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by Tolaso J Kos
Wed Nov 16, 2022 7:30 am
Forum: Meta
Topic: Forum upgrade to latest version
Replies: 0
Views: 59

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: 3632

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: 3560

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: 2847

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: 3083

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...
by Tolaso J Kos
Fri Nov 06, 2020 6:36 am
Forum: Algebraic Structures
Topic: Sum equals to zero
Replies: 1
Views: 2161

Re: Sum equals to zero

Let us suppose that $|\mathcal{G}| = \kappa$ and $x= \frac{1}{\kappa} \sum \limits_{g \in \mathcal{G}} g $. We note that for every $h \in \mathcal{G}$ the depiction $\varphi: \mathcal{G} \rightarrow \mathcal{G}$ such that $\varphi(g)=h g $ is $1-1$ and onto. Thus: \begin{align*} x^2 &=\left ( \f...
by Tolaso J Kos
Fri Nov 06, 2020 5:28 am
Forum: Algebraic Structures
Topic: Isomorphic groups
Replies: 1
Views: 2495

Re: Isomorphic groups

Using  $x^{-1}yx = y^{-1}$ or equivalently $yx = xy^{-1}$ we can write each element of  $\mathcal{Q}_{2^n}$ in the form $x^ry^s$ where $r,s \in \mathbb{N} \cup \{0\}$. Using $x^2 = y^{2^{n-2}}$ we may assume that $r\in \{0,1\}$. Using $y^{2^{n-1}} = 1$ we may also assume that $s\in \{0,1,\ldots,2^{n...
by Tolaso J Kos
Tue Jun 09, 2020 11:33 am
Forum: Functional Analysis
Topic: Inner product space
Replies: 1
Views: 2970

Re: Inner product space

Hint: Equality holds when vectors are parallel i.e, $u=kv$, $k \in \mathbb{R}^+$ because $u \cdot v= \|u \| \cdot \|v\| \cos \theta$ when $\cos \theta=1$, the equality of the Cauchy-Schwarz inequality holds.
by Tolaso J Kos
Sun Dec 15, 2019 10:51 pm
Forum: Calculus
Topic: Digamma and Trigamma series
Replies: 0
Views: 4326

Digamma and Trigamma series

Let $\psi^{(0)}$ and $\psi^{(1)}$ denote the digamma and trigamma functions respectively. Prove that:

\[\sum_{n=1}^{\infty} \left ( \psi^{(0)}(n) - \ln n + \frac{1}{2} \psi^{(1)}(n) \right ) = 1+ \frac{\gamma}{2} - \frac{\ln 2\pi}{2}\]

where $\gamma$ denotes the Euler – Mascheroni constant.
by Tolaso J Kos
Sun Oct 13, 2019 1:06 pm
Forum: Archives
Topic: Mathematical newspaper
Replies: 1
Views: 4247

Re: Mathematical newspaper

The second issue of the JoM Journal is now out. You may download it from this web address. Hope you find something interesting within its $97$ pages.