instruction stringlengths 12 30k |
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What is a set? I know that results such as [Russell's paradox][1] mean that the definition isn't as straight forward as one might expect.
[1]: http://en.wikipedia.org/wiki/Russell%27s_paradox |
The best explanation I could find was on [WikiAnswers][1]:
> Two circles intersect in at most two points, and each intersection creates one new region. (Going clockwise around the circle, the curve from each intersection to the next divides an existing region into two.)
> Since the fourth circle intersects the fi... |
The short answer, from a [paper][5] by Frank Ruskey, Carla D. Savage, and Stan Wagon is as follows:
> ... it is impossible to draw a Venn diagram with circles that will represent all the possible intersections of four (or more) sets. This is a simple consequence of the fact that circles can finitely intersect in at ... |
The short answer, from a [paper][1] by Frank Ruskey, Carla D. Savage, and Stan Wagon is as follows:
> ... it is impossible to draw a Venn diagram with circles that will represent all the possible intersections of four (or more) sets. This is a simple consequence of the fact that circles can finitely intersect in at ... |
The short answer, from a [paper][1] by Frank Ruskey, Carla D. Savage, and Stan Wagon is as follows:
> ... it is impossible to draw a Venn diagram with circles that will represent all the possible intersections of four (or more) sets. This is a simple consequence of the fact that circles can finitely intersect in at ... |
This is Problem 1.7 from <a href="http://www.colby.edu/personal/fqgouvea/deform.dvi">Gouvea's lecture notes on deformations of Galois representations</a>. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's what I've got so far, which may be wrong.
I can writ... |
How to describe the Galois group of the compositum of all quadratic extensions of Q? |
Often in my studies (economics) the assumption of a "well-behaved" function will be invoked. I don't exactly know what that entails (I think twice continuously differentiability is one of the requirements), nor do I know why this is necessary (though I imagine the why will depend on each case).
Can someone explain i... |
I am trying to show the following:
![alt text][1]
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
![alt text][2]
And then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\ga... |
###For the uninitiated
[Morse theory][1], as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of [hom... |
###For the uninitiated
[Morse theory][1], as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of [hom... |
###For the uninitiated
[Morse theory][1], as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of [hom... |
###For the uninitiated
[Morse theory][1], as many other early alebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of [hom... |
I have to divide 2860 by 3186. The question gives only 2 minutes and that division is only half part of question. Now I can't possibly make that division in or less than 2 minutes by applying traditional methods, which I can't apply on that division anyways.
So anyone can perform below division using faster techniqu... |
If this problem appears on a standardized test you may want to take into account some broader considerations... How many of these digits are even significant (relative to other quantities you will be computing with in "second half" of the problem)? What further operations will you be performing with this quantity - i.... |
As a programmer, mathematics is important basic knowledge to study some topic, especially Algorithm. Many website, and my fellows suggest me to study Discrete Mathematics before go to Algorithm, so I want to know which Discrete Mathematics is suitable for my need? |
There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why the Gamma function is "best"? In particular, I'm looking for reasons that I can explain to first-year calculus students. |
Why is the Gamma function the "best" extension of the factorial function to the reals? |
How do I go about solving $0<1−an/(mb^2)e^{−r(T−t)}<1$, where a,b,T>t>0? I have been stuck here for some time now. |
How do I go about solving $0<1−\frac{an}{mb^2}e^{−r(T−t)}<1$, where a,b,T>t>0? I have been stuck here for some time now. |
Here is one elementary thing you can't prove inside constructive mathematics, that is, without the Axiom of Choice.
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 element from it: $x \in ... |
Exercise 10 in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). That is, (using AxB to mean the direct product and A+B to be the direct sum), whether there is an isomorphism between Ax... |
Here is a classic theorem of Herstein:
$G$ is a finite group, $M$ a maximal subgroup, which is abelian. Then $G$ is solvable.
The proof is pretty easy, but it uses character theory (specifically, Frobenius' theorem on Frobenius groups). Is there a character-theory-free proof?
To get things going, note that we ... |
G a finite group, M a maximal subgroup; M abelian implies G solvable? |
For whatever reason, Nature (by which I mean integrals) seems to prefer the Gamma function as the "correct" substitute for the factorial in various integrals, which seems to come more or less from its integral definition. For example, for non-negative integers $a, b$, it's not hard to show (and there's a really nice p... |
There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is "best"? In particular, I'm looking for reasons that I can explain to first-year calculus students. |
Why Euler's Gamma function the "best" extension of the factorial function to the reals? |
Why is Euler's Gamma function the "best" extension of the factorial function to the reals? |
I have to divide 2860 by 3186. The question gives only 2 minutes and that division is only half part of question. Now I can't possibly make that division in or less than 2 minutes by applying traditional methods, which I can't apply on that division anyways.
So anyone can perform below division using faster techniqu... |
I'm trying to understand better the following addition formula: $\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{ab-\sqrt{1-a^2}\sqrt{1-b^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$
The term $ab-\sqrt{1-a^2}\sqrt{1-b^2}$ can be derived from trigonometry (since $\sin(t) = \sq... |
I'm trying to understand better the following addition formula: $$\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{ab-\sqrt{1-a^2}\sqrt{1-b^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$$
The term $ab-\sqrt{1-a^2}\sqrt{1-b^2}$ can be derived from trigonometry (since $\sin(t) = \... |
I'm trying to understand better the following addition formula: $$\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{a\sqrt{1-b^2}+b\sqrt{1-a^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$$
The term $a\sqrt{1-b^2}+b\sqrt{1-a^2}$ can be derived from trigonometry (since $\sin(t) = \... |
Here is one elementary thing you can't prove inside constructive mathematics, that is, without the Axiom of Choice.
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 element from it: $x \in ... |
I want to solve $0<1−an/(mb^2)e^{−r(T−t)}<1$, where $a, b, T, t>0$. The solution is that either $an\leq mb^2$ or $mb^2\leq an\leq mb^2e^{rT}$ and $t< T − (\ln(an) − \ln(mb^2 ))/r$.
My thoughts are that the first part $0<1−\frac{an}{mb^2}e^{−r(T−t)}$ gives me $an \leq mb^2$ because $e^{−r(T−t)}>0$, so I have the firs... |
> **Possible Duplicate:**
> [problem with inequality](http://math.stackexchange.com/questions/1482/problem-with-inequality)
<!-- End of automatically inserted text -->
How do I go about solving $0<1−\frac{an}{mb^2}e^{−r(T−t)}<1$, where a,b,T>t>0? I have been stuck here for some time now. |
Top Prime's Divisors'
Product (Plus one)'s factors are...?
Q.E.D B@%&$
[XKCD][1]
[1]: http://xkcd.com/622/ |
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/ |
There are a real numbers worth of countable models of Peano arithmetic, all elementarily equivalent to the usual model, but all pairwise nonisomorphic.
Every nonstandard example (nonstandard means not isomorphic to the usual model) is characterized by the fact that it contains "infinitely large" numbers. That is, e... |
As a follow up to [algebra-of-random-variables][1], is it possible to compute:
$z = \frac{dx}{dy}$
Where $x$ and $y$ are drawn from a known, independent probability distribution $x \in f(x)$, $y \in g(y)$? Geometrically the problem makes sense, one has a function $h(x,y) = f(x)g(y)$ that is the joint probability ... |
As a follow up to [algebra-of-random-variables][1], is it possible to compute:
$z = \frac{dx}{dy}$
Where $x$ and $y$ are drawn from a known, independent probability distribution $x \in f(x)$, $y \in g(y)$? Geometrically the problem makes sense, one has a function $h(x,y) = f(x)g(y)$ that is the joint probability ... |
SpeQ is really nice. If you close it's left/right areas it's just like Notepad, but a Notepad which can compute stuff. You can change any line displayed, press Enter at it's end and the line will be recomputed.
A lot of people recommended Python. It's is my main programming language and I love it, but SpeQ it's MUCH... |
When I was in my fourth year of high school I got a copy of _What is Mathematics?_ by Courant and Robbins. That book showed to me that Mathematics is far more than a "boring tool" to do Physics and opened up new worlds. I would recommend it to any bright high school kid with an interest in math and sciences. |
SpeQ is really nice. If you close it's left/right areas it's just like Notepad, but a Notepad which can compute stuff. You can change any line displayed, press Enter at it's end and the line will be recomputed.
Just like in Notepad, you can select any part of what's displayed and delete it. You can also paste non-ma... |
I have an application that wants controllable random functions from $\mathbb{Z}^2$ and $\mathbb{Z}^3$ to $2^{32}$ , where by controllable I basically mean seedable by some parameters (say, on the order of 3 to 5 32-bit integers) such that the same seeds will always produce the same functions. The most obvious way of d... |
'(Pseudo)-random functions' by seeding of PRNGs? |
Say we have a family {$A_\alpha$} of disjoint Baire spaces. Also suppose that each $A_\alpha$ is disjoint from the closure of the union of the other sets. Show that $\bigcup_{\alpha} A_\alpha$ is a Baire space.
I think we can prove this by transfinite induction. Suppose the property holds for all $\beta$ < $\alpha$,... |
This question was motivated by <a href="http://math.stackexchange.com/questions/1442/is-there-a-direct-proof-of-this-lcm-identity">pondering this lcm identity</a>.
Consider that gcd(1,6,15) = 1, but lcm(1,6,15)=30, but 1*6*15 = 90. (2,6,15) shows a similar phenomenon.
So what is the correct identity for n-ary gc... |
I want to solve $0<1−an/(mb^2)e^{−r(T−t)}<1$, where $r, a, b, T, t>0$. The solution is that either $an\leq mb^2$ or $mb^2\leq an\leq mb^2e^{rT}$ and $t< T − (\ln(an) − \ln(mb^2 ))/r$.
My thoughts are that the first part $0<1−\frac{an}{mb^2}e^{−r(T−t)}$ gives me $an \leq mb^2$ because $e^{−r(T−t)}>0$, so I have the f... |
As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to Algorithms, so I want to know which Discrete Mathematics book is suitable for my needs? |
What is the best book for studying discrete mathematics? |
Let's say we have a graph, with a list of edges and vertexes `(E,V)`, all the vertexes are connected to at least one edge at one end. There are many ways a complete set of [cycle basis][1] can be found out from it.
Now the issue is, is it **always** possible to find a complete set of cycle basis that each edge is sh... |
I asked this question a while back on MO : http://mathoverflow.net/questions/32689/how-should-a-homotopy-theorist-think-about-sheaf-cohomology
One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$)
I am curious, as a sort of followup if anyone can suggest... |
Any de Rham cohomology (or Dolbeault cohomology) computation is a computation in sheaf cohomology. Actually --- any computation in singular cohomology is a computation in sheaf cohomology!! ;-) We're just taking different resolutions of the appropriate constant sheaf.
IIRC, there are some good Cech cohomology comput... |
Can someone explain to me in simple terms what a non-linear transformation is in maths?
I know some single-variable calculus, but I read it has to do with multi-variable calculus, which I'm not familiar with.
If someone could explain it in simple words, that would be helpful. |
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... |
How can I calculate the length of the cathetus in a triangle if I know the length of the hypothenuse and the ratio between the two catheti?
For example:
The hypothenuse is 5cm long, and the ratio between the catheti is 4:3 - how long are the catheti or either cathetus? |
Get length of cathetus/i from length of hypothenuse and ratio between catheti? |
when trying to explain AG-codes to computer scientists, the major points of contention i am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. are there any intuitive explanations for these concepts? or at least ones that are less dependent on knowledge of algebraic-geometry/... |
This is rather scheme-y, but there's a really nice paper by [Kempf][1] (hopefully you have institutional access :() that gives a very basic and elementary proof that the higher cohomology of a quasi-coherent sheaf on an affine scheme is trivial. The first part of the paper uses nothing more than the basic properties ... |
Is this a counter example to Stromquist's Theorem? |
How can I calculate the length of the [cathetus][1] in a triangle if I know the length of the hypotenuse and the ratio between the two catheti?
For example:
The hypotenuse is 5cm long, and the ratio between the catheti is 4:3 - how long are the catheti or either cathetus?
[1]: http://en.wikipedia.org/wiki/... |
What about the square on the bottom- with it and the triangle's mutual side returned? |
What about the square on the bottom- with it and the triangle's mutual side returned?
**Edit:** Btw. Don't be disheartened that your counterexample didn't come through- looking for counterexamples to proved theorems will never give a genuine counterexample (unless ZFC turns out to be inconsistent!!;)), but it can pr... |
Pardon me for not knowing LaTeX code.
A 3x3 matrix A:<br>
[ 2 -1 1 ]<br>
| -2 3 -2 |<br>
[ -1 1 0 ]<p>
...produces this characteristic equation:<br>
λ<sup>2</sup> - 4λ + 3 = 0<p>
...these eigenvalues:<br>
λ<sub>1</sub> = 1<br>
λ<sub>2</sub> = $\dfrac{-1 + \sqrt{1... |
Let $X,Y$ be $\sigma$-finite measure spaces, and let $L^p(X) \otimes L^p(Y)$ be the algebraic tensor product. The product has a natural map into $L^p(X \times Y)$ which takes $\sum a_{ij} f_i \otimes g_j$ to the function $F(x,y) = \sum a_{ij} f_i(x) g_j(y)$. A moment's thought shows that this map is well-defined. Is... |
Is the natural map $L^p(X) \otimes L^p(Y) \to L^p(X \times Y)$ injective? |
Well, first off, a vector isn't orthogonal, a set of vectors is. Remember that for two vectors to be orthogonal, it means that they are at right angles to each other, and orthonormal means that plus they're unit vectors.
Now, if I had to say where I think your first error is, you took a 3 x 3 matrix and got a quadr... |
Let $X,Y$ be $\sigma$-finite measure spaces, and let $L^p(X) \otimes L^p(Y)$ be the algebraic tensor product. The product has a natural map into $L^p(X \times Y)$ which takes $\sum a_{ij} f_i \otimes g_j$ to the function $F(x,y) = \sum a_{ij} f_i(x) g_j(y)$. A moment's thought shows that this map is well-defined. Is... |
**Stromquist's Theorem**: If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here: www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the _corners_ of a square overlap with the curve.
I would like to sugg... |
Your characteristic polynomial is incorrect.
Go to wolframalpha.com and enter: {{2, -1, 1},{-2, 3, 2}, {-1, 1, 0}}
www.wolframalpha.com/input/?i={{2,+-1,+1},{-2,+3,+2},+{-1,+1,+0}}
You will see that the characteristic polynomial is of 3rd degree.
It must be of 3rd degree because your matrix is 3x3.
Here is wha... |
How can I calculate $27^{41} \mathrm{mod}\ 77$ as simple as possible? I already know that $27^{60}\ \mathrm{mod}\ 77 = 1$ because of [Euler’s theorem](http://en.wikipedia.org/wiki/Euler%27s_theorem):
$$ \phi(77) = \phi(7 \cdot 11) = (7-1) \cdot (11-1) = 60 $$
Can I derive the result of $27^{41} \mathrm{mod}\ 77$ ... |
I try to learn the theory of category from [The Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf). I got stacked with the first exercise (3A a).
If we have a simple graph with one vertex and 2 nodes all we know is that in category there is 2 objects and one homomorphism (beside identities). Hence:
\\[C =... |
How can I calculate $27^{41}\ \mathrm{mod}\ 77$ as simple as possible?
I already know that $27^{60}\ \mathrm{mod}\ 77 = 1$ because of [Euler’s theorem](http://en.wikipedia.org/wiki/Euler%27s_theorem):
$$ a^{\phi(n)}\ \mathrm{mod}\ n = 1 $$
and
$$ \phi(77) = \phi(7 \cdot 11) = (7-1) \cdot (11-1) = 60 $$
I als... |
A discrete one-dimensional model of optical imaging looks like this:
$I(r) = \sum_i e_i P(r - r_i)$
Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie $P(x) = ... |
If you want to avoid algebraic geometry and topology, then you will probably be forced to use algebra. This is okay, because the things you ask about have algebraic interpretations.
First of all, it probably helps to write the function field as $k(x,y),$ where $x$ and $y$ are related by some equation $f(x,y) = 0$. ... |
I try to learn the theory of category from [The Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf). I got stacked with the first exercise (3A a).
If we have a simple graph with one vertex and 2 nodes all we know is that in category there is 2 objects and one homomorphism (beside identities). Hence:
\\[C =... |
I have an application that wants controllable random functions from $\mathbb{Z}^2$ and $\mathbb{Z}^3$ to $2^{32}$ , where by controllable I basically mean seedable by some parameters (say, on the order of 3 to 5 32-bit integers) such that the same seeds will always produce the same functions. The most obvious way of d... |
How do I calculate a specific variation for a known value of the normal distribution function? |
i've sampled a real world process, network ping times. The "round-trip-time" is measured in milliseconds. Results are plotted in a histogram:
![alt text][1]
Ping times have a minimum value, but a long upper tail.
i want to know what statistical distribution this is, and how to estimate its parameters.
Even... |
How do i figure out what kind of distribution this is? |
i've sampled a real world process, network ping times. The "round-trip-time" is measured in milliseconds. Results are plotted in a histogram:
![alt text][1]
Ping times have a minimum value, but a long upper tail.
i want to know what statistical distribution this is, and how to estimate its parameters.
Even... |
Solving $x^{11} = 1$ in radicals was first accomplished by Vandermonde, whose method is described on p.24-26 of Edwards' book on Galois theory:
http://books.google.com/books?id=0bH6SUHSvloC&pg=PA24 |
Say P ~ Q (P and Q are «projectively equivalent») iff there is a projective transformation f such that f(P) = Q. Then ~ is an equivalence relation. I read that the space of inscribed n-gons modulo projective equivalence has dimension n-3. Why is this? Also, are there any related results? |
Whenever I think about how a function behaves, I always try to identify a general pattern of behavior with some common numbers (somewhere between 5 and 100 maybe) and then I try to see if anything interesting happens around 1, 0 and into negative numbers if applicable.
If that all works out, I essentially assume tha... |
Suppose I have a point *x* in R<sup>n</sup> on an [n-sphere][1]. Suppose I divide the n-sphere into 4 quadrants (I think this makes sense in *n* dimensions), how do I know which quadrant *x* lies on?
[1]: http://en.wikipedia.org/wiki/N-sphere |
Denote by $D(a_1,\dots,a_n)$ the product $\prod_{j>i}(a_j-a_i)$. Assuming that $a_i$ are integers s.t. $a_1\le a_2\le\dots\le a_n$, proove that $D(a_1,...,a_n)/D(1,...,n)$ is the number of _Gelfand-Zetlin triangles_ (that is, [triangles](http://imrn.oxfordjournals.org/content/vol2010/issue13/images/medium/rnp223ueq3.gi... |
Suppose I have a point $x \in \mathbb{R}^n$ on an [n-sphere][1]. Suppose I divide the n-sphere into 4 quadrants (I think this makes sense in $n$ dimensions), how do I know which quadrant $x$ lies on?
[1]: http://en.wikipedia.org/wiki/N-sphere |
If you want to avoid algebraic geometry and topology, then you will probably be forced to use algebra. This is okay, because the things you ask about have algebraic interpretations.
First of all, it probably helps to write the function field as $k(x,y),$ where $x$ and $y$ are related by some equation $f(x,y) = 0$. ... |
The values of a function $f(x)$ for relatively small values of its argument $x$ is *typically* a very bad predictor of asymptotic behavior of $f(x)$ for large $x$. This is true even when $f(x)$ is an [analytic function][1] which is uniquely determined by its values on any small interval $x\in[-\epsilon,\epsilon]$.
H... |
Just to make sure: you know that what you are asking for is actually Weyl's dimension formula for the highest weight representation in the case of SL(n,C) resp. SU(n)? I suppose there is no direct combinatorial proof that does not make use of this fact. |
Suppose I have a point $x \in \mathbb{R}^n$ on an [n-sphere][1]. Suppose I divide the n-sphere into 4 sections (I think this makes sense in $n$ dimensions), how do I know which section $x$ lies on?
[1]: http://en.wikipedia.org/wiki/N-sphere |
If X and Y are normed spaces I know that the space of a bounded linear functions from X to Y , B(X,Y), is complete if Y is complete. Is there an example of a normed spaces X,Y s.t. B(X,Y) is not complete? |
In DotA, there is a character called "[Axe](http://www.dotastrategy.com/hero-71-MogulKhanAxe.html)". Every time he is attacked, he has a chance to spin (17%) his blade and deal damage based on what level the skill is, 100 / 125 / 150 / 175 damage for levels 1, 2, 3, and 4. When the spin activates, it triggers the coold... |
How can I calculate this expected rate? |
Somewhat related to [this question][1], but almost infinitely more basic.
###A Confession
I am, should classification prove essential, a differential geometer and a topologist by inclination and by training: as an undergraduate I shunned any ring that wasn't $\mathbb{Z}_n$ or a ring of differential operators and h... |
Somewhat related to [this question][1], but almost infinitely more basic.
###A Confession
I am, should classification prove essential, a differential geometer and a topologist by inclination and by training: as an undergraduate I shunned any ring that wasn't $\mathbb{Z}_n$ or a ring of differential operators and h... |
Well, Qiaochu lists the affine line, so I'm going to do the projective line. (Everything here is over the complex numbers.)
The projective line $P^1$, or equivalently the Riemann sphere. This is the space of all lines through the origin in $\mathbb{C}^2$. Recall that the Riemann sphere can be obtained by gluing t... |
Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations?
I am aware of Hilbert's fou... |
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... |
I want to try to emulate what this application can do:![alt text][1]
Given the red round dots (from the mouse) it is able to solve for the bezier handles given that the tension of the curve is set to 0.6. How could I solve for the bezier handles? Is there some way I could do this with the slope or perpendicular of t... |
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