instruction stringlengths 12 30k |
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To distinguish [(general) topology][1] from more specific branches of topology? -[Differential topology][2], [algebraic topology][3], [geometric topology][4], [low-dimensional topology][5]...
[1]: http://en.wikipedia.org/wiki/General_topology
[2]: http://en.wikipedia.org/wiki/Differential_topology
[3]: htt... |
**The Problem:**
$$\frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \frac{4}{3 \cdot 4 \cdot 5 \cdot 6} + \frac{9}{5 \cdot 6 \cdot 7 \cdot 8} + \frac{16}{7 \cdot 8 \cdot 9 \cdot 10} + \cdots $$
Any smarter way to solve this may be within a minute or two?
----------
I am adding my solution,If you are a student please... |
**Question:** Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?
**Why do I care:** For a number of math kids I know, doing algebraic topology is fine until we get to homology, and... |
How to prove that in every power of 3, with natural exponent, the tens digit
is an even number?
(For example, 3 ^ 5 = 243, 4 is even.)
|
A and B are finite groups.
$H \unlhd A \times B$.
Can we find some $C \unlhd A$, $D \unlhd B$ such that $H \cong C \times D$?
I can prove it for abelian A and B.
In case the statement is not true: is it true under further assumptions about A and B, such as solvablity, nilpotency, etc? |
Finite groups: $H \unlhd A \times B$. Is $H \cong C \times D$ for some $C \unlhd A$, $D \unlhd B$? |
$A$ and $B$ are finite groups.
$H \unlhd A \times B$.
Can we find some $C \unlhd A$, $D \unlhd B$ such that $H \cong C \times D$?
In case the statement is not true: is it true under further assumptions about A and B, such as solvablity, nilpotency, etc?
Special cases I can prove:
1. $A$ and $B$ are abelian ... |
$A$ and $B$ are finite groups.
$H \leq A \times B$.
Can we find some $C \leq A$, $D \leq B$ such that $H \cong C \times D$?
In case the statement is not true: is it true under further assumptions about A and B, such as solvablity, nilpotency, etc?
Special cases I can prove:
1. $A$ and $B$ are abelian (follo... |
Finite groups: $H \leq A \times B$. Is $H \cong C \times D$ for some $C \leq A$, $D \leq B$? |
**The Problem:**
$$\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \frac{16}{7 \cdot 8 \cdot 9 \cdot 10} + \cdots$$
Any smarter way to solve this may be within a minute or two?
----------
I am adding my solution,If you are a student please don't read-on I f... |
How to sum $\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \cdots$ quickly? |
Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. One (very) simple model of this process might be as follows:
1) the probability of the particle getting kicked up in 1 second increases with th... |
$A$ and $B$ are finite groups.
$H \leq A \times B$.
Can we find some $C \leq A$, $D \leq B$ such that $H \cong C \times D$?
In case the statement is not true: is it true under further assumptions about A and B, such as solvability, nilpotency, etc?
Special cases I can prove:
1. $A$ and $B$ are abelian (foll... |
Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. One (very) simple model of this process might be as follows:
1) the probability of the particle getting kicked up in 1 second increases with th... |
Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. One (very) simple model of this process might be as follows:
1) the probability of the particle getting kicked up in 1 second increases with th... |
Here is a problem from Griffith's book Introduction to E&M.
Let $\theta(x)$ be the step function
$$\theta = \begin{cases}
0 & x \neq 0 \\
1 & x > 0
\end{cases}
$$
The question is how to prove $\frac{d\theta}{dx} = \delta(x)$.
I think since the function is discontinous at x = 0, there is no definition of ... |
How to prove $\frac{d\theta}{dx} = \delta(x)$? |
<a href = "en.wikipedia.org/wiki/Hypercube"> Wikipedia </a> says, A hypercube is an n-dimensional analog of a square/cube. What is a hypercylinder then? An n-dimensional analog of a cylinder?
<cite>Constant Approximation Algorithm for MST in Resource Constrained Wireless
Sensor Networks</cite> gives a definition,
... |
What is a hypercylinder ? |
<a href="http://en.wikipedia.org/wiki/Hypercube">Wikipedia</a> says, A hypercube is an n-dimensional analog of a square/cube. What is a hypercylinder then? An n-dimensional analog of a cylinder?
<cite>Constant Approximation Algorithm for MST in Resource Constrained Wireless
Sensor Networks</cite> gives a definition,
... |
the Fourier transformation of a scalar function with respect to one variable might be defined as
$\mathcal{F}\left[w\right](\omega )\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}w(t)e^{-\mathrm{i}\omega t}dt$
In physics, this transformation along with its generalization, the Laplace transformation, has a tre... |
Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense? |
This is a place where physicists and mathematicians would phrase the question differently. A mathematician would say that $d \theta/dx$ is undefined at $0$, and that $\delta$ is not a function. However, as Mariano says, the statement is true "in the sense of distributions".
Also, I think you have a typo; I think $\t... |
Is it possible to characterize the distributional derivative as some sort of "best linear approximation" of a distribution (ala the Frechet/Gauteaux derivatives), if viewed in the appropriate spaces?
For example, let $f \in D'$, $\phi \in D$, and let $\psi_x \in D$ be a mollified ramp function so that $\psi_x(x,y,z,... |
Is it possible to characterize the distributional derivative as some sort of "best linear approximation" of a distribution (ala the Frechet/Gauteaux derivatives), if viewed in the appropriate spaces?
For example, let $f \in D'$, $\phi \in D$, and let $\psi_x \in D$ be a mollified ramp function so that $\psi_x(x,y,z,... |
Does anybody know how to derive
$$\det(\mathbf A)\cdot \det(\mathbf D + \mathbf E \cdot \mathbf A^{-1} \cdot \mathbf B) = \det(\mathbf D)\cdot \det(\mathbf B \cdot \mathbf D^{-1} \cdot \mathbf E)$$
(most likely using [Sylvester determinant][1] theorem), where $\mathbf A$,$\mathbf B$,$\mathbf C$,$\mathbf D$,$\mat... |
Here is a problem from Griffith's book Introduction to E&M.
Let $\theta(x)$ be the step function
$$\theta = \begin{cases}
0 & x \leq 0 \\
1 & x > 0
\end{cases}
$$
The question is how to prove $\frac{d\theta}{dx} = \delta(x)$.
I think since the function is discontinous at x = 0, there is no definition of ... |
Is it possible to characterize the distributional derivative as some sort of "best linear approximation" of a distribution (*a la* the Fréchet/Gâteaux derivatives), if viewed in the appropriate spaces?
For example, let $f \in D'$, $\phi \in D$, and let $\psi_x \in D$ be a mollified ramp function so that $\psi_x(x,y,... |
Consider the life of a single dust particle that randomly gets kicked up into the air, at which point it floats around for a while before being deposited on the floor again. One (very) simple model of this process might be as follows:
1) the probability of the particle getting kicked up in 1 second increases with th... |
Does anybody know how to derive
$$\det(\mathbf A)\cdot \det(\mathbf D + \mathbf E \cdot \mathbf A^{-1} \cdot \mathbf B) = \det(\mathbf D)\cdot \det(\mathbf A + \mathbf B \cdot \mathbf D^{-1} \cdot \mathbf E)$$
(most likely using [Sylvester determinant][1] theorem), where $\mathbf A$,$\mathbf B$,$\mathbf C$,$\mat... |
These two appear in my module (without any proof):
$$\sum_{r = 1}^{n} C(n-2,r-2) = 2^{n-1}$$
$$\sum_{r = 0}^{n-1} C(n-2,r) = 2^{n-1}$$
For the first one when when $r=1 \Rightarrow C(n-2,1-2) = C(n-2,-1)$ ?!The same thing for $r=n-1$ in the second one,are they valid if yes,please explain how? |
Does these two expression valid? |
Is the equator of the Earth a line or a circle?
Well, it's a circle of course. But as you know by now, from Willie Wong's answer and from others, it's also a geodesic, which is very much like a straight line: unless you are ready for some serious digging, the equator is the shortest path between any two places on i... |
These two appear in my module (without any proof):
$$\sum_{r = 1}^{n} C(n-2,r-2) = 2^{n-1}$$
$$\sum_{r = 0}^{n-1} C(n-2,r) = 2^{n-2}$$
For the first one when when $r=1 \Rightarrow C(n-2,1-2) = C(n-2,-1)$ ?!The same thing for $r=n-1$ in the second one,are they valid if yes,please explain how? |
My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own.
The problem is that I feel like I still don't have any idea how to do algebra.... |
Intuition in algebra? |
It seems you have a kind of geometrical intuition. If that's the case, why don't you construct geometrical representations of the algebraic entities you encounter? Have you had any representation theory till now? If not, that might help you figure out why algebra is "important" or why anyone, even outside the field of ... |
Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$
For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$.
Why is the sequence $$\langle a_n\rangle$$ not equidistributed in $[0,1]$? |
I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. In addition to that there are two vertical faces $S_3$ and $S_4$, that connect between $S_1$ and $S_2$ vertically, as shown in the below diagram. The question is how to find the polyhedrons that are for... |
How to sum up this series? |
Are these two expressions valid? |
Are these two expressions for sums of binomial coefficients valid? |
First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (ie, by writting an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is consitant with the 2 and 3 demensional objects we know). In the case of a cone, this object, in 3D is a circul... |
First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (ie, by writting an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is consitant with the 2 and 3 demensional objects we know). In the case of a cone, this object, in 3D is a circul... |
For example:
What conclusions can be drawn about the relations between two objects with the same group of endomorphism?
Can we tell from End(A) if A is Abelian or not?
Does End(A) contain information about the sub-objects of A?
Any information or references to information about this is highly appreciated. |
What does the endomorphism group of an object tell us about the object in question? |
Does anybody know how to derive
$$\det(\mathbf A)\cdot \det(\mathbf D + \mathbf E \cdot \mathbf A^{-1} \cdot \mathbf B) = \det(\mathbf D)\cdot \det(\mathbf A + \mathbf B \cdot \mathbf D^{-1} \cdot \mathbf E)$$
(most likely using [Sylvester determinant][1] theorem), where $\mathbf A$,$\mathbf B$,$\mathbf C$,$\mat... |
This is a place where physicists and mathematicians would phrase the question differently. A mathematician would say that $d \theta/dx$ is undefined at $0$, and that $\delta$ is not a function. However, as Mariano says, the statement is true "in the sense of distributions".
Also, I think you have a typo; I think $\t... |
Here is a partial answer (although it uses the ring structure on the set of endomorphisms). A torsion abelian group is cyclic if and only if any two elements in $End(A)$ commute with respect to composition (in other words, $End(A)$ is a commutative ring).
http://www.springerlink.com/content/m72222448q6j7327/
Ther... |
Here is a partial answer (although it uses the ring structure on the set of endomorphisms). A torsion abelian group is cyclic if and only if any two elements in $End(A)$ commute with respect to composition (in other words, $End(A)$ is a commutative ring).
http://www.springerlink.com/content/m72222448q6j7327/
Ther... |
Is the equator of the Earth a line or a circle?
Well, it's a circle of course. But as you know by now, from Willie Wong's answer and from others, it's also a geodesic, which is very much like a straight line: unless you are ready for some serious digging, the equator is the shortest path between any two places on i... |
You might be interested in the expository article "[Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat's lemma][1]" by Anderson and Camillo. A couple of excerpts:
![alt text][2]
----------
![alt text][3]
[1]: http://books.google.com/books?id=J-tyGid... |
First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (ie, by writting an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is consitant with the 2 and 3 demensional objects we know). In the case of a cylinder, this object, in 3D is a ci... |
Is the equator of the Earth a line or a circle?
Well, it's a circle of course. But as you know by now, from Willie Wong's answer and from others, it's also a geodesic, which is very much like a straight line: unless you are ready for some serious digging, the equator is the shortest path between any two places on i... |
Let $L \subseteq M$ be A-modules and let p be an associated prime of $M/L$ and assume that the annihilator of $L$ is not contained in $p$. How would one go about proving that p is an associated prime of $M$? |
If one case is true but the other false, does that mean that the statement is true or false? |
Read the section "Rational points on conics" in the motivational chapter in the very beautiful book [Rational points on elliptic curves][0] by Silverman and Tate. Also, once you have some knowledge of elliptic curves don't miss Franz Lemmermeyer's [Conics - a poor man's ellitpic curves](http://www.fen.bilkent.edu.tr/~f... |
Read the section "Rational points on conics" in the motivational chapter in the very beautiful book [Rational points on elliptic curves][0] by Silverman and Tate and/or google "rational parametrization of the circle". Also, once you have some knowledge of elliptic curves don't miss Franz Lemmermeyer's [Conics - a poor ... |
First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (ie, by writting an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is consitant with the 2 and 3 demensional objects we know). In the case of a cylinder, this object, in 3D is a ci... |
If one case is true but the other false, does that mean that the statement is true or false?
There exists a positive integer, n, such that n^2-79n+1601 is composite
case 1 - n is even - false
case 2 - n is odd - true |
If one case is true but the other false, does that mean that the statement is true or false?
There exists a positive integer $n$ such that $n^2-79n+1601$ is composite
- Case 1 - $n$ is even - false.
- Case 2 - $n$ is odd - true. |
I need to find the domain of the Gamma function, that is to say all $z \in \mathbb{C}$, for which the integral:
$$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \mathrm dt$$
converges. I started by splitting up the integral into an integral running from $0$ to $1$ and another one from $1$ to $\infty$. I first tried to ... |
Trevor, you are confusing what is generally thought of as "Proof by cases", and a search for an example that proceeds by narrowing the possibilities/examining different cases.
A **Proof by Cases** is one that proceeds along the following lines: you want to show that a statement is true for all $x$, but you do so by ... |
I got this question as a mathematical puzzle from a friend. It looks similar to this one http://mathoverflow.net/questions/2014/if-you-break-a-stick-at-two-points-chosen-uniformly-the-probability-the-three-re
However, in this case, I have to find the expected length of the smallest segment. The two points where the ... |
If a 1 meter rope is cut at two uniformly randomly chosen points (to give three pieces), what is the average length of the smallest piece? |
Let $L \subseteq M$ be A-modules and let $\mathfrak{p}$ be an associated prime of $M/L$ and assume that the annihilator of $L$ is not contained in $\mathfrak{p}$. How would one go about proving that $\mathfrak{p}$ is an associated prime of $M$? |
Recall that $\mathfrak p$ being an associated prime of $M/L$ precisely means that
$\mathfrak p$ is a prime ideal which is the annihilator of some element of $M/L$.
If we think what this means in terms of the module $M$, it means that we can
find an element $m \in M$ such that $\mathfrak p = \{a \in A \,| \, a m \in ... |
Recall that $\mathfrak p$ being an associated prime of $M/L$ precisely means that
$\mathfrak p$ is a prime ideal which is the annihilator of some element of $M/L$.
If we think what this means in terms of the module $M$, it means that we can
find an element $m \in M$ such that $\mathfrak p = \{a \in A \,| \, a m \in ... |
A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation.
What is know about $D=(S,D)$ defined as above except that $D$ is closed under the superset operation?
Crossposted from [MO][1].
[1]: ht... |
A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation.
What is known about $D=(S,D)$ defined as above except that $D$ is closed under the superset operation?
Crossposted from [MO][1].
[1]: h... |
The proof that I know of the theorem goes like this:
1. Any module $M$ is a quotient of a free module $F$ (over any ring).
2. Any submodule $K$ of a free module $F$ over a PID $R$ is a free module, so in particular the kernel of the above quotient map is free.
3. For any free submodule $K$ of a free module $F$ ove... |
Why does the structure theorem for finitely generated modules over PIDs fail for arbitrary modules over a PID? |
Having: P = N^( CN + 1 );
How can I simplify this equation to N = ....?
I tried using logarithms but I'm stucked...
Any ideas? |
Is possible to simplify P = N^( CN + 1 ) in terms of N? |
The proof that I know of the theorem goes like this:
1. Any module $M$ is a quotient of a free module $F$ (over any ring).
2. Any submodule $K$ of a free module $F$ over a PID $R$ is a free module, so in particular the kernel of the above quotient map is free.
3. For any free submodule $K$ of a free module $F$ ove... |
The generalization of this problem to $n$ pieces is discussed extensively in David and Nagaraja's *[Order Statistics][1]* (pp. 133-135, and p. 153). I'll just summarize some of the main results; see the text itself for details.
Let $V_1, V_2, \ldots, V_n$ denote the break lengths. Then the (degenerate) joint pdf i... |
Prove that $\left \{ 0,1 \right \}^{\mathbb{N}}\sim \left \{ 0,1,2,3 \right \}^{\mathbb{N}}$ and find a direct bijection function.
I got the first part by showing that $\left \{ 0,1 \right \}^{\mathbb{N}} \subseteq \left \{ 0,1,2,3 \right \}^{\mathbb{N}} \subseteq {\mathbb{N}}^{\mathbb{N}}$, which implies that $|\le... |
Prove that $\left \{ 0,1 \right \}^{\mathbb{N}}\sim \left \{ 0,1,2,3 \right \}^{\mathbb{N}}$ and find a direct bijection function.
I got the first part by showing that $\left \{ 0,1 \right \}^{\mathbb{N}} \subseteq \left \{ 0,1,2,3 \right \}^{\mathbb{N}} \subseteq {\mathbb{N}}^{\mathbb{N}}$, which implies that $|\le... |
> But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.
Roughly speaking, geometric intuition suggests to you what should be true, and sugg... |
Currently I am implementing a DTW (Dynamic Time Warping) algorithm for my project. As some will know it has the complexity of O(n^2). Considering a sound file of length 1-h with 44100 sampling. How long will it take to compute it?
(I know we need some other knowledge such as cpu speed, length of the word we are try... |
A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation.
What's the name for the following:
>$D=(S,D)$ defined as above except that $D$ is closed under the superset operation?
Crossposted from [MO][... |
Let $\omega$ denote the countably infinite cardinality and $A$ a cardinality that is strictly larger (i.e., an uncountable one). Is it true that $A^\omega$ has strictly larger cardinality than $A$? |
Greetings!
On a test recently I ended up having to solve this for y:
$$ x = \frac{2y}{y + 1} $$
But I kept getting stuck in cirlces...
$$
\begin{aligned}
x(y + 1 ) = 2y \\
xy + x = 2y \\
\frac{xy + x}{2} = y \\
\end{aligned}
$$
That didn't get me anywhere, so then I started over and tried multip... |
I have:
$$ln(ax + b)^4$$
After calculating (u'/u) I get:
$$4/(ax+b)$$ but the solution is $$4a/(ax+b)$$
What am I doing wrong? Thanks!
PS: a and b are IR+. |
A lot of "big" names in analysis (and other fields) seem to be doing some form of combinatorics (without any order, some examples are Tim Gowers, Terence Tao and Jean Bourgain).
So, looking a bit around makes me conclude that combinatorics is a huge field. There must be one "kind" which is the most fruitful in analy... |
A lot of "big" names in analysis (and other fields) seem to be doing some form of combinatorics (without any order, some examples are Tim Gowers, Terence Tao and Jean Bourgain).
So, looking a bit around makes me conclude that combinatorics is a huge field. There must be one "kind" which is the most fruitful in analy... |
I'm studying physics, so I'm sorry if I'll write some inexact things in this post. I wish you can understand me.
If we have 1D wave equation:
$$\frac{\partial^2 \psi}{\partial x^2}=\frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2}$$
we say that it's always possible (for physics situations...) to decompose th... |
How to compute these integrals? |
> But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.
Roughly speaking, geometric intuition suggests to you what should be true, and sugg... |
Is there any intuition behind surface integrals and their applications? I get that there's some stuff with fluid flow but it doesn't really stick that well, especially when it comes to surface orientation.
Is there any surefire way to figure out the orientation of a surface? And is there any sort of intuition behind... |
First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (i.e., by writing an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is constant with the 2 and 3 dimensional objects we know). In the case of a cylinder, this object, in 3D is a ci... |
Consider the case of $R=\mathbb{Z}$ and $M=\mathbb{Z}_{p^{\infty}}$, the Prufer group (the subgroup of $\mathbb{Q}/\mathbb{Z}$ generated by $\frac{1}{p^n}$ for all $n$.
Take $F$ be freely generated on countably many generators $x_n$, with $x_n$ mapping to $\frac{1}{p^n}$. Then $K$ is freely generated by $px_1$, $x_1... |
Having: $P = N^{CN + 1}$;
How can I simplify this equation to $N = \cdots$?
I tried using logarithms but I'm stucked...
Any ideas? |
Is possible to simplify $P = N^{ CN + 1}$ in terms of $N$? |
Lets say that you accept the definition of a convex set in higher dimensions, like a sphere in $\mathbb{R}^3$. The question I seek to provide insight into is why convex functions in one variable are defined as opening up instead of down, since this seems like an arbitrary definition. This is because, depending on how y... |
I don't have enough memory to simply create diagonal matrix D x D, getting an 'out of memory' error.
Instead of performing M x D x D operations in the first multiplication, I do M x D operations, but still my code takes ages to run.
Can anybody find a more effective way to multiply these matrices in matlab?
... |
A Google search has turned up an analysis of the asymptotic behavior of the iterates of $\sin$ on [page 157][1] of de Bruijn's *Asymptotic methods in analysis*. Namely,
$$\sin^n(1)=\frac{\sqrt{3}}{\sqrt{n}}\left(1+O\left(\frac{\log(n)}{n}\right)\right),$$
which implies that your series converges.
[1]: ht... |
Consider the case of $R=\mathbb{Z}$ and $M=\mathbb{Z}_{p^{\infty}}$, the Prufer group (the subgroup of $\mathbb{Q}/\mathbb{Z}$ generated by $\frac{1}{p^n}$ for all $n$.
Take $F$ be freely generated on countably many generators $x_n$, with $x_n$ mapping to $\frac{1}{p^n}$. Then $K$ is freely generated by $px_1$, $x_1... |
Let $\mu$ be a finite positive measure on the borel sets in $\mathbb{R}$ and suppose $\mathcal{L}^{1}(\mathbb{R},\mu) \subset \mathcal{L}^{\infty}(\mathbb{R},\mu)$.
Show that there exists $c>0$ such that if A is a Borel set with $\mu(A) > 0$ then $\mu(A) \geq c$.
No idea at all. Can you please help? |
What interesting properties of convex sets are retained by star-convex sets? |
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