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by Demetres
Mon Dec 03, 2018 10:12 pm
Forum: Linear Algebra
Topic: Eigenvalues of Symmetric Matrices
Replies: 1
Views: 1679

Re: Eigenvalues of Symmetric Matrices

This follows from the following characterisation of the eigenvalues of symmetric matrices. \[\lambda_k(A) = \min_{\dim(U) = n-k+1} \max_{0 \neq x \in U} \frac{x^TAx}{\|x\|_2}\] Here the minimum is taken over all subspaces $U$ of $\mathbb{R}^n$ of dimension $n-k+1$. (I chose to write $\lambda_K(A)$ r...
by Demetres
Tue Sep 26, 2017 10:09 am
Forum: Foundation
Topic: Uncountably many disjoint subsets similar to \(\mathbb{R}\)?
Replies: 1
Views: 3204

Re: Uncountably many disjoint subsets similar to \(\mathbb{R}\)?

It remained unanswered for quite some time. Since I have been asked for it, here is a solution: The answer is affirmative. Given a sequence $a = (a_1,a_2,a_3,\ldots)$, where $a_i \in \{1,2\}$ for each $i$, we construct the subset $S_a$ of $(0,1)$ as follows: Every element of $S_a$ is of the form \[ ...
by Demetres
Thu Jul 14, 2016 10:39 am
Forum: Analysis
Topic: An interesting limit
Replies: 2
Views: 2043

Re: An interesting limit

We have also seen this here : The basic steps are as follows: We consider independent Poisson distributions \(X_1,X_2,\ldots\) with parameter 1. We know that \(Y_n = X_1 + \cdots + X_n\) is Poisson with parameter \(n\). From the central limit theorem, \((Y_n-n)/\sqrt{n}\) converges in distribution t...
by Demetres
Thu Jul 14, 2016 7:48 am
Forum: Real Analysis
Topic: Inflection Point of a function
Replies: 2
Views: 1449

Re: Inflection Point of a function

I presume you can show that \(f'(1) = 1\) showing that \(x=1\) is a stationary point. Now observe that for \(0 < x < 1\) we have \(x^x < x^1 = x\). So \(x^{x^x} < x^x\), i.e. \(f(x) < 0 = f(1).\) Also, for \(x > 1\) we have \(x^x > x\) giving \(f(x) > f(1)\). So \(x=1\) is a point of inflection.
by Demetres
Wed Jul 13, 2016 9:06 am
Forum: Real Analysis
Topic: Infinite Series
Replies: 2
Views: 1402

Re: Infinite Series

To use Stirling's approximation observe that \[ \prod_{r=1}^n \left(1 + \frac{r}{n}\right)^{1/n} = \left( \frac{(n+1)(n+2)\cdots (n+n)}{n^n}\right)^{1/n} = \left(\frac{(2n)!}{n^n(n!)}\right)^{1/n}.\] By Stirling's approximation we have \[ \frac{(2n)!}{n^n(n!)} \sim \frac{\sqrt{4\pi n}(2n/e)^{2n}}{n^...
by Demetres
Wed Jul 13, 2016 7:47 am
Forum: Calculus
Topic: floor function integral
Replies: 5
Views: 2307

Re: floor function integral

Indeed as Apostolos says, the integral diverges. What jacks's proof shows is that for every \(0 < \varepsilon < \pi/2\) we have that \(\int_{\varepsilon}^{\pi-\varepsilon} \lfloor \cot{x} \rfloor \; dx = -\pi + 2\varepsilon.\) However this does not mean that the value of the integral converges to\(−...
by Demetres
Tue Jul 12, 2016 8:09 am
Forum: Real Analysis
Topic: Positive function
Replies: 1
Views: 1185

Re: Positive function

If this is a Riemann integral, we use the fact that any Riemann integrable function has a point of continuity. (In fact the set of point of discontinuity has measure zero - this is Lebesgue's criterion.) So there is an interval \(I \subseteq [0,x]\) and a positive real number \(a\) such that \(f(t) ...
by Demetres
Tue Jul 12, 2016 7:36 am
Forum: Complex Analysis
Topic: Analytic Function
Replies: 1
Views: 1414

Re: Analytic Function

No it is not. For example \(z\) is analytic but \(\overline{z}\) is not. (As can be checked e.g. by the Cauchy-Riemann equations.)

By the way, "analytic" and "locally represented by a convergent power series" mean the same thing.
by Demetres
Tue Jul 12, 2016 7:26 am
Forum: Real Analysis
Topic: constant function
Replies: 3
Views: 1638

Re: constant function

Hi Apostole,

It is a nice proof! I have not seen it anywhere.
by Demetres
Tue Jul 12, 2016 7:25 am
Forum: Real Analysis
Topic: constant function
Replies: 3
Views: 1638

Re: constant function

Given two points \(x,y \in \mathcal{D}\) we define \[\displaystyle{D(x,y) = \frac{f(x)-f(y)}{x-y}}.\] Note that for \(x < y < z\) we have \[ D(x,y) = \alpha D(x,z) + (1-\alpha)D(z,y)\] where \(\alpha = \alpha(x,y,z) = \displaystyle{\frac{z-x}{x-y} \in (0,1)}.\) It follows that either \(D(x,z) \leqsl...