instruction stringlengths 12 30k |
|---|
I think you're trying to ask: how many such grids admit a non-trivial symmetry (in the dihedral group)?
*For fixed m*≥1, asymptotically almost all (0,1,-1)-matrices with exactly m 1's and m -1's do not admit a non-trivial symmetry (rotation/reflection). For these matrices, we can consider the X's as the 1's and... |
Let $\mathcal{F}$ be an open cover of a metric space $(X,d)$. I have a question concerning some algebraic aspects of $\mathcal{F}$ and $X$.
First of all, is it always possible to consider $(X,d)$ a subspace of some bigger space $(X^{\prime},d)$, where $d$ is the inherited metric from $X^{\prime}$? If this is possibl... |
Your first question is rather vague: you can always take $X'=X$, $d'=d$, and then $(X,d)$ is a subspace of $(X',d')$ with the inherited metric. Presumably you first of all want $X$ *properly contained* in $X'$ ("bigger" is a too vague, sorry!). But - do you also want some other property for $X'$, besides merely contain... |
First of all, you need your surfaces to be connected, otherwise the result is not true: let $X = \mathbb{T}^2 \sqcup \mathbb{S}^2$ and $Y = \mathbb{T}^2$. ($\mathbb{T}^2$ a torus, $\mathbb{S}^2$ a sphere.)
Then the connected sum $X \oplus Y$ *does* depend on the choice of the discs. It doesn't matter which disc you ... |
How does one find the solution of
$$\dfrac{dy}{dx}\left( 1-\left( 1-t\right) x-x^{2}\right) -\left( 1+h\left( 1+t\right) +x\right) y=0\quad ?$$
where $h$ is an integer constant
$($ In Roger Apéry, *Interpolations de Fractions Continues et Irrationalité de certaines Constantes*, Bull. section des sciences du C.... |
Now, I think I've got a possible real answer.
Once connection assumed, I'm going to develop my previous comment in order to prove that, at least for compact connected surfaces, connected sums do NOT depend on anything at all. Feel free to point out if I'm missing something or any mistakes in my reasonings.
The pr... |
How does one find the solution of
$$\dfrac{dy}{dx}\left( 1-\left( 1-t\right) x-x^{2}\right) -\left( 1+h\left( 1+t\right) +x\right) y=0\quad ?$$
where $h$ is an integer constant and $t$ is a real constant between $0$ and $1$.
$($ In Roger Apéry, *Interpolations de Fractions Continues et Irrationalité de certain... |
Now, I think I've got a possible real answer.
Once connection assumed, I'm going to develop my previous comment in order to prove that, at least for compact connected surfaces, connected sums do NOT depend on anything at all. Feel free to point out if I'm missing something or any mistakes in my reasonings.
The pr... |
2d rotation falls spirally inwards? |
Your first question is rather vague: you can always take $X'=X$, $d'=d$, and then $(X,d)$ is a subspace of $(X',d')$ with the inherited metric. Presumably you first of all want $X$ *properly contained* in $X'$ ("bigger" is a too vague, sorry!). But - do you also want some other property for $X'$, besides merely contain... |
Now, I think I've got a possible real answer.
Once connection assumed, I'm going to develop my previous comment in order to prove that, at least for compact connected surfaces, connected sums do NOT depend on anything at all. Feel free to point out if I'm missing something or any mistakes in my reasonings.
The pr... |
I know how to count derivatives, but I actually don't understand what are those? What do they show?
Wikipedia explanation: `Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity`. I understand nothing what is written here?
Please, if y... |
How does one find the solution of
$$\dfrac{dy}{dx}\left( 1-\left( 1-t\right) x-x^{2}\right) -\left( 1+h\left( 1+t\right) +x\right) y=0\quad ?$$
where $h$ is an integer constant and $t$ is a real constant between $0$ and $1$.
$($ In Roger Apéry, *Interpolations de Fractions Continues et Irrationalité de certain... |
According to [this article in Wikipedia][1]: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations... |
> The strength of glue is known to follow a normal distribution with a mean of 59 and a standard deviation of 3.4.
> Find the mean strength needed such that the probability that the glue is weaker than 55 is less than 5%.
To solve this, I just guessed and checked in Excel:
`=NORMDIST(55, 50.1, 3.4, 1)`
`=N... |
You can find a very nice introduction to derivatives in the videos of lectures from [Khan's Academy][1]. Khan's Academy provides a free online collection of a couple thousand videos on mathematics, science, history, and economics. It's a site worth exploring when you're looking for introductory lectures on such topics.... |
As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$.
Are there some interesting examples of rings with idempotent elements? |
I'm learning category theory by self-study. I have a couple of texts, and they both talk about how we ought to try not to think so much about the equality between objects in categories. Rather, the important relation is isomorphism.
Okay, sure.
Then they go on to talk about "Cat", the category of (small) categor... |
"Cat" modulo natural isomorphism? |
Are there any good places for me to sell off my mathematics books online especially Springer and Dover books? |
Are there any good places for me to sell off my mathematics books online especially Springer and Dover books?
(I thought perhaps this was off topic, but then I thought everybody in math probably has the problem of having bought pricey books that they would prefer to get some money back for, at one point or another.) |
In the Portuguese book *Análise Matemática* (Mathematical Analysis) by C. Sarrico, it is proved there exist
$$\displaystyle\lim_{n\rightarrow +\infty}f_{n}(x)=\lim_{n\rightarrow +\infty }e^{\log f_{n}(x)},\qquad(1)$$
where
$$f_{n}(x)=\dfrac{n!n^{x}}{x(x+1)(x+2)\cdots (x+n)}.\qquad(2)$$
And it is stated the... |
I have an image encoded in the form of a list of points, like so:
335.0 2743.0
335.0 2754.0
325.0 2754.0
304.0 2743.0
304.0 2733.0
283.0 2723.0
273.0 2702.0
273.0 2691.0
273.0 2681.0
273.0 2670.0
283.0 2670.0
294.0 2670.0
304.0 2670.0
... |
How do I apply a Gaussian Blur (low-pass filter) to an image made up from a set of points? |
Amazon will buy back a lot of textbooks at pretty fair rates. Check [this][1] out.
[1]: http://www.amazon.com/Textbooks-Trade-In/b/ref=bhp_9p2_A_txbq410_05?ie=UTF8&node=2205237011&pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-6&pf_rd_r=0RH8PF985VSEMWFKG1SV&pf_rd_t=101&pf_rd_p=1275995182&pf_rd_i=283155 |
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and the disks can be viewed as spinning fan blades
determining vectors of fluid motion tangent to the disks.
Is the r... |
Vortex Voronoi diagram? |
I know that a function of bounded variation on a closed interval can be written as a difference of increasing functions. The same is true is the function is of bounded variation on all of R, but I can't come up with a proof.
Any hints? |
Bounded variation: Jordan's theorem on all of R? |
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and the disks can be viewed as spinning fan blades
determining vectors of fluid motion tangent to the disks.
Is the r... |
You can find some nice introductory lectures on derivatives in various [videos from Khan's Academy][1]. The site has a free online collection of a couple thousand videos on mathematics, science, history, and economics. It's worth exploring when you're looking for introductory lectures on such topics.
[1]:http://www.k... |
*A school play ran for two nights, with audiences totaling 1390 adults and students. They paid $4285 for admission. One adult ticket cost $4 and one student ticket cost $2.50. The ratio of adults to students was 3:5 for the first night and 2:3 on the second night. How many students attended each night?*
I have deter... |
> A school play ran for two nights, with
> audiences totaling 1390 adults and
> students. They paid 4285 dollars for
> admission. One adult ticket cost 4 dollars
> and one student ticket cost 2.50 dollars. The
> ratio of adults to students was 3:5
> for the first night and 2:3 on the
> second night. How many stu... |
> A school play ran for two nights, with
> audiences totaling 1390 adults and
> students. They paid 4285 dollars for
> admission. One adult ticket cost 4 dollars
> and one student ticket cost 2.50 dollars. The
> ratio of adults to students was 3:5
> for the first night and 2:3 on the
> second night. How many stu... |
I have verified that the adult/student totals you got is correct.
For the second part:
Suppose a total of $x$ people (i.e. including students and adults) attended the first day and total $y$ people attended the second day.
What is the total number of students in terms of $x$ and $y$?
|
I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields
$$\mathbb{Q}\subset \mathbb{Q}(\zeta_m) \subset \mathbb{Q}(\zeta_n)$$
Now suppose we also have primes (where $(p,n)=1$)
$$(p)\subs... |
I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields
$$\mathbb{Q}\subset \mathbb{Q}(\zeta_m) \subset \mathbb{Q}(\zeta_n)$$
Now suppose we also have primes (where $(p,n)=1$)
$$(p)\subs... |
So I have to find the maximum possible error $dR$ in calculating equivalent resistance for three resistors, $\displaystyle\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}$ if the maximum error in each is 0.5%; $R_1=25\ \Omega$, $R_2=45\ \Omega$, $R_3=50\ \Omega$.
Now, originally I did $dR_1=dR_2=dR_3=0.005$, an... |
When does an orthomorphism of the cyclic group exist? |
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... |
The industry standard way of doing this is is to calculate a "mask" of sorts, and apply it to each one.
By "mask", I mean, say, a function for the current point based on the points before and after. For example:
$f(x) = \frac{1}{9}[x-2] + \frac{2}{9}[x-1] + \frac{3}{9}[x] + \frac{2}{9}[x+1] + \frac{1}{9}[x+2] $
... |
If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\backslash \left\\{ (0,0)\right\\}$, then how do I show/prove that
$d\theta =\frac{x dy - y dx}{x^2 + y^2}$?
|
How do you show that $d\theta = \frac{x dy - y dx }{x^2 + y^2}$? |
If $\mathcal{A}_m$ is the ring of integers of $\mathbb{Q}(\zeta_m)$, then $\mathfrak{B} \cap \mathcal{A}_m = \mathfrak{p}$.
You say $a \equiv b \pmod{\mathfrak{B}}$ where $a,b \in \mathbb{Q}(\zeta_m)$. Does this mean $a,b$ are algebraic integers (in $\mathcal{A}_m$)? If not, it must mean that there is some denom... |
I have been reading about Riemann Zeta function and have been thinking about it for some time.
Has anything been published regarding upper bound for the real part of zeta function zeros as the imaginary part of the zeros tend to infinity?
Thanks |
Given an affine variety, we can forget it into a variety. The other way round, given a variety $X$ we can look at
$$spec(\mathcal O_X(X)$$
Is this necessarily an affine variety (a priori its just an affien scheme)?
If so, am I right that this implys that taking spec of the global functions is left adjoint to the ... |
Turning varieties into affine varieties? |
The functor $Sch \to Ring^{op}, X \mapsto \mathcal{O}_X(X)$ is left adjoint to the functor $\text{Spec} : Ring^{op}\to Sch$.
By a variety you mean a separated integral scheme $X$ of finite type over a field $k$? Then $O_X(X)$ does not have to be of finite type over $k$. Namely, there are integral $k$-subalgebras $A,... |
If $(r, \theta)$ are polar coordinates on $\mathbb{R}^2\backslash \left\\{ (0,0)\right\\}$, then how do I show/prove that
$d\theta =\dfrac{x dy - y dx}{x^2 + y^2}$?
|
*Notation*: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the [Gauss Notation for generalized continued fractions][1].
----------
**Description of the [Bauer-Muir transformation][2]**
(Based on [pp. 76-77][3] of Lisa Lorentzen and Haakon Waadeland's [book][4] (A)... |
For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate
$$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}\qquad x>1.$$
The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{n}=\dfrac{x}{... |
What mathematical questions or areas have philosophical implications? |
What mathematical questions or areas have philosophical implications outside of mathematics? |
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 |
What are the most important results in the philosophy of mathematics? |
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and the disks can be viewed as spinning fan blades
determining vectors of fluid motion tangent to the disks.
Is the r... |
One can also see that $\mathbb{R}P^3$ is a Lie group, and that all Lie groups are orientable.
To see that $\mathbb{R}P^3$ is a Lie group, first think of $S^3$ as the collection of unit quaternions. The subset $\\{1,-1\\}$ is not only a subgroup, but it is a normal subgroup of $S^3$. Hence, $S^3/\\{1,-1\\}$ is a Li... |
Is it possible to make up a *distinct* number that is made up of four tuples. One of which indicates the direction of the arrangement of the other three.
Say, 9*4*7 = 252. 252 *is distinct* in the sense that you cannot get it as a multiplication of 3 distinct single digit numbers other than 9, 4, 7. Now, the problem... |
Here is a rather brief summary of the major positions on a couple of important philosophy of mathematics questions. First a list of questions:
- **Are we creating or discovering when we do mathematics?**
- **What are mathematical entities?**
- **How do you explain [the unreasonable effectiveness of mathematics ... |
i have lineA with points are(1,1) and (8,8) and another lineB with points(2,2) and (4,4) i like to prove lineB lies on LineA ,how can do this? |
A quick question-Something that I learned that I forgot throughout the years.
If a question asked you for a multiple of 9 and 12, which would be less than 40, would "36" be an answer, as both 9 and 12 can multiply to get 36, yet it fits under the criteria for being under 40? |
What's a Multiple? |
I've been doing some maths work using the rate of flow of liquids. I've used various models for the flow and various methods to integrate these models. The one thing that is confusing me is the difference between constants of proportionality, and constants of integration? |
What is the difference between constants of proportionality and constants of integration? |
According to Wikipedia:
> Let G be a covering group of H. The
> kernel K of the covering homomorphism
> is just the fiber over the identity in
> H and is a discrete normal subgroup of
> G.
It is easy to show that the kernel is a normal subgroup, but why is it discrete?
I know this would be true if the ide... |
Why is the kernel of a covering group discrete? |
I think Robin's result is also clear if we look at the definition of the isomorphism between De Rham and (${\cal C}^\infty$) singular cohomologies. (Or maybe we are both saying the same thing?)
Remember (see Dupont, Curvature and characteristic classes, Springer LNM 640) the definition of the isomorphism at the coch... |
Let's say you have a set of numbers S and you want to obtain subsets of S, s<sub>1</sub>, s<sub>2</sub>,...s<sub>i</sub>, where i<N.
Is there a particular operation that will group the numbers that are "close to each other"?
I will give an example to clarify the question:
Let's say you have a set S which co... |
According to Wikipedia:
> Let G be a covering group of H. The
> kernel K of the covering homomorphism
> is just the fiber over the identity in
> H and is a discrete normal subgroup of
> G.
It is easy to show that the kernel is a normal subgroup, but why is it discrete?
I know this would be true if the ide... |
By definition, if $f:Y \to X$ is a covering space and $x \in X$,
then there is some neighbourhood $U$ of $x$ such that $f^{-1}(U)$ is a union of open sets
$V_i$ such that $f$ restricted to each $V_i$ is a homeomorphism. In particular,
$f^{-1}(x) \cap V_i$ consists of a single point, and so each point of $f^{-1}(x... |
In a vector space $V$, a *density* is a function $\mu: V \times \cdots \times V \rightarrow \mathbb{R}$ satisfying the condition
$$\mu(TX_1,\ldots,TX_n)=|det T|\mu(X_1,\ldots,X_n),$$
where $T: V \rightarrow V$ is any linear map (this definition follows John M. Lee, "*Introduction to Smooth Manifolds*"). In contr... |
Is the notion of density really needed to define integration on nonorientable manifolds? |
From New Scientist some years ago: 20 teams play a round robin tournament, each gets 1 point for a win, 0 for a loss, and there are no ties. Each team's score is a square number. How many upsets occurred? An upset is defined as team A defeating team B where B scored more total points than A. |
Any curve can be considered to be made of sufficiently small straight lines. What is the name of the theorem which states this fact?
Thank you.
I asked the same question at <a href="http://news.ycombinator.com/item?id=1760209">Hacker News</a>. |
A personal favorite (albeit not ideally-worded): Jamie has a windowbox where he plants a row of iris flowers. He plants in just two colors - blue and yellow - but never plants two yellow irises next to each other (the result is just too garish). Assuming that he wants to keep the same number of flowers in his box eve... |
Allow me to preface this by saying this is not a homework problem. If I had had this thought four years ago when I was taking calculus, I probably could do it...
I'm trying to calculate the limit as n --> inf of 1-(((n-1)/n)^n) - it's a constant I'm inclined to call "natural chance of success". I have estimated thi... |
Hmmm... this question might be a bit low-brow. I'm no mathematician.
Let's say that I have an unordered sequence of fuzzy values $T = (t_1 \ldots t_n)$, $T \in (0,1]^n$. Each value is a positive indication towards some observation (hence, the values are not probabilities in that a value of 0.1 does not indicate a v... |
Allow me to preface this by saying this is not a homework problem. If I had had this thought four years ago when I was taking calculus, I probably could do it...
I'm trying to calculate the limit as $n \to \infty$ of $1-\left(\frac{n-1}{n}\right)^n$ -
- it's a constant I'm inclined to call "natural chance of succe... |
What are the most important results in the philosophy of mathematics?
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-philosophica... |
What are the most important questions or areas of study in the philosophy of mathematics? |
Suppose we have an atom at every point with integer coordinates in $\mathbb{R}^d$. Take a ($d-1$)-dimensional hyperplane going through $\mathbf{0}$ and orthogonal to $(1,1,1,\ldots)$. What is the name of the lattice formed by atoms in that plane? |
What is the name of this lattice? |
Paul **Benacerraf's identification problem** is an argument against Platonism: Many different models satisfy Peano's arithmetic axioms. Which of these are really the Natural numbers? I.e., the uniquely intended set? We are unable to pick one account to the exclusion of all the others. Adding axioms won't help. Any such... |
There's a theorem that every planar graph can be colored with 4 colors in such a way that no 2 adjacent vertices have the same color. Is the opposite true as well? |
If a graph can be colored with max 4 colors, is it planar? |
In section 4 of the article by Afred van der Poorten's [*A Proof That Euler Missed ...*][1] the following inequality is used:
$$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$$
(*In the original denoted* $\text{ord}$ $_{p}(\cdot)$ *instead of* $\nu_p(\cd... |
Any curve can be considered to be made of sufficiently small straight lines. What is the name of the theorem which states this fact?
Thank you.
I asked the same question at <a href="http://news.ycombinator.com/item?id=1760209">Hacker News</a>.
[Edit: in the Hacker News question, the OP included the following ... |
Suppose we have a linear ODE of the form dy/dt= Ay. We know that y= exp^(tA) * y(0) is a solution to the ODE, where exp^(tA) represents the matrix exponential. Suppose we also knew that this is stable in the sense of Lyapunov (i.e for any epsilon >0, there exists a delta > 0 such that if|y(0)| < delta, then |y(t)| < ... |
Hey, sorry for the non so mathematical question but that's the only place where I'd thought someone would answer this so here we go.
I'm doing a chemistry homework and I have to answer this very question:
> If 50 grams of a particular mineral containing copper nitrate (II) gave, after appropriate treatment 1.6933... |
"OK, so the actual problem is as follows. If a linear ODE $y'= Ay$ is Lyapunov stable and if $B= G^{-1}AG$ where $G$ is an invertible matrix, is $B$ also Lyapunov stable?"
That is, is the system $z'=Bz$ Lyapunov stable around the origin?
Note that the two systems are related by a change of basis. If $y=Gz$, then... |
How would you factor $6x^2 + 18x - 60$ over a set of complex numbers?
When I did it myself, I had $6(x - 1.5 + 3.25i)(x - 1.5 - 3.25i)$ .
I am not 100% sure about my answer, so I am asking anyone who knows how to do this to help me confirm the answer! (: |
$$\begin{align}
6x^2+18x-60&=6(x^2+3x-10)
\\\\
&=6(x+5)(x-2)
\end{align}$$
When I expand yours:
$$6(x - 1.5 + 3.25i)(x - 1.5 - 3.25i)=6x^2-18x+76.875$$
So, I'm not entirely sure what went wrong. How did you arrive at your answer? |
The answer is wrong (arithmetic error?). Apply the [quadratic formula][1] to $\rm\ p(x)/6 = x^2 + 3x -10\:$.
[1]:http://en.wikipedia.org/wiki/Quadratic_equation |
How would you factor $6x^2 + 18x - 60$ over a set of complex numbers?
When I did it myself, I had $6(x - 1.5 + 3.25i)(x - 1.5 - 3.25i)$ .
I am not 100% sure about my answer, so I am asking anyone who knows how to do this to help me confirm the answer! (:
<b>- EDIT -</b>
Well, first thing that I did was t... |
<B>Ahem. I can only now laugh at how blind I am to not see such a simple answer, LMAO. Thanks to Isaac ♦, Paul VanKoughnett, and Bill Dubuque!
Case closed!</b>
How would you factor $6x^2 + 18x - 60$ over a set of complex numbers?
When I did it myself, I had $6(x - 1.5 + 3.25i)(x - 1.5 - 3.25i)$ .
I am... |
$$\begin{align}
6x^2+18x-60&=6(x^2+3x-10)
\\\\
&=6(x+5)(x-2)
\end{align}$$
When I expand yours:
$$6(x - 1.5 + 3.25i)(x - 1.5 - 3.25i)=6x^2-18x+76.875$$
So, I'm not entirely sure what went wrong. How did you arrive at your answer?
----------
***edit***: Given your work, it's worth having another look. ... |
Suppose we have a linear ODE of the form $\dfrac{dy}{dt}= Ay$. We know that $y= e^{tA} y(0)$ is a solution to the ODE, where $e^{tA}$ represents the matrix exponential. Suppose we also knew that this is stable in the sense of Lyapunov (i.e for any $\varepsilon >0$, there exists a $\delta > 0$ such that if $\vert y(0... |
I was wondering what are the general spaces that the concepts limsup and liminf can apply to?
Is complete lattice one of them? Also How about metric space?
What are limsup and liminf specified with respect to? A subset? A sequence/net/filter base?
How many kinds of definitions for limsup and liminf in these vari... |
Suppose there are a finite number of disjoint unit-radii disks in the
plane, each spinning clockwise or counterclockwise at the same
angular velocity.
The plane is filled with a thin fluid layer,
and the disks can be viewed as spinning fan blades
determining vectors of fluid motion tangent to the disks.
Is the r... |
I was under the impression that all 2-forms are the wedge (^) of two 1-forms. Is it possible to have a 2-form that you can't write as A^B with A,B 1-forms? |
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