fixedpoint or fixed point or fixed-point

2013-05-22T08:56:15Z

I am unsure which is the right spelling (if there even is a ‘right’ spelling), but maybe native speakers can enlighten me: When should I use

fixed point
fixed-point
fixedpoint

when I refer to the point itself, but also in composite works (“fixed point equation”, “fixed-point juggling”, “fixed-point operator”)?

And do the same rules apply to prefixed points?

simple explaination of simplicial volume=4g-4 when genus $\ge 1$

2013-05-22T09:32:56Z

In Gromov's famous book ,it says "simplical volume of every oriented surface of genus $ \ge 1$ satisfies${\left\| {\left[ S \right]} \right\|_\Delta } = 4g - 4 = - 2\chi \left( S \right) = - 2({k_0} - {k_1} + {k_2})$(Euler characteristic in this condition is negative).One proof is to use propertionality theorem for compact locally homogeneous space V,${\left\| {\left[ V \right]} \right\|_\Delta } = cvol\left( V \right)$,where the constant c depends on the local geometry of V.But Gromov says it's elementary!Since the simplicial volume is the infimum of the number of simplices over all homotopy triangulations of V.It's easy to see that "we need at least 4g-4 simplices to triangulate S?Or for a surface with negative Euler characteristic,${k_2} \ge - 2\chi \left( S \right)$?I can't see it.Please give an explaination.

Does this qualify as "self plagiarism" or something?

2013-05-22T09:00:26Z

Over the last few years, I have been writing several papers in the same direction as part of a research program. This means that the same exact setup is introduced at the beginning of each of my papers: i.e. the basic assumptions on the categories I am using, the basic terminology and notions, etc. As you can imagine, there are only so many ways of stating the same exact assumptions over and over again in each paper and now the first one and half pages of all my papers are beginning to look very much alike. I am a little worried: is this some kind of misconduct? Even though the content of my papers is different, is this "self plagiarism" or something when the first part of the first section looks almost identical to that in other papers?

I should mention here that by "first part of the paper", I am not talking about the introduction. Obviously, each paper has different motivations and different results and hence different introductions. I am talking about the first page of the body of the paper, where you put in stuff like "Let C be a category satisfying ....yada yada yada...and we will denote this operad by this and so on..."

Probability $k$ bins are non-empty.

2013-05-21T08:54:38Z

The following problem arises in the analysis of Bloom filters.

Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at random and asks if they are all non-empty.

The main questions asked are "What is the probability that all $k$ bins in the query are non-empty?" and from there "For what $k$ is this probability minimized?". It is assumed that $k$ should be a function of $m$ and $n$.

The standard version of the analysis taught the world over and reproduced in the wikipedia page linked above contains a "now the magic occurs" step which ignores the non-independence of the bins.

Is there a clean and rigorous way of doing this analysis correctly?

Dirac measures dense in space of measures?

2013-05-22T08:42:45Z

Let $I$ be a compact interval and $\mathcal{M}(I)$ the space of (signed) Borel measures. We equip it with the weak topology, i.e. a sequence $\mu_n$ converges to zero if and only if $$ \left|\int_I f(x) \mathrm{d}\mu_n(x)\right| \longrightarrow 0$$ for all $f \in C(I)$.

Now the question is the following: Let $V \subset \mathcal{M}(I)$ be the vectorspace of all finite linear combinations of Dirac measures supported at different points in $I$. Is $V$ dense in $\mathcal{M}(I)$?

For example if $I = [0,1]$, the sequence $$ \mu_n = \frac{1}{N}\sum_{j=1}^N \delta_{j/N},$$ $\delta_{j/N}$ being the Dirac measure supported at $j/N$, weak*-converges to the Lebesgue measure as $\mu_n$ is just the approximation by Riemann sums. Hence one can easily get all measures that are absolutely continuous w.r.t. the Lebesgue measure.

However, there are more measures (singular measures) that are neither point measures nor Lebesgue measures and I don't have an idea how to reach those.

What are some applications of other fields to mathematics?

2010-02-09T17:05:20Z

It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:

What are some applications of other fields to mathematics?

Obviously the applications of physics to mathematics are ubiquitous (gauge theory is just one significant modern example, and quantum algorithms and mirror symmetry are others...the list from physics goes on). For the purposes of this question (at least) theoretical computer science is just a branch of mathematics.

So answers involving fields other than physics are of particular interest to me (and answers involving theoretical computer science are of little to no interest to me), as are answers where the application isn't bidirectional (for example, one could say that game theory is an application of mathematics to economics as much if not more than an application of economics to mathematics).

Finally (at least for the purposes of this question), anything of the form "phenomenon Y was experimentally observed and it turned out that there was a rich but hitherto unknown mathematical theory Z explaining Y" is not that interesting as an application to mathematics unless the discovery of Z has some truly special status. Something like (e.g.) symplectic geometry might fall under this (leaving aside the "experimental" bit), but is not of particular interest for reasons above.

Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

2013-05-22T08:46:19Z

Hi.

Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this convergence is that for any continuous function $\varphi\in\mathscr{C}^0([-1,1])$,

\begin{align*} \int_{-1}^1 f_n \varphi \operatorname*{\longrightarrow}_{n\rightarrow +\infty} \int^{1}_{-1} f \varphi. \end{align*}

It is easy to see that $f$ is necessarily also non-negative.

Question : do we also have $(f_n)_n \rightharpoonup f$ in $L^1([-1,1])-w$, i.e. can we replace in the previous convergence the continuous function $\varphi$ by any element of $L^\infty([-1,1])$ ?

I am quite sure that the answer is no (because there is no density of regular functions in $L^\infty$), but I did not manage to find a counterexample.

Two remarks :

1) Without the assumption of non-negativeness one may consider $f_n:= n\mathbf{1}_{[0,1/n]} - n\mathbf{1}_{[-1/n,0]}$ , which, in some sense , approximates $ " \delta_{0^+}-\delta_{0-} "$, and hence $(f_n)_n \operatorname*{\rightharpoonup}^{\mathscr{M}-\star} 0$, but it may be checked easily that $(f_n)_n$ do not converge to $0$ weakly in $L^1$ : the sequence "concentrates" the eventual discontinuity in $0$.

2) Keeping the non-negativeness but working on the open set $(-1,1)$ instead of $[-1,1]$ simplifies also the problem, since the mass may then concentrate to the boundary : $f_n:= n \mathbf{1}_{[1-1/n,1[}$ tends to $0$ in $\mathscr{M}^1(-1,1)-w\star$ (test functions in $\mathscr{C}^0_c(-1,1)$) but clearly not in $L^1-w$.

Thanks in advance for any advice !

Ayman

What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ?

2010-05-27T16:07:49Z

Let $P,Q$ be $n$ by $n$ invertible matrices. Suppose further that $P$ and $Q$ satisfies the following equation :

$$(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$$

where $\circ$ denotes the Hadamard matrix product, which is simply the entrywise product.

Then what can be said about $P$ and $Q$? More precisely, I want to know if there are additional relations between $P$ and $Q$. For example, one can show that the condition $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ implies

$$tr(P^{-1}DPE) = tr(Q^{-1}DQE)$$ for all diagonal matrices $D$ and $E$.

References in the litterature about matrices of the form $(P^{-1})^T \circ P$ would help too. Thank you, Malik

Permutations of $(Z/pZ)^*$

2013-05-20T15:08:11Z

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$. Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of $(\mathbb Z/p \mathbb Z)^*$.

Say that a map $a:(\mathbb Z/p \mathbb Z)^*\to S((\mathbb Z/p \mathbb Z)^*)$ satisfies condition (A) if, for any two distinct elements $i,j\in (\mathbb Z/p \mathbb Z)^*$ , $a(i)-a(j)\in S((\mathbb Z/p \mathbb Z)^*)$.

For example, let $a(i)(k) = ik.$ This satisfies condition (A). The same is true if we permute the functions $a'(i) = a(c(i))$, or relabel the objects $a''(i)(k) = i \cdot b(k)$, or both. Are these modifications of $a(i)(k) = ik$ the only ways to get a map satisfying condition (A)?

If $a$ satisfies (A), are there $b,c\in S((\mathbb Z/p \mathbb Z)^*)$ such that, for all $i\in (\mathbb Z/p \mathbb Z)^*$ and all $k\in (\mathbb Z/p \mathbb Z)^*$, $a(i)(k)=c(i)\cdot b(k)$, where the dot is multiplication in $\mathbb Z/p \mathbb Z$?

Note: it would probably be sufficient to prove that, if $a$ satisfies (A), then, for all $i,j\in (\mathbb Z/p \mathbb Z)^*$, $a(i)$ and $a(j)$ commute.

edit I've corrected the question -- and the paragraph before it -- thanks to comments by François Brunault and Victor Protsak, who noted that the original formulation was incorrect due to an irrelevant $b^{-1}$.

definition of a weakly doubly transitive group action

2013-05-22T07:57:34Z

I'm reading Francis M. Choucroun, "Analyse harmonique des groupes d'automorphismes d'arbres de Bruhat-Tits", Mémoires de la S. M. F., tome 58 (1994), p. 1 - 166, and he speaks of a weakly doubly transitive or weakly triply transitive group action (in French). My first thought was that maybe this means that the group acts transitively on the unordered pairs or unordered tripes, but when you actually look at the arguments involving the notion that interpretation seems not to be supported. I was wondering if anyone could clarify what the meaning of "weakly doubly transitive" is in this paper.

Exact solutions to nonlinear Klein-Gordon equation

2013-05-22T08:13:52Z

The nonlinear pde $$ \partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0 $$ has the exact solution $$ \phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i) $$ with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the dispersion relation holds $$ p^2_0=p^2+\mu^2\sqrt{\frac{\lambda}{2}}. $$ If I interpret $p_0$ as the energy, it seems that is finite. Computing the integral $$ E=\int d^Dx\left[\frac{1}{2}(\partial_t\phi)^2+\frac{1}{2}(\partial_x\phi)^2+\frac{\lambda}{4}\phi^4\right] $$ and extending the volume to infinity, this is divergent. Can one find a sound mathematical explanation for this? Is there a way to "regularize" this integral?

Thanks.

Smith Normal Form of powers of a matrix

2013-05-21T18:46:24Z

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?

The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a parameterization of the double coset space $G\backslash M/G$ and I am asking about the image of a sequence $(y^i : i \geq 1)$ under the projection $M \longrightarrow G \backslash M/G$. Clearly this factors through the quotient of $M$ under conjugation by $G$.

It is easy to produce examples to show that the characteristic polynomial is insufficient, even for 2x2 matrices.

The reason I care is that, in computing local cohomology groups for graded one dimensional rings, one often comes across a ring $S$, free as a module over a PID $R$, and an element $y \in S$ for which one wants to know all the quotients $S/(y^i)$ explicitly as an $R$-module.

If we let $A_i$ be the cokernel of $y^i$, then we have short exact sequences $ 0 \longrightarrow A_i \longrightarrow A_{i+j} \longrightarrow A_j \longrightarrow 0$ relating these quotients (at least when $det(y) \neq 0$).

I conjecture that for any $y \in M$ there exist an integer $d > 0$ and a diagonal matrix $D$ such that $SNF(y^{i+d}) = D*SNF(y^i).$

The work on the possible values of SNF(AB), given SNF(A) and SNF(B), masterfully recounted in Fulton's "Eigenvalues, invariant factors, highest weights, and Schubert calculus", Bull. AMS 37 (2000), no. 3, 209–249, is probably relevant, though $SNF(A^i)$ is in some sense the worst case, since $SNF(AB)$ is most constrained when $det(A)$ and $det(B)$ share few factors. If I understood that work, perhaps I would know that the answer is already known.

I should note that nothing crucial depends on this question: I am simply curious.

Embedded associated prime and non zero divisor

2013-05-21T06:26:27Z

$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is eqidimensional and $M$ does not have any embedded prime.

Given $x\in I$ where $I$ is an ideal of $A$ and dim $\frac{G(M)}{{x^*}G(M)}$ $< d$. Then show that x is non zero divisor of $M$.

Which Kahler Manifolds are also Einstein Manifolds?

2011-03-07T13:53:01Z

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?

What is the closed-form or the deterministic form of a quadratic form probability inequality

2013-05-21T10:30:46Z

Hello, everyone, I want to resolve one optimal problem, with the following probability inequality constraint. $Pr(h^H(W_1 - W_2 -W_3 -U)h \geq \sigma^2) \leq \rho$

where $h \sim CN(0,I) \ \text{is random vector}, I \ \text{is a identity matrix};\ w_i \in C^{L\times 1}, W_i = w_iw_i^H; \ $ $U$ is a idenpotent matrix,i.e., $U^2 = U\in H^{L \times L} \ \text{and symmetric},\text{Rank}(U)=r < L. $

I find that the probability inequality can be converted into a deterministic form using the Proposition 1.1 in link text. I want to know whether it has a closed-from expression and how to get it, or whether there is another method that can be used to convert it as a deterministic form?

P.S. I know that $h^HW_ih$ following the exponential distribution, and $h^HUh$ following the $\chi^2$ distribution with $r$ degrees of freedom. But they are not independent, because the $w_i$ is the optimal variable, and they are related with each other in the optimal problem.

In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?

2013-05-22T05:08:50Z

I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems are equivalent (to any proof of Theorem A, prepend "Assume Theorem B", and vice versa; the objection "But the proof of Theorem A doesn't really use the assumption that Theorem B holds" seems more psychological than mathematical).

One might try to formalize the notion of equivalence by considering the lengths of proofs, saying "There is a derivation of Theorem A from Theorem B that is significantly shorter than any proof of Theorem A from scratch, and vice versa", but this too is squishy, in two distinct ways: the length of a proof depends on the formalization procedure one chooses, and "significantly shorter" is vague. Moreover, it's hard to imagine how one could work with this notion of equivalence, since the totality of all short proofs is going to be hard to get a handle on, for the usual reasons.

Can one find some sort of mathematical context (a topos, perhaps?) in which there is a rigorously defined (and not vacuously true) meaning of the equivalence between Sperner and Brouwer?

(For a recent article that discusses this equivalence and gives pointers to relevant literature, see "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" by Nyman and Su in the April 2013 issue of the American Mathematical Monthly.)

Are small knots generic?

2013-05-21T08:00:40Z

A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ crossings, the proportion of small knots goes to 1 when $n$ goes to infinity?

Is deciding whether a Turing machine provably runs forever equivalent to the halting problem?

2013-05-22T02:03:49Z

Assume for this question that ZF set theory is sound.

Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF proof that M runs forever on a blank input.

It's clear that PROVELOOP is recursively-enumerable, and hence reducible to the halting problem. I can also prove that PROVELOOP is undecidable (details below). But I can't see how to prove that PROVELOOP is Turing-equivalent to the halting problem! (This is contrast to, say, the set of all descriptions of Turing machines that provably halt, which is just the same thing as the set of all descriptions of Turing machines that do halt!)

I'm guessing that there's a reduction from HALT that I haven't thought of, though it would be exciting if PROVELOOP were to have intermediate degree like the Friedberg-Muchnik languages. In any case, whatever the answer, I assume it must be known! Hence this question.

Proof that PROVELOOP is undecidable. Consider the following problem, which I'll call "Consistent Guessing" (CG). You're given as input a description of a Turing machine M. If M accepts given a blank input, then you need to accept, while if M rejects you need to reject. If M runs forever, then you can either accept or reject, but in either case you must halt.

By adapting the undecidability proof for HALT, we can easily show that CG is undecidable also. Namely, suppose P solves CG. Let Q take as input a Turing machine description $\langle M \rangle$, and solve CG for the machine $M(\langle M \rangle)$ by calling P as a subroutine. Then $Q(\langle Q \rangle)$ (i.e., Q run on its own description) must halt, accept if it rejects, and reject if it accepts.

Let's now prove that CG is Turing-reducible to PROVELOOP. Given a description of a Turing machine M for which we want to solve CG, simply create a new Turing machine M', which does the same thing as M except that if M accepts, then M' goes into an infinite loop instead. Then if M accepts, then M' loops, and moreover there's a ZF proof that M' loops. On the other hand, if M rejects, then M' also rejects, and there's no ZF proof that M' loops (by the assumption that ZF is sound). If M loops, then there might or might not be a ZF proof that M' loops -- but in any case, by calling PROVELOOP on M', we separate the case that M accepts from the case that M rejects, and therefore solve CG on M. So $CG \le_{T} PROVELOOP$, and PROVELOOP is undecidable as well.

One more note. In the comments of this blog post, Andy Drucker supplied a proof that CG is not equivalent to HALT, but rather has Friedberg-Muchnik-like intermediate status. So, the situation is

$0 \lt_{T} CG \le_{T} PROVELOOP \le_{T} HALT$

with at least one of the last two inequalities strict. Again, I'm sure this is all implicit in some computability paper from the 1960s or something, but I wouldn't know where to find it.

Are there any "related rates" calculus problems that don't feel contrived?

2011-12-10T02:09:04Z

I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the following (taken, more or less, from the textbook):

A man starts walking due north at 5 ft/sec from a Point A. Ten seconds later, a woman starts walking south at 4 ft/sec from a point 20 ft due east of Point A. How fast are they moving apart when the woman has been walking for ten seconds?

A 6' man walks away from a 20' lamppost at a speed of 5 ft/sec. How fast is the distance between the tip of his shadow and the top of the post changing when he is 40' away?

A baseball player runs from first base to second at 20 ft/sec, and simultaneously another baseball player runs from third base to home at the same speed. How fast are they approaching each other after one second?

To put my question bluntly:

Who cares?

My students do, but only because they know these questions will appear on their exams. The baseball question (or something very similar) is actually an exercise in Stewart, and I struggled in vain to imagine a situation in which the manager of a baseball team would need to know the answer.

This is in stark contrast to many other topics addressed in first-year calculus -- optimization, basic differential equations, etc. -- which are realistic models of questions of natural interest in business, biology, etc. Basically, all the related rates questions seemed to be cooked up in response to the fact that calculus students now knew a method to solve them.

My question is in the title. Can anyone share any related rates questions which don't seem quite as contrived, and which might naturally seem interesting and motivated to a typical class of college freshmen?

Thank you!

Techniques for lower-bounding angle between two eigenvectors of a matrix

2010-11-08T19:10:00Z

Are there any techniques for lower-bounding the angle between eigenvectors of a matrix? Or a lower bound on the related quantity of the condition number of the matrix of eigenvectors? In particular I'm looking for bounds that depend on the difference in the corresponding eigenvalues, with larger angles when the eigenvalues are more separated.

For symmetric matrices (and more generally normal matrices) the angles are of course all right angles. I'm looking for techniques that apply to non-normal matrices.

(The particular class of matrices that I care about is stochastic matrices with trace $n-1$ as described in my previous question http://mathoverflow.net/questions/45126/bounds-on-pk1-pk-for-n-by-n-stochastic-matrix-p-with-trace-n .)

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

2013-05-20T13:57:12Z

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

A measure of closure under sumset?

2013-05-21T23:35:34Z

Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its additive energy, defined as $E(A) = |\lbrace(a_1,a_2,a_3,a_4) \in A^4 : a_1 + a_2 = a_3 + a_4 \rbrace|$.

A related quantity that I'm interested in is: $F(A) = |\lbrace (a_1,a_2) \in A^2 : a_1 + a_2 \in A \rbrace|$. It seems to me that $F(A)$ captures the notion of "closed under sumset" more directly. How come $F(A)$ isn't studied more in additive combinatorics? What kinds of statements can one make about the relationship between $F(A)$ and $E(A)$?

In particular, I'm mostly concerned with situations when $G$ is a vector space like $\mathbb{F}^n$ for some finite field $\mathbb{F}$.

Finite rank free modules over PIDs

2013-05-22T01:49:34Z

I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules $\phi:M\rightarrow N$. Under what conditions does the kernel $\ker \phi$ have a complement $C$ in $M$ such that I can write $M=C\oplus \ker\phi$.?

I guess this question might be considered very elementary by many, so I'd also be happy to just be given a reference to a good text book. Searching Google did not turn up anything useful.

Thanks in advance for any replies.

About the curvature of a connection?

2013-05-21T19:20:46Z

In "Lectures on gauge theory and integrable systems" of M.Audin, she identifies the space of conections $\mathcal{A}$ on the trivial bundle $G\times S$ ($G$ Lie group, $S$ surface with boundary) with the vector space $\Omega^1(S,\mathfrak{g})$. Then, for each $A\in\mathcal{A}$ define $$d_A:\Omega^0(S,\mathfrak{g})\rightarrow\Omega^1(S,\mathfrak{g});\alpha\mapsto d\alpha+[A,\alpha]$$ where $\left[A,\alpha\right](X)=[A(X),\alpha]$ for tangent vector to $S$.

Similarly $$d_A:\Omega^1(S,\mathfrak{g})\rightarrow\Omega^2(S,\mathfrak{g});\phi\mapsto d\phi+[A,\phi]$$ with $\left[A,\phi\right](X,Y)=\left[A(X),\phi(Y)\right]-\left[A(Y),\phi(X)\right]$

1.- For $\alpha\in\Omega^0(S,\mathfrak{g})$, she claims $d_A\circ d_A(\alpha)=[d_AA,\alpha]$ then she defines the curvature form as $F(A)=d_AA$. Is not wrong this calculation? (I think should be $d_A\circ d_A(\alpha)=[dA+\frac{1}{2}[A,A].\alpha]$).

2.-Whit her definition of $F(A)$ she asks to show that $(T_AF)(\phi)=d_A\phi$, but I couldn't. So, I should be consider $F(A)=dA+\frac{1}{2}[A,A]?$ I like to understand this.

Iterated Tangent Category Construction

2013-05-21T23:35:11Z

We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions I need on $\mathcal{C}$ for this to make sense). In the case that $\mathcal{C}$ is commutative rings, I can think of this as being the category of square zero extensions of $A$, or equivalently, $A$-modules. Suppose I were to iterate this process. Can I think of iterations as being like higher level "approximations" (i.e. quadratic category, cubic category)? If I iterate it a countable number of times, do I obtain some sort of formal object?

One obvious initial issue with my question is to determine what one means by iterate. I imagine it might make sense to take the tangent category AGAIN at the point $A$, which corresponds to the square zero extension $A[x]/x^2$. At the first level, this obtains all $A$-modules. Also, everything in the category of $A$-modules is already an abelian group, so what do I want to mean by taking abelian group objects over $A$ (as the square-zero extension of itself)? Is it the case that not all $A$-modules are in fact abelian group objects in the overcategory over some particular module?

thanks!

When does a $W^*$-algebra have a standard Borel spectrum?

2013-05-03T12:48:21Z

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual.

This post came out a bit long, but if you're familiar with the topic, you can probably just skim through most of it - I've put the questions in bold. Some terminology:

A (commutative, as it shall always be in this post) $W^* $-algebra is a $C^* $-algebra which has a pre-dual as a Banach space. It is well known (see for example Takesaki's Theory of Operator Algebras I) that such algebras can be equivalently characterized as Von-Neumann subalgebras of the algebra of operators on a Hilbert space, or as spaces of the form $L^\infty \left (X, \mu \right) $ where $X$ is a locally compact space and $\mu$ a Radon measure.
By a standard Borel space I mean a measure (or sometimes just measurable) space which is Borel isomorphic to a complete separable metric space.

I am studying group actions on $W^* $-algebras, and I am interested in particular in the question:

When can $ \left (X, \mu \right) $ above be chosen to be a standard Borel space?

The question is important to me because if the answer is what I expect it to be, I have a very simple way of constructing concrete actions on compact measure spaces from given actions on the associated $W^* $-algebra. (I can say more about the motivation, but I don't want to burden this post with too many details.)

My hypothesis is that the answer to my question is: Exactly when $A$ is separable in the weak-* topology . Certainly the $L^\infty $ of a standard Borel space is weak-* separable: the Borel $\sigma$-algebra on a standard Borel space is countably generated, so rational linear combinations of the associated indicator functions are dense in $L^1$, making it norm-separable and hence its dual weak-* separable.

Here is how I tried to prove this: assume that $A$ is a weak-* separable $W^* $-algebra, and let $B$ be a norm-separable, norm-closed, weak-* dense sub-$C^* $-algebra (just take the norm-closed subspace generated by some weak-* dense countable subset). I denote $A$ and $B$'s Gelfand spectra (i.e., the spaces of multiplicative linear functionals on these algebras) by $X_A$ and $X_B$. Since $A \simeq C(X_A)$ and $B \simeq C(X_B)$, I know that $X_B$ is a complete separable metric space. $X$ above can be constructed as an open dense subset of $X_A$ on which $\mu$ is supported. I want to prove that there is a null set $X_A ^0$ such that $X_A-X_A ^0 $ is Borel isomorphic to $X_B$. If I prove this, I win, because removing a null subset gives an isomorphic $L^\infty$.

So basically, what I want to prove is:

Given a $W^* $-algebra $A$ and a weak-* dense subalgebra $B$, there exists a null subset $X_A ^0 $ such that $X_A-X_A ^0$ is Borel isomorphic to $X_B$ .

It may be that I also have to remove a null subset of $X_B$ - that's just as good for me, although I think it can be avoided.

To prove the last statement, I tried going through the following: I have a natural map from $X_A$ to $X_B$ given by restricting a multiplicative functional to B. This map is definitely continuous (the topology on the Gelfand spectrum is given by pointwise convergence) and, while not a completely trivial fact, it is well known that it is onto (a multiplicative linear functional on a $C^* $-subalgebra can always be extended to the entire algebra; see for example Kaniuth's A Course in Commutative Banach Algebras , theorem 4.2.17). There's no reason, of course, to think that the restriction is one to one: extensions of functionals are in general highly non-unique. However, I believe that by using weak-* density and taking away a null subset of $X_A$, it can be made onto. That would already make the map a Borel isomorphism. The following observation may or may not be helpful: asking if there is a conull subset of $X_A$ on which the restriction is one to one is the same as asking if there is a conull subset on which elements of $B$, viewed as functions, separate points.

Well, I could say more, but this post is already exceedingly long. I have been thinking and looking for information on these questions for quite some time now, so I would warmly welcome any suggestions or comments.

To recap, my questions are:

When can $(X, \mu)$ be chosen to be a standard Borel space?
Is it true that 1 is equivalent to $A$ being weak-* separable?
Is it true that the spectrum of a $W^* $-algebra is always Borel isomorphic to the spectrum of a weak-* dense subalgebra?

Constructing Polynomial Count Varieties

2013-05-21T22:18:26Z

I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$:

Are all reductive algebraic groups strongly polynomial-count?
Are products of strongly polynomial-count varieties also strongly polynomial-count? What about (disjoint) unions?
If X is strongly polynomial-count variety, and F is a finite group acting on X, is X//F also polynomial-count? More generally, is the property of strongly polynomial-count invariant under étale equivalence.
If G is reductive algebraic group acting on a variety X, and the orbit-type stratification of X consists of strongly polynomial count quasi-projective subvarieties, then is X//G also strongly polynomial-count?
Are there general conditions on a variety X and algebraic group G for X//G to be strongly polynomial count?

Basically, I would like to know if there are operations that allows one to cook up polynomial count varieties from other polynomial count varieties.

See the Appendix here for the definition of polynomial count: here

identity for number of monomials

2013-05-21T23:29:16Z

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$.

Let b(n,k) denote the number of monomials of degree kd-(n+1) in n+1 variables x_0,..x_n with all partial degrees $\leq d-2$.

Then I have observed that:

$b(n,k) = {d-1 \choose n}+ \sum_{k'=1}^{n-k} \sum_{n'=k'}^{k'+k-2} b(n',k') {d \choose n-n'}$

(where binomial coefficients are zero if they do not make sense). I was wondering if anyone can see some easy combinatorial argument for why this is true.

To provide some context, these numbers are closely related to the Hodge numbers of smooth projective hypersurfaces of degree $d$, but I am looking for an elementary argument.

Equivariant versus retractive spaces: a reference request

2013-05-21T20:35:41Z

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).

Let $R^G(\ast)$ be the category of based left $G$-spaces (where spaces are taken in $T$). Then $R^G(\ast)$ is a model category in which a fibration and weak equivalence are defined via the forgetful functor to $T$. Cofibrations are defined by the lifting property. It's well-known that this gives a model structure, so I'll take that for granted.

Let $R(BG)$ be the category of spaces containing $BG$ as a retract. Objects are spaces $Y$ equipped with maps $r:Y \to BG$, $s: BG \to Y$ such that $r\circ s : BG \to BG$ is the identity (call $r$ and $s$ structure maps). A morphism $Y \to Y'$ is a map of underlying spaces that preserves the structure maps.

Then $R(BG)$ is a model category in which a fibration, cofibration and weak equivalence are defined using the forgetful functor to $T$. This is due to Quillen.

I think the following is a folklore result:

Assertion: $R^G(\ast)$ and $R(BG)$ are Quillen equivalent.

My question: Does anyone know a concrete reference for this?

Remark: A statement suggesting that the assertion is true in the context of Waldhausen categories appears in Waldhausen's foundational paper in LNM 1126.

Categorical notions involving $\ell_p$ spaces.

2013-05-21T22:54:20Z

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One the local level, which I am not going to define here, it is enough to work with $\mathscr{L}_1$-spaces.)

Similarly, we understand 1-injectivity completely (here the injective objects are $C(X)$-spaces for $X$ extremally disconnected). Again, on the local level we can work with $\mathscr{L}_\infty$-spaces. Let me make then an assignment:

$\mathscr{L}_1$-spaces $\leftarrow $ surjective operators $\leftrightarrow $ quotients

$\mathscr{L}_\infty$-spaces $\leftarrow $ isomorphic embeddings $\leftrightarrow $ closed subspaces

Can we complete this dictionary for $p\in (1,\infty)$:

$\mathscr{L}_p$-spaces $\leftarrow $ ???

Top Questions - MathOverflow

fixedpoint or fixed point or fixed-point

simple explaination of simplicial volume=4g-4 when genus $\ge 1$

Does this qualify as "self plagiarism" or something?

Probability $k$ bins are non-empty.

Dirac measures dense in space of measures?

What are some applications of other fields to mathematics?

Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ?

Permutations of $(Z/pZ)^*$

definition of a weakly doubly transitive group action

Exact solutions to nonlinear Klein-Gordon equation

Smith Normal Form of powers of a matrix

Embedded associated prime and non zero divisor

Which Kahler Manifolds are also Einstein Manifolds?

What is the closed-form or the deterministic form of a quadratic form probability inequality

In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?

Are small knots generic?

Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?

Are there any "related rates" calculus problems that don't feel contrived?

Techniques for lower-bounding angle between two eigenvectors of a matrix

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

A measure of closure under sumset?

Finite rank free modules over PIDs

About the curvature of a connection?

Iterated Tangent Category Construction

When does a $W^*$-algebra have a standard Borel spectrum?

Constructing Polynomial Count Varieties

identity for number of monomials

Equivariant versus retractive spaces: a reference request

Categorical notions involving $\ell_p$ spaces.

Is deciding whether a Turing machine provably runs forever equivalent to the halting problem?