# All Questions

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 ...
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 $$2 \leq 2\cosh\left(\frac{1}{n}\right) = e^{1/n} + e^{-1/n} < e^{1/n} + e^{-1/(n+1)}$$ ...
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 ...
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 ...
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 ...
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 $c$
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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 ...
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 $x=0$?
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 ...
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 ...
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 ...
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 ...
1answer
85 views

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 ...
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. ...
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) - ...
1answer
127 views

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}\\ ... 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 ... 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 ... 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 ... 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), ... 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 ... 0answers 86 views ### \zeta primitive nth root of unity, help showing that \sqrt{n},\sqrt{-n}\in \mathbb{Q}(\zeta) under some conditions. Consider \zeta a primitive nth 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 ... 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 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 ... 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 )}$$satisfies the vacuum Maxwell equations where ... 1answer 60 views ### (V^*)^{\otimes n} \cong (V^{\otimes n})^* We assume that V is finite dimensional. Make \theta: (V^*)^n\to (V^{\otimes n})^* by$$ \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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