0
votes
4answers
51 views

Expectation of non-negative random variable

Let $X$ be a non-negative random variable. In a proof for $E[X]=\int_0^\infty P(X>t)dt$ from the answer of this question, we use Fubini for the middle quality. Why do we need $X$ to be ...
0
votes
0answers
34 views

A matrix transformation from R^4 to R^3 - linear algebra - how to find the image of a point

I'm trying to revise for an upcoming exam on linear algebra and have come across this question. I do not understand the line "the image of a point (x1, x2, x3, x4) can be computed from the defining ...
1
vote
1answer
11 views

transition maps of a principal bundle are smooh

A smooth principal fiber bundle is a smooth fiber bundle $\pi: E \to M$ together with a Lie group $G$ and a fiber preserving right action $E \times G \to E$ which restricts to each fiber freely and ...
0
votes
0answers
9 views

In projective geometry the dual of the cross ratio dual is an angle measurement?

I am trying to get my head around angles in projective geometry. I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley ...
0
votes
0answers
19 views

Basis of $H^1(\Omega)$ which is orthonormal wrt. $L^2(\partial\Omega)$ inner product?

Let $\Omega$ be a domain with $\partial\Omega$ bounded. Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product? ...
-8
votes
0answers
50 views

hard question, please help [on hold]

11) Assume a sorted array (A) of size n. Propose an algorithm for finding two elements x and y in A that minimize |x-y|. Your algorithm should run in O(n) time for full credit. (Note: |x-y| represents ...
-2
votes
0answers
19 views

Algorithm for vector space transformation [on hold]

In my text book I've got an example which is as follows: Create an algorithm which calculates coordinates of a point after a space transformation took place. Transformations may be scaling or ...
2
votes
2answers
48 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
1
vote
3answers
46 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
-2
votes
2answers
72 views

If we know $a^{1 / 2} + a^{-1/2}$, how can we calculate $a + a^{-1}$?

Someone could help me with this? Thanks so much! Knowing the value of $a^{1/2}+a^{-1/2}$, calculate $a+a^{-1}$.
-4
votes
2answers
47 views

How many 10-character strings can be formed using p characters [on hold]

Let $A = \{a_1, a_2, . . ., a_p\}$. In how many ways you can select 5 elements? Let $A = \{a_1, a_2, . . ., a_p\}$ is a set of $p$ symbols. In how many ways you can make codes consists of 10 symbols? ...
0
votes
1answer
35 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
2
votes
4answers
62 views

How to find the number of possible outcomes of 10 games between 20 teams?

Hi I am looking for an equation to find possible combinations in a non repeating format with a twist. Here is the example: There are 10 games between 20 teams. I have to chose 5 winners but ...
0
votes
0answers
35 views

Geometric proof of BD/BE = CD/CE ; methods for congruent triangles

It seems to me that if BAC = DEA this would be straightforward, but I lost myself in the variety of congruencies here.
-1
votes
4answers
45 views

How many ways are there to prepare one of 400 varieties of coffee in one of 7 ways?

I'm hoping someone can check my thinking: I have 400 distinct varieties of coffee. Each can be prepared in 7 ways (black, cream and sugar, etc.). How many possible combinations are there? I'm thinking ...
2
votes
0answers
38 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
4
votes
3answers
263 views

Image of open set is not open?

I'm confused by the proof that $\epsilon$-$\delta$ continuity is equivalent to open-set continuity. One can prove that a function is $\epsilon$-$\delta$-continuous if and only if the preimage of any ...
0
votes
0answers
38 views

Finding $g$ such that $(f(g(x)))'=1$ when $f:[-1,1]\to S$

This is probably a simple question, but had a little trouble figuring it out, so hopefully someone here knows how to do it. Suppose $f:[-1,1]\to S$, where $S$ is some set endowed with a nonnegative ...
0
votes
2answers
31 views

Bound on and integral

If $\alpha \in \Bbb R$, how can I show $$\int_{-M}^M \frac{1}{\sqrt{|x-\alpha|}} \, dx \le 4 \sqrt{M}$$ For $M>0$, Rewriting the integral gives $$\int_{-M+\alpha}^{M + \alpha} ...
-4
votes
0answers
27 views

homogenous differential equation with variable coefficient [on hold]

Please help to find solution of boundary value problem $$ y''+xy=0 $$ $x \in [a,b]$ with $y(a)=y(b)=0$
1
vote
0answers
39 views

How to solve $4x^2\cos y\sin y\partial{y}-3x\sin y\partial{x}+8\sin^2y\partial{y}=0$?

$$4x^2\cos y\sin ydy-3x\sin ydx+8\sin^2ydy=0$$ find the solution of this Bernoulli equation. How can I start?
1
vote
0answers
23 views

Proving a relation for representations of gauge groups

Let ${\cal G}$ be a Lie group - possibly disconnected. Let ${\mathfrak g}$ denote the corresponding Lie algebra. Let $R_k$ be a general unitary representation of ${\cal G}$ and $R$ be the adjoint ...
0
votes
0answers
31 views

How to start a proof ? What kind of mathematical tool I can use here? [on hold]

I have a set of $n$ points $\{A_1,A_2,...,A_n\}$. I draw every triangle formed with $3$ points $A$. What mathematical tool can I use to describe intersections between all these triangles ? I would ...
1
vote
1answer
32 views

Series with $n$th term having integer raised to the power of $n$ in the denominator

$$ 1+\frac{4}{6}+\frac{4\cdot5}{6\cdot9}+\frac{4\cdot5\cdot6}{6\cdot9\cdot12}+\cdots $$ I could reduce it to $n$th term being $\dfrac{(n+1)\cdot(n+2)}{n!\cdot3^n}$. Took me an hour just to get ...
4
votes
4answers
66 views

$\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ dense in $\mathbb{R}$? [duplicate]

I'm guessing $\{ a + b\sqrt{2} \ : \ a, b \in \mathbb{Z} \}$ is dense in $\mathbb{R}$. I'm having a mental block. How do you show that? (This is motivated by a different hypothesis: if $f$ is ...
0
votes
1answer
28 views

Hint on metric space

I want show that $(a_n):=d(x_n,y_n)$ converges, if $(X,d)$ is a metric space, $(a_n)$ and $(b_n)$ are cauchy sequences in $(X,d)$. Here is what i do; From the hypothesis, $(a_n)$ is bounded, because ...
1
vote
1answer
44 views

Prove that $\frac{\partial^2 f}{\partial x \partial y}=0$

Supose $g:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}$ is a function of class $C^2$, and $\frac{\partial^2g}{\partial x^2}=\frac{\partial^2g}{\partial y^2}$. If we define ...
1
vote
1answer
14 views

Elliptic functions $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $

Let $\lambda_1$ and $\lambda_2$ be complex numbers with nonreal ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $$f(z+\lambda_1)=af(z) \;\;\;\;,\;\;\;\; ...
2
votes
3answers
108 views

how to show $\mathbf{Q} $ is not free

we know torsion free plus finitely generated $\rightarrow$ free and that $\mathbf{Q}$ is torsion free is easy. But how to show Q is not finitely generated? and not free?
0
votes
0answers
30 views

Minimize total cost of one kilometer

The cost of the fuel consumption of a locomotive is proportional to the square of its speed plus 100 pounds per hour without regard to its speed. The cost of the fuel consumption is 25 pounds per hour ...
2
votes
0answers
52 views

Which convex $2n$-gons have symmetry group $D_n$ instead of $D_{2n}$?

The equilateral octagon $M$ in the first image has the same symmetry group as the small embedded square - namely the dihedral group $D_4$ - with $8$ elements and generators ${x,y}$ with $x^4 = e, y^2 ...
2
votes
3answers
38 views

Is $ x^n-y^n$ is a product of coprime factors?

In the expression: $x^n-y^n$, if $n>2$ and $x,y$ are relatively prime, are the factors $x-y$ and $ x^{n-1}+x^{n-2}y+.....$ always coprime? Why? Please exclude the cases where $x-y=\pm 1$ and $\pm ...
4
votes
0answers
39 views

Solve the given initial value problem.I need your help.

Solve the $$x'=tx^2+x-t^3\,,\quad x\left(\, 2\,\right)=1$$ I need its exact solution not a numerical solution.In fact I have to compare the exact solution with the numerical solution.I tried it but I ...
-5
votes
0answers
81 views

How to solve $234\times456\times542=$? multiplication mentally [on hold]

How to solve $234\times456\times542=$? multiplication mentally in one line answer? I would like to know the trick to find the answer of that multiplication without actually writing the whole thing ...
4
votes
3answers
230 views

how many ways to choose 3 coins?

Sorry I don't know the correct math terms here, I haven't had a math class in some time. That's probably why I have trouble finding an existing question like this, too. Let's say there are 4 differnt ...
2
votes
2answers
69 views

Is a Number Divisible by 40

One of the "shortcuts" for determining if a number is divisible by 8 is to see if the last three digits are divisible by 8. One ...
-1
votes
0answers
31 views

Bounded polyhedrons

Given a bounded polyhedron $P=P(A,b)$ and with $x$ s.t. $Ax<b$, show: $\exists \ \alpha>0 \ \ \ \ \ \text{ s.t.}\ \ \ \alpha^Tx\leq1, \ \ \ \ \forall x \in P $ How I should proceed to prove ...
1
vote
2answers
52 views

Is the preimage of a bounded set also bounded?

I need to prove the following statement: Let $f:\mathbb{C}\rightarrow\mathbb{C}$ a continuous function and $B \subseteq \mathbb{C}$ bounded, implies, that the set $A=f^{-1}(B)$ to be bounded. I do ...
0
votes
2answers
22 views

Calculus minimum cost for an open box

An open box with a squared base of volume $128 \ m^3$. The cost of the material used for the base of the box is $2$ pounds per $m^2$, and that of the material used for the lateral faces is $0.5$ ...
0
votes
0answers
18 views

maximal analytic spread

definition from Bruns-Herzog: It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true? if $ \lambda (I)= \dim R$, can one ...
0
votes
0answers
6 views

Sampling via SRSWOR and biasedness of estimates

In a survey to estimate the proportion "p" of votes that a party will poll in an election, the voter list is divided into male and female lists. A sample of 100 from each list by simple random ...
0
votes
0answers
19 views

How to express $\log^p(n)$ where $p<0$?

I know $\log^4n=\log(\log(\log(\log(n))))$ and $(\log(n))^4=\log(n)\cdot\log(n)\cdot\log(n)\cdot\log(n)$ How do you express something like $\log^pn$ where $p<0$?
5
votes
1answer
55 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
0
votes
0answers
88 views

Is there a formula telling if number is prime? [on hold]

Like the topic.. . I mean.. let's say i'm wondering if 15 is prime or not. Could i calculate it, like function roots? EDITED: I mean something like columbus8myhw said: How about: Define ...
1
vote
2answers
32 views

2-dimesional cell complexes with fundamental group isomorphic to the following.

I have been asked to give examples of 2-dimensional cell complexes whose fundamental group isomorphic to the following $$ \Bbb Z_4 * \Bbb Z_5$$and $$\Bbb Z_4\times \Bbb Z5$$ I know in the first ...
1
vote
1answer
56 views

How to solve this kind of problem?

I've just found the following problem: $\quad\quad$ $\quad\quad$ $\quad\,$ And I believe that it could be done with something in combinatorics, my feeling is that generating functions would ...
0
votes
1answer
42 views

How to define the 'error'

I have true data $G$ and wrong data $F$. Both are $n$ dimension vector. $G\in \{G_i| 0<G_i<255\}, i = 1:n$. Because the ...
0
votes
1answer
26 views

Showing convergence of a series almost everywhere

If $\sum_{k=1}^\infty a_k$ is convergent series of positive terms and $(\alpha_k)_{k\in \Bbb N}$ is a sequence of real numbers, then the series $$\sum_{k=1}^\infty\frac{a_k}{\sqrt{|x-\alpha_k|}}$$ ...
2
votes
1answer
28 views

Proof of Steinitz Theorem

I want a source containing the proof of Steinitz Isomorphism Theorem stating: For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$. Thanks!
0
votes
1answer
20 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...

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