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https://mathoverflow.net/questions/42119 | 12 | Hi,
I want to write a proof that relies on the fact that:
There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that
$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,y)) > 0$.
Note that $x < y \in \mathbb{R}$ are arbitrary.
I'm fairly sure this is true, but am having trouble com... | https://mathoverflow.net/users/10053 | Sets with equal positive measure in every interval | The basic step is to construct a nowhere dense set of positive, and controlled, measure. Then iteratively in each interval where the set is empty, you replace by another such set. See for example a paper by Erdös and Oxtoby, *Partitions of the plane into sets having positive measure in every non-null measurable set*.
... | 8 | https://mathoverflow.net/users/4600 | 42121 | 26,829 |
https://mathoverflow.net/questions/42097 | 12 | Let G be a reductive algebraic group over a field k. Let S be a maximal split torus, Z its centraliser and N its normaliser. The Weyl group W is then defined to be the quotient N(k)/Z(k). Now we cannot hope for W to be realisable as a subgroup of G, but I would like to know how close we can get.
There is a classical ... | https://mathoverflow.net/users/425 | Best approximation to the Weyl group as a subgroup of a reductive group. | Via Theorem 7.2 in Borel-Tits, Groupes reductifs (1965), Tits's lifting result for the Weyl group also applies in the non-split case (for connected groups).
This theorem states that there exists a split subgroup $F$ of $G$ such that $F$ contains the maximal split torus $S$ of $G$ and intersects each relative root gr... | 8 | https://mathoverflow.net/users/3380 | 42123 | 26,831 |
https://mathoverflow.net/questions/42106 | 6 | What is known about spaces of embeddings of contractible manifolds into Euclidean space? I am also curious about the case of small codimension (or even codimension 0). The same question about the configuration spaces in such manifolds.
| https://mathoverflow.net/users/9800 | Embedding theory for contractible manifolds | (This is by far not a complete answer, just an example.) In dimension 4, a [paper of Livingstone](http://nyjm.albany.edu:8000/PacJ/p/2003/209-2-8.pdf) (build on previous work of Lickorish) constructs some (compact with boundary) contractible 4-manifold which embeds in $\mathbb R^4$ in infinitely many (countable) distin... | 6 | https://mathoverflow.net/users/6205 | 42125 | 26,832 |
https://mathoverflow.net/questions/41804 | 2 | Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local coordinate patches of the variety). With respect to this topology the space is a Frechet space (this means that, amongst other... | https://mathoverflow.net/users/1095 | Varieties, Frechet Completions, and Regular Functions | First, $V$ needs to be the type of variety for which the ring of regular functions $O(V)$ separates points. If it doesn't, then that establishes an equivalence relation on $V$ and you might as well pass to the quotient in which points are separated. Thus $V$ is an affine algebraic variety. Moreover, if $V$ is complex, ... | 2 | https://mathoverflow.net/users/1450 | 42129 | 26,834 |
https://mathoverflow.net/questions/42127 | 12 | I learned Bezout's Theorem in class, stated for plane curves (if irreducible, sum of intersection multiplicities equals product of degrees). What is the proper general statement, for projective varieties of degree n?
I think it is something like: If finite, the sum of multiplicities equals the product of degrees.. el... | https://mathoverflow.net/users/10054 | generalization of Bezout's Theorem? | Dear unknown, the most straightforward generalization of Bézout's theorem might be the following. Consider $\mathbb P^n $, projective space over the field $k$, and
$n$ hypersurfaces $H\_1,...,H\_n$in general position in the sense that their intersection is a finite set. Then, calling $h\_i$ their local equations, Bézo... | 26 | https://mathoverflow.net/users/450 | 42131 | 26,836 |
https://mathoverflow.net/questions/42088 | 1 | If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x\_{hat} = (A' A)^{-1} (A' \* e)$ and the covariance of $x\_{hat}$ is $(A' A)^{-1}$
Now, what if I add the linear inequality constraints $Bx > c$? There are algorithms that find the answer for a given ... | https://mathoverflow.net/users/10043 | Inequality-constrained linear-regression, what is the covariance of the estimator? |
>
> Well, the vector $x$ is random right?
>
>
>
It's a parameter, so therefore fixed (yet unknown): the estimator $\hat x$ is a random variable. I would agree with Brian that a covariance matrix will not be all that useful the constraints will mean that the estimator will tend to concentrate around the edges, w... | 1 | https://mathoverflow.net/users/8019 | 42135 | 26,839 |
https://mathoverflow.net/questions/36797 | 31 | The following little trick was introduced by E. Trost
(*Eine Bemerkung zur Diophantischen Analysis*,
Elem. Math. 26 (1971), 60-61). For showing that a diophantine equation
such as $x^4 - 2y^2 = 1$ has only the trivial solution, assume that
$a^4 - 2b^2 = 1$; then the quadratic equation $a^4 t^2 - 2b^2 t - 1 = 0$
has... | https://mathoverflow.net/users/3503 | Trost's Discriminant Trick | T. Nagell in [*Norsk Mat. Forenings Skrifter.* **1**:4 (1921)]
shows that, for an odd prime $q$, the equation
$$
x^2-y^q=1 \qquad (\*)
$$
has a solution in integers $x>1$, $y>1$, then $y$ is even and $q\mid x$.
In his proof of the latter divisibility he uses a similar trick as follows.
Assuming $q\nmid x$ write ($\*$... | 16 | https://mathoverflow.net/users/4953 | 42142 | 26,841 |
https://mathoverflow.net/questions/42141 | 7 | I asked this question on [stats.stackexchange.com](https://stats.stackexchange.com/questions/1441/example-of-a-stochastic-process-that-is-1st-and-2nd-order-stationary-but-not-str) a little while back but didn't get an answer. It was suggested that I post it here at the time. There appears to be some migratory problems ... | https://mathoverflow.net/users/5378 | A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary | How about this:
Consider $X\_1,X\_2$ i.i.d. with $\mathbb{P}(X\_i = 1) = \mathbb{P}(X\_i = 0) = 0.5$. Now define $X\_3 = X\_1 + X\_2 (\text{mod }2)$. Notice that $X\_3$ is independent from $X\_1$ and from $X\_2$ individually. However the three variables $X\_1,X\_2,X\_3$ are not jointly independent.
Now consider a ... | 10 | https://mathoverflow.net/users/7631 | 42150 | 26,844 |
https://mathoverflow.net/questions/42104 | 3 | I am trying to calculate $E(\int\_0^T {W\_s ds})$, where $W\_s$ is a standard Brownian motion.
Now two approaches I can think of:
1) Take a partition of $[0,T]$. Calculate $E(\sum {W\_{t\_i}(t\_{i+1} - t\_i)})$ and take the limit as you shrink the size of the partition.
2) Calculate $\int\_0^T { E(W\_s)ds }$.
H... | https://mathoverflow.net/users/10049 | Expectation of time integral of Wiener process | For approach 2, Fubini's theorem works just as well if you show
$$ \int\_0^T E|W\_s|ds < \infty $$
which is easy. Indeed, perhaps even easier is to note
$$ \int\_0^T E(|W\_s|^2)ds = \int\_0^T s ds = \frac{1}{2}T^2 < \infty$$
and use Jensen/Hölder/Cauchy-Schwarz.
| 5 | https://mathoverflow.net/users/4832 | 42156 | 26,847 |
https://mathoverflow.net/questions/42147 | 2 | I need a name for a regular category where the inverse image maps have right adjoints.
If $\mathcal C$ is a regular category, then the poset of subobjects $\mathsf{Sub}(X)$ of any object $X$ is a semilattice and the inverse image map of any arrow $f:X\to Y$ has a left adjoint $\exists\_f:\mathsf{Sub}(X) \to \mathsf{S... | https://mathoverflow.net/users/3603 | name my cat: regular categories where inverse images also have right adjoint | From Freyd and Scedrov's book *Categories, Allegories*: a **logos** is a regular category in which $Sub(A)$ is a lattice for each object $A$, and in which the inverse-image operation $f^\*: Sub(B) \to Sub(A)$ has a right adjoint for each morphism $f: A \to B$ (page 117).
| 4 | https://mathoverflow.net/users/2926 | 42164 | 26,851 |
https://mathoverflow.net/questions/42153 | 13 | Yesterday my first work in mathematics was sent to a publisher, and of course I'm interested in its usefulness. But I know, that sometimes it is hard to get a paper, it is not available for free. I hope my paper will not be of that kind. I hope it will be simple to find it and to read it, I hope a few hundred people wi... | https://mathoverflow.net/users/8134 | Which rights do mathematicians usually have on their published works and how do they use them? | Read any contract you sign carefully, otherwise you may lose rights you wanted to keep.
In 2000, CRC Press sued Eric Weisstein because he posted free updates to the web of a mathematics book he had written and published with them.
Usually non-commercial publishers like the AMS allow you to keep any rights you want, w... | 14 | https://mathoverflow.net/users/51 | 42175 | 26,858 |
https://mathoverflow.net/questions/42190 | 0 | I am interested in solving the following equations:
$f(x) + \int\_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = 0$
and
$f(x) + \int\_{\alpha(x)}^{\beta(x)}K(x,t)u(t)dt = u(x)$
when $u(x)$ is the unknown function defined on $[0^+,\infty)$ and all other functions are known and are assumed to have convenient differenti... | https://mathoverflow.net/users/9950 | how to solve general integral equations with both variable lower and upper bounds | If $\alpha$ and $\beta$ are bounded functions, with $a\le\alpha\le\beta \le b$, then your equations are of standard type since they can be written as
$$f(x)+\int\_a^b H(x,t) u(t) dt =0 $$
where
$$ H(x,t)=K(x,t)\chi\_{[\alpha(x),\beta(x)]}(t). $$
Here $\chi\_A$ denotes the characteristic function of the set $A$.
If $\... | 1 | https://mathoverflow.net/users/7294 | 42193 | 26,867 |
https://mathoverflow.net/questions/42181 | -1 | Hi,
Do you think the following limits are correct?
$\displaystyle\lim\_{d\to\infty}\frac{\sum\limits\_{k=1}^{d} {\phi(N) \choose k} {d-1 \choose k-1}}{\phi(N)^d}=0$
$\displaystyle\lim\_{N\to\infty}\frac{\sum\limits\_{k=1}^{d} {\phi(N) \choose k} {d-1 \choose k-1}}{\phi(N)^d}=c$
I plotted the equations and gues... | https://mathoverflow.net/users/10062 | Limit involving the totient function and combination | We have $$\sum\_{k = 1}^d \binom{\phi(N)}{k} \binom{d - 1}{k - 1} = \binom{d + \phi(N) - 1}{d}$$ (this is the Vandermonde identity). Thus, with $N$ fixed the numerator of your fraction is polynomial in $d$ and so the first result follows (except for $N = 1, 2$).
Edited to add:
Okay, and the second result follows by t... | 13 | https://mathoverflow.net/users/4658 | 42194 | 26,868 |
https://mathoverflow.net/questions/42176 | 18 | Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and vindicate the topological program for complex analysis. Based on comments made in letters to Frege a major motivation f... | https://mathoverflow.net/users/2833 | What was Weierstrass's counterexample to the Dirichlet Principle? | Weierstrass simply observed that not every problem in the calculus of variations would have a solution. He considered the example
$$D[y]=\int\_{-1}^{1}x^2\left(\frac{d y}{dx}\right)^2dx\to \min,$$
where the functional $D[y]$ is minimized over continuous functions having piecewise continuous first derivatives in $[-1,1]... | 25 | https://mathoverflow.net/users/5371 | 42195 | 26,869 |
https://mathoverflow.net/questions/41958 | 30 | I think most students who first learn about (finite) groups, eventually learn about the possibility of classifying certain finite groups, and even showing certain finite groups of a given order can't be simple (I'm pretty sure every beginning algebra text has some exercises like this). Up to order 1000, I think there i... | https://mathoverflow.net/users/1446 | No simple groups of order 720? | You can only really ask for a proof that avoids a particular fact or construction, if that fact or construction is difficult enough or distinct enough from the thing that you're proving. By that principle, the comments imply that it would be enough to show that Derek Holt's subgroup $\langle a,b,c,e \rangle$ has index ... | 20 | https://mathoverflow.net/users/1450 | 42199 | 26,873 |
https://mathoverflow.net/questions/42186 | 38 | There are many equivalent ways of defining the notion of compact space, but some require some kind of choice principle to prove their equivalence. For example, a classical result is that for $X$ to be compact, it is necessary and sufficient that every ultrafilter on $X$ converge to a point in $X$. The necessity is easy... | https://mathoverflow.net/users/2926 | Does "compact iff projections are closed" require some form of choice? | Martin Escardó wrote a very nice note ["Intersections of compactly many open sets are open"](http://www.cs.bham.ac.uk/~mhe/papers/compactness-submitted.pdf) which you might want to read.
| 25 | https://mathoverflow.net/users/1176 | 42208 | 26,878 |
https://mathoverflow.net/questions/42201 | 13 | Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed geodesics projecting to each non-trivial homotopy class of $M$, and one can support an invariant probability measure on each ... | https://mathoverflow.net/users/7631 | What are the zero entropy invariant measures for an Anosov geodesic flow? | There are lots of zero entropy invariant probability measures, many more than just the obvious ones supported on periodic orbits. As you suggest in the question, one can understand the general case by just considering what happens for symbolic systems.
**Explicit example:** Let $\alpha$ be irrational and let $a\_n$ d... | 7 | https://mathoverflow.net/users/5701 | 42214 | 26,883 |
https://mathoverflow.net/questions/42215 | 25 | The classic example of a non-measurable set is described by [wikipedia](http://en.wikipedia.org/wiki/Vitali_set#Construction_and_proof). However, this particular construction is reliant on the axiom of choice; in order to choose representatives of $\mathbb{R} /\mathbb{Q}$.
"Since each element intersects [0,1], we can... | https://mathoverflow.net/users/3121 | Does constructing non-measurable sets require the axiom of choice? | In the 1960's, Bob Solovay constructed a model of ZF + the axiom of dependent choice (DC) + "all sets of reals are Lebesgue measurable." DC is a weak form of choice, sufficient for developing the "non-pathological" parts of real analysis, for example the countable additivity of Lebesgue measure (which is not provable i... | 35 | https://mathoverflow.net/users/6794 | 42220 | 26,886 |
https://mathoverflow.net/questions/42222 | 3 | Hi all,
Sorry if this question is not the right level for mathoverflow, but I already tried math.stackexchange and received no answers.
Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to ... | https://mathoverflow.net/users/7842 | Do functions defined on global elements give rise to arrows in a well-pointed topos? | Here's a counterexample. Take $\mathcal E$ to be the topos of sets and functions of some countable model of ZFC (or a suitable weaker set theory, if you're worried that ZFC might be inconsistent). This is a well-pointed topos. The natural-number object $N$ in $\mathcal E$ has a countable infinity of global elements. So... | 5 | https://mathoverflow.net/users/6794 | 42223 | 26,887 |
https://mathoverflow.net/questions/42219 | 13 | This is a question my son Bob asked me. For some sets it is relatively easy
to test for membership but a lot more difficult to find members, and for others
the reverse is true. Here is an elementary example to get the idea across. An
$m \times n$ real matrix $M$ defines a linear map $x \mapsto M x = y$, from
${\mathbb... | https://mathoverflow.net/users/7311 | Is there a name for sets for which it is easier to test membership than to find members---and vice versa? | This phenomenon occurs both positively and negatively in
many parts of logic, but to my knowledge, there is no
particular adjective that is always used in such
situations.
* In classical *computability theory*, the first phenomenon does not
occur. If one can computably test membership in a set, in
the usual Turing se... | 14 | https://mathoverflow.net/users/1946 | 42232 | 26,892 |
https://mathoverflow.net/questions/42192 | 8 | In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the particles. As a consequence, the configuration space of a N-particle system is the set of the distances on a set of N elem... | https://mathoverflow.net/users/9584 | Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension | Your question is not well stated.
In particular I did not understand why embedding into a plane is "preferred embedding".
Here are some associations...
**4-point case.**
A generic 4-point metric space can be isometrically embedded into two different model planes (i.e. simply connected surfaces of constant curvatu... | 6 | https://mathoverflow.net/users/1441 | 42233 | 26,893 |
https://mathoverflow.net/questions/42235 | 4 | Hey everyone!
Lately I remembered an exercise from an algebra class from Jacobson's book: Prove that if an element has more than one right inverse then it has infinitely many, Jacobson attributes this excercise to Kaplansky. Regardless of the solution I began to wonder:
Does anybody know any explicit examples of ri... | https://mathoverflow.net/users/9187 | Rings with right inverses | Let $M$ be a module (over some ring) such that $M$ is isomorphic to $M\oplus M$, for example an infinite-dimensional vector space over a field. Let $R$ be the ring of endomorphisms of $M$. Let $f\in R$ be projection of $M\oplus M$ on the first factor composed with an isomorphism $M\to M\oplus M$. Then $f$ has as many r... | 6 | https://mathoverflow.net/users/6666 | 42237 | 26,894 |
https://mathoverflow.net/questions/42139 | 8 | Let there be $n$ points on a unit circle. It is known they come from "normal" distribution around particular unknown direction (i.e. sum of 2 "normal" distributions on circle - one centered at point $p$ and the other at its opposite $-p$).
What is the best way to estimate this direction? By best I mean an algorithm th... | https://mathoverflow.net/users/10032 | Estimating direction from a distribution on a circle | This seems too simple to be true, but, combining some of the ideas posted earlier,
I think you could just interpret the vectors as complex numbers and take the RMS.
Squaring will turn the bimodal distribution into a unimodal one.
Then the square roots of the mean should give a good estimate of the modes of the origin... | 10 | https://mathoverflow.net/users/10075 | 42240 | 26,897 |
https://mathoverflow.net/questions/42236 | 3 | Alexander's Theorem guarantees that every oriented link is the closure of some braid. In other words, the map
$$ \displaystyle \coprod\\_n \mathcal B\_n\longrightarrow \{\text{ oriented links }\} $$
is surjective. One algorithm (I'm actually not sure that it's Alexander's original demonstration) involves choosing a... | https://mathoverflow.net/users/813 | Is a positive link the closure of a positive braid? | Rudolph proved that [positive links are strongly quasipositive](http://www.ams.org/mathscinet-getitem?mr=1734423). [This paper](http://www.ams.org/mathscinet-getitem?mr=894383) might also be relevant, which allows one to create a braid with the same number of Seifert circles and writhe. However, Yamada's algorithm does... | 5 | https://mathoverflow.net/users/1345 | 42253 | 26,905 |
https://mathoverflow.net/questions/42247 | 0 | I have Edwards and Titmarsch books on Riemann zeta function with me. I could not find (maybe I did not read through that carefully), but are there results similar to the form like the one given below:
> Is there a non-trivial zero $\rho$ starting from which $(\Im(\rho)\log(2)/2\pi )$ runs only through integer values... | https://mathoverflow.net/users/2865 | Curious about a question on zeta zeros? | Such a conjecture is false.
EDIT: A simpler argument - a more precise asymptotic for the number of
zeroes $N(t)$ of imaginary part $\le t$ (counted with multiplicity) is
$$N(t) = \frac{t}{ \pi} \log \frac{t}{2 \pi e} + o(\log t),$$
This is enough to show that, for any fixed $\epsilon > 0$,
$$N(t + \epsilon) - N(t) \s... | 6 | https://mathoverflow.net/users/nan | 42254 | 26,906 |
https://mathoverflow.net/questions/42204 | 0 | I have been reading the paper - "[Introduction to Quantum Fisher Information](http://arxiv.org/abs/1008.2417)".
In section 1.2 the author talks about the linear map $\mathbb{J}\_D$, which he defines as follows:
Let $D \in M\_n$ be a positive invertible matrix. The linear mapping $\mathbb{J}^f\_D:M\_n \to M\_n$ is def... | https://mathoverflow.net/users/10071 | Linear Mapping and integration | One way to think of the operation ${J}\_D^f$ is as follows: $M\_n$ has the structure of a Hilbert space by the inner product $\langle X, Y \rangle = {\rm Tr}(X^\* Y) = {\rm Tr}(Y X^\*)$. Then the transformations $L\_D R\_{D}^{-1}$ and ${R}\_D$ are positive operators on this Hilbert space. Since they commute, you may ta... | 0 | https://mathoverflow.net/users/9942 | 42259 | 26,910 |
https://mathoverflow.net/questions/42217 | 22 | I'm absolutely new to this stuff I'm asking about, so I hope this is not nonsense.
If X is a smooth scheme over a perfect field k, I can study its motivic cohomology in the sense of Voevodsky and Morel.
More precisely, to $\mathbb{Z}$ there is associated an Eilenberg-MacLane T-Spectrum. I'll write $H^{p,q}(X,\mathb... | https://mathoverflow.net/users/473 | Motivic Cohomology vs. Chow for singular varieties? | The answer to question 1) is yes. However, Chow groups do not form what we should call a cohomology theory, but are part of a Borel-Moore homology theory. This ambiguity comes from the fact that, by Poincaré duality, motivic cohomology agrees with motivic Borel-Moore homology for smooth schemes (up to some reindexing),... | 29 | https://mathoverflow.net/users/1017 | 42268 | 26,913 |
https://mathoverflow.net/questions/42263 | 8 | There are different strategies for numbering displayed math. The most common are
1. Number only the formulas you reference to. It makes your paper more clean and gives more freedom to the editor (i.e. making this math inline).
2. Number all displayed math. Even if you don't reference your formulas, take care of t... | https://mathoverflow.net/users/8134 | The main ideas in choosing the strategy for numbering displayed math | Fiktor, many journals explicitly indicate option (1). Some also ask to use single/double numeration, the list of references in alphabetical/appearance order, and many other things. Therefore,
>
> Strategy (4): check with the journal you are planning to publish your contribution.
>
>
>
As for (2) and (3), are ... | 3 | https://mathoverflow.net/users/4953 | 42270 | 26,914 |
https://mathoverflow.net/questions/42272 | 27 | I realise that I lack some intuition into how a curve (or surface, or whatever) looks geometrically, from just looking at the equation. Thus, I sometimes resort to some computer program (such as *Mathematica*) to draw me a picture. The problem is, all these programs require input of the form $y=f(x)$, whereas my curve ... | https://mathoverflow.net/users/1481 | Are there any good computer programs for drawing (algebraic) curves? | Mathematica does this just fine. You're looking for the command ContourPlot, as in
ContourPlot[y^3+x^3-6x^2y==0,{x,-5,5},{y,-5,5}].
A more serious issue is that if you're trying to do algebraic geometry over $\mathbb C$, the real picture isn't always terribly enlightening.
| 18 | https://mathoverflow.net/users/7399 | 42273 | 26,916 |
https://mathoverflow.net/questions/42276 | 21 | Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x\_1, \ldots, x\_n$ algebraically independent such that $R$ is integral over $k[x\_1, \ldots, x\_n]$.
Does one have a similar statement, under good assumptions, if $k$ is not a field but a ring ? In this dis... | https://mathoverflow.net/users/2330 | Noether's normalization lemma over a ring A | <http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf>
Supplementary notes from Mel Hochster's commutative algebra class. They discuss, in particular, a generalization of Noether normalization to integral domains.
| 20 | https://mathoverflow.net/users/1353 | 42281 | 26,921 |
https://mathoverflow.net/questions/42283 | 6 | **Definition 1:** Suppose $B$ is a $C^\* $-algebra. $A$ is massive $C^\* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|\_A$ is irreducible;
3. if representations $\pi$ and $\pi'$ aren't (unitary) equivalent then $\pi|\_A$ and $\pi... | https://mathoverflow.net/users/8134 | Does there exist any massive proper $C^*$-subalgebra? | It was proved by Karl H. Hofmann and Karl-H.Neeb [here](https://doi.org/10.1007/BF01270691 "Hofmann, K.H., Neeb, K.H. Epimorphisms of C^*-algebras are surjective. Arch. Math 65, 134–137 (1995). zbMATH review at https://zbmath.org/?q=an:0829.46044") (see [here](https://arxiv.org/abs/funct-an/9405003) for the preprint), ... | 8 | https://mathoverflow.net/users/8176 | 42289 | 26,923 |
https://mathoverflow.net/questions/42249 | 8 | I have often heard that Lie groups classify geometry. For example that $O(n)$ is about real manifolds, $U$ is about almost complex manifolds, $SO(n)$ about orientable real manifolds and so on.
I have also heard that manifolds can be defined very generally by patching local pieces via a pseduogroup of morphisms.
My... | https://mathoverflow.net/users/nan | How do Lie groups classify geometry? | [R.W.Sharpe, Differential Geometry - Cartan's Generalization of Klein's Erlangen Program](http://books.google.com/books?id=Ytqs4xU5QKAC&pg=PA173&dq=cartan%27s+generalization+of+klein%27s+erlangen&hl=it&ei=aGK4TPfXLIPKswbmsfyqDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q&f=false)
| 3 | https://mathoverflow.net/users/4721 | 42290 | 26,924 |
https://mathoverflow.net/questions/40860 | 10 | Note: in this post, every topological group under consideration is assumed to be Hausdorff.
Given a locally compact abelian group, one can construct its dual group, i.e. its group of (unitary) characters. This turns out to be a locally compact abelian group with respect to the compact-open topology. Pontryagin dualit... | https://mathoverflow.net/users/7392 | Which groups can be recovered from their unitary dual? | The nicest way of phrasing it is the following. Let $\mathcal H$ be the category of Hilbert spaces with unitary maps between them. For each locally compact group $G$, one can define a functor
$$Rep\_G : {\mathcal H} \to Top$$
with $Rep\_G(H) = \hom(G,U(H))$, where the space of homomorphisms is endowed with the compact-... | 8 | https://mathoverflow.net/users/8176 | 42294 | 26,927 |
https://mathoverflow.net/questions/33931 | 5 | Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the relative dimension is at least 1. Do we know anything about the singularities of Z? Z is normal more or less by construction... | https://mathoverflow.net/users/7756 | Possible singularities of the base of a Mori fiber space | Under the assumption that $X$ is $\mathbb{Q}$-factorial, section 5 of the paper <https://arxiv.org/pdf/math/0606666> addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the pair $(X,\Delta)$ is klt, so is the base $(Z,\Delta\_Z)$ for some $\Delta\_Z$. As the example of Prokhor... | 6 | https://mathoverflow.net/users/10083 | 42301 | 26,931 |
https://mathoverflow.net/questions/42174 | 20 | (**Edit:** I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first question in the case of finite type modules over a noetherian commutative ring. Mariano has given a slick negative answer... | https://mathoverflow.net/users/1 | Can a module be an extension in two really different ways? | It is worth noting some very interesting cases when the answer is yes. An amazing [result by Miyata](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-7/issue-1/Note-on-direct-summands-of-modules/10.1215/kjm/1250524308.full) states that if $R$ is Noetherian and commutative, $M,N$ are finitely gener... | 20 | https://mathoverflow.net/users/2083 | 42309 | 26,936 |
https://mathoverflow.net/questions/38931 | 2 | As an exercise, I'm trying to show that for an $(n-1)$-connected space $L$ with $\pi=\pi\_n(L)$, the map $\iota\_L:L\rightarrow K(\pi,n)$ associated to the fundamental class $\iota\_L\in H^n(L;\pi)$ induces an isomorphism $\pi\_n(L)\rightarrow \pi\_n(K(\pi,n))$. After playing with this for a while, I've found that this... | https://mathoverflow.net/users/303 | How can I show that the map L-->K(\pi_n(L),n) representing the fundamental class of an (n-1)-connected space is an isomorphism on \pi_n? | Here is a solution, based on Jeffrey's comment:
Let $f:L\rightarrow K(\pi,n)$ be the map classifying $\iota$. We will show that the induced map $f\_\*:H\_i(L)\rightarrow H\_i(K(\pi,n))$ is isomorphic for $i \lt n+1$ and epimorphic for $i=n+1$. This will imply by Whitehead's theorem that the same is true for $f\_\# :\... | 1 | https://mathoverflow.net/users/303 | 42328 | 26,945 |
https://mathoverflow.net/questions/42331 | 12 | Every symplectic form on a manifold $M^n$ determines a De Rham cohomology class in $H^2(M)$ (often a nontrivial class), and this in turn determines a class in $H\_{n-2}(M)$. What in general can be said about this class? For example, over the rationals this class is represented by a submanifold of $M$; is it possible to... | https://mathoverflow.net/users/4362 | What is the Poincare dual of a symplectic form? | One of the big advances in symplectic topology in the 90s was Donaldson's theorem that when the symplectic class is integral, high multiples of its dual are represented by symplectic submanifolds.
These submanifolds behave like hyperplane sections in algebraic geometry; for instance, they satisfy the Lefschetz hyper... | 24 | https://mathoverflow.net/users/2356 | 42334 | 26,947 |
https://mathoverflow.net/questions/42344 | 6 | I have no idea how to achieve this, any help would be greatly appreciated and very useful to me.
I have a loop in some computer code, that loops through every single combination of 7 on bits in a 64 bit integer.
For example,
```
Permutation 1: 00...001111111
Permutation 2: 00...010111111
Permutation 3: 00...0... | https://mathoverflow.net/users/9667 | Convert integer to permutation number | Suppose that your 64 bit number is $n=a\_{63} a\_{62} ... a\_0$. Let $i\_7>i\_6>\cdots>i\_1$ be those indices $j$ for which $a\_j=1$. Then
$$ whatPermutation(n)=1+{i\_7\choose 7} +{i\_6\choose 6}+\cdots+{i\_1\choose 1}. $$
For instance, if $n= 00\cdots 110001011011$, then
$$ whatPermutation(n)= 1+{11\choose 7}+{10\c... | 17 | https://mathoverflow.net/users/2807 | 42347 | 26,954 |
https://mathoverflow.net/questions/42353 | 2 | Let $V$ be a vector space with dimension greater than 1 over the field $F$ and $Sim = \{(f\in \operatorname{Lin}(V))\mapsto gf(g^{-1}) : g\in \operatorname{GL}(V)\}$, ie $Sim$ is the set of all similarity transforms on $\operatorname{Lin}(V)$. Let $Aut$ be the set of all automorphisms of the $F$-algebra $\operatorname{... | https://mathoverflow.net/users/nan | Are all automorphisms of Lin(V) given by similarity transforms? | Let $M$ denote the $F$-algebra $\hom(V, V)$. An $F$-algebra homomorphism $M \to \hom(V, V)$ is tantamount to an $M$-module structure on $V$ in the category of $F$-vector spaces. But the category of $M$-modules is equivalent to $Vect\_F$ (there is a Morita equivalence between the categories given by the functor $- \otim... | 2 | https://mathoverflow.net/users/2926 | 42355 | 26,959 |
https://mathoverflow.net/questions/42332 | 3 | I don't know anything about graph theory and I was wondering about something :
If you draw two parallel rows of n points in $\mathbb R^4$ and link each point with all the points in the opposite row except the one right in front of it, and then allow the points to move where ever, what do you get ?
n=2 gives two lin... | https://mathoverflow.net/users/5375 | On a special kind of graph connectig n point to n points. | These are the [crown graphs](http://en.wikipedia.org/wiki/Crown_graph).
| 4 | https://mathoverflow.net/users/440 | 42358 | 26,960 |
https://mathoverflow.net/questions/42356 | 6 | Recall that a prime $\mathfrak{p}$ is called nonsingular (regular) if the localization at that prime is a regular local ring. If all primes of a ring $R$ are nonsingular, $R$ is called regular. Let $S\subseteq R$ be a ring extension, and let $\mathfrak{q}$ be a prime ideal of $S$. Then $R$ is called regular above $\mat... | https://mathoverflow.net/users/1353 | Does regularity of a prime ideal in the fibre imply regularity of the prime? | First note that it is not true that regularity above $\mathfrak q$ implies regularity of the fibre.
For example, consider the map $\mathbb k[s] \to \mathbb k[t]$ given by $s \mapsto t^2$.
Each prime in $k[t]$ is regular, and so in particular each prime of $k[t]$ above
the prime $(s)$ is reqular. (In fact there is jus... | 6 | https://mathoverflow.net/users/2874 | 42360 | 26,962 |
https://mathoverflow.net/questions/42329 | 4 | It's clear that the axiom of replacement can be used to construct very large sets, such as
$$
\bigcup\_{i=0}^\infty P^i N,
$$
where $N$ is the natural numbers. I assume that it can be used to construct sets much lower in the Zermelo hierarchy, such as sets of natural numbers, but I don't know of an example. Is there a... | https://mathoverflow.net/users/3711 | Replacement and Sets of Natural Numbers | This probably isn't what you are looking for, but one can write down an explicit Diophantine equation for which ZFC proves that it has a solution, but ZFC minus replacement does not (assuming it is consistent). Namely, use Godel encoding and the solution of Hilbert's 10th problem to write down a Diophantine equation wh... | 1 | https://mathoverflow.net/users/49 | 42364 | 26,965 |
https://mathoverflow.net/questions/42339 | 3 | Let $T$ be a rooted tree with root $r$. Say an ordering $v\_1,\ldots,v\_n$ of the vertices of $T$ is a *search order* if $v\_1=r$ and for all $2 \leq i \leq n$, there is $j < i$ such that $v\_j$ is the parent of $v\_i$. In other words, parents are explored before their children in the order.
For a given search order... | https://mathoverflow.net/users/3401 | Theorems about the directed bandwidth of a rooted tree? | What you call width is known more specifically as "directed bandwidth". It's known to be NP-complete even for trees; see [Complexity Results for Bandwidth Minimization, M. R. Garey, R. L. Graham, D. S. Johnson and D. E. Knuth, SIAM Journal on Applied Mathematics Vol. 34, No. 3 (May, 1978), pp. 477-495](http://www.jstor... | 4 | https://mathoverflow.net/users/440 | 42365 | 26,966 |
https://mathoverflow.net/questions/42322 | 0 | Is every action from an amenable group amenable on a unital $C^\*$-algebra?
| https://mathoverflow.net/users/9401 | Is every action from an amenable group amenable on a unital $C^*$-algebra? | Yes it is. It follows from Theorem 3.3 of [1] and the fact that the trivial action of an amenable group on $\mathbb{C}$ is amenable. More modern reference is [2] (in particular Section 4.3).
[1] C. Anantharaman-Delaroche. Systèmes dynamiques non commutatifs et moyennabilité. Math. Ann., 279(2):297–315, 1987.
[2] N.... | 3 | https://mathoverflow.net/users/9942 | 42367 | 26,967 |
https://mathoverflow.net/questions/42366 | 4 | Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). Is it possible that $m$ is an *isomorphism* $V\otimes V \overset \sim \to V$? Yes: $V$ can be zero-dimensional, or $V$ can... | https://mathoverflow.net/users/78 | Can a commutative, associative "multiplication" on an infinite-dimensional vector space be an isomorphism? | It is impossible. Let $x$ and $y$ be linearly independent vectors in $V$. Then $x \otimes y \neq y \otimes x$ but $m(x \otimes y) = m(y \otimes x)$.
| 10 | https://mathoverflow.net/users/121 | 42368 | 26,968 |
https://mathoverflow.net/questions/42274 | 1 | This is just an extension of my previous question [Tightness of probabilty distributions](https://mathoverflow.net/questions/41259/tightness-of-probabilty-distributions)
Let $\mathcal{P}(\mathbb{N})$ be the set of all PMF's on $\mathbb{N}=\{1,2,\dots \}$. Let $E$ be a convex subset of $\mathcal{P}(\mathbb{N})$ and $Q\n... | https://mathoverflow.net/users/7699 | Showing non-attainment of supremum | I think that in your assumption, the supremum is actually attained.
Consider the set
$$\hat E:=\{tp \ :\ t\geq0 \ , \quad p\in E\ \} \cap\bar B(0,1;\ell^\alpha).$$
Since $E$ is convex, $\hat E$ is convex too (here $\bar B(0,1;\ell^\alpha)$ denotes the closed unit ball of the sequence space $\ell^\alpha$).
Moreover... | 2 | https://mathoverflow.net/users/6101 | 42376 | 26,973 |
https://mathoverflow.net/questions/42161 | 19 | **Edit:** I think LMO is correct. Massuyeau has a nice explanation [here.](http://ldtopology.wordpress.com/2010/10/14/a-problem-with-lmo/)
**Edit:** Renaud Gauthier has retracted the claim of an error in the foundations of the LMO construction, and has withdrawn both preprints from arXiv.
Original post follows:
... | https://mathoverflow.net/users/9417 | Propagation of an error in the LMO invariant? (Revision: I don't think LMO is wrong!) | Having been a part of the LMO story from its beginning, and having read and checked all relevant papers carefully at the time, and having taken part in many cross-checks that the LMO invariant passed (normalization-compatibility with Reshetikhin-Turaev, various explicit computations), and having consulted on email with... | 17 | https://mathoverflow.net/users/8899 | 42380 | 26,977 |
https://mathoverflow.net/questions/42310 | 45 | I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in Borel sets on a locally compact space $X$. I can also assume that the support of the measure is $X$, if it helps...
I... | https://mathoverflow.net/users/5301 | When is $L^2(X)$ separable? | Without loss of generality we can assume that the support of the measure equals $X$
(i.e., the measure is faithful),
because we can always pass to the subspace defined by the support of the measure.
The space $^2(X)$ is independent of the choice of a faithful measure and depends only
on the underlying [enhanced measu... | 48 | https://mathoverflow.net/users/402 | 42383 | 26,979 |
https://mathoverflow.net/questions/42371 | 6 | The adjoint of the exterior derivarive is defined by
$\delta:=(-1)^k\ast^{-1}d\ast$,
but I need a way which avoids the Hodge $\ast$ operator.
>
> Is there another definition?
>
>
>
For example, for positive definite metric there is the alternative of defining it by
$\int\_M \langle d\alpha,\beta\rangle\_... | https://mathoverflow.net/users/10095 | Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator? | On any Riemannian manifold, any differential operator has a formal adjoint.
This has nothing to do with functional analysis, it is a pure calculus fact that uses little more than Stokes theorem.
This is because taking adjoints is linear, preserves (or rather reverses)
composition and commutes with multiplication b... | 10 | https://mathoverflow.net/users/9928 | 42384 | 26,980 |
https://mathoverflow.net/questions/42391 | 10 | Dear MO Community,
Let $N$ be a prime, and let $X\_0(N)$ be the classical modular curve over $\mathbb{Q}$. We know ([1]) that, if there are noncuspidal points in $X\_0(N)(\mathbb{Q})$, then $N \in$ {${ \mbox{primes } \leq 19} $} $ \cup $ {37,43,67,163}.
The basic question of this post is:
>
> Are there simila... | https://mathoverflow.net/users/5744 | Rational Isogenies of Prime Degree | I'm glad to have stumbled on your question -- I'm actually working on something along the lines of what you're asking about now with Eric Larson. If what we think we've proven is true (and don't quote this yet, since we're not yet even done with the write-up,) then in fact you can say something stronger: you cannot hav... | 11 | https://mathoverflow.net/users/7108 | 42394 | 26,984 |
https://mathoverflow.net/questions/42298 | 10 | I have always been wondering
>
> why the term "model" is used by mathematicians (especially in mathematical logic) in a conceptually different (even opposite) way than it is used by other scientists, e.g. physicists, biologists, chemists, economists etc.
> And when has this terminology first arisen?
>
>
>
In... | https://mathoverflow.net/users/4721 | The use of the word "model" in Mathematical Logic vs the same word in Natural Sciences | * yes, it is strange, because one expects words to have similar meaning even in different contexts (but as others have noted, whether this is strange or not, there are all sorts of words that have multiple meanings, and even contradictory ones).
* yes, I agree in your assessment of how model has two, not contradictory,... | 0 | https://mathoverflow.net/users/3237 | 42397 | 26,985 |
https://mathoverflow.net/questions/42393 | 5 | Pressing the envelope, presumably the best scenario would be a simple proof of the Prime Number Theorem. After all, Wilson’s Theorem gives a necessary and sufficient condition, in terms of the Gamma Function, for a number to be a prime, and Stirling’s Formula specifies the asymptotic behaviour of the Gamma Function.
| https://mathoverflow.net/users/8649 | Has Stirling’s Formula ever been applied, with interesting consequence, to Wilson’s Theorem? | Using Robbins' [1] form of Stirling's formula,
$$\sqrt{2\pi}n^{n+1/2}\exp(-n+1/(12n+1))< n!< \sqrt{2\pi}n^{n+1/2}\exp(-n+1/(12n))$$
we get
$$\left\lceil\sqrt{2\pi}(n-1)^{n-1/2}\exp(-n-1+1/(12n-11))\right\rceil$$
$$\le (n-1)!\le$$
$$\left\lfloor\sqrt{2\pi}(n-1)^{n-1/2}\exp(-n-1+1/(12n-12))\right\rfloor$$
which i... | 36 | https://mathoverflow.net/users/6043 | 42398 | 26,986 |
https://mathoverflow.net/questions/42401 | 0 | My question is most precisely stated in the title. As an example, if we consider base 10, and k=4, then I am asking, is it possible to have a sequence of length 10^4 + 3, such that each 4 digit number appear exactly once in a consecutive block? This is related to finding a Hamiltonian cycle in a b-regular graph. But on... | https://mathoverflow.net/users/4923 | how to find a sequence of digits in base b such that each consecutive block of size k appears exactly once? | Google for "generating De Bruijn sequences".
| 5 | https://mathoverflow.net/users/1409 | 42403 | 26,990 |
https://mathoverflow.net/questions/41816 | 13 | Given three points $a,b,c$ in a (geodesic) metric space $X$, one defines a *comparison angle* $\angle(a,b,c)$ by the cosine law:
$$
\angle(a,b,c) = \arccos \frac{|ab|^2 + |ac|^2 - |bc|^2}{2\cdot|ab|\cdot|ac|}
$$
where $|ab|$, $|ac|$ and $|bc|$ are distances in $X$. In other words, $\angle(a,b,c)$ is the angle of a pla... | https://mathoverflow.net/users/4354 | Metric angles in Riemannian manifolds of low regularity | To Deane Yang, concerning the smoothness of geodesics. These issues have been studied by Hartman
and his coauthors more than 40 years ago. Together with Calabi he claimed to prove
that in a $C^{\alpha}$ continuous metric all geodesics are uniformly $C^{1,\alpha}$.
(I do not have mathscinet access now, but I think the... | 8 | https://mathoverflow.net/users/10109 | 42422 | 27,003 |
https://mathoverflow.net/questions/42404 | 2 | Assuming that one needs $k$ quantifiers to express that a graph contains an $k$-cycle, $\lfloor n/2 \rfloor$ counting quantifiers suffice to express that a graph is an $n$-cycle:
*G has exactly $n$ nodes, each node has exactly 2 neighbors, and G doesn't contain a 3-cycle, a 4-cycle, ... and an $\lfloor n/2 \rfloor$-c... | https://mathoverflow.net/users/2672 | Expressing a graph property with counting quantifiers | The previous answer showed that quantifier rank $\log\_2(k) \pm 1$ is sufficient and necessary for expressing "A is a $k$-cycle". This implies a lower bound of at least $\log\_2(k)$ quantifiers, but not an upper bound since quantifier rank counts nesting-depth of quantifiers and not the number of quantifiers. However, ... | 2 | https://mathoverflow.net/users/9355 | 42423 | 27,004 |
https://mathoverflow.net/questions/42429 | 2 | In the definition of a metric space, replace the triangle inequality by the weaker inequality
*d* (*x*, *z*) ≤ *C* max {*d* (*x*, *y*), *d* (*y*, *z*)},
where C is a positive constant (depending on the "metric", but not on the points *x*, *y*, *z*). Had structures like this ever been studied?
One can associate a... | https://mathoverflow.net/users/9937 | weak metric space | Yes, they were introduced in valuation theory by Emil Artin and remain present in many contemporary treatments, including mine: see
[http://alpha.math.uga.edu/~pete/8410Chapter1.pdf](http://alpha.math.uga.edu/%7Epete/8410Chapter1.pdf)
especially Section 1.2 and
[http://alpha.math.uga.edu/~pete/8410Chapter2.pdf](h... | 3 | https://mathoverflow.net/users/1149 | 42435 | 27,008 |
https://mathoverflow.net/questions/41305 | 15 | (Grothendieck) topoi are left-exact reflective subcategories of a category of presheaves. An important class of quasi-topoi (see: <http://ncatlab.org/nlab/show/quasitopos>) arise as the category of concrete sheaves on a concrete site. Concrete sheaves are those sheaves $X$ such that the induced map $Hom(C,X) \to Hom(\u... | https://mathoverflow.net/users/4528 | Classification of Quasi-topoi | This is only a partial answer, and you may know it already since I mentioned it recently at the nForum, but for completeness, here it is. Theorem C2.2.13 in *Sketches of an Elephant* shows that the following are equivalent for a category C:
1. C is the separated objects for a Lawvere-Tierney topology on a Grothendiec... | 8 | https://mathoverflow.net/users/49 | 42441 | 27,012 |
https://mathoverflow.net/questions/42424 | 6 | Let $M$ be an even dimensional smooth manifold.
I want to find an example $M$ satisfying the following conditions,
1. $M$ admits a Kahler structure.
2. $\omega$ is a symplectic form on $M$.
3. There is no Kahler structure $(M,\omega',J)$ such that $[\omega']=[\omega] \in H^2(M;\mathbb{R})$
(I mean, want to find... | https://mathoverflow.net/users/11705 | Examples of symplectic non-Kahler classes. | One sort of example arises from the fact that if one starts with a Kahler form $\omega$ (which represents a class of type (1,1) in the Hodge decomposition by definition of a Kahler form), then if $\phi$ is the real part of any closed form of Hodge type (2,0), $\omega+\phi$ will still be a symplectic form (it tames the ... | 3 | https://mathoverflow.net/users/424 | 42443 | 27,014 |
https://mathoverflow.net/questions/42447 | 1 | Let $\mathbb N\_{n} = \{1,2,\cdots,n\}$.
Let $S$ be of cardinality $n$ where elements of $S$ are integers from $\mathbb N\_{n}$ and at least one element of $S$ is repeated (That is at least one integer from $\mathbb N\_{n}$ is skipped. One can easily find a set $S$ with the property that:
$\displaystyle \sum\_{j \in ... | https://mathoverflow.net/users/10035 | Subset higher power sum question (related to quadratic forms) | It's a little easier to state an answer if you let $N\_n=\lbrace0,1,\dots,n-1\rbrace$.
Let $n=2^k$, let $S$ be the multiset of integers with an odd number of ones in binary, each such integer appearing with multiplicity 2. Then it works for all $i\lt k$.
E.g., $k=3$, $S=\lbrace1,2,4,7\rbrace$, each taken twice, y... | 2 | https://mathoverflow.net/users/3684 | 42451 | 27,019 |
https://mathoverflow.net/questions/42449 | 17 | How would one approach proving that every real number is a zero of some power series with rational coefficients? I suspect that it is true, but there may exist some zero of a non-analytic function that is not a zero of any analytic function. I was thinking about approaching the problem using arguments of cardinality, b... | https://mathoverflow.net/users/nan | How to prove that every real number is a zero of some power series with rational coefficients (if true) | Call your real number $\alpha$. Suppose you have found a polynomial $p$ of degree $n-1$ with rational coefficients such that $|p(\alpha)|\lt\epsilon$. Show you can find a rational $r$ such that $|p(\alpha)-r\alpha^n|\lt\epsilon/2$.
| 29 | https://mathoverflow.net/users/3684 | 42452 | 27,020 |
https://mathoverflow.net/questions/42330 | 4 | I recently learned the following formulation of the Darboux theorem in a class.
Theorem: Suppose $\omega\_t$ is a smoothly varying family of symplectic forms on a closed manifold $M$ such that the cohomology class of $\omega\_t$ is independent of $t$. Then there is a smoothly varying family of diffeomorphisms $F\_t$ ... | https://mathoverflow.net/users/4362 | Can the Darboux theorem be strengthened? | (Sorry I don't have enough reputation to make this a comment.)
I don't fully understand your question, in particular since I don't know which formulations of Darboux's theorem you are concerned with. The version I would describe as the "classical" one is that each point of a symplectic manifold admits a neighbourhood... | 5 | https://mathoverflow.net/users/477 | 42456 | 27,023 |
https://mathoverflow.net/questions/42460 | 22 | Suppose that the function $p(x)$ is defined on an open subset $U$ of $\mathbb{R}$ by a power series with real coefficients. Suppose, further, that $p$ maps rationals to rationals. Must $p$ be defined on $U$ by a rational function?
| https://mathoverflow.net/users/5229 | Is a real power series that maps rationals to rationals defined by a rational function? | No. In fact, $p(x)$ can be a complex analytic function with rational coefficients that takes any algebraic number $\alpha$ in an element of $\mathbb{Q}(\alpha)$.
(And everywhere analytic functions are not rational unless they are polynomials).
The algebraic numbers are countable, so one can find a countable sequence... | 39 | https://mathoverflow.net/users/nan | 42465 | 27,027 |
https://mathoverflow.net/questions/42466 | 6 |
>
> **Possible Duplicate:**
>
> [A learning roadmap for algebraic geometry](https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry)
>
>
>
I am a masters student and I want to study algebraic geometry.
Does there exist good good book for selfstudy of algebraic geometry?
| https://mathoverflow.net/users/10118 | Algebraic geometry | Reid's book recommended above (below,depending on your perspective) is certainly your best bet for a ground floor introduction. An older resource that's certainly worth checking out is William Fulton's *Algebraic Curves*.
In connection with Fulton's book,a new resource has recently popped up online that we ALL need ... | 14 | https://mathoverflow.net/users/3546 | 42473 | 27,032 |
https://mathoverflow.net/questions/42454 | 11 | For an example I'm trying to understand, I need to calculate some cohomology group of some $\mathbb Z$-module with coefficients in some other $\mathbb Z$-module (with no interesting actions). (In particular, letting $\mathbb C^\times$ be the multiplicative group of the complex numbers *with the discrete topology*, I wo... | https://mathoverflow.net/users/78 | Where can I easily look up / calculate (abelian) group cohomology? | This group is best understood in terms of the universal coefficient formula,
i.e., in terms of the homology of the involved group. Hence, if $A$ is any
abelian group we have $H\_1(A)=A$ and the addition map $A\times A\rightarrow A$
induces a Pontryagin product on homology making $H\_\ast(A)$ a (graded)
commutative alge... | 16 | https://mathoverflow.net/users/4008 | 42476 | 27,035 |
https://mathoverflow.net/questions/42439 | 7 | The Dedekind eta function $\eta(\tau)$ can be regarded as a formula which assigns a number to a lattice $L \subset \mathbb{C}$. The algorithm is: rotate the lattice so that one of its basis vectors lies along the real axis, then pick another basis vector $\tau$ in the upper half plane, then compute the usual formula
$$... | https://mathoverflow.net/users/401 | How can one express the Dedekind eta function as a sum over the lattice? | The functions $G\_k$ and $\Delta = \eta^{24}$ can be regarded as functions on the set of lattices because they're modular of level 1. The $\eta$ function isn't modular of level 1 (its level is 24) so there's no natural way to regard it as a function on lattices -- it's a function on lattices with additional "level stru... | 8 | https://mathoverflow.net/users/2481 | 42478 | 27,036 |
https://mathoverflow.net/questions/42479 | 3 | Recently my work has led me to consider octonion algebras. Not having much of a background with non-associative anything, I decided to check out a basic text on the subject, R.D. Schafer's *Introduction to Nonassociative Algebras*.
I was reading it happily for a good while, until I got to the discussion of alternati... | https://mathoverflow.net/users/1149 | Alternative algebras in characteristic 2, especially scalar extension | A good samaritan has delivered an answer directly to my email account. It is indeed almost obvious, as long as one ignores the bit about the multiplication operators!
Take for instance the left alternative property: for all $x,y \in A$, $[x,x,y] = 0$. As I said above, this condition is linear in $y$, so it is enough ... | 3 | https://mathoverflow.net/users/1149 | 42487 | 27,040 |
https://mathoverflow.net/questions/42482 | 1 | Is there a name for the class of finite graphs $G$ with the following property?
* Every two graphs that can be created by removing one edge from $G$ are isomorphic.
(Edited to add the word "finite.")
| https://mathoverflow.net/users/10121 | Name the class of graphs G s.t. every two graphs that can be created by removing one edge from G are isomorphic. | If all the edge-deleted subgraphs of a finite graph are isomorphic then the graph is edge-transitive. [Here](http://www.newcastle.edu.au/school-old/math-physical-science/our_staff/downloads/macdougall_jim_onereductions.pdf) is a possible reference, though you might want to look at [this](http://staff.um.edu.mt/jlau/res... | 7 | https://mathoverflow.net/users/2384 | 42494 | 27,046 |
https://mathoverflow.net/questions/42396 | 3 | I want to calculate the second homotopy group of the blow-up of the unit cotangent disk bundle of a closed surface $\Sigma$, i.e $\pi\_2\left(D^\*T\Sigma\#\overline{\mathbb{C}P^2}\right).$ Actually, I want to understand whether these symplectic manifolds are monotone. If yes, I want to find a closed monotone Lagrangian... | https://mathoverflow.net/users/10104 | Homotopy groups of the blow-ups and monotone symplectic manifolds | Write $X$ for the manifold in question (i.e. the symplectic blowup of the unit disc bundle in the cotangent bundle of a surface $\Sigma$).
If you're interested just in monotonicity, what's relevant isn't so much $\pi\_2(X)$ as the image of $\pi\_2(X)$ under the Hurewicz map to $H\_2(X;\mathbb{Z})$, which is quite a b... | 2 | https://mathoverflow.net/users/424 | 42503 | 27,054 |
https://mathoverflow.net/questions/42497 | 9 | Let $k$ be a field and let $X$ be a smooth irreducible variety over $k$.
Suppose that I know that the image of $X$ in the Grothendieck group
of varieties over $k$ is equal to that of
a) ${\mathbb A}^n$ for some $n$
b) ${\mathbb A}^{n\_1}\times {\mathbb G}\_m^{n\_2}$ for some $n\_1, n\_2$.
Does it follow that $X$ ... | https://mathoverflow.net/users/3891 | Motivic characterization of affine spaces | The answer is no: Let $X$ the projective plane $\mathbb P^2$ minus a smooth
quadric $Q$. Then $[X]=[\mathbb P^2]-[Q]=\mathbb L^2+\mathbb L+1-(\mathbb
L+1)=[\mathbb A^2]$ but $X$ is not isomorphic to $\mathbb A^2$ as its Picard group is
$\mathbb Z/2$. For b) just cross $X$ with $\mathbb G\_m^n$, the Picard group is stil... | 19 | https://mathoverflow.net/users/4008 | 42504 | 27,055 |
https://mathoverflow.net/questions/42510 | 17 | Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minim... | https://mathoverflow.net/users/1916 | How badly can Krull's Hauptidealsatz fail for non-Noetherian rings? | I think that the answer is **yes**.
Indeed, there are examples of integral domains $D$ such that every non-zero prime ideal of $D$ has infinite height.
Look at the paper
"Anti-archimedean rings and power series rings"
D.D. Anderson; B.G. Kang; M H. Park
Communications in Algebra, 1532-4125, Volume 26, Issue ... | 14 | https://mathoverflow.net/users/7460 | 42515 | 27,062 |
https://mathoverflow.net/questions/42516 | 3 | I would like to know if the the following exist or are defined
1. The Fourier transform $\mathcal{F}\left(\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}\right)$ of a fractional differential operator such as $\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}$. (I'm aware that the fractional Fourier transform exists, but this isn't qui... | https://mathoverflow.net/users/7486 | Fourier transform of fractional differential operator and Plancherel formula equivalent for fractional norms | Regarding your first question, the fractional derivative operators are in fact DEFINED by how they act on the Fourier transform side. If the Fourier Transform of $f(x)$ is $\hat f(\xi)$ then the Fourier transform of its derivative $f'(x) = Df(x)$ is $2\pi i \xi \hat f(\xi)$ and hence for a positive integer $k$, the Fou... | 6 | https://mathoverflow.net/users/7311 | 42526 | 27,072 |
https://mathoverflow.net/questions/42538 | 3 | I am talking about the principle that is to DC what the global choice is to the usual axiom of choice. Global choice involves existential quantification over classes, but global DC can be stated as a schema in first-order set theory.
$(\forall x (\phi(x) \to \exists y (\phi(y) \wedge \psi(x,y)))) \to \forall x (\phi... | https://mathoverflow.net/users/6787 | Global or Relativised Dependent Choices | Given any $x$, let $\alpha\_x$ be 0 if $\phi(x)$ fails. If it holds, for each $x'$ with $\phi(x')$ and $x'$ of set-theoretic rank less than or equal to $x$, let $\alpha^{x'}$ be the least ordinal $\alpha$ such that there is a $y$ of rank $\alpha$ with $\phi(y)\land\psi(x',y)$. Now set $\alpha\_x$ as the supremum of the... | 3 | https://mathoverflow.net/users/6085 | 42542 | 27,082 |
https://mathoverflow.net/questions/42557 | 5 | In exercise 15 of Milnor's *Topology from a Differentiable Viewpoint*, one is asked to compute the Hopf invariant of the Hopf map. The way one is supposed to do this is to compute the linking number of two of the fibres, but Milnor doesn't define the linking number in terms of an integral. He says to compute it as the ... | https://mathoverflow.net/users/1353 | Computing the hopf invariant (without integration or homology, as in Milnor) of the hopf map | If you have the Hopf link embedded in some standard way in $\mathbb{R}^3$, you can see the linking number as given by the degree of a map $S^1 \times S^1 \to S^2$ in a number of ways. For instance, the pre-image of the north pole in $S^2$ consists of pairs of points stacked vertically above each other, i.e., crossings ... | 3 | https://mathoverflow.net/users/5010 | 42559 | 27,088 |
https://mathoverflow.net/questions/42562 | 4 | Is there an intuitive way to define what the p-quotient of an integer partition is?
In response to the comments:
my vague understanding is that the p-quotient is somehow division with remainder generalized to partitions. They turn up in the classic by Littlewood that I was reading:
<http://rspa.royalsocietypublishing... | https://mathoverflow.net/users/692 | p-quotient of an integer partition | I learned the following perspective in part from Bill Doran.
Redefine a partition through its boundary, extended along the axes in both directions:
A *partition* is a bi-infinite sequence of vertical $(0,1)$ and horizontal $(1,0)$ steps $(... S\_{-2},S\_{-1},S\_0,S\_1,S\_2,...)$ whose prefix is all vertical steps,... | 8 | https://mathoverflow.net/users/2954 | 42566 | 27,091 |
https://mathoverflow.net/questions/42565 | 1 | This question is motivated by the Euclidean Traveling Salesman Problem, i.e. finding the shortest Hamiltonian path of a complete graph of N randomly placed vertices. To eliminate boundary effects I consider the problem on the unit square with periodic boundary conditions. The idea is to find a "direction" and connect p... | https://mathoverflow.net/users/9766 | Least-square fit of line with rational slope to points on a square with periodic boundary conditions | I take it that the points themselves do not have to have rational coordinates. I don't think that there would have to be a best approximation. Consider the two point set set $\{(0,0),(1,\frac{1}{\sqrt{2}})\}$. Then lines such as $y=\frac{12}{17}x$ and $y=\frac{408}{577}x$ based on convergents to $\frac{1}{\sqrt{2}}$ pr... | 0 | https://mathoverflow.net/users/8008 | 42576 | 27,093 |
https://mathoverflow.net/questions/33095 | 5 | In the [**paper**](http://arxiv.org/PS_cache/arxiv/pdf/0812/0812.3592v1.pdf), by János Kollár there is *problem 19* (page 8).
It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs.
**My question is:** *How/where is that kind of resolution used/ne... | https://mathoverflow.net/users/5506 | How/where are semi-log resolutions used? | Perhaps a little more explanation would be this:
One of the first things we learn in algebraic geometry is *normalization* and we are told that it is "harmless" to assume that something is normal since the normalization exists and canonical and all that jazz. This is fine as long as one studies a stand alone object, ... | 19 | https://mathoverflow.net/users/10076 | 42578 | 27,094 |
https://mathoverflow.net/questions/42580 | 1 | Let $X\_{p} = \mathbb{Q}\_{p} / \sim $, where $\sim$ is defined by:
$x\sim 0 \Leftrightarrow x\in \mathbb{Q}$
$X\_{p}$ is path-connected, because (unless I'm making some horrible mistake,) for any $x\in X\_{p} \backslash \mathbb{Q}$, we have that $\lbrace x,0\rbrace$ under the subspace topology is path-connected.
... | https://mathoverflow.net/users/9455 | Is this quotient space of Q_p contractible? | The answer is yes. Whenever you crunch a dense subspace $Y$ of a topological space $Z$ to a point $q$ in the quotient $Z/Y$, you have the following contracting homotopy: For any $x \in Z/Y$ and any $t \in [0,1]$, set $f(x,0) = x$ and $f(x,t) = q$ for $t > 0$. Here, $q$ is the equivalence class of zero, $Z = \mathbb{Q}\... | 7 | https://mathoverflow.net/users/121 | 42597 | 27,104 |
https://mathoverflow.net/questions/42598 | 7 | It is known, from the works of G.Margulis, etc. that lattices in semi-simple real (algebraic) groups are "often" arithmetic subgroups, as long as the split rank is high enough. Here by a lattice in a Lie group $G$ is understood a discrete subgroup $\Gamma$ in $G$ such that the measure on $\Gamma\backslash G$ induced fr... | https://mathoverflow.net/users/9246 | discrete subgroups in p-adic Lie groups? | The arithmeticity theorem of Margulis also characterizes lattices in product of simple groups over $R$ as well as $p$-adic fields. All such lattices in the higher rank case are S-arithmetic. When all but one of the factors is compact, the projection onto the non-compact factor (that you can take to be a p-adic field) g... | 10 | https://mathoverflow.net/users/3635 | 42610 | 27,114 |
https://mathoverflow.net/questions/42609 | 4 | It is well known that every stable sheaf on a K3 surface $S$ is simple but the contrary is not true. Moreover, if $M$ denotes the (coarse) moduli space of stable sheaves on $S$ with fixed Chern classes and $Spl$ the moduli space of simple sheaves on $S$ with the same Chern classes, $M$ is an open subscheme of $Spl$. Ca... | https://mathoverflow.net/users/33841 | Stable, semistable and simple sheaves on a $K3$ | $M(c\_1, c\_2)$ is **not** always dense in $Spl(c\_1, c\_2)$, in other words irreducible components of $Spl(c\_1, c\_2)$ not intersecting $M(c\_1, c\_2)$ may actually exist.
In fact, in his paper
"Moduli of simple rank-$2$ sheaves on $K3$-surfaces",
Manuscripta Math. 79 (1993), no. 3-4, 253–265,
Z. Qin constru... | 2 | https://mathoverflow.net/users/7460 | 42614 | 27,117 |
https://mathoverflow.net/questions/42615 | 5 | *[Disclaimer: this may be a very trivial question; it certainly looks like it ought to have been studied and understood. I started thinking about it this morning when writing some notes for Rellich-Kondrachov, but cannot find a simple counterexample.]*
For the time being, let us just work on $\mathbb{R}^d$ with the L... | https://mathoverflow.net/users/3948 | Is the inclusion of Lebesgue spaces compact? | Take $f\_j(x)=\sin(jx\_1)$. This is a bounded sequence in every $L^p(\Omega)$ when the domain $\Omega$ is bounded. Yet, it is non-compact in every $L^q(\Omega)$. It happens to converge weakly-star in these spaces towards $f=0$, but not strongly.
| 5 | https://mathoverflow.net/users/8799 | 42616 | 27,118 |
https://mathoverflow.net/questions/42472 | 3 | Let $X=\{x\_{1}, \cdots , x\_{n}\}$ be a set of $n$ positive integers and integer $i \ge 1$. Let’s say that the set $X$ is $i$-sum-avoiding if for any nonnegative integers $c\_{1}, \cdots, c\_{n}$ such that $\sum\_{j=1}^{n}c\_{j} = n$ and $(c\_{1},\cdots, c\_{n}) \ne (1,\cdots, 1)$, it holds that
$\displaystyle \sum\... | https://mathoverflow.net/users/10035 | A conjecture on a Subset Power Sum Problem motivated by Computer Science | Here is what I think proves that for any *i*, there is no constant *k**i* satisfying *f*(*n*,*i*)≤*n**k**i*. That is:
**Claim**. Let *i* be a positive integer. Then the function *f*(*n*,*i*) is not polynomially bounded in *n*.
**Proof**. First consider the case of *i*=1. A key observation is that if *X*={*x*1,…,*x*... | 4 | https://mathoverflow.net/users/7982 | 42620 | 27,121 |
https://mathoverflow.net/questions/42463 | 17 | I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least prerequisites. As I was looking through the papers of Bridgeland, I realized that much of the theorems are stated for Proje... | https://mathoverflow.net/users/9534 | Derived categories of coherent sheaves: suggested references? | Kapustin-Orlov'a *survey* of derived categories of coherent sheaves is pretty good,
* A. N. Kapustin, D. O. Orlov, *Lectures on mirror symmetry, derived categories, and D-branes*, Uspehi Mat. Nauk **59** (2004), no. 5(359), 101--134; translation in Russian Math. Surveys **59** (2004), no. 5, 907--940, [math.AG/030817... | 16 | https://mathoverflow.net/users/35833 | 42621 | 27,122 |
https://mathoverflow.net/questions/42590 | 5 | I'm not very experienced with respect to Category Theory. So if this question makes no sense I'm sorry. At any rate here is my question: If the existence or non-existence of specific sets can be independent of set theory, then how can it be that the category Set is complete under small limits?
For example, suppose yo... | https://mathoverflow.net/users/8843 | Independence and Category Theory | Like all fields of mathematics, Category Theory is not immune to foundational questions. Although different foundations have been presented, the most common foundation for Category Theory is within Set Theory: Category Theory starts with a given universe of sets and then develops its theory. The computation of limits, ... | 10 | https://mathoverflow.net/users/2000 | 42628 | 27,125 |
https://mathoverflow.net/questions/39300 | 2 | Assume $C$ is a $1$-category and is $M : C \to Cat$ a $1$-functor, such that for every morphism $f : x \to y$, the functor $M(f) : M(x) \to M(y)$, which is denoted by $f\_\\*$, has a left-adjoint functor, which is denoted by $f^\* : M(y) \to M(x)$. Now I expect that $x \to M(x), f \mapsto f^\*$ defines a contravariant ... | https://mathoverflow.net/users/2841 | Pointwise left-adjoint yields a pseudo-functor | Here is a way of proving the following (condition b) of Vistoli in your question). I'm assuming condition a) to be strict (the $id$ is the $id$) in here for the moment.
Let us state condition b)
>
> Consider a pseudo-functor $(-)\_\*$ with structural morphisms in the direction $(fg)\_\* \to f\_\* g\_\*$, >that is... | 2 | https://mathoverflow.net/users/4763 | 42638 | 27,132 |
https://mathoverflow.net/questions/42623 | 21 | Consider vector bundles on connected paracompact topological spaces. Such a vector bundle $E$ on $X$ is said to be invertible if there exists some other bundle $F$ whose sum with $E$ is trivial: $E\oplus F \simeq \epsilon ^N $. The terminology "invertible" (used by Tammo tom Dieck for example) comes from K-theory and i... | https://mathoverflow.net/users/450 | Are the Stiefel-Whitney classes of a vector bundle the only obstructions to its being invertible? | The answer is no. Let $G$ be a cyclic group of order $n$ not divisible by $2$, let $V$ be an irreducible $2$-dimensional representation of $G$, and consider the associated vector bundle $EG\times\_G V\to BG$ (which I'll call $V$ again). Then $V$ has trivial Stiefel-Whitney classes since $H^q(BG;\mathbb{Z}/2)=0$ if $q>0... | 21 | https://mathoverflow.net/users/437 | 42639 | 27,133 |
https://mathoverflow.net/questions/42624 | 14 | In the <http://arxiv.org/abs/math/0606464v1> I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is topologically slice but not smoothly slice (slice means zero slice genus).
Freedman has a result stating that a knot with Alexand... | https://mathoverflow.net/users/4298 | Slice knots and exotic $\mathbb R^4$ | From Jacob Rasmussen's paper "Knot polynomials and knot homologies", arXiv:math/0504045, p.13 of ArXiv version:
>
> Bob Gompf kindly pointed out another such application [of Rasmussen's $s$-invariant, a concordance invariant of knots extracted from Khovanov homology]. Namely, $s$ can also be used to give a gauge-th... | 14 | https://mathoverflow.net/users/2356 | 42649 | 27,138 |
https://mathoverflow.net/questions/42647 | 45 | Is there a special name for the class of (commutative) rings in which every non-unit is a zero divisor? The main example is $\mathbf{Z}/(n)$. Are there other natural or interesting examples?
| https://mathoverflow.net/users/532 | Rings in which every non-unit is a zero divisor | A commutative ring $A$ has the property that every non-unit is a zero divisor if and only if the canonical map $A \to T(A)$ is an isomorphism, where $T(A)$ denotes the total ring of fractions of $A$. Also, every $T(A)$ has this property. Thus probably there will be no special terminology except "total rings of fraction... | 46 | https://mathoverflow.net/users/2841 | 42657 | 27,144 |
https://mathoverflow.net/questions/42660 | 6 | Let $K$ be a field of characteristic zero. Let $G/K$ be a group scheme of finite type.
Assume that $G$ is commutative and connected. For a natural number $n$ denote by $n\_G: G\to G$ the multiplication by $n$ morphism. Is it true that $n\_G$ is surjective with finite kernel?
(I know that the answer is yes provided $G... | https://mathoverflow.net/users/8680 | Multiplication by $n$ on commutative algebraic groups | Yes, this is true. The derivative of $n\_G$ at the identity element $e$ is multiplication by $n$ on the Lie algebra (tangent space at $e$) which gives that the kernel is finite and the image of $n\_G$ has the same dimension as $G$ which implies that it is equal to $G$ as $G$ is connected.
**Addendum**: Jim may be rig... | 10 | https://mathoverflow.net/users/4008 | 42664 | 27,148 |
https://mathoverflow.net/questions/42051 | 12 | Let X be a complex hermitian manifold with hermitian form $\omega$. How can you prove that if $\omega$ has negative holomorphic sectional curvature, then its scalar curvature is negative, too?
| https://mathoverflow.net/users/9871 | Negative holomorphic sectional curvature | Here is the answer.
Let $(X,\omega)$ be a Kähler $n$-dimensional manifold. Fix a point $x\_0\in X$ an choose local holomorphic coordinates $(z\_1,\dots,z\_n)$ centered at $x\_0$ and such that $(\partial/\partial z\_1,\dots,\partial/\partial z\_n)$ is unitary at $x\_0$. Let
$$
\Theta\_{x\_0}(T\_X,\omega)=\sum\_{j,k,l,... | 9 | https://mathoverflow.net/users/9871 | 42695 | 27,170 |
https://mathoverflow.net/questions/42703 | 0 | Let $k$ be a number field, and $F/k$ a finite extension. I would like to find a countable family of extensions $k\_i/k$ of degree 2 and a place $v\_i$ of $k\_i$ such that if $v$ is the place of $k$ lying below $v\_i$, then $[k\_{v\_i}:k\_v]$=2, where $k\_{v\_i}$ and $k\_v$ are the completions of $k\_i$ and $k$ at $v\_i... | https://mathoverflow.net/users/4561 | extensions of number fields | There are infinitely many places $v$ of $k$ such that $F\_w=k\_v$ for some $w$ above $k$. For each such place, take $a\_v \in k, v(a\_v)=1$ and consider the extension $k(\sqrt{a\_v})/k$. It has the property you want at $v$ and you will get infinitely many such extensions as you vary $v$, so you are done.
| 4 | https://mathoverflow.net/users/2290 | 42705 | 27,178 |
https://mathoverflow.net/questions/42706 | 7 | Is there an algorithm which takes as input two lists of words $v\_1,...,v\_n$ and $w\_1,...,w\_n$ over an alphabet $X$ and decides if there is an infinite sequence $(k\_i)$ where $1 \leq k\_i \leq n$ for all $i$ such that $v\_{k\_1}v\_{k\_2}...=w\_{k\_1}w\_{k\_2}...$? It seems that undecidability of the original Post C... | https://mathoverflow.net/users/10159 | post correspondence problem variant | See Halava, Vesa, Harju, Tero, Karhumäki, Juhani
Decidability of the binary infinite Post correspondence problem. If the alphabet consists of $\le 2$ letters, then the problem is decidable, if the number of letters is at least 7, then the problem is undecidable. The latter result is proved in Y. Matiyasevich, G. Sénize... | 6 | https://mathoverflow.net/users/nan | 42721 | 27,187 |
https://mathoverflow.net/questions/42652 | 10 | In an earlier post (Use Lie Sub-Groups of GL(3, R) for elastic deformation ? [here](https://mathoverflow.net/questions/40470/use-lie-sub-groups-of-gl3-r-for-elastic-deformation?)), I mentioned polar decompositions as in F = RU where R in SO(3) & U in symmetric positive-semidefinite matrices. In response, I received the... | https://mathoverflow.net/users/9624 | Iwasawa Decomposition & Polar Decomposition related how ? | You can obtain the $G=KAK$ decomposition from a decomposition of the type $F=UR$. To avoid unnecessary complications, let's assume that our reductive group $G$ is a selfadjoint subgroup of $\operatorname{GL}(n,\mathbb{R})$. Then the map $g \mapsto g^{-t}$ is an involution of $G$, which is called the Cartan involution a... | 14 | https://mathoverflow.net/users/430 | 42724 | 27,189 |
https://mathoverflow.net/questions/42744 | 30 | [This](https://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735) question made me wonder about the following:
Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers?
It seems that this would require that those manifolds are not deformation equivalent.
However, there... | https://mathoverflow.net/users/10076 | Diffeomorphic Kähler manifolds with different Hodge numbers | This question was debated in another forum a few years ago. The result was [a note by Frédéric Campana](http://iecl.univ-lorraine.fr/%7EPierre-Yves.Gaillard/DIVERS/hodgenumbers.pdf) in which he describes a counterexample as a corollary of another construction. In 1986 Gang Xiao (*An example of hyperelliptic surfaces wi... | 38 | https://mathoverflow.net/users/1450 | 42746 | 27,197 |
https://mathoverflow.net/questions/42751 | 4 | I am reading [this paper of Teleman and Woodward](http://arxiv.org/abs/math/0312154).
On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}\_k$. Why is this true? Is there an easy way to see this?
$G$ is a complex reductive Lie group a... | https://mathoverflow.net/users/83 | Invariant symmetric bilinear forms and H^4 of BG | You need to combine two classical theorems:
1. if $K$ denotes the compact form of $G$, then $K \to G$ is a homotopy equivalence; this is discussed in Bröcker-tom Diecks book "Representations of compact Lie groups", in the section with "complexification" in the title. So $H^{\ast}(BG)=H^{\ast}(BK)$.
2. The Chern-Weil ho... | 6 | https://mathoverflow.net/users/9928 | 42752 | 27,201 |
https://mathoverflow.net/questions/42749 | 2 | I find in my books it is given by Bott periodicity, but this is not direct and Bott periodicity is not easy. Is there an easy and direct way to define $K^n(X)$, like $K^{-n}(X)$? I just start to learn this stuff...
| https://mathoverflow.net/users/7341 | Easy way to define positive higher K groups? | There is a definition of $K^n$ for positive $n$, without Bott periodicity. This approach goes back to Karoubi, and you can find it in his book "K-theory". The definition (for both, positive and negative $n$) uses Clifford algebras, but no Bott periodicity (Karoubi uses his new, more algebraic and direct definition to p... | 10 | https://mathoverflow.net/users/9928 | 42756 | 27,203 |
https://mathoverflow.net/questions/42726 | 12 | Given a modular form, what is the precise formulation of BSD (in particular, the residue formula for the $L$-function at special values)? And what about the special values if the $L$-function is twisted by some character? Does there exist a good reference?
| https://mathoverflow.net/users/5730 | BSD for modular forms | My comments are getting too long, so here is a tentative answer.
First, a general statement: conjectures predicting special values of $L$-functions are formulated for all motives over number fields. Because normalized eigenforms (even twisted by finite order characters $\chi$) are attached to motives (or vice-versa),... | 8 | https://mathoverflow.net/users/2284 | 42757 | 27,204 |
https://mathoverflow.net/questions/42753 | 1 | I have a random variable $X$ and I want to find the probability density function from transforming it through the Heaviside step function. So
$Y = H(X)$
where the $H$ is the Heaviside step function with
$H(0) = 1$
Given the cumulative distribution function of $X$ as $F\_X$, then I have the pdf of $Y$ as
$f\_Y... | https://mathoverflow.net/users/10009 | Heaviside Step Function of a Random Variable | I don't think there's a much nicer way. You could either say
>
> Let $G\_X(x)=\mathbb{P}(X <x)=\lim\_{(t\rightarrow x^-)}F\_X(t)$; then
> $$
> f\_Y(y)=(1-G\_X(0))\delta(y-1)+G\_X(0)\delta(y)
> $$
>
>
>
or change the random variable. Since
$$\mathbb P(X<x)=1-\mathbb P(X\ge x)=1-\mathbb P(-X\le -x)=1-F\_{-X}(-... | 2 | https://mathoverflow.net/users/4600 | 42758 | 27,205 |
https://mathoverflow.net/questions/42755 | 11 | Let $\cal{A}$ be an abelian category with enough projectives and $\mathbf{C}\_+ (\cal{A})$ the category of bounded below chain complexes.
Since Quillen (Homotopical algebra, 1.2, examples B), there is a well-known "standard" model category structure on $\mathbf{C}\_+ (\cal{A})$ taking as weak equivalences the maps in... | https://mathoverflow.net/users/1246 | Non standard (?) model category structure on (co)chain complexes. | This is well known, but formulated in a slightly different way.
Recall that a Frobenius category is an exact category which has enough injectives as well as enough projectives, and such that an object is projective if and only if it is injective (injectivity (resp. projectivity) is defined with respect to inflations ... | 20 | https://mathoverflow.net/users/1017 | 42761 | 27,207 |
https://mathoverflow.net/questions/42759 | 6 | Let $F\_1,...,F\_m$ be a partition of the 3-element subsets of $[n]$ into families such that no three subsets in any one family $F\_i$ are all contained in one 4-element subset of $[n]$. What is the minimum value of $m$?
| https://mathoverflow.net/users/10171 | partitioning the 3-sets of [n]={1,...,n} into families | Tony Huynh's update can be easily generalized to show that
$$n\geq k\left(R(\underbrace{3,3,\dots,3}\_{k-1})-1\right)+3\implies m\geq k$$ and so we get a very weak lower bound on $m$ which at least shows that $m \_{min}\to \infty$.
For an easy upper bound $m\_{min}\le \lfloor\frac{n+1}{2}\rfloor$, which you can see b... | 3 | https://mathoverflow.net/users/2384 | 42768 | 27,212 |
https://mathoverflow.net/questions/42548 | 16 | Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology.
My question: In what sense can we have a duality between the equivariant homology and cohomology, in analogue with the Poincare duality betwe... | https://mathoverflow.net/users/2555 | Poincare dual in equivariant (co)homology? | There is a paper by Brion: "Poincaré duality and equivariant (co)homology," *Michigan Math. J.* 48 (2000). As mentioned in one of the comments, he uses Borel-Moore as the homology theory.
That said, one should always be careful about the word "duality" in this context -- often people just mean there's a canonical *is... | 15 | https://mathoverflow.net/users/5081 | 42789 | 27,223 |
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