Search found 4 matches
- Mon Jan 18, 2016 4:11 am
- Forum: Combinatorics
- Topic: Sum of the chromatic numbers of a graph and its complement
- Replies: 1
- Views: 4985
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...
- Mon Jan 18, 2016 3:39 am
- Forum: Algebraic Structures
- Topic: Are they isomorphic?
- Replies: 0
- Views: 1692
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...
- Fri Jan 15, 2016 10:55 pm
- Forum: Number theory
- Topic: Exactly one square
- Replies: 2
- Views: 3320
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 \(...
- 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: 4036
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)...