Search found 597 matches

by Tolaso J Kos
Wed Nov 01, 2023 8:33 pm
Forum: Meta
Topic: Sortables captcha upgrade
Replies: 0
Views: 36743

Sortables captcha upgrade

Dear guests, we would like to inform you that we have updated the Sortables Captcha extension due to increase volume of spams. We noticed that in the last days alone $3$ spambots managed to bypass our security measures and create an account here on mathimatikoi.org. As if that weren't enough, they w...
by Tolaso J Kos
Thu Sep 21, 2023 6:43 am
Forum: Calculus
Topic: A series
Replies: 0
Views: 4658

A series

Evaluate the series

$$\mathcal{S} = \sum_{n=1}^{\infty} (-1)^{n-1} \ln \frac{n+1}{n}$$
by Tolaso J Kos
Wed Apr 12, 2023 8:01 pm
Forum: Analytic Geometry
Topic: Vector algebra
Replies: 0
Views: 5028

Vector algebra

Let $\mathbf{a} , \mathbf{b}, \mathbf{c}$ be three non coplanar vector. If $\displaystyle{\mathbf{a}' = \frac{\mathbf{b} \times \mathbf{c}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{b}' = \frac{\mathbf{c} \times \mathbf{a}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{c}' = \frac{\mathb...
by Tolaso J Kos
Sun Mar 12, 2023 3:03 pm
Forum: Calculus
Topic: An infinite product
Replies: 0
Views: 4917

An infinite product

Let $\mathcal{F}_n$ denote the $n$ -th Fibonacci number and $\mathcal{L}_n$ the $n$ – th Lucas. Prove that


$$\prod_{n=1}^{\infty} \left ( 1 + \frac{1}{\mathcal{F}_{2^n +1} \mathcal{L}_{2^n+1}} \right ) = \frac{3}{\varphi^2}$$
by Tolaso J Kos
Sun Mar 12, 2023 2:52 pm
Forum: Meta
Topic: Forum upgrade to latest version
Replies: 1
Views: 38578

Re: Forum upgrade to latest version

Greetings everyone, we are pleased to announce that the forum software has been upgraded to the latest version hardening the security of our website. You will notice that many cosmetic things have been restored to normal. This new version is compatible with php 8.2 that our server is currently runni...
by Tolaso J Kos
Wed Nov 16, 2022 7:30 am
Forum: Meta
Topic: Forum upgrade to latest version
Replies: 1
Views: 38578

Forum upgrade to latest version

Greetings, we have updated the forum to its latest version phpbb 3.3.x. You will find that many bugs have been fixed in this latest version. We would like to also inform you that the ability to add tags has been restored and now it's working flawlessly. You can select among many different tags to ca...
by Tolaso J Kos
Sun Apr 10, 2022 6:24 am
Forum: Complex Analysis
Topic: Contour integral
Replies: 1
Views: 10604

Re: Contour integral

It follows from Taylor's theorem that $f(z)=\sum \limits_{n=0}^{\infty} c_n z^n$ and that the convergence is uniform. Thus, \begin{align*} \frac{1}{2\pi i }\oint \limits_{|z|=1} \frac{\overline{f(z)}}{z-\alpha} \,\mathrm{d}z &=\frac{1}{2\pi i }\oint \limits_{|z|=1} \sum_{n=0}^{\infty} \frac{\ove...
by Tolaso J Kos
Fri Nov 06, 2020 11:59 am
Forum: Linear Algebra
Topic: Rank of product of matrices
Replies: 1
Views: 12220

Re: Rank of product of matrices

It holds that $${\rm nul} (T_1 T_2) \leq {\rm nul} (T_1) + {\rm nul} (T_2)$$ where $T_1, \; T_2$ are the corresponding linear transformations. Proof: The proof of the lemma is based on the rank - nullity theorem. Based upon the above lemma we have that \begin{align*} {\rm rank} \left ( T_1 T_2 \rig...
by Tolaso J Kos
Fri Nov 06, 2020 11:57 am
Forum: Linear Algebra
Topic: On permutation
Replies: 1
Views: 6395

Re: On permutation

The sum of $D(\sigma)$ over the even permutations minus the one over the odd permutations is the determinant of the matrix $A$ with entries $a_{i,j}=\vert i-j\vert$ and this determinant is known to be

$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$
by Tolaso J Kos
Fri Nov 06, 2020 6:50 am
Forum: Competitions
Topic: An equality with matrices
Replies: 1
Views: 9974

Re: An equality with matrices

Let $A, B$ be elements of an arbitrary associative algebra with unit. Then: \begin{align*} \left ( A^{-1} +\left ( B^{-1} - A \right )^{-1} \right )^{-1} &= \left ( A^{-1} \left ( B^{-1} - A \right )\left ( B^{-1} - A \right )^{-1} + A^{-1} A \left ( B^{-1} - A \right )^{-1} \right )^{-1} \\ &am...