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Undergraduate
By
Shiva Kintali
on July 6, 2012 | Updated Dec. 6, 2017
Linear recursion relations
Let \(a_n\) denote the Fibonacci sequence \(a_0 = 0\), \(a_1 = 1\), \(a_n = a_{n−1} + a_{n−2}\). Let \(b_n = (a_n)^2\). Prove that \(b_n\) satisfies a linear recursion relation i.e., \(b_n\) can be …
Puzzles
Puzzles
recursion
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Undergraduate
By
Shiva Kintali
on June 10, 2012 | Updated Dec. 6, 2017
Expanding expressions
You are given an algebraic expression \(E\) having variables, addition, multiplication and parenthesis. Your goal is to repeatedly expand \(E\) using the distributive law if possible. Prove that th…
Puzzles
Puzzles
convergence
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Undergraduate
By
Shiva Kintali
on June 14, 2012 | Updated Dec. 6, 2017
Connectivity of cubic graphs
Prove that if \(G\) is 3-regular graph, then its vertex-connectivity equals its edge-connectivity.
Mathematics
Graph Theory
connectivity
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Undergraduate
By
Shiva Kintali
on Aug. 12, 2012 | Updated Dec. 6, 2017
Finding perfect matching in bipartite graphs
Let \(G = (A,B)\) be a bipartite graph. Hall's theorem implies that \(G\) has a perfect matching if and only if \(|A| = |B|\) and for each \(X \subseteq A\), \(|X| \leq |\Gamma(X)|\), where …
Computer Science
Algorithms
matching
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Undergraduate
By
Shiva Kintali
on Aug. 22, 2013 | Updated Jan. 4, 2018
Graphs and Fermat's Little Theorem
Given a prime number \(n\), let \(\mathbb{Z}_n\), denote the set of congruence classes of integers modulo \(n\). Let \(a\) be a natural number having no common prime factors with \(n\); multiplication…
Mathematics
Graph Theory
Number Theory
digraphs
fermats little theorem
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Undergraduate
By
Shiva Kintali
on May 22, 2013 | Updated Dec. 6, 2017
Balls and bin game
Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following : choose one ball from the bin uniformly at random, and then put the…
Mathematics
Probability
sampling
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High School
By
shitikanth
on June 21, 2012 | Updated Dec. 6, 2017
A well known triangle
Let \(ABC\) be an acute-angled triangle. Find points \(X, Y, Z\) on sides \(BC, CA\) and \(AB\) such that the perimeter of the triangle \(XYZ\) is minimized.
Mathematics
Geometry
optimization
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Large min-degree implies perfect matching
Let \(G\) be a bipartite graph with partitions \(X\) and \(Y\) such that \(|X|=|Y|=n\). The degree of each vertex in \(G\) is at least \(n/2\). Prove that \(G\) has a perfect matching.
Mathematics
Graph Theory
matching
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Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Algebraic dual of graphs
Let \(G\) be a connected graph. An algebraic dual of \(G\) is a graph \(G'\) such that \(G\) and \(G'\) have the same set of edges, any cycle of \(G\) is a cut of \(G'\), and any cut of \(G\) is a cyc…
Mathematics
Graph Theory
matroids
0
Undergraduate
By
Shiva Kintali
on Sept. 28, 2013 | Updated Jan. 4, 2018
Task assignment using Hall's Theorem
You are given a collection of tasks each of which must be assigned a time slot in the range {\(1,\dots,n\)}. Each task \(j\) has an associated interval \(I_j = [s_j,t_j]\) and the time slot assigned t…
Mathematics
Graph Theory
assignment
halls theorem
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