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Can all 2-forms be written as the wedge of two 1-forms? |
I'm studying for an exam an I came across a problem that I am having difficultly solving.
Let $\mathcal{F}$ is a family of analytic functions on the closed unit disc, $D$.
Suppose
$\int_{D} |f|^{2} dA \le 1$
for all $f \in \mathcal{F}$.
Can I conclude that $\mathcal{F}$ is locally bounded?
|
If you expand $\lim_{n\to\infty}\Big(1-\frac{1}{n}\Big)^n,$ by using the binomial formula, you get $1 - \frac{n}{n} + \frac{n*(n-1)}{2*n^2} - \frac{n*(n-1)*(n-2)}{6*n^3} + ...$
As n goes to infinity, this approaches $1 - \frac{n}{n} + \frac{n*n}{2*n^2} - \frac{n*n*n}{6*n^3} + ... $, which equals $1 - 1 + \frac{1}... |
Can you find a 2-form not written as the wedge of two 1-forms? |
It depends upon the student's motivation. When you are hired for high school, is it by the parents when the student has a problem? And in pre-engineering is it the student? This is a big difference. If the student isn't motivated, I have no ideas. If s/he is motivated, showing what math is good for, as problems in... |
I've seen some of this stuff before, but I'm definitely not an expert. Here's my take (although I'm likely wrong on at least some of it, hopefully not too much):
- You can think of this as a matrix of 1-forms. Then, $A \wedge A$ means "matrix multiplication, where multiplication is wedge product".
- Yes, I thin... |
In practice this comes to the same algorithm as Moron's, but I'll give
my two penn'orth anyway. There are two stages: (i) find $z$ with $z^2\equiv -1$
(mod $p$), and (ii) use $z$ to find $x$ and $y$.
Stage (i). If $a$ is a quadratic nonresidue modulo $p$ then $a^{(p-1)/2}
\equiv-1$ (mod $p$) so we can take $z\equ... |
There is no silver bullet - the hardest part of the project is getting the student curious about maths again. Once you do that, teaching them the skills to pass the exams is comparatively a breeze.
Unfortunately, one is *usually* hired by parents when the student has a problem, and not before, so motivating the stud... |
Here is an explicit procedure based on the isomorphism between the de-Rham and Cech cohomologies
for smooth manifolds based on R. Bott and L.W. Tu's [book][1]: Differential forms in algebraic topology.
The description will be given for a three form but it can be generalized
along the same lines to forms of any d... |
In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this: $Hom_\mathcal{C}(colim A_i,B)=lim Hom_\mathcal{C}(A_i,B)$
I am looking for a corresponding statement for hocolims - lets say in simplicial sets, but if there are more general statements, that's even better.
E.g. I could imagine
$$... |
How does hocolim relate to Hom? |
The card game [Set][1] lead to the following question. Lets call a subset $A$ of $(\mathbb{Z}/3)^n$ dependent, if there is $\\{x,y,z\\}\subset A$ with $x+y+z=0$. (So unlike the case of linear dependence we are not allowing any coefficients here).
Let $f(n)$ denote the maximal size of a independent subset of $(\mathb... |
In a usual category $\mathcal{C}$ one can pull the colim out of the Hom like this: $Hom_\mathcal{C}(colim A_i,B)=lim Hom_\mathcal{C}(A_i,B)$
I am looking for a corresponding statement for hocolims - lets say in simplicial sets, but if there are more general statements, that's even better.
E.g. I could imagine
$$... |
You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars
the percentage of heads flipped. So if on the first trial you flip a head, you should stop and earn \$100
because you have 100% heads. If you flip a tail then a head, you could either stop and earn \$50,
or... |
Why is this coin-flipping probability problem unsolved? |
Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random?
(By reasonable I mean to exclude silly things like «generate a looong list and see that it is periodic») |
Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random?
(By reasonable I mean to exclude silly things like «generate a looong list and see that it is periodic»)
**Later:** Notice I am looking... |
If I give you the following definition of the set 'A', how could you prove it is equal the set of the natural numbers without an explicit definiton for the latter?
The set 'A' is inductively defined as follows:
i) 0 is in A; and
ii) for any 'n' a natural number, if n is in A, so is n+1
I can easil... |
I am trying to understand, in as simple terms as possible:
1. How to define integration for non-orientable manifolds, and
2. why this definition must differ from the orientable case.
In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear on w... |
I am trying to understand, in as simple terms as possible:
1. How to define integration for non-orientable manifolds, and
2. why it is impossible to do so using only differential forms.
In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear o... |
I am trying to understand, in as simple terms as possible:
1. How to define integration for non-orientable manifolds, and
2. why it is impossible to do so using only differential forms.
In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear o... |
If I give you the following definition of the set 'A', how could you prove it is equal the set of the natural numbers without an explicit definiton for the latter?
The set 'A' is inductively defined as follows:
i) 0 is in A; and
ii) for any 'n' a natural number, if n is in A, so is n+1
I can easil... |
Let $f$ be an analytic function on the open unit disk domain $D$. Suppose also that $f$ is bounded.
Since $f$ is bounded I beleive that $f$ can be continuously extended to the closed unit disk.
I know that the zeros of $f$ in the open disk $D$ are isolated. Are the zeros of $f$ in the closed unit disk also nec... |
If you have 32 2x1 dominoes then you can cover an 8x8 board easily enough; if you throw away a domino and cut a 1x1 square from each end of a diagonal of the board, can you cover the remaining shape with the remaining dominoes?
I heard this decades ago at university.
I can't do comments, so:
Ab, orpnhfr vs vg ... |
If I give you the following definition of the set $A$, how could you prove it is equal the set of the natural numbers without an explicit definiton for the latter?
The set $A$ is inductively defined as follows:
i) 0 is in $A$; and
ii) for any $n$ a natural number, if $n$ is in $A$, so is $n+1$.
I ... |
It is not true that $f$ can be extended continuously to the closed disk. The set of bounded analytic functions on the open disk is called the Hardy algebra, denoted $H^\infty$, and the set of analytic functions on the disk that have continuous extensions to the closed disk is called the disk algebra, often denoted $A(... |
It is not true that $f$ can be extended continuously to the closed disk. The set of bounded analytic functions on the open disk is called the Hardy algebra, denoted $H^\infty$, and the set of analytic functions on the disk that have continuous extensions to the closed disk is called the disk algebra, often denoted $A(... |
Provided your student has a strength, you can play to that. In some sense that's what finding "applications" is about -- appealing to someone's immediate values. But there's also a danger in that, if all you do is appeal to someone's immediate values you may be forced to build up a rather contrived and limited notion... |
Provided your student has a strength, you can play to that. In some sense that's what finding "applications" is about -- appealing to someone's immediate values. But there's also a danger in that, if all you do is appeal to someone's immediate values you may be forced to build up a rather contrived and limited notion... |
Based on [this presentation][1] (PPT; [Google-based HTML conversion][2]), I think a formula for your $f(n)$ is:
$$f(n)=\begin{cases}
\frac{2^{n+2}-4}{3} & \text{ if }n\text{ is even}
\\\\
\frac{2^{n+2}-5}{3} & \text{ if }n>1\text{ is odd}
\end{cases}$$
[1]: http://www2.truman.edu/~erickson/presentation7.ppt... |
So if you have a distribution `fx(x)` and you observed x=a, how do you calculate `E{x | x=a}`?
I am having trouble when using Bayes or finding `Fx(x|x=a)` because I end up with P(x=a) which is just a constant. Is there a simple example somewhere? Searching only leads me to bivariate distributions. |
How do you calculate conditional expectation for a single continuous r.v.? |
So if you have a distribution `fx(x)` and you observed x=a, how do you calculate `E{x | x=a} = integral (x*fx(x|x=a) from -inf to inf`, or more specifically, `fx(x|x=a)`
I am having trouble when using Bayes or finding `Fx(x|x=a)` because I end up with `P(x=a)` which is just a constant. Is there a simple example som... |
My first post here...not really a math expert, but certainly enjoy the challenge.
I working writing a random string generator and would like to know how to calculate how many possible combinations there are for a particular combination.
I am generating a string of 2numbers followed by 2 letters (lowercase) e.g. 1... |
Some nomenclature: when you say "2 number", you really mean "2 **digits**". Also, you need to specify if the digits can be anything or not (for example, do you allow leading zeroes?).
If each of the two digits can be anything, 0 through 9, and each of the letters can be anything, a through z, then your computation ... |
Can this expression be further simplified : $ \sum_{k=0}^{n-1} { n -1 \choose k } (-2)^{k} (2n - 2k)! $? This is the coefficient of $x^{2n}$ in the formal power series expansion of $(1-2x)^{n-1} \times \sum_{ k \geq 0} k! x^k$.
Motivation: I came across this when trying to solve a problem using inclusion-exclusion ... |
Can this sum be simplified? |
Can this expression be further simplified : $ \sum_{k=0}^{n-1} { n -1 \choose k } (-2)^{k} (2n - k)! $? This is the coefficient of $x^{2n}$ in the formal power series expansion of $(1-2x)^{n-1} \times \sum_{ k \geq 0} k! x^k$.
Motivation: I came across this when trying to solve a problem using inclusion-exclusion p... |
I'm currently teaching myself calculus and I'm probably trying to run before I can walk, but I've been working on this problem..
I managed to find the correct result for:
$$\int_{0}^{\infty }(2e^{-3x}+4e^{-7x})^2dx$$
by expanding it to:
$$\int_{0}^{\infty}4e^{-6x}+16e^{-10x}+16e^{-14x}dx$$
and then work... |
Theorem 2 in a [paper][1] by Borodin, Okounkov, and Olshanski states that the discrete Bessel kernel $J(x,y,\theta)$ is given by
$\sqrt{\theta} \frac{J_x J_{y+1} - J_{x+1} J_y}{x-y}$
where $J_x = J_x(2\sqrt{\theta})$ is the Bessel function of order $x$ and argument $2\sqrt{\theta}$.
Does anybody have any idea... |
I think you might find it helpful to first comprehend the essence of the matter in a slightly simpler context, e.g. see the [proof that I presented here][1]. The key ideas are already there. Generally, I recommend Kaplansky, Commutative Rings, for the circle of ideas around the generalized Nullstellensatz (Goldman, Kr... |
Why does the following integral converge?
\begin{equation}
\int_{0\leq |x_1|\leq|x_2|\leq\dots\leq |x_n|\leq 1}\frac{(1-\sum_{i=1}^n\log |x_i|)^r}{|x_1x_2\dots x_n|^{1/2}}dx_1dx_2\dots dx_n
\end{equation}
Here, $r$ is any positive real number. |
I'd like to understand why $ \int{\rho}dV = \frac{1}{c} \int{j^0}dV = \frac{1}{c} \int{j^i}dS_i $ (the second equality), where
$j^i = \rho \frac{dx^i}{dt} $ is the current density 4-vector
$\mathbf{j} = \rho \mathbf{v}$ is the current density 3-vector
$ j^i = (c\rho, \mathbf{j}) $
$\rho$ is the charge dens... |
Why $ \int{\rho}dV = \frac{1}{c} \int{j^0}dV = \frac{1}{c} \int{j^i}dS_i $ ??? |
You can show (for example, using l'Hôpital's rule) that $\frac{(1-\log(t))^r}{t^{1/2}}$ is dominated by $\frac{1}{t^{2/3}}$ for sufficiently small $t$. If you use this along with the fact that that $\sum_{i=1}^n\log|x_i|=\log|x_1\cdots x_n|$, you have your integrand dominated by $|x_1\cdots x_n|^{-2/3}$ in a neighborh... |
You can show (for example, using l'Hôpital's rule) that $\frac{(1-\log(t))^r}{t^{1/2}}$ is dominated by $\frac{1}{t^{2/3}}$ for sufficiently small $t$. If you use this along with the fact that that $\sum_{i=1}^n\log|x_i|=\log|x_1\cdots x_n|$, you have your integrand dominated by $|x_1\cdots x_n|^{-2/3}$ in a neighborh... |
One way of looking at the Vieta product
$${2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$
is as the infinite product of a series of successive 'approximations' to 2, defined by $a_0 = \sqrt{2}$, $a_{i+1} = \sqrt{2+a_i}$ (or more accurately, their ratio to their limi... |
What's the value of this Vieta-style product involving the golden ratio? |
and in the other way, orthogonal nullspace and range require symmetry?
Thanks,
R |
What is the best way to show an idempotent matrix whose nullspace and range are orthogonal must be symmetric? |
I accept Qiaochu's answer "Have you tried actually doing that?" I did try now, and now I can appreciate the challenge. :-) The paper I cited refers to another by Chow and Robbins from 1965
that has a beautiful formulation, much clearer than the cummulative binomials with which I struggled. Let me explain it, because ... |
I accept Qiaochu's answer "Have you tried actually doing that?" I did try now, and now I can appreciate the challenge. :-) The paper I cited refers to another by Chow and Robbins from 1965
that has a beautiful formulation, much clearer than the cummulative binomials with which I struggled. Let me explain it, because ... |
If you have 32 2x1 dominoes then you can cover an 8x8 board easily enough; if you throw away a domino and cut a 1x1 square from each end of a diagonal of the board, can you cover the remaining shape with the remaining dominoes?
I heard this decades ago at university.
|
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this? |
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? |
IMO there are two great references at two different levels. The first is [Brualdi][1]. This book is awesome. It is super clear, super straightfoward, and very readable by yourself.
The second is much more advanced, but even better! It is [Aigner][2]'s GTM on the subject. This book is fairly advanced, but rarely have... |
Suppose $f$ is an analytic function defined on the unit disk, $D$. I want to evaluate
$\int_{D} f(\omega) dA(\omega)$
using a change of variable. Suppose $\phi$ is a conformal map of the $D$ onto itself.
Does
$\int_{D} f(\omega) dA(\omega) = \int_{D} f(\phi(z)) |\phi^{'}(z)|^{2} dA(z) $?,
where $\ph... |
To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away from the university made my math got *extremely* rusty (so much for last-minute pre-exam cramming). While I remember va... |
I've come across a paper that mentions this as a fact...where can I find the proof? |
matrices commute=common set of eigenvectors? |
I'm trying to figure out how many ways are there to stack N cubes.
I'm generally finding 2 numbers:
1.) 6 * 24^(n-1)
and
2.) 3 * 24^(n-1)
Which is the correct answer, and why? |
I'm trying to figure out how many ways are there to stack N cubes on top of each other.
Once you get a particular stack of 4 cubes let's say, the order of the cubes doesn't matter, just which sides are being shown does.
I'm generally finding 2 numbers:
1.) 6 * 24^(n-1)
and
2.) 3 * 24^(n-1)
Which is t... |
This question is mainly to understand the meaning of my professor's correction to a proof of a theorem I gave during an oral examination.
The question was to show that $\text{diam}A = \\text{diam}\bar{A}$ where $\text{diam}$ is the diameter of a subspace of a metric space and $\bar{A}$ is the closure of the subset $... |
This question is intended to complement [What mathematical questions or areas have philosophical implications outside of mathematics?][1]
[1]: http://math.stackexchange.com/questions/6100/what-mathematical-questions-or-areas-have-philosophical-implications-outside-of-m |
If $\sum a_n$ and $\sum b_n$ both converge and one of them absolutely then the Cauchy product $\sum c_n$ converges to $\sum a_n \sum b_n$. ($c_n = \sum_{k = 0}^n a_k b_{n - k}$), by Mertens Theorem.
Now, if both converge conditionally then the product does not have to converge as $a_n = b_n = (-1)^n/n$ shows.
My qu... |
Suppose $f: \mathbb{R} \to \mathbb{R}$ is a continuous function which is 1-1, then does $f$ map Borel sets onto Borel sets? |
What doe E mean in 9.0122222900391E-5? |
In a book about radar signal processing, they are estimating the squared distance between two objects as:
$d^2 = \frac{(R_p-R_o)^2}{\sigma^2_R} + \frac{(\theta_p-\theta_o)^2}{\sigma^2_\theta}$
where p is the predicted target and o is an observation.
Is it just me or is the [dimentional analysis][1] for this e... |
Weird distance estimate? |
I am not a mathematician(IANAM), however I wish I could be.
My question: I often find this at the bottom of pages.
Page generated in 0.00013899803161621
Sometimes, I come across
Page generated in 9.0122222900391E-5
What does that time mean.
I tried searching wikipedia for E and maths but found the [... |
**Question:** Given an index number `i` and a maximum index `N`, is there a non-recursive function which can be used to determine the next "low" and "high" indexes that would be traversed *after* index `i` in a binary search of a list of `N` values?
**Background:** I am creating a binary search tree table for an emb... |
Is there a non-recursive function which can be used to build a static binary search tree? |
So you are looking for the highest and lowest valued nodes in the children of a particular node of the tree?
If so, then yes, there is a non-recursive method.
Just keep following the left pointers to get one extreme and keep following the right pointers to get the other.
Your question is a bit confusing thou... |
Let $$\pi(x) = \\{ \text{No of primes}\leq x \\}$$ One observes that $\pi(1) \mid (1)$, $\pi(6) \mid 6$, $\pi(8) \mid 8$. So does $\pi(x) \mid x$ for finitely many $x$, or is this fact true only for infinitely many $x$? |
Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? |
Let $$\pi(x) = \\{ \text{Number of primes}\leq x \\}$$ One observes that $\pi(1) \mid (1)$, $\pi(6) \mid 6$, $\pi(8) \mid 8$. So does $\pi(x) \mid x$ for finitely many $x$, or is this fact true only for infinitely many $x$? |
**Edit:** question simplified to remove confusion
Assuming a sorted list of items with indexes from 1 to `N`, and given *only* an index number `i` and the maximum index `N`, is there a simple function which will return the two possible indexes that would be reached next in a binary search of the list?
Example:
... |
Is there a simple function which can be used to determine the two "next" search indexes in a binary search? |
Let $$\pi(x) = \\{ \text{Number of primes}\leq x \\}$$ One observes that, $\pi(6) \mid 6$, $\pi(8) \mid 8$. So does $\pi(x) \mid x$ for finitely many $x$, or is this fact true only for infinitely many $x$? |
I'm trying to simplify the following booleans:
Y = ¬A¬B¬C¬D + ¬A¬B¬CD + ¬A¬BC¬D + ¬A¬BCD + A¬B¬C¬D + A¬BC¬D + ABC¬D
The solution is:
Y = ¬A¬B + ¬B¬D + AC¬D
My solution is:
Y = ¬A¬B (¬C¬D + ¬CD + C¬D + CD) + A¬B¬C¬D + A¬BC¬D + ABC¬D
= ¬A¬B (¬C (¬D + D) + C (¬D + D)) + A¬B¬C¬D + A¬BC¬D +... |
In a book about radar signal processing, they are estimating the squared distance between two objects as:
$d^2 = \frac{(R_p-R_o)^2}{\sigma^2_R} + \frac{(\theta_p-\theta_o)^2}{\sigma^2_\theta}$
where R is a radius, $\theta$ is an angle and $\sigma^2$ is the variance of the radius and angle respectivly. p is the pr... |
I agree with you. The last three terms all have A required, so anything that comes out of them (your last two terms) need an A. Are you sure the given solution is right? |
It is not true that $f$ can be extended continuously to the closed disk. The set of bounded analytic functions on the open disk is called the Hardy algebra, denoted $H^\infty$, and the set of analytic functions on the disk that have continuous extensions to the closed disk is called the disk algebra, often denoted $A(... |
I am trying to calculate the average distance between all vertices of a complete $k$-ary tree. A complete $k$-ary tree is a tree such that all vertices have $k$ children except for the leafs of the tree.
Suppose the tree has $r$ levels, where level 0 is the root. So the height of the tree is $r-1$. First i have to ... |
Suppose I have two machines, A and B. A encrypts a message m and outputs the ciphertext m^e (mod n). B outputs c such that c = m^e (mod p) and c = m^e + 1 (mod q). How can I use A and B to find p and q? I am allowed to choose m and n. |
Let $\pi(x)$ be the number of primes less than or equal to $x$.
One observes that, $\pi(6) \mid 6$, $\pi(8) \mid 8$. So does $\pi(x) \mid x$ for only finitely many $x$, or is this fact true for infinitely many $x$? |
$\pi(x) \mid x$ for how many $x$? |
Geometric multiplicity of an eigenvalue of a matrix is the dimension of the corresponding eigenspace. The algebraic multiplicity is its multiplicity as a root of the characteristic polynomial.
It is known that *the geometric multiplicity of an eigenvalue cannot be greater than the algebraic multiplicity*. This fa... |
I am trying to derive the following equation for an analytic function, $f$, defined on the unit disc, $D$
$f(z)=\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w)$.
{\bf Question}: Anyone have any ideas on how to derive this formula?
I've been able to verify that the above is true for $z =0$. In attempting to deri... |
I am trying to derive the following equation for an analytic function, $f$, defined on the unit disc, $D$
$f(z)=\int_{D} \frac{f(w)}{(1- \bar{w}z)^{2}}dA(w)$.
{\bf Question}: Anyone have any ideas on how to derive this formula?
I've been able to verify that the above is true for $z =0$. In attempting to deri... |
The formula is not even close. You are right that d should have dimensions of length and the right hand side is unitless. Worse, the formula doesn't reduce properly if the errors are zero. Are you sure it isn't d^2/\sigma_(d)^2 on the left? That would fix the units. The cosine formula gives the distance estimate y... |
Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this? |
Is this unit interval a countable disjoint union of closed sets? |
Is $[0,1]$ a countable disjoint union of closed sets? |
It seems to me not, since this would seem to imply that for all functors *F* and all objects *A* in **C** there exists a morphism *F(A) -> A* (making all functors co-pointed?). However, intuitively it seems like the identity functor acts like a terminal object; a monad *M* on **C** is a monoid on **[C, C]** where the "... |
Is the identity functor the terminal object of the category of endofunctors on C? |
Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this number, we could equivilantly express it as $\sum _{k=1}^\infty {\frac {C_k}{g^{Z(k)}}}$, where $Z(k)$ is the position o... |
Have all numbers with "sufficnetly many zeros" been proven transcendental? |
Have all numbers with "sufficiently many zeros" been proven transcendental? |
I've been self studying abstract algebra from these [notes][1]. I've ran across the following Lemma:
![alt text][2]
The confusing part is that $X$ is a "any subset" not any "subgroup", and that the term "normal set" appeared in place where I expected "normal subgroup". The definition of a "normal set" has not been ... |
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