Math by Tathagata
Tathagata Kuratowski’s first graph usually refers to the non-planar complete graph on five vertices denoted by K5 whereas Kuratowski’s second graph K3,3 is the complete bipartite graph with two sets of 3 vertices which is also non-planar. Lets discuss about Kuratowski first and second graph with their properties. What is Kuratwoski’s First Graph? Definition: Kuratowski’s first … Read more
n this page, let us prove a graph is a tree if and only if it is minimally connected. Before proving this, let us recall the definition of a tree. Definition of a Tree in Graph Theory A connected graph without any circuit is called a tree. Tree is Minimally Connected Proof Theorem: Show that … Read more
In this page, let us prove chromatic number of a cycle with n vertices is 2 if n even and 3 if n odd. Before proving this, let us recall the definition of the chromatic number of a graph. Chromatic Number of a Graph Definition: The chromatic number of a graph is the minimum number of … Read more
Discrete Math Problems: In this page, you can find a list of Discrete Math Problems which can be useful for practice for final exam. Discrete Math Practice Problems Q1: Without using truth table show the following: [Solution] Q2: Q3: State the inclusion-exclusion principle for two/three sets. Among 100 patients admitted to a hospital, 50 are … Read more
In this page, let us prove that the chromatic number of a complete graph with n vertices is n. Before proving this, let us recall the definition of the chromatic number of a graph. Chromatic Number of a Graph Definition: The chromatic number of a graph G, denoted by $\chi(G)$, is the minimum number of … Read more
Engineering Math Problems: In this page, you can find a list of Engineering Math Problems which can be useful for practice for final exam. Eng. Math II Content: Link Sequence, Series & Complex Analysis Q1: What is a bounded sequence. Show the sequence {xn} where $x_n=\dfrac{2n-1}{4n}$ is bounded. Q2: State ratio test and root test … Read more
In this page, lets learn Cauchy integral theorem and formula with examples. Cauchy’s Integral Theorem Statement If f(z) is continuous and analytic at each point within and on a closed curve C, then $\boxed{\displaystyle \int_C f(z)~dz=0}$. Cauchy’s Integral Formula If f(z) is analytic within and on a closed contour C and if $a$ is any … Read more
Cauchy-Euler equation (or Cauchy’s linear equation) is a differential equation with non-constant coefficients i.e, having variable-coefficients. It has the following form where a0, a1, …, an are real constants, and f(x) is a function of x. How to Solve Cauchy-Euler Equation? The below steps to be followed in order to solve a Cauchy-Euler equation. Solved … Read more
In this page, lets explore the variation of parameters method for solving a second order differential equation with examples. Variation of Parameters Formula Formula: Let u and v be two linearly independent solutions of y”+py’+qy=0. Then the general solution of y”+py’+qy=R obtained by using the method of variations of parameters is given by $y=u \displaystyle\int … Read more
In this page we will prove that a distributive lattice is modular. Distributive Lattice: A lattice (L, ≤) is called distributive if the distributive property holds, i.e,a∨(b∧c) = (a∨b)∧(a∨c)a∧(b∨c) = (a∧b)∨(a∧c) Modular Lattice: A lattice (L, ≤) is called modular if for all a, b, c ∈ L with a≤c we have (a∨b)∧c = a∨(b∧c). … Read more