Introduction to GAME THEORY

Game Theory Definitions

Any time we have a situation with two or more players that involves known payouts or quantifiable consequences, we can use game theory to help determine the most likely outcomes. Let's start out by defining a few terms commonly used in the study of game theory:

• Game: Any set of circumstances that has a result dependent on the actions of two or more decision-makers (players)
• Players: A strategic decision-maker within the context of the game
• Strategy: A complete plan of action a player will take given the set of circumstances that might arise within the game
• Payoff: The payout a player receives from arriving at a particular outcome (The payout can be in any quantifiable form, from dollars to utility.)
• Information set: The information available at a given point in the game (The term information set is most usually applied when the game has a sequential component.)
• Equilibrium: The point in a game where both players have made their decisions and an outcome is reached

Some important laws

1. LAW OF CONSERVATION OF ELECTRIC CHARGE: Electric charges can neither be created nor be destroyed. The total charge in an isolated system is always constant.

2. COULOMB'S LAW: The force of attraction or repulsion between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The direction of the force is along the line joining the two charges.

3. OHM'S LAW: A constant temperature the current through a conductor is directly proportional to the potential difference between the two ends of the conductor.

4. JOULE'S LAW: The heat produced in current carrying conductor is (a) directly proportional to the square of the current (b) directly proportional to its resistance and (c) directly proportional to the time of passage of current.

5. GRASSMAN'S LAW:  The eye perceives the new colour depending on the algebraic sum of the red, green and blue light fluxes. This forms the basis for the colour signal generation and is known as Grassman's law

6. BREWSTER'S LAW: The tangent of the polarizing angle is numerically equal to the refractive index of the medium.

7. LENZ'S LAW: The induced current produced in a circuit always flow in such a direction that it opposes the change or cause that produces it.

Why Should Primary School Have A Maths Club?

     Maths clubs come in all shapes and sizes and there is no one model that works for every school. however, every school should have one.

This is because they help raise the profile of maths within the school, increase the engagement of children in maths and help show that maths is a playful and diverse subject full of surprises.

A maths club can also:

• develop children’s knowledge and understanding of maths  

• strengthen the links between maths and other subjects

• provide children with opportunities to try new things

• help children apply their maths skills to other ‘real-life’ situations

• celebrate the achievement of children

• fuel a can-do approach to maths

• show children that maths is multidimensional

• develop children’s maths thinking

• promote collaborative learning between different year groups

• develop maths resilience

• boost self-confidence  

• cultivate creativity  

• help to raise standards

• increase parental engagement with the subject

Basic introduction to GROUP in abstract algebra

The definition of a group. Let G be a non-empty set and let ⋆ be a binary operation on G:
(bop) ⋆: G × G → G, (a, b) → a ⋆ b.
Then (G; ⋆) is a group if the following axioms are satisfied:
(G1) associativity: a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c for all a, b, c ∈ G
(G2) identity element: there exists e ∈ G such that a ⋆ e = e ⋆ a = a for all a ∈ G.
(G3) inverses: for any a ∈ G there exists a −1 (a inverse) ∈ G such that a ⋆ a−1 = a −1 ⋆ a = e.
If in addition the following holds
(G4) commutativity: a ⋆ b = b ⋆ a for all a, b ∈ G then (G; ⋆) is called an abelian group, or simply a commutative group.

Remarks: Note that (bop) is an essential part of the definition. and that (G2) must precede (G3) because (G3) refers back to the element e.

 Fact: if (G; ⋆) is a group then the identity e is unique and the inverse of any a in G is uniquely determined by a.

Importance of Statistics in Different Fields

Statistics plays a vital role in every field of human activity. Statistics helps in determining the existing position of per capita income, unemployment, population growth rates, housing, schooling medical facilities, etc., in a country.


(1) Business
Statistics helps businessmen to plan production according to the taste of the customers, and the quality of the products can also be checked more efficiently by using statistical methods. Thus, it can be seen that all business activities are based on statistical information.

(2) Economics

 Economics largely depends upon statistics. National income accounts are multipurpose indicators for economists and administrators, and statistical methods are used to prepare these accounts. In economics research, statistical methods are used to collect and analyze the data and test hypotheses.

(3) Mathematics

Statistics helps in describing these measurements more precisely. Statistics is a branch of applied mathematics. A large number of statistical methods like probability averages, dispersions, estimation, etc., is used in mathematics, and different techniques of pure mathematics like integration, differentiation and algebra are used in statistics.

(4) Banking

Statistics plays an important role in banking. Banks make use of statistics for a number of purposes. They work on the principle that everyone who deposits their money with the banks does not withdraw it at the same time. The bank earns profits out of these deposits by lending it to others on interest. Bankers use statistical approaches based on probability to estimate the number of deposits and their claims for a certain day.

(5) State Management (Administration)

Statistics is essential to a country. Different governmental policies are based on statistics. Statistical data are now widely used in making all administrative decisions.

(6) Accounting and Auditing

In auditing, sampling techniques are commonly used. An auditor determines the sample size to be audited on the basis of error.

(7) Natural and Social Sciences

Statistics plays a vital role in almost all the natural and social sciences. Statistical methods are commonly used for analyzing experiments results, and testing their significance in biology, physics, chemistry, mathematics, meteorology, research, chambers of commerce, sociology, business, public administration, communications and information technology, etc.

(8) Astronomy

Example: This distance of the moon from the earth is measured. Since history, astronomers have been using statistical methods like method of least squares to find the movements of stars.

Interesting things on pi

TNTET for mathematics

TNTET Exam Pattern 2019

Teachers Recruitment Board conducts TNTET exam for determining the eligibility of candidates for the post of Primary and Elementary level teachers only for schools in the state of Tamil Nadu. Given below is a detailed paper pattern for the exam:

Mode of Exam

• Offline i.e. Pen-paper based test.
• Each candidate will be allocated their own OMR sheet.
• Make sure to check that the seal of the sheet is not broken before starting the exam. 

Nature of Questions

• The questions in both the papers will be objective in nature.
• This means, all the questions in TNTET 2019 will be of Multiple Choice type (MCQs), having 4 choices.
• Candidates must select the most appropriate answer.
• The duration of Paper 1 and 2 is 3 hours each.

Total Questions | Sections

• Both the papers will have 150 questions each.
• Paper 1 will have 5 sections while Paper 2 will have 4 sections
• No restriction to move to and fro among the sections.
• All the questions are compulsory to attempt.

TNTET Marking Scheme 2019

• A uniform marking system is followed in the exam i.e. for Paper 1 and 2, each correct answer carries + 1 mark.
• As per the revised exam pattern of TNTET exam, there is no negative marking for any unanswered or incorrect answer.
• Hence, instead of skipping a question, one can use their calculated guess and mark the answer. 

TNTET Paper1 marking scheme:

                     Subjects                                       Questions       Marks

Child Development and Pedagogy
(relevant to the age group of 6 – 11 years)          30                  30

Language I 

(Tamil/Telugu/Malayalam/Kannada/Urdu)            30                  30

Language II – English                                            30                  30

Mathematics                                                           30                  30

Environmental Studies                                          30                  30

Total                                                                        150                150

TNTET Paper II Marking Scheme:

Subjects                                                               

Child Development and Pedagogy
(relevant to the age group of 11-14 years)        
Compulsory

Language
(Tamil/Telugu/Malayalam/Kannada/ Urdu)         
Compulsory

Language II - English                                                               
Compulsory

For Mathematics and Science Teacher:                

 Mathematics and Science

Above each sections carries 30 questions and 30 marks Totally 150 questions and 150 marks

Mathematics Problem Solving Strategies

When solving mathematics problems, students should be encouraged to follow a general problem solving procedure. This is summed up as follows:

1. Read the problem carefully. The first and most important step is to read the problem carefully to understand what you're asked to find out and what information you have been given. Underlining the important information is also useful so you have all the important numbers/facts to hand.

2. Choose a strategy and make a plan.

3. Carry out the plan and solve the problem.

4. Check the working out and make sure that your solution is actually answering the question.

Characteristics and qualities of Mathematics Teacher

Math Teacher Characteristics

• Sound Knowledge of Mathematics
• Engaging
• Good Motivator
• Constantly Learning
• Caring

QUALITIES OF MATHEMATICS TEACHER

Interest in mathematics: A mathematics teacher should have a full command over subject matter. It is possible when he/she has interest in mathematics.

Knowledge of different teaching methods: With a firm grip over the subject matter, a mathematics teacher should know the teaching methods. He should clearly know the aims and objectives of mathematics

Power of know the difficulties of students: It is very necessary for a mathematics teacher to know where the students are feeling difficulty in solving problems.

Presentation of subject matter: A teacher should present the subject matter skill fully making interaction with the students, introducing well methods and applying various aids.

The good teacher should participate enthusiastically in various school activities such as:

Faculty meetings

Student projects

Mathematics club

Social Events

School publications

Excursions

Educational Fairs

Educational Quiz etc.,

Books on Mathematics as Optional subject for IAS Exam

Paper-I

1. Linear Algebra - K.C. Prasad, K B Datta 
2. Calculus - Santhi Narayan Integral Calculus, , Differential Calculus , Vector Calculus 
3. Analytic Geometry - Shantinarayan, HC Sinha, DK Jha and Sharma
4. Ordinary Differential eqs:- MD Raising Lumina, Golden seris-NP Bali 
5. Dynamics, Statitics and Hydrostatics - M.Ray 
6. Vector analysis – Shantinarayan 

Paper-II

1. Algebra - K C Prasad, KB Datta 
2. Real Analysis - Shantinarayan  ,Royden 
3. Complex Analysis - GK Ranganath
4. Linear Programming - SD sharma
5. Partial Diff.eqs. - Singhania
6. Numerical analysis and Computer Progg. - V. Rajaraman, SS Shasri
7. Mechanics & Fluid dynamics - AP Mathur, Azaroff leonid

Introduction of Integral Calculus..

INTEGRAL CALCULUS

INTRODUCTION_BASICS

The word Calculus Latin word, meaning "Small Stone"

• The basic idea of Integral calculus is finding the area under a curve. 

• To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areas

• Calculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus

                                                              

Number System..

NATURAL NUMBER: The natural (or counting) numbers are 1,2,3,4,5,$1,2,3,4,5,$ etc. There are infinitely many natural numbers. The set of natural numbers,$\left\{1,2,3,4,5,\mathrm{...}\right\}$, is sometimes written Nfor short.

WHOLE NUMBER: The whole numbers are the natural numbers together with 0.

INTEGER: The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.

$\left\{\mathrm{...},-5,-4,-3,-2,-1,0,1,2,3,4,5,\mathrm{...}\right\}$

The set of integers is sometimes written J$J$ or Z for short.

RATIONAL NUMBERS: The rational numbers are those numbers which can be expressed as a ratio between two integers. Example, 1/2,1/3,1/4...

IRRATIONAL NUMBERS: An irrational number is a number that cannot be written as a ratio (or fraction).  In decimal form, it never ends or repeats. Example, 1.3333333333...

REAL NUMBERS: The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.  The real numbers are “all the numbers” on the number line.  There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers.  But, it can be proved that the infinity of the real numbers is a bigger infinity. Denoted by R.

COMPLEX NUMBER: The complex numbers are the set {a+ib | 
$a$ and 
$b$ are real numbers}, where i is the imaginary unit. Example, minus value inside the square root.

−1−−−√