0
votes
1answer
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 ...
-1
votes
2answers
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.
0
votes
1answer
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 ...
0
votes
2answers
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 $A$ ...
5
votes
0answers
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 ...
0
votes
1answer
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 ...
0
votes
1answer
26 views

Linear map - how to show this?

Assuming that I have a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ and we have $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$. Is this sufficient to conclude that $A( \lambda x+y ,x ) = \lambda ...
1
vote
1answer
8 views

Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
0
votes
2answers
28 views

Rules of i ($\sqrt -1$) to a power

$i^{2014}$ power =? A. $i^{13}$ B. $ i ^{203}$ C. $i^{726}$ D. $i^{1993}$ E. $i^{2100}$ I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if ...
0
votes
0answers
6 views

Increasing annuity problem

I learned increasing/decreasing annuity and am tackling the following problem for hours now without success. Amy deposits $Z$ into a bank account that has effective annual interest rate of $5\%$ ...
1
vote
0answers
15 views

Deleting 0's from a random mod 2 matrix

I am fairly new to optimization problems, so please forgive my lack of knowledge. That said, I'm trying to write a program that takes an NxM matrix randomly filled with 0's and 1's, then reduces this ...
1
vote
1answer
27 views

Probability Question

A casino game has two dice, each with faces numbered 1 to 6. One of the dice is fair. The other die is biased such that a 6 is twice as likely to appear on top as any one of the other faces. ...
0
votes
1answer
26 views

Conditional expectation, indication function

I am given that $X,Y$ are independent Bernoulli RVs with parameter $p\in (0,1)$. I am also told $Z=1_{(X+Y=0)}$. I am asked to find $E[X\mid Z]$ and $E[Y\mid Z]$. I can see that the expected values ...
1
vote
2answers
25 views

Clarification of the notation $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$

I have a question that uses the following notation: the function $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$ is defined by $$f(x)=\frac{2x-3}{x-3}.$$ I understand that the left side ...
3
votes
1answer
20 views

Submagmas of natural numbers

What is known about submagmas of natural numbers under addition/multiplication? For example, all subgroups of integers under addition are of the form $~n \mathbb{Z}~$. Are there similar results for ...
1
vote
2answers
26 views

Splitting of Primes in a Given Field

Find how $p=2,3,5,7$ splits in $\mathbb{Q}(\sqrt{-5})$ (i.e. find those $e_i,f_i$ for $1 \leq i \leq r$). Can somebody please explain how this is done? My attempt is the following: Let K = ...
0
votes
1answer
21 views

Probability of one device among 6 failing

A certain component of an electronic device has a probability of 0.1 of failing. If there are 6 such components in a circuit. What is the probability that at least one fails? This is not a duplicate ...
0
votes
1answer
24 views

Factorials with fractions

I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$ becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$ and then how it becomes $$ \frac {n+n^2+n}{n-1} $$ I've tried applying ...
0
votes
0answers
9 views

How do I calculate the aspect ratio of a width and height?

I am creating an image upload form. I want to restrict users from uploading images that are not of the aspect ratio of 16:9. What is the formula for calculating an aspect ratio such as this? If w = ...
0
votes
1answer
14 views

Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
1
vote
2answers
29 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
2
votes
1answer
30 views

Are adjoint functors between additive categories additive?

The question is: Prove that an equivalence between two additive categories is an additive functor. By an additive category I mean a category with zero object, that every Hom-set of morphisms is a ...
6
votes
0answers
40 views

Books to read to understand Terence Tao's Analytic Number Theory Papers

I tried to understand Terence Tao's Analytic Number Theory Papers. For example, this paper, Every Odd Number Greater Than 1 is The Sum of at Most Five Primes. Which books shall I read to prepare ...
0
votes
1answer
35 views

Simplify $|2x^{2}+3x-2| $ so we can obtain it and control it in terms of $|x-1|$

First example I worked, I had $|2x^2 + x - 3|$ after some manipulations and simplifications I obtain: $|x-1|(2|x-1|+5)$. The final answer is in terms of $|x-1|$ with multiplication between two ...
2
votes
1answer
41 views

Multiplying and adding fractions

Don't be angry with me. Just comment to delete if you think this is really a bad question. Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times ...
4
votes
2answers
40 views

How to prove $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$?

Here $D^2$ denotes the closed unit disk in $\mathbb{R}^2$. I know that $D^2$ is not homeomorphic to $\mathbb{R}^2$ as $D^2$ is compact. Intuitively I believe that $D^2\setminus\{0\}$ is not ...
-1
votes
0answers
10 views

Coset multiplication is not well defined in the case of $S_4$

I have to show that coset multiplication is not well defined in this case. I have to choose 2 cosets $aH$ and $bH$ and locate two different representatives in each coset $a, a' \in aH$ and $b,b' \in ...
0
votes
1answer
14 views

Working of selections

There are eight finalists in the 400 m athletics at the world championships. Three of the finalists are from the USA, and the others are from five different countries. The rules for allocating a lane ...
0
votes
2answers
9 views

Show that f has another inflection point and compute the (x, y)-coordinates of this other point of inflection?

Consider the function $f(x) = ax^4 − 8x^3 + b$. Assume that $(x, y) = (2, 8)$ is an inflection point of this function. Show that f has another inflection point and compute the $(x, y)$-coordinates of ...
1
vote
0answers
19 views

Infinite strings and infinite theorems - Is there a theory on these stuffs?

I can have an alphabet $\mathcal{A}$, a set of axioms $\mathcal{X}$ which are finite strings of $\mathcal{A}$ and a set of rules $\mathcal{R}$. Every finite strings produced by applying a finite ...
0
votes
0answers
6 views

Show that an affine combination can be written as a point plus a vector?

Show that an affine combination can be written as a point plus a vector:
0
votes
0answers
34 views

Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and ...
1
vote
1answer
45 views

L'Hospital Rule: Requirement that the limit exists

Could someone define what it means for a limit to "exist"? Must the limit after using L'Hospital Rule approach a specific value? What if the limit after using L'Hospital Rule approaches infinity? ...
0
votes
1answer
19 views

If a function is radial, then its Hardy-Littlewood maximal function is radial as well

I'm looking for a proof of the following statement: $$f\in L(\mathbb R^n)\ \text{ radial } \implies f^* \ \text{ radial}$$ where $f^*$ is the Hardy-Littlewood maximal function defined by: $$f^*(x)= ...
0
votes
1answer
31 views

Parametrization of $ax^2+bxy+c=0$

Can I just fix $y=t$ and use quadratic formula to get the rational points of the diophantine $$ax^2+bxy+c=0?$$ or is there another method? I feel like I am turning in circles with the quadratic ...
0
votes
1answer
16 views

Describe smallest algebra, monotone class, $\sigma$-algebra

I'm trying to understand better the concepts of monotone classes, algebras and $\sigma$-algebras so I came into the following problem. For the family $E := \{∅, \mathbb{N}, \{2\}, \{2, 4\}, \{2, 4, ...
3
votes
0answers
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.
0
votes
2answers
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?
0
votes
1answer
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?
0
votes
0answers
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 $\Bbb R^p$ ...
1
vote
0answers
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?
1
vote
0answers
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...
7
votes
1answer
38 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
0
votes
2answers
23 views

Same Arrangements of the word “MINIMUM”

In how many distinguishable ways can the seven letters in the word MINIMUM be arranged, if all the letters are used each time? My attempt: 3!(2!) = 12 ways. M has 3 choices and I has two choices. ...
3
votes
1answer
20 views

Given that there are 6 married couples. If we select only 4 people out of 12, what is the probability that none of them are married to each other? [on hold]

Please, can you help me to solve this? Given that there are 6 married couples. If we select only 4 people out of 12, what is the probability that none of them are married to each other?
0
votes
0answers
15 views

complex numbers: determining whether claims are right or not.

we should decide whether the following claims are right or not, and explain our decision. let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$ a) $w_1^{1991} + w_2^{1991} + ...
3
votes
3answers
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 ...
0
votes
0answers
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
0
votes
0answers
5 views

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 ...
0
votes
0answers
9 views

find subsequential, lim sup and lim inf

find the subsequential of {[1+(-1)^n]n + (1/n)} by using the limit theorem, I have: {2n + 1/n } as n -> infinity and n even then is 2 {0*n + 1/n} as n -> infinity and n odd then is 0 so the ...

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