# All Questions

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### Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
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### Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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### Computing the inverse explicitly (real analysis)

I have a function $\ f:\mathbb R\to\mathbb R$ such that $\ f(x,y)=(xe^y,xe^{-y})$ Let $\ a=(1,0), b=(1,1)$ and let $\ g$ be the continuous inverse of $\ f$ such that $\ g(b)=a$. Compute $\ g$ ...
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### How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
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### Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
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### Proving that a set is denumerable without using a particular theorem

this question may seem like a duplicate of another one that I asked, but it is not. In this question, I am not allowed to use the Theorem which states: Every infinite subset of a denumerable set is ...
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### Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $\mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
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### this function is in some Holder space?

Consider $\Omega$ a open, bounded, and smooth domain of $R^N$ with $N \geq 3.$ And let $f: \Omega \times R \rightarrow$ a Caratheodory function . Supoose that $f$ is locally Lipschitz. Supose that ...
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### How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
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### Show that the $n$th cyclotomic polynomial is in $\mathbb Z [t]$

For any $n$ we define the $n$th primitive root of unity to be an element $z \in \mathbb C$ such that $z^n =1$ but $z^r \neq 1$ for all $1 \le r < n$. We have proofed there are $\varphi(n)$ ...
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### Covariance of two values

A fair die is rolled twice (independently). Let X1 and X2 be the numbers resulting from the first and second rolls, respectively. Define Y=X1+X2 and Z=4⋅X1−X2. Find the covariance between Y and Z. ...
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### Help in Stats, Joint p.d.f

Let $X$ and $Y$ be random variables that have a joint p.d.f., which is given by the formula $\displaystyle p_{X,Y}(x,y)=\frac{5e^{−5x}}{x}$ when $0< y < x < \infty$, and $p_{X,Y}(x,y)=0$ for ...
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### Is every smooth function from $\mathbb{R}^n \to \mathbb R$ with compact support the laplacian of some function?

Given a smooth function $g:\mathbb R^n \to \mathbb R$ with compact support, is it true that there exists a function $u$ such that $g=\Delta u$?
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### Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$\sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right],$$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
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### Unusual 3D Packing Problem

I made up this interesting problem playing with wire sculptures: If I have a $10 \times 10 \times 10$ clear box and inside I can put wireframe unit cubes, what's the maximum number of unit edges (or ...
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### proving a natural projection is linear and finding its kernel

Let $V_i = 1,...,N$ be a collection of vector spaces over a field $F$. Consider the Cartesian product $V=V_1 \times V_2 \times ... \times V_N$ with the natural projections $\pi=V \rightarrow V_i$. ...
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### Gauss-Laguerre quadrature and error estimation

While using Gauss-Laguerre quadrature of varying orders, n, to estimate the value of integral what happens to the error as the value of n increases. would it increase or decrease and why?
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### Expected Value for coin flips for five heads.

Consider a coin that comes up heads with probability $p$ and tails with probability $1-p$ We flip this coin (independently) and stop as soon as it comes up heads for the 5th time. Let X be the random ...
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### uniformly continuous and bounded derivative

When I learn uniformly continuous function, I used to view uniformly continuous function as a function with bounded derivative. Most of time up to now it seems right. But can we prove or disprove ...
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### Why is $LG(n) \cong U(n)/O(n)$?

Let $LG(n)$ be a Lagrangian Grassmanian manifold. That is, $LG(n)$ is the set of Lagrangian subspaces of a symplectic vector space of dimension $2n$. Why is $LG(n)$ can be identified with $U(n)/O(n)$? ...
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### Show that $(\forall x)(A \lor B) \rightarrow A \lor (\forall x)B$ is, in general, NOT a theorem.

Show that $$(\forall x)(A \lor B) \rightarrow A \lor (\forall x)B$$ is, in general, NOT a theorem. My answer: First, I got the abstraction of the formula which is $p \rightarrow A \lor q$ then ...
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### Using BCR experiment

consider a random experiment of observing a mechanical or electrical unit consisting of five components and determining which components are working and which have failed. Use the BCR to find the ...
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### Which probability to calculate?

I was wondering if someone could help me with the probability question stated below. The probability that a particular moth trap $A$, collects $r$ moths overnight is given by $(1-\alpha)\alpha^r$ for ...
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### Tangent plane that passes through a point

How would I find the (a,b) that satisfies that the tangent plane to $f(x,y) = (x^2) + 2xy + (y^2)$ passes through the point $(2,1,0)$ ? I know that $f(x)= 2x + 2y$, and $F(y): 2x + 2y$. Therefore ...
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### prove following properties of self-adjoint operator

$A: V \rightarrow V$ self-adjoint; $b$ is a real number. Show 1) the minimal polynomial has distinct roots; 2) $\ker(L) = \ker(L^k)$ for $k\geq1$; 3) $\text{im}(L) = \text{im}(L^k)$ for k bigger ...
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### show that exist a left quasi-inverse of any element in J(R)

I studying for math. but this problem, I don't understand. J(R): the radical of R, written as J(R),is the set of all elements of R which annihilate all the irreducible R-module. and show that exist a ...
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### Counting processes question

Let's say that arrivals at a counter come at times of a Poisson process with rate $\lambda$. A ball that arrives to an unlocked counter is registered and then locks the counter for an amount of time ...
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### Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...