User rodrigo a. pérez - MathOverflow

Answer by Rodrigo A. Pérez for What are some applications of other fields to mathematics?

2013-05-21T18:53:10Z

There have been many theoretical advances in diffusion, traveling waves, chaos, and pattern formation derived from the Belousov–Zhabotinsky reaction.

Answer by Rodrigo A. Pérez for What are some applications of other fields to mathematics?

2013-05-21T18:49:44Z

The Lotka-Volterra predator-prey equations are a fundamental example in the qualitative theory of ODEs. Volterra originally used it to explain the large increase in the mediterranean shark population during WWI.

Answer by Rodrigo A. Pérez for objects which can't be defined without making choices but which end up independent of the choice

2013-05-21T03:51:40Z

Daniel Moskovich's notion of "support point" to anchor definitions on manifolds is interesting. I suggest a radically non-geometric example: MATROIDS.

A finite matroid is a finite set with a family of subsets satisfying a list of properties. There are different lists of properties. The subsets could be independent sets, bases, circuits, flats, etc.

Given a matroid defined by independent sets (say) there is a canonical way to find a family of subsets that form a basis (say). So the cool thing about this example is that to define a matroid you need a choice of definition.

Proof of the weak Goldbach Conjecture

2013-05-14T05:19:30Z

What are the main ideas of Harald Helfgott's proof that all odd $n \geq 5$ is the sum of 3 primes?

A function whose fixed points are the primes

2012-11-20T16:26:20Z

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points of $f$ are the primes.

Is every number preperiodic? In other words, is $f(f(\ldots(f(n)\ldots))$ eventually prime?

Answer by Rodrigo A. Pérez for Basic results with three or more hypotheses

2013-05-15T04:21:31Z

Whyburn's Theorem: Let $S$ be a planar set that is compact, connected, locally connected, nowhere dense, and such that any two components of the complement are bounded by disjoint simple closed curves. Then $S$ is homeomorphic to the Sierpinski carpet.

Answer by Rodrigo A. Pérez for The prime number $2$

2013-05-03T11:53:41Z

Because $p-1$ (an expression that appears often dealing with primes) equals 1 iff $p=2$.

Asymptotic series

2013-05-02T03:30:07Z

I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not true in general, but in this case, texts that predate Oh notation tend not to be clear).

Can you explain the idea behind asymptotic series, give an illuminating example, and/or suggest a good modern exposition of the theory?

History of the high-dimensional volume paradox

2013-04-26T01:53:47Z

Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of the ball goes to 0.

I first learnt of this in relation to Gromov. In the story I heard, he used to ask incoming students to compute the distance $(\sqrt{n}-1)/2$ from a hypercube corner to the ball, and observe them to see if they realized that the volume of the hypercube is concentrated in its corners.

Is this story correct? And is this the origin of this FFF? I could imagine a situation where several people noticed this at different times, but where the fact did not become "viral" until much more recenttly.

Answer by Rodrigo A. Pérez for 2D Problems Which are Easier to Solve in 3D

2013-04-27T04:18:20Z

Given two disjoint disks of different radii, find the intersection of their common external tangents. For lack of a better name, call this the h-center of the pair (h- for homothety?).

Problem: Given three mutually disjoint disks, the h-centers of the three pairs are colinear.

The nicest solution involves adding one dimension and inflating the disks to balls (with centers in the original plane $\Pi$). The pairs of tangents become full-fledged cones with vertices in $\Pi$, and the proof is obtained by studying a plane tangent to all three balls. It is tangent to all three cones, so it contains their three vertices, but it also intersects $\Pi$ on a straight line :)

Fixed point theorems

2013-04-10T05:36:59Z

It is surprising that fixed point theorems (FPTs) appear in so many different contexts throughout Mathematics: Applying Kakutani's FPT earned Nash a Nobel prize; I am aware of some uses in logic; and of course everyone should know Picard's Theorem in ODEs. There are also results about local and global structure OF the fixed points themselves, and quite some famous conjectures (also labeled FPT for the purpose of this question).

Many results are so far removed from my field that I am sure there are plenty of FPTs out there that I have never encountered. I know of several, and will post later if you do not beat me to them :)

Community wiki rules apply. One FPT per answer, preferably with an inspiring list of interesting applications.

Answer by Rodrigo A. Pérez for Great mathematics books by pre-modern authors

2013-04-19T03:30:27Z

F. Klein's "Development of mathematics in the 19th century". It is a history book; it is a Math book; it is a great read.

Products of matrices of a certain form

2013-04-18T15:37:34Z

Are $n \times n$ matrices of the form $$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$ studied anywhere? I am interested in the structure of the matrix obtained by multiplying a bunch of these together.

Answer by Rodrigo A. Pérez for Fixed point theorems

2013-04-10T18:19:08Z

Kakutani's FPT: Let $S$ be a non-empty, compact, convex subset of $\mathbb{R}^n$, and $\varphi:S \longrightarrow 2^S$ a set-valued function with a closed graph and the property that $\varphi(x)$ is non-empty and convex for all $x \in S$. Then $\varphi$ has a fixed point.

Application: Consider a game with finitely many players and finitely many strategies. If players are allowed to choose mixed strategies, there is always a Nash equilibrium; that is, a set of strategy choices for all players such that no player can do better by unilaterally switching to a different strategy. This is the theorem that resulted in J. Nash getting the 1994 Nobel Prize in Economics.

Answer by Rodrigo A. Pérez for Fixed point theorems

2013-04-10T05:50:38Z

Arnold's Conjecture: A Hamiltonian map on a compact symplectic manifold $(M,\omega)$ has at least as many fixed points as a function on $M$ has critical points.

Answer by Rodrigo A. Pérez for Fixed point theorems

2013-04-10T05:37:57Z

Brouwer's FPT: Every continuous function from a closed ball in $\mathbb{R}^n$ to itself has a FP.

For applications see this question.

Answer by Rodrigo A. Pérez for Fixed point theorems

2013-04-10T05:39:00Z

Banach's FPT (or contraction FPT): Every contraction in a complete metric space has a unique FP.

Application: If $f(t,y(t))$ is a real-valued function, Lipschitz continuous in $ y$ and continuous in $t$, then the initial value problem $$y'(t) = f(t,y(t)),\quad y(t_0)=y_0,\quad t \in [t_0-\varepsilon,t_0+\varepsilon]$$ has a unique solution.

Answer by Rodrigo A. Pérez for Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

2013-04-09T03:54:50Z

I just taught Leau's Flower Theorem today...

Your best bet to understand this map is to read Milnor's book. Click here for a copy of the original notes (which are less polished than the book). The chapter on parabolic points is perhaps the most pedestrian, but even that is clearer than other books!

To give you an idea of the flow of ideas, Milnor considers an analytic function $f$ with a parabolic fixed point at 0. He first assumes the multiplier is $\lambda = 1$, and proves the attraction/repulsion picture directly from the power series (call this result T1). Some corollaries follow, and then he describes the case $\lambda = {\rm e}^{2\pi{\rm i}p/q}$, which is the same as before, except that the petals are permuted by $f$ instead of determining independent basins of attraction. THIS IS YOUR MAP. The multiplier is $\lambda = {\rm e}^{2\pi{\rm i}/5}$, so the petals map counterclockwise onto each other.

The fifth iterate of your $f$ has a fixed point with multiplier 1 so each petal maps into itself. This is the situation where the Abel function makes sense. Milnor shows how to construct explicitly this coordinate change by refining the computations from the proof of T1. THIS IS THE ANSWER TO YOUR QUESTION.

Answer by Rodrigo A. Pérez for Elementary cases of Mihailescu theorem

2013-04-08T16:09:59Z

Even though you are talking about long-known results, a note with an interesting twist or new insight is worth publishing. If your focus is pedagogic you may want to consider a venue like the AMS College Math Journal.

Answer by Rodrigo A. Pérez for Uppercase Point Labels in High-School Diagrams: from Euclid?

2013-04-05T19:02:47Z

From Wikipedia: "In classical Greek, as in classical Latin, only upper-case letters existed. The lower-case Greek letters were developed much later by medieval scribes to permit a faster, more convenient cursive writing style with the use of ink and quill."

On the other hand (Wikipedia again) "the oldest surviving Latin translation of the Elements is a 12th century work by Adelard, which translates to Latin from the Arabic."

In other words, there is no clear connection between Euclid using uppercase (the only script he knew), and us using it too (what symbols did the old arab scholars use?). There is no point either in looking at "old" books like Coxeter's. The convention is surely older!

I will hazard the guess that the convention is quite old, say 1700-1800, and that it started with some random edition of Euclid that became slightly more popular than others. Which one, I do not know, but it is improbable that it is the first English translation by Billingsley (very famous, with 3D pop-out models of solids, but English was not the most influential language at the time), or Oliver Byrne's [color coded edition](http://en.wikipedia.org/wiki/Oliver_Byrne_(mathematician)#Byrne.27s_Euclid) (which is beautiful, but became well known only recently, AND does not use labels :). See also this page...

BTW, in Spanish points are also represented with capital letters.

Answer by Rodrigo A. Pérez for functions of one complex variable: geometric theory

2013-04-05T03:35:06Z

I said this before, but the best book ever (in any subject) is Ahlfors'. The chapter on the Weierstrass function (e.g.) is impeccable.

I have always claimed that the best way to study Complex Analysis correctly is to read Ahlfors for the theory, and Marsden for the worked examples and long list of computational exercises.

BTW, you will also find much interesting material in "Visual complex analysis" by T. Needham. It is full of geometric insights, although on the elementary side (and it takes loooong to get to the harder material).

Answer by Rodrigo A. Pérez for Is there a "mathematical" definition of "simplify"?

2013-04-04T23:43:50Z

What is simpler, $1+\tan^2\theta$ or $\sec^2\theta$? I prefer to put trig expressions exclusively in terms of $\sin$, $\cos$, and $\tan$, but the second expression is shorter.

What is simpler, $$x^6-20x^5+148x^4-518x^3+907x^2-758x+240$$ or

$$(x-1)(x-1)(x-2)(x-3)(x-5)(x-8)?$$

You may want your polynomials expressed in terms of the standard basis $1,x,x^2,\ldots$, or factored into linear terms; or then again, expressed in Bernstein form.

The solution in each case depends on what is required from the expression, so the answer to your question is that "simplify" is not well defined. What happens in practice is that instructors in remedial math courses teach some simplification rules so the students obtain a canonical answer that can be compared to the answer at the back of the textbook... And what dumb rules they are sometimes! I particularly mind that students learn to write $\sin(\pi/4) = \frac{\sqrt{2}}{2}$ instead of the "simpler" $\frac{1}{\sqrt{2}}$ (apparently some authors think that the students will be scared if there is a radical in the denominator). As a result the students learn not to think by themselves.

But all of this is moot... the true answer is that "simplifying" an expression may not be practical if, for instance, your expression is a word representing an element of a group whose word problem is not solvable :)

Answer by Rodrigo A. Pérez for Is there a deep reason for the fecundity of involutions?

2013-04-04T23:58:37Z

It all boils down to the fact that $\mathbb{R}$ has two ends!

In all (most) mathematical processes there is some notion of direction: counting, moving (along a curve), mapping from one space into another, reading a formula from left to right... the geometry of $\mathbb{R}$ is present always in one way or other, and the flip of the negative and positive ends usually induces some sort of involution.

Polyhedra Classification

2013-03-27T13:27:13Z

The following is inspired by this question. From time to time I search the web for tables of polyhedra, but without much success. Part of the problem is that there are many non-equivalent questions that can be asked. For example:

If $G$ is a planar graph where every edge is in a cycle, what extra conditions are needed so that $G$ realizable as a polyhedron?
How many polyhedra with $e$ edges are there?
If $G$ is a planar graph where every edge is in a cycle, what extra conditions are needed so that $G$ realizable as a polyhedron with regular faces?
How many polyhedra with $e$ edges and regular faces are there?
What are the obstructions to realizability?
etc$\ldots$

Meta-question: Where are questions like these addressed?

Answer by Rodrigo A. Pérez for How to triangulate a math reference?

2013-03-13T01:41:10Z

This risks being a useless answer, but the correct method to find references is (drum roll)...

...asking someone who knows more; perhaps by email. Even in the age of MathSciNet, Google, and MO, networking with experts is the way to go. There is someone out there who knows, or knows someone who knows, or gives you a hint to an obscure reference that may or may not have the answer. Plus you get to learn other (un)related math bits and you get to network with very knowledgeable people.

Addendum (in reply to David's comment): My point is that you cannot encode an automated database of mathematical theorems because you do not know how the search query will look like. What happens more often to me is that I find a structure in the setting I study, and notice that some property must hold. When I ask, the answer is something like "That looks like Laramie's quintionic permafrost algebras, but not quite. Your formula is equivalent to Zygyljnski's Platypus Lemma, but the indices are different."

A computer system smart enough to notice that what I describe is related to permafrost algebras would be smart enough to prove theorems of its own. I do not see that coming in the near future.

Answer by Rodrigo A. Pérez for Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems

2013-03-10T06:06:50Z

Densely scattered around the boundary of the Mandelbrot set $M$ you can find a miriad of tiny "Babybrots". We know that they are quasi-conformally isomorphic copies of $M$; that is to say, they are deformed, but not to wildly. Still, it is VERY surprising that they are visually recognizable. All the gray regions below seem to be bounded by circles and cardioids:

Surprise, surprise! The largest interior component of $M$ is indeed bounded by a cardioid, and the large disk to its left is truly round. But all other components are bounded by real algebraic curves of high degree, and are no longer true cardioids and circles:

The paper A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set by D. Giarrusso and Y. Fisher studies these boundaries as algebraic curves, and shows how to construct explicit parametrizations in the cases of period 1 (The Cardioid $C$), period 2 (The Disk), and period 3 (the two largest "disks" above and below $C$, plus the tiny "cardioid" visible in the middle of the left antenna).

Answer by Rodrigo A. Pérez for How do you prove that every curve of constant width is convex?

2013-03-09T03:47:06Z

Say that the compact set $K \subset \mathbb{R}^2$ of constant width $d$ is not convex; then there is some support line that touches two points, say $A$ and $B$ on $\partial K$. The parallel line that supports $K$ "from the other side" touches at least one point $C \in \partial K$. The distance between these two lines is the constant width $d$.

But either the distance $\overline{AC}$ or $\overline{BC}$ (or both) is strictly larger than $d$, so the width in the direction orthogonal to that line is larger than $d$ and $K$ has not constant width.

Answer by Rodrigo A. Pérez for Modern developments in finite-dimensional linear algebra

2013-02-28T03:39:27Z

From Wikipedia: "The linear-programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems."

Answer by Rodrigo A. Pérez for Untangling entwined rigid chains in 3-space

2013-02-24T15:46:50Z

You can twist one wire into a linked double noose (with free endtips):

If, in turn, you link two of these "loops", the whole thing cannot be unentangled.

Answer by Rodrigo A. Pérez for Undecidability and holomorphic functions (Reference request)

2013-02-20T14:54:07Z

The paper Constructing Non-Computable Julia Sets by Mark Braverman and Michael Yampolsky gives examples of quadratic polynomials with non-computable Julia sets. In particular, the question of deciding whether some point belongs to the Julia set is intractable.

Comment by Rodrigo A. Pérez

2013-05-19T13:53:47Z

I went over the list and oversaw his answer. I am deleting. Thanks.

Comment by Rodrigo A. Pérez

2013-05-15T19:22:54Z

@jjcale: Thanks!

Comment by Rodrigo A. Pérez

2013-05-14T16:47:10Z

Thank you!

Comment by Rodrigo A. Pérez

2013-05-14T03:40:45Z

+1 for the link. It's a great book!

Comment by Rodrigo A. Pérez

2013-05-06T02:08:46Z

@jjcale: Thanks! Do you have that data available? I only went up to 100,000 many years ago.

Comment by Rodrigo A. Pérez

2013-05-02T12:10:02Z

Basically what I was hoping for... Thanks!

Comment by Rodrigo A. Pérez

2013-04-26T17:26:12Z

That is a beautiful article, and it still leaves so many interesting questions open. Thanks!

Comment by Rodrigo A. Pérez

2013-04-23T13:46:37Z

Boy, this is the best answer I have seen in MO. I have learned a lot from other answers, but I was surprised to read more and more detail in only 7 lines; +2! (I wish). I would add that the two notations look natural enough that the reader will probably not notice if they are both used.

Comment by Rodrigo A. Pérez

2013-04-19T17:05:03Z

"Wenn die Zahl a in der Differenz der Zahlen b, c aufgeht, so werden b und c nach a congruent, im andern Falle incongruent genannt. Die Zahl a nennen wir den Modul. Jede der beiden Zahlen b, c heißt im ersteren Falle Rest, im letzteren aber Nichtrest der anderen." Although, as Chandan points out, the original WAS in latin :)

Comment by Rodrigo A. Pérez

2013-04-19T12:11:24Z

When I was an undergraduate in Mexico, an important rite of passage was attending the geometry and number theory courses of A. Barajas. He was a legend as one of the founders of Mathematics in Mexico, having worked with Einstein, and organizing the famous 1956 International Symposium on Algebraic Topology. He always started the number theory class by citing the first chapter of Disquisitiones from memory: "Wenn die zahl $a$..."

Comment by Rodrigo A. Pérez

2013-04-19T02:09:15Z

@Erik: It's the empty checkmark below the up/down vote buttons next to the answer...

Comment by Rodrigo A. Pérez

2013-04-18T22:11:00Z

Thank you Martin and Federico. $1/(1-x)$ resonates well with my problem!

Comment by Rodrigo A. Pérez

2013-04-10T14:14:33Z

@R salimi: Can you explain your notation for readers from different areas of Mathematics?

Comment by Rodrigo A. Pérez

2013-04-10T13:19:15Z

Also: Journal of Fixed Point Theory and Applications, Fixed Point Theory and Applications, Fixed Point Theory, Advances in Fixed Point Theory, and JP Journal of Fixed Point Theory and Applications.

Comment by Rodrigo A. Pérez

2013-04-10T03:44:42Z

@Ramiro: Entonces es verdad que un radical en el denominador da mas miedo :) Eso explica tambien porque escriben $2\sqrt{2}$ en lugar de $\sqrt{8}$...

User rodrigo a. p&eacute;rez - MathOverflow

Answer by Rodrigo A. Pérez for What are some applications of other fields to mathematics?

Answer by Rodrigo A. Pérez for What are some applications of other fields to mathematics?

Answer by Rodrigo A. Pérez for objects which can't be defined without making choices but which end up independent of the choice

Proof of the weak Goldbach Conjecture

A function whose fixed points are the primes

Answer by Rodrigo A. Pérez for Basic results with three or more hypotheses

Answer by Rodrigo A. Pérez for The prime number $2$

Asymptotic series

History of the high-dimensional volume paradox

Answer by Rodrigo A. Pérez for 2D Problems Which are Easier to Solve in 3D

Fixed point theorems

Answer by Rodrigo A. Pérez for Great mathematics books by pre-modern authors

Products of matrices of a certain form

Answer by Rodrigo A. Pérez for Fixed point theorems

Answer by Rodrigo A. Pérez for Fixed point theorems

Answer by Rodrigo A. Pérez for Fixed point theorems

Answer by Rodrigo A. Pérez for Fixed point theorems

Answer by Rodrigo A. Pérez for Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Answer by Rodrigo A. Pérez for Elementary cases of Mihailescu theorem

Answer by Rodrigo A. Pérez for Uppercase Point Labels in High-School Diagrams: from Euclid?

Answer by Rodrigo A. Pérez for functions of one complex variable: geometric theory

Answer by Rodrigo A. Pérez for Is there a "mathematical" definition of "simplify"?

Answer by Rodrigo A. Pérez for Is there a deep reason for the fecundity of involutions?

Polyhedra Classification

Answer by Rodrigo A. Pérez for How to triangulate a math reference?

Answer by Rodrigo A. Pérez for Are there some original papers or books related to applications of algebraic topology and algebraic geometry in complex dynamic systems

Answer by Rodrigo A. Pérez for How do you prove that every curve of constant width is convex?

Answer by Rodrigo A. Pérez for Modern developments in finite-dimensional linear algebra

Answer by Rodrigo A. Pérez for Untangling entwined rigid chains in 3-space

Answer by Rodrigo A. Pérez for Undecidability and holomorphic functions (Reference request)

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User rodrigo a. pérez - MathOverflow