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I believe you want <a href="http://mathworld.wolfram.com/SharkovskysTheorem.html">Sharkovsky's Theorem</a>.
A reasonable explanation is in:
Kaplan, Harvey. "A cartoon-assisted proof of Sarkowskiĭ's theorem."
Amer. J. Phys. 55 (1987), no. 11, 1023–1032.
MR<a href="http://www.ams.org/mathscinet-getitem?mr=917121"... |
I came across this 'paradox' -
$$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i * 2\pi i}=e^{4\pi^2}$$
I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why doesn't it work with complex numbers even though it is valid in real case? Is it related to the fact that logarithm of co... |
I came across this 'paradox' -
$$1=e^{2\pi i}\Rightarrow 1=(e^{2\pi i})^{2\pi i}=e^{2\pi i \cdot 2\pi i}=e^{-4\pi^2}$$
I realized the fallacy lies in the fact that in general $(x^y)^z\ne x^{yz}$. Why doesn't it work with complex numbers even though it is valid in real case? Is it related to the fact that logarithm ... |
Representation theory plays a big role in the group-theoretic approach to special functions.
For example, Willard Miller showed that the powerful Infeld-Hill factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This lie-theoretic approach... |
See the Monthly paper of that exact title [Period three implies chaos][1] - which made the JSTOR [All-Stars][3] list as the third most frequently accessed Monthly article. See also the this [paper][2] for some history on Sharkovsky's theorem.
[1]:http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf
[2]:http://links.j... |
See the Monthly paper of that exact title [Period three implies chaos][1] - which made the JSTOR [All-Stars][3] list as the third most frequently accessed Monthly article. See also [A simple proof of Sharkovsky's theorem][4] and this [paper][2] for some history on Sharkovsky's theorem.
[1]:http://pb.math.univ.gda.pl... |
This is a problem from "[Introduction to Mathematics - Algebra and Number Systems][1]" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.
The problem reads as follows:
>
If 3p<sup>2</sup> = q<s... |
This is a problem from "[Introduction to Mathematics - Algebra and Number Systems][1]" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would still appreciate hints rather than a complete answer.
The problem reads as follows:
>
If 3p<sup>2</sup> = q<s... |
I found [Chaos: An Introduction to Dynamical Systems][1] by Kathleen T. Alligood, Tim D. Sauer, and James A. Yorke, to be a sensible text. A small section of the text contains a brief explanation of Sharkovsky's Theorem.
In short, Sharkovsky uses this unique ordering of the natural numbers in such a way as to ens... |
Regarding [this thread][1],
it is not possible to combine complex powers in the usual way:
$$ (x^y)^z = x^{yz} $$
There was mention of multi-valued functions, is there some theory that makes this all work out correctly?
[1]: http://math.stackexchange.com/questions/2903/why-cant-i-combine-complex-powers |
How to combine complex powers? |
For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the [diagrammatic tensor notation](http://en.wikipedia.org/wiki/Penrose_graphical_notation).
<img src="https://i.stack.imgur.com/AUdLo.jpg" />
Here $\det$ denotes the determinant.
W... |
There are at least 2 places in Wikipedia saying that X_n converges to X in mean **in and only if** X_n converges to X **in probability** and X_n is uniformly integrable. See the following link for example:
<http://en.wikipedia.org/wiki/Convergence_of_random_variables#Properties_4>
However, I found a reference in Bi... |
Do convergence in distribution along with uniform integrability imply convergence in mean? |
The Gelfand transformation on $\ell^1(\mathbb Z)$ is not isometric. Do you have an example? |
There are at least 2 places in Wikipedia saying that X_n converges to X in mean **in and only if** X_n converges to X **in probability** and X_n is uniformly integrable. See the following link for example:
<http://en.wikipedia.org/wiki/Convergence_of_random_variables#Properties_4>
However, I found a reference in Bi... |
I presume that you are restricting $c$ to be real, so your question is:
>Which complex numbers $z$ can be roots of equations
$$4z^4+8x^3 -3z^2 -9z+c=0$$
as $c$ varies through the reals?
This is the same question as
>For which complex numbers is
$$g(z)=4z^4+8x^3 -3z^2 -9z$$
real?
Certainly if $z$ is real... |
The original question asked about a *theory* that would make this all work out correctly. The previous answers are all correct, but they did not point out the underlying, unifying factor which makes all these computations much simpler: the <a href="http://www.google.ca/url?sa=t&source=web&cd=3&ved=0CCMQFjAC&url=http%3... |
The original question asked about a *theory* that would make this all work out correctly. The previous answers are all correct, but they did not point out the underlying, unifying factor which makes all these computations much simpler: the <a href="http://www.google.ca/url?sa=t&source=web&cd=3&ved=0CCMQFjAC&url=http%3... |
A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A fan is a finite set $F$ of polyhedral cones, all living in the same vector space, such that
(1) if $\sigma$ is a cone in $F$, and $\t... |
I can understand how the greek alphabet came to be prominent in mathematics as the greeks had a huge influence in the math of today. Certain letters came to have certain implications about their meaning (i.e. theta is almost always an angle, never a function).
But why did x and y become to prominent? They seem like ... |
Why are x and y such common variables in todays equations? How did their use originate? |
In a proposal, assume p is the condition and q the conclusion, so the proposal is p->q.
Let ┐ be the negation on either condition or conclusion.
what is the name for the operation from p->q to ┐p->┐q?
And what is the name for the operation from p->q to p->┐q? And from p->q to ┐q->┐p?
What topics in logic cover ... |
In a proposal, assume $p$ is the condition and $q$ the conclusion, so the proposal is $p \to q$.
Let $\neg$ be the negation on either condition or conclusion.
what is the name for the operation from $p\to q$ to $\neg p\to\neg q$?
And what is the name for the operation from $p \to q$ to $p\to\neg q$? And from $p\... |
In a proposal, assume $p$ is the condition and $q$ the conclusion, so the proposal is $p \to q$.
Let $\neg$ be the negation on either condition or conclusion.
what is the name for the operation from $p\to q$ to $\neg p\to\neg q$?
And what is the name for the operation from $p \to q$ to $p\to\neg q$? And from $p\... |
Binomial coefficients: how to prove an inequality on the p-adic valuation? |
Exercise 3B.4 on <a href="http://books.google.com/books?id=pCLhYaMUg8IC&pg=PA84&lpg=PA84">page 84</a> of Isaacs's Finite Group Theory asks one to prove a precursor of the Sylow theorems for π-subgroups, and I only know how to do it using the Sylow theorems, so I have missed the point. I actually needed a similar (fals... |
Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.
A differential form then transforms essentially by multiplication with $\mathrm{det}(Df)$, while integrals transform essentially by multiplication of the integ... |
Why do differential forms and integrands have different transformation behaviours under diffeomorphisms? |
I can understand how the Greek alphabet came to be prominent in mathematics as the Greeks had a huge influence in the math of today. Certain letters came to have certain implications about their meaning (i.e. θ is almost always an angle, never a function).
But why did x and y come to prominence? They seem like 2 arb... |
Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.
A differential form then transforms essentially by multiplication with $\mathrm{det}(Df)$, while integrals transform essentially by multiplication of the integ... |
I am reading p.22 of Dan Bump's Automorphihic forms and representations. A cusp of the congruence group acting on the upper half plane is defined to be an orbit of the action of the congruence subgroup on $\mathbb{Q} \mathbb{P}^1$. It also says that intuitively, the cusps are the places where a fundamental domain of th... |
The Wikipedia statement is right, and the statement which you attribute to Billingsley is wrong, in some sense inherently so. Convergence in distribution is a notion of convergence of *measures*. A sequence of random variables can converge in distribution even if they live on totally different probability spaces, and ... |
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was the [Banach Tarski Paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox).
So I ask you which are your most su... |
Which one result in maths has surprised you the most? |
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the [Banach Tarski Paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). It states that you ca... |
When is the weighted $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)? |
I would not rate this example as surprising, but it did provoke in me a little epiphany when I finally understood it. There is a theorem of category theory that characterizes adjunctions as a pair of functors and a pair of natural transformations satisfying a bunch of equations. Now in some sense, this is a pure formal... |
When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)? |
Zero is finite. If n is finite, then n+1 is finite. Hence, by induction, all numbers are finite. What is the catch? |
Each of n people draws a card from a shuffled deck of k cards numbered from 1 to k. There are many more cards than people, say 2^n ≤ k. If everyone is honest, how can they mutually determine who is the holder of the largest card, without anyone learning additional information about the values of any cards held by oth... |
I have this wooden puzzle composed of 25 y-shaped [pentominoes][1], where the objective is to assemble them into a 5x5x5 cube. After spending quite a few hours unsuccessfully trying to solve the puzzle, I finally gave up and wrote a program to try all possible combinations using backtracking. Analyzing the results reve... |
The other solutions posted already do a good job of explaining tensors as dyads, linear transformations, or using the ellipsoid interpretation. I thought I'd give an engineering perspective.
In engineering applications, physical quantities (mass, velocity, force, etc.) are either "scalars", "vectors", or "tensors". ... |
Each of n people draws a card from a shuffled deck of k cards numbered from 1 to k. There are many more cards than people. If everyone is honest, how can they mutually determine who is the holder of the largest card, without anyone learning additional information about the values of any cards held by others? |
Let the *class number* $h(d)$ denote the number of distinct binary quadratic forms with discriminant $d < 0$.
Is there a better algorithm for $h$ than brute force?
To be precise, by brute force I meant to generate enough forms to completely cover the space and then reducing them down to see how many equivalence c... |
Each of n people draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as people. If everyone is honest, how can they mutually determine who is the holder of the largest card, without anyone learning additional information about the values of any cards held by others? |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. If everyone is honest, how can they mutually determine who is the holder of the largest card, without anyone learning additional information about the values of any cards held by others? |
Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.
A differential form then transforms essentially by multiplication with $\mathrm{det}(Df)$, while integrals transform essentially by multiplication of the integ... |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. If everyone is honest, how can they mutually determine who is the holder of the largest card, without anyone learning additional (beyond that which can now be inferred) information about t... |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. If everyone is honest, how can they mutually determine who is the holder of the largest card, without any person learning additional information (beyond that which can now be inferred) abo... |
I'm working on a game which relies on "flinging" an object. That is, click and hold on the object, and then drag and release it, and it continues on the path you were dragging it. Of course, the most well-known example of flinging is with iPhone and Android devices, where you can quickly scroll down a list by quickly s... |
Math behind a "fling"? (i.e. on a mobile touch device) |
The elementary real-valued functions are not closed under integration. (Elementary function has a precise definition -- see Risch algorithm in Wikipedia). This means there are elementary functions whose integrals are not elementary. So we can construct a larger class of functions by adjoining all the integrals of e... |
Why not compute a cubic spline from the 5 most recent samples equidistant in time? This will give you a smooth curve and you can use the parametric derivative of the spline as a velocity. http://mathworld.wolfram.com/CubicSpline.html |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. If everyone is honest, how can they mutually determine who is the holder of the largest card, without any person learning additional information (beyond that which can necessarily be infer... |
The other solutions posted already do a good job of explaining tensors as dyads, linear transformations, or using the ellipsoid interpretation. I thought I'd give an engineering perspective.
In engineering applications, physical quantities (mass, velocity, force, etc.) are either "scalars", "vectors", or "tensors". ... |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. The winner is the person who is holding the largest card. If everyone is honest, how can they mutually determine the identity of the winner, without any person learning additional informa... |
Each of n persons draws a card from a shuffled deck of k cards numbered from 1 to k. There are at least as many cards as persons. The winner is the person who is holding the largest card. If everyone is honest, how can they mutually determine the identity of the winner, without any person learning additional informa... |
I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to *you*, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than ... |
I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to *you*, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than ... |
I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to *you*, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than ... |
I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to *you*, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than ... |
I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to *you*, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than ... |
The first few chapters of Goldblatt's [Topoi: the categorial analysis of logic][1] provide another fairly elementary introduction to the basics of category theory.
[1]: http://books.google.com/books?id=VUrePAAACAAJ |
Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.
A differential $n$-form (or rather its coefficient function which is obtained by using the canonical one-chart atlas on $\mathbb R^n$) then transforms essentia... |
The [Invitation to algebraic geometry][1] by Smith et al. is also very readable.
[1]: http://books.google.com/books?id=8zQYxx7PjqsC |
In my course of linear algebra I studied that if V is a finite dimensional vector space on the complex field, then every endomorphism of V has an eigenvector. The proof is simple: taken a polynomial f that is null on the endomorphism A (it exists because of the finite dimension of V), exploiting the algebraic closure o... |
Is there something deep in the fact that an endomorphism of a finite dimensional complex vector space has an eigenvector? |
Could you explain why $de^x/dx = e^x$ "intuitively"? |
In my first attempts to understand the axiom of coice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty e... |
In my first attempts to understand the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty ... |
Well, think of [exponential growth][1] (like e.g. bacteria grow):
We know, the more bacteria exist in a colony, the faster the colony will grow. More precisely: The growth speed of the colony $B$ is proportional to it's size ... Double size, double speed.
$\frac{dB}{dt} \sim B$
Furthermore we know the growth i... |
Well, think of [exponential growth][1] (like e.g. bacteria grow):
We know, the more bacteria exist in a colony, the faster the colony will grow. More precisely: The growth speed of the colony $B$ is proportional to it's size ... Double size, double speed.
$\frac{dB}{dt} \sim B$
Furthermore we know the growth i... |
I'll act on Casebash's proposal later, but for now... care for a movie?
![tangent line][1]
(granted, I cheated and spaced the two plots of $\exp(x)$ to clearly show the moving tangent line... but you guys should be able to understand this).
[1]: https://i.stack.imgur.com/nLLL7.gif |
There are two components to the original question:
* Why do exponential functions generally (the class $x \mapsto ab^x$ for $b>0$) have the property of being eigenfunctions of the derivative operator? That is: why do these functions f have the property $\frac{\mathrm d}{\mathrm dx} f(x) = \lambda f(x)$?
* Why are... |
Well, think of [exponential growth][1] (like e.g. bacteria grow):
We know, the more bacteria exist in a colony, the faster the colony will grow. More precisely: The growth speed of the colony $B$ is proportional to it's size ... Double size, double speed.
$\frac{dB}{dt} \sim B$
Furthermore we know the growth i... |
Suppose $\frac{d}{dx}f(x)=f(x)$.
Then for small $h$, $f(x+h)=f(x)+hf(x)=f(x)(1+h)$. If we do this for a lot of small intervals of length $h$, we see $f(x+a)=(1+h)^{a/h}f(x)$. (Does this ring a bell already?)
Setting $x=0$ in the above, and fixing $f(0)=1$, we then have $f(1)=(1+h)^{1/h}$, which in limit as $h\rig... |
I'll act on Casebash's proposal later, but for now... care for a movie?
![tangent line][1]
(granted, I cheated and spaced the two plots of $\exp(x)$ to clearly show the moving tangent line... but you guys should be able to understand this).
(thanks to Stan Wagon!)
[1]: https://i.stack.imgur.com/nLLL7.gif |
In my first attempts to understand the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty ... |
First of all, you might be interested in the work of Chas and Phillips: "Self-intersection of curves on the punctured torus". I've only skimmed their paper, but they seem to be doing something closely related to what you want.
Second I want to guess, for some reason, that the average time to compute self-intersecti... |
I am interested in the Gelfand transformation
$$
\Phi\colon\ell^1(\mathbb Z)\to\mathcal C(\mathbb T),\quad a\mapsto\sum_{n\in\mathbb Z}a_n z^n.
$$
This is an injective homomorphism of Banach algebras. It is neither isometric nor surjective. However, its image---the Wiener algebra $W$ consisting of all continuous fu... |
Does the Gelfand transformation on $\ell^1(\mathbb Z)$ possess a continuous inverse on its image? |
Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$,
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example,
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$ ... |
In my first attempts to understand the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard you try, it was impossible to find one. (If you try the same for the nonempty subsets of $\mathbb N$, it is pretty ... |
I meant this as a comment to the discussion after Student's answer but it seems that I don't have the option (reputation too low?) so I'll post it as an answer. Sorry.
In order to compute quickly the inverse of 25 mod 109, note that $25=5^2$. Thus $25^{-1}=t^2$ where $t=5^{-1}$ mod 109. On the other hand, computing ... |
How to compute curvature tensors for general n-dimensions? |
I'm not very fluent in mathematical proofs. High School has, sadly, not taught me any kind of proof-theory. That's why I would like your help with my proof of
$$m = S_m (mod 3).$$
I'm going to use the very little that I know: Mathematical induction.
**Basis:**
$$A(1): S_m = 1 \wedge m = 1$$
$$m = S_m ... |
Prove that $m = S_m (mod 3)$? |
I'm not very fluent in mathematical proofs. High School has, sadly, not taught me any kind of proof-theory. That's why I would like your help with my proof of
$$m = S_m (mod 3).$$
where $S_m$ is the digit sum of m and $\wedge$ is a logical-AND.
I'm going to use the very little that I know: Mathematical indu... |
Groups can be defined abstractly as sets with a binary operation satisfying certain identities, or concretely as a collection of permutations of a set. Cayley's theorem ensures that these two definitions are equivalent: any abstract group acts as a collection of permutations of its underlying set, and this action is f... |
"Cayley's theorem" for Lie algebras? |
One can see [this reference][1] for TBL space-times.
I would like to know how the explicit expression for the function called $G$ in equations $3.108,3.108,3.110$ in the above reference is obtained.
Also it would be nice to see some further references about TBL space-times.
[1]: http://books.google.com/boo... |
Suppose we define a topology on the real numbers as that obtained by defining open sets to be those such that the complement is finite. How does one show that this topology cannot be obtained from a distance function?
Thanks in advance.
p.s: This is not a homework question. I am trying to self study point set t... |
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the [Banach Tarski Paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox). It states that you ca... |
(Looks like Niel and "J" beat me to the punch while I was composing this, but I'll post my answer anyway since I like my diagram. :) )
You can't get too much more intuitive than the Law of Exponents ($a^x a^c = a^{x+c}$) right?
To reduce symbolic (and mental) clutter for a while, let's write $k$ for $a^c$, to ge... |
What is the proper definition of a Countable Set? |
I'm quite new at measure theory and all I know I read by myself, so bear with me if this question is too trivial.
Given the definition of asymptotic, can we say that for $x \rightarrow \infty$ two almost everywhere equal functions are asymptotic? I really think that this is not the case, but I haven't been able to ... |
Are almost everywhere equal functions asymptotic? |
Is the set of continous functions from R->R path connected? |
In a differential Galois field, integration is the antiderivative operation and not the infinite summation, but I will still call it integration.
If we start out with simple elementary functions, say rational functions with integer coefficients, then there will be many functions that do not have integrals. If the ... |
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