# All Questions

16 views

### Pumping lemma to prove that a language is not context free

We've got $L = 0^{x^{2}}$. So we let $w = 0^{p^{2}}$, and we know that we can split w into $w = u\cdot v\cdot w\cdot x\cdot y$ , according to the pumping lemma for CFGs. I'd like to know how to ...
21 views

### Showing the product of two normal subgroups is normal [on hold]

Prove that if $H$ or $K$ are normal subgroups then $HK=\{hk\mid h\in H,k\in K\}$ is a subgroup. Then if both are normal subgroups, prove that HK is normal.
14 views

### Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation.

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation. I'm sorry, but this question throws me off in many ...
21 views

### $0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0.

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0. I'm kind of confused here: I had understood that if all of the eigenvalues of $A$ ...
28 views

### Constructing a Borel set A on R such that $0<m(A \cap I) < m(I)$ for all intervals $I$. [duplicate]

I need help constructing a Borel set $A$ on $\mathbb{R}$ with the following property: For every open interval $I$, $$0<m(A \cap I)< m(I)$$ A obviously needs to be dense in $\mathbb{R}$ and it ...
18 views

### Probability of Drawing a Card from a Deck (Part 2)

This is a continuation on a question I asked a few years back: Say you have a 60 card deck containing 12 red cards and 48 black cards. After drawing 7 cards, what is the probability you will have 2 ...
26 views

9 views

### Spherical electrode diffusion problem; trying to get to planar system

I'm working through the derivation of current in a spherical electrode, and so far I've been able to get it into the following, starting from Fick's 2nd Law: Any help would be appreciated.
26 views

### Number of values that satisfy $2\sin ^2(x) - 3 = 3 \cos (x), \: 90^{\circ} < x < 270^{\circ}$

Graphing this function is difficult as many overlaps exist and finding a viewing window is hard. What's a good algebraic method to solve this problem?
18 views

### Rational points of $ax^2+by^2=z^r$, $r$ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
12 views

### Range of a function composition is a subset of the range [duplicate]

Let $L:\Bbb R^n → \Bbb R^m$ and $M:\Bbb R^m → \Bbb R^p$ be linear mappings. Prove that $Range (M◦L)$ is a subspace of $Range (M)$. So I began by defining: $Range (M◦L)$ is a subset of $\Bbb R^p$ ...
17 views

### If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$.

If $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and injective, then $T^{-1}(B)$ is Borel for Borel $B$. Is it possible to prove this theorem?
16 views

### Knowing if spans overlap

Only the first checked squares are deemed to be correct. Why is D not correct? After all, the vectors do overlap on the same plane...
38 views

34 views

### Multivariable limit … no L'Hopital rule?

I am looking a bit at limits for multivariable functions by myself, and I can't figure it out; my book only mentions them shortly, but now I am looking at an "assignments for those interested" and it ...
20 views

### prove in any traingle ABC if A>B then a>b

A,B are angles and a,b are sides How do you prove this using the sine rule? if A>B do you consider separate cases when A is acute and obtuse? also how do you prove the converse if a>b then A>B
### What is the variation $Var(\hat{y} - y)$ in a linear regression model
There are $N$ data points of the form $(x_i,y_i)$ that corresponds to the model $y_i=kx_i+b+\varepsilon_i$ with $E(\varepsilon)=0$. Coefficients $k$ and $b$ are unknown but we fill the model (all ...