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I.S.I. Entrance Problems

An Hour of Beautiful Proofs

Every week we dedicate an hour to Beautiful Mathematics - the Mathematics that shows us how Beautiful is our Intellect. This week, I decided to do three beautiful proofs in this one-hour session... Proof of Fermat's Little Theorem ( via Combinatorics )It uses...

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Inequality – In Equality

Inequality – In Equality

This article aims to give you a brief overview of Inequality, which can be served as an introduction to this beautiful sub-topic of Algebra. This article doesn't aim to give a list of formulas and methodologies stuffed in single baggage, rather it is specifically...

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How to solve an Olympiad Problem (Number Theory)?

Suppose you are given a Number Theory Olympiad Problem. You have no idea how to proceed. Totally stuck! What to do? This post will help you to atleast start with something. You have something to proceed. But as we share in our classes, how to proceed towards any...

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Math Olympiad

RMO 2019 Maharashtra and Goa Problem 2 Geometry

RMO 2019 Maharashtra and Goa Problem 2 Geometry

Understand the problemGiven a circle $latex \Gamma$, let $latex P$ be a point in its interior, and let $latex l$ be a line passing through $latex P$. Construct with proof using a ruler and compass, all circles which pass through $latex P$, are tangent to $latex...

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College mathematics

An excursion in Linear Algebra

An excursion in Linear Algebra

Did you know Einstein badly needed linear algebra? We will begin from scratch in this open seminar and master useful tools on the way. The open seminar on linear algebra is coming up on 14th November 2019, (8 PM IST).

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Research Tracks

Arithmetical Dynamics: Part 6

Arithmetical Dynamics: Part 6

Consider fix point of \( R(z) = z^2 - z \) . Which is the solution of $$ R(z) = z \\ \Rightarrow z^2 - z =z \\ \Rightarrow z^2 - 2z =0 \\ \Rightarrow z(z -2) =0 $$ Now , consider the fix point of \( R^2(z) \) . \( \\ \) $$ i.e. R^2(z) = R . R(z) = R(z^2 -z)\\...

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Arithmetical Dynamics: Part 5

Arithmetical Dynamics: Part 5

And suppose that R has no periodic points of period n . Then (d, n) is one of the pairs \( (2,2) ,(2,3) ,(3,2) ,(4 ,2) \) , each such pair does arise from some R in this way . The example of such pair is $$ 1. R(z) = z +\frac {(w-1)(z^2 -1)}{2z} ; it \ has \ no \...

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Arithmetical Dynamics: Part 0

Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...

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Arithmetical Dynamics: Part 4

Arithmetical Dynamics: Part 4

\( P^m(z) = z \ and \ P^N(z)=z \ where \ m|N \Rightarrow (P^m(z) - z) | (P^N(z)-z) \) The proof of the theorem in Part 0 : Let , P be the polynomials satisfying the hypothesis of theorem 6.2.1 . Let , \( K = \{ z \in C | P^N(z) =z \} \\ \) and let \( M =\{ m \in Z : 1...

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Arithmetical Dynamics: Part 3

Arithmetical Dynamics: Part 3

Theory: Let \( \{ \zeta_1 , ......., \zeta_m \} \) be a ratinally indifferent cycle for R and let the multiplier of \( R^m \) at each point of the cycle be \( exp \frac {2 \pi i r}{q} \) where \( (r,q) =1 \) . Then \( \exists \ k \in Z \) and\( mkq \) distinct...

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Arithmetical Dynamics: Part 2

Arithmetical Dynamics: Part 2

The lower bound calculation is easy . But for the upper bound , observe that each \( z \in K \) lies in some cycle of length m(z) and we these cycles by \( C_1 , C_2 .....,C_q \) . Further , we denote the length of the cycle by \( m_j \) , so , if \( z \in C_j \) then...

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Arithmetical Dynamics: Part 1

Arithmetical Dynamics: Part 1

Definition: Suppose that \( \zeta \in C \) is a fixed point of an analytic function \( f \) . Then \( \zeta \) is : a) Super attracting if \( f^{'} (\zeta) =0 \rightarrow \) critical point of \( f \) b) Attractting if \( 0 < |f^{'}( \zeta )|< 1 \ \rightarrow \)...

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Research for School

Research for School

Research projects for school students, in mathematics and data science. For advanced learners who are in love with mathematical science.

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Arithmetical Dynamics: Two possible problems

1.1. Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s theorem. Related reading: (1) Silverman Arithmetic of Dynamical Systems: p 165. Bifurcation polynomials. Exercise 4.12 outlines some known properties and open questions (** = open...

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