# All Questions

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### Comparing variances: Cramer-Rao estimate vs. calculated variance of ML estimator.

I have a n-element sample from the Rayleigh distribution, $$f(x)=\frac{x}{a^2}e^{-\frac{x^2}{2a^2}}$$ and using the method of maximum likelihood, I find the estimator of $a$ to be ...
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### matrix calculus (differentiation of complex matrix)

I know that $f(x)=||Ax-b||_2^2$ (real matrix) has gradient $\partial f/\partial x=A^T(Ax-b)$. Now suppose $A$ is complex, then how can I prove that $\partial f/\partial x=A^*(Ax-b)$?
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### Why is the number of Hamilton circuits in $K_{n,n}$ is $\frac{(n!)^2}{2n}$

Why is the number of Hamilton circuits in $K_{n,n}$ is $\frac{(n!)^2}{2n}$? I simply saw this in some previous exam of Graph-Theory. I took for example $K_{2,2}$ and from each vertex I can go for two ...
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### How prove this $\frac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\frac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\frac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$

let $0<A,B,C<\dfrac{\pi}{2}$,and $A+B+C=\pi$,prove that $$\dfrac{\sin{(A-B)}\sin{(A-C)}}{\sin{2A}}+\dfrac{\sin{(B-C)}\sin{(B-A)}}{\sin{2B}}+\dfrac{\sin{(C-A)}\sin{(C-B)}}{\sin{2C}}\ge 0$$ my ...
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### A subset of a field that is a subfield

It can be verified that the following assertion is true: a subset $S$ of a field $F$ is a subfield if $S$ contains the additive and multiplicative identities 0 and 1, if $S$ is closed under addition, ...
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### Does a function have to be “continuous” at a point to be “defined” at the point?

I did search for whether this question was already answered but couldn't find any. Does a function have to be "continuous" at a point to be "defined" at the point? For example take the simple ...
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### Finding if the Point is in the triangle

Can You Please tell me if this is right Point(x,y,z) triangle points ABC using co-planer determinant |x-Ax y-Ay z-Az| |x-Bx y-By z-Bz| = 0 |x-Cx y-Cy z-Cz| then the point is in the tringle
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### Does $f(z)$ exist such that $f'$ and $f''$ exist in $\mathbb{C}$ but $f'''$ does not?

Is it possible to find $f(z)$ defined on $\mathbb{C}$ such that $f'$ and $f''$ exist everywhere on $\mathbb{C}$ but $f'''$ does not? I'm guessing no such $f(z)$ exists, but I don't know how to prove ...
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### Equivalent forms of the recursion theorem

I have found the next two definitions on the theorem of recursion: Definition 1: For any set $A$, any $a\in A$ and any function $g:A\times \mathbb{N} \longrightarrow A$, there exists a unique ...
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### Is this Michael subspace $M$ submetrizable?

For a more detailed discussion of Bing’s Example G in this blog, see the blog post Bing's example G. For the sake of completeness, we repeat the definition of Example G. Let $Q$ be the set of all ...
179 views

### How to convert a permutation group into linear transformation matrix?

is there any example about apply isomorphism to permutation group and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix
79 views

### A question on geometry?

I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ . Help. Thanks.
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### About the convergence of $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$

Does the series converge? $$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$$
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### Regarding orientation and orientation-reversing in local diffeomorphism

I am confused about orientation and orientation reversing in local diffeomorphism $f$ from manifold $X$ to $Y$ at some points. So, what does $f$ orientation-reversing at a point mean?
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### Local extrema of a function subject to an inequality

Preparing for my exams I came across a problem I don't know how to solve : Find the extremums of given function on domain $D$ and check if the function reaches it : $$f(x, y, z) = x + y + z$$ ...
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### $F(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)$ is integrable

I'm studying for an exam and I've encountered this exercise: Let $f(x)=x^{-0.5}\mathbb{1}_{\{0<x<1\}}$ and ${\{r_n\}}_{n=1}^{\infty}$ some enumeration of the rationals. Let: ...
145 views

### Proof regarding factorials.

Suppose $a$ and $k$ are positive integers, then how would you prove(not intuitively) that: $a!k! \leq (ak)!$ Although it is apparent that the inequality is correct, but how can I show this ...
102 views

### 3D Cartesian Transformation

I have a tetrahedron in a 3D Cartesian space. It has two orientations. I know the same three vertices positions (xyz) in the first orientation and the second orientation. I know the position of the ...
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### How many coordinates are unreachable?

I wanted to know, if a man was to go from $(0,0)$ to $(46,46)$ moving only straight and up with the following constraints:- If he walks right, he will walk atleast $4$ consecutive coordinates. If ...
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### Things that can happen to a differential equation

We have a list of things that can happen to a differential equation $y'(t)=f(t,y(t)), y(t_0) \in \mathbb{R}^n$ and $f: G \rightarrow \mathbb{R}^n$ continuous. That is given by (i) $b = \infty$ ...
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### decomposition of right Haar measure on homogeneous space

For simplicity, let $G_n=GL(n,\mathbb{R})$, $N_n$ be the upper trianguler unipotent subgroup, $P_{n-i}$ be the standard parabolic subgroup associated to partition $n=(n-i,i)$, and finally let $K=O(n)$ ...
### property of the exterior derivative $d \circ d=$ for a $\mathcal C^\infty$ function
One of the properties of the exterior derivative is that $d\circ d=0$. We're trying to prove this for the case $f\in\mathcal C^\infty (U)$ on an open set $U\subset \mathbb R^n$. The prove starts with ...