# All Questions

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### Limits of the derangements proportion within the permutations of the set $[1,n]$

Let be $D_n$ the number of derangements of a set of $n$ elements, by convention we have $D_0=1$ Ifound that $D_n=n!\sum\limits_{k=0}^{n}\frac{(-1)^k}{k!}$ For all $n\in \mathbb{N*}$, we write ...
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### Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space?

I tried starting with multiple bases of the null space and each time I was able to write the RREF form of the matrix. However, I have not been able to prove that this is true for all possible bases.
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### Find both the diagonals with area and side given [on hold]

Side of rhombus $= 65$ cm Area of rhombus $= 1024$ cm$^2$ Find the diagonals of the rhombus.
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### Length of a Chord of a circle

I was wondering about the possible values that the length of a chord of a circle can take. The Length of a chord is always greater than or equal to 0 and smaller ...
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### Example of a Lebesgue unmeasurable function f such that f*f is Lebesgue measurable

Giv an example of a Lebesgue unmeasurable function $f:[0,1]\rightarrow \mathbb{R}$ such that $f^2$ is Lebesgue measurable.
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### Smooth points of the secant variety with a given tangent space

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...
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### Another version of the Poincaré Recurrence Theorem (Proof)

The task is to prove the following version of Poincaré's Recurrence Theorem: Let $(X,\Sigma,\mu)$ be a finite measure space, $f\colon X\to X$ a measurable transformation that preserves the ...
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### What it the fourier transform of laplacian and shifted funtion?

I'm looking for the Fourier transform of $\nabla^2f(\vec{r}-\vec{a})$ I can assume that the 3D Fourier transform of $f(\vec{r})$ is $\tilde{f}(\vec{q})$ and the vector $\vec{a}$ is a const vector. ...
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### Question concerning comparison of different tetration functions

Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$ Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$ Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$? If so, is the proof elementary? (Use only ...
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### Prove or disprove that $(a_1+a_2+\ldots+a_n)\leq n\sqrt{a_1^2+\ldots+a_n^2}$, by showing that $RHS-LHS\geq 0$ if possible.

Prove or disprove that $$\left|a_1\right|+\left|a_2\right|+\ldots+\left|a_n\right|\leq n\sqrt{a_1^2+\ldots+a_n^2}$$ Where $a_1,\ldots,a_n\in\mathbb{R}$ and $n\in\mathbb{N}$. EDIT: I was hoping there ...
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### Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers. The organizer expects a margin profit ...
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### Number of Labels used in reduction of Isomorphism of Labelled Graph to Graph Isomorphism

From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , Assume that both $X$ and $X'$ have $n$ vertices. We plan to code the graph labels as suitable subgraphs which we attach to the ...
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### Geometry-||gm proof

$ABCD$ is a parallelogram in which $P$ and $Q$ are the mid points of the sides $AD$ and $BC$ respectively. If $BP$ & $QD$ intersect the diagonal $AC$ at $X$ and $Y$ respectively then prove that ...
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### Confused in some basic concept about Gauss Map $dN_p$

Here, I have some question that remain unsolved for quite a long time. My question is about the gauss map $dN_{p}$, to start the convenient expression of the symbol and formula, I have to construct ...
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### I do not understand the hypothesis of the lebesgue decomposition theorem

I do not understand the hypothesis of the lebesgue decomposition theorem. Given a mesure in a sigma-algebra i do not understand why exists a set it is concentrate on.
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### Probability and Statistics

How can I check if a Moment Generating Function is valid or not? I tried using the definition for that but it didn't help me at all.
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### Given probability distribution $f(x)=2-bx$ find $b$ and range for $x$

Suppose that the distances between houses and the center of a city are distributed with the density function: $f(x)=2-bx$, where $x$ denotes distance. If this is a proper density function, what can we ...
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### Is this complex vector bundle trivial?

Let $\Sigma$ be any Riemann surface, and let $L \rightarrow \Sigma$ be a complex line bundle (which is classified according to its degree). Then the vector bundle $L \oplus L^{-1} \rightarrow \Sigma$ ...
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### How did they calculate the possible endings?

On this link @edit you can see all the possibilities of endings. The game has six stages, on each you have 3 choices and at the end, you have 5 stages with 2 endings each. Its like: 1. > 2a 2b 2c > ...
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### Finding the formula for acceleration from $v=2s^3+5s$, where $s$ is the displacement at time $t$

This is the question: I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't ...
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### Find the sum $\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$

Find the sum $$\sum _{ k=1 }^{ 100 }{ \frac { k\cdot k! }{ { 100 }^{ k } } } \binom{100}{k}$$ When I asked my teacher how can I solve this question he responded it is very hard, you can't solve it. I ...
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### Let $B = {n \in \mathbb{Z} : n = 3j + 2; j \in \mathbb{Z}}, D = {n \in Z : n = 3j − 1; j \in \mathbb{Z}}$. Is $B = D$?

Let $B = {n \in \mathbb{Z} : n = 3j + 2; j \in \mathbb{Z}}, D = {n \in Z : n = 3j − 1; j \in \mathbb{Z}}$. Is $B = D$? How do I prove this? To me it looks to be true. But I don't know how to put it ...
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### Existence of two disjoint closed sets with zero infimal distance

Are there two closed sets $A,B\subset\mathbb{R}^2$ with the following properties? $A\cap B=\emptyset$ $\forall \epsilon>0$ there exist $a \in A$ and $b\in B$ such that $\|a-b\| < \epsilon$
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### Find $\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$

What is the value of the following sum? $$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$ where $\gcd$ is the greatest common divisor.
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### Given any 40 people, at least four of them were born in the same month of the year [on hold]

Given any 40 people, at least four of them were born in the same month of the year. Why is this true?
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### Definition of fixed point free relation

If we have such relation that for $\forall x$ $f(x)\ne x$ , how is it called in one word? I can come up with only "graph of this function is not a straight line:)" Thank you
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### Want to know what's wrong?

I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ...
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### Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
### $\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?
$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?