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User markus land - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T20:45:16Z http://mathoverflow.net/feeds/user/28048 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123958/a-formal-group-law-over-oriented-bordism A formal group law over oriented bordism Markus Land 2013-03-08T10:25:10Z 2013-04-12T20:17:14Z
My question is related to the following question by Mark Grant here on math overflow:

http://mathoverflow.net/questions/74770/formal-group-law-of-unoriented-cobordism

There it is stated that $MO_*$ has a formal group law $F_0$, universal for formal group laws in characteristic $2$ which satisfy $F(x,x) = 0$.

Of course this formal group law is classified by a map $MU_* \to MO_*$ and in view of the model of $MU$ as unitary ring spectrum and $MO$ as orthogonal ring spectrum this map is indeed induced by a canonical map of (unitary) ring spectra $$c: MU \to MO,$$

see also the answer by Neil Strickland here: http://mathoverflow.net/questions/55602/which-cohomology-theories-are-real-and-complex-orientable .

Now I have two questions:

Q1) Does the map $c$ factor over $MSO$ ?

Q2) If so, can one say anything about the corresponding formal group law over $MSO$?

In view of how I think one can construct the map $MU \to MO$ I think it should be obvious that the map factors through $MSO$, since the map $U(n) \to O(2n)$ factors over $SO(2n)$. But I have not been able to find a reference that gives the example of $MSO$ being complex oriented.

So I guess Q2) is more interesting, already for the simple reason that the ring $MSO_\ast$ is much more delicate than the ring $MO_\ast$. Moreover (as Johannes Ebert also pointed out in a comment in the link above) real oriented theories come equipped with a map from $MO$ and hence split in Eilenberg-MacLane spectra, whereas $MSO$ behaves quite differently.

In addition, I have one last question:

Supposing Q1) and Q2) are true, then as I have already mentioned that $MSO$ is complex oriented. In particular we have a notion of abstract chern classes for $MSO$-cohomology. Moreover we have a canonical map $MSO \to H\mathbb{Z}$. The composite $MU \to H\mathbb{Z}$ is just the usual orientation of integral cohomolgy, in particular the abstract chern classes of this complex oriented theory are just the usual chern classes, but since oriented real bundles have pontryagin classes I ask myself:

Q3) Is there any way to view pontryagin classes as special case of abstract chern classes?

(It seems not to be the case for the cohomology theory $MSO$ which I would have thought of first).
 http://mathoverflow.net/questions/112179/stackification-commutes-with-finite-limits stackification commutes with finite limits? Markus Land 2012-11-12T14:40:10Z 2012-11-12T20:27:33Z
Suppose we work on the Grothendieck site $\mathcal{C}$ of all schemes in the fpqc topology. If it helps it is also fine with me to work only over affine schemes. Let us denote the category of stacks over $\mathcal{C}$ by $Stacks(\mathcal{C})$. This is naturally a full subcategory of the category $Pre_{grpds}(\mathcal{C})$ of presheaves in groupoids over $\mathcal{C}$. This is equivalent to formulating stacks as being categories fibered in groupoids, fulfilling effective descent.

Now the inclusion $Stacks(\mathcal{C}) \to Pre_{grpds}(\mathcal{C})$ has a left adjoint called the stackification functor.

It is a classical fact that the sheafification functor on presheaves of sets commutes with finite limits. Is this also true for the mentioned stackification functor?

Precisely, I want to know if and if so, why, the stackification functor commutes with 2-categorical pullbacks, here is the precise setting.

Suppose we are given Hopf-algebroids $P_1,P_2,P$ and view them as presheaves in groupoids. Suppose we are given two morphisms $P \to P_i$ and we construct the 2-pullback of functors $$Q = pullback(P_1 \to P \leftarrow P_2).$$

In my situation I can show that Q is equivalent to an affine scheme, and I want to conclude that the pullback of the stackified diagram is also equivalent to an affine scheme. If I knew that stackification commuted with finite limits, this would be ok, since the (co)unit (I always mix them up) of the adjunction is an equivalence, i.e., if $X$ is a stack, and $st(X)$ is its stackification, then the natural map $X\to st(X)$ adjoint to the identity of $st(X)$ is an equivalence of stacks.

Any help and comments are appreciated, also if you could give references concerning these questions, that would be great.
 http://mathoverflow.net/questions/50519/integral-cohomology-stable-operations/50536#50536 Comment by Markus Land Markus Land 2013-03-17T10:31:29Z 2013-03-17T10:31:29Z I also believe that it is "well-known" that stable operations of degree $1$ are just given by Ext-groups of the two coefficient groups between which you want to calculate operations. Of course this implies that $H\mathbb{Z}^1H\mathbb{Z} = 0$. http://mathoverflow.net/questions/55602/which-cohomology-theories-are-real-and-complex-orientable/55605#55605 Comment by Markus Land Markus Land 2013-03-07T17:20:59Z 2013-03-07T17:20:59Z So this canonical map $MU \to MO$ will surely factor over $MSO$, right? Can one describe what formal group law it classifies? Do you know any references where this map is studied? http://mathoverflow.net/questions/112179/stackification-commutes-with-finite-limits Comment by Markus Land Markus Land 2012-11-14T22:58:40Z 2012-11-14T22:58:40Z I am not sure I know what you mean. Following Charles Rezk I wanted to define a stack over a site as a presheaf in groupoids such that it is a "homotopy sheaf" i.e. the functor evaluated on an object is given by a homotopy limit over a simplicial diagram induced by a covering in the site. Do you know what the splitting is in this viewpoint?


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Where is number theory used in the rest of mathematics? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:35:32Z http://mathoverflow.net/feeds/question/90700 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics Where is number theory used in the rest of mathematics? Tom Leinster 2012-03-09T14:50:53Z 2012-08-21T17:08:50Z
Where is number theory used in the rest of mathematics?

To put it another way: what interesting questions are there that don't appear to be about number theory, but need number theory in order to answer them?

To put it another way still: imagine a mathematician with no interest in number theory for its own sake. What are some plausible situations where they might, nevertheless, need to learn or use some number theory?

Edit It was swiftly pointed out by Vladimir Dotsenko that the borders between number theory and algebraic geometry, and between number theory and algebra, are long and interesting. One could answer the question in many ways by naming features on that part of the mathematical landscape. But I'd be most interested in hearing about uses for number theory that aren't so obviously near the borders of the subject.

Background In my own work, I often find myself needing to learn bits and pieces of other parts of mathematics. For instance, I've recently needed to learn new bits of analysis, algebra, topology, dynamical systems, geometry, and combinatorics. But I've never found myself needing to learn any number theory.

This might very well just be a consequence of the work I do. I realized, though, that (independently of my own work) I knew of no good answer to the general question in the title. Number theory has such a long and glorious history, with so many spectacular achievements and famous results, that I thought answers should be easy to come by. So I was surprised that I couldn't think of much, and I look forward to other people's answers.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90702#90702 Answer by Francesco Polizzi for Where is number theory used in the rest of mathematics? Francesco Polizzi 2012-03-09T15:15:51Z 2012-03-09T15:21:02Z
Number Theory is used in Algebraic Geometry when studying problems in characteristic $p$. But also in characteristic 0, some algebraic varieties arise from arithmetic constructions. Hilbert modular surfaces or Shimura varieties, for example. Also, the recent classification of fake projective planes by Prasad & Yeung (2007) uses Number Theory in a crucial way: in fact, this problem is equivalent to the enumeration of discrete cocompact subgroups of $PU(1,2)$. And the study of the Ring of Endomorphisms for abelian varieties needs the understanding of some Number Theory, especially number fields and quaternion algebras. I guess that there are countless examples like these.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90703#90703 Answer by Kevin Walker for Where is number theory used in the rest of mathematics? Kevin Walker 2012-03-09T15:18:38Z 2012-03-09T15:18:38Z
If you want to classify 3-manifolds, then at some point you need to start caring about number theory. Most (closed, aspherical, atoroidal) 3-manifolds are hyperbolic, and so are the quotient of hyperbolic 3-space by a Kleinian group. Understanding such groups requires number theory. (I'm not an expert on this topic, so I encourage others to edit this answer to make it better.)
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90704#90704 Answer by unknown (yahoo) for Where is number theory used in the rest of mathematics? unknown (yahoo) 2012-03-09T15:20:12Z 2012-03-09T18:40:15Z
The book The Unreasonable effectiveness of number theory mentions amongst other things that number theory crops up in stability questions in dynamical systems, e.g. the small-divisor problem in KAM-theory where diophantine approximation is used.

Another book with several applications is: Number Theory: An Introduction to Mathematics whose blurb contains "As a source for information on the 'reach' of number theory into other areas of mathematics, it is an excellent work."
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90705#90705 Answer by Timo Keller for Where is number theory used in the rest of mathematics? Timo Keller 2012-03-09T15:22:02Z 2012-03-09T15:45:44Z
It is used in homotopy theory (topological modular/automorphic forms). Class field theory is used e.g. in this article http://front.math.ucdavis.edu/math.AT/0607665 of Niko Naumann.

Also, it is a trend in algebraic geometry to reduce geometric questions to finitely generated fields, so to the realm of arithmetic geometry.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90710#90710 Answer by Vladimir Dotsenko for Where is number theory used in the rest of mathematics? Vladimir Dotsenko 2012-03-09T15:26:48Z 2012-03-09T15:26:48Z
1) For your third interpretation I have at least one relevant experience of my own - "factor systems" (some) number theorists deal with when talking about central simple algebras over number fields did contribute to the development of homological algebra in general, and learning that bit of number theory somehow expanded my homological algebra horizons a bit.

2) Sometimes, it's hard to say where the border between number theory, commutative algebra, and algebraic geometry lies. Around that "triple point" there should be many potential answers.

3) It would be interesting to see examples in the same spirit as how the $n$-factorial conjecture ended up being proved using quite advanced algebraic geometry. Maybe a good candidate along those lines is this result of Kanel-Belov and Kontsevich that uses reduction to characteristic $p$: "The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture", http://arxiv.org/abs/math/0512171
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90711#90711 Answer by Tom Goodwillie for Where is number theory used in the rest of mathematics? Tom Goodwillie 2012-03-09T15:27:32Z 2012-03-09T15:27:32Z
An old example from differential topology: In surgery theory -- the work of Kervaire and Milnor on exotic spheres -- you need to know a sufficient condition for a homogeneous quadratic equation over $\mathbb Z$ in many variables to have a nontrivial solution.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90716#90716 Answer by Denis Serre for Where is number theory used in the rest of mathematics? Denis Serre 2012-03-09T15:59:33Z 2012-03-09T15:59:33Z
Well, the Fibonacci sequence belongs to number theory. Doesn't it ? Yet it is fundamental in many parts of Mathematics, at least in sorting algorithms.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90717#90717 Answer by Denis Serre for Where is number theory used in the rest of mathematics? Denis Serre 2012-03-09T16:02:32Z 2012-03-09T16:02:32Z
Do you count the application of prime number theory and factorization in cryptography?
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90719#90719 Answer by Felipe Voloch for Where is number theory used in the rest of mathematics? Felipe Voloch 2012-03-09T16:05:13Z 2012-03-09T16:05:13Z
The study of mixing properties of commuting operators in Ergodic Theory leads to problems on unit equations.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90720#90720 Answer by Arturo Magidin for Where is number theory used in the rest of mathematics? Arturo Magidin 2012-03-09T16:09:20Z 2012-03-09T16:47:09Z
There is some interesting, recent work on the PORC conjecture (which is about Group Theory, specifically, finite $p$-groups). In some sense this is treading close to number theory even in its statement, I guess:

PORC Conjecture (Higman) Let $n$ be a fixed positive integer. The number $f(p,n)$ of nonisomorphic $p$-groups of order $p^n$ is given by a polynomial in $p$ whose coefficients depend only on the residue class of $p$ modulo some fixed $N$.

(PORC="Polynomial On Residue Classes")

The statement itself makes reference to number theory, of course; but the recent work, by du Sautoy and Vaughan-Lee (Non-PORC behaviour of a class of descendant $p$-groups, in the arXiv) delves deep into number theory and arithmetic geometry (as does a previous paper by du Sautoy associating the problem of counting nilpotent groups with elliptic curves).
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90723#90723 Answer by Timo Keller for Where is number theory used in the rest of mathematics? Timo Keller 2012-03-09T16:13:23Z 2012-03-09T16:13:23Z
The http://en.wikipedia.org/wiki/Feit-Thompson_conjecture is about a diophantine equation whose non-solvability would simplify the proof of the Feit-Thompson theorem.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90726#90726 Answer by Anthony Quas for Where is number theory used in the rest of mathematics? Anthony Quas 2012-03-09T16:18:40Z 2012-03-09T16:18:40Z
Bourgain has a nice paper, Pointwise ergodic theorems for arithmetic sets, (subsequently extended in various directions by other authors including my co-author, Máté Wierdl) on proving a version of the Birkhoff ergodic theorem where one averages along the sequence of square numbers, rather than the sequence of integers. That is: for a measure-preserving system $T\colon X\to X$ and a (square-integrable) function $f$, one considers convergence of the averages $$\frac{1}{N}\sum_{j=1}^N f(T^{j^2}x).$$

Bourgain proves using analytic number theory techniques involving exponential sums that there is convergence almost everywhere, just as in the regular Birkhoff ergodic theorem (although not to the integral as in the regular case and the convergence fails for a typical $L^1$ function, unlike the regular case).
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90742#90742 Answer by KConrad for Where is number theory used in the rest of mathematics? KConrad 2012-03-09T18:10:06Z 2012-03-10T06:08:47Z
Here are a few examples. In some, number theory provided an essential motivation. In the others, it plays a more direct role.

1) Are there nonisometric Riemannian manifolds that are isospectral (eigenvalues of the Laplacian match, including multiplicities)? An example was given by Milnor in the 1960s, which depended on prior work of Witt involving theta-functions (modular forms) of lattices. In the 1980s, Sunada created examples systematically by exploiting the analogy with the number theorist's construction of pairs of nonisomorphic number fields that have the same zeta-function. These number field pairs are found with Galois theory (find a finite group $G$ admitting a pair of nonconjugate subgroups having appropriate properties and then find a Galois extension of the rationals with Galois group isomorphic to $G$). There is a well-known analogy between Galois theory and covering spaces, and Sunada used this to translate the group-theoretic conditions for Galois groups into the setting of Riemannian manifolds. For more on this story, see the Wikipedia page here, where you'll see that the nonisometric isospectral pairs found between the work of Milnor and Sunada were closely related to other parts of number theory (quaternion algebras over the rationals).

2) Lens spaces are distinguished from each other using quadratic residues.

3) Knot theory uses continued fractions. See one of the answers to the MO question here. (Some of the other answers to that question could also be regarded as more applications of number theory, to the extent that you consider finite continued fractions to be number theory.)

4) The construction of Ramanujan graphs uses number theory. Also look here.

5) Frobenius proved that the only ${\mathbf R}$-central division algebras that are finite-dimensional are ${\mathbf R}$ and the quaternions. If you want to see infinitely many other examples of noncommutative division rings that are finite-dimensional over their centers, especially if you want examples that are more than 4-dimensional, you probably should learn number theory since the simplest examples come from cyclic Galois extensions of the rationals. Verifying the examples really work requires knowing a rational number is not a norm from a particular number field, and that amounts to showing a certain Diophantine equation has no rational solutions.

6) The classical induction theorems of Artin and Brauer about representations of finite groups were motivated by the desire to prove Artin's conjecture on Artin $L$-functions. Although number theory appears in the proof in the context of algebraic integers, the main point I want to make is that a conjecture from number theory provided an essential motivation to imagine the theorems might be true in the first place.

7) Several concepts of general importance in mathematics were originally developed within number theory. The most prominent example is ideals, which were first defined by Dedekind in his work on algebraic number theory. The first examples of finite abelian groups were unit groups mod $m$ and class groups of quadratic forms. The first finitely generated abelian groups to be studied as such were unit groups in number fields (Dirichlet's unit theorem). The first application of the pigeonhole principle was in Dirichlet's proof of the solvability of Pell's equation. The motivation for Steinitz's 1910 paper setting out a general theory of fields was Hensel's creation of $p$-adic numbers.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90743#90743 Answer by Franz Lemmermeyer for Where is number theory used in the rest of mathematics? Franz Lemmermeyer 2012-03-09T18:23:57Z 2012-03-09T18:23:57Z
Already one of the Bernoullis, I think Daniel, asked Euler what number theory was good for. He replied that solving diophantine equations such as $x^2 + y^2 = 1$ and similar ones via rational parametrizations would allow to transform integrals $\int dx/\sqrt{1-x^2}$ into rational ones. Perhaps not that exciting today, but it's a time-honoured answer.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90746#90746 Answer by Marc Palm for Where is number theory used in the rest of mathematics? Marc Palm 2012-03-09T18:37:47Z 2012-03-09T18:43:35Z
I give two application from Mathematical Physics.

Quantum chaos:

The Selberg trace formula has inspired the Gutzwiller trace formula. The upper half plane is often used as a toy model to understand Quantum Chaos: www.maths.bris.ac.uk/~majm/bib/arithmetic.pdf.

Quantum field theory:

Also I have heard repeately that automorphic forms turn up in string theory and that there is a connection between toplogical and conformal quantum field theories with the geometric Langlands program.

I am not an expert, so feel free to expand and edit.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90750#90750 Answer by Shahrooz for Where is number theory used in the rest of mathematics? Shahrooz 2012-03-09T19:10:08Z 2012-03-09T21:52:08Z
What is your idea about a problem in number theory that says:

$\frac{p^q-1}{p-1}$ never divides $\frac{q^p-1}{q-1}$ if $p,q$ are distinct primes.

This is a $\textbf{conjecture}$ and the validity of this conjecture would simplify the proof of solvability of groups of odd order, (W. Fiet, J. G. Thompson, $\textit{Pacific J. Math.}$, 13, no.3 (1963), 775-1029), rendering unnecessary the detailed use of generators and relations.

An other interesting application of number theory is in real world. Many years ago, cables used for communication. A lot of cables must be gathered near to each other for more efficiency. But for blocking the noises of each cable to the other cable, a special arrange of cables needed. For this arrangement and neighboring of cables, scientist used reminder theorem and number theory. I think first time, Bell company's scientists invented it.

Also, I think this relation between group theory, graph theory and number theory is very nice example: Suppose the order of group $G$, $|G|$, is $n=p_1^{k_1}p_2^{k_2}\ldots p_s^{k_s}$. We fix two prime numbers $p_i$ and $p_j$ of divisors of $n$ and define a graph $\Gamma(G)$ as fallow:

The vertices of $\Gamma(G)$ are the elements of group $G$ and two vertices $g_1$ and $g_2$ are adjacent if and only if $o(g_1g_2)=p_ip_j$, where $o$ means the order of element $g_1g_2$ as a group element. These graphs have very nice structures and well defined as $\textit{Prime Graph}$ of group.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90765#90765 Answer by Robert Kucharczyk for Where is number theory used in the rest of mathematics? Robert Kucharczyk 2012-03-09T21:40:40Z 2012-03-09T22:17:21Z
Assume you are a (differential) geometer and you want to construct locally symmetric spaces of higher rank. Such a space must have a (globally) symmetric space $X$ as its universal covering space, and this can be written as $X=G/K$ where $G$ is the identity component of the isometry group of $X$ and $K$ is the stabiliser of some point in $X$. To get a locally symmetric space of finite volume, you then have to find a lattice $\Gamma\subset G$, i.e. a discrete subgroup such that $\Gamma\backslash G$ has finite volume with respect to the (right-invariant) Haar measure on $G$. Then if $\Gamma$ is torsion-free you get $\Gamma\backslash X$ as a locally symmetric space.

Now how does one construct such lattices? One method is by arithmetic groups, and it is a matter of taste whether you want to consider them as objects of number theory. Suffice it to say that their study requires a lot of techniques from other areas in number theory in a broad sense. It is quite technical to define an arithmetic group, but it is easy to give some examples that already give you some flavour: $\mathrm{SL}_n(\mathbb{Z})$ as a lattice in $\mathrm{SL}_n(\mathbb{R})$, similarly $\mathrm{Sp}_{q}(\mathbb{Z})$ in $\mathrm{Sp}_q(\mathbb{R})$ or more elaborate constructions where you start from an algebraic number field and an algebra over that field and take some subgroup of the automorphism group of that algebra. Note that in the two cases I presented to you the lattices are not torsion-free, but you can find finite index subgroups which are torsion-free and therefore give you locally symmetric spaces.

As I said there is a fairly technical definition of an arithmetic lattice in a Lie group, and the first guess of everybody hearing of this for the first time is that this should be something exceptional - why should a "generic" lattice be constructible by number-theoretic methods? And indeed, the example of $\operatorname{SL}_2(\mathbb{R})$ supports that guess. The associated symmetric space $\operatorname{SL}_2(\mathbb{R})/\operatorname{SO}(2)$ is the hyperbolic plane $\mathbb{H}^2$. There are uncountably many lattices in $\operatorname{SL}_2(\mathbb{R})$ (with the associated locally symmetric spaces being nothing other than Riemann surfaces), but only countably many of them are arithmetic.

But in higher rank Lie groups, there is the following truly remarkable theorem known as Margulis arithmeticity:

Let $G$ be a connected semisimple Lie group with trivial centre and no compact factors, and assume that the real rank of $G$ is at least two. Then every irreducible lattice $\Gamma\subset G$ is arithmetic.

 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90769#90769 Answer by Robert Kucharczyk for Where is number theory used in the rest of mathematics? Robert Kucharczyk 2012-03-09T22:05:58Z 2012-03-09T22:05:58Z
Here is another example from dynamical systems. As has already been explained in some other posts, the fine structure of dynamical systems often depends on subtle number-theoretic properties of some involved constants.

Assume you have a domain $G\subseteq\mathbb{C}$ and a holomorphic map $f\colon G\to G$ with a fixed point $z\in G$, and you want to study how successive iterates of $f$ around $z$ behave. For sake of simplicity assume that $z=0$. Now locally around $0$ we can approximate $f$ by a linear function $\zeta\mapsto a\zeta$ where $a=f'(0)$, so a simple heuristic says that if we want to understand high powers of $f$, we should understand high powers of $a$. It is then quite clear that the only interesting case is $|a|=1$, so let us assume $a=\mathrm{exp}2\pi it$. Then there are some relations between the growth of the entries in the continued fraction expansion of $t$ and the behaviour of $f$ around $z=0$ under iteration.

You can read more about this in the book "Complex Dynamics in one Variable" by John Milnor. Keywords are Siegel disks and Brjuno numbers.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90771#90771 Answer by GH for Where is number theory used in the rest of mathematics? GH 2012-03-09T22:22:03Z 2012-03-09T22:22:03Z
Number theory has been used to prove many interesting results on $SO(3)$ and related Lie groups, which in turn has attractive applications for the underlying symmetric spaces. A prime example is Drinfeld's solution of the Ruziewicz problem on invariant means of the sphere, or the related recent work of Bourgain-Gamburd on the spectral gap for finitely generated subgroups of $SU(2)$. Related is the Banach-Tarski paradox on doubling the ball, or the recent result of Kiss-Laczkovich that a ball can be decomposed into 22 (or more) congruent pieces.

I also regard Gödel's incompleteness theorem as an application of number theory. Some variants of it, like Matiyasevich's theorem on diophantine equations is highly number theoretic both in its statement and proof.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90780#90780 Answer by Neal for Where is number theory used in the rest of mathematics? Neal 2012-03-10T01:00:15Z 2012-03-10T01:00:15Z
I will add to Kevin Walker's and Robert Kucharczyk's responses. I am drawing the following from a nice survey paper by Marc Lackenby, Finite covering spaces of 3-manifolds, in Proceedings of the International Congress of Mathematicians in Hyderabad, India, 2010.

There are several open questions about finite covering spaces of hyperbolic 3-manifolds, such as whether every hyperbolic 3-manifold has a finite cover that: has positive first Betti number $b_1$, admits a $\pi_1$-injective embedded surface, or fibers over the circle, for example.

Arithmetic lattices seem to be useful in achieving partial results to some of these questions because they permit the use of tools from number theory. For example, Lackenby quotes the following results about arithmetic hyperbolic manifolds (i.e., $\mathbb{H}^3$ modulo an arithmetic lattice):

Every arithmetic hyperbolic 3-manifold admits a closed orientable immersed $\pi_1$-injective surface.

Let an arithmetic hyperbolic 3-manifold $M$ have an invariant trace field $k$ and quaternion algebra $B$. If at every finite place $\nu$ where $B$ ramifies, the completion $k_\nu$ contains no quadratic extension of $\mathbb{Q}_p$ ($p$ a rational prime with $\nu$ dividng $p$), then $M$ has a finite cover with positive $b_1$.

If an arithmetic hyperbolic 3-manifold $M$ has $b_1>0$, then $M$ has finite covers with arbitrarily large $b_1$.

If an arithmetic hyperbolic 3-manifold $M$ contains a closed immersed totally geodesic surface, then $M$ has a finite cover which fibers over the circle.

One can also use number theory to construct finite covers ("congruence covers") of any hyperbolic 3-manifold through a process I do not really understand.

I understand that The Arithmetic of Hyperbolic 3-manifolds, by MacLachlan and Reid, is a good resource for the use of number theory in this subject. You might also look at the survey I mentioned above if you want something briefer.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90803#90803 Answer by Vladimir Dotsenko for Where is number theory used in the rest of mathematics? Vladimir Dotsenko 2012-03-10T08:58:46Z 2012-03-10T08:58:46Z
One thing I completely forgot of was reminded to me by the reference to Feit-Thompson conjecture: applications of number theory/basic Galois theory to characters of finite groups and to structure theory of finite groups, e.g. Burnside's theorem stating that a group of order $p^nq^m$ with $p,q$ prime is solvable.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90823#90823 Answer by Joël for Where is number theory used in the rest of mathematics? Joël 2012-03-10T14:47:45Z 2012-03-10T15:32:02Z
To develop a point already mentioned: To some extent algebraic geometry, including complex algebraic geometry, is a part of number theory. The reason for this is that any algebraic variety, say over $\mathbb{C}$, is defined by polynomial equations involving only finitely many coefficients, so the coefficients live in a ring which is a finitely generated $\mathbb{Z}$-algebra $A$, and $A$ belongs to the domain of number theory. For example, if $m$ is a maximal ideal of $A$, then $A/m$ is a finite field, and the intersection of all maximal ideals of $A$ is $(0)$, so studying "varieties" over $A$ can be reduced in principle to studying varieties over finite fields.

Applications of this method to prove results in complex algebraic geometry by means of number-theoretic methods are many. For a few, relatively elementary but still striking, examples (including the theorem of Ax-Grothendieck and the existence of a fixed point for a $p$-group acting algebraically on $\mathbb{C}^n$), see e.g. this survey paper of Serre. For a more advanced example, consider the beautiful theorem of Batyrev, that birational Calabi-Yau $n$-folds have equal Betti numbers, which is proved by reducing the question to a question over finite field solved using Deligne's proof of the (last) Weil's conjecture, one of the jewel of modern algebraic number theory.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90842#90842 Answer by Terry Tao for Where is number theory used in the rest of mathematics? Terry Tao 2012-03-10T19:10:31Z 2012-03-11T13:55:00Z
One of the first conditional proofs of the undecidability of Hilbert's tenth problem, by Davis and Putnam in 1959, assumed the existence of arbitrarily long arithmetic progressions of primes, as well as a more technical conjecture of Julia Robinson. The number-theoretic hypothesis was soon removed by Robinson (basically by replacing primes by almost primes), and the latter hypothesis removed by Matiyasevich, leading to his famous theorem. So I guess this is an example of how number theory was used in the rest of mathematics.

(The existence of arbitrarily long arithmetic progressions of primes was eventually proven several decades later, although the techniques used in the proof were mostly analytical rather than number-theoretic.)
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90891#90891 Answer by Chandan Singh Dalawat for Where is number theory used in the rest of mathematics? Chandan Singh Dalawat 2012-03-11T09:25:02Z 2012-03-11T09:37:56Z
Here is a random example (in the sense that I just happened to come across it here: http://mathoverflow.net/questions/90772/order-of-vanishing-at-the-cusps-for-the-modular-theta-function/90884#90884).

The abstract of Elkies' article (http://arxiv.org/abs/math/9906019) reads :

We use theta series and modular forms to prove that $\mathbf{Z}^n$ is the only integral unimodular lattice of rank $n$ without characteristic vectors of norm $< n$... By the work of Kronheimer and others on the Seiberg-Witten equation this yields an alternative proof of a theorem of Donaldson on the geometry of $4$-manifolds.

The paper has appeared in Math. Research Letters 2 (1995), 321--326.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90894#90894 Answer by myself for Where is number theory used in the rest of mathematics? myself 2012-03-11T10:47:16Z 2012-03-11T10:47:16Z
I don't know if the following application has been mentioned in this thread. The last step in the solution of Hilbert's third problem is to prove that $\arccos(1/3)/\pi$ is an irrational number. This proves that the cube and the regular tetrahedron have different Dehn invariants, hence are not congruent.

http://en.wikipedia.org/wiki/Hilbert%27s_third_problem
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/91142#91142 Answer by Terry Tao for Where is number theory used in the rest of mathematics? Terry Tao 2012-03-14T01:13:04Z 2012-03-14T01:13:04Z
Number theory naturally arises when analysing nonlinear partial differential equations on the torus, basically one wants to understand the extent to which nonlinear resonances between frequencies can occur, and such frequencies live on an integer lattice if the spatial domain is a torus, so one is naturally led to questions of counting lattice points in some explicit algebraic (or semi-algebraic) set, which is a problem that can often be tackled by number-theoretic methods. For instance, the basic divisor bound in number theory - that the number of divisors of a large integer n is $O(n^{o(1)})$ - already leads to some highly useful consequences for such equations; I discuss this point briefly at http://terrytao.wordpress.com/2008/09/23/the-divisor-bound/ .

(This application of number theory is not unrelated to the use of number theory to understand small divisors in dynamics, as alluded to in other responses.)
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/91144#91144 Answer by Yemon Choi for Where is number theory used in the rest of mathematics? Yemon Choi 2012-03-14T01:56:51Z 2012-03-14T01:56:51Z
This seems not quite in the spirit of the original question, but I couldn't resist mentioning some work which was presented at a recent PIMS (applied) maths colloquium here:

On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations. Kenneth D. T.-R. McLaughlin, Nigel J. E. Pitt. (arXiv 1107.1571)

My limited understanding of the story, which should be taken with a big lump of salt, is as follows:

For certain nonlinear PDEs with a periodic boundary condition, one can write down a Fourier series that represents a weak solution, and then faces the issue of determining in what sense this series converges to the solution. Bizarrely (to my eyes) the shape of the solution is different for rational and irrational times; and to understand what happens at irrational times, the authors have to deal with exponential sums of a form encountered in analytic number theory.

Furthermore, Theorem 1.5 in this paper, which demonstrates a kind of "Gibbs phenomenon" for these solutions, is proved for irrational values of $t$ satisfying a complicated condition depending on the continued fraction expansion of $t$.

(According to the speaker (McLaughlin), the collaboration started while he was sitting in number theory lectures given by the first author, reading a PDEs paper, and realizing that the sums on the page in front of him were awfully like the sums on the board.)
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/91147#91147 Answer by Vamsi for Where is number theory used in the rest of mathematics? Vamsi 2012-03-14T02:37:57Z 2012-03-14T02:37:57Z
If Arakelov geometry counts as number theory, then, http://arxiv.org/pdf/math/0401029v1.pdf demonstrates the computation of the Analytic torsion (a purely analytic object involving the product of determinants of laplacians) using the Arithmetic Riemann-Roch theorem.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/91398#91398 Answer by Orr Shalit for Where is number theory used in the rest of mathematics? Orr Shalit 2012-03-16T18:00:17Z 2012-03-16T18:00:17Z
Nonlinear PDEs were mentioned above, but there is also an old example from linear PDEs (I learned about this from Boris Paneah). In a 1939 paper (link http://www.ams.org/journals/bull/1939-45-12/S0002-9904-1939-07103-6/S0002-9904-1939-07103-6.pdf) Bourgin and Duffin study the Dirichlet problem for the wave equation on a rectangle with sides A and B. They show that the uniqueness of the problem depends on the ratio A/B being irrational. More strikingly, they show that the existence of a solution (with certain regularity) depends on how difficult it is to approximate A/B by rational numbers.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/96636#96636 Answer by Eugene for Where is number theory used in the rest of mathematics? Eugene 2012-05-11T02:02:49Z 2012-05-11T02:02:49Z
Elliptic curves are the basis for a type of public key cryptography known as elliptic curve cryptography.

Also I attended a colloquium recently that talked about the applications of elliptic curves to string theory.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/96655#96655 Answer by Tom De Medts for Where is number theory used in the rest of mathematics? Tom De Medts 2012-05-11T09:57:51Z 2012-05-11T09:57:51Z
The uniqueness of the finite Ree groups of type $^2G_2$ was established by E. Bombieri using extremely tricky number theoretical methods (involving involved elimination methods). As Stephen D. Smith wrote in his review on MathSciNet:

This result has considerable importance in the classification of finite simple groups. Ordinary mortals such as the present reviewer are overawed by the author's tour de force.

(Bombieri, Enrico; Odlyzko, A.; Hunt, D. Thompson's problem (σ2=3). Appendices by A. Odlyzko and D. Hunt, Invent. Math. 58 (1980), no. 1, 77–100.)
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/105159#105159 Answer by Sharkos for Where is number theory used in the rest of mathematics? Sharkos 2012-08-21T13:42:26Z 2012-08-21T13:42:26Z
It also finds use in the theory of information theory and algorithms (which are essentially mathematical topics); for example, number theoretic transforms are arguably the neatest way of achieving the optimal known complexity for multiplying arbitrarily long integers on a classical computer. Other topics also involving some amount of number theory are hashing, random number generation and error detection & correction.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/105163#105163 Answer by Todd Trimble for Where is number theory used in the rest of mathematics? Todd Trimble 2012-08-21T15:04:51Z 2012-08-21T15:04:51Z
Julia Robinson proved that the theory of fields is undecidable by showing that the natural numbers form a subset of the rationals definable by a first-order formula in the language of fields. The construction of this formula involved an ingenious application of number theory. For an accessible account, see the article by Flath and Wagon:

Stan Wagon and Dan Flath, How to pick out the integers in the rationals: an application of number theory to logic, Amer. Math. Monthly 98 (1991), no. 9, 812–823. MR 1132996 (93b:03076)

 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/105169#105169 Answer by unknown (google) for Where is number theory used in the rest of mathematics? unknown (google) 2012-08-21T15:45:28Z 2012-08-21T15:45:28Z
I believe that has not yet been mentionned the role of quadratic forms over integers in the classification of four manifolds (work of Freedman, Donaldson ...). It is not an example of very "high" number theory but it is a case where number theoretic objects "parametrize" topological or differential geometric objects, relation which seems unexpected at first.
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/105179#105179 Answer by Kjetil B Halvorsen for Where is number theory used in the rest of mathematics? Kjetil B Halvorsen 2012-08-21T17:08:50Z 2012-08-21T17:08:50Z
This is maybe too elementary --- very simple number theory is used in the classification of finite Markov Chains.

See for instance: Karlin & Taylor: A First Course in Stochastic Processes.

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## Oil Majors’ Costs Have Risen 66% Since 2011

##### 03 March 2017

by Nick Cunningham of Oilprice.com

The oil majors reported poor earnings for the fourth quarter of last year, but many oil executives struck an optimistic tone about the road ahead. Oil prices have stabilized and the cost cutting measures implemented over the past three years should allow companies to turn a profit even though crude trades for about half of what it did back in 2014.

The collapse of oil prices forced the majors to slash spending on exploration, cut employees, defer projects, and look for efficiencies. That allowed them to successfully lower their breakeven price for oil projects. However, some of that could be temporary, with oilfield services companies now demanding higher prices for equipment and drilling jobs, in some cases upping prices by as much as 20 percent. The result could be an uptick in the cost of producing oil for the first time in a few years. Rystad Energy estimated the average shale project could see costs rise by $1.60 per barrel, rising to$36.50.

That does not seem like the end of the world. After all, those breakeven prices are still dramatically lower than what they were back in 2014. In fact, Reuters put together a series of charts depicting the fall in costs for shale production in different parts of the United States. Every major shale basin – the Eagle Ford, the Bakken, the Niobrara, and the Midland and Delaware basins in the Permian – have seen breakeven prices fall by as much as half since 2013. The slight uptick in costs expected in 2017 is a rounding error compared to the reductions over the past half-decade.

But that is just for shale drilling. The oil majors produce most of their oil outside of the shale patch, with much of their output coming from longer-lived projects in deepwater, for example. To be sure, some of the largest oil companies have made some progress in cutting costs over the past few years, but a new report casts doubt on the industry’s track record.

According to new research from Apex Consulting Ltd., the oil majors are still spending more to develop a barrel of oil equivalent than they were before the downturn in prices – in fact, much more. Apex put together a proprietary index that measures cost pressure for the “supermajors” – ExxonMobil, Royal Dutch Shell, Chevron, Eni, Total and ConocoPhillips. Dubbed the “Supermajors’ Cost Index,” Apex concludes that the supermajors spent 66 percent more on development costs in 2015 than they did in 2011, despite the widely-touted “efficiency gains” implemented during the worst of the market slump. It is important to note that this measures “development costs,” and not exploration or operational costs.

However, performances varied by company. Eni, for example, saw its development costs decline by 32 percent between 2011 and 2015, a notable achievement. Chevron and ExxonMobil also posted efficiency gains, although more modest figures than Eni. Chevron’s costs fell 6 percent and Exxon’s were down 5 percent over the five-year period.

At the other end of the spectrum is Royal Dutch Shell, which saw development costs quadruple. ConocoPhillips and BP fared only slightly better, with costs roughly doubling over the timeframe. As a whole, the development costs for the group of “supermajors” rose 66 percent to $18.39 per barrel. After the collapse of oil prices in 2014, the cost index did decline. Oil producers squeezed their suppliers, streamlined operations, and improved drilling techniques. But costs still stood 66 percent higher than in 2011. The index points to underlying structural increases in development costs for the broader industry. At$18 per barrel, the cost figure would seem rather low. But it is important to note that this is just for “development costs,” which represent just over half of a company’s total cost. That figure excludes the cost of exploration as well as funding ongoing operations. So the “breakeven price” so often quoted in the media is actually quite a bit higher. BP, for example, recently admitted that its finances will not breakeven unless oil trades at roughly $60 per barrel. The supermajors are in a tricky position. They are trying to cut back on spending in order to fix their finances and pay down the massive pile of debt that they have accumulated in the past few years. However, their reserves will decline if they fail to replace them. Exxon, for example, only replaced 67 percent of the oil it produced in 2015. Moreover, as Apex Consulting notes, oilfield services might demand higher prices in the future as drilling activity picks up. Right now, offshore rigs are still underutilized, meaning that price inflation has yet to kick in. In other words, the decline in costs post-2014 are, at least in part, cyclical. Costs will rise again as activity picks up unless oil producers work with their suppliers to address the underlying structural costs of oil production. Link to original article: http://oilprice.com/Energy/Energy-General/Oil-Majors-Costs-Have-Risen-66-Since-2011.html ### Comments Renewable energy is coming down in price and fossil fuels are increasing in price.New wind now beats new coal plants. We are now at the break even point in price comparison. It will be interesting to see what happens next decade when renewables shift downwards even more. Oil prices are low, production prices are rising. Love this news! US oil and gas drilling is now 80% fracking/horizontal drilling and the productivity of these wells have gone up 500% since 2011. So it does not matter that it is, say, 50% more expensive to make a state of the art fracked well today. It is still far more cost effective than it was in 2011. This is was oil and gas fracking is booming again despite low oil prices of 50-55 USD per barrel. The evidence can be studied here: http://www.wtrg.com/rotaryrigs.html shows 63% annual growth in number of horizontal drilling rigs. That is a boom. And here http://www.eia.gov/petroleum/drilling/#tabs-summary-2 shows the 500% growth in productivity for nearly all rigs at the shale plays numbers in pdf files. I am all for going fully electric and renewable and to build the 100 Tesla sized giga factories that can replace gassers and diesels on a global scale. However, it will take decades to get rid of the global fleet of nearly 2 billion combustion vehicles so meanwhile we need to drill at full speed. We need to prepare for increasing wars between the free world and the part of the world that does not believe in freedom or science. Our cultural differences are growing all the time (increasing terrorism against us is evidence of that) and one way to prepare for this escalating war is by banning import of energy from all of our cultural adversaries. Another is to stop accepting immigrants from countries that are not fully democratic and science based. If people are unhappy with where they live they should do something about it in the country they were born in. We could help them with that in the form of money and weapons if their agenda is the same as ours: democracy, freedom and science as the foundation for knowledge. Instead of Exxon making$40 billion per year in profits they make \$30 billion.

Hello Change,
The war is between evidentiary reality and Republicans with Trump leading the charge against science.
The clock is rapidly advancing toward midnight because of AGW denial on the right. We all need to make monetary decisions to ditch oil use starting now. My wife and I have been doing so since the 1980s.

Many time i've said to increase ice mpg efficiency to over 100 mpg. It is doable if we go serie hybrid with exhaust heat and pressure capture instead of just crankshaft horsepower like it is the case with the volt, the bmw i3 rex and the upcoming nissan e-note. Im sick and tire of the high cost of gasoline, we need to drastically lower consumption to provoke a complete collapse of petroleum price and we gonna decrease 4x our gas bills and dramatically decrease air pollution. Here in this blog each blogger should take his responsability and stop all car buying till they release an over 100 mpg gasoline car at the same price than actual gas car. manufacturers and goverments and bank entities will never release an over 100 mpg cheap car except if we make a buying strike where we keep our car almost forever except if they decide to offer a really better car. they tried to hack us with costly battery cars and costly hybrid and the market didn't buy in great quantity because it end-up costing more. Say no to mediocrity and ask for an over 100 mpg car or keep your old car forever.

Don't worry, population decline is coming. There will be plenty of oil for the people left.

The comments to this entry are closed.


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# Triangle ABC is equilateral with sides 10 cm long. Triangle DEF is formed by drawing lines parallel to, and 1 cm away from, the sides of tri

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Triangle ABC is equilateral with sides 10 cm long. Triangle DEF is formed by drawing lines parallel to, and 1 cm away from, the sides of triangle ABC. Find the perimeter of triangle DEF.

Guest Apr 10, 2015

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zacismyname  Apr 10, 2015
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My solution assumes that DEF is bigger than ABC, though you don't actually specify this in the question.

First, look at the diagram:

With reference to this we have:

.

Alan  Apr 10, 2015
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WOW Alan,  that looks very impressive!

Melody  Apr 10, 2015
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Here's an alternative method of solution.

Assuming that the triangle DEF lies 'outside' the triangle ABC, the area of DEF is equal to the area of ABC plus the area of three trapezia.

So, calling the length of side of DEF L,

$$\frac{1}{2}L^{2}\sin 60 = \frac{1}{2}10^{2}\sin 60+\frac{3}{2}(10+L),$$

Multiplying throughout by 4,

$$L^{2}\sqrt{3}=100\sqrt{3}+6(10+L),$$

$$\sqrt{3}(L^{2}-100)=\sqrt{3}(L+10)(L-10)=6(10+L),$$

$$\sqrt{3}(L-10)=6,$$

$$L=2\sqrt{3}+10,$$

so the length of the perimeter is   $$6\sqrt{3}+30.$$

Assuming that DEF lies 'within' ABC leads to, (following the same procedure),  $$L=10-2\sqrt{3}.$$

Bertie  Apr 10, 2015
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+981
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zacismyname  Apr 10, 2015
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I like your solution, zacismyname......I would do this exactly the same way......

CPhill  Apr 11, 2015
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Hi Zacismyname,

Welcome to the forum.    :D

I think I may nickname you Zac   LOL

Melody  Apr 11, 2015


text:
Key Figures:

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7. Long life span of :
• PV Modules (up to 25 years)
• Inverters (7-10 years)


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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 5, Pages 1125–1137 (Mi izv1883)

Discrete convolution operators on the quarter plane and their indices

R. V. Duduchava

Abstract: Let $\Gamma^2=\Gamma\times\Gamma$, where $\Gamma$ is the unit circle, and let $L_2^m(\Gamma^2)$ be the Hilbert space of vector-valued functions $\varphi=(\varphi_1,…,\varphi_m)$ whose components $\varphi_k(\zeta_1,\zeta_2)$ are complex-valued square integrable functions on $\Gamma^2$. The author considers the subspace $H_2^m(\Gamma^2)$ of functions in $L_2^m(\Gamma^2)$ having analytic continuations into the torus $\{(z_1,z_2):|z_k|<1\}$; let $P$ be the projection of $L_2^m(\Gamma^2)$ onto $H_2^m(\Gamma^2)$. For a bounded measurable matrix-valued function $a(\zeta_1,\zeta_2)$ of order $m$ on $\Gamma^2$ having limits $a(\zeta\pm0,t)$ and $a(t,\zeta\pm0)$ ($\zeta\in\Gamma)$ uniform in $t\in\Gamma$, the bounded operator $T_a^2=PaP$ is defined in $H_2^m(\Gamma^2)$. In this paper a homotopy method is described for computing the index of Noetherian operators in the $C^*$-algebra generated by the operators $T_a^2$. In the case where $a(\zeta_1,\zeta_2)$ is continuous a simple formula for computing the index of $T_a^2$ is indicated.
Bibliography: 24 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:5, 1072–1084

Bibliographic databases:

UDC: 513.88
MSC: 47B35

Citation: R. V. Duduchava, “Discrete convolution operators on the quarter plane and their indices”, Izv. Akad. Nauk SSSR Ser. Mat., 41:5 (1977), 1125–1137; Math. USSR-Izv., 11:5 (1977), 1072–1084

Citation in format AMSBIB
\Bibitem{Dud77} \by R.~V.~Duduchava \paper Discrete convolution operators on the quarter plane and their indices \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1977 \vol 41 \issue 5 \pages 1125--1137 \mathnet{http://mi.mathnet.ru/izv1883} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=493484} \zmath{https://zbmath.org/?q=an:0406.47027|0426.47031} \transl \jour Math. USSR-Izv. \yr 1977 \vol 11 \issue 5 \pages 1072--1084 \crossref{https://doi.org/10.1070/IM1977v011n05ABEH001759}

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This publication is cited in the following articles:
1. Albrecht Böttcher, Hartmut Wolf, “Large Sections of Bergman Space Toeplitz Operators with Piecewise Continuous Symbols”, Math Nachr, 156:1 (1992), 129
2. Albrecht Böttcher, Bernd Silbermann, “Infinite Toeplitz and Hankel Matrices with Operator-Valued Entries”, SIAM J Math Anal, 27:3 (1996), 805
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Processing gene expression data
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Entering edit mode
3.3 years ago
Natasha ▴ 40

This is a follow up to my previous question.

I would like to implement the following steps given in the supplementary file of this study to reproduce the figure 1 displayed in the paper.

We used Affymetrix microarray data from a recent thorough analysis of the mouse and human transcriptomes [1]. We selected all 54 adult mouse non-cancer samples. The raw intensity data were transformed to normalized expression levels with the robust multi-array average (RMA) lowlevel algorithm [2] implemented in the BioConductor package [3]. We used standard settings, including perfect match (PM) only, model-based background and quantile normalization across experiments [4]. Similar results were obtained using the microarray analysis suite (MAS5) function followed by log-transformation to calculate expression levels (data not shown).

Mouse data is available on GEO with access ion number GSE1133. The data is available in different formats like CDF, CIF, GIN, PSI, SIF, PROBE, TAB, TXT. I am not sure which data format, containing the raw intensity data, has to be downloaded for implementing the procedure described above.

gene-expression • 880 views
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3.3 years ago
ATpoint 68k

It is the CEL files under GSE1133 at the bottom of the page. Under GSE1133_RAW.tar press Custom to make a selection for the samples you need. CEL files store the raw intensity values that can be processed with standard software for normalization. Please search around on how to read and handle CEL files.

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1
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3.3 years ago
c.chakraborty ▴ 170

Isn't there access to raw .CEL files for you to work on? Plus which paper, could you please share the link or doi.! I checked and there are .CEL files available for microarray analysis. If you want to analyse microarray data for gene expression analysis using, you should use the .CEL files. They are TARzipped in the supplementary files section.

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Entering edit mode

Many thanks for the response. Yes, the raw CEL files are available here. The figure 1 that I want to reproduce is available in this article. (Please find the link here) . Description of how the figure was created can be found in the supplementary. Also, figure one has been created using the data available from this study (Please find the link here).

In total 438 GSM files are listed . I am not sure how to distinguish Human and Mouse samples( I think this can be filtered using the platform id) ; cancerous and normal samples. Any suggestion on which package has to be used for RMA normalization illustrated here will be really helpful.

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Entering edit mode

I think everything prefixed MGM is mouse, and the rest 1B/ 3A is human. Simply click the GSM... links, it will tell you the organism. Check if this pattern I suggested above holds true for the majority of the samples.

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Entering edit mode

Thank you. It is mentioned that GPL1073 GNF1M platform is for Mouse (GSM18584 to 18705) GPL1074 GNF1H is for Human. (18706 to 18863)

However, I couldn't find the platform id in the CEL files .

ADD REPLY
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Entering edit mode

Go to the supplementary GSE1133_RAW.tar. Click on custom and it will lead you to all the .CEL files in this dataset. You can download whichever you need for your analysis.

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Entering edit mode

Thank you. I am trying to normalize using the following code

library(affy) %IN bioconductor package
Data <- ReadAffy() % reads all .CEL files
eset <- rma(Data) % RMA normalization

Is this right? I am trying to normalize all samples(i.e GSM18584 to 18705) together

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# Proving second value of second derivative from function and first derivative

$$Let \ f: [-1,1] \rightarrow \mathbb{R}$$ be a function, which is continuous on [-1,1] and twice differentiable on (-1,1) with: $$f(-1)=f(1)= 0 \ and \ f(0)=-3$$ Show that there exists $$c \ within \ (-1,1)$$ s.t. $$f''(c) > 3$$

My intuition was to use Rolle's theorem to show that there must be a K within (-1,1) s.t. f'(k) = 0, then use prove by contradiction and use mean value theorem to show that there is a f''(c) which cant be $$\leq 3$$ however, i am having serious trouble with this. I think i am overseeing something simple, I also figure i need to use the fact that $$f(0)=-3$$ somehow but can't work out where. Some hints would be greatly appreciated!

Hint: Instead of using the mean value theorem (in the form of Rolle's theorem) for $$f$$ on all of $$[-1,1]$$, use it separately on each of the intervals $$[-1,0]$$ and $$[0,1]$$.

• Would you then split the proof up into 2 cases, one where K (from Rolle's) is in [-1,0] and the other case when it is in [0,1] ? – Matt_G Oct 8 '18 at 5:38
• You might be able to do that, but that makes it more complicated than necessary. You don't need to use Rolle's theorem at all. – Eric Wofsey Oct 8 '18 at 5:44

$$-3=\frac{f(-1)-f(0)}{-1\ -0}=f'(a)$$ for some $$a\in(-1,0)$$ and $$3=\frac{f(1)-f(0)}{1-0}=f'(b)$$ for some $$b\in(0,1)$$ . Now $$\frac{f'(b)-f'(a)}{b-a}=f''(c)$$ for some $$c\in(a,b)$$, also $$0.Therefore $$f''(c)>3$$

One can even show that $$f''(x) \ge 6$$ for some $$c \in (-1, 1)$$ under the given conditions. (The function $$f(x) =3(x^2-1)$$ shows that this is the best possible bound.)

Assume that $$f''(x) < 6$$ for all $$x \in (-1, 1)$$.

Let $$x_0 \in (-1, 1)$$ be a point where $$f$$ assumes it's minimum, then $$f(x_0) \le -3$$ and $$f'(x_0) = 0$$. Because of the symmetry of the problem we can assume that $$x_0 \ge 0$$.

Taylor's formula applied to the interval $$[x_0, 1]$$ gives $$f(1) = f(x_0) + f'(x_0)(1-x_0) + \frac{f''(\xi)}{2}(1-x_0)^2$$ for some $$\xi \in (x_0, 1)$$, and therefore $$0 = f(1) < -3 + \frac 62 (1-x_0)^2 \le -3 + 3 = 0$$ which is a contradiction.

An alternative (mimicking the proof of Taylor's theorem) is to consider the function $$g(x) = f(x) + (x^2 -1) \cdot f(0)\, .$$ Then $$g(-1) = g(0) = g(1)$$. Repeated application of Rolle's theorem shows that $$g'$$ has two distinct zeros, and therefore $$g''(c) = 0$$ for some $$c \in (-1, 1)$$. So $$0 = g''(c) = f''(c) + 2f(0) \implies f''(c) = -2f(0) = 6 \, .$$

Note that by Taylor's theorem we have two numbers $$c_1,c_2$$ with $$-1 such that $$f(-1)=f(0)-f'(0)+\frac{f''(c_1)}{2},\,f(1)=f(0)+f'(0)+\frac{f''(c_2)}{2}$$ Adding these and noting that $$f(0)=-3, f(-1)=f(1)=0$$ we get $$\frac{f''(c_1)+f''(c_2)}{2}=6$$ It now follows via intermediate value property of derivatives that there is a $$c\in[c_1,c_2]$$ such that $$f''(c) =6$$.


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# Differential equation, limit cycles, stability, phase plane. By 11/12

1. Dec 10, 2012

### debian

1. The problem statement, all variables and given/known data

The trajectory of an arrow in space obeys the following system of equations:

$\dot{x}$ = y+$(x^2+y^2-3)^2 (x^3-x+xy^2)$

$\dot{y}$ = y+$(x^2+y^2-3)^2 (y^3-y+x^2y)$

1. Questions

a) Derive an ODE for the radial coordiante r(t) = $\sqrt[]{x^2(t)+y^2(t)}$

b) Show that the system admits two limit cycles and classify their stability

c) The target is standing at the origin (0,0) of the phase plane. Let Po=(x(0),y(0))$\neq$ (0,0) denote the initial position of the arrow at the time t = 0. Determine and plot the region in the (x,y) plane containing all the points Po such that the arrow will hit the target at some instant of time t > 0

3. The attempt at a solution

I derived ODE in radial coordiante and got:

$\dot{r}$=$(r^2-3)^2(r^2-1)r$

So we have 5 points to check the stability: $0, 1,-1,\sqrt{3},-\sqrt{3}$

Than I calculate the derivative of $\dot{r}$ and evaluate in those 5 points: I get five zeros so they are undecided.

Then I daw the plot of $\dot{r}$ and if the $\dot{r}$>0 it does mean that arrow on the phase line is to the right and if $\dot{r}$<0 the arrow is to the left.

So The phase line looks like this: <- (-\sqrt{3}) <- (-1) -> (0) <- (1) -> (\sqrt{3}) ->

Is that correct? What should I do next?

2. Dec 10, 2012

### pasmith

Are you sure these are correct? Because they tell me that
$$r\dot r = x\dot x + y\dot y$$
will not be independent of $\theta$, because the linear terms give me
$$xy + y^2 = r^2(\cos\theta\sin\theta + \sin^2\theta)$$.

Assuming that you are correct that
$$\dot r = r(r^2 - 3)^2(r^2 - 1) = f(r)$$
then since by convention $r \geq 0$, there is a fixed point at the origin and two orbits, at $r = \sqrt 3$ and $r = 1$.

To determine the stability, you look at the sign of $f'(r)$. If it's negative then the point is stable, and if it's positive then the point is unstable. If it's zero then it's inconclusive and you have to look at higher derivatives.

Alternatively, in one dimension r is increasing if $\dot r > 0$ and decreasing if $\dot r < 0$. By inspection, $\dot r < 0$ if $0 < r < 1$ and $\dot r > 0$ when $1 < r < \sqrt 3$ and $r > \sqrt 3$. So the origin is stable, the orbit at 1 is unstable, and the orbit at $\sqrt 3$ is stable from below and unstable from above.

For part (c) you need to think about what trajectories eventually end up at the origin.

3. Dec 10, 2012

### debian

There is a typo: shoul be -x instead of one of y

$\dot{x}$ = y+$(x^2+y^2-3)^2 (x^3-x+xy^2)$

$\dot{y}$ = -x+$(x^2+y^2-3)^2 (y^3-y+x^2y)$

4. Dec 10, 2012

### pasmith

Then your result for $\dot r$ is correct.


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# How do you convert or move from a linear cost function to a quadratic cost function?

I am reading a book on electricity cost modelling. I understand equation 2.7 below, which indicates that the total cost for an ith plant is a function of fixed cost(FC), fuel cost(FL), plant efficiency (af) and quantity of electricity produced (Q).

Equation 2.9 is a two-step piecewise cost function which describes the existence of two possible ranges of operation, and that producing above a threshold implies that there is an increase in the variable costs. I understand this too.

Equation 2.10 provides a more general functional form that represents a continuous and smooth version of a multiple-step piecewise linear cost function through a quadratic function. However, I do not understand how this was derived. The only difference between equation 2.7 and 2.10 seems to be the squaring of Q in equation 2.10.

My questions are:

1. Why was Q squared?
Your eq (2.10) is not more general than (2.9), but corresponds to an alternative specification. A more general version would be: $$C_i(Q_i)=FC_i+a_{1,i}Q_i+a_{2,i}Q^2_i.$$ This specification allows marginal cost to be constant as in (2.9) if $$a_{2,i}=0$$ or nonconstant as in (2.10) for $$a_{1,i}=0,a_{2,i} \neq 0.$$ It is more general because it is compatible with an increasing and decreasing marginal cost over some range of the production level. In contrast, (2.9) implies that if the cost function is increasing in $$Q_i$$, then is also has to be convex: $$C''>0.$$


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## Irregular hexagons with equal angles [closed]

Hello,

I have an irregular hexagon whose interior angles are all $2\pi/3$. That is, every pair of opposite edges are parallel. I have the distance between every pair of opposite edges, and I wish to find the perimeter of the hexagon. Intuitively, this seems possible, but I'm not sure how to do it. Also, is there a name for such a hexagon?

If that helps, I can also assume that every pair of opposite edges have the same length.

The opposite edges should not have the same lengths. For instance, the edge lengths may be $a$, $b$, $a$, $b$, $a$, $b$ in a cyclic order. You may also add up this sequence and $c$, $d$, $e$, $c$, $d$, $e$... – Ilya Bogdanov Nov 22 at 15:44


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# M11-05

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16 Sep 2014, 00:43
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Is quadrilateral $$ABCD$$ a rhombus?

(1) $$AC$$ is perpendicular to $$BD$$

(2) $$AB + CD = BC + AD$$
[Reveal] Spoiler: OA

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16 Sep 2014, 00:43
Official Solution:

Statements (1) and (2) combined are insufficient. $$ABCD$$ can be a rhombus but it can also be a kite-shaped figure. The diagonals of this figure form a cross, and S2 holds because of the symmetry ($$AB = BC$$ and $$CD = AD$$).

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14 Jun 2016, 22:53
Bunuel wrote:
Official Solution:

Statements (1) and (2) combined are insufficient. $$ABCD$$ can be a rhombus but it can also be a kite-shaped figure. The diagonals of this figure form a cross, and S2 holds because of the symmetry ($$AB = BC$$ and $$CD = AD$$).

Bunuel HI , ABCD can also be a square,correct me if i'm wrong

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15 Jun 2016, 02:39
sharma123 wrote:
Bunuel wrote:
Official Solution:

Statements (1) and (2) combined are insufficient. $$ABCD$$ can be a rhombus but it can also be a kite-shaped figure. The diagonals of this figure form a cross, and S2 holds because of the symmetry ($$AB = BC$$ and $$CD = AD$$).

Bunuel HI , ABCD can also be a square,correct me if i'm wrong

Yes, it can be a square too.
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06 Jul 2016, 03:26
I think this is a poor-quality question and I don't agree with the explanation. I dont understand how can 2nd statement be true?

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17 Aug 2016, 08:49
A kite-shaped figure is a funny explanation.

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27 Sep 2016, 08:00
Bunuel IF the condition 1 read that the AC is a perpendicular bisector of BD, would that have been enough to conclude that the figure was a rhombus ( considering square is also a special type of rhombus ) ? Will the diagonals of the a kite shaped figure also bisect each other at 90 degrees ?

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07 Oct 2016, 04:54
This is a HIGH quality question,i agree with the explanation.

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09 Nov 2016, 04:23
Bunuel wrote:
sharma123 wrote:
Bunuel wrote:
Official Solution:

Statements (1) and (2) combined are insufficient. $$ABCD$$ can be a rhombus but it can also be a kite-shaped figure. The diagonals of this figure form a cross, and S2 holds because of the symmetry ($$AB = BC$$ and $$CD = AD$$).

Bunuel HI , ABCD can also be a square,correct me if i'm wrong

Yes, it can be a square too.

Isn't it true that every square is a in fact a rhombus?

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09 Nov 2016, 07:20
yeshu_a wrote:
Isn't it true that every square is a in fact a rhombus?

Yes, every square is a rhombus.
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11 Nov 2016, 05:51
I don't understand how the first statement could be true...

How can two sides of a quadrilateral be perpendicular if they don't share any vertex? (AC and BD)

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11 Nov 2016, 07:06
pedropereira wrote:
I don't understand how the first statement could be true...

How can two sides of a quadrilateral be perpendicular if they don't share any vertex? (AC and BD)

AC and BD are diagonals, not sides.
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21 Dec 2016, 05:42
Is it sufficient for statement 1 IF IT SAYS AC and BD are perpendicular bisectors of each other?

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08 Jun 2017, 10:04
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. hi,

In a rhombus diagonals bisect each other at 90 degree. in statement one we are told that the diagonals bisect each other at 90 degree. then it must be a rhombus. even a kite shaped figure is a rhombus. correct me if im wrong.

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08 Jun 2017, 15:08
kanikas1 wrote:
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. hi,

In a rhombus diagonals bisect each other at 90 degree. in statement one we are told that the diagonals bisect each other at 90 degree. then it must be a rhombus. even a kite shaped figure is a rhombus. correct me if im wrong.

Yes, you are wrong.

RHOMBUS - a quadrilateral in which all four sides are congruent.
KITE - a quadrilateral in which each pair of consecutive sides are congruent, but opposite sides are not congruent.
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03 Aug 2017, 08:12
Those haven't understood the solution. Pls refer to the below link

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Re: M11-05   [#permalink] 03 Aug 2017, 08:12
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# M11-05

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# NAG CL Interfaced03pyc (dim1_parab_coll_interp)

Settings help

CL Name Style:

## 1Purpose

d03pyc may be used in conjunction with either d03pdc or d03pjc. It computes the solution and its first derivative at user-specified points in the spatial coordinate.

## 2Specification

 #include
 void d03pyc (Integer npde, const double u[], Integer nbkpts, const double xbkpts[], Integer npoly, Integer npts, const double xp[], Integer intpts, Integer itype, double up[], double rsave[], Integer lrsave, NagError *fail)
The function may be called by the names: d03pyc, nag_pde_dim1_parab_coll_interp or nag_pde_interp_1d_coll.

## 3Description

d03pyc is an interpolation function for evaluating the solution of a system of partial differential equations (PDEs), or the PDE components of a system of PDEs with coupled ordinary differential equations (ODEs), at a set of user-specified points. The solution of a system of equations can be computed using d03pdc or d03pjc on a set of mesh points; d03pyc can then be employed to compute the solution at a set of points other than those originally used in d03pdc or d03pjc. It can also evaluate the first derivative of the solution. Polynomial interpolation is used between each of the break-points ${\mathbf{xbkpts}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nbkpts}}$. When the derivative is needed (${\mathbf{itype}}=2$), the array ${\mathbf{xp}}\left[{\mathbf{intpts}}-1\right]$ must not contain any of the break-points, as the method, and consequently the interpolation scheme, assumes that only the solution is continuous at these points.

None.

## 5Arguments

Note: the arguments u, npts, npde, xbkpts, nbkpts, rsave and lrsave must be supplied unchanged from either d03pdc or d03pjc.
1: $\mathbf{npde}$Integer Input
On entry: the number of PDEs.
Constraint: ${\mathbf{npde}}\ge 1$.
2: $\mathbf{u}\left[{\mathbf{npde}}×{\mathbf{npts}}\right]$const double Input
Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On entry: the PDE part of the original solution returned in the argument u by the function d03pdc or d03pjc.
3: $\mathbf{nbkpts}$Integer Input
On entry: the number of break-points.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
4: $\mathbf{xbkpts}\left[{\mathbf{nbkpts}}\right]$const double Input
On entry: ${\mathbf{xbkpts}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nbkpts}}$, must contain the break-points as used by d03pdc or d03pjc.
Constraint: ${\mathbf{xbkpts}}\left[0\right]<{\mathbf{xbkpts}}\left[1\right]<\cdots <{\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$.
5: $\mathbf{npoly}$Integer Input
On entry: the degree of the Chebyshev polynomial used for approximation as used by d03pdc or d03pjc.
Constraint: $1\le {\mathbf{npoly}}\le 49$.
6: $\mathbf{npts}$Integer Input
On entry: the number of mesh points as used by d03pdc or d03pjc.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$.
7: $\mathbf{xp}\left[{\mathbf{intpts}}\right]$const double Input
On entry: ${\mathbf{xp}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{intpts}}$, must contain the spatial interpolation points.
Constraints:
• ${\mathbf{xbkpts}}\left[0\right]\le {\mathbf{xp}}\left[0\right]<{\mathbf{xp}}\left[1\right]<\cdots <{\mathbf{xp}}\left[{\mathbf{intpts}}-1\right]\le {\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$;
• if ${\mathbf{itype}}=2$, ${\mathbf{xp}}\left[\mathit{i}-1\right]\ne {\mathbf{xbkpts}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{intpts}}$ and $\mathit{j}=2,3,\dots ,{\mathbf{nbkpts}}-1$.
8: $\mathbf{intpts}$Integer Input
On entry: the number of interpolation points.
Constraint: ${\mathbf{intpts}}\ge 1$.
9: $\mathbf{itype}$Integer Input
On entry: specifies the interpolation to be performed.
${\mathbf{itype}}=1$
The solution at the interpolation points are computed.
${\mathbf{itype}}=2$
Both the solution and the first derivative at the interpolation points are computed.
Constraint: ${\mathbf{itype}}=1$ or $2$.
10: $\mathbf{up}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array up must be at least ${\mathbf{npde}}×{\mathbf{intpts}}×{\mathbf{itype}}$.
the element ${\mathbf{UP}}\left(i,j,k\right)$ is stored in the array element ${\mathbf{up}}\left[\left(k-1\right)×{\mathbf{npde}}×{\mathbf{intpts}}+\left(j-1\right)×{\mathbf{npde}}+i-1\right]$.
On exit: if ${\mathbf{itype}}=1$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$, contains the value of the solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$, at the interpolation points ${x}_{\mathit{j}}={\mathbf{xp}}\left[\mathit{j}-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{intpts}}$ and $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
If ${\mathbf{itype}}=2$, ${\mathbf{UP}}\left(\mathit{i},\mathit{j},1\right)$ contains ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{\mathrm{out}}\right)$ and ${\mathbf{UP}}\left(\mathit{i},\mathit{j},2\right)$ contains $\frac{\partial {U}_{\mathit{i}}}{\partial x}$ at these points.
11: $\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$double Communication Array
The array rsave contains information required by d03pyc as returned by d03pdc or d03pjc. The contents of rsave must not be changed from the call to d03pdc or d03pjc. Some elements of this array are overwritten on exit.
12: $\mathbf{lrsave}$Integer Input
On entry: the size of the workspace rsave, as in d03pdc or d03pjc.
13: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_EXTRAPOLATION
Extrapolation is not allowed.
NE_INCOMPAT_PARAM
On entry, ${\mathbf{itype}}=2$ and at least one interpolation point coincides with a break-point, i.e., interpolation point no $⟨\mathit{\text{value}}⟩$ with value $⟨\mathit{\text{value}}⟩$ is close to break-point $⟨\mathit{\text{value}}⟩$ with value $⟨\mathit{\text{value}}⟩$.
NE_INT
On entry, ${\mathbf{intpts}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{intpts}}\ge 1$.
On entry, ${\mathbf{itype}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{itype}}=1$ or $2$.
On entry, ${\mathbf{nbkpts}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
On entry, ${\mathbf{npde}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npde}}>0$.
On entry, ${\mathbf{npoly}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npoly}}>0$.
NE_INT_3
On entry, ${\mathbf{npts}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{nbkpts}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{npoly}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING
On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$, ${\mathbf{xbkpts}}\left[\mathit{i}-1\right]=⟨\mathit{\text{value}}⟩$, $\mathit{j}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{xbkpts}}\left[\mathit{j}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{xbkpts}}\left[0\right]<{\mathbf{xbkpts}}\left[1\right]<\cdots <{\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$.
On entry, $\mathit{i}=⟨\mathit{\text{value}}⟩$, ${\mathbf{xp}}\left[\mathit{i}-1\right]=⟨\mathit{\text{value}}⟩$, $\mathit{j}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{xp}}\left[\mathit{j}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{xbkpts}}\left[0\right]\le {\mathbf{xp}}\left[0\right]<{\mathbf{xp}}\left[1\right]<\cdots <{\mathbf{xp}}\left[{\mathbf{intpts}}-1\right]\le {\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}-1\right]$.

## 7Accuracy

See the documents for d03pdc or d03pjc.


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• CommentTimeJan 4th 2012
• (edited Jan 4th 2012)

but étale homotopy does not just handle the homotopy groups so this may be too drastic! It describes the THEORY as well so the title is now inaccurate. The subject as put forward by Artin and Mazur really centers on the profinite completion functor and other related things. there need to be links here but your change may not be optimal.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 5th 2012
• (edited Jan 5th 2012)

Like many things on the $n$Lab, this is not optimal, but for the moment I think it is better to have the redirect than to not have it.

But notice that the entry is also not just about homotopy groups. The main restriction is rather that the entry talks about the locally $\infty$-connected case (which is also the reason why the need for pro-completion disappears).

Somebody should write the generalization to the locally connected case that is usually considered in the context of étale homotopy.

(It keeps making me wonder, though: étale homotopy of topological spaces comes out wrong if the space is not locally $\infty$-connected. This makes me feel that assuming local $\infty$-connectedness generally is “good”. But maybe that is too naive.)

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeJan 5th 2012

What is ‘wrong’? Is it not just shape theory? and that corresponds to nice things for C^*-algebras.

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeJan 5th 2012
• (edited Jan 5th 2012)

Strictly étale homotopy should not make that much sense for ordinary spaces as it refers to the homotopy theory relative to the étale topology and that is not a topological notion in its origin.

But notice that the entry is also not just about homotopy groups. The main restriction is rather that the entry talks about the locally ∞-connected case

But the title refers to homotopy groups not homotopy types! The restriction is then too strong as the non-locally $\infty$-connected case is also important.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJan 5th 2012
• (edited Jan 5th 2012)

What is ‘wrong’?

The homotopy groups of a topological space are those computed by its étale homotopy only if the space is locally contractible. This is the original result of Artin-Mazur. The entry geometric homotopy groups in an (infinity,1)-topos also explains “why” this is so: only the in the locally $\infty$-connected case is the corresponding $\infty$-topos locally $\infty$-connected and then the central statement of étale homotopy / higher Galois theory simply follows from the $(\Pi \dashv Disc)$-adjunction.

Strictly étale homotopy should not make that much sense for ordinary spaces

It makes sense for every locally connected site and every locally connected topos. See

• Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995)

for a review (or, for that matter, the above $n$Lab entry).

But the title refers to homotopy groups not homotopy types!

Sure, things could be written in a better way. But for the moment I neverthess think that “étale homotopy” should redirect to that entry, because that entry discusses étale homotopy.

Eventually somebody will find the time to beautify the corresponding $n$Lab entries further. There are lots of points that would deserve improvement.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeJan 5th 2012

I’m afraid I continue to disagree. You say ’that entry discusses étale homotopy’ but it only discusses étale homotopy in a very special case, not in general. It does discuss the étale homotopy of locally $\infty$-connected sites, but the subject of étale homotopy is not restricted to such sites. so this should be clear from the entry.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 5th 2012

I’m afraid I continue to disagree.

No, I agree with that! :-) As I said in #3.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeJan 5th 2012

When I have some time I will see if I can come up with some suggestions for redistribution of material so that various other bits of work (e.g. work of Isaksen and Quick) will fit in… but it can wait!! Happy New Year.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeJan 10th 2012

@Tim: would it be better if étale homotopy redirected to shape of an (infinity,1)-topos?

Probably eventually, étale homotopy theory should be its own page. Maybe it would be worth making it a stub right now.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 10th 2012

Right, I should have done that right away. I tried to save myself the time, but that strategy didn’t work out ;-).

Here are some first words: étale homotopy.

To be expanded (by whoever feels like expanding!).

• CommentRowNumber12.
• CommentAuthorTim_Porter
• CommentTimeJan 11th 2012
• (edited Jan 11th 2012)

No. 9 I think the étale homotopy should have its own page with a link to shape etc. I have some material that can be adapted, but not for today. :-(

No. 10 Urs: that looks a good start.

• CommentRowNumber13.
• CommentAuthorTim_Porter
• CommentTimeJan 11th 2012

On the other hand, the new sentence you added at nerve theorem sort of does not make sense. I have suggested an amednment there , so check that you are ok with it.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJan 11th 2012
• (edited Jan 11th 2012)

Hi Tim,

I can live with your version.

But I am not sure why the previous version “sort of does not make sense”.

The point that I think is important here is that for paracompact spaces the limit over hypercovers that one has to take in the explicitl definition of etale homotopy type localizes at the Cech nerve of any good cover (since these always exist and are already cofibrant in the projective model structure!).

In this sense the nerve theorem is an avatar of the full definition in a simple special case.

• CommentRowNumber15.
• CommentAuthorTim_Porter
• CommentTimeJan 11th 2012

I found it did not work ’linguistically’, but I also think that the nerve theory is not central to étale homotopy. That grew out of the corresponding cohomology theory and includes profinite completion, links with Galois theory, use of pro-objects etc., and the study of objects for which the nerve theorem does not apply as they do not have good covers. They are not even topological spaces. The sentence therefore ended up being too restrictive and might have put some people off (e.g. Algebraic geometers).

It is interesting to Google on étale homotopy and see what areas it is being applied to, e.g. a recent paper (ETALE HOMOTOPY AND SUMS-OF-SQUARES FORMULAS, DANIEL DUGGER AND DANIEL C. ISAKSEN ). There is also work on étale homotopy type of Voevodsky spaces about 10 years ago, Geirion QUick’s work on profinite homotopy and étale realisations, and Jon Pridhams (http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.0928v4.pdf), and that is just picking at random. But the effect of your sentence was almost to suggest that étale homotopy theory really was about classical homotopy types. You may not have intended that, or you may feel that by going to the $\infty$-topos context they all become somehow comparable but I think the Artin-Mazur approach suitably modernised deserves a better ’press’ in the nLab than it has had so far…. hence my change.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJan 11th 2012
• (edited Jan 11th 2012)

I also think that the nerve theory is not central to étale homotopy. […] and the study of objects for which the nerve theorem does not apply as they do not have good covers. They are not even topological spaces. The sentence therefore ended up being too restrictive and might have put some people off (e.g. Algebraic geometers).

I don’t understand why you think any of this is in contradiction with a sentence that briefly asserted “that we have here a special case” of étale homotopy.

Notice also that your modification of the sentence still says that. Also this very statement is currently the lead-in at etale homotopy itself, which you called a “good start”.

So I don’t understand what you say is wrong. But since we seem to agree now on what is in the entries, it’s not so important. Let’s leave it at that.

• CommentRowNumber17.
• CommentAuthorTim_Porter
• CommentTimeJan 11th 2012

Fine by me.

• CommentRowNumber18.
• CommentTimeApr 15th 2013
• (edited Apr 15th 2013)

• Tomer Schlank, Alexei Skorobogatov, A very brief introduction to étale homotopy. In: “Torsors, étale homotopy and applications to rational points”. LMS Lecture Note Series 405, Cambridge University Press, 2013. (pdf)
• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeApr 16th 2013

Thanks. That looks useful.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJun 5th 2014

added two kinds of examples related to étale (non-)contractibiity.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJun 7th 2014

I have added some more of the fundamental references with brief comments.

In the course of this I also added a remark to analytification on Relation to etale homotopy type. (But that whole entry on analytification is in need of attenion.)

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeNov 25th 2015


text:
# How to sample uniformly from an intersection of simplices?

The title says it all. Specifically: given strictly positive real numbers $a_1,\dots,a_T$ and $b_1,\dots,b_T$, I want to sample from $$\mu:=\text{Uniform}(\{p\in[0,\infty)^T : \sum_{t=1}^T a_t p_t = 1\}\cap\{p\in[0,\infty)^T : \sum_{i=1}^T b_t p_t = 1\}),$$ assuming that the intersection is nonempty.

My best attempt so far is the following rejection sampling algorithm:

1. Set $u_t = \min\{\frac{1}{a_t},\frac{1}{b_t}\}$ for $t=1,\dots,T-1$.
2. Sample $p_t \sim \text{Uniform}(0,u_t)$ for $t=1,\dots,T-1$.
• (Justification: if $p_t>u_t$ then $\max\{a_t,b_t\} p_t > \max\{a_t,b_t\}\min\{\frac{1}{a_t},\frac{1}{b_t}\} = 1$, so $p_t$ cannot be the $t$th coordinate of a sample from $\mu$.)
3. Set $s_a = \sum_{t=1}^{T-1} a_t p_t$ and $s_b = \sum_{t=1}^{T-1} b_t p_t$.
4. If $s_a>1$ or $s_b>1$, reject.
5. If $\frac{1}{a_T}(1-s_a) \neq \frac{1}{b_T}(1-s_b)$, reject.
6. Else, set $p_T = \frac{1}{a_T}(1-s_a)$ and accept $p=(p_1,\dots,p_T)$.

The trouble is that in step 5, we will reject with probability 1.

Does anyone know how to do this properly, either using something like the above or via a totally different approach? Thanks in advance for any ideas!

I think the problem lies in step 1 & 2. Setting $u_1$ in this way is fine, but once you have already sampled $p_1$, this places further constraints on what $p_2,\dots,p_T$ can be. So you should really change step 1 to be (defining $\sum_{r=1}^{0}y_r=0$):

$$u_t=\min\left(\frac{1-\sum_{r=1}^{t-1}a_rp_r}{a_t},\frac{1-\sum_{r=1}^{t-1}b_rp_r}{b_t}\right)$$

And we have a modified version of your justification at step 2:

$$max(a_t,b_t)p_t>max(a_t,b_t)u_t=\min\left(1-\sum_{r=1}^{t-1}a_rp_r,1-\sum_{r=1}^{t-1}b_rp_r\right)$$

Which leads to the same conclusion (can't be inside the sample).

And further, you have 2 constraints, so that you should not sample $p_{T-1}$ and $p_T$. So we shall "redefine" $s_a=\sum_{t=1}^{T-2}a_tp_t$ and $s_b=\sum_{t=1}^{T-2}b_tp_t$, and we have two equations:

$$1-s_a=a_{T-1}p_{T-1}+a_Tp_T$$ $$1-s_b=b_{T-1}p_{T-1}+b_Tp_T$$

Or, in matrix form:

$$\begin{pmatrix}1-s_a\\1-s_b\end{pmatrix} = \begin{pmatrix} a_{T-1} & a_{T} \\ b_{T-1} & b_{T} \end{pmatrix} \begin{pmatrix}p_{T-1}\\p_T\end{pmatrix}$$

Which has a unique solution if $a_{T-1}b_{T}-b_{T-1}a_{T}\neq 0$. You can choose your categories for $T,T-1$ so that this condition is satisfied, because of the permutational symmetry in the problem. If you can't, then I think (but not sure) this means you really only have one constraint, and one of them can be dropped (as all of the b's are essentially functions of the a's in a similar manner to an autoregressive equation). The solution, if it exists, is given by:

$$\begin{pmatrix}p_{T-1}\\p_T\end{pmatrix} = \begin{pmatrix} a_{T-1} & a_{T} \\ b_{T-1} & b_{T} \end{pmatrix}^{-1} \begin{pmatrix}1-s_a\\1-s_b\end{pmatrix}$$ $$=\frac{1}{a_{T-1}b_{T}-b_{T-1}a_{T}} \begin{pmatrix} b_{T} & -a_{T} \\ -b_{T-1} & a_{T-1} \end{pmatrix} \begin{pmatrix}1-s_a\\1-s_b\end{pmatrix}$$ $$\implies \begin{matrix}p_{T-1} & = & \frac{(1-s_a)b_T-(1-s_b)a_{T}}{a_{T-1}b_{T}-b_{T-1}a_{T}}\\ p_{T} & = & \frac{(1-s_a)b_{T-1}-(1-s_b)a_{T-1}}{a_{T-1}b_{T}-b_{T-1}a_{T}}\end{matrix}$$

This shows that you don't need to do rejection sampling at all for this problem