Search found 308 matches

by Grigorios Kostakos
Sat Aug 11, 2018 10:29 am
Forum: Multivariate Calculus
Topic: Show that a vector field is not conservative (example)
Replies: 4
Views: 5602

Re: Show that a vector field is not conservative (example)

First we write down a useful theorem: If a continuously differentiable vector field $\overline{F}:U\subseteq{\mathbb{R}}^n\longrightarrow{\mathbb{R}}^n\,,$ where $U$ is open, is conservative, then, for every $\overline{x}\in U$, the Jacobian matrix ${\bf{D}}\overline{F}(\overline{x})$ of $\overline{...
by Grigorios Kostakos
Sat May 12, 2018 1:43 pm
Forum: Analysis
Topic: Logrithmic Integral
Replies: 3
Views: 5043

Re: Logrithmic Integral

Using that the Fourier series of $\log(\sin{x})$ on $(0,\pi)$ is(*) \begin{align} \log(\sin{x})=-\log2-\mathop{\sum}\limits_{n=1}^{\infty}\frac{\cos(2nx)}{n} \end{align} then \begin{align*} \int_{0}^{\pi}x^2\log(\sin{x})\,dx&\stackrel{(1)}{=}\int_{0}^{\pi}\Big(-x^2\log2-\mathop{\sum}\limits_{n=1...
by Grigorios Kostakos
Thu May 03, 2018 6:23 am
Forum: Complex Analysis
Topic: Complex Integral of a singularity function
Replies: 1
Views: 3173

Re: Complex Integral of a singularity function

The function $f(z)=\frac{1}{(z-2)^2(z-4)}$ is defined and is holomorphic on $\mathbb{C}\setminus\{2,4\}$. The disk $D_1=\big\{{z\in\mathbb{C}\;|\;|z|\leqslant3}\big\}$ containing the second order pole $z_1=2$, but not the simple pole $z_2=4$. By Cauchy's integral formula we have \begin{align*} \disp...
by Grigorios Kostakos
Tue May 01, 2018 4:51 pm
Forum: General Mathematics
Topic: Fibonacci closed form
Replies: 1
Views: 3734

Re: Fibonacci closed form

Because \begin{align*} \mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{\frac{1}{F_{2^{n+1}}}}{\frac{1}{F_{2^n}}}&=\mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{F_{2^n}}{F_{2^{n+1}}}\\ &=\mathop{\lim}\limits_{{n}\rightarrow{+\infty}}\frac{\phi^{2^n}-(-\phi)^{-2^n}}{\phi^{2^{n+1}}-(-...
by Grigorios Kostakos
Fri Mar 16, 2018 1:11 pm
Forum: Real Analysis
Topic: Convergence of a series
Replies: 1
Views: 3780

Re: Convergence of a series

Let $\alpha_n=\big(\frac{(2n-1)!!}{(2n)!!}\big)^2\,,\; n\in\mathbb{N}$ and $\beta_n=n\,,\; n\in\mathbb{N}$. Then $\sum_{n=1}^{\infty}\frac{1}{\beta_n}=+\infty$ and for all $n\in\mathbb{N}$ holds \begin{align*} \beta_n-\beta_{n+1}\,\frac{\alpha_{n+1}}{\alpha_n}&=-\frac{1}{4(n+1)}<0\,. \end{align*...
by Grigorios Kostakos
Wed Feb 07, 2018 10:47 pm
Forum: Real Analysis
Topic: Real analysis
Replies: 3
Views: 5339

Re: Real analysis

what can you say about A if LUB A = GLB A ? Maybe LUB is L(owest)U(pper)B(oundary) i.e. $\sup$ and GLB is G(reatest)L(ow)B(oundary) i.e. $\inf$. The main question is what kind of set is $A$? Since we talking about $\inf{A}$ and $\sup{A}$ it must be given where lies $A$ i.e. it considered as a subse...
by Grigorios Kostakos
Sun Jan 28, 2018 12:58 am
Forum: Real Analysis
Topic: Subsequences
Replies: 1
Views: 3718

Re: Subsequences

It seems that here we have an open problem. See
Salem numbers and uniform distribution modulo 1
by Grigorios Kostakos
Sat Jan 20, 2018 5:16 am
Forum: Real Analysis
Topic: Subsequences
Replies: 1
Views: 3718

Subsequences

Prove that the sequence $\alpha_n=\lfloor{\rm{e}}^n\rfloor\,,\; n\in\mathbb{N}$, where $\lfloor{\cdot}\rfloor$ is the floor function, has a subsequence with all its terms to being odd numbers and a subsequence with all its terms to being even numbers.


Note: I don't have a solution.
by Grigorios Kostakos
Wed Jan 03, 2018 11:17 pm
Forum: Linear Algebra
Topic: Linear Algebra Book Recommendation
Replies: 1
Views: 4203

Re: Linear Algebra Book Recommendation

One book worthy to read is Peter D. Lax - Linear Algebra and its applications. (Wiley-Interscience)
Also there are lot of good books applying (or compining) linear algebra to (with) geometry.
by Grigorios Kostakos
Sun Dec 17, 2017 3:51 pm
Forum: Multivariate Calculus
Topic: Calculation of the mass of solid bounded by two surfaces
Replies: 4
Views: 5628

Re: Calculation of the mass of solid bounded by two surfaces

Is there a general rule to solve this kind of problems, given two surfaces that bound a solid and its density function ? Yes, there is a general formula to calculate the mass of a solid $S$ with given density function $f:S\subset\mathbb{R}^3\longrightarrow\mathbb{R} $ : \[\displaystyle\mathop{\iiin...