now when n is any positive integer the series always terminates. But if n is not any positive integer the series does not terminate and expansion becomes infinite. This series can be expressed by another explicit formula of the form quite similar to above:

Some examples are

## Process of long division

There is a process of dividing a polynomial by another polynomial. This is called long division. This process is quite similar to the division rule of large numbers :

## Matrix algebra

Matrix algebra is an algebra which concerns with addition and multiplication of matrices. A matrix is a collection of numbers. Determinant of a 3X3 matrix can be calculated in terms of 2X2 matrix determinants.

Another application of matrix is found in solving a system of linear equations.

When we have a system of linear equations of several variables we need to find the determinant of the matrix of the coefficient of the variables.

In the example given above a system of two equations is solved by 2X2 matrix. Similar reasoning can be applied in case of more equations with more unknowns.

## Finding eigenvalue and eigenvectors

Eigenvalue and eigenvalue play an important role in quantum mechanics and linear algebra. Let us now find the eigenvector of a corresponding matrix A. The given matrix

The characteristic equation is det(A-λI ) = 0 from which we get

For eigenvectors the condition (A - λI )X = ) must be satisfied . X is the eigenvector. For the first case λ = 1 solving (A-I)X = 0 , by writing the augmented matrix a row-reducing

Thus, from back substitution , we see 0.x1 + x2 + x3 = 0 or x2 = -x3 with x3 free. Also from the first row x1 = 0 . So the eigenvector corresponding λ = 1 has the form v1 = x3

Similary for the other value of λ the other eigenvector can be found.
Matrix algebra is perhaps the most useful tool in theoretical physics. Tensors are special kind of matrices.
Lagrange intepolation formula is

## Trace and logarithm

Determinant of a 2X2 matrix is

The lograitm function of a matrix B is

Where I is the identity matrix .

## Jacobi formula

In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.
If A is a differentiable map from the real numbers to n × n matrices

By Jacobi formula for any matrix the trace identity hold.

Where adj A denotes the adjugate of matrix A. Adj of a matrix is defined as :

Magnitudes and quantities are different concepts. Magnitude is defined as whatever greater or less than something. Quantity is a particular contained in magnitude. Quantity is an instance of magnitude. What can be greater or less is surely a magnitude. Magnitude is a many-one relation whereas quantity is one-to-one relation. Many terms always belong to the same magnitude. As for example the 5 magnitude earthquake and 3 magnitude earthquake belong to the same kind of magnitude. Where quantity is one to one relation. 5 and 3 in this case are quanities or particular. This is the basic difference between qauntity and magnitude.

Mathematics is like poetry. You can only learn mathematics by doing mathematics. I have done enough mathematics in my life. There is no greater pleasure than doing or understanding mathematics. Besides mathematics is more beautiful than girls and does not cheat with us. Mathematics is probably the only language of saying the same thing in different words. Whatever let us get back to algebra for dummies :

To understand quadratic irrational we first need to learn what a continued fraction is.
A continued fraction is a finite or infinite expression a + (b + (c + (d + ... )^-11)^-1)^-1, where a, b, c, d, . . . are positive integers

Any rational number can be expressed as a terminating such expression.

To express a real number we allow the continued fraction to run on forever. Some examples are

Each number mentioned above is represented as a continued fraction. In the first two of these infinite examples, the sequences of natural numbers that appear—namely 1, 2, 2, 2, 2, . . . in the first case and 5, 3, 1, 2, 1, 2, 1, 2, . . . in the second—have the property that they are ultimately periodic (the 2 repeating indefinitely in the first case and the sequence 1, 2 repeating indefinitely in the second). Recall that, as already noted above, in the familiar decimal notation, it is the rational numbers that have (Wnite or) ultimately periodic expressions. We may regard it as a strength of the Greek 'continued-fraction' representation, on the other hand, that the rational numbers now always have a finite description. A natural question to ask, in this context, is: which numbers have an ultimately periodic continued-fraction representation? It is a remarkable theorem, first proved, to our knowledge, by the great 18thcentury mathematician Joseph C. Lagrange that the numbers whose representation in terms of continued fractions are ultimately periodic are what are called quadratic irrationals.
What is a quadratic irrational and what is its importance for Greek geometry? It is a number that can be written in the form a + √b , where a and b are fractions where b is not a perfect square. In case when a=0 some quadratic irrationals are √2, √3 , √5
Although incorporating the quadratic irrationals gets us some way towards numbers adequate for Euclidean geometry, it does not do all that is needed. In the tenth (and most diYcult) book of Euclid, numbers like √(a + √b);
are considered (with a and b positive rationals). These are not generally quadratic irrationals, but they occur, nevertheless, in ruler-and-compass constructions. Numbers sufficient for such geometric constructions would be those that can be built up from natural numbers by repeated use of the operations of addition, subtraction, multiplication, division, and the taking of square roots. But operating exclusively with such numbers gets extremely complicated, and these numbers are still too limited for considerations of Euclidean geometry that go beyond ruler-and-compass constructions. It is much more satisfactory to take the bold step. This provided the Greeks with a way of describing numbers that does turn out to be adequate for Euclidean geometry.
Ruler - compass construction is a construction involving only drawing straight lines and circles. Another problem involving this ruler-compass construction is that of squaring a cirle. The problem formally states as follows:
Using only a ruler and a compass it is not possible to square a circle or to turn the circle into a square whose perimeter is the same as that of the circumference of the circle.

## Discrete numbers in natural world

Discrete numbers are the natural numbers (0, 1, 2, 3,... ). These numbers are used for counting purpose. There is no other number in between the two consecutive numbers. These numbers are very useful in physical science , especially in physics. In physics ther are certain quantities like , spin which is expressed by these natural numbers. In quantum mechanics various quantum numbers are integer multiple of certain quantities. Positive integers are also natural numbers. Positive integers are 0 , 1, 2, 3, 4,... . These are the same as the natural numbers.

## Algebra skills

There are some basic rules for algebraic manipulations.

## Rational function

Rational functions can be expressed as a ratio of two algebraic polynomials: one being numerator and other being denominator. Such fractions are called rational. The coefficients of the polynomials need not be rational numbers , they can be taken over any field K.

There is another method partial- fraction decomposition of rational function. It is the process of decomposing a complex rational function into a sum of a polynomial and one or several fractions with simpler denominator.

An example of this decomposition can be given.

Partial fraction decomposition is very usefull in doing integration of complex functions whose usual formula is not known. As an example :

In summary the general rule of decomposition will be :

Improper rational function can be factored like this:

In some cases there are repeated roots. Rules are somewhat different in these cases:

## Middle term factor

Middle term factoring is a method to decompose a complex polynomial into simple factor expressions. It is like factoring a big number into two or three small numbers:

## Arithmetic formula

If you divide a a number by another number the result of the division will be the qoutient and the integer leftover by dividing one integer by another will be called the remainder. Here are the various terms of division process

There are relationships between arithematical mean, geometric mean and harmonic mean. Each is defined differently and has relationship with the other as

On the other hand arithmetic mean of a frequency distribution will be

## Binary arithmtic

Binary arithmetic is the arithmetic of adding and subtracting binary numbers (strings of 1 and 0).

Example can be given. Suppose we want to add 2 and 3 in terms of binary numbers then applying the rules we get

Binary subtraction is done in indirect way. First we compute 2's complement of a number then add it to the other number as follows

2's complement of a number is 1 - the number in base 2.
The usual rule for binary subtraction is

Similarly the division and multiplication can be done with binary numbers.
Similarly arithmetic with hexadecimal numbers can be carried out.
Relation between integral and summation inversion formula is

## Interval and inequality

Interval is the set of real numbers lying between two real numbers ( may be inclusive). The concept of interval has many use in measure theory. It can be used to define distance. This use can be seen in theory of relativity. There are many kinds of intervals.
A closed interval is the set of all the real numbers between two numbers including its endpoints which are those two numbers. Whereas the open interval excludes those two endpoints.
A degenerate real number is an interval with only one real number. Various kinds of interval are

A more elaborate description can be given

## Measure theory

Measure theory is related to the interval discussed above. Measure is the function that assigns real numbers to subsets of real numbers.

Also

Measure must satify these properties:

Where l(E) is the interval of set E. There is a theorem of measure as follows:

Vector algebra formula Vector algebra is a branch of algebra to deal with vectors and their addition, multiplication , etc. We will discuss vector projection, vector parallelogram law and others one by one .
Vector projection is defined by the usual dot product of two vectors. We can project one vector onto other vector in this way:

With this definition of projection comes Gram-Schmidt law.
We define projection operator by

The Gram–Schmidt process then works as follows:

Vector parallelogram law is the vector addition law. Using this law we can find the resultant of two vectors.

## Unit vector

Concept of unit vector is necessary to understand vector operation in curved and Euclidean space. A unit vector is any vector divided by its length.

After defining the unit vector we can represent any vector in the three dimensional space.

## Linear algebra

Eigenvalue can be very useful concept in quantum mechanics problem , especially solving Schrodinger equation

Addition and subtraction of two matrices can be done term by term.

The inverse of matrix is determined by this following rule

Here is an example of finding it

## Summary and linear algebra

The most common operations of vector algebra are :

The rules of matrix algebra are

Where v1, v2, v3 and u1, u2, u3 are the components of vector v and u.

## Directional cosine

Directional cosines are the angles between the vector and the coordinate axes :

## Mathematical induction

Mathematical induction is a method by which we go from special case to general case. It is the process of generalization. How can we tell that a property belong to all natural numbers? It is by mathematical induction that we can do so. The correct defintion is:
If a property belongs to n and n+1 then if it belongs to zero (0) then it belongs to all the natural numbers. This is how we go from particular case to general case.
Law of induction is not actually any theorem of mathematics. It is a principle. We can prove many general theorems and postulates using this mathematical induction principle. An example can make it clear. Sum of the first n integers. The first formula in the last section states that
---1
We first, prudently, check the first few cases.

To advance the left-hand side to the next case, we must (for this problem) add m + 1 to both sides (thus keeping the equality): The

The left-hand side is now the left-hand side of (1) for the next case n = (m + 1). How about the right-hand side? We can factor out the m + 1 from each term on the right-hand side and get

This can now be seen to be the right-hand side of the original equation 1) with n replaced by m + 1. Thus from the statement for m we have derived the statement for m + 1; we have successfully done the domino trick. Since we have already checked the formula for the first case, n = 1, we conclude that the formula is true for all positive integers n.

## Lattice

Lattice is basically a concept to be studied in abstract algebra. As abstract algebra is also a kind of algebra , it can be mentioned here. Lattice is any partially ordered set in which every two elements have a unique supremum (greatest lower bound) and a unique minimum (least upper bound). An example can be a natural numbers , partially ordered by divisibility. In this particular lattice the least upper bound is the greatest common divisor and supremum is the least common multiple.
In group theory lattice has a different meaning. In group theory a lattice in R(n) is a subgroup of additive group R(n) which is isomorphic to additive group Z(n). This lattice spans the entire real vector space R(n). In our house , especially on the floor there are sometimes seen special shapes repeated periodically over it. These are called tiling with primitive cells. A lattice can be viewed exactly like the tilings.

Formally a lattice λ in R(n) has the form :

Various lattices in Euclidean space can be visualized intuitively as follows:

When R(n) is n dimensional Euclidean space.

## Solving differential equation

Euler method is very famous. It can be used to solve differential equation. The theory is

An example can be given

This way the solution curve can be found.
All law of indices can be mentioned with a chart.

Basic algebra equations are

Percentage formula

Sometimes algebraic formulas become handy in computing number integration.

Set theory deals with concepts of sets and their properties. Using Ven diagram intersection of two or more sets can be found.

Some formulas involving planes and straight lines

Equation of plane through a point or a set of points is

The most beautiful equation in mathematics is the Riemann's formula

Where

Fibonacci Sequence is determined by this recursive forumula

The series or sequence is 1, 1, 2, 3, 5, 8, 13, 21,......
These numbers can also be found by the formula given by

Where φ is the golden ratio.

## Heron's formula

Heron's formula is a law expressing the area of a triangle.
Law of cosine implies

The altitude of the triangle has a length bsinγ and it follows that

It is worth to mention Poisson summation formula

Also Feynman path integral

## Laglands programm

Laglands program was an attempt to unify all of the continents of mathematics. In particular it connected number theory with geometry. First theorem to be stated is the Fermet's Last theorem

## Polynomial and its properties

A polynomial of one variable x is an algebraic equation of linear superposition of all powers of x up to n ( n= 1, 2,.....) . Polynomial is an useful tool in mathematics and physics. Now we will discuss some useful properties related to polynomial .
When a polynomial P(x) is divided by a linear factor (x-a) then the remainder is P(a). Proof can be given.
Let P(x) be a polynomial of degree n where n> 2 . Then P(x) = (x-a)Q(x) + R , where Q(x) is the qoutient and R(x) is the remainder. Now if we put x=a in this equation we get P(a) = (a-a)Q(x) + R or P(a) = R which is the remainder as claimed.
The factor theorem states that if the remainder of the division by the linear term x-a is zero then x-a is a factor of the polynomian and a is definitely the root of the same polynomial.
If P(a) = 0 then x-a is a factor of P(x).
The definition of logarithm is

Human beings' ear respond to the ratio of power measured in watts when the power of sounds increases. This ratio is called decibel. An increase of 10 decibal represents an increase of 100 watts. The decibel is defined as

## I did not say it would be easy , I said it would be the truth..

Fun with operators :
Operators are something that acts on other thing. The differential d/dx is an operator because it acts on functions. Operators play an essential role in quantum mechanics. The time evolution of wave function is

This is why e^[-ihH/t] is called the time evolution operator. In some cases we calculate the actual operator from power series for the exponential.

We have been working in what is called the Schrödinger picture in which the wavefunctions (or states) progress with time. There is the alternate Heisenberg picture in which the operators develop with time while the states do not change. For example, if we wish to compute the expectation value of the operator B as a function of time in the usual Schrödinger picture, we get

In the Heisenberg Picture the operator

We can use operator methods to compute uncertainty relation between two observables

similarly we can compute time derivative of expectation value of some operator.

## Axiom of Archimedes

Axiom of Archimedes is an important property of real number. Earlier Archimedes found that two magnitudes always have ratio between them so one can find a number particularly an integer which multiplied with the one will exceed the other. The exact formal statement would be :
If x and y are two elements of linearly ordered group then for every natural number n we have
x + x + ..... + x (n times) < y
or nx < y This is so called Archimedean axiom. Real numbers follow this property but the rationals do not. There is an alternative formulation of the same principle.

Let η is a small arbitrary number. Then there is a natural number n such that 1/n < η

## Some basic algebraic formulas

Algebraic formula are the basis of all algebra problems. These algebraic formulas involve two or more variables and some algebraic operations. These formulas should be better called identity.

## TENSOR ALGEBRA

Tensor algebra is a relatively new branch of mathematics dealing with tensors and tensor products. First definition of tensor should be given.

Tensors are the objects studied under linear algebra.

Bilinear Map is defined as

Inertia tensor is a three by three matrix that can be multiplied by angular velocity matrix to produce the corresponding angular momentum vector for either point mass or a rigid mass distribution.

## Lebesque Integration

If a function has two same representations then this following will hold.

Improper integral is the divergent integral

## Logistic Distribution

A random variable X is said to have a logistic distribution if its probability density function is given by

The logistic function can be used to model growth curve. Population growth curve reflects the logistic function. Let P represents population size (N is often used in ecology). The time parameter is t. The the differential equation

with r defines the growth rate and K is the carrying capacity. The solution of the equation is then

## Operator algebra

The Schrödinger equation comes directly out of our comprehending of wave packets. To get from wave packets to a differential equation, we use the new concept of (linear) operators. We determine the momentum and energy operators by demanding that, when an operator for some variable v acts on our simple wavefunction, we get v times the same wave function.

## Algebra and calculus in quantum field theory

ε is used to define the boundary condition of the denominator of the Feynman propagator (Q^2 + m^2 - iε).
Electron can interact with itself. This effect needs to be corrected in quantum electrodynamics.

Calculus is a branch of mathematics which is dependent on geometry and algebra. Fundamental theorem of calculus reflects that fact.

It is generalized to that of function of a function.
Linear algebra is such a branch of mathematics that is used almost everywhere is quantum theory and theory of relativity. It can be explained in simply with a few definitions and examples.

Tensors are entities in linear algebra. Without the understanding of linear algebra you can not understant tensor calculus.

## Magic numbers and modular arithmetic

Modular arithmetic can applied to find the solution set of an equation. Cosider an equation y^2 = x^3 - 3x + B (modulo p) . We can take any number p to find solutions of the equation. This kind of modulo equation is applied in RSA cryptography. Modular arithmetic is like a 12 hand clock. After one complete rotation the hour hand comes back to its previous position. It is a modulo 12 operation.
Likewise, we can do addition modulo any natural number N. Consider the set of all consecutive whole numbers between 0 and N − 1,
{0, 1, 2, 3, 4, ..... N-2, N-1}
If N=12, it is the set of possible hours.
For example, let’s take N = 3. Then we have the set {0, 1, 2} and addition modulo 3. For example, we have
2 + 2 = 1 (modulo 3) and so on. Euler phi-function has special properties.

## Dirichlet's theorem

If m and b are relatively prime integers with m > 0, then there exist infinitely many primes of the form km + b with k a positive integer.
We begin with the earlier treatment of the arithmetic progressions 4n + 1 and 4n + 3 by Euler. In 1737 Euler made the stunning discovery of the formula

valid for s > 1. Actually, the formula is valid for complex s with Re (s) > 1, but Euler had not considered powers ns with s complex by this time and did not need them for his purpose. Euler’s formula is a consequence of unique factorization of integers. In fact, the product for p ≤ N is

Letting N →∞, we obtain the desired formula.
Built into the formula is the result of Euclid’s that there are infinitely many primes, i.e., infinitely many primes in the arithmetic progression n. There are two ways to see this. In both cases one starts from the observation that the sum

from which it follows that the sum tends to infinity as s decreases to 1. In one case the argument continues with the observation that if there were only finitelymany primes, then

certainly have finite limit as s decreases to 1, and we arrive at a contradiction. In the other case the argument continues with the observation that the logarithm of

is comparable in size to 1/p^s , hence that

is comparable to

. Since

tends to infinity

must tends to infinity. we conclude that there are infinitely many primes. We shall return to this observation shortly in order to justify it more rigorously.

## Analyticity of complex number

"Mathematicians and physicists like holomorphic and meromorphic functions because their local properties completely determine their global properties".
A complex number has the form Z = x + iy
where i^2 = -1 and Re(z) = x and Img(Z) = y . The set of all complex numbers is denoted by C; it is also called the complex plane. Using polar coordinates, each complex number z = x + yi can be written uniquely as

Where r = (x^2 + y^2 ) ^ (1/2) .

are called the modulus and the principal argument of z, respectively. Using the Euler formula e^iϕ = cosϕ + isinϕ, each complex number z can also be uniquely represented as Z = |Z|e^eφ
A function f (z) is a function f : U -> C is called holomorphic if it is differentiable is some open interval U, That is, there exists a derivative.

Theorem: A function f(z) : U -> C is holomorphic if there exists a power series expansion as

for all z in some open neighborhood of z_0. For example polynomials are holomorphic. The power series expansion

is convergence for all z which belongs to C. Thus, the exponential function z → ez is holomorphic on the complex plane. The same is true for the functions z → sin z, cos z given by

for all z which belongs to C. Entire functions: By definition, a function f : C → C is called entire iff it is holomorphic on the complex plane. For example, polynomials, the exponential function z → ez, and the trigonometric functions z → sin z and z → cos z are entire functions.
Locally holomorphic functions: Let z_0 be a point of the complex plane. A complex-valued function f is called locally holomorphic at the point z0 iff there exists an open ball B(z_0) centered at z_0 such that the function f : B(z_0) → C is holomorphic.
A function f : U → V is called biholomorphic iff it is bijective and both f and f−1 are holomorphic. Biholomorphic maps are always angle-preserving, that is, the oriented angles between intersecting curves are preserved.

Conformal maps: Fix the point z0 of the complex plane and the complex numbers a, b with b = 0. The function f(z) := a + b(z − z0), z∈ C is the superposition of a translation, a rotation around the center z0, and a similarity transformation with respect to z0. Surely, this map is angle-preserving and a biholomorphic map f : C → C from the complex plane onto itself. Such a map is said to be a conformal map of the complex plane onto itself. We want to generalize this concept. To this end, let f : U → C (4.1) be a holomorphic function on the nonempty open subset U of the complex plane. This map is called a conformal map from U onto f(U) iff it is an angle-preserving diffeomorphism from the set U onto the set f(U). For a function (4.1) on the nonempty open subset U of the complex plane C, the following three properties are equivalent
(i) The map f is conformal from U onto f(U).
(ii) The function f : U → C is holomorphic, injective, and f(z0) = 0 for all points z0 in U.2
(ii) The set f(U) is open and the function f : U → f(U) is biholomorphic. In the case (ii), the function f looks locally like

in a sufficiently small open neighborhood of each point z0 ∈ U. Because of the condition f(z0) = 0, the map f is not locally degenerate at z0.
Integrals : It turns out that integrals of holomorphic functions reflect topological properties of the domain of definition. Let the function f : U → C be holomorphic, and let C : z = z(t), t0 ≤ t ≤ t1 be an oriented smooth curve in the set U. Define

This definition of the curve integral does care about the parametrization of the oriented smooth curve C. The integral changes sign if the orientation of the curve changes.

## Deformation Invariance of integral

if the following holds true: The two smooth curves C1 and C2 have the same initial and end point, and they can be continuously deformed into each other without leaving the set U and without changing the initial and end point. This result remains true if C1 and C2 are reasonable piecewise smooth curves (e.g., polygons).
And Couchy integral formula is

## The word residue was first used by Cauchy in 1826, but to be sure the definition there is quite complicated.

Meromorphic functions. The function f : U → C has an isolated singularity at the point z0 iff it is holomorphic in a punctured open neighborhood of z0.5 Then, there exist complex numbers . . . , a−2, a−1, a0, a1, a2, . . . such that

for all z in some punctured open neighborhood of z0.6 The number Res_z0(f) = a_1, is called the residue of the function f at the point z0. If there exists a positive integer m such that a−m = 0 and ak = 0 for all k < −m, then we say that the function f has an isolated pole of order m at the point z0. Then,

for all points z in some punctured open neighborhood of z0. The sum

is called the principle part of the function f at some point z_0.
The function f : U → C is called meromorphic on the open set U iff it is holomorphic up to isolated poles. Rational functions (i.e., quotients of polynomials) and the function tanz = sinz / cosz is meromorphic on the complex plane. The poles correspond to the zeros of the denominator. Meromorphic functions play a fundamental role in physics. The poles of meromorphic functions describe essential physical properties (e.g., the masses of elementary particles). The following theorem is called Cauchy’s residue theorem:
For a meromorphic function f: U -> C , there holds

if the following conditions are met: the circle C lies in the open set U, the function f has precisely the poles z1, . . . , zm inside the circle C and no poles on C. The integral is equal to zero if there are no poles on the closed disc bounded by the circle C. This is one of the most useful theorems in mathematics.

## Winding number

Let the functions f, g : U → R be meromorphic on the open set U, and let C be a counterclockwise oriented circle lying in U. Suppose that no zeros or singularities of f lie on C. Define

This is an integer called the winding number of the function f on the circle C. For example, if f(z) := z^n for a fixed integer n and the origin lies inside the circle C, then

Generally, the winding number tells us how often the image curve f(C) winds counterclockwise around the origin.

## A Glance at Analytic S-Matrix Theory

In the 1950s and 1960s, physicists thoroughly studied scattering processes for elementary particles by using analytic continuation for the S-matrix and the Green’s functions. If these functions have a singularity at the complex energy

then there exists an elementary particle with rest mass m_0 = E_0/c^2 and mean lifetime Δt = h/ΔE.

## Topology is precisely that mathematical discipline which allows a passage from the local to the global.

We define the mapping degree for a real conatinuous function f: [0,1] -> R with f(0) |= 0 and f(1) |= 0, we define the mapping degree deg(f) by setting

In connection with this definition there is theorem which states :
The continuous function f : [0, 1] → R has a zero if f(0)f(1) < 0.
Intuitively, the condition f(0)f(1) < 0 tells us that the function f has different signs at the two end points x = 0 and x = 1, and hence the graph of f has to

E intersect the x-axis. In terms of the mapping degree, the Bolzano theorem reads as follows. We are given a continuous function f : [0, 1] → R which does not vanish on the boundary of the interval. If deg(f) |= 0, then the function f has a zero.

Intersection numbers.
Invariance of the mapping degree under deformations of the maps. Consider a finite time interval [t0, t1]. Suppose that we are given two continuous functions f, g : [0, 1] → R. Then, we get
deg(f) = deg(g)
if there is a continuous map
H : [0, 1] × [t0, t1] → R
such that H(x, t0) = f(x) and H(x, t1) = g(x) for all x ∈ [0, 1]. Furthermore, we have to assume that H(x, t) != 0 for the boundary points x = 0,1 and all times t ∈ [t0, t1].

Local mapping degree: Consider maps as pictured in Fig. 5.7. If f(P) = P` and f preserves (resp. reverses) orientation in a sufficiently small neighborhood of the point P, then we set
degP f := 1 (resp. degP f = −1).
• Global mapping degree: If a map f : A → B globally preserves (resp. reverses) orientation, then we set
deg f := 1 (resp. deg f = −1).
If this is not the case, then we choose an image point P` and define

where the points P1, ..., Pn are precisely the preimage points of P`. Mnemonically, the global mapping degree is the sum of local mapping degrees with respect to a fixed image point P`. It turns out that, roughly speaking, this definition does not depend on the choice of the image point P`. In the plane, winding number and mapping degree coincide.

## Morphisms

Let X and Y be topological spaces. The map f : X → Y is called continuous (or a topological morphism) iff the preimage of open sets is again open. The continuous map f : X → Y is called a topological isomorphism iff it is bijective and the inverse map f−1 : Y → X is continuous, too. Topological isomorphisms are also called homeomorphisms.

## Standard examples of topological space

In order to give the reader a feeling for the universality of the notion of topological space, let us consider the following examples. • For n = 0, 1, . . . , the sets Rn and Cn and their subsets are topological spaces. • Every Hilbert space and Banach space and their subsets are topological spaces. • Every real (resp. complex) finite-dimensional manifold is a topological space. • For n = 1, 2, . . . , the n-dimensional unit sphere
Sn := {x ∈ R_n+1 : x_1^2 + x2^2 + . . . + x_n+1^2 = 1} is an n-dimensional real manifold.
• Define F(ϕ) := eiϕ for all ϕ ∈ R. The continuous map F : R → S1
sends the real line onto the unit circle S1. This is not a homeomorphism. We say that the real line R is a covering space of the unit circle.

## Energy of a non-spinning black holes

Even a black hole has energy due to its temparature. Similary a non-spinning black hole has energy due to its temperature.

The decrease of mass of a black hole is a function of time.

## Inverse pythagorian theorem

Inverse pythagorian theorem states that if a , b are two sides of a right-angle-triangle and h is its median intersecting the hypotenuse then the reciprocal of median h squared is equal to the sum of repciprocal of a^2 and reciprocal of b^2 . That is ,

let us proof this :
Consider a right angle triangle as

The area of whole is equal to the constituent part

Now, put this in Pythagoras theorem identity:

## Equations of our universe

Our universe , in every aspect, mathematical in nature. It is an apparent fact that we can see in physics. Physics has taken knowledge into a realm where nature can be seen to obey mathematics in the deppest level. Theory of general relativity has shown a gigantic model of our universe in terms of spacetime curvature or gravity. Quantum mechanics is more accurate description of small subatomic and atomic world. Even if quantum mechanics does not understand nature , nature understands quantum mechanics very well. Recently a paper written by a scientist proves that true quantum gravity theory can be found if we treat spacetime as an emergent property of a more fundamental structure which can be regarded as quantum mechanical. Here is a list of quantum field theoretical quantities.

Quark confinement is an amusing phenomena. It is related to strong interaction which is mediated by gluon or gluon field. When we try to separate two quarks which along with another quark make up the proton and neutron , we have to supply more energy. The quark model of proton and neutron is like spring mass system. As the distance between two quarks increases the energy of the system increases. As we supply more and more energy the quarks begin to be separated. But at the end a pair of quark and anti-quark gets created from the energy supplied. The new quark come inside the proton and original quark and anti-quark make a bond outside. Thus the proton structure remains intact and an extra quark and anti-quark pair is created. This is entirely a new phenomena due to stron interaction inside proton and nucleus. We can never separate two quarks at a long distance.

## Deriving Hamiltonian equations of motion

We start by defining the action. It is given by the following integral over time:

The action S, which is the time integral of the Lagrangian

The path in space and time corresponding to a given motion of the system is the one given by the bold line.
When S is used in field theories, it also contains an integration over spatial variables. Now, for simplicity, we set N=1. Varying q and p we obtain:

where the increment of the generalized velocity was re-expressed as:

For independent variations δq and δp and fixed endpoints, the stationarity of the action δS=0 leads to Hamilton’s equations of motion:

Solving the equations, for example, for the case of a simple harmonic motion (such as a spring-mass system) one can graph the corresponding phase diagram as shown below.

Order of differential equation

## Euler's sum

for any function f(x) Euler-Maclaurin's formula reads :

Geometric representation of Euler-Maclaurin formula can be given

Or alternatively

Where

Bk are Bernoulli numbers, Bm({x}) are Bernoulli polynomials and {x} = x − ⌊x⌋
For instance: f(x) = x^(m-1) gives f^m(x) = 0 {diffentiation m times). So R_m = 0;

Michael Spivey’s article ”The Euler-Maclaurin Formula and Sums of Powers” is delivered.
Claim: 1 m + 2m + · · ·(m − 1)m < mm , for m ≥ 1
Proof using Euler’s formula with R2 as error term

Error term has B2(x) = (x−1/2)2−1/12. Hence in [0,1] its minimum value is −1/12, for x = 1/2,and its maximum is 1/6, at the two endpoints. Since B2 is periodic these are also global extreme values, which gives −B2({x})/2 ≤ (−1/2)(−1/12) ≤ 1/24. This bounds the error term to

therefore

## Euler solved Bassel problem

It is the problem of summing infinite series of reciprocals of natural numbers squared. To see what Euler did we must recall Taylor's series of sine function.

Diving through by x both sides we get

Using the Weierstrass factorization theorem, it can also be shown that the left-hand side is the product of linear factors given by its roots, just as we do for finite polynomials (which Euler assumed as a heuristic for expanding an infinite degree polynomial in terms of its roots, but is in general not always true for general P(x)):

f we formally multiply out this product and collect all the x2 terms (we are allowed to do so because of Newton's identities), we see by induction that the x^2 coefficient of sin x/x is

But from the original infinite series expansion of sin x/x, the coefficient of x2 is − 1/3! = − 1/6 . These two coefficients must be equal; thus, we finaly obtain

Multiplying both sides of this equation by −π2 gives the sum of the reciprocals of the positive square integers.

## Fibonacci numbers

The nth Fibonacci number can be generated using this following formula

But in order to do this, we have to first find all Fibonacci numbers Fi for i between 1 and n − 1. However, it turns out that these numbers could also be generated in the following way. Consider the series

In words, we multiply an auxiliary variable q by the sum of all powers of the expression (q + q^2). If we open the brackets, we obtain an infinite series, whose first terms are

For example, let us compute the term with q3. It can only occur in q, q(q + q^2), and q(q + q^2)2. (Indeed, all other expressions that appear in the defining sum, such as q(q + q^2)3, will only contain powers of q greater than 3.) The first of these does not contain q^3, and each of the other two contains q^3 once. Their sum yields 2q3. We obtain in a similar way other terms of the series.
Analyzing the first terms of this series, we find that for n between 1 and 7, the coefficient in front of q^n is the nth Fibonacci number Fn. As an example, we have the term 13q^7 and F7 = 13. It turns out that this is true for all n. For this reason, mathematicians call this infinite series the generating function of the Fibonacci numbers.
Let’s go back to the numbers a_p counting the solutions of the cubic equation modulo primes.

Think of these numbers as surrogates of the Fibonacci numbers (let’s ignore the fact that the numbers ap are labeled by the prime numbers p, whereas the Fibonacci numbers Fn are labeled by all natural numbers n). It seems nearly unbelievable that there could be a formula for generating Fiboncci number. And yet, German mathematician Martin Eichler invented one in 1954.11 Namely, consider the following generating function:
---1
In words, this is q times the product of factors of the form (1−qa)2, with a going over the list of numbers of the form n and 11n, where n = 1,2,3,.... Let’s open the brackets, using the standard rules:

and then multiply all the factors. Collecting the terms, we obtain an infinite sum, which begins like this:

and the ellipses stand for the terms with the powers of q greater than 13. An astounding insight of Eichler was that for all prime numbers p, the coefficient bp is equal to a_p. In other words, a2 = b2,a3 = b3,a5 = b5,a7 = b7, and so on.
a_p is actually the difference of the expected number of solution of the equation and actual number of solutions . For example, we have seen above that there are 4 solutions modulo 5 of the above equation (1). Since 4 = 5 -1 we get a_5 = 1 . Similarly other numbers can be computed. Solution modulo x , y modulo N is a pair of numbers x and y such that the left hand side is equal to right hand side up to a number that is divisible by N.
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