1
vote
0answers
124 views

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 ...
0
votes
1answer
160 views

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 $\partial f/\partial x=A^*(Ax-b)$?
0
votes
3answers
60 views

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 ...
6
votes
1answer
111 views

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 ...
5
votes
1answer
401 views

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, ...
4
votes
6answers
5k views

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 ...
0
votes
2answers
21 views

Generelized Integral Convergence

Here what i have to solve: $$\int_{0}^2 \frac{1}{x-1} \mathrm dx$$ I have to say if that integral convergse. What I did (which is false) : $$\int_{0}^2 \frac{1}{x-1} \mathrm dx=\int_{0}^1 ...
1
vote
1answer
42 views

A form of compactness?

Let $L$ be some first-order language and let $T$ be a collection of formulas. Assume then that for all structures $M$, we have a $\phi\in T$ s.t. $M\models \phi$. I'm trying to show that this implies ...
2
votes
3answers
712 views

Number of sequences with n digits, even number of 1's

ASKED: Let $c_n$ be the number of sequences with $n$ digits from $\{1,2,3,4\} $ with an even number of $1's$. Determine $c_n$ for $n \geq 0$. GIVEN RESULT: $c_{n+1} = 3 \cdot c_n + 1 \cdot ...
0
votes
1answer
208 views

Jacobian determinant and orientation

So in Jacobian determinant, it is often said that it gives information about whether Jacobian matrix changes orientation, but I cannot get what orientation exactly in this context.
2
votes
1answer
443 views

$\mathbb Q$-basis of $\mathbb Q(\sqrt[3] 7, \sqrt[5] 3)$.

Can someone explain how I can find such a basis ? I computed that the degree of $[\mathbb Q(\sqrt[3] 7, \sqrt[5] 3):\mathbb Q] = 15$. Does this help ?
0
votes
1answer
26 views

Divide objects into parts

I want to divide x distinct objects in some specified groups.. Lets say 3 groups of a,b,c number of I am able to find when objects are similar but not in this case.
1
vote
1answer
798 views

Electric field of finite sheet: Full analytical solution of integration?

I am trying to work out the integral $$E_{z}(x,y,z)=\alpha\int\int\frac{z\, dx'\, dy'}{((x-x')^{2}+(y-y')^{2}+z{}^{2})^{3/2}}$$ with the limits $$-\frac{a}{2}\leq ...
5
votes
2answers
134 views

Hartshorne exercise 1.6.4 : Is it true that $\mathcal{O}_{P,X} \cong \mathcal{O}_{\varphi(P),\Bbb{P}^1}$?

Let us work over a fixed algebraically closed field $k$ and consider a non-singular projective curve $X$ and $\varphi : X \to \Bbb{P}^1$ a non-constant morphism. My question is: For $P \in ...
0
votes
3answers
270 views

Create animation of parametric plot

I would like to create an animation of parametric plot. With a moving point on curve. The parametric functions are: x(t) = ...
4
votes
2answers
235 views

Criterion for sum/difference of unit fractions to be in lowest terms

Pick two nonzero integers $a$ and $b$, so $(a,b)\in (\mathbb{Z}\setminus\{0\})\times(\mathbb{Z}\setminus\{0\})$. We want to add the fractions $1/a$ and $1/b$ and use the standard algorithm. First ...
3
votes
2answers
64 views

Need to find function by the data

I have found interesting sequence, but I can't find its function. Here are the input and output data: ...
1
vote
1answer
265 views

Question - Möbius inversion formula

I need your help in the next question: Prove directly from the definition the Möbius inversion formula. (Möbius function defined as follows: μ(n) = 1 if n is a square-free positive integer with ...
1
vote
0answers
183 views

Proof of that SO(3) is not simply connected.

I want to prove that $\pi_1(SO(3))\cong \mathbb{Z}/2\mathbb{Z}$. I have already proved that there exists a surjection $\mathbb{Z}/2\mathbb{Z}\rightarrow \pi_1(SO(3))$.So I want to show that ...
0
votes
0answers
88 views

Enumerating the number of subsets of size i that sum to a specific value

Suppose we are given an integer $n$ and an integer $i$ where $i \le n$. We want to find all the subsets of {1, 2, 3 ... n-1} of size $i$ that will sum to $kn$ where $k$ is a positive integer. Edit: ...
2
votes
3answers
736 views

Equilibrium point linear differential equation

as far as I know, there is a comfortable criterium to find out whether the y=0 solution of $y'=Ay$ is stable if the maximum real part of the eigenvalues of A is negative and unstable if it is ...
2
votes
0answers
101 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
1
vote
3answers
77 views

how to proof the inequality$ B^2 \leq 2AC$ using other inequality

I proofed that that $ C(x^2) - 2Bx + 2A \geq 0$ is true for all X how can I show that $B^2 \leq 2AC $ ? how do I find an X? I know that $A, B , C $ are non-negative. thanks!
0
votes
1answer
71 views

how to calculate expected value in following scenario

I have to calculate the expected value of betting in following scenario: If team A has winning probability Pa ,and someone bets X amount on this team, he gains : ...
3
votes
2answers
511 views

How to calc the square root of a number without calculator? [duplicate]

How can I find the square root of a number without using a calculator?
1
vote
2answers
188 views

Let $g(x) = \int_{0}^{2^x} \sin(t^2)\,dt $. What is the $g'(0)$?

I'm kinda stuck in this exercise : Let $\displaystyle g(x) = \int_{0}^{2^x}\sin{(t^2)\,dt}$. What is the $g'(0)$? How do I approach this kind of thing? I was thinking about Riemann sums but ...
1
vote
2answers
2k views

How to calc arc sine without a calculator?

How can I find the arc sine of a sine without using a calculator? Thank you.
1
vote
2answers
60 views

Laplace Transform of $100e^{-5t}\sin10t$

Can anybody help me with the answer of this question? $$100e^{-5t}\sin10t$$
0
votes
4answers
566 views

Topology : R open domain and closed domain

I dont understand why we can consider the domain R and $\varnothing$ as close and open domains ? I have no idea of how to demonstrates it and like to see how to do it? Edit : My definition : an open ...
1
vote
2answers
77 views

Ordered set partitions

Let $a_n$ be a number of ordered partitions of the set $\left\{1,\ldots,n\right\}$, which means that order of elements in block is not relevant, but order of blocks does matter. (so $a_n = ...
0
votes
0answers
87 views

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
2
votes
1answer
77 views

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 ...
2
votes
1answer
160 views

Moving lemma - from Liu's book

This is from Liu's book. Let $X$ be an irreducible quasiprojective variety over an infinite field. Let $D_1 \ldots , D_n$ where $n=\dim X$ be Cartier divisors on X. Show that there exists $D_i' \equiv ...
2
votes
1answer
47 views

Instead of axiomatizing ordered fields, can we axiomatize just the right half?

Since ordered fields can always be split into two halves whose only common element is $0$, namely $(\leftarrow,0]$ and $[0,\rightarrow),$ I was wondering if we can axiomatize just the right half. Note ...
0
votes
2answers
56 views

Identity Matrix Question

Let $A$ and $B$ be a $n$ by $n$ matrix such that $A^2=I, B^2=I$ and $(AB)^2=I$. Prove that $AB = BA.$ Any hints on how to attempt this question? I'm stuck. But my first approach would be, $A^2B^2=I ...
2
votes
1answer
91 views

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 ...
2
votes
1answer
61 views

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 ...
1
vote
3answers
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
2
votes
4answers
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.
2
votes
2answers
94 views

About the convergence of $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$

Does the series converge? $$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$$
1
vote
2answers
71 views

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?
0
votes
2answers
118 views

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 : $$f(x, y, z) = x + y + z$$ ...
1
vote
1answer
34 views

$F(x)=\sum_{n=1}^{\infty}2^{-n}f(x-r_n)$ 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: ...
0
votes
3answers
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 ...
0
votes
1answer
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 ...
-1
votes
2answers
97 views

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 ...
0
votes
1answer
43 views

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$ ...
1
vote
0answers
122 views

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 $K=O(n)$ ...
5
votes
1answer
157 views

Relation between quadratic refinement and quadratic form

The question in the title has now been bothering me for days. I first came across the term quadratic refinement when I read about the Kervaire invariant when reading Kervaire's 1960 paper. The ...
1
vote
1answer
131 views

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 ...

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