0
votes
0answers
4 views

Equation, Area and Circumference of A circle given equation of the tangent and center

I don't know how to solve for the Area and Circumference but I know how to solve for the equation but I just wanted to make sure... Any help and explanations would be appreciated :) Problem: A circle ...
0
votes
0answers
3 views

Concerning the problem of finding the number of invertible nxn random {1,0} matrcies

In a few more words, if we look at the space of all nxn matrices (over a field of characteristic 0) with only 1 or 0 as an element in them ("binary matrices"), how many of them are invertible for each ...
0
votes
0answers
5 views

Applications of Infinitary Matrices in Set Theory

Matrices have a natural generalization to infinitary context. There are few known applications of such matrices in set theory. For example one may use Ulam matrices to show that real-valued measurable ...
0
votes
1answer
11 views

How to prove that $\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$

It is known that $$a_n,\ b_n\ge0.$$ And they are both upper bounded. Knowing this how can one prove that $$\sup\{a_nb_n|n\in N\}\le \sup\{a_n|n\in N\}\sup\{b_n|n\in N\}$$ I don't see how to approach ...
0
votes
0answers
6 views

How are these distributions called?

Are these any common distributions? Number of dependent trials until the first success occurs Inspired by (a) in this question here: Probability Problem with $n$ keys Binomial-like distribution ...
0
votes
0answers
4 views

Relation between existense of a homomorphism and denseness of a set

Denote the pontryagin dual of a topolological group L by $\widehat L$. I encountered this claim for a discrete topological group $(G,\mathcal T)$ : Because there exists a one-to-one homomorphism ...
2
votes
2answers
13 views

How to find trigonometry function limit

What is the solution for trigonometry functions limit when we're in $\dfrac{0}{0}$ situation? $$\lim_{x\to 0} \frac{\sin^2 3x}{x^2}$$ for example
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votes
0answers
7 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
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votes
1answer
9 views

Number of disitinct subgroups of the automorphism group of the field of $3^{100}$ elements

Let $G$ be the group of all automorphisms of the field $F_{3^{100}}$ with $3^{100}$ elements . Then what is the number of distince subgroups of $G$ ? What is the order of $G$ ?
1
vote
1answer
9 views

How is this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L ...
0
votes
0answers
6 views

Probability of getting the same 10 cards after a 110 card shuffle

Consider this situation: You have two standard decks merged together (104 cards). You give 10 cards to 4 players. What is the probability that the next time you deal (after the deck is randomly ...
1
vote
1answer
13 views

Are $y_1$, $y_2$, $y_3$,

Let $v_1$, $v_2$, $v_3$, be linearly independent vectors in $\mathbb{R}^n$. Let $y_1$ = $v_2$-$v_1$, $y_2$ = $v_3$-$v_2$, $y_3$ = $v_3$-$v_1$. Are $y_1$, $y_2$, $y_3$ linearly dependent or ...
0
votes
1answer
23 views

bounds of inregral: integral (from 0 to -pi)x^2*e^{-inx}dx=integral (from pi to 0)x^2*(e^{inx})dx

$$\int_{0}^{-\pi}x^2e^{-inx} \, \text{d}x = \int_{\pi}^0 x^2e^{inx} \, \text{d}x$$ which law allows such equality? why are they equal? I was thinking it is about putting $t=x-\pi$ but during my ...
0
votes
0answers
8 views

Existence of weak Schauder-basis for concrete example.

Consider e. g. $P(X)$, the space of probability measures over some compact metrisable space, $X$. This may be viewed as a WOT*-compact subspace of some dual Banach space ...
0
votes
0answers
9 views

A different characterization of the infimum of a set

Let $E$ be a set that is bounded below. Let $l$ be a lower bound of $E$. Show that $ l = \inf E $ iff given any $\epsilon > 0$ we can always find $z \in E$ with $z < l + \epsilon $. Attempt ...
0
votes
2answers
20 views

Prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent

I am currently working on a problem that asks to prove, using Wronskian, that $e^\left(3x\right)$, $e^\left(2x\right)+x$, $e^\left(x\right)+1$ are linearly independent. I went through the general ...
1
vote
1answer
11 views

Show that an infinite set $C$ is equipotent to its cartesian product $C\times C$

So, as the title says I'd like to give a proof of the fact that if $C$ is an infinite set then it is equipotent or equivalent to its cartesian product $C\times C$ using Zorn's Lemma (and of course ...
2
votes
1answer
24 views

empty boxes puzzle

The problem is N large empty boxes (assume they are of type:1) are initially placed on a table. An unknown number of boxes (type:1) are selected and in each of them K smaller boxes (type:2) are ...
0
votes
2answers
14 views

Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
0
votes
0answers
5 views

Properties of Relations and their negations.

There are three properties of relation, 1. Reflexive 2. Symmetric 3. Transitive and if all properties are satisfy by a relation then its known as ...
1
vote
1answer
25 views

Proving that $\exists x\in[0,1]:\sum\frac{b_n^2}{|x-a_n|}<\infty$

Let $\{b_n\}$ be a sequence of positive numbers s.t $\sum b_n<\infty$ and let $\{a_n\}$ be a sequence of real numbers in $[0,1]$. Prove that $\exists ...
1
vote
1answer
16 views

Continuity of functionals in function space

I came across this problem and got confused. With the help of folks at MathStackExchange I managed to understand the following: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq ...
0
votes
0answers
7 views

the two definitions of a branch of $z^{\alpha}$ for a fixed $\alpha\in C\setminus Z$

If $\alpha\in C$ is fixed,the definition of a branch of $z^{\alpha}$ is as followed: Let $U$ be an connected open set on which there is a branch of $\log z$,then for every $f(z)$ which is branch of ...
1
vote
1answer
16 views

Calculating the degree of $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$

Let $m\in\mathbb Z$ with a prime factorization of the form $m=p\Pi p_i^{n_i}$, $p\neq p_i$. How can I calculate $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$ for a natural number $n$?
1
vote
0answers
20 views

The Kuratowski Monoid

I have been reading the paper The "Kuratowski closure compliment theorem" by" B. J. Gardener and M. Jackson".In that the author discusses the 6 different monoid structures as follows :Extremally ...
0
votes
1answer
20 views

How to combine two series of solutions into one?

I have an equation, like $$\sin\left((\beta-1)\sqrt{\xi^2-\gamma^2}\right)\sin\xi=0,$$ with $\beta>1$ and $\gamma>0$. I've found two series of solutions, corresponding to roots of each of the ...
0
votes
0answers
23 views

A complicated matrix problem

$A,B,C $ are $n×n$ matrices. These matrices satisfied following conditions. $$AB=BCA$$ $$AC=BA$$ $$BCB=A^2$$ $$A=BC-CB$$ Prove that $$AB=0$$ I find $A^3=0$, but I can not go further. I need your ...
0
votes
0answers
5 views

Test whether the proportion of days which do not have “Good” air quality has changed.

I have the following data of AQ-value $$36,52,54,60,60,31,22,13,67,59,52,54,30,12,33,39,50,51,34,56,62,51,33,39,23,12,35,28,41,54,44,66,42,33,28,22,36,57,46,39etc.$$ I have total $364$ such values in ...
0
votes
2answers
25 views

Finding limit of a function

$$\lim_{x\to0}\left(\ln(\cot x)\right)^{\tan x}$$ The answer is obviously 1 but how do I reach that conclusion without L'hospital rule
2
votes
0answers
17 views

Action on a group descends to an action on its factor group

Let $A$ and $B$ be groups and $N\unlhd A$ is a normal subgroup of $A$. Suppose that $B$ acts on $A$; that is, there exists a group homomorphism (not necessarily monomorphism) ...
0
votes
1answer
24 views

Combinatorial Problem: How many ordered pairs of distinct but overlapping ordered pairs do exist?

Suppose we have a universe of $n$ numbers. How many ordered pairs of ordered pairs, that is, $((a,b), (c,d))$ with $a\neq b$, $c \neq d$ do exist such that $(a,b)\neq (c,d)$ but either $a=c$, or ...
0
votes
3answers
18 views

continuity of functional in $C[0,1]$

I came across this problem and got confused Problem: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq t\leq1}|x(t)|$$Show $h$ is continuous in $C[0,1]$ Attempt: I am a bit confused ...
1
vote
0answers
7 views

Representing a product of orbits as a disjoint union of orbits

Let $A$ be a finite abelian group, and let $B$ and $C$ be subgroups. In the $A$-set $A/B\times A/C$, the stabilizer of any element is $B\cap C$, so we know there is a decomposition of $A$-sets like ...
0
votes
0answers
8 views

all differentials collapse of the Serre spectral sequence

Let fibration $$ SO(n)\to SO(n+1)\to S^n, $$ consider the Serre spectral sequence of cohomology $(E^{*,*}_k,d_k)$, $k\geq 2$, $E^{p,q}_2=H^p(S^n;\mathbb{Z}_2)\otimes H^q(SO(n);\mathbb{Z}_2)$. How ...
0
votes
1answer
13 views

How many possible 2-colorings of a disconnected bigraph?

Is there a relationship between the number of connected components in a bigraph and the number of possible 2-colorings? A connected bigraph (i.e. only one component) can be 2-colored in exactly two ...
3
votes
1answer
23 views

Continuum Hypothesis for closed sets

In A Beginner's Guide to Modern Set Theory [page 48], the author says: [Cantor] did prove that every closed uncountable subset of $\mathbb R$ has cardinality $2^{\aleph_0}$... ... but I cannot ...
1
vote
1answer
24 views

Evaluating $\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{{1}/{x^2}}$

$$\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{\dfrac{1}{x^2}}$$
0
votes
0answers
11 views

Find the solutions to this equation involving a normal distribution.

The equation is as follows: http://i.imgur.com/xv2zsZR.gif Some hints and comments: A-Each side is a distribution where sigma is a constant and represents the variance. B-Using simple plotting ...
1
vote
1answer
13 views

Is projective modules “graded projective”

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}- mod$ the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot ...
1
vote
3answers
36 views

Sshowing $\left(\frac{-1+\sqrt{3}i}{2}\right)^3=1$

in my textbook they asked me to show that $$\left(\frac{-1+\sqrt{3}i}{2}\right)^3=1$$ but it doesn't i think. i put down $$\begin{align} ...
2
votes
2answers
32 views

How do I show that the following series converges?

$$\sum_1^\infty \frac{(-1)^n (1 + 1/2 + \cdots+ 1/n)}n$$ I tried applying the alternating series test, but I think that fails. I don't know which other test I could use here.
1
vote
2answers
25 views

Finding the determinant of a matrix given the adjoint

My attempt: Knowing that $$A(AdjA) = IdetA$$ I took the determinant on both sides: $$det(A)det(AdjA) = det(det(A))$$ So, $$det(A)det(AdjA) = (det(A))^3$$ $$det(AdjA) = (det(A))^2$$ $$det(A) = ...
0
votes
1answer
19 views

Could there be infinite solutions to a modular linear equation?

Could there be infinite solutions to a modular linear equation of the form Ax = b mod n when solving for x?
0
votes
1answer
13 views

condition of plane to touch a given surface

Q. Show that the plane $ax+by+cz+d = 0$, touches the surface $px^2+qy^2+2z=0$, if $a^2/p + b^2/q +2cd = 0$. How to start to solve this problem?
0
votes
0answers
24 views

If $u\notin E$, then the supremum of $E\cup\{u\}$ is $\sup\{\sup E, u\}$

Let $S$ be an ordered set and $S \supset E $. Let $\alpha = \sup E \in E$. If $u \notin E$, then we have $ \sup ( E \cup \{u\} ) = \sup\{ \alpha, u \} $. Try: I know this result follows easy by ...
1
vote
2answers
62 views

What is $\frac{9}{3} - \frac{1}{2}$?

I need to compute $\frac{9}{3} - \frac{1}{2}$. I got an answer of $\frac{8}{6}$ but that is incorrect. $\frac{5}{2}$ is the correct answer. How is this possible?
1
vote
0answers
17 views

Sum of powers of a number divisible by other?

I need to find the probability that $7^m+7^n$ is divisible by $5$ where $m,n\in[1,100]$. I noticed that if the remainder when m and n are divided by 7 differs by two the last digit is zero. So I did ...
0
votes
0answers
18 views

an example of the Scott topology

Let $X=A\cup B$ where $A$ is set of all positive integers w.r.t the Scott topology and $B$ is singleton set $\{b\}$. A set $U$ is open in $X$ iff $U$ is open in $A$ or $A\subseteq U$ and $U\cap B ...
1
vote
1answer
31 views

How do I construct a nonabelian group of order 1575?

I think that it should be a semidirect product of the direct product of any two of the three groups $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/25\mathbb Z$ and the other one. But ...
0
votes
0answers
10 views

$g^{2k} \equiv g^i \pmod{p} \implies 2k \equiv i \pmod{p-1}$?

Why does $$g^{2k} \equiv g^i \pmod{p}$$ imply that $$2k \equiv i \pmod{p-1}$$ when $p$ is some prime and both $i$ and $k$ are $\in \{0, 1, \dots , p-1\}?$

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