Search found 597 matches

by Tolaso J Kos
Fri Nov 06, 2020 6:36 am
Forum: Algebraic Structures
Topic: Sum equals to zero
Replies: 1
Views: 3099

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: 3143

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: 4167

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: 6895

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: 5979

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.
by Tolaso J Kos
Sat Oct 12, 2019 12:26 pm
Forum: Blog Discussion
Topic: A logarithmic Poisson integral
Replies: 1
Views: 4727

A logarithmic Poisson integral

A logarithmic Poisson integral by Tolaso J Kos Let $a \geq 0$. We will prove that $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, \mathrm{d}x = \left\{\begin{matrix} 0 & , & \left | a \right | \leq 1 \\ 2 \pi \ln \left | a \right | &, & \text{otherwise} \end{matrix}\right.$$ ...
by Tolaso J Kos
Thu Oct 03, 2019 9:38 am
Forum: Meta
Topic: Welcome to the new and improved mathimatikoi.org
Replies: 7
Views: 31260

Re: Welcome to the new and improved mathimatikoi.org

As of today we have the ability to include xy.pic into our posts. Unfortunately, the rendering of all equations takes a little time to complete. We'll see if we can overcome this problem.
by Tolaso J Kos
Thu Oct 03, 2019 9:04 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 8372

Re: xy.jax

$$\xymatrix{ {} & {} & {} & P\ar[d]^h\ar@{-->}[ldd] & {} \\ 0 \ar[r] & \mathrm{Ker}g\ar[r]^j & A\ar[r]^g\ar@{-->}[d]_i & B\ar[r]\ar@{-->}[d]^{\theta} & 0 \\ {} & {} & P\ar@{~>}[r]_-{k} & P/\mathrm{Im}(i\circ j) & {}
}$$
by Tolaso J Kos
Thu Oct 03, 2019 9:03 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 8372

Re: xy.jax

$$\xymatrix{& {P_{1}\oplus P_{2}}\ar@{->}[dldd]|{\boxed{\pi_{1}}}\ar[drdd]^{\pi_{2}} & \\
& & \\
& M\ar[dl]_{k_{1}}\ar[dr]^{k_{2}}\ar@{=>}[dd]_f^g\ar@{-->}[uu]_{\theta} & \\
P_{1}\ar[dr]_{h_{1}} & & P_{2}\ar[dl]^{h_{2}} \\
{}& N & {}
}$$
by Tolaso J Kos
Thu Oct 03, 2019 9:02 am
Forum: LaTeX code testings
Topic: xy.jax
Replies: 4
Views: 8372

Re: xy.jax

$$\xymatrix{ && &j\ar@/_/@{->}[ddlll]\ar@/^/@{--}[ddddd] &a\ar@/^/[l]\ar@/_/@{--}[ddddd] &&& \\ \\y \ar@/_/[d]&&&&& & &u\ar@/_/[uulll]\\x\ar@/_/[ddrrr]& &&&&& & v\ar@/_/ \\\\ &&& b\ar@/^/[r] & i \ar@/...