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What are some uses for Monte Carlo simulations in mathematics? |
Couple of days ago, i actually asked this question : http://math.stackexchange.com/questions/2121/existence-of-a-normal-subgroup-with-authautg.
I was thinking about the converse of this statement. Suppose $G$ is a finite group, with subgroups $H$ and satisfies $|Aut(G)|=|Aut(H)|$, then can anything be said about, $... |
Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).
As a converse of the above, let $B$ be ... |
I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship)
A couple of things I was thinking about:
1) can create a very simple model where wind is 'one' force acting on the a uniform body.
2) model wind as vectors. This is where I am a little confused on how to start.
3) addi... |
Let $X$ denote the blowup of $\mathbb P^2$, $E$ the exceptional divisor, and $H$ the pullback of the hyperplane class. How can I compute $H^0(X,mE+nH)$, $H^1(X,mE+nH)$, and $H^2(X,mE+nH)$ for $m,n \in \mathbb Z$? If I'm working analytically, how can I think of these geometrically (e.g. "an element of $H^1(X,2E)$ is e... |
An elementary way of simplifying a trigonometric triple integral? |
What about $F = \sum_i W \cdot S_i$, where $W$ is the wind velocity vector and $S$ is a list of all the sail unit vectors?
I don't see this at all being a differential equation, but if you further specify your idea of this model, I can help you a little more. |
Suppose A is some algebraic structure, and x and y are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in A which maps x to y"?. It seems like the obvious thing to say is "x and y are symmetric in A", but I've never come across this phrasing or any other in ... |
Here is one elementary result you can't prove without the Axiom of Choice (that I used to think was not part of constructive mathematics).
Everybody knows how to prove that, if two sets $X$ and $Y$ are non-empty, then their cartesian product $X \times Y$ is non-empty too: since $X$ is non-empty, you can choose one e... |
Let p,p' = 2 mod 3 be prime.
Suppose that G is a group with the following properties:
(i) The 3-Sylow subgroup of G is cyclic.
(ii) The number of elements of order 3 in G is 2*p*p'
Claim: There exists a group which _in_addition_ satisfies either:
(iii)(a); G is simple; OR (iii)(b): 3 exactly divides |G|... |
There are $N$ bugs in a plane. All bugs are moving at the same constant (nonzero) speed, but no two bugs are moving in the same direction (velocity vectors are of the same speed, but no two are parallel).
Prove that at some point in time $N$ bugs will form **convex** polygon.
Edit: Can you loosen up any of the c... |
One (direct) way to see this is that $T$ is dominated by (a slightly modfied version of) the [Hardy-Littlewood maximal operator][1], and it is a theorem that this is a bounded operator on $L^p$ for $1<p <= \infty$. Or, directly, one can see that $T$ is bounded on $L^\infty$ and weak-bounded on $L^1$ (because if $f$ bel... |
One (direct) way to see this is that $T$ is dominated by (a slightly modfied version of) the [Hardy-Littlewood maximal operator][1], and it is a theorem that this is a bounded operator on $L^p$ for $p$ greater than 1. Or, directly, one can see that $T$ is bounded on $L^\infty$ and weak-bounded on $L^1$ (because if $f$ ... |
FIRST CASE. If you had only *one* set of values for $I$ and $R$, as you said in your answer to Rasmus, then you'd have a system of linear equations with infinitely many solutions.
For psychological reasons, let's write it this way:
$$
XA = B \ .
$$
Here $A$ is your $R$, $B$ your $I$ and $X$ your $A$. If we t... |
FIRST CASE. If you had only *one* set of values for $I$ and $R$, as you said in your answer to Rasmus, then you'd have a system of linear equations with infinitely many solutions.
For psychological reasons, let's write it this way:
$$
XA = B \ .
$$
Here $A$ is your $R$, $B$ your $I$ and $X$ your $A$. If we t... |
FIRST CASE. If you had only *one* set of values for $I$ and $R$, as you said in your answer to Rasmus, then you'd have a system of linear equations with infinitely many solutions.
For psychological reasons, let's write it this way:
$$
XA = B \ .
$$
Here $A$ is your $R$, $B$ your $I$ and $X$ your $A$. If we t... |
FIRST CASE. If you had only *one* set of values for $I$ and $R$, as you said in your answer to Rasmus, then you'd have a system of linear equations with infinitely many solutions.
For psychological reasons, let's write it this way:
$$
XA = B \ .
$$
Here $A$ is your $R$, $B$ your $I$ and $X$ your $A$. If we t... |
Suppose there are two correlated random variable and having very small correlation coefficient (order of 10^(-1)). Is it valid to approximate it as independent random variables? |
Does there exists a nonempty subset of integers $S$, such that for all $a \in S$, there exists $b,c \in S$ such that $a = b + c$, where $a, b, c$ are *distinct* integers.
Edit: As pointed out by @Captain Recursion, $n \mathbb Z$ satisfies this, but does a *finite* $S$ with this property exist?
*This question is ... |
Suppose there are two correlated random variable and having very small correlation coefficient (order of 10<sup>-1</sup>). Is it valid to approximate it as independent random variables? |
probability and statistics: Does having little correlation imply independence? |
I've got a fixed solution space defined by a minimum and maximum float which has no divisible slice.
You then have 0..N [Normal distribution][1] probability sets which I need to combine.
In the end I neeed
Method to define probability set with a numeric range (not sliceable) and 0..N Gaussian functions
A func... |
In the towers of Hanoi game, how do we know that we have the optimal algorithm for solving it? I thought about this and it seemed like any deviation from the standard strategies would be putting you back a step, but I had no idea how to demonstrate this rigorously.
All I know is that the proof involves the Lucas cor... |
How to prove the optimal Towers of Hanoi strategy? |
Answer to the latest revision:
**Yes**, there does seem to exist a finite, non-empty set S whose elements can be formed as sums of pairs of other elements of S, and which is disjoint from −S. I discovered the following set ad-hoc:
{ −6, −5, −1, 4, 9, 11, ... |
According to [XKCD][1], we have the following Haiku:
Top Prime's Divisors'
Product (Plus one)'s factors are...?
Q.E.D B@%&$
I wonder if we can edit it to make it correct
[1]: http://xkcd.com/622/ |
In the towers of Hanoi game, how do we know that we have the optimal algorithm for solving it? I thought about this and it seemed like any deviation from the standard strategies would be putting you back a step, but I had no idea how to demonstrate this rigorously.
All I know is that the proof involves the Lucas cor... |
Is it possible to find the position of a prime number online? |
One need only consult the history of algebra to find many informal
discussions on the existence, consistence etc of complex numbers.
Any *informal* attempt nowadays to justify the existence of $\mathbb C$
will face the same obstacles as it did throughout history. Nowadays
the set-theoretic foundation of algebra... |
I was wondering if the following is true:
In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?
In... |
when inf and sup of a subset achievable in itself? |
In the article about [metric space][1] on wikipedia:
>If X is a complete subset of the metric space M, then X is closed in M. Indeed, a space is complete iff it is closed in any containing metric space.
Do the second sentences say that:
in a metric space, a subset is complete if and only if the subset is closed i... |
I was wondering if the following is true:
In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?
In... |
One need only consult the history of algebra to find many informal
discussions on the existence and consistence of complex numbers.
Any *informal* attempt to justify the existence of $\mathbb C$
will face the same obstacles that existed in earlier times. Namely, the lack of any rigorous (set-theoretic) foundation... |
I am not at all sure that this would work, but it seems reasonable... (if it doesn't work, I'd like to know why)
We can very simply use Monte Carlo integration to approximate pi by setting up a known area and inscribing a circle in it, and sampling the entire area with a *uniformly distributed sequence\**, and consi... |
Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff
- The dimension of $\ker(T)$ is finite,
- The codimension of the image $\mathrm{im}(T)$ is finite,
- The image $\mathrm{im}(T)$ is closed in $Y$.
**Question:** *Is the third condition redundant?*
Some lecture not... |
Is the closedness of the image of a Fredholm operator implied by the cofinite-dimensionality of its image? |
I was wondering if the following is true:
In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?
In... |
Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff
- The dimension of $\ker(T)$ is finite,
- The codimension of the image $\mathrm{im}(T)$ is finite,
- The image $\mathrm{im}(T)$ is closed in $Y$.
**Question:** *Is the third condition redundant?*
Some lecture not... |
Is the closedness of the image of a Fredholm operator implied by the finiteness of the codimension of its image? |
There is a list at http://www.imomath.com/index.php?options=mbb|knjige&p=0 . There's also the book [Problems of number theory in mathematical competitions][1].
[1]: http://www.worldscibooks.com/mathematics/7242.html |
You could use a chi square test. Basically, this tests whether the number of draws that fall into various intervals is consistent with a uniform random distribution. See Knuth's [TAOCP volume 2, Seminumerical Algorithms][1] for an explanation of the chi-square test.
[1]: http://www.amazon.com/gp/product/02018... |
Let $X$ and $Y$ be Banach spaces. A bounded operator $T\colon X\to Y$ is called Fredholm iff
- The dimension of $\ker(T)$ is finite,
- The codimension of the image $\mathrm{im}(T)$ is finite,
- The image $\mathrm{im}(T)$ is closed in $Y$.
**Question:** *Is the third condition redundant?*
Some lecture not... |
Assume that T is injective, because if it is not we know that ker(T) is closed so we can replace T by the induced map from X/ker(T) (which is a Banach space).
Now define T' := X (+) C -> Y by T'(x, c) = T(x) + c where C is a closed complement for the range of T.
This T' is clearly bounded, linear and an isomorphi... |
Assume that $T$ is injective, because if it is not we know that $\textrm{ker}(T)$ is closed so we can replace $T$ by the induced map from $X/\textrm{ker}(T)$ (which is a Banach space).
Now define $T' := X \oplus C \to Y$ by $T'(x, c) = T(x) + c$ where $C$ is a closed complement for the range of $T$.
This $T'$ is ... |
I need to create a simple formula for determining the popularity of an item based on votes and age.
Here is my current formula, which needs some work:<br/>
30 / (days between post date and now) * (vote count) = weighted vote
Whenever a vost is cast for an item it checks if its weighted vote is > 300. If an item ... |
Let $G$ be a simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ How then one could... |
Let $G$ be a simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ Surely, we have to... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line? |
Let $G$ be a simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ Surely, we have to... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line?
*Let's try this:*
I'm looking to find a function that given an input (x) becomes closer to infinity as x grows closer to 0, and closer to 0 t... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line?
*Let's try this:*
I'm looking to find a function that given an input (x) becomes closer to infinity as x grows closer to 0, and closer to 0 t... |
Let $G$ be an infinite simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ Surely, ... |
Let $G$ be an infinite simple group. Suppose there is a subset $X$ of $G$ with
$$
X^g = gXg^{-1}=X\qquad (g \in G)
$$
which is closed under taking inverses and which even generates $G$ in finitely many steps:
$$
X^m = G
$$
for a suitable $m \ge 2.$
We somehow suspect that $X$ might be equal to $G.$ <i>If so... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line?
*Let's try this:*
I'm looking to find a function that given an input (x) becomes closer to infinity as x grows closer to 0, and closer to 0 t... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line?
*Let's try this:*
I'm looking to find a function that given an input (x) becomes closer to infinity as x grows closer to 0, and closer to 0 t... |
I saw a statement on http://en.wikipedia.org/wiki/Metric_space#Continuous_maps:
>f is continuous if and only if it is continuous on every compact subset of M1.
It is stated for the case that f is a mapping between two metric spaces.
I was wondering if the statement is true for mappings between general topologi... |
Representation theory is one of the keys to the Fourier transform on groups (and co-set spaces etc.). For this matter, I recommend this book:
[Unitary representations and harmonic analysis: an introduction
By Mitsuo Sugiura][1]
If you take a look at the [Wikipedia page][2] about representation theory, you will fi... |
Certainly, group actions used to model dynamical systems, for instance, Markov Chains, find extremely handy the results from representation theory.
For example, suppose you have a deck of cards, and every time you choose two cards at random and transpose them. How long you have to wait for the deck to be randomized? T... |
A good [Haskell link][1]
There are two descriptions that I know of. The first can easily be found by looking at wiki under Monad or consulting Harry's nice summary. The second is more interesting in my opinion.
I will assume that you don't know the definition of a monoidal action, if you do, just skip ahead.
A... |
I saw a statement on http://en.wikipedia.org/wiki/Metric_space#Continuous_maps:
>f is continuous if and only if it is continuous on every compact subset of M1.
It is stated for the case that f is a mapping between two metric spaces.
I was wondering if the statement is true for mappings between general topologi... |
One comment about your sentence "this seems to only trivially invoke representation theory".
It might be surprising, but such obvious representations are actually the source of much
interesting mathematics, and a lot of effort of representation theorists is devoted to studying them.
More precisely: the general set... |
The statement is false in general. Let $X$ be an uncountable set with the <a href="http://en.wikipedia.org/wiki/Cocountable_topology">cocountable topology</a>. Then all the countable subsets of $X$ have the discrete topology and are therefore closed but not compact, so the only compact subsets of $X$ are the finite s... |
I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem:
>Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply
>it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinite... |
What is the importance of the Collatz conjecture? |
One comment about your sentence "this seems to only trivially invoke representation theory".
It might be surprising, but such obvious representations are actually the source of much
interesting mathematics, and a lot of effort of representation theorists is devoted to studying them.
More precisely: the general set... |
I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem:
>Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply
>it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinite... |
Suppose, we have f1(n) = o(f2(n)).. (small o)
Now, we have to compare f1(f2(n)) and f2(f1(n)) asymptotically..
What can we say about it? Which one will be greater & why? |
What happens if $f(x)=x^t$ and $g(x)=e^x$
where $t$ is a positive real?
|
I have an examples book with a limit exercise that I can't understand.
The limit in quesiton is:
f(x,y)=x/(x+y) with x≠-y ; (x,y)->(0,0)lim f(x,y)
And then to solve it, it goes:
(x,y)->(0,0)lim f(x,y) = (x)->(0)lim f(x,mx) = (x->0)lim(x/(x+mx))=1/(1+m)
Can you help me understand that?
Thanks,
|
I have an examples book with a limit exercise that I can't understand.
The limit in question is:
$$f(x,y)=\frac{x}{x+y}$$
with $x\ne-y$;
$$\lim_{(x,y)->(0,0)} f(x,y)$$
And then to solve it, it goes:
$$\lim_{(x,y)->(0,0)} f(x,y)
= \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$
Can y... |
I have an examples book with a limit exercise that I can't understand.
The limit in question is:
$$f(x,y)=\frac{x}{x+y}$$
with $x\ne-y$;
$$\lim_{(x,y)\to(0,0)} f(x,y)$$
And then to solve it, it goes:
$$\lim_{(x,y)\to(0,0)} f(x,y)
= \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$
Can... |
1. The representation theory of finite groups can be used to prove results about finite groups themselves that are otherwise much harder to prove by "elementary" means. For instance, the proof of [Burnside's theorem][1] (that a group of order $p^a q^b$ is solvable). A lot of the classification proof of finite simple g... |
You could use a chi square test. Basically, this tests whether the number of draws that fall into various intervals is consistent with a uniform random distribution. See Knuth's [TAOCP volume 2, Seminumerical Algorithms][1] for an explanation of the chi-square test.
Update: If the chi square test is too complicat... |
For primes smaller than your example you can use Wolfram|Alpha, as Adam S pointed out. Wolfram|Alpha has the prime Pi function:
http://www.wolframalpha.com/input/?i=pi(55252335667) |
I have an examples book with a limit exercise that I can't understand.
The limit in question is:
$$f(x,y)=\frac{x}{x+y}$$
with $x\ne-y$;
$$\lim_{(x,y)\to(0,0)} f(x,y)$$
And then to solve it, it goes:
$$\lim_{(x,y)\to(0,0)} f(x,y)
= \lim_{x\to0} f(x,mx) =\lim_{x\to 0}\frac{x}{x+mx}=\frac{1}{1+m}.$$
Can... |
The *Mandelbrot set* is the set of points of the complex plane whos orbits do not diverge. An point $c$'s *orbit* is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.
The shape of this set is well known, and if you zoom into parts of the filaments you will find slightly deformed copies of the original shape,... |
Why is the Mandelbrot set a fractal? |
For your second question: is
$$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{y^2+y^2}=0?$$
imagine a sequence of points $(x_n,y_n)$
converging to the origin, and think whether $f(x_n,y_n)\to0$.
I would be tempted to write these points in polar form:
$x_n=r_n\cos t_n$ and $y_n=r_n\sin t_n$ as then the denominator
become... |
For your second question: is
$$\lim_{(x,y)\to(0,0)}\frac{-x^3+3xy^2}{x^2+y^2}=0?$$
imagine a sequence of points $(x_n,y_n)$
converging to the origin, and think whether $f(x_n,y_n)\to0$.
I would be tempted to write these points in polar form:
$x_n=r_n\cos t_n$ and $y_n=r_n\sin t_n$ as then the denominator
become... |
I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the chi squared test is not simple enough.)... |
I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the chi squared test is not simple enough.)... |
I have several points on a graph. The graph is of a continuous curve that flows through points (0, inf), (11, 5000), and (3, 200000). Can I find the equation for that line?
*Let's try this:*
I'm looking to find a function that given an input (x) becomes closer to infinity as x grows closer to 0, and closer to 0 t... |
consdier a cubic polynomial, p(x)=k(x-a)(x-b)(x-c) where k is some constant and a,b,c its 3 roots (not necessarily distinct, not necessarily real). It is very simple to show that if you average two roots of a cubic polynomial and compute the tangent line at their average that it will intersect the cubic polynomial at t... |
I have a non-mathematician friend who is interested in re-learning algebra. I am more than happy to help, but I am in no position to judge what is a **good** introductory text --- only to identify when a text is a very **bad** one.
My friend is interested in starting from "basics" --- he's comfortable with order-of-... |
Free introductory resources for learning algebra? |
I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the chi squared test is not simple enough.)... |
I have a function that (more or less) is supposed to select a small number m of random number from the range [1,n] (for some n >> m) and I need to test that it work. Is there an easy to implement test that will give good confidence that it is working correctly? (For reference the full on chi squared test is not simple ... |
You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?
If you list all possibilities for the sexes of two children, BB, BG, GB, GG, you see that 2 of the 3 pairs that have B (for bo... |
As a futher hint: the fundamental group of the torus $\mathbb{T}$ is generated by $a, b$ and has the relation $abAB$, where capitals denote inverses. If you remove an open disk then you get a once-holed torus $T$. Now the fundamental group is free (why?) and the boundary is the homotopic to the element $abAB$ (why?).... |
Why does the Mandelbrot set contain (slightly deformed) copies of itself? |
Say you have a very dense matrix that is 30000x30000 elements.
Solving a matrix like this in C code actually introduces an interesting problem: you cannot actually store it all in memory at once.
What algorithms exist to break down / solve a matrix "in pieces" or steps, that can be solved regardless of memory re... |
Are there three different numbers x, y, z in [0,pi/2] such that the numbers sin x, sin y, sin z, cos x, cos y, cos z can be divided into three pairs with equal sums?
[I guess the answer is no, and that the pigeon-hole principle has to be used somehow.] |
When is $n^2+n+p$ prime? |
Say you have a very dense matrix that is 30000x30000 elements.
Say you have Ax = B. You have B, and A is 30000x30000 elements. Solving: You need to find x given B.
Solving a matrix like this in C code actually introduces an interesting problem: you cannot actually store it all in memory at once.
So, I want... |
If $n>1$ is an integer, then $\sum_{k=1}^n\frac 1k$ is not an integer.
If you know [Bertrand's Postulate][1], then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by prim... |
Is there an elementary proof that $∑_{k=1}^n 1/k$ is never an integer? |
The first question that should be asked of any person presenting you with a large-ish matrix: "is the matrix dense or sparse?" In the latter case, one definitely does not need to allocate space for the zero entries, and should thus look into special techniques to storing them(which as I understand nowadays rely on a l... |
Suppose $M_1$ and $M_2$ are invariant subspaces of the unilateral shift U such that $M_1$ subset $M_2$ and $M_1$ is of codimension strictly larger than $1$ in $M_2$. Show that there exists $M$ invariant under $U$ satisfying $M_1 \subset M \subset M_2$ where the inclusions are strict. All subspaces are closed.
This p... |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that yo... |
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