instruction stringlengths 12 30k |
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This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of pi? I'm bit confused as [here][1] it states pi can be express by ≒ as it's not a rational number, but pi can also be expre... |
Consider the following assertion from [Scott Aaronson's blog][1]:
> Supposing you do prove the Riemann
> Hypothesis, it’s possible to convince
> someone of that fact, without
> revealing anything other than the fact
> that you proved it. It’s also possible
> to write the proof down in such a way
> that someone... |
Here is another result from Scott Aaronson's blog:
> If every second or so your computer’s
> memory were wiped completely clean,
> except for the input data; the clock;
> a static, unchanging program; and a
> counter that could only be set to 1,
> 2, 3, 4, or 5, it would still be
> possible (given enough time)... |
Say I have an image, with pixels that can be either 0 or 1. For simplicity, assume it's a 2D image (though I'd be interested in a 3D solution as well).
A pixel has 8 neighbors (if that's too complicated, we can drop to 4-connectedness). Two neighbouring pixels with value 1 are considered to be connected.
If I k... |
I think that complex analysis is hard because graphs of even basic functions are 4 dimensional. Does anyone have any good visual representations of basic complex functions or know of any tools for generating them? |
There is a very simple expression for the inverse of Fourier transform. What is the easiest known expression for the inverse Laplace transform?
Moreover, what is the easiest way to prove it? |
Is there any connection between exact differential equations and forms, or is the similarity in name just an accident?
|
Why are noncommutative nonassociative Hopf algebras are called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
|
Why are noncommutative groups called quantum groups? |
For $n > 1$ an integer, there are well-known formulas for volume of the balls.
What is the analogous statement in a Banach space/Hilbert Space? |
Open two browser windows side-by-side and use [wolfram alpha][1]. Recast your function f(z) as f(x+iy) and plot the [real part][2] in one window and the [imaginary part][3] in another. My example links provide plots for z<sup>3</sup>.
Not sure what I'm doing wrong, but two of my links keep vanishing (I'm guessing ma... |
Open two browser windows side-by-side and use [wolfram alpha][1]. Recast your function f(z) as f(x+iy) and plot the [real part][2] in one window and the [imaginary part][3] in another. My example links provide plots for z<sup>3</sup>.
Not sure what I'm doing wrong, but two of my links keep vanishing (I'm guessing ma... |
Ok, Ok, I know that in fact the discriminant is defined (up to sign) as a product of differences of the roots of the polynomial.
But why does it then have integral coefficients, if the polynomial you started with had integer coefficients? |
Why does the discriminant of an integral polynomial have integral coefficients? |
This looks a bit like percolation theory to me. In the 4-neighbour case, if you look at the dual of the image, the chance that an edge is connected (runs between two pixels of the same colour) is `1-2p+2p^2`.
I don't think you can get nice closed-form answer for your question, but maybe a computer can help with som... |
This is somewhere between an answer and commentary. As others have said, the question is equivalent to showing: for any prime $p > 3$, $p^2 \equiv 1 \pmod 3$ and $p^2 \equiv 1 \pmod 8$. Both of these statements are straightforward to show by just looking at the $\varphi(3) = 2$ reduced residue classes modulo $3$ and ... |
The function which is 1 on the interval `[0;1]`, and 0 elsewhere, is a non-continuous probability distribution function. The function which is 3 on `[0;1]` and -1 on (1;3], and so on and on. What kind of answer do you want? What kind of properties do you want your functions to have?
There really are too many functio... |
What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus?
Specifically, I am interested in the following areas:
* Untyped lambda calculus
* Simply-typed lambda calculus
* Other typed lambda calculi
* Church's Theory of Types (I'm not sure where this... |
As Scott himself states in comments section of [the post in question][1] (comment #9):
> (4) Width-5 branching programs can compute NC1 (Barrington 1986); corollary pointed out by Ogihara 1994 that width-5 bottleneck Turing machines can compute PSPACE
Unfortunately, I don't have any ideas how this is proved.
... |
This is still WIP. There are a few missing details, still I think it's better than nothing. Feel free to edit in the missing details.
Given a problem of `SUBSET-SUM`. We have a set of `A`={a<sub>1</sub>,a<sub>2</sub>,...,a<sub>n</sub>} numbers, and another number `s`. The question we're seeking answer to is, whether... |
There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it?
It is nice if it is $1$; we have unique factorization of all ideals; but otherwise?
|
I would read a book about Perelman's proof of the Poincaré conjecture (or even the papers themselves). Oh, you mean the book had to be written when I was starting? |
I am not a mathematician but [Flatland: A Romance of Many Dimensions][1] blew my mind. I read it when I was a college student in a class on Special Relativity and wish I had read it way earlier.
[1]: http://books.google.com/books?id=R6E0AAAAMAAJ&dq=flatland&printsec=frontcover&source=bn&hl=en&ei=eE9QTLKVF5G5rAex... |
Closed points should be thought of as being "actual points", whereas non-closed points can correspond to all sorts of different things: subvarieties, "fat points", generic points, etc. You might be interested in reading [this blog post][1] about Mumford's drawing of $\operatorname{Spec} \mathbb{Z}$.
One possible way... |
Let A and B be two matrices which can be multiplied.
Then **rank(AB) <= min(rank(A), rank(B))**
I proved **rank(AB) <= rank(B)** interpreting AB as a composition of linear maps, observing that ker(B) \subseteq ker(AB) and using the kernel-image dimension formula. This also provide, in my opinion, a nice interpret... |
How to prove and interpret rank(AB) <= min(rank(A), rank(B)) ? |
For Moebius transformations, check out [this nice YouTube video][1].
[1]: http://www.youtube.com/watch?v=JX3VmDgiFnY |
What's the difference between open and closed sets? |
Kevin Lin's answer regarding the meaning of closed points is quite reasonable, especialy in the case when the scheme in question underlies a classical variety. I want to add some additional remarks and examples for thinking about more general schemes.
Here are some tautological remarks: recall that a point x in a s... |
What is the smallest area of a parking lot in which a car (that is, a segment of) can perform a complete turn (that is, rotate 360 degrees)?
(This is obviously the Kakeya Needle Problem. Fairly easy to explain, models an almost reasonable real-life scenario, and has a very surprising answer as you probably know - th... |
I have lifted this from [Mathoverflow](http://mathoverflow.net/questions/11669/) since it belongs here.
Hi,
Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the di... |
What is the difference between matrix theory and linear algebra? |
Lifted from [Mathoverflow](http://mathoverflow.net/questions/2446):
I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc.
One suggestion per answer please. Also, please include an ex... |
Best Algebraic Geometry text book? (other than Hartshorne) |
Another vague question which didn't quite fit at [Mathoverflow](http://mathoverflow.net/questions/446):
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
(Answers at any level of sophistication are welcome.) |
Using only precalculus mathematics (including that the area of the triangle with vertices at the origin, (x1,y1), and (x2,y2) is half of the absolute value of the determinant of the 2x2 matrix of the vertices (x1,y1) and (x2,y2), 1/2 * |x1*y2 - x2*y1|) how can one prove that [the shoelace method][1] works for all non-s... |
Kevin Lin's answer regarding the meaning of closed points is quite reasonable, especialy in the case when the scheme in question underlies a classical variety. I want to add some additional remarks and examples for thinking about more general schemes.
Here are some tautological remarks: recall that a point x in a s... |
Conic sections are a frequent target for dropping when attempting to make room for other topics in advanced algebra and precalculus courses. A common argument in favor of dropping them is that typical first-year calculus doesn't use conic sections at all. Do conic sections come up in typical intro-level undergraduate... |
How many ways are there to color faces of a cube with N colors if
two colorings are the same if it's possible to rotate the cube such that
one coloring goes to another?
|
I have lifted this from [Mathoverflow](http://mathoverflow.net/questions/11669/) since it belongs here.
Hi,
Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the diff... |
I wish I'd understood the importance of inequalities earlier. I wish I'd carefully gone through the classic book [Inequalities][1] by Hardy, Littlewood, and Poyla early on. Another good book is [The Cauchy-Schwarz Masterclass][2].
You can study inequalities as a subject in their own right, often without using adva... |
Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes?
I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's some division going on and there are some fancy 'd' characters that are added in... I recall thinking that it was a lot o... |
Why are derivatives specified as d/dx? |
Closed points should be thought of as being "actual points", whereas non-closed points can correspond to all sorts of different things: subvarieties, "fat points", generic points, etc. You might be interested in reading [this blog post][1] about Mumford's drawing of $\operatorname{Spec} \mathbb{Z}[x]$.
One possible ... |
Kevin Lin's answer regarding the meaning of closed points is quite reasonable, especialy in the case when the scheme in question underlies a classical variety. I want to add some additional remarks and examples for thinking about more general schemes.
Here are some tautological remarks: recall that a point x in a s... |
I have some ploygons I would like to map onto the face of a Cone.
I can see from [this page](http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm#skip3) that I can convert the points of the polygon to cylinderical coordinates, which is almost what I want.
How do I go about modifying ... |
how do I convert from cartesian to conical coordinates? |
In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse".
Later on, we learn the power series definitions of sin and cos.
How can one prove that these two definitions are equivalent? |
I have some polygons I would like to map onto the face of a cone.
I can see from [this page](http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm#skip3) that I can convert the points of the polygon to cylinderical coordinates, which is almost what I want.
How do I go about modifying ... |
How do I convert from cartesian to conical coordinates? |
A vague question of Kevin Lin which didn't quite fit at [Mathoverflow](http://mathoverflow.net/questions/446):
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
(Answers at any level of sophistication are welcome.) |
Isn't it as simple as $sin^2(\theta)+cos^2(\theta)=1$? Here we have a right triangle. |
Suppose a finite group has an automorphism of order 3. How do you prove that it is abelian? |
Suppose we model traffic flow between two points with a directed graph. Each route has either a constant travel time or one that linearly increases with traffic. We assume that each driver wishes to minimise their own travel time and we assume that the drivers form a Nash equilibria. Can removing a route ever decrease ... |
Here is another result from [Scott Aaronson's blog][1]:
> If every second or so your computer’s
> memory were wiped completely clean,
> except for the input data; the clock;
> a static, unchanging program; and a
> counter that could only be set to 1,
> 2, 3, 4, or 5, it would still be
> possible (given enough ... |
The calculus of relations is an algebra of operations over sets of pairs of individuals, where for any relation R, we can express the relation in the usual infix manner: x R y iff (x,y) ∈ R. This allows all of the properties of relations to be expressed in equational form.
There are four fundamental relations ... |
Suppose a finite group has the property that for every $x, y$, it follows that $(xy)^3 = x^3 y^3$.
How do you prove that it is abelian? |
Conic sections should definitely be retained. If you don't cover conic sections, then what other examples can you cover?
Lines? Too simple. General curves? Insufficiently concrete.
Examples are very important for illustrating the general theory and techniques.
Also, in a multivariable calculus course, typical ... |
From [Scott Aaronson's blog][1]:
"There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof of the Riemann Hypothesis. Indeed, every formal proof of the Riemann Hypothesis with at most (say) a million symbols corre... |
All the integrals I'm familiar with have the form:
> int f(x) dx.
And I understand these as the sum of infinite tiny rectangles with an area of: f(x_i) * dx.
Is it valid to have integrals that do not have a differential, such as dx, or that have the differential elsewhere than as a factor ? Let me give couple ... |
From [Scott Aaronson's blog][1]:
> There’s a finite (and not
> unimaginably-large) set of boxes, such
> that if we knew how to pack those
> boxes into the trunk of your car, then
> we’d also know a proof of the Riemann
> Hypothesis. Indeed, every formal proof
> of the Riemann Hypothesis with at most
> (say) a... |
There is another proof that the derivative of sine is cosine that doesn't use the sandwich theorem mentioned by Qiaochu and Akhil above. Instead, one can use the definition of arcsine and the standard calculus formula for arc length in terms of an integral to show that arcsine = the integral of (1 - x^2)^(-.5). It fo... |
1) Why does a small number of states suffice?
Regardless of whether the constant is 5 or 500, its still very surprising. Thankfully, it's fairly to prove if you allow the counter to be `{1...8}` instead of `{1...5}`. [This proof is by Ben-Or and Cleve.]. Start by representing the computation as a circuit, and ignore t... |
Let f and g be two periodic functions over R with the following property: If T is a period of f, and S is a period of g, then T/S is irrational.
Can we say that f + g is not periodic? Could you give a proof or a counter example? |
Here is a simpler example. I claim that the function $h(x) = \sin x + \sin \pi x$ cannot possibly be periodic. Why? Suppose an equation of the form
$$\sin x + \sin \pi x = \sin (x+T) + \sin \pi (x+T)$$
held for all $x$ and some $T > 0$. Take the second derivative of both sides with respect to $x$ to get
$$... |
Before Hartshorne's book there was [Mumford's Red Book of Varietie][1]s. I think it is a great introductory textbook to modern algebraic geometry (scheme theory).
I found that Mumford is quite good at motivating new concepts; in particular I really enjoy his development of nonsingularity and the sheaf of differenti... |
The wiki [article][1] on eigenvectors offers the following geometrical interpretation:
> Each application of the matrix to an arbitrary vector yields a result which will have rotated towards the eigenvector with the largest eigenvalue.
Qn 1: If there is any other geometrical interpretation particularly in the con... |
Is there a geometrical interpretation to the notion of eigenvector and eigenvalues? |
Where is the flaw in this argument of a proof of that 1 =2? |
Are there any well-studied analogs to the Mandelbrot set using functions other than f(z)=z^2+c? |
How many ways are there to tile a 2 x n grid with 1 x 2 dominos?
This problem can be solved using Fibonacci numbers. Let S<sub>n</sub> be the number of valid tilings of the 2 x n grid. Each such tiling has either a vertical 1 x 2 domino or two horizontal dominos on the right. Therefore, each tiling of a 2 x (n-1) g... |
Another example: [Euler's sum of powers conjecture](http://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture), a generalization of Fermat's Last Theorem. It states:
If the equation ∑<sub>i=0</sub><sup>k</sup> x<sub>i</sub><sup>n</sup> = z<sup>n</sup> has a solution in positive integers, then n ≤ k (unless... |
I understand the mechanics of the proof of Ostrowski's Theorem, but I'm a little unclear on why one should expect valuations to be related to primes. Is this a special property of number fields and function fields, or do primes of K[x,y] correspond to valuations on K(x,y) in the same way?
I'm hoping for an answer th... |
Why should one expect valuations to be related to primes? How to treat an infinite place algebraically? |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
>The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician.... |
While it is certainly true that with the proper definition there is now 'wrong' notation, perhaps it should be mentioned that some notation is more suggestive and/or easier to work with than others, e.g. Arabic numeral vs. Roman numerals, the various symbols for the derivative, and countless others. The actual symbols ... |
Where is the flaw in this argument of a proof that 1=2? |
Is it true that there are infinitely many nonprime integers $n$ such that $3^{n-1} - 2^{n-1}$ is a multiple of $n$? |
Discrete valuations <-> points on a curve
For a nonsingular projective curve over an algebraically closed field, there is a one-one correspondence between the points on it, and the discrete valuations of the function field (i.e. all the meromorphic functions of the curve). The correspondence is point P -> the valuat... |
How many ways can a rectangle be partitioned by either vertical or horizontal lines into $ n $ sub-rectangles? At first I thought it would be:
$ f(n)=4f(n-1)-2f(n-2) where f(0)=1 and f(1)=1 $
but the recurrence relation only counts the cases which at least one side (either top, bottom, left or right of the orig... |
How many ways can a rectangle be partitioned by either vertical or horizontal lines into $ n $ sub-rectangles? At first I thought it would be:
$ f(n)=4f(n-1)-2f(n-2) where f(0)=1 and f(1)=1 $
but the recurrence relation only counts the cases in which at least one side (either top, bottom, left or right of the o... |
How many ways can a rectangle be partitioned by either vertical or horizontal lines into `n` sub-rectangles? At first I thought it would be:
f(n) = 4f(n-1) - 2f(n-2)
where f(0) = 1
and f(1) = 1
but the recurrence relation only counts the cases which at least one side (either top, bottom... |
How to find all natural X for that X^2 + (X+1)^2 is a perfect square?
|
I'd like to characterise the functions that 'have square roots' in the function composition sense. That is, can a given function f be written as f = g.g (where . is function composition)?
For instance, the function f(x) = x+10 has a square root g(x) = x+5.
Similarly, the function f(x) = 6x has a square root g(x) ... |
Characterising functions f that can be written as f = g.g? |
How many ways can a rectangle be partitioned by either vertical or horizontal lines into `n` sub-rectangles? At first I thought it would be:
f(n) = 4f(n-1) - 2f(n-2)
where f(0) = 1
and f(1) = 1
but the recurrence relation only counts the cases in which at least one side (either top, bot... |
I'd like to characterise the functions that 'have square roots' in the function composition sense. That is, can a given function f be written as f = g.g (where . is function composition)?
For instance, the function f(x) = x+10 has a square root g(x) = x+5.
Similarly, the function f(x) = 9x has a square root g(x) ... |
Are there 2 subsets, say, $A$ and $B$, of the naturals such that
$$\sum_{n\in A} f(n) = \sum_{n\in B} f(n)$$
where $f(n)=1/n^2$?
If $f(n)=1/n$ then there are many counterexamples, which is probably a consequence of the fact that the harmonic series diverges:
$$\frac23 = \frac12 + \frac16 = \frac14... |
I know volume preserving diffeomorphisms of a sphere^2 make a group sdiff(S2). I would to know if it is a Lie group, which I assume if it is that makes interpolation easier (like with rotations).
So that is one question, is it a Lie group?
My next question is how can i interpolate between two elements in the gro... |
I just came up with this problem yesterday.
**Problem**:
Assume there is an important segment of straight line `AB` that needs to be watched at all time. A watchdog can see in one direction in front of itself and must walk at a constant speed at all time. (All watchdogs don't need to have the same speed.) When it r... |
I know volume preserving diffeomorphisms of a sphere^2 make a group sdiff(S2). I would to know if it is a Lie group, which I assume if it is that makes interpolation easier (like with rotations).
So that is one question, is it a Lie group?
Also is the group path connected? If so, how can I interpolate between tw... |
Okay, so hopefully this isn't too hard or off-topic. Let's say I have a very simple lowpass filter (something that smooths out a signal), and the filter object has a position variable and a cutoff variable (between 0 and 1). In every step, a value is put into the following bit of pseudocode as "input": `position = posi... |
Suppose we have some function $f(x)$ with local extrema at $x_1, x_2, \dots$, and a second function $g(x)$ which is continuous, strictly increasing and non-zero everywhere over the range of the $x_i$. Will $g(f(x))$ have its local extrema at the same $x_i$ and no others?
If so, are there any obvious loosenings of th... |
How can I show with a heuristic argument based on a Taylor expansion that for Stratonovich
stochastic calculus the chain rule takes the form of the classical (Newtonian) one?
Concerning Ito calculus the fact that dX^2 = dt results via a Taylor expansion in Ito's lemma - this fact should stay the same with Stratono... |
Bourbaki shows in a very natural way that every continuous group isomorphism of the additive reals to the positive multiplicative reals is determined by its value at $1$, and in fact, that every such isomorphism is of the form $f_a(x)=a^x$ for $a>0$ and $a\neq 1$. We get the standard real exponential (where $a=e$) wh... |
Natural derivation of the complex exponential function? |
The term "volume preserving" sounds a bit ambiguous to me: do you mean that your map preserves the total volume or do you mean that its differential at every point preserves volume (i.e. has determinant 1)? The former is weaker than the latter, and gives you more room for interpolation.
In any case, there is a famou... |
If and only if, which direction is which? |
Can you give a definition of the Conway base-13 function better than the one actually present on wikipedia ([here][1]), which isn't clear? Maybe with some examples?
[1]: http://www.google.it/url?sa=t&source=web&cd=1&ved=0CBgQFjAA&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FConway_base_13_function&ei=empRTNK-OsqWO... |
I know that this is meant to explain variance butthe description on Wikpiedia stinks and it is not clear how you can explain variance using this technique
Can anyone explain it in a simple way? |
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