# Tagged Questions

If you already have a proof for some result, but want to ask for a different proof (using different methods).

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### Prove $f(c)\int_{a}^{b}g(x)dx=\int_{a}^{b}g(x)f(x)dx$

Assume that $f:[a,b]\rightarrow\mathbb{R}$ is continuous on $[a,b]$ and $g:[a,b]\rightarrow\mathbb{R}$ is integrable and $g(x)\geq0$ for all $x\in[a,b]$. Then there exists a $c\in(a,b)$ such that ...
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### Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...
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### Can one show that $\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||}$ is always on the range of $\cos \theta$?

A basic property of the dot product of two vectors is that $$\frac{\vec{u}\cdot \vec{v}}{||\vec{u}||*||\vec{v}||} = \cos \theta$$ Where $\theta$ is the angle between the two vectors. Since there is ...
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### Prove $T$ is invertible

If $T\in L(X,X)$ where $X$ is a Banach space and $L(X,X)$ is denoted as the space of bounded linear maps, and $\|I-T\|<1$ where $I$ is the identity operator, then $T$ is invertible? Here ...
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### Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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### Solving a Diophantine Equation with 2 variables

This is my answer for the following question: Find all natural numbers $(a,b)$ for which $a^b-b^a=1$. When $a$ or $b$ equals $1$, $(a,b)=(2,1)$ is trivial. If $a,b>1$, I generalized the problem ...
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### Direct proof of the existence of optimal memoryless deterministic policies in MDP

It is well known that (finite-state, finite-action, discrete time) MDPs admit an optimal policy that is memoryless and deterministic (sometimes called pure). The proof of this fact for ...
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### Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
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### Proving that $\sqrt{a_1^2} + \sqrt{a_2^2} +…+ \sqrt{a_n^2} > \sqrt{a_1^2 + a_2^2 +…+a_n^2}$ using Pythagoras

I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$. I'm interested in whether there's a way to use that proof with Pythagoras to prove the general ...
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### Generalisation of Binomial Theorem, Leibniz Formula and similar theorems

Since the beginning of the year, our maths teacher showed us the Binomial Theorem in $\mathbb{R}$\, then in $\mathbb{C}$\, in $M_n(\mathbb{K)}$ with two matrices which commute, and now the Leibniz ...
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### Proving finite additivity for this semi-algebra (infinite coin flips)

Background copied and pasted from another one of my questions: Background: Consider flipping a coin $n$ times. Define the sample space as $$\Omega = \{(r_1,r_2,r_3,\dots); r_i = 0 \text{ or }1\}$$ ...
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### Dummit and Foote exercise verification?

I was working on the following problem: Let $\sigma$ be the m-cycle $(1 2...m)$. Show that $\sigma^{i}$ is also an m-cycle iff $\gcd(i,m)=1$ A solution to this problem is given here. But the ...
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### How do i evaluate this sum :$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$?

How do I evaluate the following sum: $$\sum _{m=1}^{\infty } \sum _{k=1}^{\infty } \frac{m(-1)^m(-1)^k\log(m+k)}{(m+k)^3}$$ Note I used many idea such as :Hochino's Idea and taylor expansion of ...
### Tangent identity given $a + b + c = \pi$
Given that $a + b + c = \pi$, that is, three angles in a triangle - then prove that $$\tan a + \tan b + \tan c = \tan a \tan b \tan c$$ Is my solution below completely rigorous? Can I justify taking ...