2
votes
1answer
72 views

Expected value of a random variable so that

I was just shown this website so I really have no clue how things work. Let me know if I do something wrong please. Also, please bare with me because English is not my first language and I'm really ...
6
votes
2answers
230 views

Show that $2 < e^{1/(n+1)} + e^{-1/n}$

I'm trying to show $2 < e^{1/(n+1)} + e^{-1/n}$. I can show that $ 2 < e^{1/n} + e^{-1/(n+1)}$ since $$2 \leq 2\cosh\left(\frac{1}{n}\right) = e^{1/n} + e^{-1/n} < e^{1/n} + e^{-1/(n+1)}$$ ...
1
vote
1answer
416 views

How to prove uniform continuity problem!

A) $f(x)=x^3$ , give an example of an interval where $f$ is uniformly continuous and another where it is not. explain your choose of examples B) decide if $f(x)= \dfrac{1}{\sin x} - \dfrac{1}{x}$ is ...
2
votes
1answer
112 views

Does scaling lead to weak convergence to the null function?

Let $f\in L^p(\mathbb{R}^d)$, with $1<p<\infty$. Is it true that $$\lambda^{\frac{d}{p}}f(\lambda x ) \rightharpoonup 0\quad \text{ weakly in }L^p\text{ as }\lambda\to+\infty?$$ One has ...
1
vote
1answer
26 views

Every element in well-ordered set can uniquely be expressed as $y = S^n(x)$

Let $U$ be a well-ordered set. If $y \in U$ then $y$ can be expressed uniquely on the form, $y = S^n(x)$ Where $x$ is either the least element of $U$ or a limit point, $n \in \mathbb{N}$ and $S$ is ...
0
votes
1answer
23 views

finding accumulation points of a topology

in lipshutz book says I don't understand why the point $c$ isn't a limit point of $A$ since the open set $\{c,d\},\{a,c,d\},\{b,c,d,e\},X$ does contain a point of $A$ different from $c$
9
votes
3answers
192 views

Show that $2^n(\cos^n(\frac{2\pi}{9})+\cos^n(\frac{4\pi}{9})+\cos^n(\frac{8\pi}{9}))\in\mathbb{Z}$

$a_n=2^n\left[\cos^n\left(\dfrac{2\pi}{9}\right)+\cos^n\left(\dfrac{4\pi}{9}\right)+\cos^n\left(\dfrac{8\pi}{9}\right)\right]$. Show that $a_n\in\mathbb{Z}$ for all $n\in\mathbb{Z}$. Find the last ...
4
votes
2answers
422 views

equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H $. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
0
votes
1answer
31 views

Combinatorics question simple

If I have 100 people in a tennis tournament. I want to find the total number of combinations of matches of doubles. So P1&P2 on Team 1 vs P97&P98 on Team 2 count as ONE combination of matches ...
0
votes
0answers
93 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
13
votes
4answers
22k views

What does the dot product of two vectors represent?

I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, $2$ and $3$, we say that it ...
0
votes
2answers
82 views

did:$\lim_{x\to 0}\frac{\sin(x^2 + \frac{1}{x} )- \sin(\frac{1}{x}))}{x}$

Could you guys give me at least a hint at: $$\lim_{x\to 0}\frac{\sin(x^2 + \frac{1}{x}) - \sin(\frac{1}{x}))}{x}$$ ? I already tried expanding the $\sin(x^2 + \frac{1}{x})$ but got nothing. Also, ...
1
vote
1answer
45 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
1
vote
0answers
95 views

maximal irreducble subgroups of $SL(2,q)$

If $H$ is a maximal solvable irreducible subgroup of $GL(2,q)$ then intersection $H \cap SL(2,q)$ is maximal solvable irreducible subgroup of $SL(2,q)$. Why is it true? Maybe this is not true, but ...
32
votes
13answers
9k views

Algebra: What allows us to do the same thing to both sides of an equation?

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. ...
1
vote
1answer
48 views

Series expansion of $\frac{x^2}{1+ \sin x}$

For the series expansion at $x=0$ for $\dfrac{x^2}{1+ \sin x}$ WolframAlpha gives $$x^2 -x^3 +x^4-\frac{5x^5}{6}+\frac{2x^6}{3}-\frac{61x^7}{120}+O(x^8)$$ But I'm missing something in the ...
2
votes
1answer
87 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
1
vote
0answers
182 views

Existence of non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$

Show that there is no non-constant bounded analytic function on $\mathbb{C}\setminus \mathbb{Z}$. In this homework problem i tried to proceed in the following way: If possible, let $f$ be a ...
0
votes
1answer
62 views

Some questions about subspaces in Banach spaces

I just have a few question about some things in Banach spaces. Let $X$ be a separable, reflexive Banach space with basis $\{e_{i}\}$. Let $X_{n} = \text{span}\{e_{1},...,e_{n}\}$, then consider the ...
0
votes
2answers
138 views

Doob's decomposition of a brownian motion.

Let $B_n$ be a discrete Brownian motion. I need to find the Doob decomposition for ($B_n^2$). Can someone help me please. Thank you in advance.
0
votes
1answer
25 views

All matrices that share a null row space are obtainable from one another by elmentary row operations?

Given an $m\times n$ matrix $\mathbb{A}$, the set of $n$-vectors $\mathbf{x}$ that satisfy $\mathbb{A}\cdot\mathbf{x}=0$ is the null row-space of $\mathbb{A}$. The elementary row operations on ...
0
votes
1answer
175 views

exponential convergence of an infinite sum

Suppose I have a sequence of nonnegative numbers $\{x_0,x_1,\dots\}$ with the properties that The sum $x_0+x_1+\cdots$ converges. There exists some $0 \le \rho < 1$ such that ...
14
votes
3answers
958 views

Prove that when dividing a square field among three people, one person must own two points more than 1 km apart

We have a square field with a $1$ km side we need to divide among three people (it doesn't have to be fair, one of them could even get none of it!). How would I prove that at least one of the persons ...
0
votes
1answer
21 views

Relation between dense subsets in the product map and dense subsets in each component

Let $\Omega$ bea Polish space and $X_1,\dots,X_n:\Omega\rightarrow\mathbb R^d$ be Borel measurable maps. Consider now the map $X:\Omega\rightarrow(\mathbb R^d)^n$ defined by ...
2
votes
1answer
204 views

Uniform convergence

I got a task: research $$\sum_{n=1}^\infty~e^{-nx^2}\sin nx$$ for a uniform convergence. I see that $ \sup_{x\in X} |f_n(x)-f(x)|\to 0 $ when $x\ne0$. But what I must do when $x=0$?
1
vote
1answer
88 views

When is it useful to reduce mathematical objects to foundational levels and when it is not?

When is it useful to reduce mathematical objects to foundational levels and when it is not? Let's say you work in the field of computer vision, or else. How can you claim your method is optimal if ...
0
votes
2answers
1k views

Find points on a circle given arc length and radius.

I am trying to layout a circle, given the arc length l, radius r and center (cx, cy). I need to find all the n points that are on the circle. What I've tried so far: The first part is to find n: n ...
0
votes
1answer
57 views

How to find the topology of this subbase genarated?

I want to find topologies which uses this set as a subbase. two different questions says 1-X={a,b,c,d} and $\mathcal S $={{a,b},{b,c},{d}} 2- {[x,x+1]|$x\in$ R} in the first : when we take finite ...
3
votes
4answers
905 views

Lines tangent to parabola at point.

I'm struggling to figure out what I'm exactly required to do. The problem states "Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$." I believe ...
3
votes
1answer
85 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
0
votes
2answers
347 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
16
votes
8answers
697 views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

How to prove the following $$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$ I thought of separating the two integrals and use the beta or hypergeometric ...
5
votes
1answer
167 views

Undergraduate Introduction to Modular Forms

What are the best introductory texts (or lecture notes) on modular forms aimed at an advanced undergraduate audience (for a student with a course in complex analysis and two courses in algebra and ...
1
vote
2answers
42 views

help me please about adjoint of operators in L1

A : L₁→L₁ 1) A x=( x₁, x₂,.....xn , 0,0,....) 2) A x= (λ₁ x₁ ,λ₂ x₂,.....) |λ n|≤1 and λ n ∈ R I need to find adjoint of operators A in given space. ...
0
votes
1answer
200 views

Finding the connected components of topological spaces

Find the connected components of the following sets: $(a) \; A=\{(x,y):y=\sin(1/x), x\in\mathbb{R}^+\},(b)\;A\cup\{(x,y):x=0,y\in[-1,1]\},(c)\;$The Cantor Set,$(d)\;\mathbb{N}$ with the cofinite ...
2
votes
1answer
77 views

Do we sometimes have to go “each way” separately for iff proofs?

So, I often enjoy trying to prove "if and only if" statements by only using if and only if arguments. i.e. RTP: $A \Leftrightarrow D$. Proof: $A \Leftrightarrow B \Leftrightarrow C \Leftrightarrow ...
1
vote
2answers
83 views

About $\int \csc(x) \, dx$

One of the suggested proofs that I found to the $\int \csc(x) \, dx$ start with, $$\int \csc(x) \, dx= \int \csc(x) \cdot \frac{\csc(x)- \cot(x)}{\csc(x)- \cot(x)} dx = \cdots$$ By graphing the ...
1
vote
0answers
53 views

Is $1-f(x)\to 0$ equivalent with $f(x)\to 1$.

Problem. I am trying to make a proof of contradiction and have this far shown that if my assumption is true, both $$\lim_{x\to 0}(1-f(x))=0\quad \text{and}\quad \lim_{x\to 0}(1+f(x))=0$$ must be true. ...
5
votes
1answer
105 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
1
vote
1answer
127 views

Find the divergence of the following vector fields

Consider an arbitrary vector field $F$ $$\eqalign{F&=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}\\ &=F_{C_1}\hat{e}_\rho+F_{C_2}\hat{e}_{\phi}+F_{C_3}\hat{e}_{z}\\ ...
0
votes
1answer
37 views

Cauchy-Euler equation set up

I have the following 2nd ordered ODE, and I want to transform it into a cauchy-euler equation to be able to solve it. xy'' - 7xy' + 12y = 0 To be a Cauchy-Euler ...
0
votes
1answer
63 views

Discrete Random Variables Probability Exercise - How to approach this

Below is the whole exercise that I need to solve. Since this is from an online course and it's given with any other context, I need to figure what I need to learn in order to solve it. Is it ...
2
votes
1answer
92 views

The Vacisek Model and the short rate process

I am trying to do some calculations related to the Vacisek model, but I think I am mixing up concepts and I'm not getting to any solution. Let me explain what the problem is. The Vacisek model ...
0
votes
1answer
55 views

Existence of certain homogenous forms

Let $D(X,Y), E(X,Y)\in\mathbb{Z}[X,Y]$ forms of the same degree $n$ and suppose that the resultant $R=Res(D,E)$ of $D$ and $E$ is not $0$. Show that there are homogenous forms $L_0(X,Y),M_0(X,Y), ...
1
vote
1answer
180 views

Two Genius Mathematicians

This is actually a question I find really hard to answer.any hints are appreciated. By the way feel free to edit the tags as i really do not know which category is this question is in. Two genius ...
1
vote
0answers
86 views

$\zeta$ primitive $n$th root of unity, help showing that $\sqrt{n},\sqrt{-n}\in \mathbb{Q}(\zeta)$ under some conditions.

Consider $\zeta$ a primitive $n$th root of unity, show that if $n\equiv 1\mod{4}\implies \sqrt{n}\in \mathbb{Q}(\zeta)$ if $n\equiv -1\mod{4}\implies \sqrt{-n}\in \mathbb{Q}(\zeta)$. I know that ...
-1
votes
1answer
40 views

Reverse simple formula [closed]

I am needing to reverse this simple formula to get A a = variable b = float Formula a + a * b = c Example 100 + 100 * 1.5 = 250 How to get back 100
5
votes
1answer
663 views

Countable sum of measures is a measure

Prove that if $\mu_1, \mu_2, \dots$ are measures on a measurable space and $a_1, a_2, \dots \in [0,\infty)$, then $\sum_{n=1}^\infty a_n\mu_n$ is also a measure. I need some help justifying the ...
0
votes
1answer
45 views

Divergence Computation in Gauge Theories, Knots and Gravity

Hopefully this is just some minor confusion...The first exercise wants us to show that $$\vec \epsilon(t,\vec x)=\vec Ee^{-i(wt-\vec k \cdot\vec x )}$$ satisfies the vacuum Maxwell equations where ...
1
vote
1answer
60 views

$(V^*)^{\otimes n} \cong (V^{\otimes n})^*$

We assume that $V$ is finite dimensional. Make $\theta: (V^*)^n\to (V^{\otimes n})^*$ by $$ \theta(\alpha_1,\cdots,\alpha_n)(v_1 \otimes \cdots \otimes v_n ) := \prod_{i=1}^n \alpha_i(v_i). $$ Then, ...

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