0
votes
1answer
8 views

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 ...
0
votes
0answers
9 views

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.
-4
votes
1answer
20 views

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.
7
votes
6answers
161 views

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 ...
0
votes
1answer
16 views

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.
0
votes
0answers
4 views

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 ...
2
votes
1answer
133 views

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 ...
1
vote
2answers
15 views

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. ...
2
votes
0answers
5 views

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 ...
-2
votes
1answer
55 views

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 ...
1
vote
0answers
25 views

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 ...
1
vote
0answers
5 views

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 ...
0
votes
1answer
18 views

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 ...
0
votes
0answers
10 views

Non-linear simultaneous recurrence system

Suppose I have a system of discrete time, non linear recurrence equations of variables $a_1,\ldots,a_k$, and constants $C_1,\ldots,C_k$ that can be represented as: $a_i^t = ...
5
votes
5answers
557 views

Why derivative can be non-linear?

One of definitions of derivative is the slope of the tangent line. For example, $x^3$ has a quadratic derivative. How could the slope of tangent line be non-linear?
1
vote
2answers
35 views

Proof $\dbinom{n}{0} - \frac{1}{2}\dbinom{n}{1} + \cdots + (-1)^n\frac{1}{2^{n-1}}\dbinom{n}{n-1}$

For all $n \ge 1$, $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} + \frac{1}{2^2}\binom{n}{2} - \frac{1}{2^3}\binom{n}{3} + \cdots + (-1)^n\frac{1}{2^{n-1}}\binom{n}{n-1} = 0,$$ if $n$ is even               ...
0
votes
2answers
68 views

Non-abelian fundamental group on a path-connected space

I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise ...
0
votes
1answer
12 views

$X$ is a normed linear space such that for some compact $K\subseteq X$ , $\operatorname{span} K$ is dense in $X$ then is $X$ separable?

Let $X$ be a normed linear space which is separable. Then I know that there exists a compact subset $K$ of $X$ such that $\operatorname{span} K$ is dense in $X$ (in fact we can also find compact and ...
1
vote
1answer
17 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
7
votes
1answer
47 views

Prove/Disprove : Every polynomial with prime degree and coefficients in $[-1,1]$ has galois-group $S_p$

Conjecture : Let $p$ be a prime number , $f\in \mathbb Z[X]$ an irreducible polynomial with degree $p$ and coefficients in the range $[-1,1]$. Then the galois group of $f$ over $\mathbb Q$ ...
2
votes
2answers
28 views

Average number of events happening if each happens with $p=\frac{1}{n}$ and we run it $10000 n$ times.

Let an event $e$ have probability of happening $\frac{1}{n}$. Let us assume we have $m$ independent possibilities for similar events to happen. With $m>>n$. What is the average number of times ...
0
votes
0answers
6 views

The left kernel of a positive linear functional and its $w^*$-extention

Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. We let $\tilde{\phi}$ be its unique $w^*$-continuous extension on $A^{**}$. It is supposed to focus on the left kernel of ...
3
votes
0answers
40 views

Help with Proposition 13.2.9 in Ireland and Rosen

I'm currently self studying Ireland and Rosen's A Classical Introduction to Modern Number Theory and got stuck on the proof of Proposition 13.2.9. In this proof, $p$ is a prime not dividing $m$, $D, ...
1
vote
0answers
10 views

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 ...
0
votes
0answers
15 views

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.
-1
votes
0answers
16 views

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.
-1
votes
1answer
21 views

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 ...
0
votes
0answers
18 views

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 $L \oplus L^{-1} \rightarrow \Sigma$ ...
-2
votes
1answer
47 views

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 > ...
2
votes
4answers
23 views

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 ...
4
votes
0answers
37 views

Show that for $\lambda<0$ we have $\inf(\lambda A)=\lambda \sup(A)$

For $A\subset \mathbb{R}$ and $\lambda \in \mathbb{R}$ let's define: $$ \lambda A = \{\lambda a: a\in A\} $$ I have to prove that for $\lambda<0$ and bounded $A$ we have $\inf(\lambda A)=\lambda ...
1
vote
1answer
40 views

functor from complex algebraic variety to constructible function

I am reading MacPherson's paper "Chern Classes for singular varieties". Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a ...
2
votes
0answers
40 views

A kind of Sturm-Picone theorem?

My question is very simple: Suppose $u,v:(a,b)\subset \mathbb{R} \to \mathbb{R}^+$ solve \begin{equation} (p(x)u'(x))'=-q(x)f(u(x)) \end{equation} \begin{equation} (p(x)v'(x))'=-r(x)g(v(x)) ...
0
votes
5answers
62 views

Summation for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$

I know that the closed form of $\sum\limits^n_{k=1}\:k=\frac{n(n+1)}{2}$ But I'm not sure what the closed form for $\sum\limits^5_{i=2}\:\left(3i\:-\:5\right)$ would be. Any push in the right ...
2
votes
1answer
18 views

Non-Borel a.e limit of Borel functions

As a homework assignment I'm supposed to prove or disprove Borel measurability is closed under a.e convergence. I think this is not true because the Borel $\sigma$-field is not complete. However, I'm ...
0
votes
1answer
14 views

Show that $\hat{\mu}$ has minimal variance

So two independent analyses of a content in a water sample have been made using two different methods, both without systematical errors but with different standard deviations. Method $B$ is assumed to ...
2
votes
2answers
150 views

Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
1
vote
1answer
15 views

Normal real matrix

Q: Is there a normal, real matrix, which it's characteristic polynomial is $t(t-1)(t^2+1)$ ? I think that there isn't such matrix, and I prove it by: if a matrix is real than it's diagonalizable, ...
1
vote
2answers
33 views

Homogeneous localization and regularity

Let $k$ be a field, $S = k[x_0,\dots,x_r]$, $I$ a homogeneous ideal of $S$ and $R=S/I$. Let $P$ be a homogeneous prime ideal of $R$ and let $R_{(P)}$ be the homogeneous localization of $R$ at $P$. I ...
1
vote
0answers
13 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
8
votes
3answers
145 views

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 ...
1
vote
2answers
24 views

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 ...
2
votes
1answer
69 views

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 $\|a-b\| < \epsilon$
2
votes
1answer
61 views

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.
-5
votes
2answers
47 views
0
votes
0answers
34 views

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
3
votes
0answers
48 views

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 ...
0
votes
2answers
1k views

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 ...
5
votes
1answer
46 views

$\pi$ 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?
0
votes
1answer
19 views

Changing the order of a double integration ? $\int_{-5}^{5}dx\int_{-7}^{\sqrt{25-x^2}}f(x,y)dy$

I've been doing an example of changing the order of a double integral and I'm not sure if I did it right. I would really appreciate if someone would check if my solution is right and correct any ...

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