instruction stringlengths 12 30k |
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What is a primitive polynomial? |
The title really say's it all, but once again is a curve's curvature invariant under rotation and uniform scaling?
|
Is a curve's curvature invariant under rotation and uniform scaling? |
This is a followup to this question
http://math.stackexchange.com/questions/1326/is-a-curves-curvature-invariant-under-rotation-and-uniform-scaling
Where I learned that curvature is invariant to rotations.
I have learned of a version of curvature that is invariant under affine transformations
http://en.wi... |
This is a followup to [this question][1], where I learned that curvature is invariant to rotations.
I have learned of a version of curvature that is [invariant under affine transformations][2].
I am wondering if there a is a form of curvature between the two. Invariant under uniform scaling and rotation but not ... |
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... |
Nice question. Firstly, there's no solution which will always take a fixed number of steps each time: after n independent random trials (choosing a hand), there are 2<sup>n</sup> possible choices, and 2<sup>n</sup> is never divisible by 3. (Equivalently: 1/3 does not have a terminating representation in base 2.)
But... |
Nice question. Firstly, there's no solution which will always take a fixed number of steps each time: after n independent random trials (choosing a hand), there are 2<sup>n</sup> possible choices, and 2<sup>n</sup> is never divisible by 3. (Equivalently: 1/3 does not have a terminating representation in base 2.)
But... |
My favorite proof of the Pythagorean Theorem is a special case of this picture-proof of the Law of Cosines:
![Proof Without Words: The Law of Cosines][1]
Drop three perpendiculars and let the definition of cosine give the lengths of the sub-divided segments. Then, observe that like-colored rectangles have the sam... |
Given a list of coordinates of a coplanar plane (`pt1`, `pt2`, `pt3` and so on), how to compute the centroid of the coplanar plane?
One way to do it is to project the plane onto XY and YZ plane, but I don't really favor this approach as you have to check the orientation of the coplanar plane first before doing the p... |
Dot product of two vectors on plane could be defined as product of lengths
of those vectors and cosine of angle between them.
In cartesian coordinates dot product of vectors with coordinates (x1, y1) and
(x2, y2) is equal to x1*x2 + y1*y2.
How to prove it? |
Let's say we have a graph, with a list of edges and vertexes `(E,V)`, there are many ways a set of cycle bases can be found out from it.
Now the issue is, is it **always** possible to find a complete set of cycle base that
1. Connects all the vertexes and edges
2. Each edge is shared by at most 2 cycles?
|
In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? |
The dot product is invariant under rotations, we may therefore rotate our coordinate system so that v is along the x-axis. In this case, $v = (|v|, 0)$. Letting $w = (x,y)$ we have (using the definition of dot product in Cartesian coordinates) $v \cdot w = |v| x$. But what is $x$? Well, if you draw the picture and ... |
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... |
In a graph, is it always possible to construct a set of cycle basis, with each and every edge Is shared by at most 2 cycle bases? |
Proposition 0.4.5.4 in EGA appears to be a general representability theorem. It reads:
> Suppose $F$ is a contravariant functor from the category of locally ringed spaces over $S$ to the category of sets. Suppose given representable sub-functors $F_i$ of $F$, such that the morphisms $F_i \to F$ are representable by ... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.
But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré's *Elements*. This po... |
How to explain fractals to a layperson and to someone with more math training? |
I just learned (thanks to [Harry Gindi][1] and on [Qiaochu Yuan][2] MO) that the chinese remainder theorem and lagrange interpolation are really just two instances of the same thing. Similarly the method of partial fractions can be applied to rationals rather than polynomials. I find that seeing a method applied in di... |
In 1976, [Knuth and Yao][1] proved a result that may help you choose a DVD out of 5, 6, etc. (I couldn't find a good reference online.) Consider the following problem: You have a fair coin and you must write an algorithm that outputs 1 with probability p1, outputs 2 with probability p2, ..., and outputs n with probabi... |
Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you're doing number theory then you want to think of the prime numbers as acting "independently": knowing that a number is divisible by... |
I guarantee there is an easy reference on this, but for some reason I cannot find it. If you can point me to a reference or just write a short proof for me, I would be appreciative.
Given a graded ring $R_{\bullet}$ and a localization $R_{\bullet}^{*}$. We also have a graded $R_{\bullet}^{*}$-mod, $M_{\bullet}$.
... |
I guarantee there is an easy reference on this, but for some reason I cannot find it. If you can point me to a reference or just write a short proof for me, I would be appreciative.
Given a graded ring $R_{\bullet}$ and a localization $R_{\bullet}^{*}$. We also have a graded $R_{\bullet}^{*}$-mod, $M_{\bullet}$.
... |
I guarantee there is an easy reference on this, but for some reason I cannot find it. If you can point me to a reference or just write a short proof for me, I would be appreciative.
Given a graded ring $R_{\bullet}$ and a localization $R_{\bullet}^{*}$. We also have a graded $R_{\bullet}$-mod, $M_{\bullet}$.
So ... |
Struggling with yet another proof:
> Prove that, for any positive integer n: (a + b)<sup>n</sup> >= a<sup>n</sup> + b<sup>n</sup> for all a, b > 0:
I wasted 3 pages of notebook paper on this problem, and I'm getting nowhere slowly. So I need some hints.
1. What technique would you use to prove this (e.g. induc... |
It might also be helpful for you to think a little about the geometry of the inequality.
For $n=2$, find a way to put an $a \times a$ square and a $b \times b$ square into a $(a+b) \times (a+b)$ square without any
overlaps. For $n=3$, see if you can fit an $a \times a \times a$ cube and a $b \times b \times b$ cub... |
While reading one aptitude book, i came across this question :-
`35% alcohol mixed with 60% alcohol to get a 50% alcohol. In what ratio were they mixed?`
Does this question look complete? I spent 20 minutes trying to figure out what i means but still didn't understand.
Thanks in advance :) |
Given `a` liters of `35%` alcohol solution, and `b` liters of `60%` alcohol solution, what would be the alcohol ratio after we'll mix this two solutions?
The `a` liters are actually `a*35/100` alcohol, and `a*(1-(35/100))` water. Likewise the `b` liters contains `b*60/100` alcohol and `b*(1-(60/100))` water.
Mixi... |
While reading one aptitude book, i came across this question:
`35% alcohol mixed with 60% alcohol to get a 50% alcohol. In what ratio were they mixed?`
I spent 20 minutes on this question, but I couldn't even figure out what it means! Could anyone explain this to me?
|
Some time ago [Qiaochu Yuan](http://math.stackexchange.com/users/232/qiaochu-yuan) asked about [counting subsets of a set whose number of elements is divisible by 3 (or 4)](http://math.stackexchange.com/questions/918/how-do-i-count-the-subsets-of-a-set-whose-number-of-elements-is-divisible-by-3-4/).
The story become... |
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 and select one random position in that array. If the number in the array is 1, nothing is changed and otherwise the pre-c... |
Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be $\zeta_1,\zeta_2,\ldots,\zeta_{10}$, fo... |
How to solve a cyclic quintic in radicals? |
So this is fermats last theorem upside down? It occurs to me if we have two binary numbers we may add them to get another power of two,
1000000
1000000
+ --------
10000000
but if we had two numbers in base 3, say
1000000
1000000
+ -------
2000000
we would ... |
I already asked if there are more rationals than integers here...
http://math.stackexchange.com/questions/1311/are-there-more-rational-numbers-than-integers
However, there is one particular argument that I didn't give before which I still find compelling...
Every integer is also a rational. There exist (many) ... |
Relative sizes of sets of integers and rationals revisited - how do I make sense of this? |
>Is there, for example, some alternative named "size" definition consistent with the partial ordering given by the is-a-proper-subset-of operator?
I think it is important to keep in mind the context you are looking at something when you want to talk about sizes. In a topological or geometric context, if $A \subset ... |
What is the probability that the convex hull of n random points on (n-1)-dimensional sphere contains sphere's center? |
Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be $\zeta^1,\zeta^2,\ldots,\zeta^{10}$, fo... |
What is the probability that the convex hull of n+2 random points on n-dimensional sphere contains sphere's center? |
Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic? |
I already asked if there are more rationals than integers here...
http://math.stackexchange.com/questions/1311/are-there-more-rational-numbers-than-integers
However, there is one particular argument that I didn't give before which I still find compelling...
Every integer is also a rational. There exist (many) ... |
Of course, there are other notions of size. In particular, your notion of "a partial order based on inclusion of sets" is a very fruitful concept which has been used frequently. As a quick example, there is a technique in mathematical logic/set theory called "forcing" which is used to show that certain mathematical s... |
Suppose that $X$ is a topological space with a sheaf of rings $\mathcal{O}_X$. In general, the stalk at a point $p \in X$ is the direct limit of the rings $\mathcal{O}_X(U)$ for all open sets $U$ containing $p$.
Here are two questions on computing stalks - I think both should be true, since a direct limit should be ... |
Computing stalks: do direct limits behave like limits? |
Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be $\zeta^1,\zeta^2,\ldots,\zeta^{10}$, fo... |
Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic?
_Update._ There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, maybe? |
This is what I know about why someone might care about fields of positive characteristic:
1. they are useful for number theory
2. in algebraic geometry, a theory of "geometry" can be developed over them, and it's fun to see how this geometry works out
Some people might read this and think, "What more could you n... |
Why should I care about fields of positive characteristic? |
I've been looking online (and in teaching journals) for a good introduction to Algebras of Random Variables (on an undergraduate level) and their usage, and have come up short. I know I can find the probability distribution of $h(z)$ where:
$z = x + y$
If $x$ and $y$ are from known independent probability distrib... |
Algebra of Random Variables? |
This is what I know about why someone might care about fields of positive characteristic:
1. they are useful for number theory
2. in algebraic geometry, a theory of "geometry" can be developed over them, and it's fun to see how this geometry works out
Some people might read this and think, "What more could you n... |
There is a reasonably geometric proof in Section 1.4 of Noam Elkies' <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.163.1487&rep=rep1&type=pdf">The Klein Quartic in number theory</a>, but I haven't absorbed it enough to give a good summary. |
There is a reasonably geometric proof in Section 1.4 of Noam Elkies' <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.163.1487&rep=rep1&type=pdf">The Klein Quartic in number theory</a>. A very rough summary is as follows. The unique simple group $G$ of order $168$ has a $3$-dimensional irreducible re... |
Let's suppose you care about finite groups, one way or another. Then you probably care about the classification of finite simple groups. The bulk of this classification is the groups of Lie type, which were discovered by finding analogues of Lie groups over finite fields.
Finite fields are also (as one might gue... |
There is a whole [paper][1] of Serre on this, which I haven't read; I will explain briefly one of my favorite applications.
One of the really cool applications of this is to the [Ax-Grothendieck theorem][2]. Which is a 100% analytic statement: if $P: \mathbb{C}^n \to \mathbb{C}^n$ is an injective polynomial map, ... |
Suppose $f \in C(\mathbb{R}^n)$, the space of continuous $\mathbb{R}$-valued functions on $\mathbb{R}^n$. Are there conditions on $f$ that guarantee it is the pullback of a polynomial under some homeomorphism? That is, when can I find $\phi:\mathbb{R}^n \to \mathbb{R}^n$ such that $f \circ \phi \in \mathbb{R}[x_1,\ld... |
Which continuous functions are polynomials? |
<a href="http://en.wikipedia.org/wiki/List_of_finite_simple_groups">This Wikipedia article</a> states that the isomorphism type of a finite simple group is determined by its order, except that:
* L<sub>4</sub>(2) and L<sub>3</sub>(4) both have order 20160
* O<sub>2n+1</sub>(q) and S<sub>2n</sub>(q) have the same or... |
Number of finite simple groups of given order is at most 2 - is a classification-free proof possible? |
In the special case that the sample space is the non-negative integers (or a subset thereof), one can think of a probability distribution as a generating function $f(x) = \sum_{n \ge 0} a_n x^n$ where $a_n \ge 0$ and $f(1) = 1$. Then the sum of random variables corresponds to the product of generating functions, so on... |
In the special case that the sample space is the non-negative integers (or a subset thereof), one can think of a probability distribution as a generating function $f(x) = \sum_{n \ge 0} a_n x^n$ where $a_n \ge 0$ and $f(1) = 1$. Then the sum of random variables corresponds to the product of generating functions, so on... |
In the special case that the sample space is the non-negative integers (or a subset thereof), one can think of a probability distribution as a generating function $f(x) = \sum_{n \ge 0} a_n x^n$ where $a_n \ge 0$ and $f(1) = 1$. Then the sum of random variables corresponds to the product of generating functions, so on... |
Since I can't leave comments I'm writing this here. I think this question is made difficult by the condition that $\phi$ is just required homeomorphism versus say a diffeomorphism.
In the case n = 1 you can certainly come up with continuous functions that are not differentiable on a discrete set but can be pulled... |
This is an exercise from _Problems from the Book_ by Andreescu and Dospinescu. When it was <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=290872">posted on AoPS</a> a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone here can enlighten me.
**Problem:** Pr... |
Although I am not (by any stretch) an expert on finite simple groups, let me flesh out my above comment.
Consider the following **QCFSG** (i.e., "qualitative" CFSG): with only finitely many exceptions, every finite simple group has prime order, is alternating, or is one of the finitely many known infinite families o... |
Although I am not (by any stretch) an expert on finite simple groups, let me flesh out my above comment.
Consider the following **QCFSG** (i.e., "qualitative" CFSG): with only finitely many exceptions, every finite simple group has prime order, is alternating, or is one of the finitely many known infinite families o... |
How to solve this equation?
x = 10<sup>x/10</sup> |
The point is that SHORTPROOF is an NP-complete problem: given a sentence of length $n$ in the language of some formal proof system (ZFC, Peano arithmetic, etc), does it have a proof of length at most some fixed polynomial in $n$, such as $(2n)^{100}$? It's in NP because for a reasonable formal system you can check a g... |
The identity
$\displaystyle (n+1) \text{lcm} \left( {n \choose 0}, {n \choose 1}, ... {n \choose n} \right) = \text{lcm}(1, 2, ... n+1)$
is probably not well-known. The only way I know how to prove it is by using <a href="http://planetmath.org/encyclopedia/KummersTheorem.html">Kummer's theorem</a> that the power... |
Is there a direct proof of this lcm identity? |
Many people are familiar with some properties of binary relations, such as reflexivity, symmetry and transitivity.
What are the commonly studied properties of ternary (3-ary) relations?
If you could provide a motivating example of why the property is interesting that would also be helpful. |
Interesting properties of ternary relations? |
Given `a` liters of `35%` alcohol solution, and `b` liters of `60%` alcohol solution, what would be the alcohol ratio after we'll mix this two solutions?
The `a` liters are actually `a*35/100` alcohol, and `a*(1-(35/100))` water. Likewise the `b` liters contains `b*60/100` alcohol and `b*(1-(60/100))` water.
Mixi... |
It will need something clever. For n + 1 = 6, you need 6 times the lcm of 1 5 and 10 in
order to get enough powers of 2 ( and 3 ). Is there a characterization of those n + 1 where the
entire factor of n + 1 is needed, and not the lcm, for the left hand side?
Gerhard "Ask Me About System Design" Paseman, 2010.08.... |
(Idea of a proof from V.Dotsenko's paper in http://www.mccme.ru/free-books/matprosd.html (in Russian))
Consider the set of 4-tuples of points on $\mathbb{P}^1(\mathbb{F}_7)$ with cross-ratio equal to 3. This set contains exactly 28 elements. Identifying 4-tuple with its complement (recall that there are exactly 8 po... |
The [PCP theorem][1] states that:
> Every decision problem in [NP][2] has
> [probabilistically checkable proofs][3] of
> constant [query complexity][4] and
> logarithmic randomness complexity.
Can anyone give an intuitive explanation of how this can be done?
**Links**
- This is a follow up from [this
... |
I was reading [this paper][1] on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $ith$ highest bi... |
I was reading [this paper][1] on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $ith$ highest bi... |
How about probability & statistics? Not exactly physics, but lots of applications which can be demonstrated with empirical data. Any example where "taking an average" seems reasonable is amenable to finding a distribution. Many examples: frequencies of arrival (traffic, say) as Poisson or negative binomial; arrival ... |
I was reading [this paper][1] on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $ith$ highest bi... |
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... |
I am not an expert on this topic, so forgive me if this is obvious. I see what you have done, but isn't this a rather "vacuous" category? Products and coproducts satisfy universal properties in that we can factor through them with respect to other morphisms. For example, if there are arrows from B and C each to A, t... |
Let $O(f(n))$ be the minimum time complexity for output a real sequence $\{a_i\}$.
The input of the algorithm is a integer $n$. It output the finite sequence $a_1, a_2, \ldots, a_n$. (Clearly $f(n)\geq n$.)
What can we say about the time complexity of $\max(a_1, a_2, \ldots, a_n)$? Is there always an algorithm to... |
Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is:
[Conceptual Mathematics][1]
An added benefit is that it is written by an expert!
[1]: http://www.amazon.com/Conceptual-Mathematics-First-Introduct... |
...and encoding it as a probability distribution.
Suppose we have a sequence of non-negative integers that is periodic with period N:
$A_{1},A_{2},...,A_{N},A_{1}...$
Each $A_{k}$ takes on a value no greater than some constant B:
$0 \leq A_{k} \leq B$
We then take this sequence and do a simple convolutio... |
Let *C* = {0, 1, 2, …, ℵ<sub>0</sub>, ℵ<sub>1</sub>, ℵ<sub>2</sub>, …}. What is |*C*|? Or is it even well-defined? |
I was reading [this paper][1] on position auctions for web ads. Basically, there are N slots each with an expected number of clicks (in a particular time period) $x$. Each agent makes a bid $B_i$ of how much they are willing to pay per click. The bids are put in decreasing order and agent who makes the $i^\text{th}$ hi... |
[This page][1] gives a few examples of Venn diagrams for 4 sets. Some examples:
![alt text][2]
![alt text][3]
Thinking about it for a little, it is impossible to partition the plane into the 16 segments required for a complete 4-set Venn diagram using only circles as we could do for <4 sets. Yet it is doable w... |
Why can a venn diagram for 4+ sets not be constructed using circles? |
[This page][1] gives a few examples of Venn diagrams for 4 sets. Some examples:
![alt text][2]
![alt text][3]
Thinking about it for a little, it is impossible to partition the plane into the 16 segments required for a complete 4-set Venn diagram using only circles as we could do for <4 sets. Yet it is doable w... |
I want to solve 0<1−an/(mb^2)exp{−r(T−t)}<1, where a, b, T, t>0. The solution is that either an<=mb^2$ or $mb^2\leq an\leq mb^2e^{rT}$ and $t< T − (ln(an) − ln(mb2 ))/r$.
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 first part. Th... |
I want to solve $0<1−an/(mb^2)exp^{−r(T−t)}<1$, where $a, b, T, t>0$. The solution is that either $an<=mb^2$ or $mb^2\leq an\leq mb^2e^{rT}$ and $t< T − (ln(an) − ln(mb2 ))/r$.
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 first pa... |
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... |
Is there a collective term for the XY, YZ and ZX planes in 3D co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that. |
Transitive action on the vertices is usually the definition of "looking the same from each vertex". So maybe you have two different groups in mind:
1. The automorphism group of the combinatorics of the tiling; the abstract structure of vertices, faces, edges, etc.
2. The group of Euclidean motions that leave th... |
Given `a` liters of `35%` alcohol solution, and `b` liters of `60%` alcohol solution, what would be the alcohol ratio after we'll mix this two solutions?
The `a` liters are actually `a*35/100` alcohol, and `a*(1-(35/100))` water. Likewise the `b` liters contains `b*60/100` alcohol and `b*(1-(60/100))` water.
Mixi... |
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