2
votes
0answers
12 views

Tangent of Circles

$k_1$ is a circle with center $O_1$ and radius $r_1$. Similar for $k_2(O_2;r_2)$. $r_1 < r_2$. $AB$ and $CD$ are tangent lines to $k_1$ and $k_2$. Prove that $AP=DQ$.
0
votes
0answers
16 views

Improper integral test

I am looking for a reference for this fact (or a proof): The Improper integral $\int_{1}^\infty f(x) dx $, where $f$ is positive and continuous, exists if and only if $\lim_{x\to \infty}\frac{\log ...
0
votes
0answers
18 views

True or false: sets, subsets, and topologies in $\mathbb R$

I am pondering the following statements about sets, subsets and topologies in $\mathbb R$. The empty set is a closed subset of $\mathbb R$ regardless of the topology on $\mathbb R$. Any open ...
0
votes
0answers
2 views

Restriction, measurability

I have a question about $\sigma$-algebra generated by functions. Let $(S,\Sigma)$ be a measurable space and $u_{n}, n \in \mathbb{N}$ be $\Sigma$-measurable $\mathbb{R}$-valued functions on $S$. $A ...
0
votes
0answers
6 views

Compute a sum with finite and infinite elements

I would like to compute the following summation: $$ s = \sum_{i=1}^n a_i \, \Phi^{-1}(u_i) $$ where $\Phi^{-1}$ is the inverse of the standard Gaussian distribution function, $a_i$ are some real ...
0
votes
0answers
4 views

Linear operator differentiation on a torus

I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence. 4.1. Linear involutions. We start characterizing the linear involutions $R \! : ...
2
votes
0answers
10 views

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length.

Prove that cube connot be tiled with $n>1$ cubes, such that all of them have different side length. I believe this is not hard problem, but I just do not have an idea how to start. I tried to ...
0
votes
0answers
2 views

Difference of two measurements (=means) from two normal distributions

I need help to understand which statistical test can be applied to test whether two subsequent measurements (from two different instruments measuring the same quantity) are signifcantly different from ...
0
votes
0answers
2 views

Adjustment coefficient problem

Claims arrive at an insurance company as a Poisson process {$N(t) : t \ge 0$} at rate $\lambda > 0$ and $X_i$ is the claim size of the $ith$ claim. I assume that {$X_i, i=1,2,...$} is iid ...
0
votes
1answer
12 views

Show that this integral is finite $\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx$

Let $p > -1$ and $r \in \mathbb{N}$, show that $$\lim_n \int_0^n x^p (\ln x)^r \left(1 - \frac{x}{n} \right)^n dx = \int_0^\infty x^p (\ln x)^r e^{-x} dx$$ and that this integral is finite. To ...
0
votes
0answers
16 views

How can $n^5+4$ be a perfect square?

How can one find all $n \in \mathbb{N}$ such that $n^5+4$ is a perfect square? I see that $n^5=(x+2)(x-2)$ here im suck can someone help ?
1
vote
0answers
11 views

Find $\int \limits_0^1 \int \limits_x^1 \arctan \bigg(\frac yx \bigg) \, \, \, dx \, \, dy$

Find $$\int \limits_0^1 \int \limits_x^1 \arctan \bigg(\frac yx \bigg)dx \, \, dy$$ So obviously using cylindrical is the way to go to give $\theta r$ inside the integral (after considering the ...
0
votes
3answers
22 views

How to find $\int_{0}^{x}\sqrt{t^2-2|t|+1}dt$

How to find $\int_{0}^{x}\sqrt{t^2-2|t|+1}dt$ where x>0. I tried to know how will the Wolfram treat with absolute term and how to find this integral, but I found that the Wolfram couldn't ...
0
votes
0answers
7 views

Expectation subscript notation

http://www.stat.cmu.edu/~larry/=stat705/Lecture8.pdf At the bottom of page 1 in this pdf, they say the expectation is taken over theta, but I don't understand why the expectation isn't over X when the ...
0
votes
1answer
15 views

Solving the problem with Fourier

I want to solve the following problem: $$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ $u$ bounded I have done the ...
0
votes
0answers
1 view

Unsure how to get the PDF from a CDF (Xi ~U(0,θ)) so that I can use it to prove an estimator is biased

Fx max (t) = (t / θ) ^ n for 0 < t < θ Now part of my confusion is what to differentiate with respect to as t is a constant to get the PDF? If I differentiate the CDF with respect to x, I'll ...
0
votes
0answers
3 views

Expectation of bernoulli trials

Could someone let me know if this looks correct? Let $X$ denote the number of successes in n bernoulli trials and let $Y$ denote the corresponding number of failures. Find an expression for ...
0
votes
0answers
7 views

Set notation and translation

I've been checking my understanding of sets by setting up the following for myself: $\{ 1, 3, 5, 7, ... \}$ = the set of all natural odd numbers $\{ ..., -4, -2, 0, 2, 4, ... \}$ = the set of all ...
1
vote
1answer
6 views

Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
0
votes
0answers
6 views

Non homeomorphic spaces with same homology groups [duplicate]

Is it possible for two spaces X and Y to have the same homology groups with X not homeomorhpic to Y.
0
votes
0answers
5 views

Is there a ring - homomorphism $\mathbb{F}_p \rightarrow \mathbb{F}_q $ (p,q prime , $p \not= q$ )?

So we have two prime fields and seek a homomorphism between them. I assume that i have to find a homomorphism that is valid for all p,q prime , $p \not= q$, not just one for each choice. I would say ...
0
votes
0answers
11 views

Injectivity Unclear

Let $R=K[x_1,...,x_n]/I$ and $m$ be maximal ideal of $R.$ Let $(s_1,...,s_d)$ be a base of $m/m^2$ where $dim(R_m)=dim(m/m^2)=d.$ Then by Kunz Chapter V.5.10 the canonical epimorphism ...
0
votes
0answers
3 views

Relationship between inductive reasoning and first order reasoning

I know what is induction and tableau reasoning. I happen to see that if reasoning is done via induction, then the reasoning is not first order. Why inductive reasoning and first order reasoning are ...
1
vote
0answers
8 views

prove if $Z \subset \mathbb{R}^2$ has zero-content and $U \subset Z$, then $U$ has zero-content

I have to prove if $Z \subset \mathbb{R}^2$ has zero-content and $U \subset Z$, then $U$ has zero-content. The definition of zero-content that my textbook states is this. A set $Z \subset ...
0
votes
1answer
9 views

Finding the average temperature over a certain amount of time

I'm assuming you use some time of equation such as the Mean Value Theorem of Integrals to find the average here, but the equation is not represented in the form of an integral, so I'm not entirely ...
-2
votes
0answers
10 views

Question of probability .

the probab. that any one of the men A1,A2,A3,A4 is alive after 95 yr.s of age is 1/2 . the probab. that A1 will die at the age of 95 and will be the first to die is?
0
votes
1answer
13 views

Quadratic equations

The width of a rectangular TV screen is 22.9 in longer then the height. Of the diagonal is 60 in find dimensions of the screen. 1. Using Pythagorean theorem express the given information in the ...
6
votes
2answers
25 views

Sumo Wrestler partial round robin tournament

A coworker of mine is a big fan of Sumo. He recently came to me with a problem he's been wondering about for years: In a professional sumo tournament there are 42 wrestlers. A tournament lasts 15 ...
3
votes
1answer
27 views

Error in linear algebra proof

In "The linear algebra a beginning graduate student ought to know" by Golan there is a proof that if $V$ is a vector space of over a field $F$ and $D$ is a maximal linearly-independent subset of $V$ ...
1
vote
3answers
17 views

Is the function max{x,y} defined if x and y take equal values?

If x and y take the same values, will the function return a result? I am asking this as maximum means greatest of two values. So if both the values are equal, the existance of the function confuses ...
1
vote
1answer
10 views

The geometry meaning of Riemann–Stieltjes integral

Maybe my question seems so strange but I want to know what is the geometry meaning of Riemann stieltjes integral ??
2
votes
1answer
11 views

What questions are independent from the axiom of constructibility?

Wikipedia gives a list of statements true in L which would be true also for set theory if the axiom of constructibility (V=L) holds. However I wonder about the converse: Are there any important open ...
0
votes
1answer
21 views

$X_n\rightarrow X$ in probability, but $\mathbb{E}(X_n)$ does not converge to $\mathbb{E}(X)$ [duplicate]

What is an example of a sequence $X_1,X_2,...$ such that $X_n\rightarrow X$ in probability, but $\mathbb{E}(X_n)$ does not converge to $\mathbb{E}(X)$?
0
votes
1answer
21 views

In a connected graph, if the maximum path you could make is of length 100, and there are two paths of length 100, aren't they the same path?

Here's the question: Let G be a connected graph. (Remember that this means that every two vertices of G can be joined by a path starting at one and ending at the other.) Suppose also that G ...
0
votes
1answer
9 views

metric on the set of complex sequences

Let X be the set of complex sequences $(a_n)_{n\in\mathbb{N}}\in \mathbb{C}$. Show that the transformation: $$ d((a_n), (b_n)) := \sum_{n=0}^\infty \frac{1}{2^{n+1}} \frac{|a_n - b_n|}{1 + |a_n - ...
1
vote
0answers
6 views

Proving a.s. convergence for martingales

Let $ε_n, n > 1$, and $V_n, n > 0$, be independent random variables, with $P(ε_n = 1) = P(ε_n = −1) = 1/2$, $P(V_n = 1) = p_n, P(V_n = 0) = 1 − p_n$, for all n. Define $X_n$ inductively by $X_0 ...
1
vote
1answer
9 views

xor-ing vectors

This question might be wrong on mathematics, but I don't know where else to put it. I have a given equation, and there is one calculation step, that I don't understand. I thought, I have to xor ...
0
votes
1answer
14 views

Canonical Jordan Form

Hello I have a lot of trouble trying to put this matrix in Jordan form. I do not really understand Jordan form theory so any help or comment would be much appreciated. $$\begin{matrix} ...
1
vote
0answers
14 views

Express one constant as a linear combination of two other constants?

Let a,b,n,m,o be nonzero rational scalars. How do I express o as a linear combination of n and m with coefficients a and b? Explicitly how to find a,b such that an + bm = o? I know how to do linear ...
0
votes
1answer
10 views

Finding volume of a cone given density

Let $C$ be the solid cone with the boundary surfaces $x^2 +y^2 = z^2$ and $z = 0$. The density of the solid at point $(x,y,z)$ is $z$. Find the volume of the solid using the integrals in both the ...
7
votes
0answers
52 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...
1
vote
1answer
10 views

verifying properties of relations to test equivalence

We are doing some more with relations and this time we are given a relation and told that it is not equivalent. We need to find out which property it does not fulfill. So we at least know there is ...
-1
votes
0answers
15 views

Rational,and irrational number between any 2 real numbers.How to prove?

If $x,y$ are 2 real numbers such that $x < y$ how to to prove there is an $r$, belongs to the set of rational numbers, and a $i$, belongs to the set of irrational numbers and hence many more ...
1
vote
0answers
9 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
0
votes
0answers
10 views

Line integral simplification

Suppose I have a line integral of the form $$ I = \int_C {\bf \ddot{x}} \cdot \mathrm{d}{\bf x} $$ where ${\bf x} \in \mathbb{R}^n$ and $C$ is some curve. Suppose I also know that ${\bf \ddot{x}} = ...
0
votes
0answers
7 views

Isomorphism from $SL_2(F_2)$ to Sym(3)

Can you help me find a isomorphism from $SL_2(F_2)$ to Sym(3). I supose that I have to use vectors. ex. $v_1 (1,0), v_2(1,1), v_3 (0,1)$, but I don't remeber how to do it.
2
votes
1answer
15 views

$p$-group acting on a finite set

Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$. Any thoughts? I'm stumped ...
0
votes
0answers
18 views

How fast things con/diverge.

Is there some kind of standard listing for how fast common series converge/ diverge? For instance, is it true that this order holds (diverges faster closer to the top)? Pretend there's a ...
-1
votes
1answer
10 views

Using var[ r ] = E[ r^2 ] − E[ r ]^2, obtain the formula σ^2 = w1^2σ1^2 + w2^2σ2^2 + 2w1w1σ12

Able to solve it using var(x)= E[(x-E(x))^2] but cannot get anywhere with this one
1
vote
0answers
11 views

Fourier inversion Lemma (Lars Hörmander)

I always like to have more than one proof for the same theorem. The other day I was browsing through my copy of Lars Hörmander's book on PDE (volume 1). When proving the fourier inversion formula (on ...

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