Search found 4 matches

by Vangelis Mouroukos
Mon Jan 18, 2016 4:11 am
Forum: Combinatorics
Topic: Sum of the chromatic numbers of a graph and its complement
Replies: 1
Views: 3407

Re: Sum of the chromatic numbers of a graph and its compleme

Let \( k = \chi(G) \) and \( \bar{k} = \chi(\bar{G}) \). Consider a \(k\)-coloring of \(G\) and denote by \(n_i \) the number of vertices given the color \(i\) for \( i = 1,2, \ldots, k \). Let \(j \in \left\{ {1,2, \ldots ,k} \right\}\) be such that \({n_j} = \max \left\{ {{n_1},{n_2}, \ldots ,{n_k...
by Vangelis Mouroukos
Mon Jan 18, 2016 3:39 am
Forum: Algebraic Structures
Topic: Are they isomorphic?
Replies: 0
Views: 1642

Are they isomorphic?

Let \(G\) and \(H\) be groups such that there exist monomorphisms \(\varphi : G \longrightarrow H\) and \(\psi : H \longrightarrow G\). In each of the following cases, determine whether the groups \(G\) and \(H\) are isomorphic: (a) \(G\) and \(H\) are abelian groups. (b) \(G\) and \(H\) are finitel...
by Vangelis Mouroukos
Fri Jan 15, 2016 10:55 pm
Forum: Number theory
Topic: Exactly one square
Replies: 2
Views: 3079

Re: Exactly one square

First of all, we note that \(x\) and \(y\) cannot be both squares. Indeed, suppose that \(x\) is a square. Since \(xy+x=x(y+1) \) is a square, \(y+1\) must also be a square. But then (since \(y >0\) ) \(y \) cannot be a square. Suppose now that \(x\) and \(y\) are positive integers such that both \(...
by Vangelis Mouroukos
Sun Nov 15, 2015 10:38 am
Forum: Real Analysis
Topic: \(\sum_{n=1}^{+\infty}({-1})^{n+1}\frac{\sin{n}}{n}\)
Replies: 4
Views: 3921

Re: \(\sum_{n=1}^{+\infty}({-1})^{n+1}\frac{\sin{n}}{n}\)

Another approach uses the power series expansion of the complex logarithm: For all \(z \in \Bbb{C}\) with \(|z| < 1\), we have \[ \boxed{\displaystyle \log(1+z) = \sum_{n=1}^{\infty} (-1)^{n+1}\frac{z^n}{n}}\quad {\color {red} {(1)} }.\] Substitute \(z = e^{i\theta} \), with \(\theta \in (-\pi, \pi)...