instruction stringlengths 12 30k |
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Are translations of a polynomial linearly independent? |
Is it even worth the theorem below?
For every positive integer n, there is a real number r,
and 1/(12n 1)<r<1/(12n), such that:
n! = sqrt (2.n.pi). [(n/e)^n].(e^r)
I saw this statement on some sites, but got no further details. |
n! = sqrt (2.n.pi). [(n/e)^n].(e^r)? |
Is it even worth the theorem below?
For every positive integer n, there is a real number r,
and 1/(12n 1)<r<1/(12n), such that: n! = sqrt (2.n.pi). [(n/e)^n].(e^r)
I saw this statement on some sites, but got no further details.
I think the statement refers to an exact value of n factorial, not an approximation... |
Is it even worth the theorem below?
For every positive integer n, there is a real number r,
and 1/(12n+1) < r <1/(12n), such that:
n! = sqrt (2.n.pi). [(n/e)^n].(e^r)
I saw this statement on some sites, but got no further details.
I think the statement refers to an exact value of n factorial, not an approx... |
Is it even worth the theorem below?
For every positive integer $n$, there is a real number $r$,
and $\frac{1}{12n+1} \lt r \lt\frac{1}{12n}$, such that:
$$ n! = \sqrt{2n\pi}\left(\frac{n}{e}\right)^n e^r.$$
I saw this statement on some sites, but got no further details.
I think the statement refers to an exac... |
$n! =\sqrt{2n\pi}\left(\frac{n}{e}\right)^n e^r$? |
When did 'none' come to be a number? My guess is that it was when merchants or scribes first codified symbolic addition and subtraction to calculate quantities of food and supplies on hand. Intuitively, they must have known that 'none' would be the result of subtracting 100 bushels of wheat from same. And that adding 1... |
The problem was that math started out of necessity. Shepherds needed to count how many sheep they own, parents needed to count how many children they had, etc. For most simple mathematics, zero is simply a place-holder. Who needs to count zero sheep? Even the most basic forms of counting--using your fingers--who ha... |
What is the value of n $\in mathbb{Z} for which the function $f(x) = \frac{sin nx} { \sin \biggl( \frac{x}{n} biggr) } \text { has } 4\pi $ as period?
Also could it be possible to solve this if we need $x\pi$ as period ?I am interested in learning the general approach for this particular type of the problem. |
Can we call the [greatest integer function][1] as a periodic function with no fundamental period or is it just non-periodic.
Please explain your answer.
----------
To my understanding, if we consider $f(x) = [x]$ now, $f(3) = [3] = 3$ and $f(3+0.5) = [3.5] = 3$ So, can't we say that it is periodic (**A constan... |
What is the value of n $\in \mathbb{Z} $ for which the function $\displaystyle f(x) = \frac{sin nx} { \sin \biggl( \frac{x}{n} \biggr) } \text { has } 4\pi $ as period?
Also could it be possible to solve this if we need $x\pi$ as period ?I am interested in learning the general approach for this particular type of th... |
Let $B$ be the unit ball of $\ell^2(\mathbb{N})$. For each $x=(x_1,x_2,\cdots)\in B$, let
$$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define $T:B\to 2^B$ by $$T(x)=B(f(x),r(x))\cap B, \mbox{ where }r(x)=\frac{1}{2}(\|x-f(x)\|).$$
Is it true that $$D(Tx,Ty)\le \|x-y\| \mbox{ for all }x,y\in B?$$
Here $D$ is the Hausdorff met... |
Can we call the [greatest integer function][1] as a periodic function with no fundamental period or is it just non-periodic.
Please explain your answer.
----------
To my understanding, if we consider $f(x) = [x]$ now, $f(3) = [3] = 3$ and $f(3+0.5) = [3.5] = 3$ So, can't we say that it is periodic (**A constan... |
What is the value of n $\in \mathbb{Z} $ for which the function $\displaystyle f(x) = \frac{\sin nx} { \sin \biggl( \frac{x}{n} \biggr) } \text { has } 4\pi $ as period?
Also could it be possible to solve this if we need $x\pi$ as period ?I am interested in learning the general approach for this particular type of t... |
There have been some confusing comments regarding dependence on $x$ or on $c$, so let me try to put it all together.
You are correct that $\delta$ should not depend on $x$. However, when one is proving that $f(x)$ is continuous *at $c$*, then $\delta$ is allowed to depend on both $\epsilon$ *and* $c$.
Remember ... |
Let $B$ be the unit ball of $\ell^2(\mathbb{N})$, i.e. $B=\lbrace x\in \ell^2(\mathbb{N}): \|x\|\le 1\rbrace.$ For each $x=(x_1,x_2,\cdots)\in B$, let
$$f(x)=(1-\|x\|,x_1,x_2,\cdots).$$ Define $T:B\to 2^B$ by $$T(x)=B(f(x),r(x))\cap B, \mbox{ where }r(x)=\frac{1}{2}(\|x-f(x)\|).$$
Is it true that $$D(Tx,Ty)\le \|x-y... |
Question: Let $V \subset M(n,n,\mathbb{R})$ be the set of all symmetric, real $(n \times n)$ matrices, that is $a_{ij} = a_{ji}$ for all $i,j$. Show that $V$ is a subspace of $M(n,n,\mathbb{R})$ and calculate dim$(V)$.
My attempt so far:
First part: To show that $V$ is a subspace I need to show:
(a) $ 0 \in V$ an... |
To write that the matrix is the zero matrix, you should write "let $a_{ij}=0$ for all $i$ and $j$", not "$a_{ij}\in 0$". (Nothing is an element of $0$).
For (b): No, notice that the $n$ is fixed. You are only considering matrices that are symmetric of a *fixed size*. If $n=2$, then you only consider $2\times 2$ mat... |
13pi/4=2pi+5pi/4
k=1
angle=5pi/4
Am i right that k=2pi? |
$$13\pi\frac4=2\pi+5\frac\pi\frac4$$
k=1
angle=5\pi\frac4
Am i right that k=2pi? |
$$ \frac{13*\pi}{4}=(2\pi)+\frac{5\pi}{4}$$
k=1
angle=5\pi\frac4
Am i right that k=2pi? |
$$ \frac{13*\pi}{4}=(2\pi)+\frac{5\pi}{4}$$
$k=1
\text{angle}=\frac{5\pi}{4}$
Am I right that $k=2\pi$? |
The angle $\frac{5\pi}{4}$ needs to be larger than 0 and lower than $2\pi$
$ \frac{13*\pi}{4}=(2\pi)+\frac{5\pi}{4}$
Am i right that k=$2\pi$?
|
I heard that they are difficult. Is this true? Are they like the qualifying exams in grad school? For example, is the probability exam and the financial math exam comparable to qualifying exams (e.g. needs months and months of study)? |
Are the actuarial exams hard? |
Given an angle $\theta$ (in radians), write $\theta = 2\pi t + \phi$, where $\phi$ is between $0$ and $2\pi$, $0\leq \phi\lt 2\pi$, and $t$ is an integer. Call $k=2\pi t$ the number of "full circles" or "full turns" corresponding to the angle.
If $\theta = \frac{13\pi}{4}$,
am I right that k=$2\pi$?
|
Can n! be a perfect square when n is an integer greater than 1?
|
Can n! be a perfect square when n is an integer greater than 1? |
Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic oscillator. Even if this is not the strongest spectral theorem, it still seems useful enough that there should be many nic... |
What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for? |
I'm not quite sure where such a concept fits. Suppose $X$ is a topological space. I know then that the diagonal $\Delta=\{(x,x)\ | x\in X\}$, so $\Delta\subseteq X\times X$. What then would a neighborhood of the diagonal be comprised of?
By this I mean, suppose $E\in\mathscr{N}_\Delta$, where $\mathscr{N}_\Delta$ i... |
What exactly is a neighborhood of the diagonal on a topological space? |
I'm not quite sure where such a concept fits. Suppose $X$ is a topological space. I know then that the diagonal $\Delta=\{(x,x)\ | x\in X\}$, so $\Delta\subseteq X\times X$. What then would a neighborhood of the diagonal be comprised of?
By this I mean, suppose $E\in\mathscr{N}_\Delta$, where $\mathscr{N}_\Delta$ i... |
If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ?
Isometry sounds more geometrical than topological. What would be a good name for "piecewise topology" ? If you take a topological space and break it up and p... |
Maybe a definition of functional calculus? Using quantum mechanics notation, if $A$ is self-adjoint and compact, then $A = \sum \lambda_k |k\rangle\langle k|$, which means that for $f:\mathbb{R}\to\mathbb{R}$ we can define $f(A) = \sum f(\lambda_k) |k\rangle\langle k|$.
This allows us to give a simple demonstration... |
Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $/pi$ or $e$. An excerpt from my textbook:
> In 1748, Leonard Euler used the sum of the infinite series of $e$ (mentioned in the bo... |
Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. An excerpt from my textbook:
> In 1748, Leonard Euler used the sum of the infinite series of $e$ (mentioned in the bo... |
Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. An excerpt from my textbook:
> In 1748, Leonard Euler used the sum of the infinite series of $e$ (mentioned in the bo... |
There are a couple of reasons why people compute $\pi$ and $e$ to so many digits.
One is simply that it is a way to test hardware. The values are known (or can be known), and you want to see if your hardware is fast and accurate with these computations.
Another is that there are actually a lot of questions about ... |
There are a couple of reasons why people compute $\pi$ and $e$ to so many digits.
One is simply that it is a way to test hardware. The values are known (or can be known), and you want to see if your hardware is fast and accurate with these computations.
Another is that there are actually a lot of questions about ... |
Theorem: If a function f is analytis throughout a simply connected domain D, then for every closed contour C lying in D, the integral is o along the contour.
This is a theorem from the text I am using without complete proof. The problem is that when the closed contour has an infinite number of self-intersection poin... |
I'd just like to give two quotes from the book [*The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing*][1] that might help explain motivation. Here is the one from their chapter on computing constants to 10,000 digits:
> While such an exercise might seem frivolous, the fact is we learned a
> l... |
I want to solve equations like this (mod 2^n):
3x+4y+13z=3 (mod 16)
x+5y+3z=5 (mod 16)
4x+7y+11z=12 (mod 16)
Since we are working over a ring and not a field, Gaussian elimination doesn't work. So how can I still solve these types of equations?
|
I want to solve equations like this (mod $2^n$):
$$\begin{array}{rcrcrcr} 3x&+&4y&+&13z&=&3&\pmod{16} \\ x&+&5y&+&3z&=&5&\pmod{16} \\ 4x&+&7y&+&11z&=&12&\pmod{16}\end{array}$$
Since we are working over a ring and not a field, Gaussian elimination doesn't work. So how can I still solve these types of equations?
|
You can find a proof in [Section IV.6][1] of J.B. Conway's *Functions of one complex variable*.
[1]: http://books.google.com/books?id=9LtfZr1snG0C&lpg=PP1&dq=conway%20complex&pg=PA87#v=onepage&q&f=false |
What is the spectral theorem for compact operators good for? Here are some examples. (I am ignoring the self-adjoint aspects, since they don't really play
a role in the theorem. And it is valid for more general spaces than Hilbert spaces too, so I will also ignore that part, in the sense that I won't pay too much a... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( *any* spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not in... |
There are a couple of reasons why people compute $\pi$ and $e$ to so many digits.
One is simply that it is a way to test hardware. The values are known (or can be known), and you want to see if your hardware is fast and accurate with these computations.
Another is that there are actually a lot of questions about ... |
Just curious. You pay electronically, and you get a string of numbers. My question, what is the likelihood for a person to type in the same serial number to say, top up their mobile credit? What is the range of probability electronic payment companies are using?Then, does electronic payment in China have longer serial ... |
What probability methodology is electronic payment using? |
If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n-$th term test (checking whether the $n-$th term of a series goes to zero) is sufficient to check convergence of a series in $K$.
I was wondering if we have any results that guarantee, under additional hypothesis, whether a serie... |
i wanted to ask these questions here, feeling that this place place is without doubt is better than anywhere else,feeling hesitant and shy to ask such question.<br><br>
for several reasons, the environment which i live in doesn't encourage the learning of natural sciences, it permits it but doesn't put efforts in teac... |
If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a series in $K$.
I was wondering if we have any results that guarantee, under additional hypothesis, whether a series ... |
i wanted to ask these questions here, feeling that this place place is without doubt is better than anywhere else,feeling hesitant and shy to ask such question.<br><br>
for several reasons, the environment which i live in doesn't encourage the learning of natural sciences, it permits it but doesn't put efforts in teac... |
I wanted to ask these questions here, feeling that this place place is without doubt is better than anywhere else, feeling hesitant and shy to ask such question.
For several reasons, the environment which I live in doesn't encourage the learning of natural sciences, it permits it but doesn't put efforts in teaching ... |
Let $\rho : S_n \to \mathrm{GL}(\mathbb{C}^n)$, where $\rho(\sigma)(x_1, \ldots, x_n) = (x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)})$. How can you prove that $W = \{ (x_1, \ldots, x_n) : x_1 + \cdots + x_n = 0 \} \subset \mathbb{C}^n$ is an irreducible representation? |
Let's assume that all the integers are in fact at least $1$.
Let $\max(P) = \prod_i s_i$. If $\max(P) \neq \min(P)$ then $\max(P)-1$ must be representable. Now $(s_1 - 1) \prod_{i>1} s_i = \prod_i s_i - \prod_{i>1} s_i$, so there must be an index $I$ such that $S_i = \{1\}$ for $i \neq I$. $S_I$ itself is of course ... |
A <a href="http://en.wikipedia.org/wiki/Doubly_stochastic_matrix">doubly-stochastic matrix</a> is an $n\times n$ matrix $P$ such that
$\displaystyle\sum_{i=1}^n{p_{ij}}=1$
and
$\displaystyle\sum_{j=1}^n{p_{ij}}=1$
where $p_{ij}\ge 0$.
Can someone suggest an algorithm for generating these matrices uniform... |
How do I Generate Doubly-Stochastic Matrices Uniform Randomly? |
Suppose I take $n$ sets $\{S_i\}$ where each $S_i$ is some subset of integers. Let $P$ indicate a set of integers obtainable by drawing exactly one factor from each $\{S_i\}$ and multiplying them together. How can I choose $S_i$'s so that
1. $P$ contains every integer between $\max(P)$ and $\min(P)$
2. Each element... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( *any* spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not in... |
I have a homework problem in my textbook that has stumped me so far. There is a similar one to it that has not been assigned and has an answer in the back of the textbook. It reads: How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?
The answer is 609638400... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( *any* spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not in... |
First consider the position of the men, then the position of the women.
Place the eight men in a row. How many possible ways are there ? It will be $8! = 40320$
Now as no two women stand next to each other,lets imagine the condition:
> * M * M * M * M * M * M * M * M *
Hence now we ne... |
First consider the position of the men, then the position of the women.
How many possible ways are there to place the eight men in a row? It will be $ ^8P_8 = 8! = 40320$
Now as `no two women stand next to each other`,we can imagine the situation as
> * M * M * M * M * M * M * M * M *
... |
How can I obtain from a differential equation a stochastic version? |
For this type of problem first consider the position of the men, then the position of the women.In general first consider the the group with more people.
How many possible ways are there to arrange eight men in a row? It will be $ ^8P_8 = 8! = 40320$
Now as `no two women stand next to each other`,we can imagine t... |
For this type of problem first consider the position of the men, then the position of the women.In general first consider the the group with more people.
How many possible ways are there to arrange eight men in a row? It will be $ ^8P_8 = 8! = 40320$
Now as `no two women stand next to each other`,we can imagine t... |
I wasn't sure that you wanted a proof that used that fact from Apostol.
One easy method *not* using the result you quote from Apostol is as follows:
$$
\sum_{p\le x} \frac{1}{\sqrt{p}} > \sum_{p \le x} \frac{1}{\sqrt{x}} = \frac{\pi(x)}{\sqrt{x}} > c \frac{\sqrt{x}}{\log{x}}
$$
where $c$ is a constant you can ge... |
Original post on Mathoverflow [here](http://mathoverflow.net/questions/47487/probability-of-the-maximum-levy-stable-random-variable-in-a-list-being-greater).
Given a list of identical and independently distributed [Levy Stable](http://en.wikipedia.org/wiki/Stable_distribution) random variables, $(X_0, X_1, \dots, X_... |
Is there a "SETI at Home" style initiative for calculating constants with distributed computing? |
For what values of $x = (x_0 x_1 ... x_n)$ the following inequality holds for all positive values ($a_i > 0$) of the vector $a = (a_0 a_1 ... a_n)$: $a x^T \ge 0$? |
While learning *topology* one learns about compact set. The standard definition is:
- A set $X$ is said to be compact *if open cover has a finite subcover.*
Since $[0,1]$ is compact, if we take a open cover for this we should be able to get a finite subcover. I know, that $(0,1)$ is not compact, so there must... |
Isn't this clear from $(0,1) = (0,1/2) \cup (0,2/3) \cup (0,3/4) \cup \cdots$? |
I have a plane defined through a point P and 2 3d-vectors Xdir and yDir.
I wish to convert coordinates of points on this plane between local 2d-parametric and world 3d coordinate systems.
I know the conversion from 2d Parametric to 3d is
C(u, v) = P + u*XDir + v*YDir
however i have been unable to find a wa... |
Given a plane and a 3d-Vertex, what are U and V? |
For what values of $x = (x_0 x_1 ... x_n)$ the following inequality holds for all positive values of the vector $a = (a_0 a_1 ... a_n)$: $a x^T \ge 0$? |
Hi i have a problem with quadratic splnes, i am supposed to find $
S_1 $ and $S_2$ that interpolates the following points S(-1)=0 S(0)=1 S(1)=2, and at the same time we want to find S such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x.$ is minimal. The answer is on the form
$
S_1(x)=a_1 x^2 +b_1x +c_1$ on ... |
Hi I have a problem with quadratic splines, I am supposed to find $
S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ is minimal. The answer is on the form
$S_1(x)=a_1 x^2 +b_1x +c_1$ on... |
Let $S$ be a collection of disjoint sets. Let a binary operation '*' defined $\forall x,y$ each belonging to two different sets or a same set in $S$ with the property that $z=x*y$ belongs to some set in $S$ . Let $\forall A_1,A_2 \in S$ a binary operation $ \otimes$ is defined between $A_1,A_2$ as $B=A_1 \otimes A_2$... |
Question: Let $V \subset M(n,n,\mathbb{R})$ be the set of all symmetric, real $(n \times n)$ matrices, that is $a_{ij} = a_{ji}$ for all $i,j$. Show that $V$ is a subspace of $M(n,n,\mathbb{R})$ and calculate dim$(V)$.
My attempt so far:
First part: To show that $V$ is a subspace I need to show:
(a) $ 0 \in V$ an... |
In a 2-d plane, let $A$ be the set of all points inside a circle $C$ and $P$ be the set of all points on the perimeter of the circle $C$. Is there any generalized measure in set theory which distinguishes between two such sets ? |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( *any* spanning tree would do), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not in... |
Suppose a <a href="http://en.wikipedia.org/wiki/Parabolic_mirror">parabolic mirror</a> of a telescope is large enough (or equivalently that the camera <a href="http://en.wikipedia.org/wiki/Shutter_speed">exposure time</a> is short enough) that you need to take into account the transit time of the light signal from hitt... |
I know that there was a guy that could get 100 pi digits before de computers be able to get lot of thousands.
How did the guy did that? |
The <a href="http://stackoverflow.com/questions/4324490/gaussian-distribution-hash-tables">original question</a> I posted on StackOverflow. I think it's more mathematically inclined so I posted it here again. In terms of math.
There's a class of students. Each students has a score between 0 and 300 (gaussian, with s... |
Let us assume that we are given a closed, orientable 2D manifold embedded in $R^3$, and lets call it $M$.
I think it is clear that in a coordinate neighborhood $(U, \phi)$ it is possible to at each point define a unit normal vector $N_p$ by looking at the tangent space $T_pM$ as a subspace of $T_pR^3$ and considerin... |
You question is a bit underdefined, but in general what you're looking for is index (order?) statistics. For example, if the number of classes is to be $2$, you need to cut them at the median. If it's $3$, you cut them at the $1/3$ and $2/3$ order statistics, and so on. The classes will be entirely evenly-spread.
Th... |
Statisticians would usually refer to your "classes" as "bins".
Your requirements for "minimum number of bind" and "minimum size of the bins" are in opposition, and you would have to clarify how the conflict should be resolved. For instance, if one arrangement has 10 bins with 5 marks each, and another has 5 bins wi... |
Hi I have a problem with quadratic splines, I am supposed to find $
S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ is minimal. The answer is on the form
$S_1(x)=a_1 x^2 +b_1x +c_1$ on... |
The area of A is positive, while the area of P is zero. Did you mean to distinguish between the interior of the circle and the closure of that set (i. e. the union of the perimeter and the interior)? |
Suppose a <a href="http://en.wikipedia.org/wiki/Parabolic_mirror">parabolic mirror</a> of a telescope is large enough (or equivalently that the camera <a href="http://en.wikipedia.org/wiki/Shutter_speed">exposure time</a> is short enough) that you need to take into account the transit time of the light signal from hitt... |
Suppose a <a href="http://en.wikipedia.org/wiki/Parabolic_mirror">parabolic mirror</a> of a telescope is large enough (or equivalently that the camera <a href="http://en.wikipedia.org/wiki/Shutter_speed">exposure time</a> is short enough) that you need to take into account the transit time of the light signal from hitt... |
Can you divide an infinite set, into an infinite number of infinite sets? |
Suppose a <a href="http://en.wikipedia.org/wiki/Parabolic_mirror">parabolic mirror</a> of a telescope is large enough (or equivalently that the camera <a href="http://en.wikipedia.org/wiki/Shutter_speed">exposure time</a> is short enough) that you need to take into account the transit time of the light signal from hitt... |
This question is motivated by [this recent question][1], which asked for a characterisation of groups with at most one subgroup of each finite index. Arturo Magidin's answer showed that every such finite group is cyclic, but the questioner, Louis Burkill, added in comments that he is really interested in the infinite ... |
If G is residually cyclic, does G have at most one subgroup of each finite index? |
Can you partition an infinite set, into an infinite number of infinite sets? |
How do we show the following?
Let $X$ be a topological space and let $x \in X$. Show that if $x$ has a countable neighborhood basis in $X$ then $x$ has a countable neighborhood basis in $\beta X$. Here $\beta X$ denotes the Stone-Cech compactification of $X$. |
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