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201,250 |
I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
|
No. If one selects a number $k$ at random from $1$ to a large number $n$, then for any fixed $h$, the random variables $\sin((k+1)^2), \dots, \sin((k+h)^2)$ asymptotically have mean zero, variance 1/2, and covariances 0, from standard Weyl sum estimates. Hence the variance of $\sum_{i=1}^h \sin((k+i)^2)$ is asymptotically $h/2$, which goes to infinity as $h \to \infty$. On the other hand, if the partial sums of $\sin(k^2)$ were bounded, then this variance would have to be bounded also. [Exercise: what part of the above argument breaks down when working with $\sin(k)$ instead of $\sin(k^2)$?] It may be possible to push this argument to show that the partial sums have to fluctuate by $\gg \sqrt{n}$ infinitely often, but I haven't checked this (certainly a lower bound of $\gg n^\varepsilon$ for some small $\varepsilon>0$ should be possible from the above argument, perhaps contingent on some conjecture about the irrationality measure of $\pi$). Heuristically, the law of the iterated logarithm suggests that the sum can occasionally get as large as $\gg \sqrt{n \log\log n}$, but no larger.
|
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201,424 |
Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere differentiable?
|
The image of an interval under a Lipschitz map has finite $1$-dimensional Hausdorff measure. EDIT: Here's a corrected version of Pablo Shmerkin's construction. Suppose $f: \mathbb R \to \mathbb R^d$ is differentiable. For positive integers $m,n$ let $A_{m,n} = \{x: |y -x| \le 1/n \implies \|f(y) - f(x)\| \le m |y - x|\}$.
For $k \in \mathbb Z$ let $A_{m,n,k} = A_{m,n} \cap [(k-1)/n, k/n]$. Then
$\bigcup_{m,n,k} A_{m,n,k} = \mathbb R$, and $f$ is Lipschitz on $A_{m,n,k}$ with Lipschitz constant $m$. We conclude that $f(\mathbb R)$ has $\sigma$-finite $1$-dimensional Hausdorff measure, which in particular implies that it has $2$-dimensional Lebesgue measure $0$.
|
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201,708 |
Does there exist a probability distribution on $\mathbb{Z}$ such that
for every integer $n\geq 1$, the probability that a random integer $x$
is divisible by $n$ equals $1/n$? Henry Cohn has an argument why this is not possible, but it is not
completely rigorous. First, it is easy to see that we can assume that
the distribution is supported on the positive integers. Let $p_n$ be
the probability of $n$. For any function $f$ on the positive integers
for which we get convergence, we have (by the assumption on $p_n$)
$$ \sum_k p_k\sum_{n|k}f(n) = \sum_n \frac{f(n)}{n}. $$
Let $g(k)=\sum_{n|k} f(n)$. By Möbius inversion,
$f(n) = \sum_{k|n}g(k)\mu(n/k)$. Writing $n=mk$, the first equation
becomes
$$ \sum_k p_k g(k) =\sum_k g(k)\sum_m \frac{\mu(m)}{mk}. $$
This should hold for all $g$ for which $\sum_n f(n)/n$ converges
absolutely, so it should follow that
$$ p_k = \frac 1k\sum_m \frac{\mu(m)}{m}. $$
This is nonsense since first of all, $\sum_m \mu(m)/m =0$ (equivalent
to the prime number theorem), and even if we didn't know that, there's
no way $p_k$ can be proportional to $1/k$ since $\sum 1/k=\infty$. This argument is not completely rigorous since we have interchanged
sums and equated coefficients of $g(k)$ without justification. It's
also a problem that $\sum_m \mu(m)/m$ is conditionally convergent.
|
No. Let's restrict our attention to $\mathbb{N}$. The hypotheses imply that if $q$ is a prime, then the probability that a random positive integer is not divisible by $q$ is $1 - \frac{1}{q}$. They also imply that these events are independent. Now let $n$ be a positive integer. If $q_1, q_2, \dots$ is an enumeration of the primes not dividing $n$ it follows that $$p_n \le \prod_{i=1}^m \left( 1 - \frac{1}{q_i} \right)$$ for all $m$. But taking $m \to \infty$ the RHS approaches $0$; contradiction. Note that this argument does not require the prime number theorem; we just need to know that the harmonic series diverges. Edit: Here is a generalization which more completely rescues Henry Cohn's argument. Generalize the condition to being that the probability of a positive integer being divisible by $n$ is $\frac{1}{n^s}$, for some real number $s > 0$. This is equivalent to requiring that the probability is 1) multiplicative in $n$ and 2) monotonically decreasing. It follows that if $q$ is prime, then the probability that the exponent of $q$ in the prime factorization of a random positive integer is exactly $k$ is $$\frac{1}{q^{ks}} - \frac{1}{q^{(k+1)s}} = \frac{1}{q^{ks}} \left( 1 - \frac{1}{q^s} \right).$$ We again have that for different primes $q$ these events are independent. Now, if $q_1, q_2, \dots$ is an enumeration of the primes and $n = \prod q_i^{k_i}$ is a positive integer, it follows that $$p_n \le \prod_{i=1}^m \frac{1}{q_i^{k_i s}} \left( 1 - \frac{1}{q_i^s} \right).$$ for all $m$. If $s \le 1$ then the RHS converges to $0$ as $m \to \infty$ (this, again, does not require the prime number theorem) and we get a contradiction. If $s > 1$ then the RHS converges to $$p_n = \frac{1}{n^s \zeta(s)}$$ and this is an equality because the RHS is the probability that a random positive integer has the same prime factorization as $n$. There is a straightforward generalization where $\mathbb{N}$ is replaced by the set of nonzero ideals in the ring of integers $\mathcal{O}_K$ of a number field $K$ and the probability of an ideal being divisible by an ideal $I$ is $\frac{1}{N(I)^s}$, where we get the Dedekind zeta function instead.
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201,718 |
The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me while I explain the intended meaning. I have taught multi-variable calculus several times. As usual of such repetition, I had the feeling that I know the concepts involved and how they are connected to each other and so on. But, when last week I was preparing for one of my sessions - in which I decided to use a bathymetric map (depth contours) rather than a topographic map (height contours) - a problem occurred to me for the first time. Imagining myself swimming to the shore while looking at the bathymetric map, it seemed "obvious" that if I wanted to take the shortest path the to shore (from where I was), moving in the opposite direction of the gradient would not be my choice! Prompted by my observation, I came to this quite "recent" paper " When Does Water Find the Shortest Path Downhill? The Geometry of Steepest Descent Curves " addressing the very same problem that whether gradient curves are geodesics. Now here is the question: Do you know any personal (or historical) examples of such "teaching-based research"? And, here is why I think the question is suitable for MO: Many mathematician friends of mine, for obvious reasons, prefer spending their time on research rather than teaching. Having a collection of such examples could be encouraging in particular for early career mathematicians. There is a recent movement to encourage " teaching inquiry " the point of which is to "teach students to ask and explore mathematical questions". For that aim, it seems that lecturers should be ready to be faced with some problems never posed before in the subjects that are too familiar to them, and better, be ready to pose such genuine questions in such contexts. Finally, it goes without saying that, it is a habit of mind to pose such questions in everyday research practice. It seems that what makes it difficult in teaching is rooted in an all-knowing feeling. If we know how to bypass such feeling, we could understand how students might develop such a habit beyond procedural fluency and conceptual understanding.
|
The first time I taught forcing, I wanted to mention, as motivation, the fact that the independence of the continuum hypothesis (CH) or even of the axiom of constructibility (V=L) cannot be proved by the method of inner models. That fact was proved in Cohen's book, "Set Theory and the Continuum Hypothesis" under the assumption that there is a standard model (i.e., a transitive set model with standard $\in$) of ZF, and Cohen mentions that one could prove the same fact under the strictly weaker assumption that ZF is consistent. So I wanted to show my class the proof under this weaker hypothesis, but I couldn't figure out how to do the proof. That's a good thing, because, in fact, the conclusion doesn't follow from that weaker hypothesis. Discovering that and analyzing the situation a little further, I found that the fact in question is equivalent to the $\omega$-consistency of ZF, which is strictly stronger than consistency and strictly weaker than existence of a standard model. This work was too complicated to present in that class, but it became one of my early papers, "On the inadequacy of inner models" [J Symbolic Logic 37 (1972) 569-571].
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201,728 |
A brief description: I have written a paper which contains a new result which I believe is somewhat important but not vital to the field. It is a generalization of an existing proof to get significant new information, in a framework that did not exist at the time of the original paper (not by me). I do not believe the result can be recovered from the original result, only from the proof, which is lengthy. The paper as it stands is self-contained but a significant portion is a reproduction of the original paper (recovering additional information from various lemmas). Some lemmas are new but many are old. A copy of it which does not reproduce original work is only about 5 pages long but is very difficult to read. It is my adviser's opinion (and I agree) that neither form of the paper is suitable for publication, for different reasons. We both believe the result on its own is significant enough, though. So, the question: would posting the long form on the arXiv as is be appropriate? To head off the obvious question: yes, the author of the original paper is prominently pointed out in the abstract and the introduction, where it is pointed out that we follow the original proof closely in most cases.
|
Based on what you say, your paper would be valuable and useful for the mathematical community. So I think you should put it on the arXiv, with the remark in the comment field that the paper is not intended for publication. Update. I meant "the paper is not intended for publication in a journal". Thanks for the comments and the (unexpectedly) large number of upvotes!
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201,853 |
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. Whether it is of closed form or not. see) My question is why differential Galois theory is not widely used in differential geometry. It is plausible that we can solve some problems of differential/integral geometry using this set of theory. I have read some answers provided here like Why do we need admissible isomorphisms for differential Galois theory? and other stuffs. I have read Kaplansky's and Buium's books. My question follows: So what is the major 'pullback' in this theory that prevents its wide application to other situations rather than discrete geometry (e.g. Diophantine geometry)? My original question on Mathematics Stack Exchange is: Why differential Galois theory is not widely used? which yields no satisfying answers.
|
The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a representation of the differential Galois group, in the same way that a sheaf is a more complicated than just a Galois representation, but I think it is cut from the same cloth. A D-module describes not just the solutions of a differential equation but also how they behave at singularities. D-modules are used in many different algebraic geometry situations. While differential Galois theory may seem analytic it is actually much more algebraic. For instance, in analysis and differential geometry you tend to care how large things are, while in algebra you don't, and differential Galois theory says nothing about size. In algebra you hope for exact solutions, while in analysis approximate solutions are usually good enough, and differential Galois theory good for describing exact solutions. In differential geometry you often have great freedom in gluing together local pieces to get a global structure, where in algebra local pieces are rigid and hard to glue together, and differential Galois theory describes rigid structures where one tiny piece controls everything.
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201,855 |
Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$? In $L_p[0,1]$ with $1<p<\infty$ any monotone basis is unconditional so one cannot expect better than $1+\varepsilon$ in general. What about $L_1[0,1]$, is every basis equivalent with a monotone basis? Edit: The original question has a negative answer, thank you for the reference. What about the $L_p[0,1]$ case? Any known results in this direction?
|
The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a representation of the differential Galois group, in the same way that a sheaf is a more complicated than just a Galois representation, but I think it is cut from the same cloth. A D-module describes not just the solutions of a differential equation but also how they behave at singularities. D-modules are used in many different algebraic geometry situations. While differential Galois theory may seem analytic it is actually much more algebraic. For instance, in analysis and differential geometry you tend to care how large things are, while in algebra you don't, and differential Galois theory says nothing about size. In algebra you hope for exact solutions, while in analysis approximate solutions are usually good enough, and differential Galois theory good for describing exact solutions. In differential geometry you often have great freedom in gluing together local pieces to get a global structure, where in algebra local pieces are rigid and hard to glue together, and differential Galois theory describes rigid structures where one tiny piece controls everything.
|
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202,923 |
While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$. Let $r_p$ be an integer with $0\leq r_p < p$ and $r_p^2 \equiv -1 \mod p$. How then is $\frac{r_p}{p} \in \mathbb{Q}$ distributed in $[0,1]$? Naively I would guess this is uniform distribution. How can we prove that? Edit I noticed in the comments, it might be simpler to ask about the equidistribution of $$\{ \tfrac{1}{\sqrt{p}}(a,b): a^2 + b^2 = p\} \subset S^1$$ still in the case $p = 4k+1$.
|
The equidistribution of the roots of quadratic congruences $\pmod p$ (such as $x^2+1$ in the question) was established in a famous paper of Duke, Friedlander and Iwaniec . The proof uses sieve ideas as well as ideas from the theory of modular forms.
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202,979 |
I am asking if this variant of the weak Goldbach Conjecture is already known. Let $N$ be an odd number. Does there exist prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we find $p_1$, $p_2$ and $p_3$ so that they are small enough? For example, can we prove that for large enough $N$, we can find such a triplet that all of them are smaller than $N$?
|
Yes - the standard proof of Vinogradov's result by means of the circle method gives this result. You just need to examine an integral
$$\int_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^2 \widehat{f}(-\alpha) e(-\alpha N) d\alpha$$
instead of
$$\int_{\mathbb{R}/\mathbb{Z}} (\widehat{f}(\alpha))^3 e(-\alpha N) d\alpha.$$
Here $\widehat{f}(\alpha) = \sum_n \Lambda(n) e(\alpha n) \eta(n/N)$, where $\eta$ is any weight supported in $\lbrack 0,1\rbrack$.
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203,505 |
Let $P(x)$ be a non-constant polynomial with real coefficients. Can natural density of $$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$ be positive?
|
No. There are two cases. Firstly, suppose that one of the non-constant coefficients of $P$ is irrational. Then, by the Weyl equidistribution theorem, $\lfloor P(n) \rfloor$ is equidistributed mod $W$ for any modulus $W$, which already limits the natural density of the prime-producing $n$ to be at most $\phi(W)/W$ for any $W$, which implies zero density by taking $W$ to be a product of all the primes less than a large threshold $w$. If the non-constant coefficients are all rational, then by passing to a suitable arithmetic progression one can make them all integer, at which point one may as well make the constant coefficient integer as well. Then one can sieve using the Chebotarev density theorem (or Landau prime ideal theorem ) as in David's answer. (One should probably get an upper bound of $O(x/\log x)$ for the number of $n \leq x$ with $P(n)$ prime by this method, where the implied constants depend on the coefficients of $P$ of course.)
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203,519 |
My question is somehow related to the one here First Collision Time for k Random Walkers on a Torus but, unfortunately, the answer does not cover my concern. My problem is: consider $n$ walkers on the cycle $\mathbb{Z}/k$ ($n < k$). At each step, one walker is selected with probability $1/n$ and moves by one unit counter-clockwise; the other walkers remain at their locations. The steps are independent. I would like to have some information on the first time $T$ until two walkers collide (go into to the same site); e.g., expectation, asymptotic behaviour (e.g. $k,n \rightarrow \infty$ in some proportion), etc. When $n = 2$, this reduces to a single random walker on the cycle moving clockwise, or counter-clockwise with probability $1/2$, and the time $T$ is simply the hitting time of the site $0$. But for arbitrary $n$, this approach does not seem to work ... Do you have any ideas, or references to similar problems ? Thank you.
|
No. There are two cases. Firstly, suppose that one of the non-constant coefficients of $P$ is irrational. Then, by the Weyl equidistribution theorem, $\lfloor P(n) \rfloor$ is equidistributed mod $W$ for any modulus $W$, which already limits the natural density of the prime-producing $n$ to be at most $\phi(W)/W$ for any $W$, which implies zero density by taking $W$ to be a product of all the primes less than a large threshold $w$. If the non-constant coefficients are all rational, then by passing to a suitable arithmetic progression one can make them all integer, at which point one may as well make the constant coefficient integer as well. Then one can sieve using the Chebotarev density theorem (or Landau prime ideal theorem ) as in David's answer. (One should probably get an upper bound of $O(x/\log x)$ for the number of $n \leq x$ with $P(n)$ prime by this method, where the implied constants depend on the coefficients of $P$ of course.)
|
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203,602 |
Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups (as in Serre), algebraic groups and étale cohomology.
|
You have given no indication as to your background, so the following imagines you don’t know anything. I have purposely left interesting things out as this is designed to get you from 0 to DL theory. Essentially this is what I would do if I could have my time over again, precisely in this order. EDIT: Assuming the background specified you can miss out anything labelled with a $\star$. If you wanted to go into less detail then you could also not read Chapter 7 of Carter's book and instead just read Chapter 4 of Geck. However I would advise reading the stuff about Harish-Chandra induction in Digne and Michel before going on to read about Deligne-Lusztig characters. It's good to think of it as generalising HC induction. Character Theory of Finite Groups ($\star$) First 3 chapters of Ledermann’s “Introduction to group characters”. It is short and contains the Mackey formula, which is an important idea later on. Chapter 28 of James and Liebeck’s “Representations and characters of groups”. There they compute the generic character table of $\mathrm{GL}_2(q)$ in a completely elementary way. It’s important to have an idea of what you’re trying to achieve with DL theory and it will be a good example to test things against later. EDIT: As pointed out by Nick Gill, Isaacs' book “Character theory of finite groups” is brilliant if you want to go deeper into character theory but it is more demanding than Ledermann. Serre's “Linear representations of finite groups” is also quite good but again does not go into as much detail as Isaacs and as far as I can remember doesn't do the Mackey formula. I personally like James and Liebeck's book but they again don't do the Mackey formula. Connected Reductive Algebraic Groups There are many great books on linear algebraic groups (they almost all bear this name) however I would suggest the following. ($\star$) First 2 chapters of Springer’s “Linear algebraic groups (Second Edition)” skipping everything in sections 1.9 and 2.5 (except Lemma 1.9.1, which is very important). The stuff about being defined over an arbitrary field is useful when thinking about finite reductive groups. ($\star$) Chapter 3 to 12 of Malle and Testerman’s “Linear algebraic groups and finite groups of Lie type” to get a general overview of the structure theory of connected reductive algebraic groups. It’s reasonably concise and gives you more detail than something like the intro to Digne and Michel or Carter’s book. Finite Reductive Groups Chapter 4 of Geck’s “An introduction to algebraic geometry and algebraic groups” up to section 4.3.10. This is, in my opinion, the most clear and straight forward presentation of Frobenius endomorphisms. It is much less confusing than Chapter 3 of Digne and Michel’s book. You should also see the relationship to Springer’s notions of being defined over an arbitrary ground field. The first chapter of this book is also a more gentle introduction to algebraic geometry than Springer if you wanted it but the more general framework in Springer will serve you well in the future. Character Theory without Geometry Chapters 4 to 9 of Digne and Michel’s “Representations of finite groups of Lie type”. Digne and Michel’s book is often hard work but completely worth it. These sections are very well written and emphasise the Mackey formula, which is very important for the modern view point on the subject. I would, however, advise keeping the errata list close at hand http://www.lamfa.u-picardie.fr/digne/errata.pdf At this point it’s not really necessary but if you are interested in learning more about Harish-Chandra theory then you might want to look at §11 and §67 of Curtis and Reiner’s phenomenal “Methods of representation theory Vol. I and Vol. II“. This will give you a lot of information about the decomposition and behaviour of the Harish-Chandra induction of the trivial character from a maximally split torus. This is the prototype for looking at Deligne—Lusztig characters and the theory of Hecke algebras plays a crucial role in the general theory. Deligne—Lusztig Characters ($\star$)? Chapter 7 in Carter’s “Finite groups of Lie type“ is still, in my opinion, a great reference for learning about the Deligne—Lusztig virtual characters $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$. Comparing with Digne and Michel you should think of the compactly supported cohomology groups being the correct bimodule generalising the Harish-Chandra bimodule. Finish reading Chapter 4 of Geck’s book. Similar material is covered there but in a less involved way and he computes the generic character table of ${}^2\textsf{B}_2(q)$ using Deligne—Lusztig characters, which is a great example. ($\star$) Another good source is Srinivasan’s “Representations of finite Chevalley groups“. Chapter 5 of this book is a great place to look if you want to understand a little more about the definition of $\ell$-adic cohomology and so forth. She also gives this nice proof in Theorem 5.13 counting the number of F-stable maximal tori using $\ell$-adic cohomology. Deligne—Lusztig Induction and Restriction Chapters 11 and 12 in Digne and Michel’s book. Notice how slick the Mackey formula and Alvis—Curtis duality makes things. For instance, compare the proof of Proposition 12.17 in Digne—Michel to the proof of Theorem 7.5.1 in Carter. Chapter 4 of Carter’s book. This will give you a bit more of an idea of duality. This is done quite quickly in Digne—Michel’s book and is one of the more difficult things to wrap your head around, in my opinion. Chapter 13 of Digne—Michel to get an idea of Lusztig series and the Jordan decomposition of characters. §15.4 of Digne-Michel now gives you a way to construct all the irreducible characters of $\mathrm{GL}_n(q)$ and $\mathrm{U}_n(q)$ as explicit linear combinations of $R_{\mathbf{T}}^{\mathbf{G}}(\theta)$'s and explains the role of the Lusztig series. Look also at the example of $\mathrm{GL}_2(q)$ in §15.9. Go back and compare with the computation in James and Liebeck, especially look at what is going on in Proposition 28.14. I would say that, at this point, you should really go back and read Deligne—Lusztig’s original paper from 1976. You will be able to truly appreciate just how wonderfully elegant and well written it really is. It is a true highlight of mathematics. From this point on it’s a bit about what you want to know and what you plan to do with Deligne—Lusztig theory. For information about unipotent characters and their classification look at: Carter, “On the representation theory of the finite groups of Lie type over an algebraically closed field of characteristic 0” in Algebra IX. Contains lots of information about the parameterisation of unipotent characters and more about the relationship to intersection cohomology. This is essentially a survey of Lusztig’s orange book. Geck, “Finite groups of Lie type”. This is a survey article contained in the book “Representations of reductive groups” which is edited by Carter and Geck. It also contains lots of other nice survey articles. This article by Geck is shorter than that by Carter and gives a bit more in the direction of character sheaves. Lusztig, “Representations of finite Chevalley groups“. This is published in the CBMS Regional Conference Series in Mathematics. It comes before the orange book and contains a lot of nice things. For more hands on stuff with Deligne-Lusztig varieties, cohomology groups and modular representation theory check out: Bonnafè, “Representations of $\mathrm{SL}_2(q)$”. This book is great to understand a bit more the geometry of things and get your hands dirty with modules. You have to know a bit to see how it fits in to the general theory, so it’s better to approach it later if you want to understand general stuff. However I highly recommend it! It's also useful as a gateway to understanding the modern aspects of the modular representation theory of these groups, where more emphasis is necessarily placed on the structure of the $\ell$-adic cohomology groups as modules. For information about values of Green functions and the generalised Springer correspondence have a look at: Shoji, “Green functions of reductive groups over a finite field”. This is again in the Arcata Conference proceedings from 1986. This gives a good overview of the relationship between the Springer correspondence and explains how you can compute values of Green functions. With this, together with what you have learnt above, you could compute the generic character table of any finite general linear or finite unitary group. This necessarily involves more geometry. Shoji, “Geometry of orbits and Springer correspondence”. This is in Astérisque (1988). This is a good introduction to the generalised Springer correspondence, which is a key ingredient in determining the values of Green functions and, more generally, generalised Green functions (which arise from character sheaves). Geck, "Some applications of CHEVIE to the theory of algebraic groups". This is a nice paper considering computational aspects of really computing things. It covers similar material to Shoji in his Arcata conference proceedings article but minimises the geometry and IC's, so is a bit easier to read. There really isn't a good reference for character sheaves except Lusztig's original articles but this is not bad: Lusztig, “Introduction to character sheaves” published in The Arcata Conference proceedings from 1986. This gives a little snapshot of character sheaves, which is quite nice. In the end, I hope some of this advice helps and good luck!
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203,836 |
I saw a very remarkable asymptotic formula (or a conjecture?) for the volume of of the unitary group $ U(n)$ which is the following: $$\log[\mathrm{Volume}(U(n))] \sim_{n\rightarrow \infty} \frac{n^2}{2} \log(n) + \sqrt{2\pi}\log (n) -\sum_{g\geq 2}n^{2-2g} \chi(\mathcal{M}_{g}) $$
where the volume is calculated with respect to the Killing metric and $\chi(\mathcal{M}_{g})$ is Euler characteristic of the moduli space of Riemann surfaces of genus $g$.
Can someone explain why it should be true? Is it a conjecture?
|
As indicated by Igor Rivin, the volume of the unitary group is given by
$vol(U(N))=(2\pi)^{(N^2+N)/2}/\prod_{k=1}^{N-1} k!$. The denominator is the Barnes G-function, which is well-known : http://en.wikipedia.org/wiki/Barnes_G-function and in particular has a known Stirling-like asymptotic expansion for large $N$: $\log(\prod_{k=1}^{N-1} k!) \sim$
$\frac{N^2}{2} \log N - \frac{1}{12} \log N - \frac{3}{4}N^2
+\frac{N}{2} \log(2 \pi)
+ \zeta'(-1) + \sum_{g \geq 2} \frac{B_{2g}}{2g(2g-2)} N^{2-2g}$. Comparing with the Harer-Zagier formula $\chi(M_g)=\frac{B_{2g}}{2g(2g-2)}$, we obtain $vol(U(N))\sim$
$ - \frac{N^2}{2} \log(N) + \frac{N^2}{2}(\log(2 \pi)+\frac{3}{2}) + \frac{1}{12}
\log(N) - \zeta'(-1)-\sum_{g \geq 2} \chi(M_g) N^{2-2g}$ which, up to probable typos and forgotten terms, is the asymptotic expansion of the question. Of course, in such a proof, the fact that unitary groups and moduli spaces of Riemann surfaces are related appears as a coincidence: essentially we have just taken two places in mathematics where Bernoulli numbers appear. We can ask if there is a more direct intrinsic explanation of this relation. I don't think that such explanation is known at the level of rigorous mathematics but one is known at the level of theoretical physics. On general grounds, it is expected that gauge theories of group $U(N)$ are related in the large $N$ limit to a form of string theory. The first argument in this direction was given by 't Hooft in the 70's and is the observation that Feynman diagrams in perturbative $U(N)$ gauge theory can be rewritten as double-line graphs, or ribbon graphs, that it is possible to obtain closed surfaces from ribbon graphs by gluing disks along their boundary components, and that in some appropriate limit the series of Feynman diagrams organizes as a genus expansion of these surfaces. In fact, it is possible to prove the Harer-Zagier relation along these lines by describing the moduli space of Riemann surfaces in terms of ribbon graphs, interpreting these ribbon graphs as the perturbative expansion of some $N$ by $N$ matrix model, solving this matrix model, which gives something containing the $\Gamma$ function and then expanding the solution. In this proof, which can be found in an appendix to Kontsevich's paper on Witten's conjecture, http://www.ihes.fr/~maxim/TEXTS/intersection_theory_6.pdf , the Bernoulli numbers appearing in the Harer-Zagier formula really comes from the Stirling expansion of the $\Gamma$-function. Making 't Hooft idea concrete is one of the main theme of modern string theory and can go under various more or less general and more or less precise names: gauge/gravity duality, AdS/CFT correspondence, open/closed duality, holographic relation... One explicit example of that is the Gopakumar-Vafa correspondence asserting the equivalence of Chern-Simons theory of group $U(N)$ and level $k$ on the 3-sphere with the A-model of the topological string, i.e. Gromow-Witten theory, on the resolved conifold, i.e. the total space of $\mathcal{O(-1)}\oplus {\mathcal{O(-1)}}$ over $\mathbb{P}^1$, with $\mathbb{P}^1$ being of volume $t=\frac{2 \pi N}{k+N}$ and with a string coupling constant
$g_s = \frac{2 \pi}{k+N}$. As the volume of $U(N)$ appears explicitely in the one-loop perturbative expansion of Chern-Simons theory on the 3-sphere, it is possible to "explain" the asymptotics expansion of these volumes in terms of moduli spaces of Riemann surfaces.
Of course, all that is not a proof and the direct matching of the two sides of the equalities is often used as a support of physicists conjectures but I wanted to mention it because it is a natural circle of ideas in which the formula of the question appears naturally.
|
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|
203,970 |
Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. Does it follow that $F$ is a constant function? This is clearly false for $n=2$. I am wondering if it is true for sufficiently large $n$. ADDED LATER The $\mathbb R$-valued examples in Cranch's answer may be combined into a single example $v\mapsto v\otimes v$ with values in $\mathbb R^n\otimes \mathbb R^n$, or $n\times n$ matrices. Its image generates the group of symmetric matrices with integer trace. It seems reasonable to expect that every continuous real-valued example comes from this one -- in other words has the form $v\mapsto B(v,v)$ for symmetric bilinear $B$. Maybe this can be worked out using Sawin's suggestion about representations of $O(n)$. But I was also curious about the general case, where the target group might not be (uniquely) divisible.
|
Given a vector $u$ and an orthonormal basis $x_1,\ldots,x_n$, we have $||u||^2 = \left<u,x_1\right>^2 + \cdots + \left<u,x_n\right>^2$. But that means that, if you choose a nonzero $u$, then the function $F(x) = \left<u,x\right>^2$ gives a counterexample.
|
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|
204,020 |
Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$? My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly overlooking. Can anyone provide one? Of course, any proof would be welcome! Any partial results would also be of interest.
|
If $p$ is an odd prime (EDIT: other than 5) for which $-1$ is a quadratic residue mod $p$, and $A$ is the set of non-zero quadratic non-residues mod $p$, then $A+A$ is all of ${\bf Z}/p{\bf Z}$, whilst $A+AA$ is ${\bf Z}/p{\bf Z} \backslash \{0\}$. So counterexamples exist in finite fields, which rules out some methods of proof (e.g. "Ruzsa calculus" by itself will be insufficient). Unfortunately, this example does not appear to be adaptable to the reals (for which $-1$ is certainly not a square, and for which there are no large multiplicative subgroups). Actually it looks difficult to build an example in the complex numbers (or any other characteristic zero field); I don't even see a way to construct an (EDIT: arbitrarily large) finite set $A$ obeying the weaker inequality $|A+AA| < \frac{|A| (|A|+1)}{2}$. One may indeed conjecture (in the spirit of the Erdos-Szemeredi sum-product conjecture) that one always has $|A+AA| \geq \frac{|A| (|A|+1)}{2}$ (EDIT: for sufficiently large $A$), but this is well beyond our current technology to prove. (EDIT: as noted in comments, there are small counterexamples obeying the weaker inequality, although they do not give counterexamples to the original inequality.)
|
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|
204,051 |
Being far from the field of quantum groups, I have nevertheless made in the past several (unsuccessful) attempts to understand their definition and basic properties. The goal of this post is to try to improve the situation. The definition of the quantum group I saw is that it is a Hopf algebra given by some explicit generators and relations. Though I have heard that at least the case of the quantum $sl_2$ was motivated by physics, this was not explicit enough. If someone defined the classical (i.e. non-quantum) Lie algebra $gl_n$ or the symmetric group $S_n$ using generators and relations rather than operators acting on a vector space or on a finite set respectively, such a definition would be equally unclear to me. The abstract approach to quantum groups as deformations of a universal enveloping algebras in the class of Hopf algebras is very useful, but still not very intuitive and explicit. While any clarifying remarks would be appreciated, I can ask the following more specific questions. (1) In simple examples of quantum groups, such as quantum $sl_2, sl_n$, are there "natural" examples of their representations, like the standard representation of the classical $sl_n$, its dual representation and their tensor powers? (2) Are there mathematical problems which are not about quantum groups, but whose solution does require this notion?
|
In algebraic combinatorics, there is an important concept of a "$q$-analogue". Surprisingly often when you have a counting problem with a good integer answer, you realize that it can be refined to a (finite) generating function with an equally good polynomial answer. A simple example of this is the $q$-analogue of the number of permutations, which is of course $n!$. If you define the weight of a permutation to be $q^k$, where $k$ the inversion number of the permutation, then the total weight is then the important and beautiful formula $$1(q+1)(q^2+q+1)\cdots(q^{n-1}+\ldots+1).$$ This expression is called a $q$-factorial the factors are called $q$-integers. Interesting q-analogues usually involve $q$-integers. Note that $q$-integers are closely related to cyclotomic polynomials: Every $q$-integer is a (unique) product of cyclotomic polynomials, and every cyclotomic polynomial is a (unique) ratio of products of $q$-integers. Gaussian binomial coefficients are among the most important $q$-analogues. The best way to think of a quantum group $U_q(\mathfrak{g})$ is that it is an algebraic $q$-analogue of a simple Lie group $G$ or its universal enveloping algebra $U(\mathfrak{g})$. For generic values of $q$, it has exactly the same (names of) representations as $G$ that tensor in the same way, plus (depending on conventions) possibly some other representations that are less important. But, what has changed is the positions of the sub-representations and the specific representation matrices. In general, when you see expressions such as integers, binomial coefficients, and factorials in the formulas for representation matrices, you see $q$-integers, Gaussian binomial coefficients, and $q$-factorials in the quantum group version. The only asterisk to this is the preferred convention of using centered Laurent polynomials (which may have half-integer exponents) rather than standard polynomials with non-negative integer exponents. The only non-generic values of $q$ for quantum groups are roots of unity. In this case a new and also fundamental effect appears: The representation theory acquires features shared with representations of algebraic groups in positive characteristic. They have been used to strengthen or at least clarify the representation theory of algebraic groups. The main application of quantum groups: Topological invariants. Eventually when studying the representation theory of a Lie group, you consider tensor networks, i.e., invariant tensors combined with tensor products and contractions. Because of the extra non-commutativity of a quantum group (or any non-commutative, non-cocommutative Hopf algebra), an invariant tensor network of a quantum group needs to be embedded in $\mathbb{R}^3$ in order to be interpreted or evaluated as an algebraic expression. And then the remarkable outcome is that you obtain the Jones polynomial, when the quantum group is $U_q(\text{sl}(2))$, and its well-known generalizations for other quantum groups. Quantum groups are the main algebraic way to understand the quantum polynomial invariants of knots and links; and quantum groups at roots of unity are the main algebraic way to understand the corresponding 3-manifold invariants. In fact, this is closely connected to why they were first defined. In response to Semyon's question in the comments: The concept of a $q$-analogue in combinatorics has never been entirely rigorous, and if anything the construction of quantum groups has been clarifying. The rough idea is that a counting problem in combinatorics is interesting when it has a "nice" answer, which often (but not by any means always) means an efficient product formula. So then a $q$-analogue is a weighted enumeration or finite generating function in which every weight is a power of $q$, and the enumeration still has all favorable numerical properties, and $q$-integers or cyclotomic factors arise. In the case of quantum groups, first of all they are Hopf algebras. A Hopf algebra is an algebra together with all necessary extra apparatus to define the tensor product of two representations as a representation (i.e., comultiplication) and the dual of a representation as a representation (i.e., the antipode map). A universal enveloping algebra $U(\mathfrak{g})$ is of course a Hopf algebra. In this case, any deformation of $U(\mathfrak{g})$ as a Hopf algebra is potentially interesting. There is a cohomology result that if $\mathfrak{g}$ is complex and simplex, then there is only one non-trivial deformation, and you might as well call its parameter $q$ with $q=1$ at the undeformed point. (Sometimes the logarithm of $q$ is used and is called $h$, in reference to Planck's constant.) Since this is the only deformation, it is an analogue of some kind, and it is interesting. Moreover, there is a parametrization of the deformation so that $q$-integers (or quantum integers, in centered form) and cyclotomic polynomials arise in the structure of the Hopf algebra and its representations. The analogue thus deserves to be called a $q$-analogue. Theo's explanation illustrates this more explicitly. The quantum plane is a non-commutative algebraic space (in the sense that you can interpret it as a purely formal "Spec" of the quantum plane ring) that is associated to the non-cocommutative algebraic group version of $U_q(\text{sl}(2))$. So, then, in the ring of the quantum plane, if you just expand $(x+y)^n$, you get a $q$-analogue of the binomial coefficient theorem using Gaussian binomial coefficients. (Where the $q$-exponent of a word in $x$ and $y$ is its inversion number, just as with the $q$-enumeration of permutations.) This is one of many examples where $q$-analogues that were considered long before quantum groups appear in the theory of quantum groups. As for knots and links: In order for $U(\mathfrak{g})$ to have a non-trivial deformation as a Hopf algebra, you have to allow comultiplication to be non-commutative, even though $U(\mathfrak{g})$ itself is cocommutative. So then if $V$ and $W$ are two representations, $V \otimes W$ and $W \otimes V$ are not isomorphic via the usual switching map $v \otimes w \mapsto w \otimes v$, because that switching map is not in the category. (Due to non-cocommutativity, it is not equivariant, i.e., not an intertwiner.) However, $V \otimes W$ and $W \otimes V$ are still isomorphic, just by an adjusted version of the switching map. (You see the same theme in the category of super vector spaces, where there is a sign correction when $v$ and $w$ both have odd degree.) However, there are two natural, in-category deformations of the switching map, not just one. It so happens that they should be interpreted as left- and right- half twists in a braid group, so that you get braid representations and ultimately knot and link invariants. The point is that ordinary tensors (and tensor networks) live in spaces that have natural actions of symmetric groups, because you can permute indices of tensors. The whole theme of quantum group definitions is deformations, and it so happens that the symmetric group action deforms into a braid group action. This explains topological invariants such as the Jones polynomial in a one- and two-dimensional sort of way, from braids to diagrams of knots to knots themselves. It is more satisfying to have a more intrinsically 3-dimensional definition. (Actually, what counts as intrinsically 3-dimensional is somewhat debatable, but never mind that.) This is why Witten provided a "definition" of the Jones polynomial and related invariants using Chern-Simons quantum field theory. It is not really a rigorous definition, but it is very credible as a "physics definition" or even an incomplete, but maybe one-day rigorous, mathematical definition. This leads to the basic association between Chern-Simons quantum field theory and quantum groups, that they are two ways to describe the same topological invariants.
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|
204,106 |
I would like to know what the definition of a short proof is. In Lance Fortnow’s article “ The Status of the P Versus NP Problem ”, Communications of the ACM, Vol. 52 No. 9, he says, If a formula θ is not a tautology, we can give an easy proof of that fact by exhibiting an assignment of the variables that makes θ false. But if θ were indeed a tautology, we don’t expect short proofs. If one could prove there are no short proofs of tautology that would imply P ≠ NP. I have tried to find a definition of a “short proof”, but have not been able to.
|
The statement you quoted is somewhat sloppy, since there is no precise notion of a short proof for a single formula. There is, however, a notion of short proofs for a class $C$ of formulas, when the class contains formulas of arbitrarily high length. One says that $C$ admits short proofs if there is a polynomial $p(x)$ such that, for every natural number $n$, all formulas in $C$ of length $n$ have proofs of length at most $p(n)$.
|
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|
204,110 |
Let $\mathsf{A}$ be an Abelian category (perhaps vector spaces or modules over your favorite ring), and let $\mathsf{A}(x,y)$ denote the set of morphisms in $\mathsf{A}$ from an object $x$ to another object $y$. Does there exist a non-trivial partial order on the morphism-sets of $\mathsf{A}$ which behaves well with respect to composition and happens to be monotone with rank, whenever rank is defined? More specifically, do there exist partial orders $<$ so that for $f,f':x \to y$ and $g, g':y \to z$, we have: $f < f'$ and $g < g'$ implies $gf < g'f'$ in $\mathsf{A}(x,z)$, and if $f < f'$ then there is a monomorphism from the image of $f$ to image of $f'$? Probably the answer is no, but I'm not sure how to prove that no such thing could possibly exist --- even in the category of vector spaces over $\mathbb{R}$ or $\mathbb{C}$.
|
The statement you quoted is somewhat sloppy, since there is no precise notion of a short proof for a single formula. There is, however, a notion of short proofs for a class $C$ of formulas, when the class contains formulas of arbitrarily high length. One says that $C$ admits short proofs if there is a polynomial $p(x)$ such that, for every natural number $n$, all formulas in $C$ of length $n$ have proofs of length at most $p(n)$.
|
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|
204,167 |
This is a follow-up to Dan Ramras' answer of this question . The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here ). The weak Hausdorff rather than the Hausdorff property should be required of spaces [...] in order to validate some of the limit arguments used [...]. I was not able to figure out which type of "limit arguments" really would have needed the weak Hausdorff condition instead of the regular one used in the original paper and would be happy to understand the necessity of the correction, ideally by means of an explicit example in the paper.
|
I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better
behaved in CGWH than in CGH. Specifically, CGWH is closed
under pushouts, one leg of which is the inclusion of a closed
subspace. CGH does not have such nice behavior, and pushouts
like that are used all over The Geometry of Iterated Loop Spaces,
specifically in the construction of a monad from an operad and
in the use of geometric realizations of simplicial spaces.
|
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|
204,460 |
Given matrices $$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$ where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following? $$\det \big( I + e^{A_1}e^{A_2}\ldots e^{A_N} \big) \ge 0$$ This seems to be true numerically. Update1 : As was shown in below, the above inequality is related to another conjecture $\det(1+e^M)\ge 0$, given a $2n\times 2n$ real matrix $M$ that fulfills $\eta M \eta =-M^T$ and $\eta=diag(1_n, -1_n)$. The answers of Christian and Will, although inspiring, did not really disprove this conjecture as I understood. Update2 : Thank you all for the fruitful discussion. I attached my Python script down here. If you run it for several times you will observe $\det(1+e^{A_1}\ldots e^{A_N})$ seems to be always larger than zero (the conjecture), $M = \log(e^{A_1}\ldots e^{A_N})$ are sometimes indeed pure real and it fulfills the condition mentioned in update1 . In this case the eigenvalues of $M$ are either pure real or in complex conjugate pairs. Thus it is easy to show $\det(1+e^M)=\prod_l(1+e^
{\lambda_l})\ge 0$, However, sometimes the matrix $M = \log(e^{A_1}\ldots e^{A_2})$ can be complex and they are indeed in the form written down by Suvrit. In this case, it seems that the eigenvalues of $M$ will contain two sets of complex values: $\pm a+i\pi$ and $\pm b + i\pi$. Therefore, $\det(1+e^{M})\ge 0$ still holds because $(1-e^a)(1-e^{-a})(1-e^{b})(1-e^{-b})\ge 0$. Update3 : Thank you GH from MO, Terry and all others. I am glad this was finally solved. One more question: how should I cite this result in a future academic publication ? Update4 : Please see the publication out of this question at arXiv:1506.05349 .
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Here are some ideas how to decide the conjecture. (EDIT: In fact these ideas lead to a proof of the conjecture as Terry Tao explained in two comments below.) As Christian Remling and Will Sawin showed, the conjecture is equivalent to $\det(I+T)\geq 0$ for any $T\in\mathrm{SO}^0(n,n)$. We can assume that $-1$ is not an eigenvalue of $T$. Up to conjugacy, $T$ is a sum of indecomposable blocks as in Theorem 1 of Nishikawa's 1983 paper , and then $\det(I+T)$ is the product of the determinants of the corresponding blocks of $I+T$. Hence, by the idea of jjcale, we can forget about the blocks that are of exponential type. By page 83 in Djoković's 1980 paper , the remaining blocks are of type $\Gamma_m(\lambda,\lambda^{-1})$ with $\lambda<0$ and $\lambda\neq -1$, which in turn are described on page 77 of the same paper. Such a block contributes $(1+\lambda)^{2m+2}/\lambda^{m+1}$ to $\det(I+T)$, hence we can forget about the blocks where $m$ is odd. To summarize, we can assume that $T$ is composed of $(2m+2)\times(2m+2)$ blocks of type $\Gamma_m(\lambda,\lambda^{-1})$ with $\lambda<0$ and $\lambda\neq -1$ and $m$ even. The conjecture is true if and only if the number of such blocks is always even. For this, the explicit description of $\mathrm{SO}^0(n,n)$ on page 64 of Nishikawa's 1983 paper might be useful (see also page 68 how to use this criterion for $m=1$). Based on this, I verified by hand that one cannot have a single block for $m=2$, which also shows that the smallest possible counterexample to the conjecture is of size $10\times 10$ (i.e. $n\geq 5$). Added 1. Terry Tao realized and kindly added that in the remaining case we are done. Read his comments below. To summarize and streamline his ideas, we have in this case
\begin{align*}\det(I_{2n}+T)
&=\det(I_n+A)\det(I_n+A^{*-1})\\
&=\det(A)\det(I_n+A^{-1})\det(I_n+A^{*-1})\\
&=\det(A+A^{*-1})\frac{\det(I_n+A^{-1})^2}{\det(I_n+A^{-1}A^{*-1})},
\end{align*}
where $(A+A^{*-1})/2$ can be described as the restriction of $T$ to a totally positive subspace followed by the orthogonal projection to this subspace. Now we have $\det(A+A^{*-1})>0$ by $T\in\mathrm{SO}^0(n,n)$, while the fraction on the right is clearly positive, hence we conclude $\det(I_{2n}+T)>0$. Added 2. Terry Tao wrote a great blog entry on this topic. Added 3. Let me add a variation on Terry's original argument. Djoković defines $\mathrm{SO}(n,n)$ via $J:=\begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix}$, while Nishikawa defines it via $K:=\begin{pmatrix} I_n & 0 \\ 0 & -I_n\end{pmatrix}$. These two matrices are connected via $J=M^*KM$, where $M:=\frac{1}{\sqrt{2}}\begin{pmatrix} I_n & I_n\\ -I_n & I_n\end{pmatrix}$, hence any matrix $T$ in Djoković's $\mathrm{SO}(n,n)$ corresponds to $MTM^*$ in Nishikawa's $\mathrm{SO}(n,n)$. We need to examine the case of $T = \begin{pmatrix} A & 0 \\ 0 & A^{*-1} \end{pmatrix}$, which corresponds to $MTM^*=\frac{1}{2}\begin{pmatrix} A+A^{*-1} & -A+A^{*-1} \\ -A+A^{*-1} & A+A^{*-1} \end{pmatrix}$. This lies in Nishikawa's $\mathrm{SO}^0(n,n)$, whence $\det(A+A^{*-1})>0$.
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|
205,843 |
We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$ Given $A,B\subseteq \mathbb{N}$ we set $A+B = \{a+b: a\in A, b\in B\}$. Are there $A, B\subseteq \mathbb{N}$ with lower density 0, but $A+B$ does not have lower density 0?
|
It is relatively easy to prove that the set of perfect squares has asymptotic density equal to $0$. Then either the set $Q_2 := \{x^2+y^2: x,y \in \mathbf N\}$ has positive lower asymptotic density, and we're done, or the lower density of $Q_2$ is zero, and then we just consider that $\mathbf N = Q_2 + Q_2$ (by Lagrange's four-squares theorem). By the way, it follows, e.g., from E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate , Arch. Math. Phys. 13 (1908), 305-312 that the asymptotic density of $Q_2$ is actually zero, but this is more than what you need to answer the question in the OP.
|
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|
206,195 |
Let $D=(d_1,d_2,\dots,d_k)$ be a sequence of correlated random variables. $D$ is "everywhere $r$-probably increasing" if the event $d_j > d_i$ has probability $\geq r$ for all $j > i$. Fix $r \geq 1/2$. How long can a random sequence of integers in $\{1, 2, \dots, n\}$ be, and still be $r$-probably increasing? Perhaps surprisingly, the answer is not $O(n)$. For example, the random sequence which is $(1,1,1,2,2,2,3,3,3)$ with 50% chance and $(1,2,3,1,2,3,1,2,3)$ with 50% chance is everywhere $1/2$-probably increasing. It's easy to generalize this to get, when $r=1/2$, a sequence of length $n^2$. With a bit more work, when $r=a/b$, one can get a sequence of length $\sim n^{b/a}$. I conjecture the answer is $O(n^{1/r})$. But I can't find any relevant papers, and frustratingly, I can't even prove any upper bound for any $r<1$.
|
I adapt an argument from this blog post of mine , exploiting the $\ell^2$ boundedness of the discrete Hilbert transform (i.e. Hilbert's inequality ), to obtain an exponential upper bound. I don't see any obvious way to improve this to a polynomial bound (EDIT: A Whitney decomposition seems to do the trick, see below). The argument is inspired by a stability result of Hrushovski (based on de Finetti's theorem) which shows that some finite bound is possible, although it is not easy to extract a quantitative bound from Hrushovski's argument (and if one did, it would surely be worse than exponential); see Proposition 2.25 of that paper. Suppose that ${\bf P}(d_j > d_i) \geq r$ for some $r>1/2$ and all $1 \le i < j \leq k$. Then of course
$${\bf P}( d_j > d_i) - {\bf P}( d_j < d_i ) \geq 2r-1$$
for all $1 \le i < j \le k$. We expand this as
$$ \sum_{1 \leq a < b \leq n} {\bf P}( d_j = b \wedge d_i = a ) - {\bf P}( d_j = a \wedge d_i = b ) \geq 2r-1.$$
Multiply by the positive quantity $\frac{1}{j-i}$ and sum to conclude that
$$ \sum_{1 \leq a < b \leq n} \sum_{1 \leq i < j \leq k} \frac{1}{j-i} [{\bf P}( d_j = b \wedge d_i = a ) - {\bf P}( d_j = a \wedge d_i = b )] \gg (2r-1) k \log k \qquad (1).$$
The LHS can be rearranged as
$$ \sum_{1 \leq a < b \leq n} \sum_{1 \leq i,j \leq k: i \neq j} \frac{1}{j-i} {\bf P}(d_j = b \wedge d_i = a ) $$
and rearranged further as
$$ {\bf E} \sum_{1 \leq a < b \leq n} \sum_{1 \leq i,j \leq k: i \neq j} \frac{1}{j-i} 1_{d_j = b} 1_{d_i = a}.$$
By Hilbert's inequality, we have
$$
\sum_{1 \leq i,j \leq k: i \neq j} \frac{1}{j-i} 1_{d_j = b} 1_{d_i = a} \ll (\sum_{1 \leq j \leq k} 1_{d_j=b})^{1/2} (\sum_{1 \leq i \leq k} 1_{d_i=a})^{1/2}$$
and
$$ {\bf E} \sum_{1 \leq a < b \leq n} \sum_{1 \leq j \leq k} 1_{d_j=b}, {\bf E} \sum_{1 \leq a < b \leq n} \sum_{1 \leq i \leq k} 1_{d_i=a} \ll k n $$
so by Cauchy-Schwarz
$$
{\bf E} \sum_{1 \leq a < b \leq n} \sum_{1 \leq i,j \leq k: i \neq j} \frac{1}{j-i} 1_{d_j = b} 1_{d_i = a} \ll kn $$
and hence
$$ kn \gg (2r-1) k \log k$$
leading to the exponential upper bound
$$ k \ll \exp( O( \frac{n}{2r-1} ) ).$$ EDIT: Looks like one can improve this to the polynomial bound $k \ll n^{O(1/(2r-1))}$ using the following standard Whitney decomposition trick (used for instance to prove the Rademacher-Menshov theorem or the Christ-Kiselev lemma). Firstly, without loss of generality we may take $n$ to be a power of 2. Then observe that if $1 \leq a < b \leq n$, then there is a unique pair of distinct dyadic intervals $I,J$ in $\{1,\dots,n\}$ with the same parent such that $a \in I$ and $b \in J$; let's call such pairs "adjacent". As such, the LHS of (1) can now be rearranged as $$ {\bf E}\sum_{2^l < n} \sum_{I,J: |I|=|J|=2^l, \hbox{adjacent}} \sum_{1 \leq i,j \leq k: i \neq j} \frac{1}{j-i} 1_J(d_j) 1_I(d_i).$$ We apply Hilbert's inequality to bound this by
$$ \pi {\bf E} \sum_{2^l < n} \sum_{I,J: |I|=|J|=2^l, \hbox{adjacent}} (\sum_j 1_J(d_j))^{1/2} (\sum_i 1_I(d_i))^{1/2}$$
which by Cauchy-Schwarz and the disjointness of the $I,J$ can be bounded by
$$ \pi \sum_{2^l < n} k^{1/2} k^{1/2} \ll k \log n$$
leading to
$$ k \log n \gg (2r-1) k \log k $$
and thus $k \ll n^{O(1/(2r-1))}$.
|
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|
206,645 |
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in integers $x>1, y>1, n>2$, except when $n=4, x=7$ and $n=5, x=3$. Since neither I have been able to spot a copy of Ljunggren's paper online (and even if I had made every effort to acquire an electronic copy of it, it all would have been to no avail because I don't read Norwegian) nor I have access to Ribenboim's edition of the Collected Papers of these authors, would anybody here be so kind as to explain in some detail how it was that Ljunggren proved this notable result? Unfortunately, Nagell doesn't say much about Ljunggren's demonstration in his review of 1 ... Please, let me thank you in advance for your learned replies! References [ 1 ] W. Ljunggren, Some theorems on indeterminate equations of the form $\frac{x^{n}-1}{x-1} = y^{q}$ (In Norwegian). Norsk. Mat. Tidsskr. 25 (1943), pp. 17--20.
|
I went to my office today and scanned the Ljunggren's paper that OP asked for. I provide some bibliographical information first: Wilhelm Ljunggren, Noen setninger om ubestemte likninger av formen $\frac{x^n - 1}{x-1}=y^q$ , Norsk. Mat. Tidsskrift, 25 (1943), 17 -- 20 ( = Collected Papers of W. Ljunggren edited by P. Ribenboim, Volume 1, #14, p. 363 -- 366). I put the paper in the web space I have. EDIT: Here is another scan ( Internet Archive ) Since the paper is so small and I have been thinking about Brahmagupta-Pell equations recently, I decided to translate the part of the paper relevant to the OP. This was surprisingly easy (Two Ronnies and some knowledge of German finally paid off!). What follows is the functional translation of nearly half the paper; it goes without saying that what follows is due to Ljunggren while the errors in translation rest with me. We owe to T. Nagell [N] the following theorem: The diophantine equation $$ \frac{x^n - 1}{x - 1} = y^2 \qquad (n > 2)\tag{1}$$ has only a finite number of solutions in integers $x$ and $y$. The possible solutions are found at the end of this work. Specifically, (1) is impossible with $|x| > 1$ if $n$ does not have one of the following four forms: $1^\circ. n = 4$; $2^\circ. n = p$; $3^\circ. n = p^2$; $4^\circ. n = p^2 q$ with $p$ and $q$ distinct primes, $q \equiv 1 \bmod{24 p}$ and $q < p^2 - 3$. I will first show how one can deduce the following theorem about (1) using the theorems of K. Mahler [M]: Theorem 1. The diophantine equation (1) is impossible with $|x| > 1$ in all cases except $n = 4, x = 7$ and $n = 5, x = 3$. We need the following theorem of K. Mahler: Let $D$ be a natural number that is not a perfect square. Furthermore, let $A$ be a square-free divisor of $2D$ ($A \neq 0$). Then, the solutions of the equation $$ x^2 - Dy^2 = A\tag{2}$$ are given by the following formula where $m$ an odd positive integer:
\begin{align*}
\pm x_m &= \frac{(u+v\sqrt{D})^m+(u-v\sqrt{D})^m}{2|A|^{\frac{m-1}{2}}}\\
\pm y_m &= \frac{(u+v\sqrt{D})^m-(u-v\sqrt{D})^m}{2|A|^{\frac{m-1}{2}}\sqrt{D}}
\end{align*} Here $u$ and $v$ are natural numbers that satisfy (2) with $|y_{\min}| = v$. Furthermore, if we let $\mathfrak{n}(D, A)$ denote the set of odd integers $m$ such that the pair $(x_m, y_m)$ solves (2) and the set of prime divisors of $y_m$ is contained in those of $D$, then, either $\mathfrak{n}(D, A)$ is empty or $\mathfrak{n}(D, A) = \{1\}$ or $\mathfrak{n}(D, A) = \{1, 3\}$. According to Nagell, it is enough to look at odd $n$. We assume now that $x > 1$. The equation (1) can be written in the form: $$ [(x - 1)y]^2 - x(x - 1) \left[x^{\frac{n-1}{2}}\right]^2 = - (x - 1).$$ Here evidently Mahler’s theorem applies; we have $D = x(x - 1)$ and $A = - (x - 1)$ and further we have that $u = x - 1$ and $v = 1$. This gives $$x^{\frac{n-1}{2}} = 1 \qquad \text{(impossible)}$$ or $$ x^{\frac{n-1}{2}} = 4x - 3.$$ For $x > 1$, it follows from the last equation that $x = 3$ with $n = 5$. Suppose secondly that $x < - 1$. We put $x_1 = -x$ with $x_1 > 1$. The equation (1) is of the form \begin{align*}
\frac{x_1^n + 1}{x_1 + 1} &= y^2 \qquad\text{ or } \\
[(x_1+1)y]^2 - x_1(x_1+1)&\left[x_1^\frac{n-1}{2}\right]^2 = x_1 + 1
\end{align*} Here we have $x_1^{\frac{n-1}{2}} = 1$ (impossible) or $x_1^{\frac{n-1}{2}} = 4x_1 + 3$ which is also impossible for $x_1 > 1$. […] References. [N] = Nagell: (1) in the above linked pdf, [M] = K. Mahler: (1) in the above linked pdf. [M] K. Mahler: Über den grössten Primteiler spezieller Polynome zweiten Grades, Arch. Math. Naturvidensk., 41 (1935), pp. 3-26 [N] Nagell, T., Sur l'équation indéterminée $\frac{x^n−1}{x−1} = y^q$, Norsk. Mat. Forenings Skrifter, Serie I, nr. 3, (1921). Remarks. The paper of K. Mahler is available from here . Incidentally, this archive of K. Mahler’s collected papers deserves to be more well known. There are some simple details that need to be put into the proof but I will leave that to you. Feel free to ask if necessary and I can add some calculations/clarifications. :-) I must note that Ljunggren’s and Mahler’s papers are both so clearly written that you must read them entirely!
|
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|
207,321 |
I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193{,}707{,}721\times 761{,}838{,}257{,}287$ . There doesn't seem to be a historical record of how Cole achieved this; all we have I could find his statement that it took "three years of Sundays". Ira Glass's guest, Paul Hoffman, suggests that this was done by trial division. But this is nuts, unless I am missing something. Three years of Sundays is $156$ days. If he works $10$ hours a day, that's $93{,}600$ minutes. There are $10{,}749{,}692$ primes up to $193{,}707{,}721$ . So that is more than $100$ trial divisions a minute. Worse than that, existing prime tables didn't go high enough: According to Chapter XIII of Dickson's History of the Theory of Numbers , existing tables of primes only ran to something like $10{,}000{,}000$ ( $664{,}579$ primes), so for the vast majority of the trial divisions, he'd have to find the primes first. (Lehmer, in 1914, went up to $10{,}006{,}721$ .) But I'm puzzled thinking what else Cole could have done. I skimmed Chapter XIV in Dickson. The methods which seem to have existed at the time are: Various ways to speed up trial division for the first $1000$ 's of prime numbers. That only helps at the start. Writing $N$ as $x^2-y^2$ . But $y$ would be $380{,}822{,}274{,}783$ , which is an even larger search. Since $2$ is a square modulo $N$ (namely, $(2^{34})^2 \equiv 2 \mod N$ ), we know that all prime factors must be $\pm 1 \bmod 8$ , which cuts the time in half. But that's only a factor of $2$ . Since $N \equiv 3 \mod 4$ , there must be a prime factor which is $3 \bmod 4$ , so we could try only checking those primes. But this turns out to make things worse, since the SMALL factor is the one which is $1 \bmod 4$ . If we could write $N$ as a sum of squares in two ways, we'd be done. But $N$ isn't a sum of squares, since it is $3 \bmod 4$ . Generalizations to other positive definite quadratic forms were known at the time, but how would Cole know which quadratic form to try? A variant of the above would be to use the quadratic form $2x^2-y^2$ , since we already have one solution. Dickson doesn't mention any work using mixed signature forms, but it would work. And since $\mathbb{Z}[\sqrt{2}]$ is a PID, there must be a second way to write $N$ as $2x^2-y^2$ , not related to the previous by units of $\mathbb{Z}[\sqrt{2}]$ . I'm not sure how large this second solution is. So, my question is: How could someone find the prime factors of $N$ in $100{,}000$ minutes of hand computation?
|
The paper by Cole "On the factoring of large numbers." BAMS (1903) discusses this.
|
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207,365 |
Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$. Q . Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a (closed) geodesic on $S$? Here the metric on $S$ is inherited from $\mathbb{R}^3$.
The curve $\gamma$ could be knotted, but it is non-self-intersecting.
I am seeking $S$ homeomorphic to a sphere, i.e., genus-zero. One can construct an appropriate surface patch locally in a neighborhood of each point
$x \in \gamma$, but it is unclear to me how to argue that these patches
can be completed to a genus-$0$ embedded surface $S$. Revision . Andy Putman and Igor Rivin both answered the original question: No if $\gamma$ is knotted.
So I have revised the question to restrict $\gamma$ to be unknotted.
|
Edit: See the end for a summary of this answer I disagree with the statement "One can construct an appropriate surface patch locally in a neighborhood of each point". In fact, there are local obstructions to the existence of the desired surface. Let $\gamma:(-\epsilon,\epsilon) \rightarrow \mathbb{R}^3$ be an embedded arc parameterized proportional to arc length and let $S \subset \mathbb{R}^3$ be a smooth surface containing $\gamma$. Then $\gamma$ is a geodesic on $S$ if and only if $\gamma''(t)$ is orthogonal to the tangent plane of $S$ for all $t$. The tangent planes of $S$ thus give a smoothly varying family of planes in the restriction to $\gamma$ of the tangent bundle of $\mathbb{R}^3$ which are orthogonal to $\gamma''$. Such a family of planes need not exist. The problem arises at points where $\gamma''(t)=0$; it is clear what the tangent plane to $S$ must be elsewhere. For example, let $\gamma_1:(-\epsilon,\epsilon)\rightarrow \mathbb{R}^3$ be a smooth embedded curve with the following properties. For all $t$, we have $\|\gamma_1'(t)\| = 1$. In particular, $\gamma_1$ is parameterized proportional to arc length. For $-\epsilon<t\leq 0$, we have $\gamma_1(t) = (0,0,t)$. For $0<t<\epsilon$, we have $\gamma_1''(t) \neq 0$ and $\gamma_1(t) \in \{(x,y,z)\text{ $|$ }z > 0\}$. Such curves are easy to construct. Now, of course, we might have already found a problematic curve, but let's assume that we haven't, so there exists a surface $S_1$ in $\mathbb{R}^3$ containing $\gamma_1$ such that $\gamma_1$ is a geodesic in $S_1$. For $0<t<\epsilon$, let $n_1(t) \in \mathbb{P}(\mathbb{R}^3)$ be the line in the direction $\gamma_1''(t)$. We know that the tangent plane to $S_1$ at $\gamma_1(t)$ is the orthogonal complement of $n_1(t)$. This implies that $v_1:=\lim_{t \mapsto 0^+} n_1(t)$ exists: it is the orthogonal complement to the tangent plane to $S_1$ at $(0,0,0)$. The key point here is that the tangent plane to $S_1$ at $(0,0,0)$ is uniquely determined by $\gamma_1$. It is clear that $v_1 \neq [0,0,1]$. Let $M:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the orthogonal linear map obtained by composing the reflection in the $z$-axis with a small rotation in the $xy$-plane.
Define $\gamma_2:(-\epsilon,\epsilon) \rightarrow \mathbb{R}^3$ via the formula
$$\gamma_2(t) = M(\gamma_1(-t)).$$
The following then hold. For all $t$, we have $\|\gamma_2'(t)\| = 1$. In particular, $\gamma_2$ is parameterized proportional to arc length. For $0 \leq t < \epsilon$, we have $\gamma_2(t) = (0,0,t)$. For $-\epsilon<t<0$, we have $\gamma_2''(t) \neq 0$ and $\gamma_2(t) \in \{(x,y,z)\text{ $|$ }z < 0\}$. For $-\epsilon<t<0$, define $n_2(t) \in \mathbb{P}(\mathbb{R}^3)$ to be the line in the direction
$\gamma_2''(t)$. Then $v_2:=\lim_{t \mapsto 0^-}n_2(t)$ exists and is different from $v_1$ (this is the point
of the rotation in the $xy$-plane in the definition of $M$). Now define $\gamma:(-\epsilon,\epsilon) \rightarrow \mathbb{R}^3$ via the formula
$$\gamma(t) = \begin{cases}
\gamma_2(t) & \text{if $-\epsilon<t\leq 0$},\\
\gamma_1(t) & \text{if $0<t<\epsilon$}. \end{cases}$$
The second condition on $\gamma_1$ and $\gamma_2$ implies that $\gamma$ is a smooth curve. For $-\epsilon<t<\epsilon$
satisfying $t \neq 0$, we have $\gamma''(t) \neq 0$; define $n(t) \in \mathbb{P}(\mathbb{R}^3)$ to be the line
in the direction of $\gamma''(t)$. We then have
$$\lim_{t \mapsto 0^+} n(t) = v_1$$
and
$$\lim_{t \mapsto 0^-} n(t) = v_2.$$
These are different, so $\lim_{t \mapsto 0} n(t)$ does not exist. This implies that $\gamma$ cannot possibly
be a geodesic in any surface. Of course, $\gamma$ is not a closed curve, but it is easy to close it up to a simple closed unknotted curve. In this edit, I'll comment on further obstructions. Let's assume that the desired family of planes on $\gamma$ exists. Using a tubular neighborhood, it is easy to find a small "strip" around $\gamma$ on which $\gamma$ is a geodesic. As David points out, this strip might be a Mobius band, which is bad, so let's assume that it is an annulus $A$ with boundary components $\alpha_1$ and $\alpha_2$. There is now a new obstruction: the linking number of $\alpha_1$ with $\gamma$ might be nonzero (nb: it is an easy exercise to see that the linking numbers of $\alpha_1$ and $\alpha_2$ with $\gamma$ must be the same, though since these loops are unoriented these linking numbers are only well-defined up to signs). This is a problem because we want $\alpha_1$ to bound a disc in $\mathbb{R}^3 \setminus \gamma$, which since $\gamma$ is an unknot is homeomorphic to the result of removing a point from an open disc cross $S^1$; in particular, $\pi_1(\mathbb{R}^3 \setminus \gamma) = \mathbb{Z}$. The linking number of $\alpha_1$ with $\gamma$ is the image of $\alpha_1$ in $\pi_1(\mathbb{R}^3 \setminus \gamma)$. So we have to assume that this linking number is $0$. Letting $U$ be a closed tubular neighborhood of $\gamma$ containing $A$ with $\partial A \subset \partial U$, we have $\pi_1(\mathbb{R}^3 \setminus \gamma) = \pi_1(\mathbb{R}^3 \setminus U)$. We deduce that $\alpha_1$ is nullhomotopic in $\mathbb{R}^3 \setminus U$. Applying Dehn's Lemma (which is overkill in this situation, but why not?), we see that $\alpha_1$ bounds a disc $D_1$ in $\mathbb{R}^3 \setminus U$. We now want $\alpha_2$ to bound a disc in $\mathbb{R}^3 \setminus (A \cup D_1)$, but this is easy since $A \cup D_1$ is homeomorphic to a closed disc, so $\mathbb{R}^3 \setminus (A \cup D_1)$ is homotopy equivalent to $\mathbb{R}^3 \setminus \{x_0\}$ for a point $x_0 \in A \cup D_1$; in particular, $\pi_1(\mathbb{R}^3 \setminus (A \cup D_1)) = 0$. I want to close with one further remark about the above. Assuming it exists, the family of planes that was the input to the above construction is unique exactly when the set of points on $\gamma$ where $\gamma'' \neq 0$ is dense. But if there exists an interval on which $\gamma''=0$, then we can use that interval to introduce as many half twists as we need to the planes to get rid of the Mobius band and linking number obstructions. SUMMARY OF ANSWER Let me summarize the answer, which gives a
set of necessary and sufficient conditions for the existence of the
sphere (whose logic is, alas, a little complicated). First, there is a
"local" obstruction that must be satisfied for the desired sphere to exist. It
can be defined as follows. Let $U \subset \gamma$ be the set of all
points where $\gamma''$ is nonzero. Define
$\phi:U \rightarrow \mathbb{P}(\mathbb{R}^3)$ to take $u \in U$ to the
line in the direction of $\gamma''(t)$. Then for a sphere to exist,
the function $\phi$ must be able to be extended to a function
$\widehat{\phi}:\gamma \rightarrow \mathbb{P}(\mathbb{R}^3)$ such
that $\widehat{\phi}(x)$ is orthogonal to $\gamma'(x)$ for all $x$. This
is a vacuous condition if $\gamma''$ never vanishes. Now assume that such a $\widehat{\phi}$ exists. If $U$ is not dense
in $\gamma$, then no further conditions are needed: the sphere exists. Otherwise, two further conditions are needed. Observe that in
this case, by the way, the extension $\widehat{\phi}$ is unique. The
first is that there exits a continuous function
$\widehat{\psi}:\gamma \rightarrow S^2$ such that
$\widehat{\phi}(x)$ is the line in the direction $\widehat{\psi}(x)$
for all $x \in \gamma$. This condition is vacuously satisfied if
$\gamma''$ never vanishes (just take $\widehat{\psi}(x)$ to be the
unit vector in the direction of $\gamma''(x)$). The purpose of
this condition is to ensure that the "strip" defined in the answer is
not a Mobius band. If this condition holds,
then we can define the "winding number" of $\widehat{\psi}$ around
$\gamma$ since $\widehat{\psi}(x)$ lies in the orthogonal complement
of $\gamma'(x)$, which is a great circle in $S^2$. The second
needed condition is that this winding number vanishes.
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|
207,438 |
Let $w$ be a group word with two variables $x$ and $y$.
Is the sentence $(\forall x)(\exists y)w=1$
true in every group if it is true
in every finite group?
The same question about the sentence $(\exists x)(\forall y)w=1$.
|
The answer is Yes for the second question, about $(\exists x)(\forall y)w=1$. Following Christian Remling's idea:
If a sentence like
$$\exists x(\forall y)(yxy^{-1}x^2y^{-9}\dots=1)$$
holds in all finite groups then it holds in $\mathbb Z/n\mathbb Z$
where it just says (for certain constants $a,b,c,d$)
$$
(\exists x)(\forall y)((a-b)x+(c-d)y=0).
$$
The only way this can be true is if $c=d$.
So the exponents of $y$ in $w$ add up to 0 .
In that case, the sentence is true in all groups because we can take $x=e$, the group identity (called 1 by the OP). The answer is also Yes on Question 1. If $\forall x\exists y (w=1) $ holds in
$\mathbb Z/n\mathbb Z$ then there it says $ ax=by $, i.e., $ b $ divides all $ ax $, so $ b $ divides $ a $. But then in any group given $ x $ we can take $ y=x^{-a/b} $. On the other hand, Wikipedia gives the following $\Pi^0_2$ sentence where the answer is No: given two elements of order 2, either they are conjugate or there is a non-trivial element commuting with both of them .
|
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|
207,477 |
It would seem that John Nash and his wife Alicia died tragically in a car accident on May 23, 2015 ( reference ). My condolences to his family and friends. Maybe this is an appropriate time to ask a question about John Nash's work which has been on my mind for awhile. John Nash's best known work to the world at large involves his contributions to game theory, but to many geometers his work on embeddings of Riemannian manifolds is really his crown jewel. An excerpt from a note by Gromov : When I started studying Nash’s 1956 and 1966 papers (it was at Rokhlin’s
seminar ≈1968), his proof has stricken me as convincing as lifting oneself by the
hair. Under a pressure by Rokhlin, I plodded on, and, eventually, got the gist
of it... Trying to reconstruct the proof and being unable to do this, I found out that my ”formalization by definitions” was incomplete and my argument, as stated in 1972 was invalid (for non-compact manifolds). When I simplified everything up
and wrote down the proof with a meticulous care, I realized that it was almost
line for line the same as in the 1956 paper by Nash - his reasoning turned out
to be a stable fixed point in the ”space of ideas”! (I was neither the first nor
the last to generalize/simplify/improve Nash, but his proof remains unrivaled.) So I'm wondering if anyone can comment on the legacy of Nash's work in geometry today. Have his ideas been absorbed into a larger theory? Have his techniques found applications outside of manifold embeddings? Perhaps this is a good place to comment on other parts of his mathematical legacy as well, if anyone would like to.
|
Nash's major contributions, as far as I know, are the following: His work on game theory. EDIT: This is viewed by many mathematicians as being more important to economics than mathematics. However, see the answer by Gil Kalai (someone else whose views should be taken much more seriously than mine). His famous work on the existence of smooth isometric embeddings of Riemannian manifolds into Euclidean space. As Denis Serre mentions, he developed in this paper what is now known as the Nash-Moser implicit function theorem, which has been used in other applications. His work on regularity of solutions to elliptic and parabolic PDE's, which were also obtained, I believe independently by Moser and DiGorgi. This is perhaps his most cited work. His theorem about how any Riemannian manifold has a $C^1$ isometric embedding as a codimension 2 submanfold of Euclidean space. Kuiper showed that the embedding could be actually a hypersurface (codimension 1). This is perhaps his most spectacular theorem.
Recently, De Lellis, László Székelyhidi, and their collaborators have used Nash's original "twist" construction to obtain new results on Onsager's conjecture, which is about the apparently totally unrelated topic of fluid dynamics. Nash's influence on mathematics (and on my own work) is enormous.
|
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|
207,589 |
(First posted on math.SE , with no answers.) That is: For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$? The discussion at this math.SE question has narrowed it down to the following two cases: either $n = k-1$, in which case $k = 2, 4, 8$, realized by the complex, quaternionic, and octonionic Hopf fibrations, or $n = 3k-3$, in which case $k \ge 4$ is even. I am moderately confident that the second case doesn't occur, but don't know how to rule it out. Here's what I can show about it: such a submersion gives rise to a smooth fiber bundle $F \to S^{4k-3} \to S^k$ where $F$ is a smooth frameable closed manifold of dimension $3k-3$. Taking homotopy fibers gives a map $\Omega S^k \to F$ whose homotopy fiber is $(4k-5)$-connected, hence which induces an isomorphism on homotopy and on cohomology up to degree $4k-5$. This determines the cohomology of $F$ as a ring: $F$ has the cohomology of $S^{k-1} \times S^{2k-2}$. When $k = 2$ Mike Miller showed that $F$ must in fact be homeomorphic to $S^1 \times S^2$ and then gets a contradiction from looking at homotopy groups. When $k \ge 4$ we also know that $F$ is simply connected. Aside from knowing whether it's possible to rule out the last case, I'd also be interested in a simpler argument that $k$ must be even. The argument I gave passes through both the topological Poincaré conjecture and Adams' solution to the Hopf invariant $1$ problem...
|
In most cases $\pi_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitrarily high dimensions and cannot be a manifold. The only exceptions with $n>0$ have $n=k-1$.
|
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|
207,590 |
It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus in his Zur Geschichte der Mathematik in Alterthum und Mittelalter (p. 163): At this point, there is a mistake to be corrected, a mistake that is reinforced by false nomenclature and therefore, or so I fear, impossible to weed out. In education, one designates linear equations of the form $ax+by=c$, that are to be solved in integers $x,y$, as Diophantine . Now, not only did Diophantus not know the solution method of these equations, that in the West was first obtained by his commentator Bachet, but the very problem would have been utterly alien to him, since he never fixes the condition that his solutions should be integral, but is completely satisfied with rational solutions. (Italics are mine.) Now, in connection with this passage, I have the following questions: When did people start talking/writing about "Diophantine equations"? And also: Were Diophantine equations originally considered as "polynomial equations to be solved in rational numbers", in accordance with Diophantus' own preference, or was the mistake that Hankel aims to correct made from the beginning?
|
In most cases $\pi_{n+k}(S^k)$ is a finite group, so that the homotopy fiber of any map $S^{n+k}\to S^k$ is rationally equivalent to $\Omega S^k\times S^{n+k}$ and therefore has homology in arbitrarily high dimensions and cannot be a manifold. The only exceptions with $n>0$ have $n=k-1$.
|
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|
207,668 |
I have two questions regarding to $p$-groups. A $p$-group $G$ is said to be extraspecial of $G'=Z(G)$ has order $p$. Hence extraspecial groups are examples of $p$-groups with cyclic center. Of course there are many other $p$-groups with cyclic center that are not extraspecial. I would to know if there is any classification of $p$-groups with cyclic center. Is there any classification of $p$-groups of order $p^n$ and nilpotency class $k$ for suitable fixed $k$ and $n$?
|
Every $p$-group is a homomorphic image of a $p$-group with cyclic center of order $p$, so a classification (whatever that means) of $p$-groups with cyclic center would (more-or-less) yield a construction for all $p$-groups, and I would not hold my breath waiting for that. To see why a $p$-group $P$ is a homomorphic image of a $p$-group $G$ with center of order $p$, let $G$ be the regular wreath product of a cyclic group of order $p$ with $P$. Thus $G$ has an elementary abelian subgroup $E$ of order $p^{|P|}$, where $P$ permutes the cyclic factors of $E$ the way it permutes its own elements by right multiplication, and thus $P$ acts faithfully on $E$. Also, $G$ is the semidirect product of $E$ by $P$. It is easy to see that $E \cap {\bf Z}(G)$ has order $p$, so I need to show that every element of ${\bf Z}(G)$ lies in $E$. If $z \in {\bf Z}(G)$, write $z = au$, where $a \in E$ and $u \in P$. Since $z$ centralizes $E$ and $a$ centralizes $E$, it follows that $u$ centralizes $E$ and thus $u = 1$ by the faithfulness of the action. Thus $z = a \in E$, as required.
|
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|
207,845 |
I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture)
For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , x_n, y_1,y_2,\ldots, y_n>0$ such that for all
$k\in\{1,\ldots, n-1\}$
it holds $\sum_{i_1<\ldots<i_k} x_{i_1}\, x_{i_2}\ldots x_{i_k}\le \sum_{i_1<\ldots<i_k} y_{i_1}\, y_{i_2}\ldots y_{i_k}$ and $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2 \,y_3\ldots y_n$ it follows $\sum_{i=1}^n (\log x_i)^2\le \sum_{i=1}^n (\log y_i)^2$ Replacing the assumption $x_1\, x_2\, x_3 \ldots x_n=y_1\, y_2\, y_3\ldots y_n$ by
$x_1\, x_2\, x_3 \ldots x_n\le y_1\, y_2\, y_3\ldots y_n$ easily admits counterexamples. Proofs are known for $n\in \{1,2,3,4\}$. More information can be found at https://www.uni-due.de/mathematik/ag_neff/log_conjecture Immediately after Lev Borisov's sketch of a proof idea below, Lev Borisov, Suvrit Sra, Christian Thiel and myself agreed to work out the details and to write together a complete and self-contained paper on the sum of squared logarithm conjecture and relations to other topics which can be found at: http://arxiv.org/abs/1508.04039 ${}{}{}$ As announced in my first post, the prize winner is Lev Borisov.
|
Is there anything wrong with the following argument? First of all, by scaling all $x_i$ and all $y_i$ by a positive constant, we may safely assume that $\prod_i x_i = \prod_i y_i =1$. The result would now follow from the following more general conjecture. $\bf Conjecture:$ For ${\bf a}=(a_1,\ldots,a_{n-1})\in (\mathbb R_{>0})^{n-1}$ consider
$h_{\bf a}(z) = z^n+a_{n-1}z^{n-1}+\cdots+a_1z+1$. Define the function $f:(\mathbb R_{>0})^{n-1}\to {\mathbb R}$ by
$$
f(a_1,\ldots,a_{n-1}) = \sum_{z\vert h_{\bf a}(z)=0} (\log(-z))^2
$$
where the branch of $\log$ is picked as usual on the complement of ${\mathbb R}_{\leq 0}$ in $\mathbb C$. Then all partial derivatives $\frac{\partial f}{\partial a_k}$ are positive. First, a few comments. The function $f$ is well-defined because none of the roots
of $h_a(z)$ are positive reals. In addition, $f$ is real-valued, because roots of $h_a(z)$ come in complex conjugate pairs for which values of $\log(-z)$ are complex conjugates.
The consequence of this conjecture is that if ${\bf a}\leq {\bf b}$ coordinate-wise,
then $f({\bf a})\leq f({\bf b})$. Applied to the case when roots of $h_{\bf a}(z)$
and $h_{\bf b}(z)$ are real numbers $-x_i$ and $-y_i$, we get the original conjecture. Now, let me describe what I think is the proof of this new conjecture. First of all, as standard, I can write the function $f(\bf a)$ by a contour integral. In a neighborhood of fixed ${\bf a}$ for any large enough $R$ and small enough $\epsilon>0$ there holds
$$
f({\bf a}) = \frac 1{2\pi i}\int_{C} (\log(-z))^2 \frac {h_{\bf a}'(z)}{h_{\bf a}(z)}\,dz
$$
over the contour $C$ on the union of $\mathbb C$ with two copies of $\mathbb R_{>0}$
which I will call "north" and "south" shores (so that $\log(-t)=\log t -\pi i$ on the north shore and $\log(-t)=\log t +\pi i$ on the south shore).
The contour $C$ is the union of the following four pieces $C_\epsilon$, $C_R$, $C_+$, $C_-$. $C_\epsilon$ is a circle of radius $\epsilon$ around $z=0$ traveled from $\epsilon$-south
to $\epsilon$-north clockwise. $C_R$ is a circle of radius $R$ around $z=0$ traveled from $R$-north to $R$-south counterclockwise. $C_+$ is the line segment $[\epsilon,R]$-north. $C_-$ is the line segment $[R,\epsilon]$-south. The derivative $\frac{\partial f({\bf a})}{\partial a_k}$ is the integral of the
derivative, so we get:
$$
\frac{\partial f({\bf a})}{\partial a_k}= \frac 1{2\pi i}\int_{C} (\log(-z))^2 \frac \partial{\partial a_k}\frac {h_{\bf a}'(z)}{h_{\bf a}(z)}\,dz
$$
$$
=
\frac 1{2\pi i}\int_{C} (\log(-z))^2 \Big(\frac {z^k}
{h_{\bf a}(z)}
\Big)'\,dz
=
-\frac 1{2\pi i}\int_{C} \Big((\log(-z))^2 \Big)'\frac {z^k}
{h_{\bf a}(z)}
\,dz
$$
$$
=-\frac 1{\pi i}\int_{C} \log(-z)\frac {z^{k-1}}
{h_{\bf a}(z)}\,dz = -\frac 1\pi {\rm Im}\Big(\int_{C} \log(-z)\frac {z^{k-1}}
{h_{\bf a}(z)}\,dz\Big)
$$
We can take a limit as $R\to +\infty$ and $\epsilon\to 0$. Since $k\leq {n-1}$ the integral over $C_R$ goes to zero (the length is $2\pi R$ and the size of the function is
$O(R^{-2}\log R)$). The integral over $C_\epsilon$ also goes to zero, because the $k>=1$,
so the function is $O(\log \epsilon)$ and the length is $2\pi\epsilon$. So we get
$$
\frac{\partial f({\bf a})}{\partial a_k}
= -\lim_{\epsilon\to 0^+}\frac 1\pi {\rm Im}\Big( \int_{[\epsilon,+\infty]-{\rm north}\cup [+\infty,\epsilon]-{\rm south}}
\log(-z)\frac {z^{k-1}}
{h_{\bf a}(z)}\,dz\Big)
$$
$$
=-\lim_{\epsilon\to 0^+}\frac 1\pi
\int_{\epsilon}^{+\infty}{\rm Im}\Big(
(\log(t)-\pi i)\frac {t^{k-1}}
{h_{\bf a}(t)} -(\log(t)+\pi i)\frac {t^{k-1}}
{h_{\bf a}(t)}\Big)\,dt
$$
$$
=2\lim_{\epsilon \to 0^+}
\int_{\epsilon}^{+\infty}
\frac {t^{k-1}}{h_{\bf a}(t)}\,dt
>0.$$
This finishes the proof of the new conjecture, and consequently of the old conjecture. Remark:
I am guessing that there is a simpler argument for
$$
\frac{\partial f({\bf a})}{\partial a_k} = 2\int_{0}^{+\infty}
\frac {t^{k-1}}{h_{\bf a}(t)}\,dt
$$
but I am just writing the first thing that came to my mind.
|
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|
208,071 |
Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$ In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient conditions for $\sum a_nz^n$ to be in $\mathbb{C}(z)$ with all poles simple?
|
The function
$$ f(z)=\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} $$
defines a holomorphic function for $|z|<2$, and it satisfies
$$ f(2z) = f(z)+\frac{z}{1-z} $$
for $|z|<1$. Based on this identity, it is easy to prove that $f(z)$ extends to a meromorphic function on $\mathbb{C}$, and the set of poles is $\{2^n:\ n=1,2,\dots\}$. In particular, $f(z)$ does not define a rational function, because its meromorphic extension to $\mathbb{C}$ has infinitely many poles. Regarding your second question, I recommend the work of Dwork (with which I am not familiar), e.g. (8) in Alain Robert's article "Des adèles: pourquoi" , and Lemma 9 in Tao's blog . See also Remark 2 below. Remark 1. A more direct proof of the above claims follows from the identity
$$ \sum_{m=1}^\infty\frac{z}{2^m-z} = \sum_{n=1}^\infty \frac{z^n}{2^n-1},\qquad |z|<2. $$
Indeed, left hand side defines a meromorphic function on $\mathbb{C}$ with pole set $\{2^m:\ m=1,2,\dots\}$. Remark 2. One can give a different, number theoretic proof using Eisenstein's theorem on algebraic functions (the proof was published by Heine because of Eisenstein's early death). Indeed, the Taylor coefficients of $f(z)$ around the origin are rational, but their denominators $2^n-1$ are not supported on finitely many primes by Fermat's little theorem. (As Gerald Edgar remarked below, this argument proves that $f(z)$ is not even algebraic.)
|
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|
208,112 |
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. Sometimes, it goes the other way, i.e. algebraic topology is used to attack some problem from algebra. Here are two examples I particularly like: 1) Schreiers theorem: Every subgroup of a free group is free: This reduces to study coverings of some wedge of circles. 2) Group cohomology: E.g. questions like "Does every finite group have nontrivial cohomology in infinitely many degrees?" - see Non-vanishing of group cohomology in sufficiently high degree . Such questions can be phrased purely algebraically, but still the best way to think about them is via topology. I am interested in seeing more examples of this kind. What is your favourite application of topology in algebra?
|
Theorem (Arnold - 1970): The algebraic function defined by the solutions of the equation $$\ \ z^n+a_1z^{n-1}+\cdots +a_{n-1}z+a_n=0\ \ \ ,$$ cannot be written as a composition of polynomial functions of any number of
variables and algebraic functions of less than $\phi(n)$
variables, where $\phi(n)$
is
$n$
minus the number
of ones appearing in the binary representation of the number
$n$. The proof is essentially a clever application of the computation of the mod. 2 cohomology ring of the braid group $B_n$ by Fuchs. (And I seem to remember Vershinin explaining that Arnold asked Fuchs to compute this ring for this very reason).
|
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|
208,341 |
Recently I discovered the differential identity $$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$ valid for any odd natural number $k$; for instance $\frac{d^6}{dx^6} (1+x^2)^{5/2} = \frac{225}{(1+x^2)^{7/2}}$. This identity was surprising at first, since usually the repeated application of the product rule and chain rule leads to far messier expressions than this, but there are now several proofs that adequately explain this identity (collected at this blog post of mine ). There is also the more general identity $$ |\frac{d}{dx}|^{2s-1} (1+x^2)^{s-1} = \frac{2^{2s-1}\Gamma(s)}{\Gamma(1-s)} (1+x^2)^{-s}$$ valid for any complex $s$ (if everything is interpreted distributionally), which is related to the isomorphisms between principal series representations of $PGL_2({\bf R})$. The purpose of my question here is not to ask for more proofs of this identity (but you are welcome to visit the above-mentioned blog post to contribute another proof, if you wish). Instead, I am asking as to whether this identity (or something close to it) already appears in the literature - I find it hard to believe that such a simple identity has been missed for centuries, given that it could easily have been discovered and proven by (say) Euler. The closest match that I know of so far are the Rodrigues formulae for the classical orthogonal polynomials, but I was not quite able to place the above identity as a special case of these formulae (the exponents don't quite match up).
|
It's the Rodrigues formula for Gegenbauer polynomials $C_n^{(\alpha)}(x)$, in the special case $\alpha=-n/2$ when the polynomial is just unity. The general formula reads $$C_n^{(\alpha)}(x)=\frac{(-2)^n}{n!}\frac{\Gamma(n+\alpha)\Gamma(n+2\alpha)}{\Gamma(\alpha)\Gamma(2n+2\alpha)}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].$$ Substitution of $\alpha=-n/2$, with $C_n^{(-n/2)}(x)=1$ for $n$ even, gives $$1=\frac{(-2)^n(n/2)!}{n!(n-1)!}\lim_{\alpha\rightarrow-n/2}\frac{\Gamma(n+2\alpha)}{\Gamma(\alpha)}(1-x^2)^{(n+1)/2}\frac{d^n}{dx^n}\left[(1-x^2)^{(n-1)/2}\right]$$ $$\qquad = (-1)^{n/2}\left(\frac{1}{(n-1)!!}\right)^2(1-x^2)^{(n+1)/2}\frac{d^n}{dx^n}\left[(1-x^2)^{(n-1)/2}\right],$$ or with $x\to ix$, $$\frac{d^n}{dx^n}\left[(1+x^2)^{(n-1)/2}\right]=[(n-1)!!]^2(1+x^2)^{-(n+1)/2},$$ which is the desired expression.
|
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|
208,574 |
When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e., when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm interested in solving the diophantine equation $x^3-2y^3=1$ using this specific method. Is there any way I can use Skolem's p-adic method here?
|
The only solutions of $x^3-2y^3=1$ are $(x,y)=(1,0)$ and $(x,y)=(-1,-1)$.
I don't know whether there's a nice Skolem-style proof, but here
this happens to be unnecessary because $(1,0)$ and $(-1,-1)$ are the
only rational solutions and this can be proved by a Fermat-style descent:
the Weierstrass form is $Y^2 = X^3 - 27$, and there's a $2$-torsion point
at $(X,Y)=(3,0)$. One could also use descent via $3$-isogeny to
$Y^2 = X^3+1$, which has $6$ rational points, at $\infty$, $(-1,0)$,
$(0,\pm1)$, and $(2,\pm3)$. ADDED LATER : 1) As I already reported in a comment,
the result on $x^3 - 2y^3 = 1$ turns out to be due to Euler himself.
I found the reference in Dickson's History of the Theory of Numbers,
Vol II on page 572: it is Theorem 247 in Euler's Elements of Algebra ,
see p.456 ff. of this
English translation (Google Books scan of a Harvard library book from 1829).
It looks like Euler chose to use a 3-descent (presumably because
it was in the context of equations of the form $ax^3+bx^2+cx+d = y^3$),
even though a 2-descent was also available. 2) Meanwhile Rene Schoof notes that his book Catalan's Conjecture reproduces a 3-adic proof using Skolem's method,
"from Bill McCallum's 1977 honours project at the University of Sydney".
See Proposition 4.1, pages 17-19. [The $\root 3 \of 4$ coefficient of
$(\root 3 \of 2 - 1)^n$ is $0 \bmod 3$ iff $n = 3k$ or $n = 3k+1$,
and in both cases it vanishes mod $3^e$ iff $k$ does (each $e=1,2,3,\ldots$,
by induction on $e$), whence the known zeros for $k=0$ are the only ones.]
In the first paragraph of page 17, Schoof cites Euler's proof by descent,
which he gives later in the book in an Appendix.
|
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|
208,645 |
In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$. So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd prime $p$, there is not any divisor $a$ of $p^4+1$, where $1<a<p^4+1$ and $a=kp+1$, for some $k>0$.
Could you help me about this question? If it is true how we can prove it? Also when I checked the same fact for $p^8+1$, we get many counterexamples. What is the difference between $4$ and $8$?
|
There's none indeed. Lemma : if $1<m<n$ are coprime integers then $mn+1$ does not divide $n^4+1$. First observe that for any $m,n$, of $mn+1$ divides $n^4+1$, then it divides $n^4m^4+m^4=((nm)^4-1)+1+m^4$, and since $mn+1$ clearly divides $((nm)^4-1)$, we deduce that it also divides $1+m^4$. To prove the lemma, assume the contrary. As we have just seen, $mn+1$ divides $m^4+1$ as well. Write $(m^4+1)/(mn+1)=k$, and $k=(\ell m+r)$ with $0\le r\le m-1$. Then $m^4+1=(\ell m+r)((mn+1)=mN'+r$, so $r=1$. So $(\ell m+1)$ divides $m^4+1$. Then $m$ and $\ell$ are coprime: indeed, we have $(m\ell +1)(mn+1)=m^4+1$, so $\ell(mn+1)-m^3=-n$. If a prime $p$ were dividing both $m$ and $\ell$ then it would also divide $n$, contradicting that $m$ and $n$ are coprime, so $m$ and $\ell$ are indeed coprime. We have $\ell<n$ because otherwise
$$m^4+1=(mn+1)(m\ell+1)\ge (mn+1)^2> (m^2+1)^2>m^4+1.$$ So we found a new pair $(m,\ell)$ with $\max(m,\ell)<n$, with $m\ell+1$ dividing both $m^4+1$ and $\ell^4+1$. So, assuming that $n$ is minimal, we're done unless $\ell=1$. This happens if $m+1$ divides $m^4+1$, and since $m+1$ divides $m^4-1$ as well, if this occurs then $m+1=2$, contradicting $m>1$. (Note: without the coprime assumption the conclusion fails, as $(m,n)=(m,m^3)$ for $m\ge 2$, e.g. $(m,n)=(2,8)$, satisfies $mn+1|n^4+1$.) Proposition If $p$ is prime then $p^4+1$ has no divisor of the form $kp+1$ except $1$ and $p^4+1$. Proof: write $p^4+1=(kp+1)(k'p+\ell)$ with $0\le\ell\le p-1$; then $\ell=1$. So exchanging $k$ and $k'$ if necessary we can suppose $k\le k'$.
If by contradiction $k$ is divisible by $p$ then $k'\ge k\ge p$, so $p^4+1\ge (p^2+1)^2>p^4+1$, contradiction. So $k$ and $p$ are coprime, and the lemma yields a contradiction.
|
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|
208,820 |
Question : Should we post on arXiv only papers in publishable shape (or very close)? This question should be distinguished from the following: Should one post a paper on the arXiv if it is not intended to be published? in the sense that a paper which has not a publishable "contents" can have a publishable "shape". Moreover, a paper which is intended to be published can be not yet in a publishable shape. Sometimes it happens that we start to write a paper, we develop some interesting new ideas, but then we do not continue the paper for several possible reasons: change of the research subject loss of motivation for this problem too difficult $\dots$ Sometimes also we just want to share the current state of our work through a draft, even if all the proofs are not yet complete. Whatever the reasons, the ideas contained in such drafts can be interesting and useful for the community, regardless of the state of advancement of the paper. So after this explanation: Should we definitely not post such a paper on arXiv, even if its state is clearly specified at the beginning ?
|
The documentation of arXiv has this to say (my emphasis): Inappropriate format. arXiv accepts only submissions in the form of an article that would be refereeable by a conventional publication venue. This excludes abstract-only submissions, submissions without references, book announcements or reviews, reports that do not contain original or substantive research, papers that contain inflammatory or fictitious content, papers that use highly dramatic and mis-representative titles/abstracts/introductions, or papers in need of significant review and revision. Source: http://arxiv.org/help/moderation
|
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|
209,058 |
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & a_{34} \\
a_{14} & a_{24} & a_{34} & a_{44} \\
\end{array}
\right)
$$
contains exactly 21 minors of order 2, but there is a linear combination of them, namely
$$
-(a_{13} a_{24}-a_{14} a_{23})+(a_{12} a_{34}-a_{14} a_{23})-(a_{12} a_{34}-a_{13} a_{24})\, ,
$$
which vanishes, as opposed as to the case of symmetric $3\times 3$ symmetric matrices, whose 6 minors of order two are linearly independent. BIG QUESTION : Why is that? I mean, what is the theory behind such a phenomenon? PHILOSOPHICAL QUESTION: What is the "true number" of minors (from the example above), 20 or 21? I already gave myself an explanation, but I still cannot see the big picture. I would be grateful if anyone pointed out a reference, sparing me the efforts of reinventing the wheel. If an $n\times n$ matrix $A$ is regarded as an element of $V\otimes_{\mathbb{K}} V^\ast$, with $V\equiv \mathbb{K}^n$, then there is an obvious way to extend $A$ to a $\mathbb{K}$-linear map
$$
A^{(k)}:\bigwedge^kV\longrightarrow\bigwedge^kV^\ast\, .
$$
If $A$ is symmetric, then so is $A^{(k)}$, i.e., there is a (polynomial of degree $k$) map
$$
S^2(V^\ast)\ni A\stackrel{p}{\longmapsto} A^{(k)}\in W^k:=S^2\left( \bigwedge^kV ^\ast \right)\, .
$$
Observe that $W_k$ has dimension $\frac{{n\choose k}\left({n\choose k}+1\right)}{2}$, which is 21 for $n=4$ and $k=2$. So, I was led to identify $W^k$ with the space of "formal minors" of order $k$ of a $n\times n$ matrix (indeed, the entries of $A^{(k)}$ are precisely such minors). How to explain now the dependency of three of them? There is a linear map
$$
S^2\left( \bigwedge^2V ^\ast \right)\ni\rho\odot\eta\stackrel{\epsilon}{\longmapsto}\rho\wedge\eta\in \bigwedge^4V ^\ast\equiv\mathbb{K}\, ,
$$
and it can be proved that $\epsilon\circ p=0$, i.e., that $p$ takes its values in the subspace
$$
W^2_0:=\ker\epsilon\, ,
$$
which has dimension 20 (vanishing of the above linear combination corresponds precisely to the equation $\epsilon(p(A))=0$). SIDE QUESTION: How to define this $W_0^k$ for arbitrary $n$ and $k$? It should be the "linear envelope" of the image of $p$: but how to exhibit it explicitly? Probably, the theory of representations may answer this: $W^k$ is not always an irreducible $\mathfrak{gl}(n,\mathbb{K})$-module, and $W_0^k$ is the only irreducible component which contains the image of $p$. Provided this guess is true, it doesn't help me finding the expression of $W_0^k$ in terms of tensors on $V$.
|
This is about your specific question. For any vector space $V$ of dimension $n$, one has canonical decompositions
$$
S^2(S^2V^*) \simeq S^4(V^*)\oplus K(V^*)
$$
and
$$
S^2(\Lambda^2V^*) \simeq \Lambda^4(V^*)\oplus K(V^*),
$$
where $K(V^*)$ is an irreducible $\mathrm{GL}(V)$-module of dimension $n^2(n^2{-}1)/12$. It follows that, up to a constant multiple, there is only one nontrivial $\mathrm{GL}(V)$-equivariant linear mapping
$$
S^2(S^2V^*) \longrightarrow S^2(\Lambda^2V^*),
$$
and hence that the number of linearly independent minors of order $2$ of an $n$-by-$n$ symmetric matrix is $n^2(n^2{-}1)/12$. You can answer similar questions by comparing $S^k(S^2V^*)$ with $S^2(\Lambda^kV^*)$ for higher values of $k$. This kind of question goes by the name 'plethysm' in representation theory. For example, when $k=3$, we find that $S^3(S^2V^*)$ and $S^2(\Lambda^3V^*)$ share only one irreducible $\mathrm{GL}(V)$-constituent, which has dimension
$$
\frac{(n{-}2)(n{-}1)^2n^2(n{+}1)}{144},
$$
so this is the number of linearly independent $3$-by-$3$ minors of an $n$-by-$n$ symmetric matrix. Update: Here is a conjectural answer to the big question: Recall that, for a vector space $V$ of dimension $n$, a fundamental system of irreducible $\mathrm{SL}(V)$ modules is given by $\Lambda^kV^*$ for $0<k<n$. If, as is traditional, one assigns highest weights so that $\Lambda^kV^*$ has highest weight $\alpha_k$, then any irreducible representation $W$ of $\mathrm{SL}(V)$ is determined by its highest weight $w = m_1\,\alpha_1 + \cdots + m_{n-1}\,\alpha_{n-1}$, where the $m_i$ are nonnegative integers. Using this nomenclature, $S^2(V^*)$, which is irreducible, has highest weight $2\,\alpha_1$. The following Conjecture is supported by explicit computations for $0<k< n\le 9$. (Presumably, this conjecture (if true) would be easy to prove for an expert combinatorialist or representation theorist, but I am neither.) Conjecture: The $\mathrm{SL}(V)$-modules $S^k(S^2V^*)$ and $S^2(\Lambda^kV^*)$ have only one common irreducible $\mathrm{SL}(V)$-constituent, namely, the irreducible $\mathrm{SL}(V)$-module $W$ of highest weight $2\,\alpha_k$. Since, by the Weyl Character formula, the dimension of the $W$ with highest weight $2\,\alpha_k$ is
$$
\frac1{k{+}1}{n\choose k}{{n+1}\choose k},
$$
it would follow from the Conjecture that this is the number of linearly independent $k$-by-$k$ minors of a symmetric $n$-by-$n$ matrix.
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209,243 |
I came across this apparent random question in some math questions website. At first, I thought it was easy to show that there are no non-trivial integer solutions to this equation, but then I realized that the question is far beyond what I can answer. Irrationality and transcendence of $\pi$ play no role I think, because if we change $\pi$ with $\log_2(9)$ there are non-trivial solutions. My question is, Do you know some tool to attack this kind of problem? I'll immediately delete this question if you think it's outside the scope of MathOverflow, but I know there are really knowledgeable people here who could say something about this, and my "innocent" curiosity for this question made me post it here. Thank you for your attention. PS: more accurate tags for the question are also welcomed.
|
(Turning comments into an answer, as requested) This follows from Schanuel's conjecture but it's probably hard to prove unconditionally. Apply Schanuel to $2\pi i,\log n,\log m$. The last two numbers are linearly independent over $\mathbb Q$ because of your hypothesis and the fact that $\pi$ is irrational. Then all three numbers are linearly independent over $\mathbb Q$ since $2\pi i$ is not real. Finally, the exponentials of all three numbers are rational, so Schanuel implies that the three numbers are algebraically independent over $\mathbb Q$ which is stronger than what you want.
|
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209,465 |
I work in PDEs. I have now written 3 papers. I find my style is of the form: introduction, statement of results, paragraphs to introduce something, lemma, more text, lemma, more text, lemma, more text, theorem, concluding remarks (I missed the proofs). I am getting sick of this type of writing. I want to write my next paper with more style and elegance than something which looks like the output from a workhouse. For example I have seen papers where they come up with some interesting result which they don't put into a lemma but instead put it in a paragraph. When I cite such a thing, I say something like "see the paragraph on page 10 of [1]" so I think it is a bad idea to put results outside environments. Does anyone have any ideas? Basically I want to write a more unpredictable paper instead of the usual routine which I wish to take myself out of. This is good because readers will also find it more interesting.
|
Since the OP is looking for an alternative text structure , here's a possibility that I've seen in Dieudonné's multi-volume Eléments d'Analyse ( Treatise on Analysis ). He does not always put statements of lemmas or theorems, proofs and discussion in separate environments. Instead, breaks the text into numbered Chapters, Sections, and Subsections in the following way. Chapters and Sections have titles, while Subsections are anonymous and referred to only by number, like ( 23.3.10 ), with equations even further subnumbered, like ( 21.4.4.1 ). The point is that each Subsection is short enough to contain only one main idea, which could be a statement with or without proof, a remark, a definition, some motivational paragraphs, or an example. A definition or a statement could be highlighted in italics, but not in a separate environment. It seems to me a decent compromise between citability and a more flexible text structure. As long as the Subsections are short enough, with the most important text somehow highlighted, the reader does not have to spend too much time looking for the information that is being referred to.
|
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|
210,089 |
A paper by Gabor Etesi was published that purports to solve a major outstanding problem: Complex structure on the six dimensional sphere from a spontaneous symmetry breaking Journ. Math. Phys. 56, 043508-1-043508-21 (2015) journal version , current arXiv version . Since this is obviously an important and groundbreaking
result (if true), published in a physical journal, I
am interested whether it is accepted by mathematicians.
|
I am Gabor Etesi, the author of the current paper "Complex structure on the six dimensional sphere from a spontaneous symmetry breaking", Journ. Math. Phys. 56, 043508-1-043508-21 (2015). First of all I would like to thank for the interest in my work on this classical problem. Because I have been asked by Andre Henriques, hereby I confirm that this published 2015 version is indeed completely independent of the wrong and withdrawn 2005 version available on arXiv. Therefore it is absolutely unnecessary to spend time with that version. This new published 2015 version is self-contained and constructs a complex structure on $S^6$ using the following two simple observations: (i) A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field $J$ is mathematically the same as a Higgs field $\Phi$. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined; (ii) A complex structure on $S^6$ is then constructed as the Fourier expansion of the usual Samelson complex structure (regarded as spontaneously broken vacuum solution in the sense of item (i) above) on the exceptional Lie group $G_2$. The mathematical theory of this Fourier expansion is itself very useful and is contained in the text. Please note that this is a freshly new approach to this old problem, apparently without a predecessor. Finally, I would like to kindly ask everybody to read the paper first, before sending negative comments. I am ready to explain some details of the construction however please understand that I cannot take part in an infinitely long discussion. (Recently I have been working on different stuffs.) Thanks again, Gabor
|
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|
210,167 |
While trying to answer this MSE question , I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of positive integers:
$$\begin{align}
\arctan\varphi&=2\arctan1-\frac12\,\arctan2\\
\arctan\varphi^3&=\arctan1+\frac12\,\arctan2\\
\arctan\varphi^5&=4\arctan1-\frac32\,\arctan2\\
\arctan\varphi^7&=3\arctan1+\frac12\,\arctan2-\arctan5\\
\arctan\varphi^9&=\arctan1-\frac12\,\arctan2+\arctan4\\
\arctan\varphi^{11}&=5\arctan1+\frac12\,\arctan2-\arctan5-\arctan34\\
\arctan\varphi^{13}&=3\arctan1-\frac12\,\arctan2+\arctan4-\arctan89\\
\arctan\varphi^{15}&=-2\arctan1+\frac32\,\arctan2+\arctan11
\end{align}$$
I was not able to find such a representation for $\arctan\varphi^{17}$ though. Question 1. Can we prove that it does not exist? I also could not find such a representation for any positive even power. Question 2. Can we prove that it does not exist for any positive even power? Question 3. Is there a simple way to determine if such a representation exists for a given power?
|
Q1, Q2: Such a representation exists for all odd powers of $\varphi$ as we can show by induction. Using the arctangent identity, let us first write: $\arctan \varphi^{2n+1} = \arctan\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v} - \arctan v$ Next, we set the argument $\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v}$ equal to $\varphi^{2n-1}$ in order to set up an induction and solve for $v$. After moving terms around and simplifying via identities involving $\varphi$, we end up with $v = \frac{\varphi^{2n-1}-\varphi^{2n+1}}{1+\varphi^{4n}} = -\frac{1}{L_{2n}}$ where $L_n$ is the $n^{th}$ Lucas number. As $\arctan\left(-\frac{1}{x}\right) = -(2\arctan 1-\arctan x)$, this proves the claim for odd powers of $\varphi$. Q3: For even powers if we set up the analogous equation: $\arctan \varphi^{2n} = \arctan\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} - \arctan v$ However in this case, setting $\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} = \varphi^{2n-2}$ and solving for $v$ gives $v=\frac{-1}{F_{2n-1}\sqrt{5}}$. This implies there cannot be a rational expression for $\arctan\varphi^{2n}$ in terms of $\arctan\varphi^{2n-2}$ and arctangents of integers for, if there were such an expression, the arctangent identity could then be applied to these terms to express $\frac{-1}{F_{2n-1}\sqrt{5}}$ as a rational number. Similar problems with $\sqrt{5}$ appearing in the expression for $v$ occur if we try setting the argument $\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} = \varphi^k$ for other powers $k$ such as $k=2n-1$, so if any $\arctan \varphi^{2n}$ happen to be rational combinations of arctangents of integers, such expressions ought to be unrelated to one another unlike the situation for odd powers. As a side note, this difference between the even and odd powers reminds me of the situation of values of the $\zeta$ function at integers where the $\zeta(2n)$ have simple closed form expressions while it is unknown if any $\zeta(2n+1)$ have a closed form expression. Perhaps someone with more familiarity of $\zeta$ values can comment on whether we might expect such similar behavior here.
|
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|
210,239 |
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here). Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very likely true that
$$\det \left(A^{\frac{1}{2}}(A+B)A^{\frac{1}{2}}+B^{\frac{1}{2}}(A+B)B^{\frac{1}{2}}\right) \ge \det(A+B)^2. $$
Here $A^{\frac{1}{2}}$ is the unique positive definite square root of $A$.
I am able to confirm the $3\times 3$ case. Comments: Only recently did I notice that the majorization $\lambda\left(A^{\frac{1}{2}}(A+B)A^{\frac{1}{2}}+B^{\frac{1}{2}}(A+B)B^{\frac{1}{2}}\right) \prec \lambda(A+B)^2$ follows immediately by THEOREM 2
of [R.B. Bapat, V.S. Sunder, On majorization and Schur products, Linear Algebra Appl. 72 (1985)
107–117.] http://www.sciencedirect.com/science/article/pii/0024379585901478
|
Let $C := A^{1/2} (A+B) A^{1/2} + B^{1/2} (A+B) B^{1/2}$; this is a positive semi-definite matrix with the same trace as $(A+B)^2$. We show that the eigenvalues of $C$ are majorised by the eigenvalues of $(A+B)^2$, that is to say that the sum of the top $k$ eigenvalues of $C$ is at most the sum of the top $k$ eigenvalues of $(A+B)^2$ for any $k$. By the Schur concavity of $(\lambda_1,\dots,\lambda_n) \mapsto \lambda_1 \dots \lambda_n$, this gives the claimed determinantal inequality. The sum of the top $k$ eigenvalues of $C$ can be written as
$$ \hbox{tr}( C P_V )$$
where $V$ is the $k$-dimensional space spanned by the top $k$ eigenvectors of $C$. This can be rearranged as
$$ \hbox{tr}( (A+B) (A^{1/2} P_V A^{1/2} + B^{1/2} P_V B^{1/2}) ). \quad\quad (*)$$ We can conjugate $A+B$ to be a diagonal matrix $\hbox{diag}(\lambda_1,\dots,\lambda_n)$ with $\lambda_1 \geq \dots \geq \lambda_n \geq 0$. In particular we have $A+B \leq \lambda_k I + D$ in the sense of positive definite matrices, where $D := \hbox{diag}(\lambda_1-\lambda_k, \dots, \lambda_{k-1}-\lambda_k, 0, \dots, 0)$. Using the fact that $\hbox{tr}(XZ) \leq \hbox{tr}(YZ)$ whenever $X,Y,Z$ are positive semi-definite with $X \leq Y$, we can bound (*) by $$ \hbox{tr}( (\lambda_k I + D) (A^{1/2} P_V A^{1/2} + B^{1/2} P_V B^{1/2}) )$$ which rearranges as $$ \lambda_k \hbox{tr}( (A+B) P_V ) + \hbox{tr}( P_V (A^{1/2} D A^{1/2} + B^{1/2} D B^{1/2}) ).$$
Using $A+B \leq \lambda_k I + D$ for the first term and $P_V \leq I$ for the second term, this is bounded by
$$ \lambda_k^2 \hbox{tr}( P_V ) + \lambda_k \hbox{tr}( D P_V ) + \hbox{tr}( A^{1/2} D A^{1/2} + B^{1/2} D B^{1/2} ).$$
For the second term we use $P_V \leq 1$, and for the third term we use the cyclic property of trace to bound by
$$ \lambda_k^2 \hbox{tr}( P_V ) + \lambda_k \hbox{tr}( D ) + \hbox{tr}( (A+B) D ).$$
For the first term we write $\hbox{tr}(P_V) = k = \hbox{tr}(P_W)$, where $W$ is the span of the first $k$ basis vectors $e_1,\dots,e_k$. For the third term we use $A+B \leq \lambda_k I + D$ to bound the above by
$$ \lambda_k^2 \hbox{tr}( P_W ) + \lambda_k \hbox{tr}( D ) + \hbox{tr}( (\lambda_k I + D) D ).$$
Since $D = P_W D P_W$, we can collect terms to obtain
$$ \hbox{tr}( P_W (\lambda_k I + D)^2 P_W ).$$
But by construction, $P_W (\lambda_k I + D)^2 P_W = \hbox{diag}( \lambda_1^2, \dots, \lambda_k^2, 0, \dots, 0 )$, so we have bounded (*) by the sum of the top $k$ eigenvalues of $(A+B)^2$, as required.
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|
210,291 |
One way to phrase the
" concentration-of-measure "
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator." 1 Q . How could one explain/justify this intuitively—perhaps just verbally—to a mathematically literate but naive audience
(say, advanced undergraduate math majors)? That "most of the mass is close to the equator, for any equator"
seems almost contradictory (imagining orthogonal equatorial hyperplanes), or at the
least, superficially quite puzzling.
Can one only gain intuition via working
through details of the Brunn–Minkowski theorem or the isoperimetric inequality ? 1 Boáz Klartag, in a book review in the AMS Bulletin , July 2015, p.540.
According to the Wikipedia article, the idea goes back to Paul Lévy.
|
As I see it, the key intuition is passing from the equator orthogonal to a single vector to looking at a whole orthonormal basis. Suppose we pick a random unit vector $(x_1,\dots,x_n)$. What we want to know is why $x_1$ is probably near zero, since this is equivalent to being near the equator relative to the first basis vector. But this feels intuitively obvious to me: all the coordinates have the same distribution, and they surely can't all be large, so they had better all be small. To be a little more precise, we have $x_1^2+\dots+x_n^2=1$, and each coordinate has the same distribution, so the expected value of $x_1^2$ is $1/n$. Now we can just apply Markov's inequality. For example, the probability that $|x_1|$ is at least $1/n^{1/4}$ must be at most $1/n^{1/2}$, since otherwise the expected value of $x_1^2$ would be too large. (This is not so different from Bjørn and Dustin's answers, but expressed in a less sophisticated way.)
|
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|
210,340 |
How would you go about writing an abstract for a Math paper? I know that an abstract is supposed to "advertise" the paper. However, I do not really know how to get started. Could someone tell me how they go about writing an abstract?
|
Avoid notation if possible. Notation makes it really hard to search electronically. Put the subject in context, e.g., "In a recent paper, T. Lehrer introduced the concept of left-bifurcled rectangles. He conjectured no such rectangles exist when the number of bifurcles $n$ is odd." State your results, in non-technical language, if possible. "In this paper we show the existence of left-bifurcled rectangles for all prime $n$." Mention a technique, if there is a new one: "Our methods involve analytic and algebraic topology of locally euclidean metrizations of infinitely differentiable Riemannian manifolds". Never, ever, ever, cite papers in the bibliography by giving citation numbers; the abstract is an independent entity that should stand on its own.
|
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|
210,432 |
Reverse mathematics (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the FOM mailing list ) that $Con(PA)$ is equivalent to a variant of Bolzano-Weierstrass over the rationals between 0 and 1 inclusive (something like: every sequence has a subsequence $\{q_i\}$ which is Cauchy, in the that sense $\forall i,j \geq n, |q_i - q_j| < 1/n$). Apparently a very similar result is given in Simpson's book and "it is clear to the experts" how to get to Friedman's claim. (As an aside, I find this such an amazing result it should be written up for the average mathematician, and not buried in a vaguely equivalent form in a book that is hard to get one's hands on.) The reason I bring up Bolzano-Weierstrass is that Todd Trimble, in a nice answer on Ways to prove the fundamental theorem of algebra , uses B-W to prove (as the key tool among other, elementary considerations) the fundamental theorem of algebra. Todd then had a look to see, at my behest, if the RM strength of FTA was known. He came up blank, so I ask here: How close in reverse mathematical strength are the Bolzano-Weierstrass statement from Friedman's claim and the fundamental theorem of algebra? If they are the same, then we find ourselves in the amazing situation that the consistency of PA is equivalent to a theorem that we all would use with no qualms whatsoever. However, I have a vague feeling that FTA is strictly weaker than BW (as used here), but cannot make this precise.
|
Tanaka and Yamazaki (in the volume Reverse Mathematics 2001 , see review ) show that a substantial portion of field theory can be done in the weak base theory RCA$_0$, by proving in RCA$_0$ the fundamental theorem of algebra as well as quantifier elimination for the theory of real closed fields. So the FTA is weaker than BW.
|
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210,644 |
My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to generalize the notion of covering spaces and algebraic curves to include primes in rings? Today we call p-adic fields local and number fields global, which suggests a very geometric interpretation of their relationship. I can certainly see the parallels more and more as I learn more of the theory, but I am curious as to when this "realization" that geometric ideas were all over number theory rose to prominence among number theorists. Does it go all the way back to classical number theory (i.e. Euler and Gauss), or perhaps does it begin with Hensel and Hasse? Any resources on the matter would also be appreciated.
|
Treating number and function fields on the same footing or (for instance) the idea that ramification in algebraic number theory and in the theory of covering of Riemann or analytic surfaces are two incarnations of the same mathematical phenomenon are classical ideas of the German school of the second half of the 19th century. It is very present in the research as well as expository material of Kronecker, Dedekind and Weber (see for instance the algebraic proof of Riemann-Roch by the last two). In fact, it is so ubiquitous in Kronecker's work that in some of his results on elliptic curves, it is often hard to ascertain if the elliptic curve he is studying is supposed to be defined over $\mathbb C$, $\bar{\mathbb Q}$, the ring of integers of a number field or over $\bar{\mathbb F}_p$ (or over all four depending on where you find yourself in the article). This is why the introduction of SGA1 says that the aim of the volume is to study the fundamental group in a "kroneckerian" way. At any rate, the analogy was so well-known to Hilbert that Takagi actually says in his memoirs that Hilbert had a negative influence on his definition and study of ray class field: Hilbert always wanted Takagi's theorem to make sense for Riemann surfaces and so was asking Takagi to only consider extension of number fields unramified everywhere. In the 1920s and 1930s (so to mathematicians like Artin, Hasse or Weil), this was thus very common knowledge. The revolutionary idea of Weil, in fact, is not at all the idea that arithmetic and geometry should be unified or satisfy deep analogies, it was the idea that they should be unified by topological means (at a low level, by systematically putting Zariski's topology to the forefront, at a high level, by introducing the idea that the rationality of the Zeta function and Riemann's hypothesis for varieties over function fields of positive characteristics were the consequences of the Lefschetz formula on a to be defined cohomology theory). A fortiori , the idea of viewing arithmetic through a geometric lens should certainly not be credited to Grothendieck, whose contribution (at least, the first and most relevant to the question) was the much more precise and technical insight that combining Serre's idea of studying varieties through the cohomology of coherent sheaves on the Zariski topology and Nagata's and Chevalley's generalization of affine varieties to spectrum of arbitrary rings, one would get the language required to carry over Weil's program. I cannot resist concluding with the following anecdote of Serre. In a talk he gave in Orsay in Autumn 2014 on group theory, he started by explaining that finite group theory should be of interest to many different kind of mathematicians, if only because the Galois group of an extension of number fields or the fundamental group of a topological space are examples of finite groups. In fact, he continued, these two kind of groups are the same thing and (quoting from memory and in my translation) "that they are the same thing is due to German mathematicians of the late 19th century, of course, except in Orsay where it is due to Grothendieck."
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210,779 |
Is the Solar System stable? You can see this Wikipedia page. In May 2015 I was at the conference of Cedric Villani at Sharif university of technology with this title: "Of planets, stars and eternity (stabilization and long-time behavior in classical celestial mechanics)"
, at the end of this conference one of the students asked him this question and he laughed strangely(!) with no convincing answer! Edit : The purpose of "long-time" is timescale more than Lyapunov time , hence billions of years.
|
Due to chaotic behaviour of the Solar System, it is not possible to precisely predict the evolution of the Solar System over 5 Gyr and the question of its long-term stability can only be answered in a statistical sense. For example, in http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html (Existence of collisional trajectories of Mercury, Mars and Venus with the Earth, by
J. Laskar and M. Gastineau) 2501 orbits with different initial conditions all consistent with our present knowledge of the parameters of the Solar System were
traced out in computer simulations up to 5 Gyr. The main finding of the paper is that one percent of the solutions lead to a large enough increase in Mercury's eccentricity to allow its collisions with Venus or the Sun. Probably the most surprising result of the paper (see also http://arxiv.org/abs/1209.5996 ) is that in a pure Newtonian world the probability of collisions within 5 Gyr grows to 60 percent and therefore general relativity is crucial for long-term stability of the inner solar system. Many questions remain, however, about reliability of the present day consensus that the odds for the catastrophic destabilization of the inner planets are in the order of a few percent. I do not know if the effects of galactic tidal perturbations or possible perturbations from passing stars are taken into account. Also different numerical algorithms lead to statistically different results (see, for example, http://arxiv.org/abs/1506.07602 ). Some interesting historical background of solar system stability studies can be found in http://arxiv.org/abs/1411.4930 (Michel Henon and the Stability of the Solar System, by Jacques Laskar).
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211,507 |
In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" because "why would it not be". In such a situation, I have heard the argument: The function is clearly measurable, because the axiom of choice was not used to define it. This argument makes some sense, because (as far as I know, I am not an expert) there are models of ZF (no C), where every function on $\mathbb{R}^n$ is Lebesgue measurable. So, suppose that we have a function $f$ in our model of ZFC constructed without the axiom of choice, then it is also a function in the model of ZF constructed above. Hence it is measurable there, and "clearly" all functions that are measurable in the model above are also measurable in our model. But the question is: The bold statement above is very "meta". So how rigorous is this argument? Can it be made rigorous? /Edit: I changed "Borel"- to "Lebesgue"-Measurable.
|
The bold statement is not true in the generality in which you
state it. Nevertheless, something very like it is true, if one
adopts the perspective and philosophy of large cardinal set theory
and restricts the kinds of definitions that are considered. First, let's get a little more clear on what you mean. One does
not formally use axioms at all in a definition , but rather in a
proof. To define an object means to provide a statement
$\varphi(x)$ that one and only one object satisfies. What one
means by not using an axiom in a definition, is that one can
prove, without using that axiom, that there is such a unique
object fulfilling the definition. Perhaps one has in mind a
constructive procedure, but this is really just a sequence of such
definitions, and such a construction does not use the axiom of
choice, if at every step of the construction, the definition used
at that step is a definition in any model of ZF. One can easily make a counterexample, now, by the definition: let
$f$ be the characteristic function of the least non-measurable set
of reals in the constructible universe $L$, using the canonical
definable well-ordering of $L$ . This definition does not use the axiom of choice, since it is
sensible as a definition in any model of ZF, and picks out a
unique function on the reals in any model of ZF. But it is not
necessarily true in ZF that this function is measurable, since if
the axiom of constructibility holds, that is, if we are living in
$L$, then $f$ is definitely non-measurable. Meanwhile, it is
consistent with ZFC that the set of all reals in $L$ is countable
in $V$, and in this case, the function $f$ is the characteristic
of a countable set, and hence measurable in $V$. So the
definition, which did not use the axiom of choice, sometimes
defines a measurable function and sometimes does not, in the
various ZF worlds. Let's give another concrete counterexample. The canonical
well-ordering of the reals in the constructible universe $L$, mentioned by Andres,
is a definable subset of the real plane $A\subset\mathbb{R}^2$,
which in $L$ has complexity $\Delta^1_2$ in the descriptive-set-theoretic
projective hierarchy. Thus, in our current universe $V$, the set $A$ has complexity at worst $\Sigma^1_2$, and so it arises from a certain definable
closed subset of $\mathbb{R}^4$ by projecting onto $\mathbb{R}^3$,
taking the complement, and then projecting to $\mathbb{R}^2$. So $A$ is definable in a highly concrete manner, without
making any use of the axiom of choice. Nevertheless, it is not
necessarily true that the resulting set is measurable, since
inside the constructible universe itself, the resulting set is not
measurable; in contrast, it is also consistent with ZF that there
are only countably many constructible reals, and in this case the
set $A$ would be countable and hence measurable. So the
measurability of the set $A$ is not determined, despite the simple
definition. At the end of your post, you seem to suggest that, ("clearly") if
a definition defines a measurable set in some model of ZF, then it
defines a measurable set (in our current ZFC universe). But this
is not quite right. One can write down a definition $\varphi(x)$
that ZF proves defines a unique set of reals, but the set of reals
defined is measurable in an inner model and non-measurable in a
larger model. Lastly, let me explain the sense in which your bold statement is
on the right track. One of the truly surprising and remarkable
discoveries of large cardinal set theory is that the existence of
large cardinals has effects on fundamental mathematical truth at
the level of descriptive set theory. In particular, the existence
of sufficient large cardinals implies that every projectively
definable set of reals is Lebesgue measurable. If there is a
supercompact cardinal, and much less suffices, as explained in the
article Saharon Shelah, Hugh Woodin, Large Cardinals Imply That
Every Reasonably Definable Set of Reals Is Lebesgue Measurable,
Isreal Journal of Mathematics, vol. 70, (1990) pp. 381-394 ( reviewed by J. Bagaria in BSL 8:4(2002) pp. 543-545 , as linked to by Andres in the comments), then every
set of reals in $L(\mathbb{R})$ is Lebesgue measurable. The universe $L(\mathbb{R})$ consists of those sets that are constructible relative to reals. So, if you assume large cardinals, and you define a set of reals
by a definition that is absolute to $L(\mathbb{R})$ — and
this is very likely the case if your definition works in ZF and
does not involve set theory explicitly — then your set is
Lebesgue measurable. In particular, assuming that there are sufficient large cardinals, then every projective set of reals is Lebesgue measurable, and this may provide a soft sufficient criterion. The projective statements are those that can be expressed using quantifiers only over the reals and the integers, with the usual algebraic and order structure. Alternatively, the projective sets are those that you get by closing the Borel sets under continuous images and complements. Let me point out that this kind of consequence of large cardinals
is often pointed to by large cardinal set theorists as evidence
that the large cardinal axioms themselves are on the right track,
since they provide a such a rich, coherent and desirable structure
theory for our everyday mathematics. We infinitely prefer the
smooth and elegant descriptive set theory of large cardinals to
the awkward land of counterexamples provided by the axiom of
constructibility $V=L$.
|
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"https://mathoverflow.net/questions/211507",
"https://mathoverflow.net",
"https://mathoverflow.net/users/16702/"
]
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|
212,364 |
This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at math.stackexchange ( https://math.stackexchange.com/questions/1356066/global-structure-of-the-gromov-hausdorff-space ), but received no answers even after bountying. Background. If $A, B$ are compact subsets of a metric space $M$ , the Hausdorff distance between $A$ and $B$ is the worst worst best distance between their points: $$d_H(A, B)=\max\{\sup\{\inf\{d(a, b): b\in B\}: a\in A\}, \sup\{\inf\{d(a, b): a\in A\}: b\in B\}\}.$$ For two compact metric spcaes $X, Y$ , their Gromov-Hausdorff distance is the infimum (in fact, minimum) over all isometric embeddings of $X, Y$ into $Z$ via $f, g$ of $d_H(f(X), g(Y))$ . The Gromov-Hausdorff space $\mathcal{GH}$ is then the "set" of isometry classes of compact metric spaces, with the metric $d_{GH}$ .) Question. How homogeneous is $\mathcal{GH}$ ? For example: while distinct points in $\mathcal{GH}$ are in fact distinguishable in a broad sense, it's not clear that distinct points can always be distinguished just by the metric structure of $\mathcal{GH}$ , so it's reasonable to ask: Are there any nontrivial self-isometries of $\mathcal{GH}$ ? There are of course lots of related questions one can ask (e.g., autohomeomorphisms instead of isometries); I'm happy for any information about the homogeneity of $\mathcal{GH}$ . Please feel free to add tags if more are appropriate.
|
To answer the main question -- there are no nontrivial self-isometries of $\mathcal{GH}$. I can give a proof of this, but as it is getting rather long, I will state some facts in $\mathcal{GH}$ without proof for now, and will come back and provide provide proofs or references. First, a bit of notation. Let ${\rm diam}(A)=\sup\{d(a,b)\colon a,b\in A\}$ denote the diameter of a metric space $A$. For any $\lambda > 0$ use $\lambda A$ to denote the space with the same points as $A$, but with scaled metric given by $d_\lambda(a,b)=\lambda d(a,b)$. Let $\Delta_n$ denote the (n-1)-simplex - a space consisting of $n$ points all at unit distance from each other. So, $\Delta_1$ is the space consisting of a single point, and $\lambda\Delta_n$ is a space consisting of $n$ points all at distance $\lambda$ from each other. Now, the following are standard,
\begin{align}
&d_{GH}(\Delta_1,A)=\frac12{\rm diam}(A),\\
&d_{GH}(A,B)\le\frac12\max({\rm diam}(A),{\rm diam}(B)),\\
&d_{GH}(\lambda A,\mu A)=\frac12\lvert\lambda-\mu\rvert{\rm diam}(A).
\end{align}
For reference, I am using Some Properties of Gromov–Hausdorff Distances by Fecundo Mémoli for the standard properties of $\mathcal{GH}$. Then, the one-point space $\Delta_1$ is distinguished just in terms of the metric on $\mathcal{GH}$ as follows, 1. $A$ is isometric to $\Delta_1$ if and only if $d_{GH}(B,C)\le\max(d_{GH}(A,B),d_{GH}(A,C))$ for all $B,C\in\mathcal{GH}$. That this inequality holds for $A=\Delta_1$ follows from the first two standard properties above. On the other hand, if $A$ contains more than one point, so ${\rm diam}(A)>0$, we can take $B=\Delta_1$ and $C=\lambda A$ for any $\lambda > 1$,
$$
\begin{eqnarray}
d_{GH}(B,C)&=&d_{GH}(\Delta_1,\lambda A)=\frac\lambda2{\rm diam}(A) \\
&>&\frac12\max(1,\lambda-1){\rm diam}(A)=\max(d_{GH}(A,B),d_{GH}(A,C))
\end{eqnarray}.
$$
So, statement 1 holds in both directions. Therefore, any isometry fixes $\Delta_1$ and, by the first standard property of $\mathcal{GH}$ above, it preserves the diameter of spaces. Next, the simplexes $\Delta_n$ are distinguished in terms of the metric as follows, 2. The following are equivalent for any $A\in\mathcal{GH}$. ${\rm diam}(A)=1$ and $d_{GH}(B,C)\le\max(d_{GH}(A,B),d_{GH}(A,C))$ for all $B,C\in\mathcal{GH}$ with diameter less than or equal to $1$. $A=\Delta_n$ for some $n\ge2$. The proof of this is given below. So, an $\iota$ is an isometry on $\mathcal{GH}$ maps the set of simplifies $\{\Delta_n\colon n\ge2\}$ to itself.
Now, the finite spaces in $\mathcal{GH}$ can be identified. 3. For any $n\ge2$ and $A\in\mathcal{GH}$, the following are equivalent. $d_{GH}(A,B)=d_{GH}(\Delta_n,B)$ for all $B\in\mathcal{GH}$ with $d_{GH}(\Delta_n,B)={\rm diag}(B)\ge\max({\rm diag}(A),1)$. $A$ is a finite space $n$ or fewer points. Therefore, if $\mathcal{GH}_n$ denotes the finite metric spaces with $n$ or fewer points, (2) and (3) imply that $\iota$ permutes $\{\mathcal{GH}_n\colon n\ge2\}$. As it must preserve the inclusions $\mathcal{GH}_n\subset\mathcal{GH}_{n+1}$, it follows that $\iota$ maps $\mathcal{GH}_n$ to itself for each $n\ge2$. Now, fix $n\ge2$, let $N=n(n-1)/2$, and $S$ be the subset of $\mathbb{R}^N$ consisting of points $\mathbf x=(x_{ij})_{1\le i < j\le n}$ with $x_{ij}>0$ and $x_{ik}\le x_{ij}+x_{jk}$ for all distinct $i,j$ (here, I am using $x_{ij}\equiv x_{ji}$ whenever $i > j$. Let $G$ be the group of linear transformations of $\mathbb{R}^N$ mapping $\mathbf x$ to $g_
\sigma(\mathbf x)=(x_{\sigma(i)\sigma(j)})$ for permutations $\sigma\in S_n$. Then, $S$ is a region in $\mathbb{R}^N$ with nonempty interior bounded by a finite set of hyperplanes, and maps in $G$ take the interior of $S$ into itself. We can define $\theta\colon S\to\mathcal{GH}$ by letting $\theta(\mathbf x)$ be the space with $n$ points $a_1,\ldots,a_n$ and $d(a_i,a_j)=x_{ij}$, and define a (continuous multivalued) function $f\colon S\to S$ by $\theta\circ f=\iota\circ\theta$. Then, $\theta$ maps $S$ onto the metric spaces with $n$ points, the values of $f(\mathbf x)$ are orbits of $G$, and the fact that $\iota$ is an isometry means that if $\mathbf y=f(\mathbf x)$ and $\mathbf y^\prime=f(\mathbf x^\prime)$, then $\min_g\lVert g(\mathbf y)-\mathbf y^\prime\rVert=\min_g\lVert g(\mathbf x)-\mathbf x^\prime\rVert$, with the minimum taken over $g\in G$ and using the $\ell_\infty$ norm on $\mathbb{R}^N$. Now, let $X\subset\mathbb{R}^N$ consist of the fixed points of elements of $G$, which is a finite union of hyperplanes, and $S^\prime=S\setminus\ X$. Note that $f$ maps $S^\prime$ into itself -- suppose that $\mathbf y = f(\mathbf x)\in X$ for some $\mathbf x\in S^\prime$. Then, $g(\mathbf y)=\mathbf y$ for some $g\in G$. Choosing $f(\mathbf x^\prime)=\mathbf y^\prime$ arbitrarily close to $\mathbf y$ with $h(\mathbf y^\prime)\not=\mathbf y^\prime$ (all $h\in G$), we have $\mathbf x^\prime$ and and $g(\mathbf x^\prime)$ arbitrarily close to $\mathbf x$, so $g(\mathbf x)=\mathbf x$, contradicting the assumption.
So, in the neighbourhood of any point in $S^\prime$, we can take a continuous branch of $f$, in which case $f$ is locally an isometry under the $\ell_\infty$ norm, which is only the case if $f(\mathbf x)=P\mathbf x+\mathbf b$ (where $P$ permutes and possibly flips the signs of elements of $\mathbf x$, and $\mathbf b\in\mathbb{R}^N$), with $P$ and $\mathbf b$ constant over each component of $S^\prime$. As $\mathbf x\to 0$, $\theta(\mathbf x)$ tends to $\Delta_1$, from which we see that $\mathbf b=0$. So, we have $f(\mathbf x)=P\mathbf x$, and as the components of $f(\mathbf x)$ are positive, $P$ is a permutation matrix. In order that $f$ is continuous across the hyperplanes in $X$, we see that $P$ is constant over all of $S^\prime$ (choosing continuous branches of $f$ across each hyperplane). Then, $P^{-1}gP\in G$, for all $g\in G$, as $f$ is invariant under the action of $G$. So, $P$ is in the normalizer of $G$. Now, it can be seen that centraliser of $G$ in the group of permutations (acting on $\mathbb{R}^N$ by permuting the elements) is trivial, which implies that its normaliser is itself ( Permutation Groups , see the comment preceding Theorem 4.2B). Hence $P\in G$, so $\iota$ acts trivially on the spaces with $n$ points. As the finite spaces are dense in $\mathcal{GH}$, $\iota$ is trivial. I'll now give a proof of statement (2) above, for which the following alternative formulation of the Gromov-Hausdorff distance will be useful. A correspondence , $R$, between two sets $A$ and $B$ is a subset of $A\times B$ such that, for each $a\in A$ there exists $b\in B$ such that $(a,b)\in R$ and, for each $b\in B$, there is an $a\in A$ with $(a,b)\in R$. The set of correspondences between $A$ and $B$ is denoted by $\mathcal R(A,B)$. If $A,B$ are metric spaces then the discrepancy of $R$ is,
$$
{\rm dis}(R)=\sup\left\{\lvert d(a_1,a_2)-d(b_1,b_2)\rvert\colon (a_1,b_1),(a_2,b_2)\in R\right\}.
$$
The Gromov-Hausdorff distance is the infimum of ${\rm dis}(R)/2$ taken over $R\in\mathcal R(A,B)$. Now, lets prove (2), starting with the case where $A=\Delta_n$, some $n\ge2$, for which we need to prove
$$
d_{GH}(B,C)\le\max(d_{GH}(\Delta_n,B),d_{GH}(\Delta_n,C))
$$
whenever $B,C$ have diameter bounded by $1$.
As we have, $d_{GH}(B,C)\le1/2$, the required inequality is trivial unless $d_{GH}(\Delta_n,B)$ and $d_{GH}(\Delta_n,C)$ are strictly less than $1/2$, so we suppose that this is the case. Denote the points of $\Delta_n$ by $p_1,p_2,\ldots,p_n$. If $R\in\mathcal R(\Delta_n,B)$ is such that ${\rm dis}(R)/2 < 1/2$, then letting $B_i$ consist of the points $b\in B$ with $(p_i,b)\in R$, the sets $B_1,\ldots,B_n$ cover $B$. For any $b,b^\prime\in B_i$ then $d(b,b^\prime)=\lvert d(p_i,p_i)-d(b,b^\prime)\rvert\le{\rm dis}(R)$, so the $B_i$ have diameters bounded by ${\rm dis}(R)$. Also, for any $i\not=j$, if $b\in B_i\cap B_j$ then ${\rm dis}(R)\ge\lvert d(p_i,p_j)-d(b,b)\rvert=1$, giving a contradiction. So, the $B_i$ are disjoint sets covering $B$. Similarly, if $S\in\mathcal R(\Delta_n,C)$ has ${\rm dis}(S)/2 < 1/2$ then we can partition $C$ into $n$ sets, $C_i$, of diameter bounded by ${\rm dis}(S)$. Defining $T=\bigcup_{i=1}^n(B_i\times C_i)\in\mathcal R(B,C)$, it can be seen that ${\rm dis}(T)\le\max({\rm dis}(R),{\rm dis}(S))$, from which the required inequality follows. Now, we prove the converse - if $A$ has diameter $1$ and is not isometric to $\Delta_n$ for any $n$, then we can find spaces $B,C$ of diameter $1$ with $d_{GH}(B,C)=1/2$ and with $d_{GH}(A,B)$, $d_{GH}(A,C)$ strictly less than $1/2$.
I'll consider first the case where $A$ is finite with $m\ge2$ points, so $A=\{a_1,\ldots,a_m\}$. As $A$ is not isometric to $\Delta_m$, there must exist two points separated by less than unit distance. Wlog, take $d(a_{m-1},a_m)=x < 1$. Then, $m > 2$, otherwise $A$ would have diameter $x < 1$. We can define $R\in\mathcal R(A,\Delta_{m-1})$ to be $\{(a_i,p_i)\colon i=1,\ldots,m-1\}\cup\{(a_m,p_{m-1})\}$, which has discrepancy bounded by the max of $\lvert d(a_i,a_j)-1\rvert$ over $i\not=j$ and $d(a_{m-1},a)m)=x$. So, $d_{GH}(A,\Delta_{m-1}) < 1/2$. Next, we can define $R\in\mathcal R(A,\Delta_m)$ to be the collection of pairs $(a_i,p_i)$ for $i=1,\ldots,m$. Its discrepancy is the max of $d(a_i,a_j)-1$ over $i\not=j$, which is strictly less than $1$, so $d_{GH}(A,\Delta_m) < 1/2$. However, $d_{GH}(\Delta_{m-1},\Delta_m)=1/2$, so $B=\Delta_{m-1}$ and $C=\Delta_m$ satisfies the desired properties. Now, suppose that $A$ is not a finite space. For any $m\ge2$ there exists a collection $a_1,a_2,\ldots,a_m$ of $m$ distinct points in $A$. Then, by compactness, we can cover $A$ with a finite collection of nonempty sets $A_1,\ldots,A_r$ of diameter bounded by $1/2$. Set $A_{r+i}=\{a_i\}$ for $i=1,\ldots,m$. Let $S=\{s_1,\ldots,s_{m+r}\}$ be the finite space with $d(s_i,s_j)=1$ for $i,j > r$ and with distance $1/2$ between all other pairs of points. The correspondence $R=\bigcup_{i=1}^{m+r}(A_i\times\{s_i\})$ has discrepancy
$$
{\rm dis}(R)=\max\left\{1/2,1-d(a_i,a_j)\colon 1\le i < j\le m\right\} < 1.
$$
So, $S$ is finite with diameter $1$, contains a subset isometric to $\Delta_m$, and $d_{GH}(A,S) < 1/2$. Now, we can let $B$ be any finite set with diameter $1$ and $d_{GH}(A,B) < 1/2$. If $m$ greater than the number of points in $B$, let $C$ be a finite set with diameter $1$, a subset isometric to $\Delta_m$, and $d_{GH}(A,C) < 1/2$. Then, $d_{GH}(B,C)=1/2$, and satisfies the required properties, proving (2).
|
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"source": [
"https://mathoverflow.net/questions/212364",
"https://mathoverflow.net",
"https://mathoverflow.net/users/8133/"
]
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|
212,550 |
I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible. I have found some related questions on stackoverflow but those algorithms are not an exact solution more random distributions. For special number of points (inscribed Platonic solids) it is clear but how about 5 points for example. I would be grateful for hints to the literature. Thank you everyone for your time.
|
There is considerable literature on this question, and closely related variations.
See: The Thompson problem : Which configurations of $n$ electrons on a sphere minimize the electrostatic potential energy? The Tammes problem : Which configurations of $n$ points on a sphere maximize the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere: (Image from Paul Sutcliffe .) According to Musin, Oleg R., and Alexey S. Tarasov. "The Tammes problem for $N=14$." arXiv:1410.2536 Abstract (2014). the Tammes problem is solved exactly for For $n=3,4,6,12$ by L. Fejes Toth (1943). For $n=5,7,8,9$ by Schütte and van der Waerden (1951). For $n=10,11$ by Danzer (1963). Added ( 8Sep15 ): Exact radius for $n=10$ by Sugimoto & Tanemura. For $n=24$ by Robinson (1961). For $n=13, 14$ by Musin and Tarasov (2014). Fig.1 from Musin & Tarasov: $n=14$. Added ( 8Sep15 ): The exact radius for $n=10$ was just found: Teruhisa Sugimoto, Masaharu Tanemura.
"Exact value of Tammes problem for N=10." Sep 2015. arXiv 1509.01768 Abstract . Fig.1b from Sugimoto & Tanemura. Added ( 31Dec2017 ) in response to a question by @R_Berger:
For $n=20$, the best arrangement for the Tammes problem is not
the dodecahedron's vertices.
The optimal is unknown, but this beats the dodecahedron: Coordinates from Neil Sloane link ,
due to R.H. Hardin, N.J.A. Sloane & W.D. Smith (1994).
|
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"https://mathoverflow.net/questions/212550",
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|
213,072 |
EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$ . The natural Laplace operator associated to the metric is denoted by $\Delta$ . Are there $n$ vector fields $X_{1},X_{2}, \ldots, X_{n}$ such that $\Delta=\sum \partial^{2}/\partial X _{i}^{2}$ ? Are there some local obstructions?(However our question search for global vector fields $X_{i}s$ ) The obvious motivation for this question is the usual metric on $\mathbb{R}^{n}$
|
As Raziel wrote, the local question is whether one can find a local basis of orthonormal vector fields that are divergence-free. It's true that, in dimension $2$, this can only be done if the metric is locally flat, which is the local obstruction. This is because this is an overdetermined problem; one has two equations for a single unknown. However, in higher dimensions, it is not clear that there is a local obstruction because this is a system of $n$ first-order PDE for $\tfrac12n(n{-}1)$ unknowns, so, for $n=3$, this is a determined system while, when $n>3$ it is underdetermined. One can prove that the system for $n=3$ is always locally solvable in the real-analytic case (even though the determined system cannot be written in Cauchy-Kowalevski form, even locally), so there cannot be any local obstruction that is computed on the basis of some kind of curvature condition or identity. (Presumably, it is also always locally solvable in the smooth case as well, but that would require further study.) Remark: It is interesting to note that, if $(M^3,g)$ is real-analytic and possesses a real-analytic orthonormal frame field $X = (X_1,X_2,X_3)$ where each of the $X_i$ are divergence free, then $(M,g)$ can be isometrically embedded as a hypersurface in a Calabi-Yau surface $S$ in such a way that $X_i = I_i(N)$ where $N$ is the oriented unit normal and $I_1$, $I_2$, and $I_3$ are the orthogonal parallel complex structures that define the Calabi-Yau structure. I expect that, in dimension $n>3$, the problem is so underdetermined that it is always locally solvable, though I have not yet carried out the analysis. However, see below, where I do complete the analysis in the real-analytic case. Global solvability is, of course, much harder, and it's conceivable that there are counterexamples, even for metrics on $S^3$, though I don't know of one. The analysis via exterior differential systems: At the OP's request, I will sketch the EDS analysis of this system. I don't have time to put in all the details, and, in any case, they won't make sense to anyone who doesn't already know Cartan-Kähler theory, but for those who do know this theory, the following explains the proof of the following results: When $n=2$, local solutions exist if and only if the metric is flat. When $n=3$, local solutions always exist if the metric is real-analytic. Moreover, the local solutions depend on 2 arbitrary functions of two variables. If the metric is real-analytic and the scalar curvature is positive, then every local solution is also real analytic. When $n>3$, local solutions always exist if the metric is real-analytic, and the local solutions depend on ${n\choose2}{-}n$ functions of $n$ variables. In particular, (2) and (3) imply that there are no curvature-type obstructions to local solvability when $n>2$. Whether we have solvability in the smooth case when $n>2$ will require further study. Here is the argument: Let $(M^n,g)$ be a Riemannian manifold, and let $\pi:F\to M$ be the orthonormal frame bundle, so that an element $f\in F$ is an $n$-tuple $f = (e_1,\ldots,e_n)$ where $e_1,\ldots,e_n$ is an orthonormal basis of $T_xM$ where $x = \pi(f)$. We define the canonical $1$-forms $\omega_1,\ldots,\omega_n$ on $F$ so that the equation
$$
\pi'(v) = \omega_1(v)\,e_1 + \cdots \omega_n(v)\,e_n
$$
holds for all $v\in T_fF$ where $f = (e_1,\ldots,e_n)$. A standard result (cf. Kobayashi & Nomizu) then says that there exist unique $1$-forms $\phi_{ij}=-\phi_{ji}$ (where the indices run from $1$ to $n$) satisfying the first structure equations $$
\mathrm{d}\omega_i = -\phi_{ij}\wedge\omega_j\,,
$$
that the forms $\omega_i$ and $\phi_{ij}$ ($i<j$) give a basis for the $1$-forms on $F$,
and that the $\phi_{ij}$ satisfy the second structure equations $$
\mathrm{d}\phi_{ij}
= -\phi_{ik}\wedge\phi_{kj} + \tfrac12\,R_{ijkl}\,\omega_k\wedge\omega_l
$$
for some unique functions $R_{ijkl}=-R_{ijlk}$ on $F$. A local orthonormal coframing $X = (X_1,\ldots,X_n)$ defined on an open set $U\subset M$
is simply a section of $F$ over $U$, and it satisfies $X^*\omega_i = \xi_i$ where the $\xi_i$ are the $1$-forms on $U$ dual to the $X_i$. The volume form of the metric is, up to a sign, the wedge product of the $\xi_i$, and so the condition that the $X_i$ be divergence free is that
$$
\mathrm{d}\left(\xi_1\wedge\cdots\wedge\widehat{\xi_i}\wedge\cdots\wedge\xi_n\right) = 0
$$
for all $i = 1,\ldots,n$. In other words, defining the $(n{-}1)$-forms
$$
\Omega_i
= (-1)^{i-1}\,\omega_1\wedge\cdots\wedge\widehat{\omega_i}\wedge\cdots\wedge\omega_n\,,
$$
we are requiring that $\mathrm{d}\left(X^*\Omega_i\right)=X^*\left(\mathrm{d}\Omega_i\right) = 0$, i.e., that the image of the section $X$ in $F$ should be an integral manifold of the differential ideal $\mathcal{I}$ on $F$ generated by the $n$ $n$-forms $\mathrm{d}\Omega_i$. Unfortunately, $\mathcal{I}$ is not involutive. However, it turns out that $\mathcal{I}$ can be enlarged to an ideal $\mathcal{I}_+$ as follows: For $i<j$, define the $(n{-}2)$-forms
$$
\Omega_{ij}
= (-1)^{i+j-1}\,\omega_1\wedge\cdots\wedge\widehat{\omega_i}\wedge
\cdots\wedge\widehat{\omega_j}\wedge\cdots\wedge\omega_n\,,
$$
and set $\Omega_{ii}=0$ while $\Omega_{ji}=-\Omega_{ij}$. Now define the $(n{-}1)$-form
$$
\Upsilon = \phi_{ij}\wedge\Omega_{ij}\,.
$$
It is not hard to show that $\mathrm{d}\Omega_i = \pm\omega_i\wedge\Upsilon$, and, from this, one concludes that $X_i$ is divergence free for all $i$ if and only iff $X^*(\Upsilon)=0$. Thus, we can let $\mathcal{I}_+$ be the differential ideal generated by $\Upsilon$ (i.e., the exterior ideal generated by $\Upsilon$ and $\mathrm{d}\Upsilon$) and look for integral manifolds of this ideal instead. When $n=2$, $\Upsilon = 2\phi_{12}$, so $\mathrm{d}\Upsilon
= 2R_{1212}\,\omega_1\wedge\omega_2 = 2K\,\mathrm{d}A$, so there are no sections $X$ that are integral manifolds unless $K=0$. (When $K=0$, of course, the sections that are integral manifolds of $\phi_{12}$ are exactly the parallel sections.) When $n>2$, the structure equations show that $\mathcal{I}_+$, which encodes a system of $n{+}1$ first-order equations on the orthonormal coframing $X$, is involutive, with the Cartan characters of a regular flag being $s_i = 0$ for $i<n{-}2$, $s_{n-2}=1$, $s_{n-1} = n{-}1$, and $s_n = {n\choose2}-n$. Now apply Cartan-Kähler.
|
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"https://mathoverflow.net/users/36688/"
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|
213,266 |
Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has an idea of how to extend an existing tool, what (non-existent) tool would be useful, or which piece of documentation should be (re)written. Due to lack of time & energy and often also programming expertise, these ideas rarely materialize. On the other hand, every now and then I meet programmers with a strong interest in mathematics (who are often actually trained mathematicians), and who are looking for a software project to work on. However, normally they don't really know what's needed and end up doing a non-mathematical project. This gave me the idea to ask the mathematical community to compile a wish list for mathematical software. Wishes can be very small or or something bigger. Just try to make sure that it's realistic and maybe also give an explanation why you consider your project as interesting. And if you happen to be a programmer fulfilling one of the wishes, please leave a comment. It would be great if you could also include an estimate on how complex your project and what the math/coding ratio is -- but this is optional. tl;dr What software tool would you like to see created? What existent software tool would you like to see extended by what feature? What piece of documentation is missing or should be updated/extended? One suggestion per answer, please.
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I think some aspects of math would be revolutionized by having a good math search engine. Recently, a question was asked on Meta.MathStackExchange about what they perceived as the greatest problems facing the site. The biggest response was that there was no search engine that indexed mathematics. This is partly reasonable, since math is stored and documented in $\TeX$ and this can be taken as a standard. But this is also problematic, as there are multiple noncanonical ways to do things in $\TeX$. I would be remiss if I didn't say there are very many other challenging aspects of this. As an example use case, I often have to look things up in the Gradshteyn and Ryzhik Table of Integrals and Series. It would be remarkable if there were a reasonable way to search for my expressions within the book. Even if I had to attempt multiple searches, it would almost certainly be faster. Taking it up a step, it would be great to search through TeX on the arXiv for certain expressions as well. I think that even a relatively mediocre math search engine would be a handy start.⠀⠀⠀⠀⠀
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|
214,526 |
Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?
|
No, a path-connected Hausdorff space is arc-connected, whence it would be of (at least) continuum cardinality provided it has more than one point. This follows from a more general (and deep) result that a Peano space (a compact, connected, locally connected, and metrizable space) is arc-connected if it is path-connected, together with the observation that a Hausdorff space that is the continuous image of the unit interval is a Peano space. See this section of the nLab, and references therein.
|
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|
214,677 |
The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the Dedekind completion of the ordered set $\mathbb{Q}$ with its usual ordering. When these two constructions are performed internally in a topos, they generally yield different results, and often it is the Dedekind construction that gives a more useful answer. In particular, for any topological space $X$, the Dedekind real number object in $\mathrm{Sh}(X)$ is the sheaf of continuous $\mathbb{R}$-valued functions on $X$, where $\mathbb{R}$ has its usual topology. (And this generalizes to some big toposes too.) Roughly, this is because the definition of "a Dedekind real number" is constructively equivalent to "a point of the locale of formal real numbers", where the latter locale can be defined constructively, and classically turns out to be equivalent to the usual topological space $\mathbb{R}$. However, the Cauchy definition of $\mathbb{R}$ has other generalizations: we can complete with respect to a non-Archimedean absolute value instead, obtaining the $p$-adic numbers $\mathbb{Q}_p$. We can do this internally in a topos too, but as with $\mathbb{R}$, in general we will not get the "right" answer. My question is, is there any way to construct "the $p$-adic numbers" constructively which, when interpreted in $\mathrm{Sh}(X)$, will yield the sheaf of continuous $\mathbb{Q}_p$-valued functions on $X$, where $\mathbb{Q}_p$ has its usual topology? In particular, is there a "locale of formal $p$-adic numbers" that can be defined constructively and that classically is equivalent to the usual topological space $\mathbb{Q}_p$?
|
Yes there is: the formal locale of p-adic integer is simply defined as the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a pro-finite locale). So internally in any topos a continuous function with values in $\mathbb{Z}_p$ corresponds to an element of the projective limit of the $\mathbb{Z}/p^k\mathbb{Z}$ (as a set this time) You can then define the locale $\mathbb{Q}_p$ as $\bigcup \frac{1}{p^k} \mathbb{Z}_p$ (this is a sequence of spaces where each space is open in the next so everything goes well...) and internally, the ring of continuous functions with values in $\mathbb{Q}_p$ is just $\mathbb{Z}_p [1/p]$ for the previously defined $\mathbb{Z}_p$. It turns out that, in this precise case, this coincides with the sequential completion (easy to see from the projective limit description...). This is a strange coincidence, but you can relate this to the fact that on a locally connected space the sequential completion corresponds to locally constant functions, but on a locally connected space, continuous functions with values in $\mathbb{Z}_p$ or $\mathbb{Q}_p$ are locally constant functions anyway because of the total disconnectedness. There is in fact way more to it: Steve Vickers has show how to define more generally the localic completion of a metric set (a set with a distance), and at least when the distance is symmetric this locale is essentially the Classifying space for the theory of regular Cauchy filter. Moreover it appears that this completion is stable under pullback along geometric morphism (this is not completely trivial, I prove it in my paper linked below) and because of this you will trivially have that if your metric set is the pullback of a metric set in the base topos then the points of its localic completion will be exactly the continuous function with values in the localic completion computed in the base topos. So a general consequence to remember is that (constructively) "completion by Cauchy filters always give you the correct set of points" so in any case if you have any valuation on a fields (note that over a non boolean topos even if the fields is $\mathbb{Q}$ it might not be equivalent to $|\_ |_p$ for some $p$) the "correct" completion (the one corresponding to the Dedekind completion) is the Cauchy filter completion.
For metric space there is also a notion of "Cauchy approximation" which gives the same completion and is a bit more 'sequential': they are decreasing sequences of open ball whose diameter goes to zero at a controlled speed. References for localic completion: Vickers wrote several paper on the subject, the more relevant is I think "Localic completion of quasimetric spaces" and after that there is two other on the non-symmetric case "Localic completion of generalized metric spaces" I & II. His presentation is rather different from what I add in mind, so I will look if there is something older more simple. This paper of mine where I define the completion of a symmetric metric locale and study all the geometricity properties (stability under pull-back) in section 3.3 . It is exactly what you need but because it deals with metric locales instead of metric sets there is some annoying additional complication...
|
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|
214,727 |
Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in dimension 7.) Yes, the topological category is fine, to avoid the smooth Poincaré conjecture.
|
In dimension 4, we have the following:
Simply-connectedness implies that $H_1(M)=0$. The condition that $M$ be a rational homology sphere implies that $H_2(M), H_3(M)$ are finitely generated torsion groups. It follows that
$H^3(M) = Ext(H_2(M),\mathbb{Z})$, which is noncanonically isomorphic to $H_2(M)$ again (that's true for finitely generated torsion groups). But Poincare duality tells us that $H^3(M)=H_1(M) =0$, so $H_2(M)=0$. Similarly, we can obtain $H_3(M)=0$. It follows that $M$ is already a homology sphere. In dimension 5, there's the first counterexample: The so-called Wu manifold $SU(3)/SO(3)$ has homology groups $\mathbb{Z}, 0, \mathbb{Z}/2, 0, 0, \mathbb{Z}$, so rationally, it is a homology sphere.
|
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|
214,728 |
This will not be altogether unrelated to this earlier question . For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of real numbers, if $\displaystyle f\mapsto \lim_{n \to\infty} \sum_{i=1}^n a_{f(i)}$ remains the same for all $f\in C$ then it remains the same for all bijections $f$ from $\{1,2,3,\ldots\}$ to itself? Thus if you have a conditionally convergent series, then at least one of the rearrangements that change the value of the sum is within $C$.
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Update. A research collaboration growing out of this question and some of its answers has now resulted in the following article, providing an account of the rearrangement number: A. Blass, J. Brendle, W. Brian, J. D. Hamkins, M. Hardy, and P. B. Larson, The rearrangement number , manuscript under review. Abstract. How many permutations of the natural numbers are needed so that
every conditionally convergent series of real numbers can be
rearranged to no longer converge to the same sum? We show that the
minimum number of permutations needed for this purpose, which we
call the rearrangement number, is uncountable, but whether it equals
the cardinal of the continuum is independent of the usual axioms of
set theory. We compare the rearrangement number with several
natural variants, for example one obtained by requiring the
rearranged series to still converge but to a new, finite limit. We
also compare the rearrangement number with several well-studied
cardinal characteristics of the continuum. We present some new
forcing constructions designed to add permutations that rearrange
series from the ground model in particular ways, thereby obtaining
consistency results going beyond those that follow from comparisons
with familiar cardinal characteristics. Finally we deal briefly
with some variants concerning rearrangements by a special sort of
permutations and with rearranging some divergent series to become
(conditionally) convergent. Original answer. $\newcommand\N{\mathbb{N}}\newcommand\P{\mathbb{P}}$This is a great question. Let me focus
on the title question, namely, the question of how many functions
one might need to ensure your rearrangement property. There are a
few things one can say. Theorem. No countable family of functions suffice to ensure
your rearrangement property. Specifically, for any countably many
permutations $f_n:\N\to\N$, there is a series $\sum_k a_k$ of real
numbers, such that $\sum_k a_{f_n(k)}=0$ for every $f_n$, but
$\sum_k a_k=0$ is only conditionally convergent. Proof. Indeed, I claim further that for any given series
$\sum_k b_k$, we may create a new series $\sum_k a_k$ by padding
the original series with zeros, but maintaining the order of the
nonzero terms, in such a way that for every function $f_n$, the
nonzero terms of $\sum_k a_{f_n(k)}$ appear in exactly the same
order as the original series $\sum_k b_k$, except for at most $n$
values. So all these series have the same sum, even though the
original series may be only conditionally convergent. So fix any series $\sum_k b_k$, and any countably many
permutations $f_n:\N\to\N$. We may start by defining $a_0=b_0$. At
stage $n$, we will have specified finitely many $a_k$, in such a
way that the nonzero entries specified so far include $b_0,
b_1,\ldots,b_n$, in that order, but we have padded with a possibly
very large number of zeroes in between, and furthermore such that
for every $m<n$, the nonzero values of $a_{f_m(k)}$ that have been
specified also agree with $b_0,\ldots,b_n$, in that order (except for $m$ values at most), but with the zeroes possibly inserted differently. At step $n$, consider the functions $f_m$ for $m\leq n$, and find
some index $k_n$ that is sufficiently large so that $k_n$ and
$f_m(k_n)$ are larger than any index we have yet used, for all
$m\leq n$. Let $a_{k_n}=b_n$, and pad the sequence with zeros
$a_k=0$ at all the other indices up to $k_n$. Since we add the new
term $b_n$ such a far distance out, the rearranged non-zero terms
$a_{f_m(k)}$ maintain the same order of $\sum_n b_n$ as far as we
have yet specified them (except possibly for the errors that we present before stage $m$, when the function $f_m$ began to be considered). It follows that all the particular rearrangements $\sum_k
a_{f_n(k)}$ using functions $f_n$ have the same value as $\sum_k
a_k$, which agrees with $\sum_k b_k$. But if $\sum_k b_k$ is only
conditionally convergent, then of course there is some other
rearrangement with a different sum. QED I take this theorem to suggest a new cardinal characteristic of
the continuum. Specifically, let us define $\kappa$ to be the
size of the smallest family $\cal C$ of permutations that have
your desired rearrangement property. So $\kappa$ is the answer to the title question of "how many?" because fewer than $\kappa$ are insufficient, by definition, but there is a family of $\kappa$ many permutations that work. So far, we've proved that
$\kappa$ is uncountable, and clearly it is at most continuum. Corollary. If the continuum hypothesis holds, then
$\kappa=\frak{c}$ is the continuum. This conclusion is also consistent with the failure of the
continuum hypothesis. Theorem. It is relatively consistent with ZFC that no family of fewer than continuum many permutations has your rearrangement property, even when the continuum is large. Indeed, Martin's axiom implies $\kappa=\frak{c}$. Proof. Assume that Martin's axiom holds, and that $\cal C$ is
a family of fewer than the continuum many permutations
$f:\N\to\N$. Fix any series $\sum_k b_k$, and let $\P$ be the
partial order consisting of pairs $(s,F)$, where $s$ is a finite
sequence of real numbers, whose nonzero values agree with a finite
initial segment of those appearing in $\sum_k b_k$, but possibly
padded with extra zeros, and $F$ is a finite subset of $\cal C$.
The order is $(s,F)\leq (s',F')$ just in case $s\supseteq s'$ and
$F\supseteq F'$, so that $(s,F)$ specifies more of the sequence
and $F$ mentions more functions, and furthermore, for any $f\in
F$, the nonzero portion of $s$ beyond $s'$ appears in the same
order in $s$ as it does in the rearrangement of $s$ by $f$. In
other words, once we add a function $f$ to $F$, then we will
insist that all further extensions of $s$ respect $f$. As a forcing notion, $\P$ has the countable chain condition,
because any two conditions $(s,F)$, $(s,F')$ with the same first
part are compatible, simply by using $(s,F\cup F')$, and there are
only countably many ways to pad a finite initial segment of
$\sum_n b_n$ with zeros. So Martin's axiom will apply to this
forcing notion. For each $m$, the collection of conditions $(s,F)$ for which $s$
includes the first $m$ terms of $\sum_k b_k$ is dense, since the
construction in the main theorem above shows how to handle
finitely many functions at once. Also, for any particular
$f\in\cal C$, it is dense for it to be added to the second
coordinate. Since we have thus specified fewer than continuum many dense sets,
by Martin's axiom there is a filter in $\P$ that meets all these
dense sets. Such a filter amounts to a series $\sum_k a_k$, which
agrees fully on its nonzero terms with the series $\sum_k b_k$, in
the same order, and such that for every $f\in \cal C$, the
rearrangement $\sum_k a_{f(k)}$ also agrees with $\sum_k b_k$
except on finitely many terms. So all these rearrangements have the same sum as $\sum_k b_k$,
even if this series is only conditionally convergent. QED It remains to see whether a small family can ever suffice. Question. Is it consistent with ZFC that $\kappa$ is less
than the continuum? That is, is it consistent with ZFC that a small family can
suffice?
|
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|
214,927 |
Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post , for the poster, and cash prizes offered for identifying the formulas.) This demonstrates that sometimes (but certainly not always) a major research progress, even areas, can be represented by a single formula. Naturally, following Alon's poster, I thought about representing other people's works through formulas. (My own work, Doron Zeilberger's, etc. Maybe I will pursue this in some future posts.) But I think it will be very useful to collect major formulas representing major research in combinatorics. The Question The question collects important formulas representing major progress in combinatorics. The rules are: Rules 1) one formula per answer 2) Present the formula explicitly (not just by name or by a link or reference), and briefly explain the formula and its importance, again not just link or reference. (But then you may add links and references.) 3) Formulas should represent important research level mathematics. (So, say $\sum {{n} \choose {k}}^2 = {{2n} \choose {n}}$ is too elementary.) 4) The formula should be explicit as possible, moving from the formula to the theory it represent should also be explicit, and explaining the formula and its importance at least in rough terms should be feasible. 5) I am a little hesitant if classic formulas like $V-E+F=2$ are qualified. An important distinction: Most of the formulas represent definite results, namely these formulas will not become obsolete by new discoveries. (Although refined formulas are certainly possible.) A few of the formulas, that I also very much welcome, represent state of the art regarding important combinatorial parameters. For example, the best known upper and lower bounds for diagonal Ramsey's numbers. In the lists and pictures below an asterisk is added to those formulas. The Formulas (so far) In order to make the question a more useful source, I list all the formulas in categories with links to answer (updated Feb. 6 '17). Basic enumeration: The exponential formula ; inclusion exclusion ; Burnside and Polya ; Lagrange inversion ; generating function for Fibonacci; generating function for Catalan ; Stirling formula ; Enumeration and algebraic combinatorics: The hook formula ; Sums of tableaux numbers squared , Plane partitions ; MacMahon Master Theorem ; Alternating sign matrices ; Erdos-Szekeres ; Ramanujan-Hardy asymptotic formula for the number of partitions ; $\zeta(3)$ ; Shuffles ; umbral compositional identity ; Jack polynomials ; Roger-Ramanujan ; Littlewood-Ricardson ; Geometric combinatorics: Dehn-Somerville relations ; Zaslavsky's formula ; Erhard polynomials; Minkowski's theorem . Graph theory: Tutte's golden identity ; Chromatic number of Kneser's graph ; (NEW) Tutte's formula for rooted planar maps ; matrix-tree formula ; Hoffman bound ; Expansion and eigenvalues ; Shannon capacity of the pentagon ; Probability: Self avoiding planar walks ; longest monotone sequences (average); longest monotone sequences (distribution); Designs : Fisher inequality ; Permanents: VanderWaerden conjecture ; Coding theory: MacWilliams formula ; Extremal combinatorics: Erdos-Sauer bound ; Ramsey theory : Diagonal Ramsey numbers ( ) ; Infinitary combinatorics: Shelah's formula ( ) ; A formula in choiceless set theory . Additive combinatorics : sum-product estimates (*) ; Algorithms: QuickSort . (Larger formulas): Series multisection ; Faà di Bruno's formula ; Jacobi triple product formula ; A formula related to Alon's Combinatorial Nullstellensatz ; The combinatorics underlying iterated derivatives (infinitesimal Lie generators) for compositional inversion and flow maps for vector fields (also related to the enumerative geometry of associahedra) ; some mysterious identities involving mock modular forms and partial theta functions ; Formulas added after October 2015: Hall and Rota Mobius function formula ; Kruskal-Katona theorem ; Best known bounds for 3-AP free subsets of $[n]$ (*); After October 2016: Abel's binomial identity ; Upper and lower bounds for binary codes ;
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The Hook Formula . If $\lambda$ is a partition of $n$ then the number of standard Young tableaux of shape $\lambda$ is $$f^\lambda = \frac{n!}{\prod_{\alpha \in [\lambda]} h_\alpha} $$ where $h_\alpha$ is the hook-length of the box $\alpha$ in the Young diagram $[\lambda]$ of $\lambda$, as shown below for $(5,4,2,1)$. The special case $\lambda = (n,n)$ gives the Catalan numbers: $$f^{(n,n)} = C_n = \frac{(2n)!}{(n+1)!n!} = \frac{1}{n+1} \binom{2n}{n}. $$ If $m_k>m_{k-1}>\dots>m_1$ are hook-lengths in the first column of Young diagram $\lambda$, i.e. lengths of rows are $0<m_1\leqslant m_2-1 \leqslant m_3-2\leqslant \dots \leqslant m_k-(k-1)$, then equivalent form is
$$
f^{\lambda}=\frac{n!}{\prod m_i!}\prod_{1\leqslant i<j\leqslant k} (m_j-m_i).
$$ This formula for $f^{\lambda}$ was established by G. Frobenius (Uber die charaktere der symmetrischer gruppe, Preuss. &ad. Wk. sitz. (1900), 516–534.) and A. Young (Quantitative substitutional analysis II, Proc. London Math. Sot., Ser. 1, 35 (1902), 361–397). Equivalence follows from the observation that product of hook lengths in $j$-th row equals $m_j!/\prod_{i<j} (m_j-m_i)$. The Hook Formula was first proved by Frame, Robinson and Thrall . It is important as a unifying result in enumerative combinatorics. It also gave another early indication (after Nakayama's Conjecture) of the importance of hooks, $p$-cores and $p$-quotients to the representation theory of the symmetric group.
|
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|
215,102 |
In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups . Namely, he said that a proof that relied on the CFSG and said so was ok, but a proof that relied on the ATLAS was not so ok, because the content has not been completely independently verified, and has as its basis old computer and other calculations that have only been done once. Given that (numerous?) little errors have been found over time , it would be good to have a definitive record as to which bits have either been calculated or proved elsewhere, or formally verified in the case of the computer calculations, and if so where and by whom (with code for the latter case). Note that merely being able to do some calculations in GAP is not quite enough, since, as the documentation says Part of the constructions have been documented in the literature on almost simple groups, or the results have been used in such publications, see for example the references in [CCNPW85] and [BN95]. where CCNPW85 is the ATLAS and BN95 is Breuer and Norton's Improvements to the Atlas (in an appendix of Atlas of Brauer Characters ). EDIT Since it may not have been clear, I was after statements modeled on the following: "All results about classical groups of Lie type are well-known and documented elsewhere" "All results about conjugacy classes of the sporadic groups except Janko 4 [say] are calculated afresh and contained in X computer package" "The results on [blah] about [some group] are only contained in the ATLAS, and no papers or independently written software have reproved/recalculated them" If it's easier to specify what is only in the ATLAS then that would be good, since clearly a lot of the classical material would be known and calculated long before. EDIT May 2017 In a recent talk ( 50 years ago: a great time for number theory — the first few minutes only before the main talk) Serre mentions his comments discussed here, the fact people got worked up about it, and the paper Breuer, Malle, and O'Brien - Reliability and reproducibility of Atlas information in Farrokh Shirjian's answer , which he feels addresses his complaints.
|
Unlike Dima, I am inclined to agree with Serre on this point. Although most of the facts recorded in the ATLAS have been proved elsewhere or, in the case of all of the character tables except for those of the very large groups like the Monster, can be easily recomputed in GAP or Magma using standard algorithms for finite groups, it can be very difficult in some cases to track down alternative proofs. I have recently completed a book (co-authored with John Bray and Colva Roney-Dougal) calculating complete lists of maximal subgroups of almost simple classical groups in dimensions up to $12$, and we were confronted with this problem. Although we cited the ATLAS many times, we tried hard to provide alternative citations or, for facts that could be easily checked by computer, we provided code to do this. In fact nearly all of the facts we required were either about maximal subgroups of groups in the ATLAS or involved entries in character tables. For the sporadic groups there were virtually always alternative papers to cite, which were generally also cited in the ATLAS. We had more difficulties with things like maximal subgroups of almost simple extensions of some of the more complicated classical groups in the ATLAS, like $U_4(3)$. For these we could not always find alternative sources that gave precise enough information, and we were told informally that some of the information had been originally calculated by unidentified research students or PostDocs. So we tried hard to re-prove these facts. Having said that, we found remarkably few errors in the ATLAS. I think there might have been one or two very small and minor inaccuracies in some of the structure descriptions, which we reported to the authors, and I think they might have known about them already. I see that there is an "ATLAS 30 Years On" conference coming up in Princeton in November 2015, so perhaps this will lead to more discusssion of these questions. I should also reinforce the point made by David Roberts that one needs to be very cautious when using Computer Algebra Systems, such as GAP and Magma, to verify facts contained in sources like the ATLAS, because it is possible that the code used is itself relying on these sources. For example $\mathtt{ MaximalSubgroups}$ in Magma will generally look up the maximal subgroups of the group's composition factors in a database, which will have been constructed using the ATLAS. However, a default use of $\mathtt{CharacterTable}$ on a finite group will use a general purpose algorithm (such as Dixon-Schneider), which does not rely on properties of specific (simple) groups.
|
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|
215,377 |
Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution $A^*(AB^*)^{1/2}$, and I am hoping that the cubic and higher dimensional versions are always solvable.
|
Here is an argument showing that the answer is 'yes'. I'll let you check the details and that this result generalizes to all higher degrees. Consider the map $f_{ABC}:\mathrm{U}(n)\to\mathrm{U}(n)$ defined by
$$
f_{ABC}(X) = XAXBXC.
$$
Since the image of this map is compact, if this map were not onto, it would have to have topological degree equal to zero. Next, since $\mathrm{U}(n)$ is connected, the map $f_{ABC}:\mathrm{U}(n)\to\mathrm{U}(n)$ defined by
$$
f_{ABC}(X) = XAXBXC
$$
is homotopic to the map $f_{III}$, and hence has the same degree as $f_{III}$. Thus, it suffices to show that the map $f_{III}$ has nonzero topological degree to show that $f_{ABC}$ is surjective. I claim that the map $f_{III}$ has topological degree $3^n$. To prove this, it suffices to compute the local degrees around the pre-images of a regular value. Let $Y = \mathrm{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})$, where $0<\theta_1<\theta_2<\cdots<\theta_n<\pi$. Then $Y$ has distinct eigenvalues. Hence, any solution $X$ to $X^3 = Y$ has distinct eigenvalues and has the same eigendirections as $Y$. Thus, $X$, too, must be diagonal and must be of the form
$$
X = \mathrm{diag}(e^{i\tau_1},\ldots,e^{i\tau_n})
$$
where $3\tau_k \equiv \theta_k\ \mathrm{mod}\ 2\pi$ for $k=1,\ldots, n$. Thus, there are $3^n$ solutions $X$ to $X^3=Y$. I claim that $Y$ is a regular value of the map $f_3:\mathrm{U}(n)\to \mathrm{U}(n)$ defined by $f_3(X)=X^3$ and that $f_3$ is orientation preserving at each solution $X$ of $X^3=Y$. This follows from the following computation: Consider the pullback under $f_3$ of the canonical left-invariant form $g^{-1}\mathrm{d} g$.
$$
f_3^*(g^{-1}\mathrm{d}g) = g^{-3}\mathrm{d}(g^3)
= (I + \mathrm{Ad}(g^{-1})+\mathrm{Ad}(g^{-2}))(g^{-1}\mathrm{d}g)
$$
When one computes the determinant of $\bigl(I + \mathrm{Ad}(X^{-1})+\mathrm{Ad}(X^{-2})\bigr):{\frak{u}}(n)\to {\frak{u}}(n)$ for each solution $X$ of $X^3=Y$, one finds that, because $Y$ has distinct eigenvalues, this determinant is a positive number. Thus, $Y$ is a regular value of $f_3$, and each of the $3^n$ solutions $X$ to $X^3=Y$ contributes a $+1$ to the topological degree in the usual degree formula. (By a similar argument, the map $f_k:\mathrm{U}(n)\to\mathrm{U}(n)$ defined by $f(X) = X^k$ has topological degree $k^n$, which is nonzero, so it is necessarily surjective. This answers the higher degree cases as well.)
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215,573 |
Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function
$$dist^2\colon M\times M\to \mathbb{R}$$
given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this function is infinitely smooth near the diagonal. Now fix a point $a\in M$. Consider the Taylor series of $dist^2$ at $(a,a)$ in some coordinate system (say normal). Can one compute explicitly its coefficients up to the third order?
|
Fix a point $x_0 \in M$. Then let $x = \exp_{x_0}(t v)$ and $y=\exp_{x_0}(t w)$, with $v,w \in T_{x_0}M$. Then we have the following formula for the distance squared between two geodesic emanating from $x_0$ $$ d^2(\exp_{x_0}(t v),\exp_{x_0}(t w)) = |v-w|^2t^2-\frac{1}{3}R(v,w,w,v) t^4 + O(t^5)$$ whre $R$ is the Riemann curvature tensor. From here you can derive an expression where you follow the two geodesics for different times (just rescale $w \to s/t w$), that is $$ d^2(\exp_{x_0}(t v),\exp_{x_0}(s w)) = |v|^2t^2 +|w|s^2 -2 g(v,w) ts -\frac{1}{3}R(v,w,w,v) s^2 t^2 + O(|t^2+s^2|^{5/2})$$ Take now $|v| = |w|=1$, then $(t,v)$ and $(s,w)$ (both $\in [0,\epsilon) \times \mathbb{S}_{x_0}^{n-1}$) are polar coordinates of $x,y$. More explicitly, setting $t = d(x_0,x)$ and $s = d(x_0,y)$, you have $$ d^2(x,y) = t^2 + s^2 - 2\cos\theta t s - \tfrac{1}{3} K_\sigma (1-\cos\theta^2)t^2 s^2 + O(|t^2+s^2|^{5/2})$$ where $\cos\theta = g(v,w)$ is the angle between the two vectors and $K_\sigma$ is the sectional curvature of the plane $\sigma = \mathrm{span}(v,w)$. As you can see, this can be used effectively as a purely metric definition of sectional curvature, with no connection or covariant derivative whatsoever.
|
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215,664 |
[EDITED mostly to report on the answer by Kevin Costello (and to improve the gp code at the end)] I thank Nicolas Dupont for the following question
(and for permission to disseminate it further): I have a playlist with, say, $N$ pieces of music.
While using the shuffle option (each such piece is played randomly
at each step), I realized that, generally speaking, I have to hear
quite a lot of times the same piece before the last one appears.
It makes me think of the following question: At the moment the last non already heard piece is played, what is the
max, in average, of number of times the same piece has already been played? I have not previously enountered this variant of the
coupon
collector's problem. If it is new, then (thinking of the original
real-world context origin of the problem) I propose to call it the "coupon
collector's earworm ". Is this in fact a new question?
If not, what is known about it already? Let us call this expected value $M_N$. Nicolas observes that $M_1=1$
and $M_2=2$ (see below), and asks: I doubt there is such an easy formula for general $N$ -
would you be able to find some information on it,
e.g. its asymptotic behavior? [Nicely answered by Kevin Costello :
$M_N$ is asymptotic to $e \log N$ as $N \rightarrow \infty$.
Moreover, the maximal multiplicity is within $o(\log N)$ of $e \log N$
with probability $\rightarrow 1$. I don't recall any other instance
of a naturally-arising asymptotic growth rate of $e \log N$...] Recall that the standard coupon collector's problem asks for the expected value
of the total number of shuffles until each piece has appeared at least once.
It is well known that the answer is $N H_N$ where $H_N = \sum_{i=1}^N 1/i$
is the $N$-th harmonic number. Hence the expected value of the average
number of plays per track is $H_N$, which grows as $\log N + O(1)$.
The expected maximum value $M_N$ must be at least as large $-$ in fact
larger once $N>1$, because one track is heard only once, so the average
over the others is $(N H_N-1) / (N-1)$. One might guess that $M_N$
is not that much larger, because typically it's only the last few tracks
that take most of shuffles to appear. But it doesn't seem that easy to get
at the difference between the expected maximum and the expected average,
even asymptotically. Unlike the standard coupon collector's problem,
here an exact answer seems hopeless once $N$ is at all large (see below),
so I ask: How does the difference $M_N - H_N$ behave
asymptotically as $N \rightarrow \infty$? [ Looks like this was a red herring : "One might guess" plausibly that
$M_N - H_N$ is dwarfed by $H_N$ for large $N$, but by Kevin Costello 's
answer $M_N - H_N$ asymptotically exceeds $H_N$ by a factor $e - 1$,
and that factor is more complicated than
$\lim_{N\rightarrow\infty} M_N/H_N = e$,
so analyzing the difference $M_N-H_N$ is likely not a fruitful approach.] Here are the few other things I know about this: @ For each $N>1$, the expected value of the maximum count
is given by the convergent $(N-1)$-fold sum
$$
M_N = \sum_{a_1,\ldots,a_{N-1} \geq 1}
N^{-\!\sum_i \! a_i} \Bigl(\sum_i a_i\Bigr)! \frac{\max_i a_i}{\prod_i a_i!}.
$$
Indeed, we may assume without loss of generality that the $N$-th track
is heard last; conditional on this assumption, the probability that
the $i$-th track will be heard $a_i$ times for each $i<N$ is
$N^{-\!\sum_i \! a_i}$ times the multinomial coefficient
$(\sum_i a_i)! \big/ \prod_i a_i!$. Numerically, these values are
$$
2.00000, \quad
2.84610+, \quad
3.49914-, \quad
4.02595\!-
$$
for $N=2,3,4,5$. @ A closed form for $M_N$ is available for $N \leq 3$ and probably
not beyond. Trivially $M_1 = 1$; and N.Dupont already obtained
the value $M_2 = 2$ by evaluating $M_2 = \sum_{a \geq 1} a/2^a$.
But for $N=1$ and $N=2$ the problem reduces to the classical
coupon collector's problem. Already for $N=3$ we have a surprise:
$M_3 = 3/2 + (3/\sqrt{5})$, which has an elementary but somewhat tricky proof.
For $N=4$, I get
$$
M_4 = \frac73 - \sqrt{3} + \frac4\pi \int_{x_0}^\infty
\frac{(2x+1) \, dx}{(x-1) \sqrt{4x^3-(4x-1)^2}}
$$
where $x_0 = 3.43968\!+$ is the largest of the roots (all real) of
the cubic $4x^3-(4x-1)^2$. I don't expect this to simplify further:
the integral is the period over an elliptic curve of a differential with two
simple poles that do not differ by a torsion point. In general
one can reduce the $(N-1)$-fold sum to an $(N-2)$-fold one
(which is one route to the value of $M_3$ and $M_4$), or to an
$(N-3)$-fold integral, but probably not beyond. @ It's not too hard to simulate this process even for considerably larger $N$.
In GP one can get a single sample of the distribution from the code try(N) = v=vector(N); while(!vecmin(v),v[random(N)+1]++); vecmax(v) [ turns out that one doesn't need to call vecmin each turn : try(N, m,i)= v=vector(N); m=N; while(m, i=random(N)+1; v[i]++; if(v[i]==1,m--)); vecmax(v) does the same thing in $\rho+O(1)$ operations per shuffle rather then
$\rho+O(N)$, where $\rho$ is the cost of one random(N) call. ] So for example sum(n=1,10^4,try(100)) / 10000. averages 1000 samples for $M_{100}$; this takes a few seconds, and
seems to give about $11.7$.
|
For the asymptotic case: Let $t_1=n \log n - Cn$ and $t_2 = n \log n + Cn$, where $C$ is slowly tending to infinity. It is a classic result that as $C$ tends to infinity the probability all coupons are collected at time $t_1$ tends to $0$, and the probability all coupons are collected at time $t_2$ tends to $1$. So the number being asked for in the original post is with high probability sandwiched between the most collected coupon at time $t_1$ and the most collected coupon at time $t_2$. This has been studied a fair amount, though often in language of load-balancing and balls-in-bins ("if you toss $m$ balls in $n$ bins, what is the typical number of balls in the bin with the most balls"). In particular, there's a nice analysis of Raab and Steger that uses the second moment method to give a tight concentration on this. It follows from Theorem $1$ in their paper that at time $c n \log n$, the most common coupon has almost surely been collected $(d_c+o(1)) \log n$ times, where $d_c$ is the larger real root of
$$1+x(\log c - \log x +1)-c=0$$ In our case, we have $d_1=e$, so the most common song will have been heard $(e+o(1))\log n = (e+o(1))H_n$ times. In response to the comment: The first moment part of the calculation also comes with come concentration estimates for free. At time $t_1$, the probability that we've seen some coupon at least $C \log n$ times can be bounded by
\begin{eqnarray*}
& & \sum_{j=C \log n}^{t_1} n \binom{t_1}{j} n^{-j} (1-\frac{1}{n})^{t_1-j} \\
&\leq& \sum_{j=C \log n}^{t_1} n \left(\frac{e t_1}{ nj}\right)^j (1-\frac{1}{n})^{n \log n + o(\log n)} (1-\frac{1}{n})^{-j} \\
&=& n^{o(1)} \sum_{j=C \log n}^{t_1} \left(\frac{e \log n(1+o(1))}{j}\right)^j \\
&\leq& n^{o(1)} \sum_{j=C \log n}^{t_1} \left(\frac{e}{C}+o(1)\right)^j
\end{eqnarray*} Which should decay rapidly enough in $C$ to guarantee the mean lies close to the concentration. The linked bounds for the probability of the completion time lying outside $[t_1, t_2]$ also decay quite rapidly (e.g. cardinal's answer, and David's subsequent comments on it).
|
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215,739 |
Let's say you are a prospective mathematician with some addled ideas about cardinality. If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :) If you thought that natural numbers and reals had the same cardinality - measure theory would almost surely break down, and your assumption would conflict with any number of "completeness theorems" in analysis (like the Baire Category Theorem for instance). However, let's say you concluded that there were only three types of cardinality - finite, countably infinite, and uncountable. Would this erroneous belief conflict with any major theorems in analysis, algebra or geometry ? Would any fields of math - outside set theory - be clearly incompatible with your assumption ? PS: Apologies for the provocative title. Hope the question is clear.
|
$\newcommand\ZFC{\text{ZFC}}$Perhaps it would be useful to mention that set theorists have, of course,
studied numerous weaker set theories, including some extremely
weak theories, which do not give rise to higher cardinalities. One
may interpret your question as: to what extent do these weak set
theories serve as a foundation of mathematics? To be sure, set theorists generally study these weak theories not
as foundational theories, but rather because they want to
undertake certain set-theoretic constructions in some much
stronger theory, but the objects appearing in the construction are
transitive sets satisfying these weaker theories, and so they need
to know, for example, whether those objects are themselves closed
under certain constructions. If those constructions can be
undertaken in the weak theory, then they are. To give a few examples, the theory known as $\ZFC^-$, which is
basically $\ZFC$ without the power set axiom (but see my
recent paper, What is the theory ZFC without power set? for what this means exactly), is
widely used in set theory and has an enormous number of natural
models, including the universe $H_{\kappa^+}$, in which every set
has cardinality at most $\kappa$ and $P(\kappa)$ does not exist as
a set, but only as a class. For example, in the universe
$H_{\omega_1}$, the theory $\ZFC^-$ holds, and every set is
countable. This is a very rich universe in which to undertake
classical mathematics: you have all the reals individually, but
you cannot form them into a set; but you can still consider
(definable) functions on the reals and so on. You just cannot put
them all together into a set. The theory known as Kripke-Platek set theory $\text{KP}$ is another intensely studied theory, particularly for
those doing set theory with the constructible universe and admissible set theory , and knowing what can be proved in
$\text{KP}$ and what cannot is very important in that area. Even Zermelo set theory itself can be considered as a kind of example, since it does not prove
the existence of uncountable cardinals beyond the $\aleph_n$ for
$n<\omega$, because the rank-initial segment of the universe
$V_{\omega+\omega}$ is easily seen to be a model of Zermelo set
theory. So one could count this as a case of a weak theory that
does not prove a huge number of different infinities. Perhaps this perspective on your question reveals that there is
really a continuum of such kind of answers. The really weak set
theories such as $\text{KP}$ and $\ZFC^-$ cannot prove even that
uncountable cardinals exist, but then slightly stronger theories,
which become true in $H_{\omega_2}$ or $H_{\omega_3}$, can prove a
few more uncountable cardinals. Zermelo's theory provides more,
but still only countably many uncountable cardinals. The $\ZFC$
theory of course then explodes with an enormous number of
different uncountable cardinals. But let me say that this process continues strictly past $\ZFC$, for large cardinal set theorists look upon $\ZFC$
itself as a weak theory, in precisely this sense, because it
cannot prove the existence of measurable or supercompact
cardinals (and many others), for example, and so one must continue up the large cardinal
hierarchy in order to get the kinds of infinities that we like.
Set theorists consider theories all along the large cardinal
hierarchy, with the stronger theories giving us more and stronger
axioms of the higher infinite. At every step of this entire hierarchy, starting from the very
weak theories I mentioned and continuing into the large cardinal
hierarchy, there are fundamental set-theoretic assertions that are
provable by the stronger theory but not provable by the weaker
theory. Meanwhile, despite the fact that some every-day mathematical
objects have distinct uncountable cardinalities (and so the weak
set theories cannot prove they exist), nevertheless it is quite
surprising how close an approximation one can get just in
second-order number theory, where in a sense every object is
countable. The work of reverse mathematics generally takes place in the
context of second-order number theory, and seeks to find exactly
the theory that is necessary in order to prove each of the
classical theorems of mathematics. (Thus, they try to prove the
axioms from the theorem, rather than the other way.) They have
numerous examples of which classical theorems you can prove and
exactly what theory (provably so!) you need to do it.
|
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215,862 |
Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad g(x) = \int_c^d f(x,y) dy$$
is not real-analytic on $(a,b)$? Edit: What about an example of bounded $f$ satisfying the above?
|
$$\int_0^1 \sqrt{x^2+y}\; dy = \dfrac{2}{3} \left((x^2+1)^{3/2} - |x|^3\right)$$
for $x \in (-1,1)$.
|
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216,184 |
What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian geometry and earlier works of Mochizuki, but what's the order to study those material? I think I had seen somewhere a complete list of papers to read from beginning to end in order to come to a level of understanding to tackle the original four papers about IUT theory, but I can't find it.
|
According to Mochizuki himself , the essential prerequisites for the IUTeich papers are: Semi-graphs of Anabelioids (sections 1 to 6) The Geometry of Frobenioids I: The General Theory (complete) The Geometry of Frobenioids II: Poly-Frobenioids (sections 1 to 3) The Etale Theta Function and its Frobenioid-theoretic Manifestations (complete) Topics in Absolute Anabelian Geometry I: Generalities (sections 1 and 4) Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms (section 3) Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (sections 1 to 5) Arithmetic Elliptic Curves in General Position (complete) While other sources also recommend: The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve A Survey of the Hodge-Arakelov Theory of Elliptic Curves I A Survey of the Hodge-Arakelov Theory of Elliptic Curves II Particularly interesting is Fesenko's recent extended remarks on IUT (and learning IUT): Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions There's also an introductory paper by Yuichiro Hoshi, but at least for the moment it is avaible in japanese only Yuichiro Hoshi, Introduction to inter-universal Teichmüller theory As for the (considerable) gap between Hartshorne and Mochizuki's work, the references on each paper are quite concrete and helpful (see for example the ones on Topics in Absolute Anabelian Geometry I for a good sample).
|
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217,792 |
I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different. My goal is to study derived algebraic geometry, where derived schemes are built out of simplicial commutative rings rather than ordinary commutative rings as in algebraic geometry (there's also a variant using commutative ring spectra, which I don't know anything about). Anyways, since the category of simplicial rings form a model category, we can apply homotopy theoretic methods to study derived schemes. I thought the first thing I should do is study simplicial homotopy theory, in order to learn about model categories and simplicial objects. So I started reading Simplicial Homotopy Theory by Goerss and Jardine. How should I study this book? There are very few exercises, unlike standard graduate textbooks like Hartshorne, and a lot of the proofs are simplex/diagram chasing, so I decided to skip a lot of the proofs and read the book casually. A big disadvantage to this method is that I don't understand anything at a deep level and I'm only familiar with a few buzzwords. But I feel overwhelmed by the amount of prerequisite material I need to understand to learn DAG, because most of it is written in the language of $\infty$-categories. So what should I do? How can I get to "research level mathematics"? EDIT: I'm a senior math major and I've taken the graduate algebraic geometry and algebraic topology sequences. I've also studied some deformation theory.
|
I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lurie's Higher algebra .) The plan is based on what worked best for myself, and it's certainly possible that you may prefer to jump into Higher Topos Theory as Yonatan suggested. 0) First of all, make sure you have a solid grounding in basic category theory. For this, read the first two chapters of the excellent lecture notes of Schapira . I would strongly recommend reading chapters 3 and 4 as well, but these can be skipped for now. Then read chapters I and II of Gabriel-Zisman, Calculus of fractions and homotopy theory , to learn about the theory of localization of categories. 1) The next step is to learn the basics of abstract homotopy theory. I recommend working through Cisinski's notes . This will take you through simplicial sets, model categories, a beautiful construction of the Quillen and Joyal model structures (which present $\infty$-groupoids and $\infty$-categories, respectively), and the fundamental constructions of $\infty$-category theory (functor categories, homotopy (co)limits, fibred categories, prestacks, etc.). Supplement the section "Catégories de modèles" with chapter I of Quillen's lecture notes Homotopical algebra . Then read about stable $\infty$-categories and symmetric monoidal $\infty$-categories in these notes from a mini-course by Cisinski. (By the way, these ones are in English and also summarize very briefly some of the material from the longer course notes). These notes are very brief, so you will have to supplement them with the notes of Joyal . It may also be helpful to have a look at the first chapter of Lurie's Higher algebra and the notes of Moritz Groth . 2) At this point you are ready to learn some derived commutative algebra: Read lecture 4 of part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry together with section 3 of Lurie's thesis . Supplement this with section 2.2.2 of Toen-Vezzosi's HAG II , referring to chapter 1.2 when necessary. This material is at the heart of derived algebraic geometry: the cotangent complex, infinitesimal extensions, Postnikov towers of simplicial commutative rings, etc. Other helpful things to look at are Schwede's Diplomarbeit and Quillen's Homology of commutative rings . 3) Before learning about derived stacks, I would strongly recommend working through these notes of Toen about classical algebraic stacks, from a homotopy theoretic perspective. There are also these notes of Preygel . This will make it a lot easier to understand what comes next. Then, read Lurie's On $\infty$-topoi . It will be helpful to consult sections 15-20 of Cisinski's Bourbaki talk , section 40 of Joyal's notes on quasi-categories, and Rezk's notes . For a summary of this material, see lecture 2 of Moerdijk-Toen. 4) Finally, read about derived stacks in lecture 5 of Moerdijk-Toen and section 5 of Lurie's thesis. Again, chapters 1.3, 1.4, and 2.2 of HAG II will be very helpful references. See also Gaitsgory's notes (he works with commutative connective dg-algebras instead of simplicial commutative rings, but this makes little difference). His notes on quasi-coherent sheaves in DAG are also very good. 5) At this point, you know the definitions of objects in derived algebraic geometry. To get some experience working with them, I would recommend reading some of the following papers: Antieau-Gepner, Brauer groups and étale cohomology in derived algebraic geometry , arXiv:1210.0290 Bhatt, p-adic derived de Rham cohomology , arXiv:1204.6560 . Bhatt-Scholze, Projectivity of the Witt vector affine Grassmannian , arXiv:1507.06490 . Gaitsgory-Rozenblyum, A study in derived algebraic geometry , link Kerz-Strunk-Tamme, Algebraic K-theory and descent for blow-ups , arXiv:1611.08466 . Toen, Derived Azumaya algebras and generators for twisted derived categories , arXiv:1002.2599 . Toen, Proper lci morphisms preserve perfect complexes , arXiv:1210.2827 . Toen-Vaquie, Moduli of objects in dg-categories , arXiv:math/0503269 .
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217,854 |
Motivation: Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. Many more such numbers are described for example in the wonderful book "Mathematical Constants" by Steven R. Finch. The question: I am interested in theorems where a "special" rational number makes a surprising appearance as an answer to a natural question. By a special rational number I mean one with a large denominator (and preferably also a large numerator, to rule out numbers which are simply the reciprocals of large integers, but I'll consider exceptions to this rule). Please provide examples. For illustration, here are a couple of nice examples I'm aware of: The average geodesic distance between two random points of the Sierpinski gasket of unit side lengths is equal to $\frac{466}{885}$. This is also equivalent to a natural discrete math fact about the analysis of algorithms, namely that the average number of moves in the Tower of Hanoi game with $n$ disks connecting a randomly chosen initial state to a randomly chosen terminal state with a shortest number of moves, is asymptotically equal to $\frac{466}{885}\times 2^n$. See here and here for more information. The answer to the title question of the recent paper "" The density of primes dividing a term in the Somos-5 sequence " by Davis, Kotsonis and Rouse is $\frac{5087}{10752}$. Rules: 1) I won't try to define how large the denominator and numerator need to be to for the rational number to qualify as "special". A good answer will maximize the ratio of the number's information theoretic content to the information theoretic content of the statement of the question it answers. (E.g., a number like 34/57 may qualify if the question it answers is simple enough.) Really simple fractions like $3/4$, $22/7$ obviously do not qualify. 2) The question the number answers needs to be natural. Again, it's impossible to define what this means, but avoid answers in the style of "what is the rational number with smallest denominator solving the Diophantine equation [some arbitrary-sounding, unmotivated equation]". Edit: a lot of great answers so far, thanks everyone. To clarify my question a bit, while all the answers posted so far represent very beautiful mathematics and some (like Richard Stanley's and Max Alekseyev's answers) are truly astonishing, my favorite type of answers involve questions that are conceptual in nature (e.g., longest increasing subsequences, tower of Hanoi, Markov spectrum, critical exponents in percolation) rather than purely computational (e.g., compute some integral or infinite series) and to which the answer is an exotic rational number. (Note that someone edited my original question changing "exotic" to "special"; that is fine, but "exotic" does a better job of signaling that numbers like 1/4 and 2 are not really what I had in mind. That is, 2 is indeed quite a special number, but I doubt anyone would consider it exotic.)
|
We have
$$\int\limits_0^\infty {\frac{{\sin x}}{x}dx} = \int\limits_0^\infty {\frac{{\sin x}}{x}\frac{{\sin \left( {{x/3}} \right)}}{{{x/3}}}dx} = \ldots = \int\limits_0^\infty {\frac{{\sin x}}{x}\cdots\frac{{\sin \left( {{x/{13}}} \right)}}{{{x/{13}}}}dx} = \frac{\pi }{2}$$ But
$$\int\limits_0^\infty {\frac{{\sin x}}{x}\cdots\frac{{\sin \left( {{x/{15}}} \right)}}{{{x/{15}}}}dx} = \frac{{467807924713440738696537864469}}{{935615849440640907310521750000}} \cdot \pi$$ See http://link.springer.com/article/10.1023%2FA%3A1011497229317
|
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217,969 |
Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the weak homotopy type of a finite CW-complex, (i.e. when is it a finite space)? By "when", I mean what adjectives do I have to add to make this true, e.g. finite type, separated, smooth... I'm also interested in question 1.) for when $X$ is affine. If you happen to know a reference also, that would be fantastic. Thanks! P.S. I'm aware of this mathoverflow question: How to prove that a projective variety is a finite CW complex? However, it addresses only the case of varieties, unless I am missing something..
|
Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of Lojasiewicz , this is true for any semi-algebraic set (one can even get this for subanalytic sets, by a result of Hironaka, in Triangulation
of algebraic sets , Proc. Amer. Math. Soc. Inst. Algebra Geom. Arcata(1974)); however, the case of (possibly singular) algebraic varieties goes back to the early times of Algebraic Topology: e.g. these papers of van der Waerden and of Lefschetz and Whitehead ). If you only are interested in weak homotopy types, it follows from Lurie's proper base change theorem that considering complex points satisfies proper (hyper)descent (this is Prop. 3.21 in this paper of A. Blanc , which is now published in Compositio Math.). Using Hironaka's resolution of singularities theorem, this implies that, for any scheme of finite type $X$, the space $X(\mathbf{C})$ is a finite homotopy colimit of spaces of the form $Y(\mathbf{C})$ with $Y$ affine and smooth (using Mayer-Vietoris-like homotopy pushouts associated to blow-ups and to coverings by Zariski open subschemes). A smooth affine algebraic variety has the homotopy type of a finite CW-complex: this follows from Morse theory, as can be seen from (the proof of) Theorem 7.2, page 39 in Milnor's book Morse Theory . From all this, we get that a sufficient condition for $X(\mathbf{C})$ to have the weak homotopy type of a finite CW-complex is to be of finite type, while a sufficient condition to get the homotopy type of a finite CW-complex is to be separated of finite type. A sufficient condition to get an actual finite triangulated space is to be proper. If we drop the assumption that the scheme is of finite type, I don't see how we can control/define what happens unless we work with pro-homotopy types of some sort.
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"https://mathoverflow.net/questions/217969",
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|
218,113 |
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
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Let $A$ be the additive group of bounded sequences of elements of $\mathbb{Z}[\sqrt{2}]$. Clearly $A\cong A\oplus\mathbb{Z}[\sqrt{2}]\cong A\oplus\mathbb{Z}^2$ as abelian groups, so we just need to show that $A\not\cong A\oplus\mathbb{Z}$. Let $A_i\cong\mathbb{Z}[\sqrt 2]$ be the subgroup of $A$ consisting of sequences whose terms are all zero, apart from possibly the $i$th term. Lemma 1. If $\varphi:A\to\mathbb{Z}$ is a group homomorphism, then the restriction of $\varphi$ to $A_i$ is zero for all but finitely many $i$. Proof. If not, we can choose $i_0<i_1<\dots$ so that the restriction of $\varphi$ to $A_{i_k}$ is nonzero for all $k$. The intersection of $A_{i_k}$ with $\ker(\varphi)$ has rank at most one, so we can inductively choose $x_k\in A_{i_k}$ so that $\varphi(x_k)\neq0$, $\vert x_k\vert<1$, and $x_k$ is divisible by a larger power of $2$ than any of $\varphi(x_0),\dots,\varphi(x_{k-1})$. Consider the sequences in $A$ whose $i_k$th term is either $x_k$ or $0$, with all other terms zero. Since there are uncountably many such sequences, $\varphi$ must agree on two of them. Taking the difference of these two, we get a non-zero sequence in $\ker(\varphi)$ whose first non-zero term, in the $i_k$th place for some $k$, is $\pm x_k$ and with all other terms divisible by a higher power of $2$ than $\varphi(x_k)$. But this is a contradiction, since $\varphi(x_k)=\pm\varphi(y)$ where $y$ is the sequence obtained by removing the first non-zero term. $\square$ Remark. The same proof works if we replace $A$ by the group of sequences that tend to zero, or the group of sequences such that $\sum_ia_i$ is absolutely convergent, by replacing the condition $\vert x_k\vert<1$ by sharper inequalities. Lemma 2. If $\varphi:A\to\mathbb{Z}$ is a group homomorphism whose kernel contains every $A_i$, then $\varphi=0$. Proof. Let $A'\leq A$ be the subgroup consisting of sequences such that $\sum_ia_i$ is absolutely convergent. Suppose $\varphi(a)\neq0$ where $a\in A$ is the sequence $(a_0,a_1,\dots)$, but that every sequence with finitely many nonzero terms is in $\ker(\varphi)$. Define a homomorphism $\theta:A'\to A$ by
$$\theta(y_0,y_1,y_2,\dots)=(y_0a_0,(y_0+y_1)a_1,(y_0+y_1+y_2)a_2,\dots).$$
Then if $e(k)\in A'$ is the sequence which is zero except that the $k$th term is $1$, then $\varphi\theta\left(e(k)\right)=\varphi(a)\neq0$ for every $k$, contradicting the version of Lemma 1 that applies to $A'$. $\square$ The proofs of Lemmas 1 and 2 are adapted from well-known proofs of the corresponding facts for the "Baer-Specker group" (the group of sequences of integers). These were first proved (I think) by Specker, but the particular proofs that I've adapted are due (I think) to Sasiada and Łos respectively. There are other proofs, and from Yves de Cornulier's comments it seems that at least some of those can also be adapted for what we need. Prop. 3. Every group homomorphism $\varphi:A\to A$ is determined by the compositions $\varphi_{ij}:A_j\to A\stackrel{\varphi}{\to}A\to A_i$, where for each $i$, all but finitely many $\varphi_{ij}$ are zero. Proof. Since $A$ is a subgroup of a direct product of copies of $\mathbb{Z}$ in an obvious way, this follows immediately from Lemmas 1 and 2. $\square$ In other words, this means that if we think of sequences as infinite column vectors, we can represent $\varphi$ as an infinite matrix of homomorphisms $\varphi_{ij}:\mathbb{Z}[\sqrt 2]\to\mathbb{Z}[\sqrt 2]$, with finitely many nonzero entries in each row. Lemma 4. Let $\vartheta:\mathbb{Z}[\sqrt 2]\to\mathbb{Z}[\sqrt 2]$ be a group homomorphism that is not a $\mathbb{Z}[\sqrt 2]$-module homomorphism. Then for any $\epsilon>0$ and $N>0$ there is some $x\in\mathbb{Z}[\sqrt 2]$ with $\vert x\vert<\epsilon$ and $\vert\vartheta(x)\vert>N$. Proof. Straightforward. $\square$ Lemma 5. If $\varphi:A\to A$ is a group endomorphism, and $\varphi_{ij}$ are as above, then for all but finitely many $j$, all the $\varphi_{ij}$ are $\mathbb{Z}[\sqrt 2]$-module homomorphisms. Proof. Suppose not. Then because, by Lemma 1, for each $i$ all but finitely many $\varphi_{ij}$ are zero, we can choose $(i_0,j_0),(i_1,j_1),\dots$ so that $\varphi_{i_kj_k}$ is not a $\mathbb{Z}[\sqrt 2]$-module homomorphism for any $k$, and such that $\varphi_{i_kj_l}=0$ for $k<l$. Using Lemma 4, we can construct a bounded sequence $a=(a_j)$ of elements of $\mathbb{Z}[\sqrt 2]$ inductively so that $a_j=0$ for $j\not\in\{j_0,j_1,\dots\}$ and
$$\vert\varphi_{i_kj_k}(a_{j_k})+\sum_{l<k}\varphi_{i_kj_l}(a_{j_l})\vert>k.$$
But this contradicts the fact that $\varphi(a)$ is bounded. $\square$ Theorem 6. $A\not\cong A\oplus\mathbb{Z}$. Proof. Suppose there were such an isomorphism. Then there would be an injective map $\varphi:A\to A$ with $A/\varphi(A)\cong\mathbb{Z}$. By Lemma 5, $\varphi$ is described by a matrix $(\varphi_{ij})$ with only finitely many columns containing entries that are not $\mathbb{Z}[\sqrt 2]$-module homomorphisms. So for sufficiently large $n$, if $A[n]\leq A$ consists of the sequences whose first $n$ terms are zero, then the restriction of $\varphi$ to $A[n]$ is a $\mathbb{Z}[\sqrt 2]$-module homomorphism, and so $A/\varphi(A[n])$ is a $\mathbb{Z}[\sqrt 2]$-module. But $A/\varphi(A[n])\cong\mathbb{Z}^{2n+1}$ as an abelian group, which is impossible, since $A/\varphi(A[n])\otimes_{\mathbb{Z}}\mathbb{Q}$ is a vector space over $\mathbb{Q}(\sqrt 2)$ and so has even dimension over $\mathbb{Q}$. $\square$ It's obvious that $A\cong A\oplus A$, and if $B=A\oplus\mathbb{Z}$ then it follows that $A\cong B\oplus B$ and $B\cong B\oplus B\oplus B$, so $B$ is an example of an abelian group $B$ with $B\cong B\oplus B\oplus B\not\cong B\oplus B$, I think rather simpler to describe than other examples I know of. Edit (7 April 2016) I've just discovered that this question was answered by Eklof and Shelah in 1985. They reference Shabbagh for asking the question. The link I've given, from Google books, only gives a few pages of the paper, and I haven't yet got hold of a full copy, but their example seems to be more complicated (at least to describe) than mine. I've also realized that my example, if you let $B=A\oplus\mathbb{Z}$, gives an example of non-isomorphic abelian groups $A$ and $B$ with $A\oplus A\cong B\oplus B$, which is one of Kaplansky's "test problems" for abelian groups in his famous 1954 book on Infinite Abelian Groups, which also seems to be simpler to describe than other examples that I know of.
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|
218,192 |
To me, as an non-expert in the field, it seems as if numeric mathematics should have lost its importance because nowadays symbolic calculations or calculations with unlimited precision are generally available. So, just out of curiosity, I would like to know, whether my impression is wrong and what current hot research-topics of practical relevance in numeric mathematics are.
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No, research in numerical mathematics is still very relevant today. One of the main challenges is big data : scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sparse matrices of dimensions 100k or 1M, for instance. Using traditional methods such as Gaussian elimination will take ages even on modern computers, and require way more RAM than they have. To obtain faster methods, one has to understand the structure of the problem and exploit, for instance, knowledge of the behaviour of similar smaller problems to construct better approximations. There is a lot of research on algorithms that can scale up to huge sizes and their mathematical properties. To mention just one example, designing randomized numerical algorithms and proving their effectiveness requires deep mathematical results. For another extreme case, in some quantum chemistry applications, one has to compute approximate eigenpairs in a context in which a single vector of the dimension wouldn't fit in RAM using normal storage. $n=10^{30}$ isn't out of the question, for instance. Another observation is that most problems aren't solved with infinite precision even on today's computers. High precision or exact rational arithmetic is a cause of major slowdowns of a very large factor on modern architectures, and doesn't really solve the issue: if you are using an unstable algorithm, things will go wrong even if you throw 500 digits of precision in it. For instance, the error in your measurements and input data will be amplified to a factor to which you have no more significant digits. People agree that the solution is designing more stable algorithms, not raising the precision. For instance, by exploiting symmetries and hidden structures in the data.
|
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"source": [
"https://mathoverflow.net/questions/218192",
"https://mathoverflow.net",
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]
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|
218,207 |
Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or Sullivan's classic textbooks - that last three of which I have read. But what information about a space is contained in its p-completions? Whilst I have a good handle on the information that may be gleamed from study of the p-localizations, the slightly more abstract nature of the completion leaves me unsure of how to interpret its properties. I am aware of the various arithmetic squares and have no trouble with the concepts or algebra, I just lack any concrete examples of the theory of completions yielding any useful information about the homotopy type of a given space.
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First one should separate between the property and being $p$-complete and process of $p$-completion. In the classical setting, the $p$-completion functor is not so well-behaved for general spaces. For example, the $p$-completion of a space need not be $p$-complete. One way to remedy this is to notice that $p$-completion is not really a functor that should take values in spaces. To understand why, consider the analogous case of groups. The pro-$p$ completion of a group should really be consider as a pro-finite group, as apposed to an ordinary group. This additional structure can be encoded either via a suitable topology on the group, or by replacing the pro-finite group with its inverse system of finite (continuous) quotients. The latter description turns out to fit in a more general categorical context. The collection of "inverse systems of finite groups" can be organized into a category, which is called the pro-category of the category of finite groups. This is a general categorical construction which associates to a category $C$ the category $Pro(C)$ whose objects are inverse systems of objects in $C$ and whose morphisms are suitably defined. We have a natural fully-faithful embedding $C \longrightarrow Pro(C)$ which exhibits $Pro(C)$ as the free category generated from $C$ under cofiltered limits. Furthermore, if $C$ has finite limits then $Pro(C)$ has all small limits. Now, given categories $C,D$ which have finite limits, and a functor $f:C \longrightarrow D$ which preserves finite limits, we obtain an induced functor $Pro(f):Pro(C) \longrightarrow Pro(D)$ which preserves all limits. Under suitable additional conditions (for example, if $C,D$ and $f$ are accessible), the functor $Pro(f)$ will admit a left adjoint $G: Pro(D) \longrightarrow Pro(C)$. A classical example is when $C$ is the category of finite groups, $D$ is the category of all groups, and $f: C \longrightarrow D$ is the natural inclusion. In this case, the corresponding left adjoint $G: Pro(D) \longrightarrow Pro(C)$, when restricted to $D$, is exactly the pro-finite completion functor. If we replace $C$ with the category of finite $p$-groups then we obtain the pro-$p$-completion functor. A similar situation occurs with spaces. Recall that a $p$-finite space is a space with finitely many connected components, each of which has finitely many non-trivial homotopy groups, and all the homotopy groups are finite $p$-groups. Let $\mathcal{S}_p$ be the $\infty$-category of $p$-finite spaces, $\mathcal{S}$ the $\infty$-category of spaces and $f: \mathcal{S}_p \longrightarrow \mathcal{S}$ the natural inclusion. The induced left adjoint $G:Pro(\mathcal{S}) \longrightarrow Pro(\mathcal{S}_p)$, when restricted to $\mathcal{S} \subseteq Pro(\mathcal{S})$, is in some sense the more correct version of the $p$-completion functor. In particular, if $X$ is a space, then the $p$-completion should really be considered as an inverse system of $p$-finite spaces, and not a single space. The inverse limit of this system then coincides with the classical $p$-completion. However, for many reasons it is better to consider the inverse system itself. For example, unlike the classical $p$-completion functor, the functor $G:Pro(\mathcal{S}) \longrightarrow Pro(\mathcal{S}_p)$ is a localization functor with respect to $\mathbb{Z}/p$-cohomology (of pro-spaces). As such, the functor $G$ is idempotent, in the sense that $G(G(X)) = G(X)$, a property that is not shared by the classical $p$-completion functor. Furthermore, the answer to the question "what information on $X$ is contained in $G(X)$" has a precise answer now. It is exactly all the information concerning maps from $X$ to $p$-finite spaces. In light of the enhanced version of the $p$-completion functor, one might ask what does it mean for a space to be $p$-complete. Going back to the situation of groups, one may observe that some groups have the property that they are isomorphic to the underline discrete group of their pro-$p$ completion. In terms of pro-objects, some groups are isomorphic to the inverse limit of their pro-$p$-completion, realized in the category of groups. For example, the group $\mathbb{Z}_p$ of $p$-adic integers has this property. In this case, the group itself is completely determined by its $p$-finite quotients. Similarly, a space is $p$-complete when it is equivalent to the realization of its (enhanced) $p$-completion in the $\infty$-category of spaces. This property has several equivalent manifestations. One of them is the following. For each space $X$, we may consider the cochain complex $C^*(X,\overline{\mathbb{F}}_p)$ with values in the algebraic closure $\overline{\mathbb{F}}_p$ of the finite field $\mathbb{F}_p$. It turns out that $C^*(X,\overline{\mathbb{F}}_p)$ carries a natural structure of an $E_\infty$-algebra over $\overline{\mathbb{F}}_p$. The construction $X \mapsto C^*(X,\overline{\mathbb{F}}_p)$ can then be considered as a functor from spaces to the opposide category of $E_\infty$-algebras over $\overline{\mathbb{F}}_p$. This functor admits a right adjoint, sending an $E_\infty$-algebra $R$ to the mapping space $Map_{E_\infty-Alg}(R,\overline{\mathbb{F}}_p)$. For every space $X$ we then obtain a unit map $X \longrightarrow Map_{E_\infty-Alg}(C^*(X,\overline{\mathbb{F}}_p),\overline{\mathbb{F}}_p)$. It turns out that $X$ is $p$-complete precisely when this unit map is an equivalence. This means that the functor $X \mapsto C^*(X,\overline{\mathbb{F}}_p)$ is fully-faithful when restricted to $p$-complete spaces and we can hence consider $p$-complete spaces as a suitable full sub-category of the opposite category of $E_\infty$-algebras. In addition to the conceptual importance of this result, it also has practical applications. For example, it means that we may construct an Adams-type spectral sequence to compute the homotopy groups of $X$ by resolving $C^*(X,\overline{\mathbb{F}}_p)$ into free $E_\infty$-algebras.
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|
218,518 |
Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can tell after some googling, a new family of integral representations of $\zeta(2n+1)$. I was told I should post this here by an algebraic number theorist at my university, to see if it was a known result or not. Feel free to look at Equations (4), (5), and (7), to see if you recognize them before reading the entire text. Most of this text is for the special case of $\zeta(3)$, but all results generalize quite readily to the case of $\zeta(2n+1)$, which I discuss at the end. Beginning of setup As a preliminary step, we will split the sum $\zeta(3)$ into even and odd parts. Observe that $$
\sum_{n=1}^\infty \frac{1}{(2n)^3} = \frac{1}{8} \sum_{n=1}^\infty \frac{1}{n^3} = \frac{1}{8}\zeta(3)
$$
Therefore $$
\sum_{n=1}^\infty \frac{1}{(2n+1)^3} = \frac{7}{8}\zeta(3) \tag{1}
$$
We now write $$
\frac{1}{(2n+1)^3} = \left(\int_0^1 x^{2n} dx\right)^3 = \int_0^1 \int_0^1 \int_0^1 x^{2n} y^{2n} z^{2n} \tag{2} dx dy dz
$$
and plug this triple integral into Equation (1), obtaining $$
\zeta(3) = \frac{8}{7} \sum_{n=0}^\infty \left[ \int_0^1 \int_0^1 \int_0^1 x^{2n} y^{2n} z^{2n} dxdydz \right]
$$
Using the absolute convergence of the geometric series on $(0,1)$, we can interchange the sum and integrals, obtaining $$
\zeta(3) = \frac{8}{7} \int_0^1 \int_0^1 \int_0^1 \frac{1}{1-x^2y^2z^2} dxdydz \tag{3}
$$ End of setup With this part out of the way, I will now describe the integral representations that form the meat of my question. Consider the $u$-substitution for the integral given by Equation (3) $$
x = \frac{\sinh{u}}{\cosh{v}}, \qquad y = \frac{\sinh{v}}{\cosh{w}}, \qquad z = \frac{\sinh{w}}{\cosh{u}}
$$
Some computation shows that $$
dxdydz = [1-(\tanh{u}\tanh{v}\tanh{w})^2]dudvdw
$$
But dividing both sides by $1-(\tanh{u}\tanh{v}\tanh{w})^2$ and rewriting the left-hand side in terms of $(x,y,z)$ gives $$
\frac{1}{1-x^2y^2z^2}dxdydz = dudvdw
$$
After changing limits, the transformed integral then reads $$
\zeta(3) = \frac{8}{7} \int_0^\infty \int_0^{\sinh^{-1}(\cosh(w))} \int_0^{\sinh^{-1}(\cosh(v))} dudvdw \\
= \frac{8}{7} \int_0^\infty \int_0^{\sinh^{-1}(\cosh(w))} \sinh^{-1}(\cosh(v)) dv dw \tag{4}
$$ But the most interesting part of the above is the following generalization. Let $f : (0,\infty) \to (0,\infty)$ be a function satisfying the following three properties. $f$ is surjective, with $\lim\limits_{x\to 0} f(x) = 0$ and $\lim\limits_{x \to \infty} f(x) = \infty$ $f$ is invertible $f$ is differentiable Now consider the $u$-substitution $$
x = \frac{\sinh{f(u)}}{\cosh{f(v)}}, \qquad y = \frac{\sinh{f(v)}}{\cosh{f(w)}}, \qquad z = \frac{\sinh{f(w)}}{\cosh{f(u)}}
$$
We then find that $$
dxdydz = f'(u)f'(v)f'(w)[1-(\tanh{f(u)}\tanh{f(v)}\tanh{f(w)})^2]dudvdw
$$
and hence $$
\zeta(3) = \frac{8}{7} \int_0^\infty \int_0^{g(w)} \int_0^{g(v)} f'(u)f'(v)f'(w) dudvdw \tag{5}
$$
where $g(x) = f^{-1}(\sinh^{-1}(\cosh(f(x))))$. All of the above can be generalized to $\zeta(2n+1)$. Equation (3) becomes $$
\zeta(2n+1) = \frac{2^{2n+1}}{2^{2n+1}-1} \int_0^1 \int_0^1 \cdots \int_0^1 \frac{1}{1-x_1^2 x_2^2 \cdots x_{2n+1}^2} dx_1 dx_2 \cdots dx_{2n+1} \tag{6}
$$
Our $u$-substitution becomes (where $u_{(2n+1)+1} = u_1$) $$
x_i = \frac{\sinh(f(u_i))}{\cosh(f(u_{i+1}))}
$$
which transforms Equation (6) to $$
\zeta(2n+1) = \frac{2^{2n+1}}{2^{2n+1}-1} \int_0^\infty \int_0^{g(u_{2n})} \int_0^{g(u_{2n-1})} \cdots \int_0^{g(u_1)} f'(u_1)f'(u_2)\cdots f'(u_{2n+1}) du_1 du_2 \cdots du_{2n+1} \tag{7}
$$ My questions are then: Are Equations (4), (5), and (7) known results? Equation (4) is a volume integral. Exploring the region of integration in Mathematica numerically, it looks like an octant of a hyperbolic cube with vertices at infinity. Is this in fact the case?
|
As indicated in my comment, some of these integrals are essentially known, and involve the hyperbolic "Beukers-Kolk-Calabi" change of variables. In particular, in this paper , Z. Silagadze shows in (27) that
\begin{equation*}
\zeta(n) = \frac{2^n}{2^n-1}\int_0^1\cdots\int_0^1 \frac{dx_1\cdots dx_n}{1-x_1^2\cdots x_n^2},
\end{equation*}
which clearly yields (7) above. The hyperbolic change of variables is used later in that paper to obtain a related formula over the $2n$-simplex $\Delta_{2n}$, namely
\begin{equation*}
\zeta(2n+1) = -\frac{1}{2n}\frac{2^{2n+1}}{2^{2n+1}-1}\int_{\Delta_{2n}} \log(\tan x_1 \tan x_2\ldots \tan x_{2n})dx_1\cdots dx_{2n}.
\end{equation*}
This integral is amenable to further interesting modifications. EDIT: The arXiv paper cited above has been published in the journal "Resonance" Another paper by the same author (Z. Silagadze) comments on these integrals. The "Beukers-Kolk-Calabi" change of variables was also considered in this paper . This paper (cited below by Benjamin Dickman) is also relevant here. The relation of these representations to Amoebas is quite interesting, and worthy of further exploration.
|
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|
219,109 |
Let $(M,g)$ be a compact manifold without boundary. Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known? An important example is the $n$-sphere with its standard metric. To find eigenvalues, we embed $S^n$ inside $\mathbb{R}^{n+1}-\{0\}$ in the usual way, consider a positive homogeneous function $f\in C^\infty(\mathbb{R}^{n+1}-\{0\})$ of degree $s$, and then take the restriction to the sphere of the Laplacian $\Delta$ on $\mathbb{R}^{n+1}-\{0\}$ applied to the function $|x|^{-s} f$. The result is that if $f$ is harmonic relative to the Laplacian on $\mathbb{R}^{n+1}-\{0\}$, then the restriction to $S^n$ of $\Delta(|x|^{-s} f)$ is a scalar multiple of the restriction of $f$ to $S^n$, with the scalar being $s(s+n-2)$. One sees very quickly that for more complicated manifolds, such a method does not apply. Various authors comment that the spectrum of the Laplacian is not easy to determine explicitly, and much of the literature seems to be consumed only with estimates for certain eigenvalues of the Laplacian given various constraints on the geometry of $(M,g)$. Are there other interesting manifolds for which the spectrum of the Laplacian is known? In particular, are they known for ellipsoids?
|
Besse (1978, p.202) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine ( 1863 , §19 ; 1878 , §128 ), this gives $\mathbf{RP}^n$ , $\mathbf{CP}^n$ , $\mathbf{HP}^n$ and $\mathbf{OP}^2$ . Edit: Also, for $M$ a compact semisimple Lie group it is well known (due apparently to Freudenthal (1954) ) 1 that the Laplacian (= Casimir) acts on the $\lambda$ -subspace in the Peter-Weyl decomposition $L^2(M)=\smash{\bigoplus_\lambda V_\lambda^{\phantom*}\!\otimes V_\lambda^*}$ by the scalar $c(\lambda):=\smash{\|\lambda + \rho\|^2-\|\rho\|^2}$ ; so these are its eigenvalues. $(\lambda$ : dominant weight; $\rho=\frac12\!\sum\limits_{\alpha > 0}\alpha$ ; $\|\cdot\|$ : Killing norm.) Further edit: The literature contains quite a few more cases than the answers so far. As no single source or search word easily returns them, I list here what I found (others’ answers not repeated): First, the Casimir method above extends to give the spectrum of the normal metric on $G/H$ ( $G$ compact semisimple, $H$ closed). In fact, by Frobenius reciprocity, $V_\lambda$ occurs in $L^2(G/H) = \operatorname{Ind}_H^G1$ with multiplicity equal to the dimension of $V_\lambda{}^H=\{H$ -fixed vectors in $V_\lambda\}$ . So the eigenvalues are exactly all $c(\lambda)$ for $\lambda$ such that $V_\lambda{}^H\ne0$ . After spheres, this method was applied to: Stiefel manifolds $\mathrm{SO}_n\,/\,\mathrm{SO}_{n-m}$ by Levine (1969, p.519) , Gelbart (1974) , Strichartz (1975) . CROSSes ( $\mathbf{RP}^n$ , $\mathbf{CP}^n$ , $\mathbf{HP}^n$ , $\mathbf{OP}^2$ ) by Berger & al. (1971, pp.159-173) , Cahn & Wolf (1976) . Flag manifolds $G\,/\,T$ ( $T$ : maximal torus) by Yamaguchi (1979, p.110) . Grassmannians $\mathrm{Gr}_2(\mathbf R^n)$ by Strese (1980, p.78) and Tsukamoto (1981) . Aloff-Wallach spaces $\mathrm{SU}_3\,/\,\mathrm S^1$ by Urakawa (1984, p.984) and Joe et al. (2001, p.417) . Symmetric spaces $\mathrm{SU}_n\,/\,\mathrm{SO}_n$ by Gurarie (1992, p.253) . Grassmannians $\mathrm{Gr}_n(\mathbf C^{n+m})$ and $\mathrm{SU}_{n+m}\,/\,\mathrm{SU}_n\times\mathrm{SU}_m$ by Ben Halima (2007, pp.546, 549) . Secondly, some cases yield to other methods: Lens spaces $\mathrm S^{2n-1}\,/\,\mathbf Z_p$ by Sakai (1976, p.256) . Hopf manifolds $M_\alpha$ by Bedford & Suwa (1976, p.261) . Berger spheres (total spaces of the Hopf fibration $\mathrm S^1\to\mathrm S^{2n+1}\to \mathbf{CP}^n$ with rescaled fiber) by Tanno (1979, p.184) . Jensen spheres (total spaces of the Hopf fibration $\mathrm S^3\to\mathrm S^{4n+3}\to \mathbf{HP}^n$ with rescaled fiber) by Tanno (1980, p.103) and Nilsson & Pope (1983, p.68) . Grassmannians $\mathrm{Gr}_2(\mathbf C^n)$ by Sumitomo & Tandai (1985, p.153) . Riemannian two-step nilmanifolds $G\,/\,\Gamma$ by Pesce (1993) . 1 Note added: Rogawski–Varadarajan ( 2012 , p. 690) attribute the formula for $c(\lambda)$ to Casimir–van der Waerden ( 1935 ; note the reviewer). However, I’m not sure I can find it there...?
|
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|
219,118 |
I'm curious what languages contribute the largest fraction of published research mathematics. That is, for a given language the percent of new research being published in that language. I'm especially curious to see how you come up with such numbers. From some googling I managed to get a very crude estimate of total publication volume: 28643 arXiv articles in 2014 . Looks like arXiv sometimes has articles in other languages but I can't search by language. Edit: Thanks S. Carnahan for pointing out that I could search the full text by common words from different languages!
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This question has no definite answer if time frame is not specified. The situation in 20s century changed very quickly. In the first half of the century, German and French dominated.
(More German than French). In the second half, it is clearly English, and one does not need any research to see this. But English dominates more and more. In the 1970-s and 1980-s about 1/3 of all papers was published in Russian. (I did count this. There was a Russian counterpart of Math Rev, using Cyrillic for Russian papers, so it was easy to count).
Not anymore. Until the late 70s, most German papers were published in German. Not anymore.
My guess would be that in 21st century an overwhelming majority of publications are in English, and there are some in French and Chinese. Proportion of other languages is negligibe. EDIT. In the discussion, mostly arXiv is discussed, so you probably mean very recent papers. I follow the arxiv since its inception, and my impression is that less than
1% of all papers are written in French or Russian. I have not seen any in German or other languages, including Chinese. EDIT2. One also has to specify what kinds of books and papers are we talking about. Of course, textbooks and lecture courses are published in every language. Because TEACHING in most countries is in the native language, especially undergraduate teaching. Same applies to many publications intended for more general audience, publications related to education and history, popular mathematics etc. In the arXiv they are in the sections "History" or "General math". All non-English papers on the arXiv that I know are in "History".
I will not be surprised if statistics from the arXiv will be very different from that in Mathscinet, even if one restricts Mathscinet to the recent years when arXiv exists. In fact arXiv is not a good representative for the whole mathematics. In some areas, almost all papers go to the arXiv, in other areas, only a small proportion. EDIT3. OK, in Mathscinet they indicate the language:-) Since 2000:
70K in Chinese, 66K in Russian, 21K in French, 5K in Spanish and Japanese each, 4K in German and Italian each, 1K in Portuguese. Out of 1,358,623 total. Zentralblatt seems to have a similar feature but I could not make it work.
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219,132 |
Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic ( http://cantorsattic.info/Upper_attic ) Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type B cardinals can coexist), I define: *Type A > Type B if smallest Type A cardinal has higher cardinality than smallest Type B *Type A = Type B if smallest Type A and Type B have same cardinalities *Type A $\perp$ Type B if the ordering in the sense above is undecidable So, for instance, Inaccesssible < Hyper Inaccessible < Mahlo What is known about the ordering of large cardinals from ( http://cantorsattic.info/Upper_attic ) in this sense ? I am particularly interested in the "large" large cardinals from measurable upwards.
For eg: How would one order measurable, extendible, huge and rank-into-rank ? Motivation: I understand researchers are mostly focused on consistency strength, but I am interested in the intuitive notion of getting "much bigger infinities" from each successive large cardinal axiom.
|
The usual relations to consider in the large cardinal hierarchy
are Direct implication: every A cardinal is also a B cardinal Consistency strength implication: if ZFC + there is an A cardinal
is consistent, then so is ZFC + there is a B cardinal. Your concept, however, is focused on the least instance of the
large cardinal notion, and this is also studied. In broad terms, the large cardinal hierarchy is roughly linear,
with the stronger cardinals being stronger with respect to all
three of these relations. In most instances, we have that every A
cardinal (the stronger notion) is also a B cardinal, as well as a
limit of B cardinals, and so we get also the consistency
implication and the least A cardinal is strictly larger than the
least B cardinal. However, there are some notable deviations from this. These
deviations come in two types. First, there are the instances where a large cardinal concept A
has stronger consistency strength than B, but the least instance
of A is definitely less than the least instance of B. For example,
a superstrong cardinal has higher consistency strength
than a mere strong cardinal, since if $\kappa$ is superstrong, then $V_\kappa\models$ ZFC + there
is a proper class of strong cardinals, but the least superstrong
cardinal is definitely less than the least strong cardinal. This
is simply because superstrongness is witnessed by a single object,
and strong cardinals are $\Sigma_2$ reflecting, and therefore
reflect the least instance below. There are numerous similar instances of this. Any time a large
cardinal notion is witnessed by a single object or is witnessed
inside some $V_\theta$ — and this would include weakly
compact, Ramsey, measurable, superstrong, almost huge, huge,
rank-to-rank and others — then the least instance of that cardinal will be
less than the least $\Sigma_2$-reflecting cardinal and indeed less
than the least $\Sigma_2$-correct cardinal. But $\Sigma_2$ correct
cardinals provably exist in ZFC, and therefore have very low
consistency strength. So we have numerous interesting instances where your $<$ order
does not align with consistency strength: The least almost huge cardinal is strictly less than the least
strong cardinal. The least rank-to-rank cardinal is strictly less than the least
strongly unfoldable cardinal. The least $5$-huge cardinal is strictly less than the least
uplifting cardinal. There are hundreds of other similar examples. You can invent them yourself! Meanwhile, second, there are examples of your $\perp$ situation,
where the size of the smallest instance is not yet settled. This
phenomenon is known as the "identity-crises" phenomenon, named by
Magidor when he proved that the least measurable can be the same
as the least strongly compact, or strictly less, depending on the
model of set theory. Many further instances of this are now known,
some of which appear in my paper: B. Cody, M. Gitik, J. D. Hamkins, and J. A. Schanker, The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact , Archive for Mathematical Logic, pp. 1-20, 2015. This paper provides many instances of your $\perp$ situation, where the question of whether the least A cardinal is smaller than or the same size as the least B cardinal is not settled in ZFC. Finally, let me qualify my remark that the large cardinal hierarchy is roughly linear. The hierarchy is indeed mainly linear, but one sometimes hears stronger assertions of linearity, as something that we know and which needs explanation, but I don't feel these knowledge claims are justified. Of course, the identity crises phenomenon provides instances of non-linearity in the direct implication hierarchy, and so when large cardinal set theorists assert that the large cardinal hierarchy is linear, they are speaking of the consistency strength order. So let me mention a few cases where we simply don't yet know linearity: A supercompact cardinal versus a strongly compact plus an inaccessible above. A supercompact cardinal versus a proper class of strongly compact cardinals. A Laver-indestructible weakly compact cardinals versus a strongly compact cardinal. A cardinal $\kappa$ that is $\kappa^+$-supercompact versus $\kappa$ is $\kappa^{++}$-strongly compact. A PFA cardinal versus a strongly compact cardinal. And many others. My perspective is this. Because we have essentially no method for proving non-linearity in the consistency strength hierarchy, it is not surprising that we see only instances of linearity, and this may be a case of confirmation bias. But don't get me wrong: of course I agree that the consistency strength hierarchy is mainly linear in broad strokes.
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219,196 |
Let $q$ be a number. Let us consider the $q^2-1$-th line of the Pascal triangle (i.e. numbers ${{q^2-1} \choose i}$, $i=0,1,...q^2-1$). We have $q^2$ numbers. Let us form naively a $q \times q$ matrix from them. Example for $q=2$: \[\begin{pmatrix} 1 & 3 \\
3 & 1 \end{pmatrix}\] And let us take its determinant: $-8$ if $q=2$. How to prove that if $q$ is prime then $q$ enters in this determinant with
$(q+4)(q-1)/2 $-th exponent? Example: $q=2$, $(q+4)(q-1)/2 =3$. Really, $8=2^3$. Case $q=3$: the binomial coefficients are $1,8,28,56,70,56,28,8,1$. The matrix: \[\begin{pmatrix} 1 & 8 & 28 \\
56 & 70 & 56 \\
28 & 8 & 1 \end{pmatrix}\] The determinant is $2 \cdot 3^7 \cdot 7$. Really, $(q+4)(q-1)/2 =7$.
|
Looking at Krattenthaler's famous "determinant calculus" survey (also, suggested by Per Alexandersson ) we find Theorem 26 in there, which answers a more general question, and should (most likely) yield the claim in the OP. Theorem Let $q$ be a nonnegative integer, and let $L_1,\ldots,L_q$ and $A, B$ be indeterminates. Then,
\begin{equation*}
\det\left( \binom{BL_i+A}{L_i+j}\right) = \frac{\prod_{i < j}(L_i-L_j)}{\prod_{i=1}^q(L_i+q)!}\prod_{i=1}^q\frac{(BL_i+A)!}{((B-1)L_i+A-1)!}\prod_{i=1}^q(A-Bi+1)_{i-1},
\end{equation*}
where $(a)_k$ is the usual Pochhammer symbol . The matrix in the OP is given by
\begin{equation*}
M_{ij} = \binom{q^2-1}{q(i-1)+j-1},
\end{equation*}
thus we can select $A=q^2-1$, $B=0$, $L_i = q(i-1)-1$, and apply the above result, and hopefully things fall into place.
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219,201 |
Let $\Lambda(x,y)$ be the count of totatives of $x$ that are less than or equal to $y$. I am asking for the following result to be verified, (particularly the final proposal), I have found no counterexamples and believe the reasoning to be correct. The first part of this is an interpretation of $\phi(x)$ and the second part uses those observations to construct a formula for $\Lambda(x,y)$. Let $\phi(x)$ be the number of totatives of a natural number $x$. Euler's product formula for $\phi(x)$ is; $\phi(x) = x \prod_{p|x}(1-\frac{1}{p})$ This statement can be interpreted as a probabilistic result as follows. Probabilistic intepretation of $\phi(x)$ Let a $p$-interval be a interval of the form $\lbrace a_{1+kp}, a_{2+kp}, ..., a_{p+kp} \rbrace$ where: a) $a_i=i$, $\forall i \in \mathbb{N}$ b) $p$ is any prime number, and gives the size of the interval. c) $k$ can be chosen to position this interval such that it includes any natural number we want. A $p$-interval is a complete residue class for $p$, which gives rise to the following probabilities: For a randomly selected number $x$ in a $p$-interval, the probability that $x \equiv 0 ($mod $p)$ is $\frac{1}{p}$. For a randomly selected number $x$ in a $p$-interval, the probability that $x \not\equiv 0 ($mod $p)$ is $(1-\frac{1}{p})$. Another useful property of $p$-intervals is that if we have an interval $I$ of the natural number line, and $I$ is the union of a discrete amount of $p$-intervals for some prime $p$, therefore call $I$ by the new notation $I_p$, then the following probabilities are true. For a randomly selected number $x$ in an interval $I_p$, the probability that $x \equiv 0 ($mod $p)$ is $\frac{1}{p}$. For a randomly selected number $x$ in an interval $I_p$, the probability that $x \not\equiv 0 ($mod $p)$ is $(1-\frac{1}{p})$. Now consider an interval $I$ that is expressible as both $I_{p_1}$ and $I_{p_2}$ It is easy to show that; $P(x \not\equiv 0 ($mod $ p_1)$ for $ x\in I) \perp P(x \not\equiv 0 ($mod $ p_2)$ for $ x \in I)$, that is; the two probabilities are independent of each other. The independence result can be extended to include any number of unique prime numbers, that is; the divisibility of any natural number by a prime number is independent of its divisibility by any other prime number. What this mean in practice is that the probability that a randomly selected number $x$ from an interval $I$, where $I$ is expressable as $I_{p_a}$, $I_{p_b}$, ..., $I_{p_z}$ (So $I$ is the union of a discrete amount of $p$-intervals for each $p=p_a, p_b, ..., p_z$); the probability that $x$ is not equal to $0$ modulo $p_a,p_b,..., p_z$ is just $(1-\frac{1}{p_a}) \times (1-\frac{1}{p_b}) \times$ ... $ \times (1-\frac{1}{p_z})$. So to conclude, the count of totatives of some $x$ is just $x$ times the probability that a randomly selected number $y\leq x$ is not equal to $0$ modulo any prime divisor of $x$. And this probability is $\prod_{p|x}(1-\frac{1}{p})$. To summarise Euler's probability , $\prod_{p|x}(1-\frac{1}{p})$ is a result of the following three probabilistic observations: i) For a randomly selected number $x$ in an interval $I_p$, the probability that $x \not\equiv 0 ($mod $p)$ is $(1-\frac{1}{p})$. ii) For interval $I$ expressible as both $I_{p_1}$ and $I_{p_2}$, $P(x \not\equiv 0 ($mod $ p_1)$ for $ x\in I) \perp P(x \not\equiv 0 ($mod $ p_2)$ for $ x \in I)$, that is; the two probabilities are independent of each other. iii) The interval $[1,x]$ is expressible as $I_p$ for all prime divisors of $x$. When considering $\Lambda(x,y)$, we can build an expression for it that is a result of the three probabilistic observations required to construct Euler's product formula for $\phi(x)$. Constructing $\Lambda(x,y)$ To construct $\Lambda(x,y)$ let the expected value of $\Lambda(x,y)$ be $\frac{y}{x}\times \phi(x)$. That is; Let $\Lambda_E(x,y) =\frac{y}{x}\times \phi(x)$ This value may be expected because it is simply the Euler probability for $x$ multiplied by the amount of numbers in the interval $[1,y]$ being $y$. Therefore this expected value relies on the following three conditions: i) For a randomly selected number $x$ in an interval $I_p$, the probability that $x \not\equiv 0 ($mod $p)$ is $(1-\frac{1}{p})$. ii) For interval $I$ expressible as both $I_{p_1}$ and $I_{p_2}$, $P(x \not\equiv 0 ($mod $ p_1)$ for $ x\in I) \perp P(x \not\equiv 0 ($mod $ p_2)$ for $ x \in I)$, that is; the two probabilities are independent of each other. iii) The interval $[1,y]$ is expressable as $I_p$ for all prime divisors of $x$. Note that the first two conditions are mathematical facts. The third condition is not always true. To refine condition iii) into a statement of mathematical fact about the relationship between a general natural number $y$ and its divisibility by the prime factors of some general number $x$, we would need to find a deeper global statement. If the refined global statement was a statement of equivalence then arguably it will be highly lengthy and chaotic because of the infinitely chaotic distribution of prime numbers to be described in order to scribe the relationship between divisibility properties of general $x$ and $y$. Therefore, we are forced to make use of fuzzier mathematics. Not only that, but until we find deeper global statements that can replace condition iii) then it may be more fruitful to compensate for condition iii) locally; that is in terms of the chosen $x$ and $y$ values, and with functions which cannot yet be generalized and require direct computation. Now less philosophy and more mathematics... As the third condition is not always true, we expect some degree of variance $V$ such that $\Lambda(x,y) = \Lambda_E(x,y) \pm V$. In order to calculate the variance, we will first make some definitions. For any natural number $x$, let $x_\flat$ (or $x$ flat) be the product of prime divisors of $x$.
Also, let $l_\flat=$gcd$(x_\flat,y_\flat)$ and $x_\sharp = \frac{x_\flat}{l_\flat}$. This gives us the notation to talk about prime divisors of $x$ which do not divide $y$. These are important for calculating $\Lambda(x,y)$ because the variance $V$ is a consequence of $[1,y]$ not being expressable as $I_p$, where $p$ is any of those prime divisors. Note that: For each $p$ that divides $x_\sharp$, there exists a $\zeta_p$ such that $p$ divides a natural number in the interval $[y-\zeta_p, y+ \zeta_p]$. Particularly, there exists a natural number in the interval $[y-\zeta_{x_\sharp}, y+ \zeta_{x_\sharp}]$ that is divisible by $x_\sharp$ for some $\zeta_{x_\sharp}$ It is obvious that $\zeta_{x_\sharp} \leq x_\sharp - 1$. Therefore my proposal is that the variance $V$ belongs in the range $0\leq V \leq \frac{x_\sharp -1}{x_\sharp}\phi(x_\sharp)$. This proposal comes from the fact that we are essentially adding or subtracting totatives of $x_\sharp$ from the interval $[y\pm \zeta_{x_\sharp},y]$ (either + or -, and not caring about the direction of the interval). So $\Lambda(x,y) = \frac{y}{x}\phi(x) \pm V$ where $0 \leq V \leq \frac{x_\sharp -1}{x_\sharp}\phi(x_\sharp)$ Or more simply; $\Lambda(x,y) = \frac{y}{x}\phi(x) \pm V$ where $0 \leq V \leq \phi(x_\sharp)$ (I believe there is room for improvement).
|
Looking at Krattenthaler's famous "determinant calculus" survey (also, suggested by Per Alexandersson ) we find Theorem 26 in there, which answers a more general question, and should (most likely) yield the claim in the OP. Theorem Let $q$ be a nonnegative integer, and let $L_1,\ldots,L_q$ and $A, B$ be indeterminates. Then,
\begin{equation*}
\det\left( \binom{BL_i+A}{L_i+j}\right) = \frac{\prod_{i < j}(L_i-L_j)}{\prod_{i=1}^q(L_i+q)!}\prod_{i=1}^q\frac{(BL_i+A)!}{((B-1)L_i+A-1)!}\prod_{i=1}^q(A-Bi+1)_{i-1},
\end{equation*}
where $(a)_k$ is the usual Pochhammer symbol . The matrix in the OP is given by
\begin{equation*}
M_{ij} = \binom{q^2-1}{q(i-1)+j-1},
\end{equation*}
thus we can select $A=q^2-1$, $B=0$, $L_i = q(i-1)-1$, and apply the above result, and hopefully things fall into place.
|
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219,264 |
(Crossposted from math.stackexchange by suggestion) On page 12 of Shinichi Mochizuki's " On the Verification of Inter-universal Teichmuller Theory: A Progress Repor ", he writes "The representation-theoretic approach exemplified by the Langlands
program does indeed constitute one major current of research in modern
number theory. On the other hand, my understanding is that the idea
that every essential phenomenon in number theory may in fact be
incorporated into, or somehow regarded as a special case of, this
representation-theoretic approach is simply not consistent with the
actual content of various important phenomena in number theory." Does anyone know what these 'various important phenomena' are that he's referring to?
|
As noted by Lucia, large parts of number theory are completely "beyond the scope of the Langlands program": most of analytic number theory, obviously, is, but also many important, active and beautiful subfields of algebraic number theory -- for a list of examples, see for instance the list of publications of Bjorn Poonen, which cover a large scope of subjects in algebraic number theory, but touch the Langlands program only tangentially. To make sense of the proposed quote of Mochizuki, it is therefore necessary to give a much more restrictive interpretation of "number theory",
which I imagine would be in this context the set of number-theoretic questions that can be solved or at least attacked by understanding the Galois groups of number fields, together with all its attached tractors of decomposition groups, inertia subgroups, and Frobenius elements . This set contains all the reciprocity laws, proved or conjectured, and many diophantine equations and problems, from Fermat's last theorem to Mordell's conjecture and its generalizations, through Birch and Swinnerton-Dyer. My guess is that Mochizuki means that even in this relatively restricted sense, many phenomenons of number theory lie beyond the scope of Langlands. In saying so, he is following an idea that Grothendieck tried to disseminate in the 80's ("avec force", as Deligne said), which I will try to summarize as follows. The Langlands program tries to understand Galois groups by looking at their representations, that is, their action over vector spaces over fields or slightly more generally over modules over commutative rings. At any rate, these objects form an abelian category (even Tannakian), a linear object. Grothendieck's theory of Motives
should be attached to this general program, because the motives also
are "linear objects", and as Langlands himself famously argued in his famous 1979 paper "Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen". But, Grothendieck argues, there are many other kind of objects on which we may let Galois group act in order to study them in a different direction than with representations: sets, for instance, as in the version of Galois theory developed in SGA 1 (1960), or non-abelian (profinite) groups, as the étale fundamental groups of various variety over $\mathbb Q$: to this aspect belong all the theory of Grothendieck-Teichmüller, the Dessins d'enfants, the anabelian geometry. That is of course a subject in which Mochizuki has himself proved some remarkable results, and which he claims to have enormously developed up to the point of deducing the ABC conjecture. An example of a number theoretic (in the restricted sense mentioned above) problem that arguably lies beyond the Langlands program but that
the practicians of anabelian geometry plans to study successfully is Mordell's conjecture (proved by Faltings, it is true, with methods not so far from the Langlands program, at least in that they study "linear objects", namely abelian varieties) and all its generalizations (still open). For a very interesting, if speculative, discussion of these questions, see "Galois Theory and Diophantine Geometry" by Minhyong Kim.
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219,414 |
Let $p$ be a prime congruent to $1$ mod. 8. If $p= 17$ one has : $p+ 8 = 5 ^2$. If $p= 41$ one has : $p+ 8 = 7 ^2$. If $p= 73$ one has : $p+ 8 = 9 ^2$. If $p= 89$ one has : $p+ 32 = 11 ^2$. If $p= 97$ one has : $p+ 128 = 15 ^2$. If $p= 113$ one has : $p+ 8 = 11 ^2$. If $p= 137$ one has : $p+ 32 = 13 ^2$. If $p= 193$ one has : $p+ 32 = 15 ^2$. If $p= 233$ one has : $p+ 128 = 19 ^2$. For $p=241$, there is no value $s\leq 4000$ for which $p+2^{2s+1}$ is a square. Question 1 : Is there a simple way to prove that no such $s$ can exist ? Question 2 : Can one estimate the density of those primes $p\equiv 1$ mod. 8 that can be written under the form
$$p=m^2-2^{2s+1}\ \ \ ?$$
|
To answer your first question: there is indeed no $s$ such that
$241+2^{2s+1}$ is a perfect square. -- Proof: $2^{2s+1}$ is always
congruent to either $2$, $8$ or $32$ modulo $63$, which makes
$241+2^{2s+1}$ congruent to either $21$, $54$ or $60$ modulo $63$.
However none of these values is a quadratic residue modulo $63$,
and thus $241+2^{2s+1}$ cannot be a perfect square, as claimed.
|
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219,534 |
Good epigraphs may attract more readers. Sometimes it is necessary.
Usually epigraphs are interesting but not intriguing. To pick up an epigraph is some kind of nearly mathematical problem: it should be unexpectedly relevant to the content. What successful solutions are known for you?
What epigraphs attracted your attention? Please post only epigraphs because quotes were collected in Famous mathematical quotes . There are certain common Privileges of a Writer,
the Benefit whereof, I hope, there will be no Reason to doubt;
Particularly, that where I am not understood, it shall be concluded,
that something very useful and profound is coucht underneath.
(JONATHAN SWIFT, Tale of a Tub, Preface 1704) [Taken from Knuth, D. E. The art of computer programming. Volume 3: Sorting and searching .]
|
The most interesting epigraphs I have seen in mathematical books are in: Bender and Orszag, Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory. Every chapter is decorated by an epigraph from Sherlock Holmes. For example: The triumphant vindication of bold theories - are these not the pride and justification of our life's work? (Conan Doyle, The Valley of Fear) Just feel like re-reading Sherlock Holmes:-) Reed and Simon, Methods of Mathematical physics, especially volume 1. For example, the chapter on Unbounded Operators has this: I tell them that if they will occupy themselves with the study of mathematics, they will find that it is the best remedy against the lusts of the flesh. (Th. Mann, Magic Mountain). But my favorite one is the following, from Kirillov, What's a number?: Examiner: What is a multiple root of a polynomial? Student: Well, this is when we plug a number to a polynomial and obtain zero;
plug it again and obtain zero again... And this happens $k$ times. But on the $(k+1)$ -st time we do not obtain zero. Cannot help citing one more. Brocker, Lander, Differentiable germs and catastrophes:
|
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219,590 |
Today I started reading Maddy's Believing the axioms . As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity. Apparently, the reason why we usually take it is because it makes sets well-founded and makes $\in$ -induction work, or because it puts all sets into a hierarchy (namely $V$ ). However, these reasons sound to me more like "we take this, because it's convenient". Another reason commonly given is "It's difficult to think of a set which is an element of itself". This is not a good reason, because many things are difficult to think of, and also one could argue that a set represented by $\{\{\{...\}\}\}$ should do the trick. That brings me to my question: Are there any "philosophical" reasons to believe that the axiom of regularity holds? I understand that this question is quite vague and maybe too broad, but I will be thankful for any responses. ${}{}$
|
Regularity (aka Foundation) can be seen philosophically as an axiom of restriction . It is not necessarily saying “all the things you consider as sets must be well-founded”. It can be read saying “for the purposes of this set theory, we restrict our universe of discourse to just the well-founded objects”. It’s clarifying what we mean by sets , in a similar way as the extensionality axiom does. You may find this explanation unsatisfying, since it’s fairly similar to what Maddy gives. But the point is that if you are philosophically unsure about it, the question to ask is not “Are all sets really well-founded?” but “Is it really convenient/harmless/natural to restrict attention to the well-founded sets?” A precise statement which can be seen as justifying this is the fact that within (ZF – Regularity), one can prove that the class of well-founded objects is a model of ZF. Edit: see this followup question and its answer for: a rather stronger sense in which regularity is harmless, in the presence of choice: ‘Over (ZFC – regularity), regularity has no new purely structural consequences’ a counter-observation that in the absence of choice, over (ZF – regularity), it’s not so clearly harmless; it has consequences that can be stated in purely structural terms, such as ‘every set is isomorphic to the set of the children of some element in some well-founded extensional relation’.
|
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219,638 |
Background Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by Tim Gowers and is in line with other forms of Internet mathematical research activity which include MathOverflow. Former and current projects The polymath wiki page gives a description and links to former polymath projects and much additional information. So far, there were about 10 polymath projects of which 6-7 led to intensive research, and among those 3-4 were successful. (There were several MathOverflow questions motivated by running polymath projects, especially questions related to polymath5 .) Those projects ran over Gowers's blog ( polymath1 , polymath5 and others), Tao's blog ( polymath8 ), the Polymath Blog (administered by Tao, Gowers, Nielsen, and me) ( polymath4 and polymath7 ), and my blog ( polymath3 ). Updates (Before Nov 2016) There were a couple of additional polymath-type projects. (Nov '15, 2016) Currently, polymath10 on Erdős-Rado delta system conjecture is running on my blog .(New, Dec 29, '15) Terry Tao posted (on behalf of Dinesh Thakur) an interesting proposal for a polymath project regarding identities for irreducible polynomials Update: problem solved by David Speyer. ( January 31, 2016) Tim Gowers launched on his blog polymath11 on Frankl's union-closed conjecture. Updates (Before January 2018) : Timothy Chow launched polymath12 on Rota's basis conjecture (February 24, 2017). (It was proposed as an answer to this question here .) (May 14, 2017) Tim Gowers is running a polymath-like project polymath13 on "Intransitive dices". (Dec 24 2017) A spontaneous polymath project, polymath14, over Tao's blog: A problem was posed by Apoorva Khare was presented and discussed and openly and collectively solved . (And the paper arxived .) Update (January 25,2018) A new polymath project is emerging on Tao's blog: Polymath proposal: upper bounding the de Bruijn-Newman constant . Update: This is polymath15 which seems very active and quite successful. ( wikipage ) Updates (April 14, 2018, June, 2019) Dustin Mixon and Aubrey de Grey have launched Polymath16 over at Dustin’s blog. The project is devoted to the chromatic number of the plane ( Wikipage ) following Aubrey de Grey's example showing that the chromatic number of the plane is at least 5. See also a proposal post and discussion thread over the polymath blog , and a proposal over here . Polymath 16 was now concluded . Update, June 2019 Terry Tao initiated a sort of polymath attempt to solve the following problem conditioned on some conjectures from arithmetic algebraic geometry: Is there any polynomials $P$ of two variables with rational coefficients, such that the map $ P: \mathbb Q \times \mathbb Q \to \mathbb Q$ is a bijection? This is a famous 9-years old open question on MathOverflow . Update, March 2020 : On Terry Tao's blog, Polymath proposal: clearinghouse for crowdsourcing COVID-19 data and data cleaning requests . The proposal is to: (a) a collection of public data sets relating to the COVID-19 pandemic, (b) requests for such data sets, (c) requests for data cleaning of such sets, and (d) submissions of cleaned data sets. (Proposed by Chris Strohmeier after discussions among several mathematicians.) Update (January 12, 2021) A polyTCS blog-based project was launched a year ago by Rupei Xu and Chloe Yang. It contains several interesting proposals. Former proposals for future projects There were also 10-20 additional serious proposals. A few proposals of various nature (from which polymath5 was selected) are gathered in this post on Gowers's blog, and several that appeared on various places are summarized on the polymath Wiki and also on the polymath blog . The polymath projects so far consisted of an attempt to solve a specific open problem but some of the proposals were of different nature. More background So far, polymath projects, while getting considerable attention and drawing enthusiasm, (and some controversy ,) were limited in scope within mathematics and among mathematicians. In most cases a small team of participants were the devoted contributed and in some cases those devoted participants were experts in the relevant area. Thus projects may apply primarily to experts in a specific field of mathematics. In all existing examples the project itself had some general appeal. For a polymath project, in addition to the main task of trying to reach or at least greatly advance the goals of the specific project there are secondary goals of trying to understand the advantages and limitation of the polymath concept itself, and of trying to openly record the thought process of different participants towards the specific goal. The question The question is simple: Make additional proposals for polymath projects. Summary of proposals (updated: January 12, 2021) The LogRank conjecture . Proposed by Arul. The circulant Hadamard matrix conjecture . Proposed by Richard Stanley. Finding combinatorial models for the Kronecker coefficients . Proposed by Per Alexandersson. Eight lonely runners . Proposed by Mark Lewko. A problem by Ruzsa: Finding the slowest possible exponential growth rate of a mapping $f:N→Z$ that is not a polynomial and yet shares with (integer) polynomials the congruence-preserving property $n−m∣f(n)−f(m)$ . Proposed by Vesselin Dimitrov. Finding the Matrix Multiplication Exponent ω. (Running time of best algorithm for matrix multiplication.) Proposed by Ryan O'Donnell. The Moser Worm problem and Bellman's Lost in a forest problem . Proposed by Philip Gibbs. Rational Simplex Conjecture ( by Cheeger and Simons). Proposed by Sasha Kolpakov. Proving that for every integer $m$ with $|m| \le c(\sqrt{n}/2)^n$ there is an $n \times n$ 0-1 matrix matrix whose determinant equals $m$ . Proposed by Gerhard Paseman. Proving or disproving that the Euler's constant is irrational. Proposed by Sylvain JULIEN. The Greedy Superstring Conjecture. Proposed by Laszlo Kozma. Understanding the behavior and structure of covering arrays. Proposed by Ryan. The group isomorphism problem , proposed by Arul based on an early proposal by Lipton . Frankl's union closed set conjecture (Proposed by
Dominic van der Zypen; Also one of the proposals by Gowers in this post ). ( Launched ) Komlos's conjecture in Discrepancy Theory . Proposed by Arul. Rota's Basis Conjecture. Proposed by Timothy Chow. Launched on the polymath blog. To show that $2^n+5$ composite for almost all positive integers $n$ . (Might be too hard.) Proposed by me. To prove a remarkable combinatorial identity on certain Permanents . Proposed by me. Update, Aug 6, 2016: settled! Real world applications of large cardinals Proposed by Joseph van Name.
There were a few more proposals in comments. A project around a cluster of tiling problems. In particular: Is the Heech number bounded for polygonal monotiles? Is it decidable to determine if a single given polygonal tile can tile the plane monohedrally? Even for a single polyomino? Proposed by Joseph O'Rourke To prove that $\sum \frac{\sin (2^n)}{n}$ is a convergent series. Proposed by
JAck D'aurizio The Nakayama conjecture and the finitistic dimension conjecture (major problems from the intersection of representation theory of finite dimensional algebras) and homological algebra. Proposed by Mare. Major questions in the field of stereotype spaces and their applications , proposed by Sergei Akbarov. The Erdos-Straus conjecture , proposed by Amit Maurya The Collatz conjecture , proposed by Amit Maurya. Indecomposability of image transformations , proposed by Włodzimierz Holsztyński Is there a degree seven polynomial with integer coefficients such that (1) all of its roots are distinct integers, and (2) all of its derivative's roots are integers? , Proposed by Benjamin Dickman. The Cartan determinant conjecture for quiver algebras , proposed by Mare. The number of limit cycles of a polynomial vector field , Proposed by Ali Taghavi. Small unit-distance graphs with chromatic number 5 , proposed by Noam Elkies. Became Polymath16, see above. (new) Lower bounds for average kissing numbers of non-overlapping spheres of different radii Proposed by Sasha Kolkapov. (new) A uniformly distributed random variable decomposition conjecture proposed by Sil. (new) The 3ᵈ conjecture and the flag-number conjecture proposed by me. Proposed rules (shortened): All areas of mathematics including applied mathematics are welcome. Please do explain what the project is explicitly and in some details (not just link to a paper/wilipedea). Even if the project appeals to experts try to give a few sentences for a wide audience. Please offer projects that you genuinely think to be potentially suitable for a polymath project. (Added) Criteria that were proposed for a polymath project. Joel David Hamkins asked for some criteria that have been proposed for what kind of problem would make a good polymath project? I don't think we have a clear picture on criteria for good polymath projects and there could be good projects of various kind. But the criteria for the first project are described by Gowers (I modified the wording to make them not specific in one sentence), and they seem like good criteria for a first project
in a new field be it algebraic geometry, algebraic topology, group theory, logic, or set theory (to mention a few popular MO tags). " I wanted to choose a genuine research problem in my own area of mathematics, rather than something with a completely elementary statement or, say, a recreational problem, just to show that I mean this as a serious attempt to do real mathematics and not just an amusing way of looking at things I don’t really care about. This means that in order to have a reasonable chance of making a substantial contribution, you probably have to be a fairly experienced [researcher in the field of research]. So I’m not expecting a collaboration between thousands of people, but I can think of far more than three people who are suitably qualified. Other criteria were that I didn’t want to choose a famous unsolved problem, or a problem where I had no idea whatever where to start. For a first attempt, it seemed a better idea to choose a problem that I’d love to solve, about which I already have some ideas, but in which I don’t (yet) have a significant emotional investment. Does the problem split naturally into subtasks? That is, is it parallelizable? I’m actually not completely sure that that’s what I’m aiming for. ... I’m interested in the question of whether it is possible for lots of people to solve one single problem rather than lots of people to solve one problem each. However, my contention would be that any reasonably complex solution to a problem is somewhat parallelizable and becomes increasingly so as one thinks about it."
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The circulant Hadamard matrix conjecture states that for $n>4$ there
does not exist a sequence $(a_1,\dots,a_n)$ of $\pm 1$'s that is
orthogonal to every proper cyclic shift of itself. It has a similar
flavor to the Erdős discrepancy problem that was the topic of
Polymath5. Terry Tao says the following on his blog about the
circulant Hadamard matrix conjecture: "One may have to wait for (or to
encourage) a further advance in this area (which would be more or less
an exact analogue of the situation with Polymath5 and the Erdos
discrepancy problem)."
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|
220,032 |
Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the category of Hilbert spaces and continuous linear maps. Now, dagger categories are evil! For example, here's a quote from the Nlab : "Note that regarded as an extra structure on categories, a †-structure is evil, since it imposes equations on objects." This is not just a little thing. It's a fundamental philosophical problem with category theory: category theory does not seem to be able to cope with this rather important category-theory-related notion which are dagger categories. Also, because of this buit-in evilness, there's a whole bunch of very familiar concepts that one cannot apply to dagger categories without rethinking everything very, very carefully from the beginning: equivalences, limits, colimits, adjoint functors, duals, algebra objects, etc... Here's another quote : "It is possible that this problem will force a change in thinking in either the concept of the principle of equivalence or our thinking in quantum theory." and yet another one : "Often concepts violating the principle of equivalence (like the concept of “strict monoidal category”) have equivalence-invariant counterparts (like the concept of “monoidal category”). But in this particular case there appears to be no known way to express the idea without equations between objects." Now here's my question: Question: Can one improve the above "there appears to be no known way to express the idea without equations between objects" to "there is no way to express the idea without equations between objects" ? I expect dagger categories to be truly evil (in some yet-to-be-defined technical sense of "evil"). Having a proof of that fact might be very informative, and would at least force us to define the term "evil" at a mathematical level of precision.
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I will have another go at arguing that dagger-categories are not evil. Let’s look at a simpler case first. Consider the property “$1 \in X$” on sets. As a property of abstract sets, this is evil: it’s not invariant under isomorphism, e.g. any iso $\{1,2\} \cong \{2,3\}$. But it is manifestly non-evil as a property of, say, “sets equipped with an injection to $\mathbb{N}$”. Looking at this gives a non-evil structure on abstract sets: “an injection $i : X \to \mathbb{N}$, such that $1 \in \mathrm{im}\ i$”. Ah (you may say) so if this is non-evil, it can’t reflect our original idea correctly: it would have to transfer along that $\{1,2\} \cong \{2,3\}$. But that’s not such a clear-cut complaint. As a structure on abstract sets, it does transfer along that isomorphism. But we were thinking of them from the start not just as abstract sets, but as subsets of $\mathbb{N}$, i.e. as already equipped implicitly with injections to $\mathbb{N}$. Considered as such, one of them contains $1$ and the other doesn’t; and there’s no isomorphism between them which commutes with those injections. Summing up: “containing 1” is certainly non-evil as a property of “sets with a mono to $\mathbb{N}$”. This induces a non-evil property/structure on abstract sets which you may or may not agree matches our original idea, because it requires considering different monos to $\mathbb{N}$ besides the ones we were already (implicitly) thinking of. Now, back to dagger-categories. It seems reasonable that dagger-categories are evil when regarded as structure on categories . The post by Peter Selinger linked by Simon Henry argues this quite persuasively: proving a specific no-go theorem, showing there can exist no notion of dagger-structure satisfying certain desirable properties. However, structure on categories is not the only way to look at dagger-categories. They can instead be seen as structure on “pairs of categories connected by a faithful and essentially surjective functor $i : \mathbf{C}_u \to \mathbf{C}$”. (Full definition below.) You may be thinking: this can’t be right, because it induces a non-evil structure on categories (“equip with a fully faithful inclusion from some other category, and then the dagger-structure”) which would violate Selinger’s no-go argument. However, it doesn’t: this structure doesn’t allow a definition of unitary maps in the sense Selinger’s argument assumes. Given $A, B \in \mathbf{C}_u$, we can say a map $iA \to iB$ is unitary if it’s the image of some map $A \to B$. But given just $A, B \in \mathbf{C}$, this unitariness isn’t well-defined for $f : A \to B$; different ways of expressing $A$ as $iA'$ and $B$ as $iB'$ might give different answers as to whether $f$ is unitary. Expressed in this form, transferring the “weak dagger structure” on $\mathbf{fdHilb}$ along the equivalence to $\mathbf{fdVect}$ yields a weak dagger structure where the functor $i$ is not injective on objects. Selinger’s argument shows that something like this is unavoidable. Summing up again: dagger-structure is certainly not evil when viewed as a structure on “categories with a distinguished faithful, ess. surj. inclusion”. This gives a definition of weak dagger structure as a non-evil structure on categories, which you may or may not accept, because it requires us to loosen up our original expectation that the “subcategory” of unitary maps should be literally bijective on objects, i.e. to go beyond the kind of “subcategories” we were originally thinking of. Full definition: a weak dagger category may be taken to consist of: a category $\newcommand{\C}{\mathbf{C}}\C$; a groupoid $\C_u$, with a faithful and essentially surjective functor $i : \C_u \to \C$; a functor $\dagger : \newcommand{\op}{\mathrm{op}}\C^\op \to \C$; a natural isomorphism $\varphi : \dagger \cdot i^\op \cong i \cdot (-)^{-1} : \C_u^\op \to \C$; a natural isomorphism $\psi : \dagger \cdot \dagger^\op \cong 1_\C$; such that for all $A \in \C_u$, $\psi_{iA} = \varphi_A (\varphi_{A}^\dagger)^{-1} : (iA)^{\dagger \dagger} \to iA$, or equivalently, as natural transformations, $\psi \cdot i = (\varphi \cdot (-)^{-1})(\dagger \cdot \varphi^\op)^{-1} : \dagger \cdot \dagger^\op \cdot i \to i$; and such that for any $A, B \in \C_u$, the image of $\C_u(A,B) \to \C(iA,iB)$ consists of all $u$ such that $\varphi_A \cdot u^\dagger \cdot \varphi_B^{-1}$ is a 2-sided inverse for $u$. Given this, I claim: each component is “non-evil” as structure on the earlier components (this can be made precise as a lifting property for forgetful functors between 2-categories); strict dagger categories are precisely weak dagger categories such that $i$ is bijective on objects and $\varphi_A = 1_{iA}$ for all $A \in \C_u$; given any weak dagger cat with $i$ bijective on objects, one can modify $\dagger$, $\varphi$, and $\psi$ to obtain a strict dagger-structure, equivalent to the original, with the equivalence acting trivially on $\C$, $\C_u$, and $i$; therefore, given a category $\C$ equipped with a distinguished all-objects subgroupoid $\C_u$, “strict dagger structures on $\C$ with unitaries $\C_u$” correspond to “weak dagger structures on $\C_u \to \C$”; so strict dagger structure is non-evil as a structure on “cats with a distinguished all-objects subgroupoid”; if we drop the last component of the definition, we similarly get a non-evil structure of “strict dagger structures on $\C$ with unitaries including at least $\C_u$”; given any weak dagger cat $\C$, there is an equivalent strict one, with the same unitary category $\C_u$, but with new ambient category given by the objects of $\C_u$ with the morphisms of $\C$. So overall, weak dagger categories do not give us anything essentially different from strict ones. (I’m pretty sure that I’m remembering most of the ideas here from somewhere, but I can’t find where. The nearest I can find is this post by Mike Shulman on that same categories list thread. Better references very welcome.)
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|
220,220 |
I have recently been wondering if the following is consistent with ZFC: For every infinite ordinal $\alpha$: $|V_\alpha\cap L|=|\alpha|$. Intuitively, this states that for $L$ is very "thin", in that it doesn't branch off too much from the set of ordinals as we climb cumulative hierarchy. I have heard that it is possible to have it for $V_{\omega+1}$, which is essentially having $\Bbb R^L$ countable. I suspect that this is consistent, and we can construct a model of this starting from $V=L$ and adding a proper class of generic sets giving bijections between parts of $L$ and ordinals, but I have no real background in forcing to see if that makes any sense. Thanks in advance.
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Claim: $|V_\alpha \cap L| = |\alpha|$ for every $\alpha \geq \omega$ implies that $0^\#$ exists. Proof: Let's assume, toward contradiction, that $0^\#$ doesn't exist that $|V_\alpha \cap L| = |\alpha|$ for every $\alpha \geq \omega$. Let $\mu$ be a singular cardinal in $V$. Let's apply the assumption $|V_\alpha \cap L| = |\alpha|$ for $\alpha = \mu + 1$. In $V$, $\mu = |\mu + 1|=|V_{\mu + 1} \cap L| \geq |\mathcal{P}^L(\mu)| \geq |(\mu^{+})^L|$. In particular, $\mu^{+} > (\mu^{+})^L$. Therefore, the cofinality of $(\mu^{+})^L$ in $V$ is strictly below $\mu$. Applying Jensen's covering theorem, we conclude that the cofinality of $(\mu^{+})^L$ in $L$ is strictly below $\mu$ - a contradiction to the regularity of $(\mu^{+})^L$ in $L$. By the way, if we restrict the values of $\alpha$ for which we want $|V_\alpha \cap L| = |\alpha|$ to be $\leq \aleph_{\omega}$, the consistency strength drops considerably: Claim: $|V_\alpha \cap L| = |\alpha|$ for every $\omega \leq \alpha \leq \aleph_\omega$ is equiconsistent with the existence of $\omega$ inaccessible cardinals. Proof: For the first direction, force over $L$ with an iteration of Levi collapses in order to make the $n$-th inaccessible in $L$ equal to the $\aleph_n$ of the generic extension. For the second direction, note that the assumption implies that for all $1\leq n < \omega$, $\aleph_n$ is a $\beth$-fixed point in $L$, since for every infinite $\beta < \aleph_n$, $|V_\beta \cap L| = |\beth_\beta^L| < \aleph_n$. In particular, $\aleph_n$ is a limit cardinal in $L$. It is regular in $V$ and thus also in $L$ and therefore it is inaccessible.
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220,396 |
Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$ , for each fixed $y\in[0,1]$ ) partition the domain of $f$ . I am curious for set theoretic or point set topology criteria for which partitions of $[0,1]$ could be the level sets for a continuous function. That is, can someone fill in the blank in the following "theorem". THEOREM : Let $\mathscr{P}$ be a partition of $[0,1]$ . The following are equivalent. There is some continuous function $f:[0,1]\to[0,1]$ such that the collection of level sets of $f$ is exactly the partition $\mathscr{P}$ . The partition $\mathscr{P}$ satisfies the property __________. Even in the case where all the parts of $\mathscr{P}$ are finite, the picture seems pretty mysterious to me. For example, it is perfectly fine to have $\mathscr{P}$ consist of all pairs, and one singleton (for example the level sets of $f(x)=(x-1/2)^2$ ), but impossible for $\mathscr{P}$ to consist of all pairs except for two singletons. Can someone see a non-analytic over-arching principle which discerns the the first case from the second? As a side note, the question can obviously be posed with the $[0,1]$ as domain and codomain replaced by arbitrary topological spaces $V$ and $W$ . EDIT: Fixed the counter-example above.
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Any map $[0,1]\to[0,1]$ is a quotient map onto its image, and the image must be either a point or a closed interval. So a partition comes from such a map iff the quotient of $[0,1]$ by the equivalence relation associated to the partition is homeomorphic to either a point or an interval. In particular, given any characterization of an interval up to homeomorphism (among all spaces which are continuous images of an interval, which can themselves be nicely characterized if you assume they are Hausdorff), you get a characterization of such partitions. Here is one such characterization, though maybe not a very easy one to apply. It is known that any compact connected metrizable space with at most two non-cut-points is homeomorphic to either a point or $[0,1]$, and any Hausdorff quotient of $[0,1]$ is metrizable. So a quotient of $[0,1]$ is homeomorphic to either a point or $[0,1]$ iff it is Hausdorff and has at most two non-cut-points. These conditions can be stated explicitly in terms of the partition as follows. Say that an element $A\in\mathscr{P}$ is a cut-set for $\mathscr{P}$ if $\mathscr{P}\setminus\{A\}$ can be split into two disjoint nonempty subpartitions $\mathscr{Q}$ and $\mathscr{R}$ such that $\bigcup\mathscr{Q}$ and $\bigcup\mathscr{R}$ are both open subsets of $[0,1]$. Then a partition $\mathscr{P}$ is the level sets of a continuous map $[0,1]\to[0,1]$ if and only if: The equivalence relation associated to $\mathscr{P}$ is closed as a subset of $[0,1]\times[0,1]$, and All but at most two elements of $\mathscr{P}$ are cut-sets. (Condition (1) is equivalent to the quotient being Hausdorff, and condition (2) is equivalent to the quotient having at most two non-cut-points.)
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220,662 |
In Weil cohomology, a nice curve has cohomology up to degree 2, or more generally a nice $n$-dimensional variety has cohomology up to degree $2n$. I know that this was motivated at least in part by a desire to extend cohomology of complex manifolds to algebraic varieties over other fields, and since complex curves are two-dimensional manifolds, curves should have cohomology up to degree 2. My question is if this is just an artifact of the motivation, or if there is a more 'intrinsic' reason why cohomology should work out that way? Or put differently, should one expect there to be cohomology theories where, say, curves act like they're five-dimensional?
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This is a very subtle question, I think. First of all, the sense in which an $n$-dimensional algebraic variety $X$ acts as if it is "cohomologically $2n$-dimensional" is quite complicated--for example, unless one uses some notion of cohomology with compact support, an affine $n$-dimensional variety looks $n$-dimensional, not $2n$-dimensional (just as over $\mathbb{C}$, affine varieties have the homotopy types of $n$-dimensional CW complexes). So this is really something about proper varieties, or about the more complicated notion of cohomology with compact support. Second, this is just a fact about algebraically closed fields. For example, if $k$ is a finite field and $X$ is an $n$-dimensional smooth projective $k$-variety, $X$ "looks" like it is $2n+1$-dimensional (indeed, it looks a lot like a $2n+1$-manifold fibered over a circle). For example, there is an arithmetic Poincare duality between $H^i$ and $H^{2n+1-i}$. (Of course the reason for this is that $k$ has cohomological dimension $1$ and has reasonable duality properties.) Nonetheless, let me try my hand at some kind of explanation. I think the proof of Grothendieck's vanishing theorem gives a reasonable explanation of why Zariski cohomology of $X$ vanishes in degree greater than $\dim(X)$. Typically when one defines a fancier cohomology theory (e.g. etale or crystalline cohomology) one chooses a topology on some category of schemes which is finer than the Zariski topology. Call such a topology $\tau$. As $\tau$ is finer than the Zariski topology, there is a forgetful functor $f_*$ from sheaves on the topology $\tau$ to Zariski sheaves. Grothendieck teaches us to think of this as a morphism $$f: X_\tau\to X_\text{Zar}.$$ (Maybe a good analogy is that there is a natural continuous map of spaces $$g: X(\mathbb{C})^{\text{an}}\to X_{\text{Zar}}$$ if $X$ is a finite-type $\mathbb{C}$-scheme.) For all of the topologies $\tau$ which we like to use to define Weil cohomology theories, the morphism $f$ has cohomological dimension $n$, in the sense that the right-derived functor of $f_*$, denoted $Rf_*$, sends certain sheaves to complexes with cohomology concentrated in degrees $[0, n]$. For example, if $\tau$ is the etale topology, and we are working over an algebraically closed field, $Rf_*\mathcal{F}$ is concentrated in degrees $[0,n]$ if $\mathcal{F}$ is constructible. Or if $\tau$ is the crystalline site (over a field of characteristic zero), $$Rf_*\mathcal{O}_X\simeq \Omega^\bullet_{X, dR}$$
which is concentrated in degrees $[0,n]$ (and there is a similar formula with any flat vector bundle in place of $\mathcal{O}_X$). So the fact that varieties in these topologies are $2n$-dimensional comes from the formula $$2n=n+n,$$ where the first $n$ comes from $Rf_*$ and the second comes from the cohomological dimension of the Zariski topological space. Likewise, the honest continuous map $g: X(\mathbb{C})^{\text{an}}\to X_{\text{Zar}}$ has cohomological dimension $n$. Of course (1) proving that $f$ has the claimed cohomological dimension is usually quite involved, and (2) we like these cohomology theories because they behave in ways which match our intuitions. So it's unclear to me whether this is really a non-anthropological answer to your question. But I think the pattern (of finding a site over the Zariski site which has relative cohomological dimension $n$) is ubiquitous enough to be worth commenting on.
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220,796 |
The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated to the prime element $\pi$ (i.e. such that $F^{\mathrm{ab}}=F_\pi\cdot F^{\mathrm{un}}$ with the usual notations). The idea to consider 1-dim. formal groups over the ring of integers $\mathcal{O}_F$ is a deus ex machina for me, and I wonder if anyone can explain Lubin-Tate's motivation to consider such a thing? Related, on page 50 of J.S. Milne's online notes on the class field theory, he offers the speculation that the motivation comes from complex multiplication of elliptic curves and how one might try to get an analogue of the theory for local fields. But this requires again that it is somehow natural to consider formal groups as an analogue which I think still needs a motivation. What is the motivation to consider formal groups a la Lubin-Tate theory? Is there a way to motivate their construction?
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Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening. Since I had read and enjoyed Lazard’s paper on one-dimensional formal group (laws), which dealt with the case of a base field of characteristic $p$, I decided to look at formal groups over $p$-adic rings. For whatever reason, I wanted to know about the endomorphism rings of these things, and gradually recognized the similarity between, on the one hand, the case of elliptic curves and their supersingular reduction mod $p$, when that phenomenon did occur, and, on the other hand, formal groups over $p$-adic integer rings of higher height than $1$. I had taken, or sat in on, Tate’s first course on Arithmetic on Elliptic Curves, and was primed for all of this. In addition, I was aware of Weierstrass Preparation, and the power it gave to anyone who wielded it. And in the attempt to prove a certain result for my thesis, I had thought of looking at the torsion points on a formal group, and I suppose it was clear to me that they formed a module over the endomorphism ring. Please note that it was not my idea at all to use them as a representation module for the Galois group. But Tate was looking over my shoulder at all times, and no doubt he saw all sorts of things that I was not considering. At the time of submission of my thesis, I did not have a construction of formal groups of height $h$ with endomorphism ring $\mathfrak o$ equal to the integers of a local field $k$ of degree $h$ over $\Bbb Q_p$. Only for the unramified case, and I used extremely tiresome degree-by-degree methods based on the techniques of Lazard. Some while after my thesis, I was on a bus from Brunswick to Boston, and found not only that I could construct formal groups in all cases that had this maximal endomorphism structure, but that one of them could take the polynomial form $\pi x+x^q$. Tate told me that when he saw this, Everything Fell Into Place. The result was the wonderful and beautiful first Lemma in our paper, for which I can claim absolutely no responsibility. My recollection, always undependable, is that the rest of the paper came together fairly rapidly. Remember that Tate was already a master of all aspects of Class Field Theory. But if the endomorphism ring of your formal group is $\mathfrak o$ and the Tate module of the formal group is a rank-one module over this endomorphism ring, can the isomorphism between the Galois group of $k(F[p^\infty]])$ over $k$ and the subgroup $\mathfrak o^*\subset k^*$ fail to make you think of the reciprocity map?
|
{
"source": [
"https://mathoverflow.net/questions/220796",
"https://mathoverflow.net",
"https://mathoverflow.net/users/44812/"
]
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|
220,827 |
Consider Lemma 1 from Beilinson's paper " Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra ", as follows. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: \mathcal{C} \to \mathcal{D}$ an exact functor, $\{X_i\}$ be a family of objects of $\mathcal{C}$. Let us assume that $\{X_i\}$ generates $\mathcal{C}$, $\{F(X_i)\}$ generates $\mathcal{D}$, and for any pair $X_i$, $X_j$ from the family $F: \text{Hom}^\bullet (X_i , X_j) \to \text{Hom}^\bullet(F(X_i), F(X_j))$ is an isomorphism. Then $F$ is an equivalence of categories. My question is, what is the correct way to think about it/the intuition for this? Thanks.
|
I think some of the commenters have forgotten the time when they found vector space linear algebra understandable, but triangulated categories confusing. For someone in such a state, a useful tool to help understand statements about triangulated categories is passage to the Grothendieck group. Recall that this is done by taking the abelian group generated by the objects in the category $\mathcal{C}$, divided by the relation in which the sum of three elements of a triangle is zero. This gives an abelian group, and I'll tensor it with a field to get a vector space $[\mathcal{C}]$. Supposing in addition that Hom in the category gives finite dimensional vector spaces, $[\mathcal{C}]$ gets a bilinear form by $B(X, Y) = \dim Hom(X, Y)$. For instance, in this case the thus "decategorified" statement is as follows: Let $C$ and $D$ be vector spaces equipped with bilinear forms, and $F: C \to D$ a linear map. Say $X_i$ is a spanning set of vectors for $C$ and $F(X_i)$ spans $D$. Suppose $F$ is "an isometry on the spanning set", i.e., $B(X_i, X_j) = B(F(X_i), F(X_j))$. Then $F$ is an isometry. This is a statement more accessible to intuition. And, as a proof of the original statement necessarily "decategorifies" to a proof of the decategorified one, often one can proceed in reverse and first prove the decategorified statement and then try to lift.
|
{
"source": [
"https://mathoverflow.net/questions/220827",
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|
221,351 |
I asked the following question ( https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con ) on math.stackexchange.com and received no answers, so I thought I would ask it here. I've asked several people in my department who were all stumped by the question. The question is: why is every finite subgroup of $\operatorname{GL}_n(\mathbb{Q})$ conjugate to a finite subgroup of $\operatorname{GL}_n(\mathbb{Z})$? Note that at least for $n=2$ the question of isomorphism is much easier, since one can (with some effort) work out exactly which finite groups can be subgroups of $\operatorname{GL}_2(\mathbb{Q})$. Further, there are isomorphic finite subgroups of $\operatorname{GL}_2(\mathbb{Q})$ that are not conjugate to each other. For example, the group generated by $-I_{2 \times 2}$ and $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ are both isomorphic to $C_2$, but they cannot be conjugate to each other because the eigenvalues of the two generators are different. If there is a relatively simple proof, that would be ideal, but a reference with a potentially long proof is fine as well. Thanks for any assistance.
|
This argument is fairly standard, but it is quicker to repeat it than to find a reference: Let $G$ be a finite subgroup of $GL_n(\mathbb{Q})$. Set $\Lambda = \sum_{g \in G} g \cdot \mathbb{Z}^n \subset \mathbb{Q}^n$. Then $\Lambda$ is a finitely generated torsion free abelian group, hence isomorphic to $\mathbb{Z}^r$ for some $r$. Since $\mathbb{Z}^n \subseteq \Lambda \subset \mathbb{Q}^n$, we have $r=n$. Clearly, $g \Lambda = \Lambda$ for all $g \in G$. Let $h \in GL_n(\mathbb{Q})$ take the standard basis to a basis of $\Lambda$, so $h \mathbb{Z}^n = \Lambda$ and $\mathbb{Z}^n = h^{-1} \Lambda$. Then $h^{-1} g h$ takes $h^{-1} \Lambda = \mathbb{Z}^n$ to itself for all $g \in G$, so $h^{-1} G h \subset GL_n(\mathbb{Z})$.
|
{
"source": [
"https://mathoverflow.net/questions/221351",
"https://mathoverflow.net",
"https://mathoverflow.net/users/10898/"
]
}
|
221,431 |
It is not difficult to tile the plane with incongruent triangles.
One could tile with equilateral triangles, and then partition
each equilateral into three triangles, displacing their common
centerpoint so that no two triangles are congruent (left below). Q1 .
Is it possible to tile the plane with isosceles triangles,
no two of which are congruent? It is easy to tile the plane with congruent isosceles triangles,
as illustrated right above.
But I don't see how to achieve a tiling with incongruent isosceles triangles. Perhaps it is easier to answer this question: Q2 .
Is it possible to tile the plane with equilateral triangles,
no two of which are congruent? Added 14Feb2020 : Q2 has been answered (negatively) in two papers
(independently). These results were presaged
(by @Wojowu) below . (1) Pach, János, and Gábor Tardos. "Tiling the plane with equilateral triangles." arXiv:1805.08840 abstract (2018). Corollary 4 . There is no tiling of the plane with pairwise noncongruent equilateral triangles
whose side lengths are bounded from below by a positive constant. (2) Richter, Christian, and Melchior Wirth. "Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles." Discrete & Computational Geometry 63, no. 1 (2020): 169-181. Springer link . Question Q1 was inspired by (but not addressed in) this paper: Malkevitch, J. "Convex isosceles triangle polyhedra." Geombinatorics 10 (2001): 122-132.
|
Q1: Yes. Any acute non-isosceles triangle can be tiled by
three pairwise incongruent isosceles triangles, by connecting each vertex to
the circumcenter. Start from some isosceles $T_0$ with repeated side $s$;
inscribe $T_0$ into a larger triangle $T_1$ such that $T_1 - T_0$ is
the union of three acute, non-isosceles triangles with circumradii
distinct from each other and from $s$;
likewise inscribe $T_1$ into $T_2$, and $T_2$ into $T_3$, etc.,
tiling the complement of $T_0$ with ever-larger acute, non-isosceles triangles
with all circumradii pairwise distinct and different from $s$.
Now connect each of these triangles' vertices to its circumcenter
to obtain a tiling of the plane by isosceles triangles any two of which
have distinct repeated sides, and thus a fortiori are not congruent, QEF . [ Joseph O'Rourke asks how to find $T_k$ so that the three components of
$T_k-T_{k-1}$ are acute and avoid circmuradius coincidences.
One way is to deform the triangle, call it $T'_k$, each of whose sides
contains a vertex of $T_{k-1}$ and is parallel to the opposite side
of $T_{k-1}$ (so $T_{k-1}$ is the median triangle of $T'_k$).
Then each component of $T'_k - T_{k-1}$ is congruent to $T_{k-1}$,
and thus acute. Now form $T_k$ by slightly moving each vertex of $T'_k$,
keeping all angles acute but removing any coincidences among the
circumradii and $s$. While you're at it, you can make sure that
none of the angles is $30^\circ$ if you don't accept an equilateral
triangle as isosceles.]
|
{
"source": [
"https://mathoverflow.net/questions/221431",
"https://mathoverflow.net",
"https://mathoverflow.net/users/6094/"
]
}
|
221,856 |
At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\to \mathbb R$ linear with $f\le p|_L$. Then there is a linear $F:X\to\mathbb R$ with $F\le p$ and $F|_L=f$ . However the theorem is true if the majorant $p$ is merely convex. This version has a very similar proof as the classical statement and several advantages. For instance, there is no need to introduce the new notion of sublinearity and the result is even interesting for $X=\mathbb R$. The only reference I know is the book of Barbu und
Precupanu Convexity and Optimization in Banach Spaces . Two questions: Who first observed that sublinearity can be replaced by convexity? Is there any (e.g. pedagocial) reason to prefer the sublinear version?
|
If $p$ is convex, then $P(x)=\inf_{t>0}t^{-1}p(tx)$ is sublinear, isn't it? Also, if a linear functional is dominated by $p$, it is also dominated by $P$. Finally, $P\le p$. So there is no non-trivial gain in generality whatsoever unless you start talking about extending non-linear functionals but then you should restate the question accordingly.
|
{
"source": [
"https://mathoverflow.net/questions/221856",
"https://mathoverflow.net",
"https://mathoverflow.net/users/21051/"
]
}
|
221,882 |
I would like to understand what kind of stochastic process are Ito Processes. According to Kuo[p. 102] an Ito Process is a stochastic process of the form $$dX_t=g(t)dt+f(t)dW(t),$$ where $W(t)$ is a wiener process, $f\in \mathcal{L}_\text{ad}(\Omega,L^2[a,b])$ and $g \in \mathcal{L}_\text{ad}(\Omega,L^1[a,b])$. $\mathcal{L}_\text{ad}(\Omega,L^2[a,b])$ is explained in in Notation 5.1.1 ( Kuo[p. 61] ). I have tried to understand it too a degree to describe it here but failed. The same is true for $\mathcal{L}_\text{ad}(\Omega,L^1[a,b])$, which is explained in Notation 7.4.1 ( Kuo[p. 102] ). I would like to know if there is an easier equivalent set of conditions. For all the examples of Ito Processes that I have seen, $g(t)=h(t)X_t$ with $h:\mathcal{X} \rightarrow \mathbb{R}^{n \times n}$, with $\mathcal{X}$ being the index set of $X_t$, and $f:\mathcal{X} \rightarrow \mathbb{R}^{n \times n}$. In particular, I am interested if $h(t)$ can be a stochastic process.
|
If $p$ is convex, then $P(x)=\inf_{t>0}t^{-1}p(tx)$ is sublinear, isn't it? Also, if a linear functional is dominated by $p$, it is also dominated by $P$. Finally, $P\le p$. So there is no non-trivial gain in generality whatsoever unless you start talking about extending non-linear functionals but then you should restate the question accordingly.
|
{
"source": [
"https://mathoverflow.net/questions/221882",
"https://mathoverflow.net",
"https://mathoverflow.net/users/76238/"
]
}
|
221,920 |
Have there been any studies of publication rates in Mathematics? We are trying to construct a workload model for the Faculty of Science and Engineering at my institution. Part of this involves assigning a fixed number of "points" for each published paper. It seems that our colleagues in some of the sciences publish many more papers than we do in Mathematics, which leaves us asking for the number of points per paper to be far higher in Mathematics than elsewhere. But we need to be able to back up our impressions with facts. What I would like to do is to get some idea of how many papers one might expect a research mathematician to publish over, say, a five-year period. I recognize that there are a lot of problems here with the words "expect" and "research mathematician", not to mention problems with counting a 100-page paper on the same footing as a 5-page paper, or a paper in a "top" journal on the same footing as a paper in a not-so-top journal; I want to stay away from all those subjective and opinion-based issues. I would like to know whether there are any publically-available figures along the following lines: pick a university where faculty are expected to be engaged in research; find out how many publications each member of the Math Department has had over (say) a five-year period; publish the median, or some other measure of the distribution of the publication numbers (not the mean, which could be skewed by a small number of members publishing a large number of papers). I'm aware of the paper by Jerrold Grossman, Patterns of collaboration in mathematical research , SIAM News 35 (2002), but that's a study of all papers listed in Math Reviews, which includes people who published a paper or two and then left research mathematics for other fields. I'm really interested only in people who are employed by departments where publication in refereed journals is expected.
|
Here is the AMS culture statement on publication rates in mathematics. Even the best young mathematicians publish average of two or fewer articles per year.
|
{
"source": [
"https://mathoverflow.net/questions/221920",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3684/"
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|
222,154 |
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-trivial isometry group?
|
In genus 2, every surface has a symmetry, namely a hyperelliptic involution. In higher genus, generic surfaces will not have any symmetries. If a surface has a non-trivial symmetry group, then the quotient by the symmetry group will be an orbifold, and the moduli space of hyperbolic structures on this orbifold will have strictly smaller dimension. Hence moduli space will not be covered by these smaller dimensional Teichmüller spaces. One possible obstruction: suppose a surface admits a symmetry (not hyperelliptic in genus 2), then there will be simple closed geodesics which get moved by an isometry, and hence there are simple closed curves which have the same lengths. So if a surface has all simple closed curves of different lengths, then it cannot admit any non-trivial symmetries. One could probably also state this for curves of some bounded length depending on the genus and injectivity radius. To see that some curve must be sent to a distinct curve, suppose not. Then every non-separating curve intersects another curve in a single point. Then this point must be fixed by the isometry, and in fact the isometry must be an involution fixing this point. Arguing like this, one can conclude that the isometry is a hyperelliptic involution. However, in genus $>2$, there are always curves not fixed by any given hyperelliptic.
|
{
"source": [
"https://mathoverflow.net/questions/222154",
"https://mathoverflow.net",
"https://mathoverflow.net/users/82102/"
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|
222,159 |
Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that: $$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$ is a solid of constant width with a finite symmetry group? The analogous result is true in two dimensions, with an explicit degree- $8$ example given in this paper . If we revolve this curve about its axis of symmetry (or equivalently replace every instance of $y^2$ with $y^2 + z^2$ ), we would obtain a degree- $8$ surface of constant width depicted below: This has an infinite symmetry group isomorphic to $O(1)$ , so does not answer the question. Similarly, the sphere: $$ f(x, y, z) = x^2 + y^2 + z^2 - 1 \leq 0 $$ has symmetry group $O(3)$ , which is again infinite. Does there exist an algebraic solid of constant width and finite automorphism group?
|
There exist many algebraic surfaces of constant breadth with no continuous symmetries, even ones with no symmetries at all. To see this, consider the properties of the support parametrization: The support parametrization. Let $B\subset\mathbb{R}^3$ be a smooth, strongly convex surface, i.e., the outward Gauss map $u:B\to S^2$ is a diffeomorphism. Since it is a diffeomorphism, there is a smooth inverse $b:S^2 \to B$ that is also a diffeomorphism. Since, for $u\in S^2$ , the (supporting) tangent plane to $B$ at $b(u)$ is orthogonal to $u$ , it can be written in the form $x\cdot u = p(u)$ , where $p: S^2\to \mathbb{R}$ is a smooth function on $S^2$ . Then, of course, one has $b(u)\cdot u = p(u)$ and one also has $\mathrm{d}b\cdot u = 0$ (since the differential of $b$ at $u\in S^2$ has its image being the subspace parallel to the tangent plane to $B$ at $b(u)$ , which, as we have seen, is perpendicular to $u$ ). The two equations $b(u)\cdot u = p(u)$ and $\mathrm{d}b\cdot u = 0$ then determine $b$ in terms of $p$ : One has the vector equation $$
b = p\,u + \nabla p,
$$ where $\nabla p$ is the gradient vector field of $p$ as a function on $S^2$ . The function $p$ is called the support function of $B$ , and the above parametrization is called the support parametrization . (Of course, there is nothing special about $3$ dimensions here, the support parametrization works for strongly convex hypersurfaces in all dimensions.) Surfaces of Constant Breadth. Now, consider the two supporting tangent planes to $B$ that are perpendicular to $u$ : They are the tangent plane at $b(u)$ and the tangent plane at $b(-u)$ . They have equations $x\cdot u = p(u)$ and $x\cdot (-u) = p(-u)$ respectively. In particular, the distance between these two planes is $p(u)+p(-u)$ . Thus, $B$ has constant breadth $2d$ if and only if its support function $p$ satisfies $p(u)+p(-u) = 2d$ . In particular, the function $p_0=p-d$ must be an *odd* function on $S^2$ , i.e., $p_0(-u) = -p_0(u)$ for all $u\in S^2$ . Conversely, if $p_0:S^2\to\mathbb{R}$ is an odd function, then $p = p_0+d$ for $d>0$ sufficiently large is then the support function of a strongly convex body of constant breadth $2d$ . Alternatively, for all sufficiently small $\epsilon>0$ , the function $p = 1 + \epsilon p_0$ is the support function of a strongly convex surface $B_\epsilon$ of constant breadth $2$ . An algebraic example. To construct an algebraic example, it suffices to take $p_0$ to be an odd algebraic function on $S^2$ . Taking $p_0$ to be the restriction of a linear function in $\mathbb{R}^3$ to $S^2$ will only give a surface that is a translation of the round $S^2$ , so we need to look at something else. The next simplest choice would be $p_0 = xyz$ , where $S^2$ is defined by $x^2+y^2+z^2=1$ . For $\epsilon^2<1$ , the surface $B_\epsilon=b_\epsilon(S^2)$ is a surface of constant breadth $2$ that is smooth, strongly convex, and algebraic. (When $\epsilon^2=1$ , the surface $B_1$ is still strictly convex and algebraic and has breadth $2$ , but it is not smooth. When $\epsilon^2>1$ the image $b_\epsilon(S^2)$ is not a convex surface.) When $\epsilon\not=0$ , it is easy to show that $B_\epsilon$ has the symmetry of the function $xyz$ but no more symmetry than that, i.e., the symmetry group will have (finite) order $24$ (in fact, it is the symmetry group of the regular tetrahedron.) It is algebraic because it is parametrized by algebraic functions on the algebraic surface $S^2$ . It is therefore (a component of) the zero locus of an irreducible polynomial function $f_\epsilon(x,y,z)$ on $\mathbb{R}^3$ . A calculation using elimination theory (carried out by MAPLE) shows that $f_\epsilon$ has degree $20$ (when $\epsilon\not=0$ ) and, for most values of $\epsilon$ , has more than 1000 terms. It is not obvious that the zero locus of $f_\epsilon$ has no other (real) components other than $B_\epsilon$ . Thus, I don't know that the interior of $B_\epsilon$ (which makes sense as long as $\epsilon^2\le 1$ ) can be characterized as the set where $f_\epsilon$ is negative. However, I don't know why you want this; it doesn't seem natural to me. Surely, you only really need it to be algebraic. In any case, one has an explicit parametrization of $B_\epsilon$ using any explicit parametrization of $S^2$ . It is easy to draw pictures using graphical software. Here are what the surfaces $B_{1/2}$ and $B_1$ look like: Added comment: David Speyer has shown that $f_\epsilon$ cannot change sign anywhere other than across the image $b_\epsilon(S^2)$ , since, as he shows, this is the only two-dimensional component of the real locus of $f_\epsilon=0$ . More generally, taking $p = 1+p_0$ , where $p_0 = \lambda_1\lambda_2\lambda_3$ and where each $\lambda_i$ is a linear function of $x$ , $y$ , and $z$ , produces a $7$ -parameter family of distinct algebraic surfaces of constant breadth $2$ , and the generic one of these has no nontrivial symmetries. (In fact, one only gets continuous symmetries when $p_0 = \lambda^3$ for some linear function $\lambda$ .)
|
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"https://mathoverflow.net/questions/222159",
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|
222,172 |
I am looking for general principles or specific answers to this generic example. Assume a 2d grid with no boundaries and a roving dot (ant/drunk guy/particle) that is initially located at some starting position S. There's an equal probability of going up (u), down (d), left (l) or right (r) from any position on the grid. A block is defined by analogy to a city block, and 2 adjacent points as well as 2 points that are linked by a u+r path (for example) are both said to be distant by 1 block. How would I approach computing the likelihood that my roving dot is at least n blocks away from startig point x after i moves? For example, what is the mathematical approach to determining the likelihood that the roving dot is 4 or more blocks away from S after 10 moves?
|
There exist many algebraic surfaces of constant breadth with no continuous symmetries, even ones with no symmetries at all. To see this, consider the properties of the support parametrization: The support parametrization. Let $B\subset\mathbb{R}^3$ be a smooth, strongly convex surface, i.e., the outward Gauss map $u:B\to S^2$ is a diffeomorphism. Since it is a diffeomorphism, there is a smooth inverse $b:S^2 \to B$ that is also a diffeomorphism. Since, for $u\in S^2$ , the (supporting) tangent plane to $B$ at $b(u)$ is orthogonal to $u$ , it can be written in the form $x\cdot u = p(u)$ , where $p: S^2\to \mathbb{R}$ is a smooth function on $S^2$ . Then, of course, one has $b(u)\cdot u = p(u)$ and one also has $\mathrm{d}b\cdot u = 0$ (since the differential of $b$ at $u\in S^2$ has its image being the subspace parallel to the tangent plane to $B$ at $b(u)$ , which, as we have seen, is perpendicular to $u$ ). The two equations $b(u)\cdot u = p(u)$ and $\mathrm{d}b\cdot u = 0$ then determine $b$ in terms of $p$ : One has the vector equation $$
b = p\,u + \nabla p,
$$ where $\nabla p$ is the gradient vector field of $p$ as a function on $S^2$ . The function $p$ is called the support function of $B$ , and the above parametrization is called the support parametrization . (Of course, there is nothing special about $3$ dimensions here, the support parametrization works for strongly convex hypersurfaces in all dimensions.) Surfaces of Constant Breadth. Now, consider the two supporting tangent planes to $B$ that are perpendicular to $u$ : They are the tangent plane at $b(u)$ and the tangent plane at $b(-u)$ . They have equations $x\cdot u = p(u)$ and $x\cdot (-u) = p(-u)$ respectively. In particular, the distance between these two planes is $p(u)+p(-u)$ . Thus, $B$ has constant breadth $2d$ if and only if its support function $p$ satisfies $p(u)+p(-u) = 2d$ . In particular, the function $p_0=p-d$ must be an *odd* function on $S^2$ , i.e., $p_0(-u) = -p_0(u)$ for all $u\in S^2$ . Conversely, if $p_0:S^2\to\mathbb{R}$ is an odd function, then $p = p_0+d$ for $d>0$ sufficiently large is then the support function of a strongly convex body of constant breadth $2d$ . Alternatively, for all sufficiently small $\epsilon>0$ , the function $p = 1 + \epsilon p_0$ is the support function of a strongly convex surface $B_\epsilon$ of constant breadth $2$ . An algebraic example. To construct an algebraic example, it suffices to take $p_0$ to be an odd algebraic function on $S^2$ . Taking $p_0$ to be the restriction of a linear function in $\mathbb{R}^3$ to $S^2$ will only give a surface that is a translation of the round $S^2$ , so we need to look at something else. The next simplest choice would be $p_0 = xyz$ , where $S^2$ is defined by $x^2+y^2+z^2=1$ . For $\epsilon^2<1$ , the surface $B_\epsilon=b_\epsilon(S^2)$ is a surface of constant breadth $2$ that is smooth, strongly convex, and algebraic. (When $\epsilon^2=1$ , the surface $B_1$ is still strictly convex and algebraic and has breadth $2$ , but it is not smooth. When $\epsilon^2>1$ the image $b_\epsilon(S^2)$ is not a convex surface.) When $\epsilon\not=0$ , it is easy to show that $B_\epsilon$ has the symmetry of the function $xyz$ but no more symmetry than that, i.e., the symmetry group will have (finite) order $24$ (in fact, it is the symmetry group of the regular tetrahedron.) It is algebraic because it is parametrized by algebraic functions on the algebraic surface $S^2$ . It is therefore (a component of) the zero locus of an irreducible polynomial function $f_\epsilon(x,y,z)$ on $\mathbb{R}^3$ . A calculation using elimination theory (carried out by MAPLE) shows that $f_\epsilon$ has degree $20$ (when $\epsilon\not=0$ ) and, for most values of $\epsilon$ , has more than 1000 terms. It is not obvious that the zero locus of $f_\epsilon$ has no other (real) components other than $B_\epsilon$ . Thus, I don't know that the interior of $B_\epsilon$ (which makes sense as long as $\epsilon^2\le 1$ ) can be characterized as the set where $f_\epsilon$ is negative. However, I don't know why you want this; it doesn't seem natural to me. Surely, you only really need it to be algebraic. In any case, one has an explicit parametrization of $B_\epsilon$ using any explicit parametrization of $S^2$ . It is easy to draw pictures using graphical software. Here are what the surfaces $B_{1/2}$ and $B_1$ look like: Added comment: David Speyer has shown that $f_\epsilon$ cannot change sign anywhere other than across the image $b_\epsilon(S^2)$ , since, as he shows, this is the only two-dimensional component of the real locus of $f_\epsilon=0$ . More generally, taking $p = 1+p_0$ , where $p_0 = \lambda_1\lambda_2\lambda_3$ and where each $\lambda_i$ is a linear function of $x$ , $y$ , and $z$ , produces a $7$ -parameter family of distinct algebraic surfaces of constant breadth $2$ , and the generic one of these has no nontrivial symmetries. (In fact, one only gets continuous symmetries when $p_0 = \lambda^3$ for some linear function $\lambda$ .)
|
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"https://mathoverflow.net/questions/222172",
"https://mathoverflow.net",
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|
222,516 |
Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread ). The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also Hausdorff. A minimal Hausdorff topology is such that no strictly coarser topology is Hausdorff. The trivial topology is compact. Every topology that is coarser than a compact topology is also compact. A maximal compact topology is such that no strictly finer topology is compact. A compact Hausdorff topology is both minimal Hausdorff and maximal compact.
A minimal Hausdorff topology need not be compact and a maximal compact topology need not be Hausdorff (Steen & Seebach, Examples 99 and 100). Question: It seems that there is some duality between Hausdorffness and compactness? Can this kind of duality be stated more explicitly (e.g. in some category theoretic formulation)? Is a compact topology on a fixed set $X$ some "dual" of a Hausdorff topology on $X$? Here it is stated that a Hausdorff topology need not contain a minimal Hausdorff topology. If there is some duality these notions then we could also automatically deduce that a compact topology must not be contained in a maximal compact topology.
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There are several ways I think of expressing this 'duality'. But before describing this, maybe it would help to explain a sense in which 'existence' (at least one element, or totality of a relation) is dual to 'uniqueness' (at most one element, or well-definedness of a relation). So let's consider the category whose objects are sets and whose morphisms are relations $R: A \to B$: let $R(a, b)$ denote the truth value of $(a, b) \in R$, and for relations $R: A \to B$ and $S: B \to C$, define the composite $SR: A \to B$ by the rule $SR(a, c) = \exists_{b: B} R(a, b) \wedge S(b, c)$. The identity relation $1_A: A \to A$ is the diagonal subset of $A \times A$, defined by $1_A(a, b) \Leftrightarrow a = b$. Technically this "category" is a 2-category, where 2-cells are given by inclusions $R \subseteq S$ between relations of the same type $A \to B$; we will write 2-cells as $R \leq S$. Even more, we have a $\dagger$-operation which takes a relation $R: A \to B$ to its opposite $R^{op}: B \to A$, where $R(a, b) \Leftrightarrow R^{op}(b, a)$. Altogether the structure one obtains is what is known as in categorical literature as an allegory , or as a bicategory of relations : there are various axiomatic frameworks for describing categories of relations. There are also various notions of 'duality' in such a situation. One is by reversing the direction of 1-cells (called 'op'), another is by reversing the direction of 2-cells (called 'co'), and a third is by reversing directions of both (called 'co-op'). In this 2-categorical context, we may categorically express the condition that a relation $R: A \to B$ is well-defined or functional (for all $a \in A$ there exists at most one $b \in B$ such that $R(a, b)$ is true) by the condition $R \circ R^{op} \leq 1_B$. The co-op dual of this condition is $1_A \leq R^{op} \circ R$, which translates to saying that to each $a \in A$ there exists at least one $b \in B$ such that $R(a, b)$, or that $R$ is a total relation . If we have both conditions $R \circ R^{op} \leq 1_B$ and $1_A \leq R^{op} \circ R$, then the relation is a function; in other words, a relation $R$ is a function iff considered as a 1-cell in the 2-category $\mathbf{Rel}$, it has a right adjoint (which is necessarily $R^{op}$; this is a good exercise). Onto the topology: if $X$ is a topological space and $\beta X$ is the set of ultrafilters on the underlying set of $X$, then there is a convergence relation $\gamma: \beta X \to X$ where $\gamma(U, x)$ (an ultrafilter $U$ converges to $x$) means that the filter of neighborhoods of $x$ is contained in $U$. In fact the very notion of topological space can be expressed in terms of ultrafilter convergence, as explored in the notion of 'relational $\beta$-module' for which you can find an account at the nLab . A space $X$ is compact iff every ultrafilter on $X$ converges to at least one point. This is the same as saying that the convergence relation $\gamma: \beta X \to X$ is total. A space $X$ is Hausdorff iff every ultrafilter converges to at most one point. This is the same as saying that $\gamma: \beta X \to X$ is well-defined. A space is compact Hausdorff iff its convergence relation $\gamma$ is a function: this is an important ingredient in the theorem that compact Hausdorff spaces are exactly algebras of the ultrafilter monad . Summarizing: In the 2-category of sets and relations, the condition of compactness on the convergence relation of a topological space is co-op dual to the condition of Hausdorffness. There are various other ways in which the duality between compactness and Hausdorffness manifests itself. One is that $X$ is compact if every projection map $X \times Y \to Y$ is closed, whereas $X$ is Hausdorff if the diagonal $X \to X \times X$ is closed (projections and diagonals being the two ingredients of product structures) -- although it would take some time to elaborate a sense in which these properties should be seen as "dual".
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222,856 |
There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here . My concern in this question is slightly different. In the course of a discussion I am having, the question has come up of the extent to which the perceived quality of a journal is a good reflection of the quality of its papers. The suggestion has been made that because authors tend to submit their best work to the best journals, that makes it easy for those journals to select papers that are on average of a high standard, but it doesn't necessarily solve the reverse problem -- that they miss other papers that are also very important. (Note that the situation more generally in science is different, because there is a tendency for prestigious journals to value papers that make exciting claims, and not to check too hard that those claims are actually correct. So there one has errors of Type I and Type II, so to speak.) I am therefore interested to know of examples of papers that are very important, but are published in middle-ranking journals. I am more interested in recent papers than in historical examples, since it is the current journal system that we are discussing. Just in case it doesn't go without saying, please do not nominate a paper that you yourself have written...
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One case in point may be Frey, Gerhard: Links between stable elliptic curves and certain Diophantine equations ,
Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1. This is the paper where Frey establishes the link between modularity and Fermat's Last Theorem.
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222,863 |
I would like to prove presence of a spectral gap for the transfer (Ruelle-Perron-Frobenius) operator for some non-uniformly hyperbolic dynamical system on the unit interval. Suppose that I know how to prove presence of a spectral gap for the transfer operator of the induced system. Is there some standard approach (and a good reference for it) that would allow to get presence of a spectral gap for the original system? As far as I understand usually inducing is used together with coupling-like approach and the spectrum of the transfer operator does not appear explicitly.
There are some papers where people work with spectral gaps for transfer operators for induced systems but those that I have seen are focused on more delicate problems (linear response, for example) and it is not that easy to extract the essence of the argument from them.
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One case in point may be Frey, Gerhard: Links between stable elliptic curves and certain Diophantine equations ,
Ann. Univ. Sarav. Ser. Math. 1 (1986), no. 1. This is the paper where Frey establishes the link between modularity and Fermat's Last Theorem.
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223,277 |
In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$: $$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})$$ where $$A(x)=\int^x_0 \frac{1}{1+t^2}\log \frac{4}{1+t^2}dt$$ In this sense, the result is conjectured to hold for $2<s\in \mathbb{N}$, with the $A(x)$ replaced by more complicated functions $A_m(x)$ This might seem rather unenlightening, but we can also state Zagier's result like this: $$\zeta_K(2)=\text{the volume of a hyperbolic manifold}$$ This amazing fact doesn't seem to have a direct analogue for $\zeta_K(2m)$ with $m \neq 1$ What I'd like to know is if there is any big picture explanation for the appearance of hyperbolic manifolds in this context. Zagier's calculation is quite geometrical, but as far as I understand gives no clear explanation of "what the manifold is doing here". Don Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions (1986)
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As ThiKu mentions, the connection between $\zeta_K(2)$ and hyperbolic manifolds is that the volume formula for arithmetic hyperbolic manifolds is given by an explicit formula involving $\zeta_K(2)$. This formula is due to Borel in the case of arithmetic manifolds arising from quaternion algebras. A more general formula was later given by Prasad . In order to get some intuition for general hyperbolic surfaces or $3$-manifolds, it might help to start by recalling the modular surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ where $\mathfrak h_2$ denotes the hyperbolic plane. It is known that $\mathrm{vol}(\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2)=\frac{2}{\pi}\zeta(2)=\frac{\pi}{3}$. (See for instance Section 2 of these notes by Garret or these notes by Venkatesh.) So in this case we see the appearance of a zeta value in the context of the volume of a hyperbolic surface. Here is an admittedly circuitous way to construct the hyperbolic surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ that will provide some intuition for what happens more generally. Consider the quaternion algebra $B=\mathrm{M}_2(\mathbb Q)$ and the maximal order $\mathcal O=\mathrm{M}_2(\mathbb Z)$ inside this quaternion algebra. Let $\mathcal O^1=\mathrm{SL}_2(\mathbb Z)$ denote the multiplicative group of elements of $\mathcal O$ having determinant $1$ and let $\Gamma$ denote the image of $\mathcal O^1$ inside $\mathrm{SL}_2(\mathbb R)$ where we make use of the map $B\hookrightarrow B\otimes_{\mathbb Q} \mathbb R\cong \mathrm{M}_2(\mathbb R)$. The group $\Gamma$ is a discrete subgroup of $\mathrm{SL}_2(\mathbb R)$ which has finite volume, and as $\mathrm{SL}_2(\mathbb R)$ is the group of orientation preserving isometries of $\mathfrak h_2$, we get our quotient surface $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$. The connection which Zagier makes between zeta values and manifolds ultimately arises from a "number field" analog of the above. (A very interesting fact that is not directly related to your question by which I cannot resist mentioning is that many aspects of the geometry of manifolds defined in this sort of arithmetic manner (i.e., via the construction given below) are directly related to quantities of number theoretic interest. For instance, the lengths of closed geodesics on $\mathrm{SL}_2(\mathbb Z)\backslash \mathfrak h_2$ correspond to regulators of real quadratic fields and their multiplicity to the class number of the associated real quadratic field. This connection generalizes to manifolds constructed from number fields other than $\mathbb Q$ as well.) Let $K$ be a number field with $r_1$ real places and $r_2$ complex places and $B$ be a quaternion algebra over $K$ which is not totally definite. Let $s$ denote the number of real places of $K$ that split in $B$. Thus $$B\otimes_{\mathbb Q} \mathbb R\cong \mathbb H^{r_1-s}\times \mathrm{M}_2(\mathbb R)^{s} \times \mathrm{M}_2(\mathbb C)^{r_2}.$$ Let $\mathcal O$ be a maximal order of $B$ and $\mathcal O^1$ the multiplicative subgroup consisting of elements with reduced norm $1$. (If $B$ was a matrix algebra then the reduced norm and determinant would coincide.) Let $\Gamma_\mathcal{O}$ denote the image of $\mathcal{O}^1$ in the group $$G_{s,r_2}=\mathrm{SL}_2(\mathbb R)^{s}\times \mathrm{SL}_2(\mathbb C)^{r_2}.$$ The group $G_{s,r_2}$ is the group of orientation preserving isometries of $\mathfrak h_2^{s}\times \mathfrak h_3^{r_2}$ which preserves factors. Here $\mathfrak h_3$ is hyperbolic $3$-space. The group $\Gamma_\mathcal O$ is a discrete subgroup of $G_{s,r_2}$ which is cocompact if $B$ is a division algebra and has covolume given by the formula $$\mathrm{vol}(\Gamma_\mathcal O\backslash \mathfrak h_2^{s}\times \mathfrak h_3^{r_2})=\frac{2(4\pi)^sd_K^{3/2}\zeta_K(2)}{(4\pi^2)^{r_1}(8\pi^2)^{r_2}}\prod_{\mathfrak p\in\mathrm{Ram}_f(B)}\left(N(\mathfrak p)-1\right),$$ where $d_K$ is the absolute value of the discriminant of $K$ and $\mathrm{Ram}_f(B)$ is the set of finite primes of $K$ which ramify in $B$. This formula is proven in Section 7.3 of Borel's paper. In particular note the appearance of $\zeta_K(2)$ in the formula. The following two special cases of all of this are worth noting: When $s=1$ and $r_2=0$ the manifolds (or orbifolds) constructed are arithmetic hyperbolic surfaces . When $s=0$ and $r_2=1$ the manifolds (or orbifolds) constructed are arithmetic hyperbolic $3$-manifolds (or $3$-orbifolds ). Finally, note that more general zeta values like $\zeta_K(2m)$ do appear in the volume formulas for different types of arithmetic manifolds. This is already the case for higher dimensional arithmetic hyperbolic manifolds. See for instance some of the formulas in this paper by Belolipetsky and Emery. (As was mentioned above, these zeta values arise when one works out the relevant case of Prasad's general volume formula.)
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