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https://mathoverflow.net/questions/49979 | 3 | Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is
$\rho\_k = (2p-1)^k$
If I take an exponential moving average of this series with weighting parameter $\lambda$, what does the distribution of values of the new ser... | https://mathoverflow.net/users/11727 | Statistics of a simple Markov chain | The answer is certainly not "Gaussian". What you describe is often called Bernoulli convolutions and, even in the independent case (in your setting, $p=1/2$), the limiting object is quite complicated and interesting since it involves some deep number theoretic properties of $\lambda$.
To begin with, let $(X\_n)\_{n\i... | 3 | https://mathoverflow.net/users/4661 | 50002 | 31,420 |
https://mathoverflow.net/questions/49940 | 11 | Let $(e\_j)\_{j\in\mathbb N}$ be an orthonormal basis of a Hilbert space $V$. Let $T:V\to V$ be continuous, selfadjoint linear operator.
Assume that for all $i,j\in\mathbb N$ the number $\langle Te\_i,e\_j\rangle$ is rational.
Let $\lambda\in\mathbb C$. Suppose there exists an eigenvector $v\in V$ with $Tv=\lambda v$.... | https://mathoverflow.net/users/nan | Algebraicity of Eigenvectors in a Hilbert space | The answer is no.
**Update.** Here is an explicit example (which also preserves the $\mathbb Q$-span of the basis). Let $V=\ell\_2$ over $\mathbb R$ and $(e\_j)$ the standard basis. Define $T:V\to V$ by
$$
\begin{aligned}
(Tx)\_1 &= x\_1+3x\_2+x\_3 \\
(Tx)\_2 &= 3x\_1+10x\_2+6x\_3+x\_4 \\
(Tx)\_n &= x\_{n-2}+6x\_{... | 15 | https://mathoverflow.net/users/4354 | 50005 | 31,422 |
https://mathoverflow.net/questions/50006 | 4 | *First a note of caution:* I am a physicist with a rudimentary knowledge of algebraic geometry picked up here and there. So don't assume I know anything besides basic properties of sheaves and try to give as simple answers as possible. Also, if my questions don't make sense for any reason, try to point me in the right ... | https://mathoverflow.net/users/5708 | What is ample generator of a Picard group? | First of all, the Picard group of a variety is not always monogenerated, so that the notion of "the ample generator" you are referring to surely concerns a restricted class of varieties.
Furthermore, an ample line bundle (or invertible sheaf) is a line bundle $L$ which satisfies any of the following properties :
... | 14 | https://mathoverflow.net/users/5659 | 50009 | 31,425 |
https://mathoverflow.net/questions/49981 | 4 | please note that this question deals with *undirected* knots/links!
The most generic cubic skein relation for a knot polynome would be
$$S^2=wvS+w/S+w^2(u-v)I-u\cdot\infty$$
where $w^3$ is one positive writhe unit. The form is fairly obvious from
some self-consistency demands. Now since 20 years or so I try to pr... | https://mathoverflow.net/users/11504 | Cubic skein relations | First off I think there's at least one other knot polynomial satisfying these skein relations: the Reshetikhin-Turaev invariant coming from the 133-dimensional representation of E7. Unfortunately for you, I don't think anyone's ever given an elementary description of that knot polynomial. On the other hand, it would be... | 7 | https://mathoverflow.net/users/22 | 50010 | 31,426 |
https://mathoverflow.net/questions/49831 | 12 | ZF (if consistent) has models where $\omega$ supports no non-principal ultrafilters and others where it supports $2^c$ such. Can other cardinals occur? Or does the existence of just one somehow entail the existence of $2^c$.
| https://mathoverflow.net/users/10909 | ZF and and the number of non-principal ultrafilters | Much of the following is David Feldman's argument in one of the comments, but with the topology replaced by combinatorics. There is, however, one slightly tricky point, arising from the fact that, in the absence of choice, the existence of a surjection from one set to another does not imply that the former set has at l... | 10 | https://mathoverflow.net/users/6794 | 50013 | 31,428 |
https://mathoverflow.net/questions/50004 | 16 | From what I have been reading I understand that it is a part of the Langlands program to express Zeta-function of a Shimura variety associated to the group G in terms of L-functions attached to G. I am not sure to what extent this already has been proven.
In the cases I "know", which are the modular curves and Shimur... | https://mathoverflow.net/users/2260 | L-functions and higher-dimensional Eichler-Shimura relation | Surprisingly, the case of modular curves is misleading! General theory of correspondences, plus the theory of the mod $p$ reduction of curves like $X\_0(Np)$ ($p$ doesn't divide $N$) give a relationship between the Hecke operator $T\_p$ and the Frobenius endomorphism $Frob\_p$ acting on, say, the Jacobian of a modular ... | 24 | https://mathoverflow.net/users/1384 | 50015 | 31,430 |
https://mathoverflow.net/questions/50033 | 29 | Burnside's Lemma states that, given a set $X$ acted on by a group $G$,
$$|X/G|=\frac{1}{|G|}\sum\_{g\in G}|X^g|$$
where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element ... | https://mathoverflow.net/users/1916 | Intuitive explanation of Burnside's Lemma | I'm not sure I'd call this a categorification, but the way I think of Burnside's Lemma is as follows.
Consider the subset $Z \subset G \times X$ consisting of pairs $(g,x)$ such that $g\cdot x =x$, where by $\cdot$ I just mean the action of $G$ on $X$.
The cartesian product $G \times X$ comes with the two surjectio... | 44 | https://mathoverflow.net/users/394 | 50036 | 31,443 |
https://mathoverflow.net/questions/50048 | 10 | Call a function of the following form a *beep*: $e^{-(\frac{x-\alpha}{\beta})^2}\sin(\rho x+\theta)$. Given a real-valued function $f\in L^2(R)$ and a number $n$, I'm interested in the approximating $f$ as closely as possible by a linear combination of $n$ beeps.
Does this particular type of non-linear regression pr... | https://mathoverflow.net/users/10909 | Non-linear "Fourier analysis" | Yes, <http://en.wikipedia.org/wiki/Matching_pursuit#References>
Yes, <http://en.wikipedia.org/wiki/Tanh-sinh_quadrature>
No, I don't think so.
Yes, <http://en.wikipedia.org/wiki/Matching_pursuit#Properties>
Yes, it gives an inner product.
No, I don't think so.
| 2 | https://mathoverflow.net/users/nan | 50052 | 31,452 |
https://mathoverflow.net/questions/50040 | 14 | I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.
It is striking to me that both developments arise, essentially, by studying spaces of functions that satisfy non-linear elliptic equatio... | https://mathoverflow.net/users/11743 | When can Witten-esque moduli spaces be used to define invariants of geometric structures? | One general lesson we can learn from the work of Witten and others is that supersymmetric field theories are a systematic source of deep invariants (in the form of moduli spaces from the classical theories and meaningful linearizations of them from the quantum theories, to oversimplify egregiously). In other words one ... | 20 | https://mathoverflow.net/users/582 | 50069 | 31,467 |
https://mathoverflow.net/questions/50049 | 11 | In a Secret Santa game, each of $n$ players puts their name into a hat and then each player picks a name from the hat, who they buy a Christmas present for. Obviously, if someone picks their own name then they put it back and draw again (if they're the only person remaining in the hat then everyone heaves an exasperate... | https://mathoverflow.net/users/11727 | Secret Santa (expected no of cycles in a random permutation) | One way to calculate the expected number of cycles of length $k$ is by the inclusion-exclusion formula. You can get the set of permuations of length $n$ with no fixed points by taking the set of permutations, subtracting the multiset of permutations with a fixed point at $i$ (for all $i$), adding the multiset of permua... | 14 | https://mathoverflow.net/users/2294 | 50076 | 31,470 |
https://mathoverflow.net/questions/50045 | 0 | I have seen the coradical filtration of a coalgebra $C$ defined as follows:
$C\_0 = \text{sum of all simple subcoalgebras of }C$;
for any $n\geq 1$, let $C\_n$ be $\Delta^{-1}\left(C\otimes C\_{n-1}+C\_{n-1}\otimes C\right)$.
Now I am trying to prove that this really is a coalgebra filtration, and something seems... | https://mathoverflow.net/users/2530 | How is the coradical filtration defined? | I am voting to close this. $C\_n$ should be defined as $\Delta^{-1}\left(C\otimes C\_0+C\_{n-1}\otimes C\right)$. My wrong definition was due to my bad memory. Sorry for spamming.
| 0 | https://mathoverflow.net/users/2530 | 50081 | 31,474 |
https://mathoverflow.net/questions/50080 | 13 | I've been thinking about surgery on complex manifolds. Not very seriously, but just to the point that I think it's odd how there's almost no mention of it in the literature. I figure there's something I'm missing.
In "Complex manifolds and deformations complex structures" Kodaira defines a surgery operation which tak... | https://mathoverflow.net/users/4054 | Surgery in complex geometry | There is an operation in algebraic geometry, called a *flip*, which is a (kind of a special) surgery, so one could say that you hear about it, but under a different name. You can see the definition of a flip on page 41 of [Birational geometry of algebraic varieties by János Kollár and Shigefumi Mori](http://books.googl... | 9 | https://mathoverflow.net/users/10076 | 50083 | 31,475 |
https://mathoverflow.net/questions/50071 | 9 | Hello,
we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following thought:
Let
$$D = \{\text{all possible nonnull finite digit strings}\},$$
$$D' = \{\text{all things in D that do not... | https://mathoverflow.net/users/7607 | Prepending strings to primes. | Two comments:
(1) It is impossible that $m(A,B)= \infty$. The sequence you are interested in is $B$, $10^k A+B$, $(10^{2k}+10^k)A+B$, $(10^{2k}+10^{2k}+10^k)A+B$, etcetera. Let $p$ be a prime dividing $B$. We claim that there is some $n$ such that $p$ divides $(10^{nk}+\cdots +10^{2k}+10^k)A+B$. Proof: if $p=2$ or $5... | 13 | https://mathoverflow.net/users/297 | 50086 | 31,476 |
https://mathoverflow.net/questions/50090 | 8 | Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind of geometric invariant?
I tried a set of convex point sets formed by centroids of vertices of 8 dimensional 7-simplex, ... | https://mathoverflow.net/users/7655 | Do singular values of a point set determine its shape? | If I understand you correctly, you have certain sets of points which you would like to count up to permutations of the coordinates. This is the same as counting lists of points up to permutations of the coordinates and the order of the list. If the set of points is written as a matrix, this is the same as counting such... | 7 | https://mathoverflow.net/users/5963 | 50098 | 31,484 |
https://mathoverflow.net/questions/50102 | 0 | Let us define a "scattered" function as a function f with the following property:
For any a, there exists an positive number e such that for any a' not equal to a such that |a'-a|=e. In other words, when we consider the graph (x,f(x)), there is a square centered on (x,f(x)) which contains no other points.
The questio... | https://mathoverflow.net/users/9712 | Constructing a "Scattered" function over the reals | I'm not clear on your question, but I think that you may be trying to say that every point on the graph of $f$ is isolated in the plane. Is that right? In this case, we can associate to each point $x$ in the domain of $f$ an open ball inside your square containing $(x,f(x))$ and no other points of the graph, such that ... | 3 | https://mathoverflow.net/users/1946 | 50105 | 31,488 |
https://mathoverflow.net/questions/50119 | 7 | Let $G$ be an infinite locally finite connected graph with finitely many [ends](http://en.wikipedia.org/wiki/End_(topology)). A real-valued function $f : G \to \mathbb{R}$ is **harmonic** if
$$f(v) = \frac{1}{d\_v} \sum\_{v \sim w} f(w)$$
where $v \sim w$ means that $v, w$ are connected by an edge. Playing around ... | https://mathoverflow.net/users/290 | Reference request: discrete harmonic functions and ends of graphs | Life is much more complicated than that. In nice situations (for instance, if your graph is $\delta$-hyperbolic), then you can attach a more refined boundary than just the ends and (if you are lucky) solve the Dirchlet problem. A lot depends on what kinds of regularity conditions you assign to functions on the boundary... | 12 | https://mathoverflow.net/users/317 | 50121 | 31,499 |
https://mathoverflow.net/questions/31025 | 26 | Hello!
I read through parts of Khovanov/Rozansky's paper on the categorification of the HOMFLY polynomial using Matrix Factorizations. Technically, I can follow (though it seems to me that quite a lot of details are missing and tedious to fill in) - intuitively, however, I have no idea why one is lead to consider mat... | https://mathoverflow.net/users/3108 | Intuitive explanation for the use of matrix factorizations in knot theory | Many knot homologies are expected to have Floer-theoretic interpretations.
However, in Floer theory often the chain "complex" $CF(L\_0,L\_1)$ is not a complex but
rather an a-infty bimodule over a pair of curved a-infty algebras; matrix factorizations are more or less a special case of these, where the curvature of t... | 11 | https://mathoverflow.net/users/6223 | 50127 | 31,503 |
https://mathoverflow.net/questions/50126 | 8 | My question is: is it true that $\zeta\_n$ is in $K$ (a number field) iff $\zeta\_n$ is in all but finitely many of the $K\_{\mathfrak{p}}$?
| https://mathoverflow.net/users/5309 | If the n-th root of unity exists locally, does it exist globally? | The answer is *yes*, and this follows (as BCnrd points out in the comments above) immediately from the Chebotarev Density Theorem. See e.g. Corollary 9 [here](http://alpha.math.uga.edu/%7Epete/8430notes5.pdf).
More generally, let $K$ be any global field and $f \in K[t]$ be any irreducible, separable polynomial of deg... | 6 | https://mathoverflow.net/users/1149 | 50133 | 31,505 |
https://mathoverflow.net/questions/50120 | 2 | Hi Everyone,
Assume that we have a real symmetric matrix $H$, which can be written in the form $H=D \cdot B$, where $D$ is a positive diagonal matrix, and $B$ is a diagonally dominant matrix.
All elements of all matrices are positive real numbers.
We know that real symmetric matrices have real eigenvalues, and that d... | https://mathoverflow.net/users/11763 | Eigenvalues of Matrix Product | The answer is **Yes**. Write $B=D^{-1}H$. Thus $B$ is the product of two Hermitian matrices, ones of which ($D$) being positive definite. It is a classical fact (see my book on Matrices, 2nd edition, Prop. 6.1) that this product is diagonalisable with real eigenvalues of the same signs as those of $H$. The $B$ has real... | 4 | https://mathoverflow.net/users/8799 | 50138 | 31,507 |
https://mathoverflow.net/questions/50065 | 6 | $T$ is a set of some prime numbers. $S$ is the multiplictive set generated by $T$.
How to compute the Pontryagin dual of $S^{-1}Z$ ($Z$ is integal ring and $S^{-1}Z$ is localization of $Z$ at $S$).
If $T$ is an empty set, then the Pontryagin dual of $S^{-1}Z=Z$ is $R/Z$.
If $T$ is the set of all prime numbers, the Pont... | https://mathoverflow.net/users/11750 | pontryagin dual of the group S^{-1}Z | Let $\mathbf A\_T$ denote the restricted direct product $\mathbf R\times \prod'\_{p\in T}\mathbf Q\_p$ (relative to the subgroups $\mathbf Z\_p$, $p\in T$).
The OP asked whether the Pontryagin dual of $S^{-1}\mathbf Z$ is isomorphic to $\mathbf A\_T/(S^{-1}\mathbf Z)$. Let's take it up in two steps:
Step 1: $\mathb... | 1 | https://mathoverflow.net/users/9672 | 50142 | 31,510 |
https://mathoverflow.net/questions/50151 | 3 | Hi guys,
I have a simple question
Linear movement can be found in 1D, 2D and 3D world objects
Rotation can be found in 2D and 3D world objects.
Now, are there any kind of motion can *only* be found in 3D world objects? This kind of motion should be as simple as linear movement or rotation, and very basic that o... | https://mathoverflow.net/users/171 | Which motion is exclusive in 3D or higher dimensions? | I interpret your question to be asking for a description of 1-parameter groups of isometries.
The generic kind of motion in 3 dimensions is a screw-motion, which is translation along an axis combined with rotation about the same axis. Up to changing the speed of a parametrization, translations and rotations in 2 and ... | 18 | https://mathoverflow.net/users/9062 | 50157 | 31,515 |
https://mathoverflow.net/questions/48586 | 6 | I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant homology $H\_\ast^G(X)= H\_\ast(X\times\_G EG)$. In all of the examples I know, either the calculation is degenerate, for exa... | https://mathoverflow.net/users/2051 | Simple examples of equivariant homology and bordism | Sorry this got too long for a comment, so I post it as an answer.
A CW-complex equipped with a $G$- action is not the right thing to consider. A $G$-CW complex is a space, that can be built using blocks of the form $G/H \times (D^n,S^{n-1})$ (in the same way you build a CW-complex). A $G$-CW complex has better prope... | 2 | https://mathoverflow.net/users/3969 | 50158 | 31,516 |
https://mathoverflow.net/questions/50075 | 9 | An imprecise version of the question is that when $A$ and $B$ are regular rings, is $A \otimes B$ also regular? Please allow me to put more restrictions, here I am only interested in the case when $A$ and $B$ are finitely generated $k$-algebras. When $k$ is perfect, the answer is yes, see <http://arxiv.org/abs/math/021... | https://mathoverflow.net/users/1877 | Is the tensor product of regular rings still regular | This is inspired by Tom Goodwillie's answer.
>
> Let $X$ be an algebraic variety over a field $k$ (i.e. a scheme of finite type over $k$). If $X\times\_k X$ is regular, then $X$ is smooth over $k$.
>
>
>
Proof. The first projection $X\times\_k X\to X$ is faithfully flat, so the regularity of $X\times\_k X$ i... | 10 | https://mathoverflow.net/users/3485 | 50162 | 31,517 |
https://mathoverflow.net/questions/50165 | 1 | Suppose $T$ is a first-order theory whose signature contains $(+,\cdot,0,1,<)$ as well as a unary predicate $R(x)$. Suppose every finite subset $S \subseteq T$ has a model in which the set of elements satisfying $R(x)$ forms a substructure isomorphic to the field of real numbers. Does it follow that $T$ itself has a mo... | https://mathoverflow.net/users/8049 | Compactness theorem with preserved substructure | No. Suppose the signature of T contains a distinguished symbol $\omega$, and $T$ contains the statements $R(\omega)$ and the infinitely many statements $1+\cdots+1<\omega$. Then any finite subset of $T$ has a model where $R$ is isomorphic to the reals and $\omega$ is interpreted as some large enough real. But in any mo... | 5 | https://mathoverflow.net/users/8991 | 50170 | 31,522 |
https://mathoverflow.net/questions/49571 | 10 | One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a\_0 x^4 + 4a\_1 x^3 y + 6a\_2x^2y^2 + 4a\_3xy^3 + a\_4 y^4$$
the ring of invariants is generated by the following functions,
$$g\_2(a) = a\_0a\_4 - 4a\_1 a\_3 + 3a\_2^2$$
and
$$g\_3(a) =... | https://mathoverflow.net/users/2678 | Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$ | The description of orbits in the $d=4$ and $n=2$ is not difficult.
It is better to work projectively i.e. look at the orbits in the
projective space $P(\mathbb{C}^5)$
of nonzero binary quartics up to a constant.
A nonzero binary quartic $F$ can be written as a product of linear forms and therefore
corresponds to a co... | 15 | https://mathoverflow.net/users/7410 | 50171 | 31,523 |
https://mathoverflow.net/questions/50152 | 3 | Let $M$ be a $n$ dimensional manifold. Let $Aut(M)$ be the group of homeomorphisms
of $M$ viewed as a topological group under the compact-open topology.
What can we say in general about $Aut(M)$?
1.For example, what is $\pi\_0(Aut(M))?$
2.Is $Aut(M)$ homotopy equivalent to a space which can be describe
in terms... | https://mathoverflow.net/users/11765 | On the group of homeomorphisms of a manifold | A lot has been said about homeomorphism and diffeomorphism groups of spaces (more diffeo), starting with von Neumann and Ulam in the mid 40s (probably before then, as well), and until today. I would advise looking at papers by D.B.A. Epstein in the late sixties, John Mather in the early seventies, and a cool paper by D... | 6 | https://mathoverflow.net/users/11142 | 50172 | 31,524 |
https://mathoverflow.net/questions/50153 | 3 | The only examples of ultranets/ultrafilters described in Bourbaki and Willard are the trivial ones (generated by a single point). I know that their existence relies in general on the axiom of choice or the ultrafilter lemma.
I have two questions:
has any non-trivial ultranet been constructed?
if not:
is there a... | https://mathoverflow.net/users/10299 | How to make an ultranet | It is not possible to prove in ZF alone that there is a nonprincipal ultrafilter. This was established by Andreas Blass in 1977. [*A model without ultrafilters*,
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 329–331, [MR0476510](http://www.ams.org/mathscinet-getitem?mr=476510)]
| 7 | https://mathoverflow.net/users/2000 | 50175 | 31,526 |
https://mathoverflow.net/questions/50144 | 3 | According the the original definition by Smythe, a homology boundary link $L\subseteq S^3$ with $m$ components is a link which sarisfies one of the following equivalent conditions:
(1): The fundamental group $\pi\_1 (S^3\setminus L)$ of the link complement surjects onto the free group $F\_m$ of rank $m$;
(2): There... | https://mathoverflow.net/users/6206 | Homology boundary links | This is easy.
>
> if (1) holds one may construct a family of disjoint surfaces that represent a basis of $H\_2(M;\partial M;\Bbb Z)$
>
>
>
Indeed; but in order to do so, one has to identify the free group $F\_m$ with the fundamental group of the wedge of $m$ circles. This identification depends on a choice of ... | 3 | https://mathoverflow.net/users/10819 | 50181 | 31,531 |
https://mathoverflow.net/questions/50132 | 9 | To begin with, observe that the notion of a rectifiable curve makes sense in, say, a smooth or a PL-manifold but not in merely a topological manifold. Indeed if $f:[0,1]\rightarrow U\subset{\Bbb R}^n$, $U$ open, represents a rectifiable curve in $U$, and $g:U\rightarrow V$ a mere homeomorphism, we would generally have ... | https://mathoverflow.net/users/10909 | Manifolds with rectifiable curves | This is just Lipschitz structure: only locally Lipschitz maps preserve rectifiability of all curves.
What is wrong with the $x\mapsto x^{1/3}$ map is explained in Tapio Rajala's answer. (A more explicit example is the path $t\mapsto t^2\sin(1/t)$, $t\in[0,1]$, and you can make a non self-overlapping example if you go... | 7 | https://mathoverflow.net/users/4354 | 50202 | 31,540 |
https://mathoverflow.net/questions/49628 | 2 | A *topological bistellar flip* is the term used by [Dougherty, Faber, and Murphy](https://doi.org/10.1007/s00454-004-1097-3 "Dougherty, Randall; Faber, Vance; Murphy, Michael. Unflippable tetrahedral complexes. Discrete Comput. Geom. 32, No. 3, 309–315 (2004). zbMATH review at https://zbmath.org/?q=an:1138.52304") to d... | https://mathoverflow.net/users/11621 | When are nontopological bistellar flips manifold-preserving? | By a topological (2,2)-flip Dougherty, Faber, and Murphy mean a bistellar move on
2-manifolds (and not on 3-manifolds). So it looks like the example
>
> on a 3-simplex, a (2,2) non-topological flip would map one edge of the tetrahedron into the edge it does not intersect.
>
>
>
is supposed to be about trying ... | 4 | https://mathoverflow.net/users/10819 | 50203 | 31,541 |
https://mathoverflow.net/questions/50195 | 5 | The title says it all. Singular moduli of the j-function satisfy polynomials, but as the class number grows, these polynomial coefficients become very large. Weber functions are modular (not over the full modular group), and their values also satisfy polynomials. But the Weber polynomials tend to have much smaller coef... | https://mathoverflow.net/users/6769 | Why Are Weber Polynomial Coefficients Smaller than Hilbert Polynomial Coefficients? | Simply because they satisfy an equation of the form $P(f)-fj$ for some polynomial $P$. This immediately implies that the height of $f(z)$ will be around $1/deg(P)$ of the height of $j(z)$, or more precisely, asymptotic to it as the discriminant of $z$ goes to infinity.
See A. Enge and F. Morain's "Comparing invariant... | 8 | https://mathoverflow.net/users/2024 | 50204 | 31,542 |
https://mathoverflow.net/questions/50201 | 2 | Hi,
I've been struggling with this for awhile ( <http://en.wikipedia.org/wiki/Banach%27s_matchbox_problem>)
and I put together this little bit of Python code
```
def foo(n=3):
from numpy import rand
x = [1,]*n # one box of matches with elements 1
y = [0,]*n # the other box with elements 0
c=[]
whi... | https://mathoverflow.net/users/11780 | Question about Banach's matchbox problem. | Your mistake has to do with the definition of the problem. The man does not stop taking matches as soon as one of the boxes is empty (as in your code). He stops taking matches after he *picks a box and finds it empty*. This means that when Box A is empty, he doesn't immediately stop -- he continues until he picks Box A... | 8 | https://mathoverflow.net/users/673 | 50207 | 31,544 |
https://mathoverflow.net/questions/50194 | 3 | Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know
that $\mu$ is unique up to a scalar in $\mathbf{R}\_{>0}$. I don't know so much about unitary representations of groups but for the sake of convenience let us make the following definition:
Let $(V,\langle\ ,\ \rangle)$ be an Hilbert spa... | https://mathoverflow.net/users/11765 | Irreducible unitary representations of locally compact groups | I'm not quite sure if this is the answer that you looking for but anyway he we go. For a locally compact group you are going to generally want to look at strongly continuous representation. By this is mean endow $B(H)$, the bounded operators on a hilbert space $H$ with the topology of point-wise norm convergence. And o... | 3 | https://mathoverflow.net/users/5732 | 50210 | 31,546 |
https://mathoverflow.net/questions/50183 | 14 | This question stems from the paper "Computably categorical fields via Fermat's Last Theorem," by Russell Miller and Hans Schoutens (available online at <http://qcpages.qc.cuny.edu/~rmiller/Fermat.pdf>). In this paper, they construct a field $F$ by starting with the field $\mathbb{Q}(x\_0, x\_1, x\_2, . . .)$ of infinit... | https://mathoverflow.net/users/8133 | Fermat's Last Theorem and Computability Theory | Actually, I am harder-pressed to say anything the supposedly simpler question. For Question 1, I would just add that we need the existence of an infinite *computably enumerable* set of primes satisfying FLT, and that this ought to suffice. (Since the set P of primes NOT satisfying FLT is obviously $\exists$-definable a... | 10 | https://mathoverflow.net/users/11244 | 50215 | 31,549 |
https://mathoverflow.net/questions/50128 | 4 | Let $G$ be group and let $X$ be a topological space on which $G$ acts continuously. Now let us consider the following two properties relative to the group action:
(a) For every compact subset $K\subseteq X$ we have that
$|\lbrace g\in G:gK\cap K\neq\emptyset\rbrace|<\infty$
(b) For all sequence $\{g\_n\}\_{n\geq 1... | https://mathoverflow.net/users/11765 | A question about group action on topological space | Let's assume that $X$ is locally compact and instead of assuming that $G$ is a discrete group, assume more generally that it is locally compact and the action of $G$ on $X$ is continuous (i.e., the map $(g,x) \mapsto gx$ is continuous). Then the natural generalization of (a) is that if $K$ is compact, then the set $((K... | 6 | https://mathoverflow.net/users/7311 | 50219 | 31,552 |
https://mathoverflow.net/questions/50221 | 6 | Do we have examples of a contractible **bounded** open set $D\subseteq\mathbf{C}^n$ such that $Hol(D)$ (the group of biholomorphisms $f:D\rightarrow D$) acts transitively on $D$ but such that there exists no symmetry at a given point $x\in D$ (so at all points by homogeneity). By a symmetry at $x$ I mean an element $s\... | https://mathoverflow.net/users/11765 | A bounded homogeneous space which fails to be symmetric? | There are some examples given by Pjateckiĭ-Šapiro in
Classification of bounded homogeneous regions in n-dimensional complex space. Dokl. Akad. Nauk SSSR 141 1961 316–319.
and
On bounded homogeneous domains in an n-dimensional complex space. Izv. Akad. Nauk SSSR Ser. Mat. 26 1962 107–124.
(I have a vague memory tha... | 7 | https://mathoverflow.net/users/51 | 50222 | 31,553 |
https://mathoverflow.net/questions/50154 | 2 | Let $X$ be a Markov process (in continuous or discrete time) and define an event
$$
R(T,A) = (\exists t\leq T: X\_t \in A).
$$
I have seen in one paper that
$$
\Pr[R(\infty,A)] = \sup\limits\_{\tau} \mathbb{E}[I\_A(X\_\tau)],
$$
where $I\_A(x) = 1$ for $x\in A$ and $I\_A(x)=0$ otherwise is an indicator function and su... | https://mathoverflow.net/users/11768 | Reachability for Markov process | It is right, and not so obvious.
The question of whether or not a Markov process hits
particular sets is usually studied using the concept of capacity.
For a continuous time parameter Markov process taking values in
a general topological state space, this leads to non-trivial problems of measurability.
For instance... | 6 | https://mathoverflow.net/users/nan | 50241 | 31,560 |
https://mathoverflow.net/questions/50196 | 0 | **Background/motivation:** A model for the **classical** propositional calculus is a boolean function *b(S)* which assigns 1 or 0 to each (modal-free) sentence *S* according to the usual rules. I'm looking at models for propositional **modal** logic, where a modal model is simply a collection of classical models as poi... | https://mathoverflow.net/users/8224 | How many models are there, for a particular propositional modal logic? | You've shown that there are $2^c$ models $B$ but only $c$ corresponding functions $f$, so indeed many $B$'s must yield the same $f$. Since you say you don't see how it's possible for even two $B$'s to yield the same $f$, here's an example. Consider the countably many sentences $S$ such that neither $S$ nor its negation... | 4 | https://mathoverflow.net/users/6794 | 50244 | 31,561 |
https://mathoverflow.net/questions/45126 | 10 | **The problem:**
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a stochastic matrix.
Let $b(n,k)$ denote the suprememum, over all such matrices $P$ and initial distributions, of the ... | https://mathoverflow.net/users/10614 | Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$ | OK, I think I have a full answer at this point, so let me post it.
>
> Step 1. (algebra).
>
>
>
If $P$ is an $n\times n$ stochastic matrix and $\lambda$ is an eigenvalue of $P$ with $|\lambda|=1$, then $\lambda^k=1$ for some $k\le n$ and $1,\lambda,\lambda^2,\dots\,\lambda^{k-1}$ are eigenvalues of $P$.
Inde... | 10 | https://mathoverflow.net/users/1131 | 50249 | 31,565 |
https://mathoverflow.net/questions/50245 | 48 | David Feldman [asked](https://mathoverflow.net/questions/50186/) whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly predicted that this question would be essentially as hard as the Riemann hypothe... | https://mathoverflow.net/users/1464 | The Hardy Z-function and failure of the Riemann hypothesis | This doesn't quite sound like the right conjecture here, because Z is known to go to infinity on the average, by Selberg's central limit theorem (see my [blog post](http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/) on this topic). But this is easy to fi... | 35 | https://mathoverflow.net/users/766 | 50250 | 31,566 |
https://mathoverflow.net/questions/50237 | 3 | I saw a statement about the Birkhoff center. Namely let $X$ be a compact metric space and $f:X\to X$ be a homeomorphism on $X$. Then for each ordinal $\alpha$ we define
1. for $\alpha=0$, let $\Omega\_0(f)=X$,
2. for a successor ordinal $\alpha=\beta'$, let $\Omega\_{\alpha}(f)=\Omega(f,\Omega\_\beta(f))$,
3. for a l... | https://mathoverflow.net/users/11028 | center depth of Birkhoff center | First, to attempt to answer Gerry's question: I think that if $C$ is a closed set for which $f(C) = C$, then $\Omega(f, C)$ is supposed to be the set of non-wandering points of the restriction of $f$ to $C$, that is
$$\Omega(f, C) = \{x \in C: \forall U \in Nbhd(x) \forall n\_0 \exists n \geq n\_0 s.t. f^{(n)}(U) \c... | 3 | https://mathoverflow.net/users/2926 | 50254 | 31,568 |
https://mathoverflow.net/questions/50251 | 6 | Does anyone have access to Hamilton's 1978 Cornell preprint 'Deformation Theory of Foliations'. It is widely quoted but I couldn't find any online copy.
| https://mathoverflow.net/users/8941 | Preprint of Hamilton on deformations of foliations | I have a printed copy, but I don't think it was ever published or put online. If you contact me, I can make a copy and send it to you.
| 3 | https://mathoverflow.net/users/11800 | 50260 | 31,572 |
https://mathoverflow.net/questions/50233 | 11 | Let $F$ be the free group on (say) two generators, $a$ and $b$. Let $A$ and $B$ be (freely reduced) elements of $F$. Let $W(X, Y)$ denote a word on the words $X, Y$.
-Is it ever true that the equation $W(a, b) = W(A, B)$, has finitely many non-conjugate solutions (by conjugate solution I mean there exists a word $V$ ... | https://mathoverflow.net/users/6503 | Solutions to some equations in a free group | Let $G=\langle x,y,c\mid w(x,y)=c\rangle$ (which, in this case, happens to be a free group). The set of solutions to the system that you are looking for corresponds precisely to the set of homomorphisms $f:G\to F$ satisfying $f(c)=w(a,b)$. The basic idea of Sela's approach to the study of equations over $F$ (see [here]... | 11 | https://mathoverflow.net/users/1463 | 50265 | 31,575 |
https://mathoverflow.net/questions/50261 | 6 | Suppose that $A$ is a real square matrix with all diagonal entries $1$, all off-diagonal entries non-positive, and all column sums positive and non-zero. Does it follow that $\det(A)\neq0$? Is this just an exercise? Are these matrices well-known?
| https://mathoverflow.net/users/2225 | Determinants of "almost identity" matrices. | A nonzero row vector $v$ has a nonzero coordinate of greatest magnitude. $vA$ has a nonzero entry in that coordinate by the triangle inequality, hence is not the $0$ vector.
| 19 | https://mathoverflow.net/users/2954 | 50270 | 31,578 |
https://mathoverflow.net/questions/50277 | 2 | Does there exists an example of an extension of topological groups $1 \to N \to E \to G \to 1$ admitting a section $s:G \to E$ which is continuous (or continuous in a neighbourhood of identity) and satisfy the propery that $s(x^{-1}) =s(x)^{-1}$. One can see that the extension $0 \to \mathbb{Z} \to \mathbb{R} \to \math... | https://mathoverflow.net/users/11932 | Sections of topological group extensions | A very easy example (inspired by an answer by Laurent Moret-Bailly that he removed immediately after posting): Take $0 \to 3\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/3\mathbb{Z} \to 0$ and define a section $s: \mathbb{Z}/3\mathbb{Z} \to \mathbb{Z}$ by $s(0) = 0$, $s(1) = 1$ and $s(2) = -1$.
Here's an example from func... | 10 | https://mathoverflow.net/users/11081 | 50279 | 31,582 |
https://mathoverflow.net/questions/50291 | 0 | Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize $0\vec{y}$, subject to $\vec{y}\matrix{A}\geq\vec{c}, \vec{y}\geq 0$". I'm particularly bothered by the "minimize $0$" part of ... | https://mathoverflow.net/users/5534 | Degenerate case of linear programming duality? | Yes, the duality theorem holds. "Minimize 0" makes your life much easier, because you know exactly what the optimal value of the dual program is...
| 1 | https://mathoverflow.net/users/11142 | 50292 | 31,587 |
https://mathoverflow.net/questions/50253 | 6 | For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum\_{i\_1 = 1}^n \sum\_{i\_2 = i\_1+1}^n \cdots \sum\_{i\_k=i\_{k-1}+1}^n \frac{1}{i\_1 i\_2 \cdots i\_k}.$$
I found a couple of references (by [Adamchik](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.... | https://mathoverflow.net/users/9716 | Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups? | One can rephrase Qiaochu's argument without using symmetric functions as follows:
A standard property of $\left[ n+1\atop k+1\right]$ is the generating function
$$ \sum\_{k=0}^{n+1}\left[ n+1\atop k\right]t^k = t(t+1)(t+2)\cdots (t+n). $$
Equating coefficients of $t^{k+1}$ gives the result
$$ S(n,k) =\frac{1}{n!}\lef... | 7 | https://mathoverflow.net/users/2807 | 50293 | 31,588 |
https://mathoverflow.net/questions/50294 | 6 | I've come across the following puzzle:
>
> You're on an island, on which there is
> a fence (which is a simple closed
> contour). You need to determine
> whether you're inside or outside the
> fence.
>
>
>
Now if you had the function defining the contour as well as the point you're in (e.g. you have a GPS)... | https://mathoverflow.net/users/5230 | Point in Polygon algorithm from the viewpoint of a robot | Yes, this is correct, the distance you will traverse will be 6.28 meters greater (if you are outside) and 6.28 meters shorter if you are inside, so you better have a very accurate instrument. The relevant result is Steiner's formula for the measures of parallel sets (see Santalo's book on Integral Geometry and Geometri... | 6 | https://mathoverflow.net/users/11142 | 50295 | 31,589 |
https://mathoverflow.net/questions/50305 | 12 | What is the main use of indiscernibles in model theory? reading through Chang and Keisler's Model Theory it seems that the main motivation for indicernibles is for getting many non-isomorphic models for a theory (like the theory of dense linear order without endpoint).
Also, can you recommend the best source for readin... | https://mathoverflow.net/users/10708 | Use of indiscernibles in model theory | Eran,
As far as I know, indiscernibility is used in two ways in model theory. One, as you say, is to obtain many non-isomorphic models. This is for sure the classical use of indiscernibility.
Another, more modern one, is to have access to tools such as Ramsey's theorem and its uncountable version, the Erdős-Rado th... | 14 | https://mathoverflow.net/users/6085 | 50308 | 31,594 |
https://mathoverflow.net/questions/50300 | 13 | I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and smoothness, the situation becomes very complicated.
My question is, other than the family of surfaces from Delaunay, is ther... | https://mathoverflow.net/users/11262 | Is there a complete classification of constant mean curvature surfaces? | Karsten Grosse-Brauckmann, Rob Kusner, and John Sullivan have written on the classification of embedded CMC surfaces for quite some time. I think a reasonable place to read about this program is [this survey by Rob Kusner](http://arxiv.org/abs/math/0207160). You might also want to look at some of their other papers on ... | 8 | https://mathoverflow.net/users/353 | 50311 | 31,597 |
https://mathoverflow.net/questions/50288 | 1 | Is $e^{(rA)} = (e^{A})^r$ when $r \in \mathbb{R}$ and $A$ is an element of a Banach algebra?
Clearly if $n$ is an integer, then
$e^{(nA)} = e^{A+A \cdots +A} = e^{A}e^{A}\cdots e^{A} = (e^{A})^n$,
where the second equality follows from the Baker-Hausdorff lemma and the fact that [A,A]=0. On the other hand, I thin... | https://mathoverflow.net/users/7671 | Is exp(rA) = (exp(A))^r for real r and A in a Banach space? | A particular case of a Banach algebra is the one-dimensional case $\mathbb C$, right?
And according to Pietro, if you can prove this in that case, you get the general case by saying "functional calculus". (At least for $A$ where it applies, such as normal $A$.)
But is it true for $\mathbb C$?
$A = 2\pi i$, $r=... | 5 | https://mathoverflow.net/users/454 | 50314 | 31,600 |
https://mathoverflow.net/questions/50322 | 1 | Let $f$ and $g$ be positive definite forms in the polynomial ring ${\mathbb{R}}[x\_0,\ldots, x\_n]$ such that $\deg(g)$ divides $\deg(f)$. A generalization of a theorem by Reznick is that $g^N f$ is a sum of squares for large $N$.
A corollary of the Representation Theorem says that if $V$ is an affine real variety w... | https://mathoverflow.net/users/nan | Removing a hypersurface when applying the Representation theorem to prove Positivstellensatz with uniform denominators | The real point of $g = 0$ are the empty set, but $g = 0$ is still a non-trivial hypersurface over $\mathbb R$, just with no real points. (The fact that is has no real points is the reason why the real points of $V$ are compact.)
| 1 | https://mathoverflow.net/users/2874 | 50324 | 31,604 |
https://mathoverflow.net/questions/50329 | 0 | Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int\_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
\displaystyle\prod\_{k = 1}^{n} \frac{\Gamma(\frac{z}{2^k}+\frac{1}{2})}{\Gamma(\frac{1}{2})} = & \frac{\Gamma(z+1)}{2... | https://mathoverflow.net/users/10632 | Generalizations of a product formula for the gamma function | Certainly not. Let all $H\_i$ tend to zero. By continuity yu get
$$
1=\frac{\Gamma(z+1)}{2^{2z}},
$$
which is false.
| 4 | https://mathoverflow.net/users/nan | 50332 | 31,609 |
https://mathoverflow.net/questions/50334 | 1 | I have a Bernoulli trial with success rate $p$ and failure rate $1-p$ the odds of $k$ successes is $\binom{N}{k} p^k (1-p)^{N-k}$. I need to evaluate an integral
$$ \int\_0^1 dp p^k (1-p)^{N-k} = \frac{k!(N-k)!}{(N+1)!} $$
This was done with Mathematica, but you can use induction. To avoid mindlessly integrating, **can... | https://mathoverflow.net/users/1358 | How to reading of an integral? Bernoulli trials with variable success rate, p | [More or less, yes](https://math.stackexchange.com/questions/3528/beta-function-derivation).
| 1 | https://mathoverflow.net/users/290 | 50335 | 31,610 |
https://mathoverflow.net/questions/50315 | 4 | We are over some field $k$ of characteristic $0$. The general linear group $\mathrm{GL}\_n$ canonically acts from the left and from the right on the space $\mathrm{M}\_n$ of all $n\times n$-matrices, and thus also acts from the left and from the right on the ring $k\left[\mathrm{M}\_n\right]$ of polynomial functions on... | https://mathoverflow.net/users/2530 | Is the space of polynomial functions on M_n a faithful U(gl_n)-module? | The usual way to prove something like this is by taking associated graded; if you have a filtered map between two filtered rings, then if the induced map of associated gradeds is injective, the original map was injective (the other way is wildly false).
So what is this associated graded of this map for the PBW filtra... | 7 | https://mathoverflow.net/users/66 | 50354 | 31,620 |
https://mathoverflow.net/questions/50319 | 4 | In plane for a smooth non self intersecting curve $C$ the tubular neighbourhood can be constructed with non intersecting starigh line segments normal to the curve from points of the curve.
If we use the fact that the $\varepsilon$-tubular neighbourhood is constructed diffeomorphically by extending the normals we could ... | https://mathoverflow.net/users/11804 | Tubular neighbourhood of a smooth curve | As shown in Richard Palais' answer, the radius $\epsilon$ of the larger tubular neighborhood can be much smaller than $1/K$.
On the other hand, at least for smooth embeddings of $\mathbb{S}^1$, the best uniform radius $\epsilon$ is also never larger that $1/K$. So in this sense yours is a necessary condition (with we... | 2 | https://mathoverflow.net/users/6101 | 50361 | 31,623 |
https://mathoverflow.net/questions/50360 | 9 | I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can do is ask it here...
Let $A$ be a finite-dimensional $k$-algebra, where $k$ is a field. Is it true that the Jacobson r... | https://mathoverflow.net/users/2530 | Jacobson radical = intersection of all maximal two-sided ideals | Yes, this is true; it's essentially just a restatement of Artin-Wedderburn. All you need to do is note that by Artin-Wedderburn, a finite dimensional algebra with trivial Jacobson radical is a sum of matrix algebras over division rings (where it's obvious that the intersection of all maximal ideals is trivial); for an ... | 6 | https://mathoverflow.net/users/66 | 50369 | 31,627 |
https://mathoverflow.net/questions/50371 | 3 | Suppose that $f : X \rightarrow Y$ is mapping between topological spaces that is not continuous at $x\_0$. Then there is an open set $V$ in $Y$ containing $f(x\_0)$ such that for any open set $U$ containing $x\_0$, there is some $x\_U \in U$ with $f(x\_U) \notin V$. By picking one from each $U$ we can build a net $x\_U... | https://mathoverflow.net/users/4903 | Nets and the Axiom of Choice | The axiom of choice is not needed if $X$ is $T\_1$; i.e., singleton sets are closed.
Suppose $f:X\to Y$ is not continuous at $x\_0$. Let $D$ be the collection of all $(U,x)$ s.t. $U$ is an open set containing both $x\_0$ and $x$ and $x\_0\not=x$. Say that $(U,x) \le (V,y)$ provided $V\subset U$ and $x$ is not in $V$.... | 10 | https://mathoverflow.net/users/2554 | 50377 | 31,630 |
https://mathoverflow.net/questions/50359 | 9 | Let $X$ be a smooth quasi-projective variety (so irreducible) over $\mathbf{C}$. We may think of $X$ as a complex manifold which we denote by $X^{an}$. Of course the topology on $X^{an}$ is finer than the Zarisiki topology on $X$. Now let us suppose that we have a **surjective finite unramified analytic cover**
$f:Y\... | https://mathoverflow.net/users/11765 | Finite unramified analytic coverings vs finite etale coverings | (Using the notation from the question) $\mathcal F$ is a coherent sheaf of $\mathcal O\_{\overline X}$-algebras. Then $Z={\rm Spec}\_{\overline X}\, \mathcal F\to \overline{X}$ is a finite morphism between projective schemes. Looking at the construction of ${\rm Spec}\_{\overline X} \mathcal F$ should tell you that $\o... | 5 | https://mathoverflow.net/users/10076 | 50386 | 31,638 |
https://mathoverflow.net/questions/50328 | 5 | I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic?
The reason I'm asking this is because I was wondering whether or not there were any disconnected real topological vector spaces.
| https://mathoverflow.net/users/36720 | Are all topological (finite-dim) real vector spaces homeomorphic to a coordinate space? | Any (Hausdorff) topological real vector space of dimension $n<\infty$ is homeomorphic to $\mathbf{R}^n$ with the standard topology, see e.g. Rudin, Functional analysis, theorem 1.21.
Here are some comments:
1. For some reason it is stated there for complex vector spaces, but, as remarked after the theorem, the proo... | 12 | https://mathoverflow.net/users/2349 | 50391 | 31,643 |
https://mathoverflow.net/questions/50362 | 4 | It is quite known fact that the determinant of arbitrary symmetric matrix is an irreducible polynomial in algebra $\mathbb C [x\_{ij}, 1\leq i,j\leq n]$ ($x\_{ij}=x\_{ji}$) (see this: atlas.mat.ub.es/personals/sombra/papers/cayley/cayley.ps ).
Is there any geometric proof of this statement like the proof of irreduci... | https://mathoverflow.net/users/11072 | Irreducibility of determinant of symmetric matrix | I don't know a proof based on biduality, but here is a short geometric proof, which generalizes to other similar situations (all degenerate matrices, singular hypersurfaces and many other examples considered in the book by Gelfand, Kapranov and Zelevinsky).
In all these cases the discriminant variety admits a ``canon... | 8 | https://mathoverflow.net/users/2349 | 50393 | 31,644 |
https://mathoverflow.net/questions/50382 | 8 | Hello!
I have written a program in Mathematica 7, which calculates for a (finite abstract) simplicial complex all its homology groups. I would really like to test it on the projective spaces, but cannot find a way to triangulate them. There seem to be very few articles on this matter and none of them states directly ... | https://mathoverflow.net/users/11317 | How to triangulate real projective spaces (as simplicial complexes in Mathematica)? | If you want code, see the patch for Sage at <http://trac.sagemath.org/sage_trac/attachment/ticket/9125/trac_9125-projective-space.patch>. This patch also includes references for triangulations of $\mathbf{R}P^n$, but I'll summarize the results here, too. For $n=3$, every triangulation must have at least 11 vertices, an... | 14 | https://mathoverflow.net/users/4194 | 50394 | 31,645 |
https://mathoverflow.net/questions/50392 | 3 | Since the theorems of (PA + "there is a nonstandard number") are recursively enumerable, by the
[Low Basis Theorem](http://en.wikipedia.org/wiki/Low_basis_theorem), [WKL0](http://en.wikipedia.org/wiki/Reverse_mathematics#Weak_K.C3.B6nig.27s_lemma_WKL0)'s proof of the completeness theorem gives a nonstandard model o... | https://mathoverflow.net/users/nan | Turing degrees of nonstandard models of PA | b) is impossible, because the only low computably dominated degree is $\mathbf 0$ (see Soare's book *Recursively Enumerable Sets and Degrees*) and there are no computable nonstandard models of PA.
a) (minimal) and c) (K-trivial) are also impossible. See Theorem 4.2 of Csima/Harizanov/Hirschfeldt/Shore, *Bounding homo... | 7 | https://mathoverflow.net/users/4600 | 50397 | 31,648 |
https://mathoverflow.net/questions/50406 | 3 | This is a question about support of modules under extension of scalars.
Let $f \colon A \to B$ be a homomorphism of commutative rings (with unity), and let $M$ be a finitely generated $A$-module.
Recall that the *support* of $M$ is the set of prime ideals $\mathfrak{p}$ of $A$ such that the localization $M\_{\mathfr... | https://mathoverflow.net/users/5792 | Extension of scalars and support of a non-finitely generated module | Take $A = \mathbb Z$, $M = \mathbb Q$, $B = \mathbb Z/p\mathbb Z$, were $p$ is a prime.
| 6 | https://mathoverflow.net/users/4790 | 50407 | 31,654 |
https://mathoverflow.net/questions/50402 | 3 |
>
> **Possible Duplicate:**
>
> [Multiplicative order of zeros of the Artin-Schreier Polynomial](https://mathoverflow.net/questions/46133/multiplicative-order-of-zeros-of-the-artin-schreier-polynomial)
>
>
>
I will be grateful for any reference to some literature on the following question (to the best of my ... | https://mathoverflow.net/users/11832 | Finite fields: Is multiplicative order of $x^p - x - 1$ equal to $\frac{p^p-1}{p-1}$ over GF(p)? | see papers of mine (on Bell numbers) with coauthors and papers of sam wagstaff.
| 1 | https://mathoverflow.net/users/11016 | 50409 | 31,655 |
https://mathoverflow.net/questions/50410 | 3 | Forgive me if this is a well-known observation/result, but I'm quite new to graduate-level algebra and I was wondering if there are generalisations of the constructions I describe below.
It's straightforward to see that the functor "adjoin an indeterminate to a ring" is the unit of the adjunction $F : \mathbf{Ring} \... | https://mathoverflow.net/users/11640 | Indeterminates and forgetful functors | In your examples, there is an evident underlying functor $U: C \to Set$ which allows you to talk about points as elements of the underlying set. So a pointed $C$-object would be an object of the comma category $1 \downarrow U$. There is an evident projection $\pi: 1 \downarrow U \to C$. A left adjoint to this functor w... | 7 | https://mathoverflow.net/users/2926 | 50420 | 31,659 |
https://mathoverflow.net/questions/50422 | 1 | A [computer program](http://nezumi-lab.org/blog/?cat=11) ouputs the digits of $\pi$ by evaluating the recurrence relation
$a\_{n+1} = a\_n + sin \ a\_n$
with $a\_0 = \frac{6}{5}$
Does the sequence actually converge or is this just coincidence?
---
I would like to answer this by rewriting it as a differentia... | https://mathoverflow.net/users/nan | Verifying a sequence that converges to pi | Regarding the second question, no.
But you can write $\Delta a\_x=\sin a\_x$.
| 0 | https://mathoverflow.net/users/10059 | 50429 | 31,665 |
https://mathoverflow.net/questions/50458 | 3 | The topology on a topological group is generated by a family of pseudometrics. The only proof I know passes through uniform spaces (by which I mean the entourage definition): A topological group has a uniformity and by a theorem of Weil, every uniformity comes from a family of pseudometrics. Is there a direct construct... | https://mathoverflow.net/users/3711 | Topological Groups and Families of Pseudometrics | The basic strategy should consist of two steps: (1) topological groups are completely regular, and (2) completely regular spaces are subspaces of products of pseudometric spaces. Of course, this is not hugely different from the proof which passes through uniform structures, but certainly you can bypass that language.
... | 6 | https://mathoverflow.net/users/2926 | 50464 | 31,685 |
https://mathoverflow.net/questions/50452 | 32 | Thanks to Freudenthal we know that $\pi\_{n+k}(S^n)$ is independent of $n$ as soon as $n \ge k+2$. However, I was looking at the [table on Wikipedia](http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_homotopy_groups) of some of the homotopy groups of spheres and noticed that $\pi\_2^S$ (the second stable... | https://mathoverflow.net/users/6936 | Early stabilization in the homotopy groups of spheres | Nice question! Funny enough, the answer is that you've already found *all* of the examples of early stabilization (excepting, of course, the fact that you didn't mention $\pi\_n S^n$ stabilizing early). This is true even if you ignore odd torsion.
The "pesky copy of ℤ" is indeed related to the Hopf invariant. The fac... | 39 | https://mathoverflow.net/users/360 | 50470 | 31,689 |
https://mathoverflow.net/questions/50482 | 1 | In Folland's "real\_analysis: modern techniques and their applications", second edition, page 70, line 4 of proof of Theorem 2.40 (a), the author asserts the inequality $m(U\_j) < m(T\_j)+\epsilon2^{-j}$. I think this comes from $m(i$-th side of $U\_j)\leq m(i$-th side of $T\_j)+$a small number, $i=1,...,n$, by Theorem... | https://mathoverflow.net/users/5072 | A question in a proof on approximating n-dimensional Lebesgue measurable set by open set | If some rectangle of measure zero has a side of infinite measure, divide that side into a countable number of sides of finite measure. In other words, you may arrange things so that none of the $T\_j$ has any side of infinite measure.
| 2 | https://mathoverflow.net/users/802 | 50484 | 31,694 |
https://mathoverflow.net/questions/50465 | 12 | Physicists computing multiloop Feynman diagrams have introduced various
techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines
1) $DoT(r)=0$, r rational
2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,
3) $DoT(\zeta(k))=k$,
4) $DoT( a \cdot b)= DoT(a)... | https://mathoverflow.net/users/10475 | Degree of Transcendentality and Feynman Diagrams | Your *transcendentality* reminds me about the Institute of Algebraic Meditation at Höör (Sweden). To be honest, your definition corresponds to what is known as the *weight* of a (multiple) zeta value (see Michael Hoffman's <http://www.usna.edu/Users/math/meh/mult.html>, especially the [references on MZVs](http://www.us... | 9 | https://mathoverflow.net/users/4953 | 50489 | 31,698 |
https://mathoverflow.net/questions/50432 | 10 | Recall that Zariski's Main Theorem states that if $f: X \to Y$ is a quasi-finite, separated, and finitely presented morphism into a quasi-compact separated scheme $Y$, then there is a factorization of $f$ into an open immersion followed by a finite morphism. In EGA IV-8, this is proved by reducing to the case of $Y$ th... | https://mathoverflow.net/users/344 | A noetherian proof of Zariski's Main Theorem? | I would suggest chapter IV of the 1970 book "Anneaux Locaux Henséliens", by Michel Raynaud published in Springer Lecture Notes in Math no. 169. It gives a very general proof, way simpler than the one in EGA IV and, in my opinion, very readable. The proof is based in a paper by Peskine from 1966. The proof in Raynaud's ... | 6 | https://mathoverflow.net/users/6348 | 50502 | 31,706 |
https://mathoverflow.net/questions/50490 | 2 | Over a base field $k$, linear $k$-groups stand for affine algebraic $k$-groups. For simplicity take $k$ to be a field of characteristic zero, as in this case one has the correspondence between connected linear $k$-groups and finite dimensional Lie $k$-algebras.
Let $\mathfrak{g}$ be the Lie algebra of a connected sem... | https://mathoverflow.net/users/9246 | on a characterization of parabolic subgroups | (Edit)
In the given generality, I'm not sure the question has much hope for a tidy answer.
Consider $G=GL(V)$ and write $V = W \oplus W'$ where $W$ has dimension 2. So you get an embedding $H=GL(W) \to G$ in a natural way ($H$ acts trivially on $W'$).
The stabilizer $Q$ in $H$ of a line $L \subset W$ is a parabo... | 2 | https://mathoverflow.net/users/4653 | 50504 | 31,707 |
https://mathoverflow.net/questions/50475 | 7 | If $\alpha$ is a (limit) ordinal, then a subset $S\subseteq\alpha$ is club if $\alpha$ is closed as a subset of $\alpha$ under the order topology and unbounded in $\alpha$. The set of all sets containing a club forms a filter on the subsets of $\alpha$, called the club filter.
This definition can be extended in the ... | https://mathoverflow.net/users/8133 | Generalization of the club filter | Here's a little piece of an answer: Section 3 of Solovay's paper "The independence of DC from AD" (Cabal Seminar 76-77, Springer Lecture Notes in Math 689 (1978) pp. 171-183) has a construction of a normal, countably complete ultrafilter $U$ on the set of countable subsets of the real line. The construction assumes $AD... | 6 | https://mathoverflow.net/users/6794 | 50532 | 31,724 |
https://mathoverflow.net/questions/50519 | 42 | There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its uses and constructions, and I am at a loss as to imagine some chain level construction of such an operation, other than ... | https://mathoverflow.net/users/3901 | Integral cohomology (stable) operations | $HZ^nHZ$ is trivial for $n<0$. $HZ^0HZ$ is infinite cyclic generated by the identity operation. For $n>0$ the group is finite. So you know everything if you know what's going on locally at each prime. For $n>0$ the $p$-primary part is not just finite but killed by $p$, which means that you can extract it from the Steen... | 31 | https://mathoverflow.net/users/6666 | 50536 | 31,727 |
https://mathoverflow.net/questions/50541 | 3 | Let $G$ be a locally compact second countable topological group and $H$ be a closed subgroup of $G$.
Let $\pi: G\to G/H$ be the quotient map. For the Borel $\sigma$-algebra $\mathcal A$ on $G/H$. Is it true
that $\pi^{-1}(\mathcal A)$ is the Borel subsets of $G$ consisting of right $H$ orbits?
| https://mathoverflow.net/users/11056 | quotient sigma-algebra on quotient space of locally compact groups | **Yes**. $\DeclareMathOperator{\calB}{\mathcal{B}}$
Since the projection $\pi: G \to G/H$ is continuous, pre-images of Borel sets are Borel, so we have that $\pi^{-1}{(\mathcal{A})}$ is contained in the $\sigma$-algebra $\calB\_{H}$ of right $H$-invariant Borel-sets on $G$.
To see the reverse inclusion $\calB\_{H} ... | 4 | https://mathoverflow.net/users/11081 | 50546 | 31,733 |
https://mathoverflow.net/questions/50516 | 28 | Clearly the etale fundamental group of $\mathbb{P}^1\_{\mathbb{C}} \setminus \{a\_1,...,a\_r\}$ doesn't depend on the $a\_i$'s, because it is the profinite completion of the topological fundamental group. Does the same hold for when I replace $\mathbb{C}$ by a finite field? How about an algebraically closed field of po... | https://mathoverflow.net/users/5309 | Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points? | It is a result of Tamagawa that for two affine curves $C\_1, C\_2$ over finite fields $k\_1,k\_2$ any continuous isomorphism $\pi\_1(C\_1)\rightarrow \pi\_1(C\_2)$ arises from an isomorphism of schemes $C\_1\rightarrow C\_2$. Hence, if $\pi\_1( \mathbb{P}^1\setminus\{a\_1,\ldots, a\_r\})$ were independent of the choice... | 28 | https://mathoverflow.net/users/259 | 50555 | 31,736 |
https://mathoverflow.net/questions/50550 | 10 | This question is in response to one of the questions asked [here](https://mathoverflow.net/questions/50483/why-does-nobody-care-about-heuristics-in-logic). The OP wanted to know if the percentage of statements provable from ZFC tended to some value, and if so, what it was. In particular, the OP seemed interested in det... | https://mathoverflow.net/users/11318 | Asymptotic density of provable statements in ZFC | You can make $T\_n(n)/n$ converge to any value you like by choosing a suitably silly Gödel numbering. Partition $\mathbb{N}$ (computably) into a set $A$ of density $p$, and set $B$ of density $1-p$, and an infinite set $C$ of density 0. Arrange your numbering scheme so that $A$ corresponds exactly to the ZFC axioms, $B... | 20 | https://mathoverflow.net/users/11771 | 50556 | 31,737 |
https://mathoverflow.net/questions/50474 | 7 | For a symplectic 4-manifold, it is possible for two symplectic submanifolds to intersect negatively? Actually it is an exercise in the 4-manifold book by Gompf and Stipsicz to find symplectic planes in $\mathbb{R}^4$ with negative intersection. Now I want to know the closed case. Does this happen also for closed symple... | https://mathoverflow.net/users/11846 | Negative intersection of symplectic submanifolds | The answer to this question is YES. I assume you want $A$ and $B$ to be connected.
Already in the case of four manifolds two symplectic surfaces can have negative intersection.
To construct an example, we use that if two symplectic surfaces in a $4$-fold intersect positively, you can always smoothen the neighborhood... | 5 | https://mathoverflow.net/users/943 | 50560 | 31,739 |
https://mathoverflow.net/questions/50561 | 0 | Greetings,
We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We would like to make several cuts at certain heights. Each cut (a plane perpendicular to the Z- axis) may create one or ... | https://mathoverflow.net/users/11868 | 3d width and cross section | These are applied computational geometry questions, not really mathematics research questions.
<http://en.wikipedia.org/wiki/Computational_geometry>
Maybe try asking on stackoverflow or on a blender or CAD forum or something.
<http://www.blender.org/>
| 0 | https://mathoverflow.net/users/11853 | 50562 | 31,740 |
https://mathoverflow.net/questions/50539 | 4 | Let $T\_t$ be the geodesic flow on a surface $S$ of constant negative curvature, and let $M(f,t) := \langle \bar f \cdot (f \circ T\_t) \rangle$, where $\langle f \rangle := \int\_S f(x) d\mu(x)$ and where $\mu$ is the natural invariant measure.
A [1984 paper by Collet, Epstein, and Gallavotti](https://projecteuclid.... | https://mathoverflow.net/users/1847 | Question about an early result on the mixing of geodesic flows | Dear Steve,
In general the "precise" rate of mixing depends on the class of functions one considers. In particular, the authors of the paper you quoted treat only "analytic" functions (with some fixed band of analyticity) and they show that you get a rate of $t^{b(\xi)}e^{-t/2}$. In this sense, the rate of mixing is ... | 5 | https://mathoverflow.net/users/1568 | 50564 | 31,741 |
https://mathoverflow.net/questions/50563 | 3 | A set $S$ (of natural numbers) is (semi)decidable if its (semi)characteristic function is effectively calculable.
From a set theoretic point of view, the semicharacteristic function of a set is just a particular subset of its characteristic function. Consider the subsets of the characteristic function of a given set ... | https://mathoverflow.net/users/6466 | Semidecidable sets | Any finite modification of a computable function is computable, so the only instance of your phenomenon is the set $S=\mathbb{N}$, for which the semi-characteristic function is the same as the characteristic function.
Meanwhile, a modified version of your question is answered by the concept of a *simple* set, a c.e.... | 8 | https://mathoverflow.net/users/1946 | 50568 | 31,742 |
https://mathoverflow.net/questions/50548 | 10 | An integer-valued polynomial is a polynomial with real coefficients mapping integers to integers. It is well known that all such polynomials $h(x)$ are generated as an additive group by the binomial coefficients $\binom{x}{n}$. My question concerns the problem of approximating an arbitrary polynomial $f$ with real coef... | https://mathoverflow.net/users/5229 | Approximating polynomials in R[x] using integer-valued polynomials |
>
> I would be grateful if you could spell out some details in an answer.
>
>
>
OK, let's do the details.
Start with any continuous function $f$ on $[-0.9,0.9]$ that is $0$ in some open neighborhood of the origin. Let $g$ be its even part. Then $f-g$ is odd and also vanishes near the origin. Thus, we can wri... | 10 | https://mathoverflow.net/users/1131 | 50571 | 31,743 |
https://mathoverflow.net/questions/50566 | 2 | In "Linear algebraic groups, 2nd ed. T.A.Spinger, Birkhauser" 8.4.5, one finds a characterization of parabolic subgroups via co-characters, as follows:
for simplicity, assume that $k$ is a base field of characteristic zero which is algebraically closed, with linear groups standing for affine algebraic groups over $k$... | https://mathoverflow.net/users/9246 | relate parabolic subalgebras to gradings? | Recall that when $\mathfrak{g}$ is a Lie algebra over a field $k$, a *derivation* of $\mathfrak{g}$ is a $k$-linear map $D: \mathfrak{g} \rightarrow \mathfrak{g}$ such that
$$D( [X,Y] ) = [DX, Y] + [X, DY],$$
for every $X,Y \in \mathfrak{g}$.
Given a $Z$-grading on a Lie algebra $\mathfrak{g}$ (i.e., a grading on the... | 7 | https://mathoverflow.net/users/3545 | 50585 | 31,747 |
https://mathoverflow.net/questions/50578 | 2 | Let X be a curve over $F\_{p^2}$ defined by equation $X^{p+1}-Y^p-Y$.
In general ideal $I$ of $F(X)$ is should not be principal. But for some $\alpha$ $I^\alpha$ becomes principal. Do you know how I calculate this $\alpha$?
| https://mathoverflow.net/users/4246 | What is the Picard group of Hermitian curve | The zeta function of $X/\mathbb{F}\_{p^2}$ is $P(t)/((1-t)(1-p^2t))$, where $P(t)=(1+pt)^{2g}, g=p(p-1)/2$ is the genus of $X$. Finally, from the general theory of curves over finite fields, the class number of $X$ is $\alpha=P(1)$ which in this case is $\alpha = (p+1)^{2g}$.
Edit: Actually the class group is $(\math... | 7 | https://mathoverflow.net/users/2290 | 50589 | 31,750 |
https://mathoverflow.net/questions/50599 | 6 | Is the projection map $\amalg\_{i \in \mathbb{Z}} X \to X$ from an infinite disjoint union of copies of a scheme $X$ to $X$ an etale map, using the definition in EGA IV 17?
According to EGA, an etale map of schemes $f:X \to Y$ is locally of finite presentation and formally etale. Hence I don't think etale maps need to ... | https://mathoverflow.net/users/19943 | is quasicompact part of the definition of etale in EGA | No, quasi-compact is not part of the definition of etale. Yes, your map from the infinite disjoint union to $X$ is etale.
| 7 | https://mathoverflow.net/users/1114 | 50601 | 31,758 |
https://mathoverflow.net/questions/50605 | 3 | Given a category ${\cal C}$, a functor $F:{\cal C}\rightarrow {\rm Sets}$,
an object $A$ in ${\cal C}$ and an element $x\in F(A)$, one may consider the smallest
subfunctor $F\_x$ of $F$ that contains $x$. Explicitly,
$F\_x(B) = \{ F(f)(x) \}\_{f\in {\rm Hom}(A,B)}$ and
$F\_x(f) = F(f)|\_{F\_x(B)}$ for $f:B\rightarro... | https://mathoverflow.net/users/10909 | A weak Yoneda-type lemma for certain nonrepresentable functors? | Up to isomorphism, they are precisely quotients of representables. Indeed, these $F\_x$ arise as image (epi-mono) factorizations of the classifying map $\theta\_x: \hom(A, -) \to F$ of $x \in F(A)$:
$$\theta\_x = (\hom(A, -) \stackrel{epi}{\to} F\_x \stackrel{mono}{\to} F)$$
since the image of the map $\theta\_x$... | 5 | https://mathoverflow.net/users/2926 | 50608 | 31,761 |
https://mathoverflow.net/questions/50613 | -8 | Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra $g$ of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most 4.
What are all types of $g$ such that:
1) $a+b$ can be a root and $a+2b$ is not a root?
2) $a+2b$ can be a root a... | https://mathoverflow.net/users/40886 | Lia algebra strings | 3) is only possible if $a$ is 3 times longer than $b$, so it only happens in $G\_2$.
2) is only possible if $a$ is 2 times longer than $b$ (or if $a=b+a'$ for $a'$ a root 3 times longer), so it happens in all non-simply laced types ($B,C,F$ and $G$).
1) happens in every simple root system other than $A\_1$, since ... | 3 | https://mathoverflow.net/users/66 | 50622 | 31,769 |
https://mathoverflow.net/questions/50522 | 1 | We can "travel" on all the vector space $V =GF(2)^n$ by doing the following
(a) choose a primitive polynomial $P(t)$ of degree $n$ over $GF(2)$.
(b) change vector $ X = (x\_1, \ldots,x\_{n-1}) \in V$ into vector $Y = (y\_1, \ldots, y\_{n-1}) \in V$.
(c) repeat until $V$ is exhausted (2^n times)
where
$y\_1+y\... | https://mathoverflow.net/users/11016 | obtaining all vectors of given length and with with $+-1$ entries from a given one | Simply represent each vector as a binary integer:
$v\_0 = "0 0 0 ... 0 0 0" = b\_{n-1} b\_{n-2}... b\_2 b\_1 b\_0 = 0$
so the binary digit sequence representing $i$ is a decimal number $d$
$$d = \sum\_{j=0}^{j=n-1} b\_j \cdot 2^{j}$$
Given such a binary representation, you can transform the binary digits into t... | 0 | https://mathoverflow.net/users/8676 | 50623 | 31,770 |
https://mathoverflow.net/questions/50609 | 3 | This question might have an easy answer, but my research is far from the region of topology that makes use of classifying spaces of categories, so I can't find it.
For (possibly infinite) integers $0 \leq k\_1 \leq k\_2 \leq \infty$, define a category $X(k\_1,k\_2)$ as follows. There are objects $S\_n$ for any finite... | https://mathoverflow.net/users/11878 | Classifying space of variant on category of simplices | Yes.
Claim 1: $X(k,\infty)$ is contractible.
Proof: There is a functor $X(k,\infty)\times X(k,\infty)\to X(k,\infty)$ given by laying two ordered sets end to end. It admits a natural map from each of the two projection functors, and this implies that the resulting map of spaces (realizations of nerves) is homotop... | 4 | https://mathoverflow.net/users/6666 | 50628 | 31,773 |
https://mathoverflow.net/questions/50634 | 6 | We see the following in an answer to [This Question](https://mathoverflow.net/questions/50557/-) : (mangled by me for my purposes, all errors my fault)
---
Dedekind stated (in 1871) that the ring of algebraic integers is a Bezout Domain. He calls it an important but difficult theorem, and only presents it as an e... | https://mathoverflow.net/users/8008 | Constructive Bezout cofactors in the ring of algebraic integers | This is completely constructive. Given elements $a, b \in O\_K$, the ring of integers in some number field, let $A = (a,b)$ denote the ideal generated by these elements. Compute
the class number $h$ of $K$ (or, if you want, the order you are working in), compute
a generator $c$ with $A^h = (c)$, and set $L = K(\gamma)$... | 5 | https://mathoverflow.net/users/3503 | 50642 | 31,783 |
https://mathoverflow.net/questions/36455 | 9 | Let $G$ be a finite group. I wonder whether the following statement is true, known and written down:
There is a t-structure on the stable $G$-equivariant homotopy category such that the associated heart is isomorphic to the category of Mackey functors (on $B\_G$).
I feel like someone has told me so, but I can't fin... | https://mathoverflow.net/users/2146 | A heart for stable equivariant homotopy theory | Since G is finite, there is no problem with just repeating the proof in the
case $G=e$, using $Z$-graded homotopy group functors on the orbit category. Take
$D^{\leq n}$ to be the spectra whose homotopy groups $\pi\_q(X^H)$ are zero for $q>n$,
and dually for $D^{\geq n}$. The intersection for $n\leq 0$ and $n\geq 0$ ... | 13 | https://mathoverflow.net/users/14447 | 50666 | 31,800 |
https://mathoverflow.net/questions/50591 | 5 |
>
> Suppose $n$ quadric hypersurfaces cut
> out $2^n$ distinct points
> $p\_1,\ldots,p\_{2^n}$ in
> $\mathbb{P}^n$. What is the maximal
> number of points $p\_i$ a quadric can
> contain without containing all of
> them?
>
>
>
For n=2, there is of course a quadric going through any three points and avoiding... | https://mathoverflow.net/users/3996 | Quadrics containing many points in special position | To complement the answers of Sasha and Damiano I would like to give one (completley non-generic) example of a very specific configuration when a quadric contains exactly
$3\cdot 2^{n-2}$ points.
Take in $\mathbb C^n$ the collection of $2^n$ points $(\pm 1,..., \pm 1)$. It is clear that this collection is the interse... | 4 | https://mathoverflow.net/users/943 | 50673 | 31,805 |
https://mathoverflow.net/questions/50665 | 6 | ... curious to me, that is.
Suppose two module filtrations $$ \cdots < A\_3 < A\_2 < A\_1 < \cdots $$ and $$ \cdots < B\_3 < B\_2 < B\_1 < \cdots $$ are comparable in the sense that for all $j$, $ B\_{j+1} < A\_{j} < B\_{j-1} $; then there are natural complexes
$$ \cdots \to \frac{A\_3}{B\_4} \to \frac{A\_2}{B\_3} \t... | https://mathoverflow.net/users/1631 | A curious construction of a chain complex and its homology | I don't know that this qualifies as an answer, but I wanted to point out that your isomorphism arises as the boundary map in a short exact sequence of chain complexes. Note that
\[
\cdots\to \frac{B\_2}{B\_4} \to \frac{B\_1}{B\_3}\to\frac{B\_0}{B\_2}\to\cdots
\]
is a long exact sequence. When viewed as a chain comp... | 7 | https://mathoverflow.net/users/5963 | 50681 | 31,808 |
https://mathoverflow.net/questions/50649 | 13 | One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum\_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum\_{l \geq 0} \binom{l + \chi(C)-1}{l} q^{2l} $$
First of all, I'm not sure what "sufficiently nice" means here. I'm guessing any CW complex ... | https://mathoverflow.net/users/1358 | "C choose k" where C is topological space | I presume you are asking is whether one can make sense of $\binom{X}{k}$ as a space in such a way that it relates to the formula you are asking about.
The answer is yes.
First let $l = 1$. For a space $X$ define $\binom{X}{k}$ to be the configuration space of subsets of X
having cardinality $k$. Then $\binom{X}{1}... | 12 | https://mathoverflow.net/users/8032 | 50696 | 31,816 |
https://mathoverflow.net/questions/50632 | 3 | Does anyone know a proof of the 3 square theorem which uses Cassels' Lemma: n can be expressed in 3 squares of rationals iff n can be expressed in 3 squares of integers.
| https://mathoverflow.net/users/11030 | 3 square theorem | Probably worth pointing out a certain distinction. The quoted property is Pete L. Clark's ADC property, see:
[Must a ring which admits a Euclidean quadratic form be Euclidean?](https://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean)
For positive forms in three var... | 7 | https://mathoverflow.net/users/3324 | 50698 | 31,817 |
https://mathoverflow.net/questions/50657 | 10 | This is a follow up of an interesting [recent question](https://mathoverflow.net/questions/50548/approximating-polynomials-in-rx-using-integer-valued-polynomials) on the topic. The answer given there by fedia shows that the matter is rich and complicated, and I can't resist to submit here a further question.
>
> **... | https://mathoverflow.net/users/6101 | The closures in $C^0(\mathbb{R}, \mathbb{R})$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients | I'd say that the closure of the integer-valued polynomials (IVP) by uniform convergence on bounded sets is the whole set $C^0\big((\mathbb{Z},\mathbb{R}),(\mathbb{Z},\mathbb{R})\big)$ of continuous functions mapping integers to integers.
Let $r\le s\in\ \mathbb{N}$. Then $\big|\binom x s \big| < 1 $ holds for all $x ... | 4 | https://mathoverflow.net/users/6101 | 50708 | 31,820 |
https://mathoverflow.net/questions/50703 | 6 | More explicitly, if $M\_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), how do i properly define multiplication? e.g. suppose $A\_0,A\_1,B\_0,B\_1 \in M\_{2 \times 2}(\mathbb{R})$, and let $f... | https://mathoverflow.net/users/11906 | Defining Multiplication in Polynomials over Rings of Matrices | I have two answers for you, depending on what you have in mind.
You want to add an $x$ to the ring of 2x2 matrices, such that while $x$ commutes with multiples of the identity, it doesn't commute with anything else. You can adjoin a noncommuting indeterminate by using what's called the free product. You take the two ... | 9 | https://mathoverflow.net/users/3711 | 50721 | 31,826 |
https://mathoverflow.net/questions/50710 | 1 | Define $J\colon W\_0^{1,2}(\Omega)\to \mathbb{R}$ by $J(u)=\int\_\Omega u^{p+1}dx$ for $p\in (1,\frac{n+2}{n-2})$.
Is $J'(u)(v)=\int\_\Omega (p+1)u^pvdx$?
| https://mathoverflow.net/users/11909 | Derivative of functional | Yes.
Start with the case of $J$ on $L^{p+1}(\Omega)$, here with any measure space $\Omega$ and $p>1$. Integrate the first order Taylor expansion for $(u+v)^{p+1}$ at $u$ and get
$$\bigg|\, J(u+v)-J(u)-(p+1)\int\_\Omega |u|^p v\, dx\, \bigg| \le p(p+1)\int\_\Omega \big(|u|+|v|\big)^ {p-1} |v|^2 \, dx.$$
Use the Hoe... | 8 | https://mathoverflow.net/users/6101 | 50722 | 31,827 |
https://mathoverflow.net/questions/50302 | 33 | By definition, a von Neumann algebra is a C\*‑algebra A
that admits a predual, i.e., a Banach space Z such that
Z\* is isomorphic to the underlying Banach space of A.
(We require that isomorphisms in the category of Banach
spaces preserve the norm.)
Moreover, a morphism of von Neumann algebras is a morphism
f: A→B of... | https://mathoverflow.net/users/402 | Can we recover a von Neumann algebra from its predual? |
>
> In particular, can we dualize the product on a von Neumann algebra using some kind of tensor product?
>
>
>
The answer to this is explored in a number of papers. AFAIK, it was first considered by Quigg in "Approximately periodic functionals on C\*-algebras and von Neumann algebras.", <http://www.ams.org/math... | 22 | https://mathoverflow.net/users/406 | 50731 | 31,832 |
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