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https://mathoverflow.net/questions/42774 | 8 | Given a set $A\subset \mathbb{R}^n$ such that $A\cap (x+\mathbb{Z}^n)\ne \emptyset$ for any $x\in \mathbb{R}^n$ (that is, $p(A)=\mathbb{T}^n$ for the projection $p:\mathbb{R}^n\rightarrow \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$). Is it true that supremum of Eucledian distances between points of $A$ is not less then $\s... | https://mathoverflow.net/users/4312 | minimal diameter of full preimage of torus | The second claim is false for $n=3$. Choose $\varepsilon$ small and $\delta\ll\varepsilon$. Let $A$ be the set of all points $(x,y,z)\in\mathbb R^3$ satisfying the following inequalities:
$$
\begin{cases}
-1.5+\varepsilon+\delta &\le x+y+z &\le 1.5+\varepsilon \\
-1.5+\delta &\le x+y-z &\le 1.5 \\
-1.5+\delta &\le x... | 4 | https://mathoverflow.net/users/4354 | 42792 | 27,224 |
https://mathoverflow.net/questions/42791 | 3 | I need an algorithm to solve a Quadratic Programming optimization problem where the unknowns are allowed to be negative.
I have an implementation of the Philip Wolfe simplex algorithm based on his article <http://pages.cs.wisc.edu/~brecht/cs838docs/wolfe-qp.pdf>, but it assumes x >= 0. In other places describing simila... | https://mathoverflow.net/users/10176 | Quadratic optimization without non-negativity restriction | You could replace each unrestricted variable x by (y-z) where y and z are each restricted to be positive.
| 1 | https://mathoverflow.net/users/9501 | 42793 | 27,225 |
https://mathoverflow.net/questions/42780 | 5 |
>
> **Question 1.** Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example Hausdorff) dimensions of $(X,a)$ and $(X,b)$ are equal?
>
>
> **Question 2.** Is there a notion of a dimension for to... | https://mathoverflow.net/users/2631 | How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces? | **Q2**. If you want that dimension for topological spaces to agree with Hausdorff dimension (for example) in case of metric space, then NO. For any $0 \le s \le \infty$, there is a metric on the Cantor set so that the Hausdorff dimension is $s$.
**Another topological result**. Let $X$ be a separable metrizable space.... | 5 | https://mathoverflow.net/users/454 | 42804 | 27,229 |
https://mathoverflow.net/questions/42809 | 19 | Let $k \ge 4$ be an even integer, and let $d$ be the dimension of the space $M\_k(\operatorname{SL}\_2(\mathbb{Z}))$ of modular forms of level 1 and weight $k$. Then the space of Hecke operators acting on $M\_k$ also has dimension $d$. Is it spanned by $T\_1, \dots, T\_d$?
Equivalently (more explicitly but also more ... | https://mathoverflow.net/users/2481 | How many Hecke operators span the level 1 Hecke algebra? | The answer is yes when $k$ is a multiple of $4$. There is a unique form
of weight $k$ of the form $f\_k=1+a\_dq^d+\cdots$. When $k$ is a multiple
of $4$ this is the theta series for a putative extremal even unimodular
lattice of rank $2k$. Theorem 20 in chapter 7 of Conway and Sloane's
*Sphere Packings, Lattices and Gr... | 12 | https://mathoverflow.net/users/4213 | 42811 | 27,232 |
https://mathoverflow.net/questions/42808 | 19 | This is partly inspired by answers to the question:
[Question about Hodge number](https://mathoverflow.net/questions/42709/question-about-hodge-number/42735#42735) .
Is there a family of compact complex manifolds, where the general fibres are
Kähler, but for which $E\_1$ degeneration of the Hodge to de Rham spectral se... | https://mathoverflow.net/users/4144 | Deformations of Kähler manifolds where Hodge decomposition fails? | This is known, for projective (even Moishezon)
manifolds as shown by Dan Popovici in his
paper <http://arxiv.org/abs/1003.3605>
For general Kaehler manifold, this is conjectured.
Popovici has proved that a property of "strong Gauduchon"
is preserved in limits <http://arxiv.org/abs/1009.5408>
and (I think) there are n... | 12 | https://mathoverflow.net/users/3377 | 42823 | 27,238 |
https://mathoverflow.net/questions/42818 | 6 | At a high level, my question is the following: given a set of $k$ vectors in Euclidean space which are pairwise "almost orthogonal", can one find a set of $k$ orthogonal vectors which are pairwise close to the original ones? This could be seen as a stable version of Gram-Schmidt orthogonalization, in which, under the p... | https://mathoverflow.net/users/10183 | Stable orthogonalization procedure | Use a Procrustes rotation of the standard basis vectors onto your vectors. This gives the set of orthogonal vectors with the smallest sum of squares of distances to your vectors.
<http://en.wikipedia.org/wiki/Orthogonal_Procrustes_problem>
"The orthogonal Procrustes problem is a matrix approximation problem in line... | 4 | https://mathoverflow.net/users/9501 | 42824 | 27,239 |
https://mathoverflow.net/questions/42837 | 7 | Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit
$\lim\_{X\to\infty} \pi(X)^{-1}\sum\_{p\leq X} f(T^{p}x)$
exist almost everywhere? Is it constant almost everywhere? Here the sum runs over primes, and $\pi(X)$ is the prime counting ... | https://mathoverflow.net/users/1464 | A non-standard ergodic limit | Here is a partial answer: Mate Wierdl proved that the limit exists almost everywhere if $f \in L^r (\mu)$ for some $r>1$. See "Pointwise Ergodic Theorem along the Prime Numbers".
Also, there is a recent article by Trevor Wooley and Tamar Ziegler ("Multiple Recurrence and Convergence along the Primes") which proves $L... | 3 | https://mathoverflow.net/users/7392 | 42842 | 27,251 |
https://mathoverflow.net/questions/42678 | 1 | ¿Could somebody tell me how can i write a zero order bessel function in an Hermite-Gauss basis?
Thanks
| https://mathoverflow.net/users/10155 | Bessel functions in an Hermite-Gauss basis | The Hermite-Gauss functions (suitably normalized) are orthonormal for an inner product defined by integration, just like $\{\sin(nx),\cos(mx)\}$ as $n,m$ run over positive integers. The coefficients of an expansion of a $J$-Bessel function in terms of Hermite-Gauss functions are computed by integrating the Bessel funct... | 2 | https://mathoverflow.net/users/6756 | 42843 | 27,252 |
https://mathoverflow.net/questions/42847 | 12 | What kind of categories $C$ have the property that each slice category $C/c$ is a topos? Obviously, topoi have this property, but, the converse is not true. An example is the category $EtTop$ of topological spaces and only local homeomorphisms. $EtTop/X \cong Sh(X)$, but $EtTop$ is very far from being a topos! It doesn... | https://mathoverflow.net/users/4528 | Locally a topos | The question of local toposes and similar categories was discussed a couple of years ago on the categories list by Peter Johnstone and others, if I recall correctly. I don't know anywhere that they appear in print, but I don't know the topos theory literature nearly well enough to be an authoratitive source on this.
... | 12 | https://mathoverflow.net/users/2273 | 42858 | 27,261 |
https://mathoverflow.net/questions/42851 | 2 | The [ETCS](http://ncatlab.org/nlab/show/ETCS) axioms give conditions on a category for it to be a category of sets. These axioms can be [written out in first order language](http://ncatlab.org/nlab/show/fully+formal+ETCS), resulting in a finite axiomatisation of the category of sets. Given a model of ZFC one can form t... | https://mathoverflow.net/users/4177 | Can we define geometric morphisms (between ETCS categories) elementarily? | Yes, it is possible. Precisely, we can write down a first-order theory for which a model is a pair of ETCS-models and a geometric morphism between them (am I right in thinking this is what you're asking for?).
To do this, on top of axiomatising “a pair of models of ETCS”, you add some extra function symbols for the a... | 6 | https://mathoverflow.net/users/2273 | 42860 | 27,263 |
https://mathoverflow.net/questions/42853 | 15 | Let $K$ be the imaginary quadratic field obtained by joining $\sqrt{-1}$ to the field of rational numbers $Q$. I would like to describe the extension $K^{ab}/Q^{ab}$, where for $F$ a number field, $F^{ab}$ denotes its maximal abelian extension (everything is taking place inside a big fixed field...).
More precisely I... | https://mathoverflow.net/users/10189 | An explicit computation in class field theory | Given that you want to know the structure of the Galois group and ramification, I think that you are best off working with the kernel of the norm map between connected components of idele class groups, as you yourself suggest.
These groups are very explicit: for $K := \mathbb Q(i)$, one obtains $\hat{\mathcal O}\_K^{... | 16 | https://mathoverflow.net/users/2874 | 42861 | 27,264 |
https://mathoverflow.net/questions/42778 | 0 | Suppose we are given a representation of a finite series of natural numbers:
$\sum\_{i=0}^N{c\_i x^i}$
The representation is essentially an expression that is a rational function of two polynomials.
Is it possible to add/subtract this series repeatedly to get a result that contains only part of the series?
> ... | https://mathoverflow.net/users/3647 | Piece of a sequence | I'm not sure I fully understand the problem, but here goes. Start with $10+200x$. Multiply by $20x$ to get $200x+4000x^2$. Subtract $(10+200x)(400x^2)$ to get $200x-80000x^3$. Add $(10+200x)(8000x^3)$ to get $200x+1600000x^4$. Keep going. The result of this infinite procedure is, in some sense, $200x$, and you don't ev... | 5 | https://mathoverflow.net/users/3684 | 42862 | 27,265 |
https://mathoverflow.net/questions/42820 | 2 | Is there a way to simplify the following expression:
$\lgroup{\int^A\_0 x(s)ds}\rgroup ^2$
I'm looking for an expression that can possibly get rid of the squared term, so that I can have just an integral of the first order.
| https://mathoverflow.net/users/10184 | Expressions for the Square of an Integral | I'm not sure about simplifying, but you can easily write your objective functional in Bolza form like this:
$$
\begin{align}
&\min\_{u(t) \in \Omega(t)} \, J = z(T)^2 + \int\_{0}^{T} s(t)u(t)dt \\
s.t. &\frac{dz(t)}{dt} = r(t)u(t),\quad z(0) = 0
\end{align}
$$
| 1 | https://mathoverflow.net/users/7851 | 42869 | 27,269 |
https://mathoverflow.net/questions/42875 | 3 | At first, if a group G is an infinite loop space (all are based), then `\pi_0(G)` must be Abelian. Therefore, if G is discret, then it must be Abelian. In fact, any Abelian group does be infinite loop space, by the EM space construction. But we have non-Abelian examples, the infinite groups U and O are infinite loop sp... | https://mathoverflow.net/users/7341 | Which Groups are Infinite Loop Spaces? | Edit: I just reread the question, and it says "*if* a group is an infinite loop space..." I realise the first paragraph of my answer is incorrect. The rest still stands.
---
Firstly, $\pi\_0(G)$ does not have to be abelian - it is $\pi\_1(G,e)$ which is abelian, as the Eckmann-Hilton argument shows.
The coeffic... | 0 | https://mathoverflow.net/users/4177 | 42877 | 27,273 |
https://mathoverflow.net/questions/14803 | 4 | There are nonequivalent geometries, nonequivalent groups finite and infinite, nonequivalent logics ( fregean and nofregean <http://www.formalontology.it/suszkor.htm>), even nonequivalent logicians;-)
**Are there nonequivalent randomnesses?**
The main two theories we know dealing with randomness and probability is ... | https://mathoverflow.net/users/3811 | Are there nonequivalent randomnesses? | A different answer from the ones so far: Quantum randomness is another kind of randomness that is a generalization of traditional randomness, i.e. classical or non-quantum probability. I think that it fits the question because you could likewise say that non-Euclidean geometry, interpreted as not-necessarily-Euclidean ... | 5 | https://mathoverflow.net/users/1450 | 42885 | 27,280 |
https://mathoverflow.net/questions/42888 | 5 | It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve quotients). The usual solution, if one is needed, is to change to the category of k-spaces and k-continuous maps. There are other... | https://mathoverflow.net/users/4177 | Categories with products that preserve quotients | Note: I have edited this answer further because I was being silly before (unnecessarily restrictive).
I take it you mean categories $E$ for which $- \times -: E \times E \to E$ preserves quotients. The word 'quotient' may be slightly ambiguous because sometimes people use it to mean'coequalizer', and sometimes just ... | 3 | https://mathoverflow.net/users/2926 | 42891 | 27,284 |
https://mathoverflow.net/questions/39823 | 13 | I am a graduate student trying to get involved in Ramsey theory. My question comes from:
Erdős on graphs: his legacy of unsolved problems
By Fan R. K. Chung, Paul Erdős, Ronald L. Graham
p.14 of this book is available as a google ebook.
They quote Erdos in a 1980/1981 paper
"Faudree, Shelp, Rousseau, and I n... | https://mathoverflow.net/users/9486 | Differences of near diagonal Ramsey numbers. |
>
> **Edit**: Erdős got three things wrong. First of all, it wasn't Faudree, Shelp, and Rousseau, it was Faudree, Shelp, and Burr. Second, it wasn't "recently", it was in the future (with respect to the quote you provide)! Third, they didn't prove that $(R(n+1,n)-R(n,n))/n \to \infty$, but only that $R(n+1,n)-R(n,n) ... | 7 | https://mathoverflow.net/users/3401 | 42893 | 27,285 |
https://mathoverflow.net/questions/42832 | 3 | Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t\_0\in I$ and $\xi\_0\in T\_{c(t\_0)}Q$ be arbitrary. I am looking for a citeable reference for the following statement: There is a unique $W^{1,2}$-vector field $\xi$ along $c$ satisfying
(i) $\nabla\_{\do... | https://mathoverflow.net/users/3509 | parallel transport along $W^{1,2}$-curves | Since $W^{1,2} \subset C^0$ and the zero-th order term of (i) depends linearly on $\dot{c} \in L^2$, the usual rewrite-as-integral-equation proof seems to work and is rather straightforward. I don't recall seeing this written down anywhere, but it's easy to verify and summarize.
| 3 | https://mathoverflow.net/users/613 | 42898 | 27,288 |
https://mathoverflow.net/questions/42900 | 19 | I have often read that the Riemann hypothesis is somewhat a statement like:
>
> The primes are as regularly distributed as we can hope for.
>
>
>
For example $\pi(x) = Li(x)+ O(x^{\sigma+\epsilon})$ for any $\epsilon>0$ as long as there are no zeros of $\zeta$ for any
$s \in \mathbb{C}$ with $\Re s > \sigma$. ... | https://mathoverflow.net/users/3757 | Refinements of the Riemann hypothesis | Yes, your question is imprecise. If we knew exactly where the zeta zeroes were, we could answer any question about the primes that could be formulated by means of the explicit formulae. In crude terms, the primes are obtained from the zeroes by an integral transform. Specific questions can depend, for example, on ratio... | 7 | https://mathoverflow.net/users/6153 | 42901 | 27,289 |
https://mathoverflow.net/questions/42836 | 18 | Does the braid group $B\_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex?
**Update 1.** During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are a-T-menable. I seem to remember that somebody proved that they do act properly by isometries on CAT(0) cube compl... | https://mathoverflow.net/users/nan | Braid groups acting on CAT(0)-complexes | As Sam Nead says, $B\_n$ contains $\mathbb{Z}^2 \* \mathbb{Z}$, and you can find an example the way he suggests.
If you'd like something much more explicit, you can simply take the first three standard generators $\sigma\_1$, $\sigma\_2$ and $\sigma\_3$, and then it follows from the solution (by Crisp and Paris) of a... | 9 | https://mathoverflow.net/users/1335 | 42905 | 27,291 |
https://mathoverflow.net/questions/42873 | 5 | Recently during a lecture, my professor mentioned that forcing over any poset which is countable, separative, and atomless, is essentially the same as forcing over the Cohen poset, that is to say results in adding a Cohen real.
My question is: Are there any other similar characterizations of "commonly used" forcing ... | https://mathoverflow.net/users/8843 | Strange question about Hechler | Here are a few additional examples of the kind you seek,
and in fact each of them directly generalizes the
characterization you mention of the forcing to add a Cohen
real. However, I know of no such characterization of Hechler forcing.
* The collapse forcing
$\text{Coll}(\omega,\theta)$ is, up to forcing equivalence... | 8 | https://mathoverflow.net/users/1946 | 42917 | 27,297 |
https://mathoverflow.net/questions/42915 | 5 | In any sort of type theory, there are a bunch of rules for constructing derivations of typing judgments such as $x:A,\; y:B(x) \;\vdash\; z:C(x,y)$. (I intend to include also judgments of the form $B:\mathrm{Type}$.) It's certainly possible to get to the same typing judgment using different derivations; for instance I ... | https://mathoverflow.net/users/49 | Can a typing judgment admit essentially different derivations? | This property is called "coherence", and no, it doesn't always hold.
Establishing this property holds for a given semantics of proofs is a proof obligation. An example of when it doesn't arises with coercive subtyping -- if the diagrams corresponding to possible coercions do not all commute, then the semantics is no... | 8 | https://mathoverflow.net/users/1610 | 42921 | 27,300 |
https://mathoverflow.net/questions/42923 | 6 | Specifically, is it the case that (for $a,b\in\omega^\omega$) $a$ $\leq\_T$ $b$ implies $a$ $\leq\_c$ $b$?
I suspect it might be trivial, but not knowing much Recursion Theory, it's hard to see how it could.
Thank you in advance.
| https://mathoverflow.net/users/3462 | How does the Constructibility Degree of a real compare with its Turing Degree? | Yes, if $a\le\_T b$ then $a$ is first-order definable from $b$; in particular $a\in L(b)$ so $a\le\_c b$.
| 6 | https://mathoverflow.net/users/4600 | 42924 | 27,301 |
https://mathoverflow.net/questions/42553 | 13 | I'm finally at the end of Milnor's "On manifolds homeomorphic to the 7-sphere", and I stumbled upon something I cant figure out...
For those with the reference I'm talking about "lemma 5", it goes something like this, you have two $\mathbb{S}^3$ bundles over $\mathbb{S}^4$, we want to obtain the total space of this b... | https://mathoverflow.net/users/9187 | Morse Theory and Exotic Spheres | Milnor didn't explain the formula as much as maybe he should have, but the point is that the real part of a unit-length quaternion is invariant under both conjugation and inversion. Let $$r = ||u|| \qquad \hat{u} = u/r,$$
so that
$$v' = \hat{u}^h v \hat{u}^j \qquad u' = \hat{u}r \qquad ||u'|| = ||u''|| = 1/r.$$
Thus
$$... | 10 | https://mathoverflow.net/users/1450 | 42930 | 27,302 |
https://mathoverflow.net/questions/19313 | 12 | Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean [combinatorial model category](http://ncatlab.org/nlab/show/combinatorial+model+category). Consider a Quillen pair
$$ L: M\rightleftarrows N: R.$$
I want the following result:
>
> There exists a set of maps $S$ in $M$, such that $L$ and $R$ d... | https://mathoverflow.net/users/437 | How to localize a model category with respect to a class of maps created by a left Quillen functor | I suspect that you already have one, but here is a proof. I will assume that $M$ and $N$ are combinatorial and that $M$ is left proper (otherwise, I don't think that the literature contains a general construction of the left Bousfield localizations of $M$ by any small set of maps). Everything needed for a quick proof i... | 11 | https://mathoverflow.net/users/1017 | 42934 | 27,305 |
https://mathoverflow.net/questions/42928 | 6 | Let $G$ be a finitely generated group with the natural word length function ($|x|$ is the length of the shortest word in generators of $G$ representing $x$). We call a partial left invariant order $\le $ on $G$ *word order* if whenever $a\le b\le c$ we have $|b|\le C(|a|+|c|)$ for some constant $C$. Say, the standard o... | https://mathoverflow.net/users/nan | orders and length functions on finitely generated groups | If $G$ is a finitely generated infinite group and $\leq$ is a linear word order, then for each $a, c \in G$ there are only finitely many elements $b \in G$ such that $a \leq b \leq c$. From this it follows that $(G, \leq)$ is order isomorphic to $\mathbb Z$. If $\leq$ is also left invariant, then this isomorphism must ... | 10 | https://mathoverflow.net/users/6460 | 42953 | 27,317 |
https://mathoverflow.net/questions/42966 | 17 | If $X$ a topological space one says that $X$ is *universally closed* if for every Hausdorff space $Y$ and every (continuous) map $f:X\rightarrow Y$, the image of $X$ is a closed subset of $Y$.
It is clear that every compact space is universally closed, but are there non compact universally closed spaces?
| https://mathoverflow.net/users/10217 | Topological spaces whose continuous image is always closed | If $Z$ is not compact, and $X=\{p\}\cup Z$ is the space whose nonempty open sets are of the form $\{p\}\cup V$ with $V$ open in $Z$, then $X$ is not compact, but every continuous function from $X$ to a Hausdorff space is constant.
| 15 | https://mathoverflow.net/users/1119 | 42973 | 27,330 |
https://mathoverflow.net/questions/41011 | 20 | What is the indefinite sum of the tangent function, that is, the function $T$ for which
$\Delta\_x T = T(x + 1) - T(x) = \tan(x)$
Of course, there are infinitely many answers, who all differ by a function of period 1. Ideally, I would like the solution to be of the form
$T(x) = $ nice\_function$(x)$ + possibly\_... | https://mathoverflow.net/users/7540 | What is the indefinite sum of tan(x)? | I add more details for the solution in the distinguished answer due to Anixx. First, we need the **digamma** function
<http://en.wikipedia.org/wiki/Digamma_function>
which we will call $\Psi(x)$. Important properties (from that web page) are: $\Psi(x)$ is analytic in the complex plane except at the nonpositive in... | 33 | https://mathoverflow.net/users/454 | 42975 | 27,332 |
https://mathoverflow.net/questions/42969 | 0 | Given an $m\times n$ 0-1 matrix A, I am interested in an efficient algorithm to locate all copies of a given $p\times q$ 0-1 submatrix B within it, where a permutation of rows and columns is allowed, i.e. find all collections of row indices $r\_1, r\_2,\ldots, r\_p$ and column indices $c\_1, c\_2,\ldots, c\_q$ (with $r... | https://mathoverflow.net/users/10219 | Locating a submatrix within a matrix | Note that the problem of deciding whether there exist such row indices and column indices is already NP-complete. This is because the case where *B* is a square matrix entirely consisting of 1s is identical to the Balanced Complete Bipartite Subgraph problem, which is known to be NP-complete [Joh87].
[Joh87] David S.... | 3 | https://mathoverflow.net/users/7982 | 42993 | 27,344 |
https://mathoverflow.net/questions/41315 | 4 | Consider the Banach algebra $W^+=\ell^1(\mathbb{Z}^+)$, viewed upon as the analytic functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|=\sum\_{k\ge0}|a\_k|<\infty$$ where
$$f(z)=\sum a\_kz^k$$
is the Taylor expansion of $f$. Clearly, $W^+\subset H^\infty(\mathbb{D})$. Now, it is well known that $H^\infty$ adm... | https://mathoverflow.net/users/8294 | Factorization in the Wiener algebra on the unit disc. | There is no factorization in Wiener algebra, it is easy to construct a counterexample.
Namely, if $B$ is a Blaschke factor with zeroes $z\_n$, $z\_n\to 1$ (of course $\sum(1-|z\_n|) <\infty$) and $g= (z-1)^2$ then
$
f= Bg$ has $C^1$ boundary values, and so is in the Wiener algebra.
On the other hand, $B$ is an i... | 2 | https://mathoverflow.net/users/10223 | 42996 | 27,346 |
https://mathoverflow.net/questions/42728 | 14 | If someone hands you a prime number $p$, and an algebraic number $x$ inside the Hasse-Weil bound, is there a normalized newform (say of weight two) so that $a\_p=x$, where $a\_p$ is the $p$th Fourier coefficient?
| https://mathoverflow.net/users/5730 | Fourier coefficient of a modular form | Some Remarks.
I parse the problem in the following way:
Start with a totally real algebraic integer $\alpha$ such that every conjugate of $\alpha$ has absolute value at most
$2 \sqrt{p}$. Then does there exist a normalized cuspidal Hecke eigenform $f$ of weight $2$ with $a\_p = \alpha$?
First, here is a heuristic rea... | 9 | https://mathoverflow.net/users/nan | 43016 | 27,355 |
https://mathoverflow.net/questions/43027 | 10 | Given a complete graph of n vertices (no three of which are no collinear) in the plane and straight edges, what is the maximal possible number of "incidental intersections" of edges, i.e., number of non-vertices at which two distinct edges intersect each other, not counting multiplicity?
This is a question that I pos... | https://mathoverflow.net/users/9102 | "incidental" intersections of a complete graph in the plane | Assuming straight line segment edges, any 4 vertices determine 6 edges with at most one non-vertex intersection, so you can't have more than $n$-choose-4. You will have $n$-choose-4 if no vertex is interior to any triangle of vertices, which is to say if all $n$ vertices lie on the boundary.
| 8 | https://mathoverflow.net/users/3684 | 43034 | 27,368 |
https://mathoverflow.net/questions/42976 | 6 | ### Motivation
I was re-reading parts of Grothendieck-Murre, and these questions came up naturally.
The situation in chapter 9 is that $S'=Spec(A)$ where $A$ is a complete local noetherian ring of dimension $2$ with an algebraically closed residue field, and $D$ is some divisor in $A$. Then they take a "desingulari... | https://mathoverflow.net/users/5756 | Relation between tame fundamental group w.r.t. to D, and the fundamental group of the complement of D | Your question as stated does not strictly speaking make sense. In Grothendieck-Murre sect.2.4 the tame fundamental group $\pi\_1^D(S,\xi)$ is only defined when $S$ is normal and $D$ is a DNC. Although in 2.2.2 they define tamely ramified covers in greater generality, they observe in Remark 2.2.3(4) that this is "certai... | 5 | https://mathoverflow.net/users/5480 | 43041 | 27,374 |
https://mathoverflow.net/questions/42959 | 4 | Let $X$ be a projective variety. Symmetric product of $X$ is the quotient of the product $X^n$ by the action of the symmetric group $\Sigma\_n$ permuting the factors.
When does it exist (as an algebraic variety)?
| https://mathoverflow.net/users/2234 | Symmetric products of projective varieties | To fix ideas, let $K$ be a field and $X/K$ be a seperated $K$-scheme of finite type. Let $G$ be a finite group operating on $X$ via $K$-morphisms. The operation is said to be admissible provided every orbit of $G$ is contained in an open affine subset of $X$.
If the operation is admissible, then
there is a pair $(Y, ... | 4 | https://mathoverflow.net/users/8680 | 43043 | 27,375 |
https://mathoverflow.net/questions/43042 | 5 | Given a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ and a sequence of groups $G\_i$, one can define its ultraproduct as:
$$ ^\*\prod\_{i\in \mathbb{N}}G\_i:=\{(x\_i)\_{i \in \mathbb{N}}| x\_i\in G\_i\}/\sim$$, where $(x\_i)\_i\sim (y\_i)\_i$, iff $x\_i=y\_i$ $\mu$-almost everywhere.
Suppose you are given also g... | https://mathoverflow.net/users/3969 | What is known about the ultra-inverse limit? | Either the even numbers or the odd numbers are $\mu$-large, so the condition degenerates to ultraproduct.
| 7 | https://mathoverflow.net/users/408 | 43046 | 27,377 |
https://mathoverflow.net/questions/43002 | 23 | Let $G$ be a topological group, and $\pi\_1(G,e)$ its fundamental group at the identity. If $G$ is the trivial group then $G \cong \pi\_1(G,e)$ as abstract groups. My question is:
If $G$ is a non-trivial topological group can $G \cong \pi\_1(G,e)$ as abstract groups?
About all I know now is that $G$ would have to ... | https://mathoverflow.net/users/5795 | Fundamental groups of topological groups. | Here is an example: a product of infinitely many $\mathbb{RP}^\infty$'s.
The crucial thing thing to see is that $\mathbb{RP}^\infty$ (or, easier to see, its universal cover $S^\infty$) has a group structure whose underlying group is a vector space of dimension $2^{\aleph\_0}$. This is not hard: the total space $S^\i... | 43 | https://mathoverflow.net/users/2926 | 43047 | 27,378 |
https://mathoverflow.net/questions/43054 | 10 | Let $R$ be a dvr with residue field $k$ and quotient field $K$. Define $S=Spec(R)$. Let
$A/K$ be an abelian variety. To my knowledge the Neron model of $A$ is a group scheme
${\cal N}/S$ with generic fibre $A$, which represents the functor
$$Y\mapsto Mor\_K(Y\times\_S Spec(K), A)$$
on the category of smooth $S$-scheme... | https://mathoverflow.net/users/8680 | Basic properties of Neron models | 1) Yes, that is part of Neron's theorem. (There also exist Neron models for semiabelian varieties, which are not of finite type in general.)
2) Yes, for the reason you give ($A$ is connected). When people say "connected component of the Neron model" they generally mean the open subgroup scheme which is in every fibre... | 13 | https://mathoverflow.net/users/5480 | 43056 | 27,383 |
https://mathoverflow.net/questions/42707 | 13 | I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one from other. But my question is the following :
"What is the most strict definition of Calabi-Yau Manifolds"
By that ... | https://mathoverflow.net/users/9534 | Calabi - Yau Manifolds | There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can be found in Joyce's book:
[Compact Manifolds with Special Holonomy](http://books.google.ca/books?id=c3P-YUD8GZQC&dq=jo... | 10 | https://mathoverflow.net/users/6871 | 43062 | 27,385 |
https://mathoverflow.net/questions/43058 | 1 | I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation gradient, R in S0(3), & U in GL+(3,R)/S0(3) = stretches & shears.
I think that:
(1) GL+(3,R)/SO(3) being diffeomo... | https://mathoverflow.net/users/9624 | Understanding manifold GL+(3,R)/SO(3) ? | $GL^+(3,R)/SO(3)$ is the space of 3 dimensional positive definite symmetric matrices because the [polar decomposition](http://en.wikipedia.org/wiki/Polar_decomposition) of $g \in GL^+(3,R)$ is $ g = o p $ , $o \in SO(3)$ and $p$ is positive definite symmetric .
The wikipedia page treats the complex case, but by repea... | 3 | https://mathoverflow.net/users/1059 | 43068 | 27,387 |
https://mathoverflow.net/questions/43065 | 1 | Let $M$ be a smooth manifold of 2n-dim, $v$ be a map from $M$ to the matrix of order $m\times m$.
We call $p\in M$ is the singularity, if $v(p)$ is non-invertible. Suppose $v$ is smooth and the singularity is submanifold. Let $C$ be a connected component of singularity, $U$ is the tubular neighborhood of $C$. How to co... | https://mathoverflow.net/users/3896 | A question about to computing a integration | I think your guess is correct and the integral is zero, at least if you assume that $M$ is a closed oriented manifold and that U contains all singularities. Here is the proof, assuming closedness of M. Let $N = M - U$, a compact manifold with boundary. Let $G=Gl\_n (C)$ (in your problem, it does not matter whether you ... | 3 | https://mathoverflow.net/users/9928 | 43071 | 27,388 |
https://mathoverflow.net/questions/43069 | 9 | Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T\_0$-spaces with, say, 5 points?
In know that a $T\_0$-topology on a finite set is equivalent to a partial ordering, and [wikipedia](http://en.wikipedia.org/wiki/Part... | https://mathoverflow.net/users/1291 | Is there a list of all connected T_0-spaces with 5 points? | There is a Java applet that displays all 5-element connected posets at
<http://www1.chapman.edu/~jipsen/gap/posets.html>.
| 9 | https://mathoverflow.net/users/2807 | 43074 | 27,391 |
https://mathoverflow.net/questions/42971 | 3 | In John Steel's paper "The derived model theorem",
<http://math.berkeley.edu/~steel/papers/dm.ps>
John Steel asserts that it is clear that $\mathrm{Hom}^{Y}\_{\kappa}$ is closed downward under continuous reducibility. Unfortunately that is not clear to me and I was wondering if anyone could help me understand it.
... | https://mathoverflow.net/users/7966 | Question about John Steel's "The derived model theorem" | Rupert, I will explain the argument for $Y=\omega$ (this makes no difference, but you may find it easier to visualize) when the continuous function is particularly nice (in a way I will make precise. The general case requires a slight adaptation).
First, some basic background: The topology on $\omega^\omega$ is the ... | 5 | https://mathoverflow.net/users/6085 | 43075 | 27,392 |
https://mathoverflow.net/questions/41628 | 2 | In the Hopf algebra $SL\_q(N)$, it can be shown, using direct calculations, that $S(u^1\_i)u^j\_1 = q^{-1}u^j\_1S(u^1\_i)$. Can anyone see a more elegant way of establishing this?
Moreover, does anyone know of a similar relation in the more general case of $S(u^1\_r)u^i\_j$?
Edit (referneces): By $SL\_q(N)$ I mean ... | https://mathoverflow.net/users/1095 | The relation $S(u^1_i)u^j_1 = q^{-1}u^j_1S(u^1_i)$ | This is a reasonably known result. That $S(u^1\_i)u^j\_1 = q^{-1}u^j\_1S(u^1\_i)$, was originally proven (to the best of my knowdledge) in FRT's '89 paper "Quantum Groups and Lie Algebras" - the paper is in Russian though. The only English write up of the proof that I known is in Theorem 1 of Vainermann and Podkolzin's... | 2 | https://mathoverflow.net/users/1867 | 43089 | 27,401 |
https://mathoverflow.net/questions/43070 | 3 | Is there any statistic that can tell how "hubby" is a graph?
By this I mean a number that is small when a graph has no hubs, that is, when all nodes are more or less equal degree-wise, and big when there are hubs, nodes that concentrate most of the connectivity.
I expect it to be zero (or minimal) for a fully conn... | https://mathoverflow.net/users/757 | Hubbiness of a graph | Here are some concepts that might be helpful to you:
1. A vertex $v$ of a connected graph $G$ is an *articulation point* if the removal of $v$ from $G$ causes $G$ to be disconnected. My interpretation is that an articulation point corresponds to a "hub". This may or may not match your intuition. On the one hand, the ... | 2 | https://mathoverflow.net/users/9840 | 43094 | 27,406 |
https://mathoverflow.net/questions/43083 | 11 | What is a good introduction in gradient flows in metric spaces?
I know the book *Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré*, but is too hard for an introduction (for me).
I'm looking for something with a similar content.
| https://mathoverflow.net/users/5295 | Textbooks or notes on gradient flows in metric spaces | Here are some links to the online lecture notes which are hopefully more accessible than the book you mentioned:
* [Lecture Notes on Gradient Flows and Optimal Transport](https://arxiv.org/abs/1009.3737) by S. Daneri
* [An Introduction to Gradient Flows in Metric Spaces](http://igk.math.uni-bielefeld.de/study-materia... | 11 | https://mathoverflow.net/users/2149 | 43097 | 27,407 |
https://mathoverflow.net/questions/43103 | 10 | if $x=d(n)$ is the number of divisors of $n$, what is the tightest lower-bound for $n$ only given $x$?
<http://en.wikipedia.org/wiki/Highly_composite_number>
| https://mathoverflow.net/users/10246 | What is the lower bound for highly composite numbers? | I will start off with the simplest type, $$ d(n) \leq \sqrt{3 n} $$ and $$ d(n) \leq 48 \left(\frac{n}{2520}\right)^{1/3} $$ and
$$ d(n) \leq 576 \left(\frac{n}{21621600}\right)^{1/4}. $$
The first one has equality only at $n = 12,$ second only at $n =2520,$ third only at $n= 21621600.$ Instead of continuing with fract... | 21 | https://mathoverflow.net/users/3324 | 43105 | 27,410 |
https://mathoverflow.net/questions/43015 | 12 | A (standard, real-valued) Brownian motion $W = \{W(t): t \geq 0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t \geq 0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost sure... | https://mathoverflow.net/users/10227 | The conditions in the definition of Brownian motion | No, it is not true that a process *W* satisfying the properties (1), (3) and (4) has to be a Brownian motion. We can construct a counter-example as follows.
This construction is rather contrived, and I don't know if there's any simple examples.
Start with a standard Brownian motion *W*. The idea is to apply a small b... | 11 | https://mathoverflow.net/users/1004 | 43111 | 27,414 |
https://mathoverflow.net/questions/43057 | 5 | Hi all,
I am looking to do some linguistic analysis of informal proofs. Therefore I am on a search for a collection of entry level proofs written in a clear, uninvolved style. I have one recommendation for Hardy and Wright's "An Introduction to the Theory of Numbers," and was wondering if there is something else you ... | https://mathoverflow.net/users/10234 | Looking for a collection of entry level proofs | You can try Aigner and Ziegler's book [Proofs from the book](http://rads.stackoverflow.com/amzn/click/3540636986)
| 2 | https://mathoverflow.net/users/5372 | 43115 | 27,418 |
https://mathoverflow.net/questions/43081 | 4 | The Kirchhoff's theorem is a classical result for counting the number of spanning trees in a graph.
However, what are the best known upper bounds on the number of spanning trees in a graph in terms of structural parameters (e.g., number of vertices, degrees, etc.) instead of algebraic quantities?
| https://mathoverflow.net/users/nan | Number of spanning trees: bounds from structural parameters | I had an email discussion with Russell Lyons a few years ago about maximizing the number of spanning trees among all graphs with a given number of vertices and edges. He had a simple argument for an upper bound of $(2e/v)^{v-1}$. There's an even simpler argument for an upper bound of $e\choose v-1$. Russell thought the... | 5 | https://mathoverflow.net/users/3684 | 43116 | 27,419 |
https://mathoverflow.net/questions/43125 | 1 | Does anybody in here know how to get hold of this article:
"Tutte, W.T., A Theory of 3-connected graphs, Indag. Math. 23 (1961) 441-455"
or have it on paper?
| https://mathoverflow.net/users/1539 | A Theory of 3-connected graphs | It is in the volume "Selected Papers of W.T. Tutte" published by the Charles Babbage Research Centre about 20 years ago.
(Strangely the reference is slightly different, but the title and page numbers are identical.)
| 2 | https://mathoverflow.net/users/1492 | 43128 | 27,424 |
https://mathoverflow.net/questions/43124 | 19 | Let $A\in\mathcal M\_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi\_1,\ldots,\xi\_k)$. We can view it as a smooth function $A:\mathbb R^k\to\mathcal M\_n$.
>
> 1. What conditions should the matrix $A$ satisfy so that its eigenvalues
> $\lambda\_i(\xi\... | https://mathoverflow.net/users/10095 | Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters | The fact that the entries of the matrix are real does seem to help. The state of the art is the following.
* The spectrum is continuous functions of $\xi$. However, it is not always possible to label the eigenvalues so that they individually are continuous functions.
* When the multiplicities $m\_1,\ldots,m\_r$ do no... | 19 | https://mathoverflow.net/users/8799 | 43133 | 27,426 |
https://mathoverflow.net/questions/43095 | 8 | Fix positive integers $k, N$ and let $\omega$ be a Dirichlet character mod $N$.
Let $f\in S\_k(N,\phi)$ be a normalized newform (i.e. of weight $k$, level $N$ and character $\phi$) with fourier expansion $\sum\_{n\geq 1} a(n)q^n$. In her paper 'Newforms and Functional Equations', Winnie Li showed that if $q$ is a pr... | https://mathoverflow.net/users/nan | Hilbert Modular Newforms | If I've understood your question correctly, you're right that $C(q,f)\not=0$ always and there is a natural representation-theoretic proof of this result (before I start let me say that I don't know how to get these gothic $q$s and $N$s as in your question, so I am just using usual $q$s and $N$s, but they are ideals of ... | 10 | https://mathoverflow.net/users/1384 | 43134 | 27,427 |
https://mathoverflow.net/questions/43131 | 7 | I saw a statement somewhere that for the Hirzebruch surfaces $F\_n:=\mathbb{P}\_{\mathbb{P}^1}(\mathcal{O}\oplus\mathcal{O}(n))$, $F\_n$ and $F\_m$ are symplectormorphic when $m$ and $n$ have the same parity.
My question is: Why is this true?
I can see that they are diffeomorphic by Freedman's Theorem: computing t... | https://mathoverflow.net/users/1657 | Why are the following varieties symplectomorphic? | In order to obtain an explicit description of the diffeomorphism, one can use the following argument.
Take $B=\mathbb{C}^{n-1}$, with coordinates $t\_1, \ldots, t\_{n-1}$, and consider the complex space
$\mathcal{X}$ obtained glueing $\mathbb{P}^1 \times \mathbb{C} \times B$ with $\mathbb{P}^1 \times \mathbb{C} \time... | 10 | https://mathoverflow.net/users/7460 | 43142 | 27,431 |
https://mathoverflow.net/questions/43139 | 6 | The title says it all, what's a good dense open of $\bar{M}\_g,n(X,\beta)$ which play the role of ${M}\_g$ in $\bar{M}\_g$?
My first (naive) guess is maps from a genus $g$ smooth curve to $X$ which represents the class $\beta$. But I'm a little bit concerned, is it dense for sure? Could it happen that in some cases o... | https://mathoverflow.net/users/1657 | What's a good dense open of $\bar{M}_g,n(X,\beta)$? | If $X$ is convex and $g = 0$, then you can take smooth curves with distinct marked point, this will be dense. However, in general the locus of smooth curves is not dense. An easy example is that of degree 1 unpointed maps from a curve of genus $g > 0$ to $\mathbb P^1$; such a curve must consist of a copy of $\mathbb P^... | 9 | https://mathoverflow.net/users/4790 | 43143 | 27,432 |
https://mathoverflow.net/questions/43147 | 29 | I have a basic question that others have definitely considered.
Often there are papers that originally appeared in a language that one might not understand (and I mean a natural language here). I would like to cite the original paper, because that is where the credit belongs. But on the other hand, doing so violates ... | https://mathoverflow.net/users/8430 | Citing papers that are in a language that you do not read. | I think a common-sense approach is to cite the original paper (whatever the language) in order to give credit and attribution but only rely on arguments from papers you can understand in your proofs (so you don't violate the golden rule).
Regarding reviewers, the worst that can happen (I think) is that you use a cru... | 48 | https://mathoverflow.net/users/2284 | 43151 | 27,437 |
https://mathoverflow.net/questions/42922 | 3 | The following problem arises when we try to bound the expected offline optimal value of a simple online assignment problem with random values and unit weights, by its deterministic approximation.
The Problem
-----------
Consider a sequence $\{X\_i\}\_{i=1}^n$ of non-negative integrable i.i.d. random variables with ... | https://mathoverflow.net/users/10203 | An Upper Bound for the Average of Top Order Statistics | Write
$$
E[T\_k]=E[E[T\_k|X\_{(n-k)}]]
$$
The distribution of $X\_{(n-k+1)},\dots,X\_{(n)}$ given $X\_{(n-k)}=x$ is the same as the conditional distribution of a monotone arrangement of $n-k$ independent r.v.'s with cdf $F$ given they all are not less than $x$. (This can be proved easily e.g. by using the quantile tran... | 3 | https://mathoverflow.net/users/8146 | 43159 | 27,443 |
https://mathoverflow.net/questions/43172 | 0 | I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this:
"For any permutation matrix $P$ in $G$ there exists an orthogonal matrix $Q$ such that $Q^{-1}PQ=A$, where $A$ is a block-diagonal matrix representing a ... | https://mathoverflow.net/users/4078 | Morphisms between representations | Isn't this just a fancy way of talking about the cycle decomposition of a permutation? The language you use is a little imprecise. But, firstly, by change of basis P can be made into a block matrix form, with each block corresponding to a cycle. And secondly one needs only to think of a cyclic permutation in matrix for... | 2 | https://mathoverflow.net/users/6153 | 43178 | 27,455 |
https://mathoverflow.net/questions/42731 | 2 | Let $V$ be a Hausdorff locally convex topological vector space over the field $\mathbb{K}$.
Let $B$ be a subset of $V$ such that
$\;$ for all functions $c : B\to \mathbb{K}$, if $\displaystyle\sum\_{b\in B} \; c(b)\cdot b = 0$, then $c$ is identically zero
and $f : B\times V \to \mathbb{K}$ be a function such t... | https://mathoverflow.net/users/nan | Are coordinate functions on topological vector spaces always continuous? | Here is a non orthogonal example that is however natural and probably simpler than the other one I gave. Let $b\_1, b\_2,...$ be the character basis for $L\_2(-\pi,\pi)$. Let $b\_0$ be an $L\_1$ function whose Fourier series does not converge in the $L\_1$ norm. Then $b\_0,b\_1,b\_2,...$ is countably linearly independe... | 2 | https://mathoverflow.net/users/2554 | 43184 | 27,458 |
https://mathoverflow.net/questions/43180 | 3 | Hello,
I'm trying to bound an integral. I have a function $A(\nu) = | 1 + \exp(-I \nu) |$ (with $I$ being the imaginary unit) and I want to show that the condition (Paley-Wiener criterion for causality) applies
$$\int\_{-\infty}^{\infty} \frac{|\log(A(\omega))|}{1+\omega^2} \mathrm{d}\omega < \infty$$
(log is the n... | https://mathoverflow.net/users/10256 | Integration problem: $\int_{-\pi}^{\pi} | \log( | 1 + \exp(- I \nu ) | ) | \mathrm{d}\nu < \infty$ | You want to show that
$$\int\_{-\pi}^\pi|\log|1+e^{-it}||dt$$
is finite. Now
$$|1+e^{-it}|=|e^{it/2}+e^{-it/2}|=2\cos(t/2)$$
so your integral is
$$\int\_{-\pi}^\pi|\log|2\cos(t/2)||dt
=2\int\_0^\pi|\log|2\cos(t/2)||dt.$$
Replacing $t$ by $\pi-2$ in the last integral gives
$$2\int\_0^\pi|\log|2\sin(t/2)||dt.$$
The integ... | 3 | https://mathoverflow.net/users/4213 | 43185 | 27,459 |
https://mathoverflow.net/questions/43186 | 0 | Hi all,
what are the best strategies to find cutting edge papers and books on a field of mathematics?
..
Example:
2-3 years ago I had to analyze a time series. I found a paper and showed that to a mathematician who referred me to the REAL state-of-the art method how to do it.
Then I read the book ‘Analysis... | https://mathoverflow.net/users/10258 | finding cutting edge papers and books | One method that is maybe more indirect is to regularly browse through the new articles posted on the arxiv in the section of your interests. I use an RSS reader for this, but you can also subscribe to an e-mail list I think.
Also, conferences for example are always a good way to keep up to date of what's going on in ... | 4 | https://mathoverflow.net/users/9545 | 43188 | 27,461 |
https://mathoverflow.net/questions/43175 | 5 | Suppose I have a family of Dirac operators over a compact base space B. From the paper of Atiyah and Singer about skew adjoint Fredholm operators we know that it has an index in $K^1(B)$.
Suppose furthermore I know "a lot" about these Dirac operators (like their spectrum, eigenspaces etc.) and $B$ is a simple space l... | https://mathoverflow.net/users/3816 | index of a family of Dirac operators in $K^1$ | Whether the following is useful might depend on your concrete example. Because you are mentioning $K^1 $ instead of $K^0$, I assume that your Dirac operator is ungraded (if it is graded, the index should be in $K^0$). The graded case is the ordinary Atiyah-Singer family index theorem. There is a version for the ungrade... | 5 | https://mathoverflow.net/users/9928 | 43191 | 27,463 |
https://mathoverflow.net/questions/43024 | 8 | Hi,
Is the modular curve defined as the quotient of the upper half-plane by an arithmetic group $ \Gamma $ always a moduli space of elliptic curves with extra structure? I know this is true for $ \Gamma\_0(N), \Gamma\_1(N), \Gamma(N) $, but I'm interested in some of other groups, particularly those of the form $ \Gam... | https://mathoverflow.net/users/4192 | Modular Curves as Moduli Spaces of Elliptic Curves | For the groups $\Gamma$ in Conway-Norton, there is always a moduli problem of $\Gamma$-structures, but since the groups always contain $\Gamma\_0(N)$ for some $N$, you won't be able to construct a universal family (because there is a $-1$ automorphism in the way). However, you will sometimes get a ``relatively represen... | 11 | https://mathoverflow.net/users/121 | 43194 | 27,465 |
https://mathoverflow.net/questions/43138 | 10 | Let $A \otimes B$ be the algebraic tensor of two $C^{\ast}$ -algebras, and an element x in $A\otimes B$ is positive if $x=yy^{\ast}$. Then is it always possible to write x in the form $x=\sum a\_i\otimes b\_i$, where $a\_i$ and $b\_i$ are positive elements?
| https://mathoverflow.net/users/9858 | positive elements in tensor products | I think the answer is no. The matrix
$$
a=\begin{bmatrix}
1&0&0&1\\
0&0&0&0\\
0&0&0&0\\
1&0&0&1
\end{bmatrix}
$$
is positive in $M\_4(\mathbb{C})$. When we see this algebra as $M\_2(\mathbb{C})\otimes M\_2(\mathbb{C})$, it cannot be obtained as a sum of elementary tensors with positive entries. .
*(ok, several hours ... | 18 | https://mathoverflow.net/users/3698 | 43198 | 27,468 |
https://mathoverflow.net/questions/43211 | 5 | In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian varieties
$0 \to F \to \text{Alb}(X) \to \text{Jac}(C) \to 0$
with $F$ being isogenous to $E$.
Assume that $X$ is ... | https://mathoverflow.net/users/5197 | an exercise about elliptic surface in Beauville's book | Donu Arapura is right:
If you want $F$ isomorphic to $E$ you can do the following:
Consider the product $E \times P^1$. You find $F \simeq Alb(X) \simeq E$.
If you want $F$ and $E$ to be non isomorphic, one can do as follows.
Let $c$ be a 2-torsion point of $E$. Form the quotient $X$ of $E \times E$ by the involuti... | 8 | https://mathoverflow.net/users/5273 | 43232 | 27,491 |
https://mathoverflow.net/questions/43104 | 2 | I need to bound the expectation of a nonnegative random variable that satisfies a Poisson-type tail bound:
$\mathbb{P}( X \geq t ) \leq \min( d \cdot (\frac{a}{t} )^{t}, \ 1)$ for $t > 0$
where $a > 0$ and $d \geq 3$. My guess for the mean:
$\mathbb{E} X \leq {\rm const} \cdot \max( a,\ \frac{\log d}{\log \log d... | https://mathoverflow.net/users/10247 | Expectation of RVs with Poisson-type decay | It's a bit late now (so maybe I made a trivial mistake), but it seems to me that the statement is actually false. Take some large $d$ and set $a=\log d / \log \log d$. Let $t=Ka=K\log d / \log \log d$ for $K$ to fixed soon. Then the tail bound is
$$\mathbb{P}(X\ge t) \le d K^{-K\log d / \log \log d}=exp(\log d - \frac{... | 0 | https://mathoverflow.net/users/1061 | 43266 | 27,510 |
https://mathoverflow.net/questions/43209 | 7 | Dear all,
I have a probably rather simple question: Suppose we have a Matrix $ M\in SL\_2(\mathbb{Q}) $. Does the group $ M^{-1} SL\_2(\mathbb{Z}) M \cap SL\_2(\mathbb{Z})$ then always have finite index in $SL\_2(\mathbb{Z})$? Why? Why not?
I really was not able to solve this problem!
All the best
Karl
| https://mathoverflow.net/users/10264 | Has a conjugation of SL2(Z) finite index in SL2(Z)? (Modular group) | I, for one, am less than thrilled with snobbish kibitzing in the comments. Just answer the question already instead of dropping hints and passing judgment.
The answer is yes for $\text{SL}(n,\mathbb{Z})$. Let $d$ be the product of the denominators in the matrices $M$ and $M^{-1}$. Let $\Gamma\_d \subseteq \text{SL}(n... | 22 | https://mathoverflow.net/users/1450 | 43272 | 27,513 |
https://mathoverflow.net/questions/43252 | 6 | In [A New Kind of Science: Open Problems and Projects](http://www.wolframscience.com/openproblems/NKSOpenProblems.pdf)(pg. 36).
>
> How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? The symbolic forms of the Ackermann function with a fi... | https://mathoverflow.net/users/nan | Do maps have flows? | In general, there are obstructions for a map being the "time one map" of a flow (see [this question](https://mathoverflow.net/questions/31050/homomorphisms-from-r-to-diffeor-or-fractional-iterations/31056#31056)).
However, and I am not quite sure this is what you are looking for, there is a general procedure to cons... | 8 | https://mathoverflow.net/users/5753 | 43276 | 27,517 |
https://mathoverflow.net/questions/43269 | 2 | Can anyone give me the reference for this statement?:
Let $M$ be a closed oriented smooth 4-manifold. Any element $a\in H\_2(M)$ can be represented by a smoothly embedded, oriented surface.
I found this statement and the proof at Saveliev's book, Lectures on the topology of 3-manifold, but I think it is not a compl... | https://mathoverflow.net/users/6569 | Reference for the proof of this statement? | This result actually holds in all dimensions.
>
> Let $M^n$ be a closed smooth manifold $M^n$ of any dimension $n\geqslant 3$. Every element $\alpha \in H\_2(M,\mathbb Z)$ is represented by a smoothly embedded closed surface.
>
>
>
You can prove it as follows:
1. Take a cycle $a\_1\sigma\_1 + \ldots + a\_n... | 7 | https://mathoverflow.net/users/6205 | 43278 | 27,519 |
https://mathoverflow.net/questions/43281 | 14 | I have some vague sense that certain types of categories are related to certain types of logic. I've been meaning to learn more about this, so I thought I'd ask about the simplest case, propositional logic. In particular, I'd be interested in a statement of the completeness theorem for propositional logic using these i... | https://mathoverflow.net/users/290 | Propositional logic with categories | Qiaochu, let me see if this answers your question:
**Proposition:** Suppose $B$ is a cartesian closed category with finite coproducts such that the canonical double dual embedding
$$b \to (b \Rightarrow 0) \Rightarrow 0$$
is an isomorphism (excluded middle). Then $B$ is equivalent (as a category) to a poset, an... | 11 | https://mathoverflow.net/users/2926 | 43285 | 27,523 |
https://mathoverflow.net/questions/41292 | 0 | If you have non-Arch. local field F and E its finite extension, I am just wondering if anybody has any idea about the action of $\operatorname{Gal}(E/F)= \operatorname{Aut}\_F(E)$ on the lines in $k^2\_E$, Where $k\_E$ is a residue field of the extension?
Note:
* $\{ \text{Adjacent points} \} \simeq \mathbb{P}^1\_k... | https://mathoverflow.net/users/9842 | Galois Action on the lines in the k^{2}_{E} | The action is fairly straightforward. Any line in $k\_E^2$ can be described by an equation of the form $ax+by=c$, where $a,b,c \in k\_E$, and $a$ and $b$ are not both zero. You can describe the action of $\operatorname{Gal}(E/F)$ on $k\_E$ by taking reduction modulo the maximal ideal. Alternatively, you can look at the... | 2 | https://mathoverflow.net/users/121 | 43290 | 27,526 |
https://mathoverflow.net/questions/43221 | 12 | Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real) Lie algebra and let $\Lambda \mathfrak{g}^{\ast}$ be the Chevalley-Eilenberg complex (i.e. the
exterior algebra,
wit... | https://mathoverflow.net/users/9928 | Lie's third theorem via differential graded algebras? | The details are here:
Marius Crainic, Rui Fernandes, *Integrability of Lie brackets*
<http://arxiv.org/abs/math/0105033>
(To connect this to your question, notice that a morphism $T X \to \mathfrak{g}$ of Lie algebroids, which is the language they use, is dually the same as a morphism $\Omega^\bullet(X) \leftarrow ... | 5 | https://mathoverflow.net/users/381 | 43291 | 27,527 |
https://mathoverflow.net/questions/43164 | 10 | Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda\_n$ of the matrix $M\_n=(m\_{ij})\_{1\le i,j\le n}$ with the elements $m\_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?
It is not difficult to show that
$$ c\sqrt n \le \lambda\_n \le C\sqrt{n\log n} $$
for appropriate positive ... | https://mathoverflow.net/users/9924 | The largest eigenvalue of a "hyperbolic" matrix | I think, $C\sqrt{n}$ is an upper bound aswell. Take vector $x=(x\_1,\dots,x\_n)$ with $x\_j=j^{-1/2}$. Then $(Ax)\_i$ behaves like $C\sqrt{n/i}=C\sqrt{n}x\_i$. But we know that if $(Ax)\_i\leq C x\_i$ for vector $x$ with positive coordinates, then the largest eigenvalue of $A$ does not exceed $C$ (kind of Perron-Froben... | 6 | https://mathoverflow.net/users/4312 | 43293 | 27,528 |
https://mathoverflow.net/questions/43019 | 3 | For a harmonic function $\Phi$ on a simply connected subset $\Gamma$ of $\mathbb{R}^3$, define a **guide curve** $\gamma: I \mapsto \Gamma$ of $\Phi$ as a simple regular $C^1$ curve such that
* all point in $\gamma(I)$ are critical points of $\Phi$, and
* for all points $p$ in $\gamma(I)$ there exists a neighborhood ... | https://mathoverflow.net/users/6302 | Geometrical structure of critical points of harmonic functions | [This is an answer made of two comments and an example]
Since the laplacian is elliptic with real-analytic coefficients, a harmonic function $f$ is real-analytic in its domain of definition. Hence the set $C$ of critical points of $f$ is a real-analytic subset of $R^3$, and as such it admits a locally finite partitio... | 4 | https://mathoverflow.net/users/6451 | 43295 | 27,530 |
https://mathoverflow.net/questions/43249 | 35 | In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be conceived of as submanifolds (of the target manifold in a sigma model), ... | https://mathoverflow.net/users/6005 | What are D-branes, really? | There is an *abstract algebraic* formulation of QFT: this says that an $n$-dimensional QFT is a consistent assignment of spaces of states and of maps between them to $n$-dimensional *cobordisms*.
If one allows cobordisms with boundary here, one speaks of *open-closed QFT*. A **D-brane** in this context is the type of... | 21 | https://mathoverflow.net/users/381 | 43299 | 27,533 |
https://mathoverflow.net/questions/43007 | 2 | In maximum likelihood estimation, one typically needs to compute the log (natural log) of probability values. When a probability, say $p(x)$, becomes so close to zero, $log(p(x))$ returns -Inf. What is the usual trick to avoid these cases?
| https://mathoverflow.net/users/5223 | Numeric problem when evaluating log of a pdf | Work with the logs of probability directly, rather than trying to compute the probability and then compute the log. You can do arithmetic with the logs, as well; multiplication becomes addition, of course. Addition is somewhat more complicated, but it's not too hard to work out how to do it without taking the exponenti... | 5 | https://mathoverflow.net/users/5010 | 43301 | 27,535 |
https://mathoverflow.net/questions/43311 | 6 | I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please tell me a good refernce on theses topics, if there is some.
| https://mathoverflow.net/users/9492 | sheaves and cosheaves | **edit:** I was assuming you wanted an equalizer sheaf property, but this is not the definition of cosheaf, see comments - the following has nothing to do with cosheaves then!
If by "cosheaf" you mean a covariant functor from the opens of a space to sets/groups/etc., you could look at Moerdijk/MacLane's ["Sheaves in ... | 2 | https://mathoverflow.net/users/733 | 43312 | 27,541 |
https://mathoverflow.net/questions/43298 | 15 | In the Wikipedia article on Hilbert's Nullstensatz,
<http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz>
the following application of the Weak Nullstensatz is mentioned:
>
> Commuting matrices
>
>
> The fact that commuting matrices have a common eigenvector – and hence by induction stabilize a common flag a... | https://mathoverflow.net/users/5273 | Commuting Matrices and the Weak Nullstellensatz | My "solution" is a bit strange, but I hope it is correct.
We regard $V$ as a module over $A$. $X = {\rm Spec}\ A$ is zero-dimensional. Then (as a sheaf on $X$) $V$ decomposes into a direct sum of sheaves over the finitely many points of $X$. This corresponds to a decomposition $V$ into subspaces corresponding to diff... | 6 | https://mathoverflow.net/users/3847 | 43315 | 27,542 |
https://mathoverflow.net/questions/43323 | 4 | I am talking about the principle that says that every set is the image of a [projective](http://planetmath.org/?op=getobj&from=objects&id=6437) set. For every set $x$ there is a surjection $f:y \twoheadrightarrow x$, such that for any set $u$ and function $j:y \to u$ and surjection $g:v \twoheadrightarrow u$ there is a... | https://mathoverflow.net/users/6787 | Existence of enough projectives in the category of sets | I haven't heard any new information about "enough projective sets" in the classical ZF context. Your formulation, however, looks weaker. "Enough projective sets" should say that for every $x$ there is a surjection $f:y \to x$ such that every surjection from any $w$ to $y$ splits. That implies your formulation (by takin... | 2 | https://mathoverflow.net/users/6794 | 43330 | 27,551 |
https://mathoverflow.net/questions/43313 | 45 | Every now and then I attempt to understand better quantum mechanics and quantum field theory, but for a variety of possible reasons, I find it very difficult to read any kind of physicist account, even when the physicist is trying to be mathematically respectable. (I am not trying to be disrespectful or controversial h... | https://mathoverflow.net/users/2926 | Good references for Rigged Hilbert spaces? | Some time ago I was interested in rigged Hilbert space to get a better understanding of quantum physics. On that occasion I collected some references on this subject, see below. It's quite comprehensive. A good starting point for an overview could be the works of Madrid and Gadella. Note that there are different versio... | 31 | https://mathoverflow.net/users/7538 | 43332 | 27,552 |
https://mathoverflow.net/questions/43346 | 6 | Can any one help me in proving the following equality:
$$n^n= \sum\_{i=1}^n {n \choose i}\cdot i^{i-1}\cdot (n-i)^{n-i}$$
I tried some different ideas but none of them worked!
| https://mathoverflow.net/users/10286 | Combinatorial equation | Your equation can be written as an equation for exponential generating functions: $f(x) = g(x)(f(x)+1)$, where $$f(x) = \sum\_{n\ge1}n^nx^n/n!$$ and $$g(x) = \sum\_{n\ge1}n^{n-1}x^n/n!$$
We can see that for those $f(x)$ and $g(x)$, we have $f(x) = xg'(x)$. If we then solve the differential equation $$xg'(x) = \frac{g... | 6 | https://mathoverflow.net/users/400 | 43349 | 27,559 |
https://mathoverflow.net/questions/43338 | 17 | Let X be a compact symplectic manifold with a form $\omega$. And $X \times X$ is equipped with the symplectic form $(\omega,-\omega)$. The diagonal $\Delta:X \mapsto X \times X$ is a Lagrangian submanifold. So, in this question, [Hochschild (co)homology of Fukaya categories and (quantum) (co)homology](https://mathoverf... | https://mathoverflow.net/users/6986 | Comparison between Hamiltonian Floer cohomology and Lagrangian Floer cohomology of the diagonal | Let $f:X\to X$ be a (Hamiltonian) symplectomorphism. The claim is that the fixed point Floer homology of $f$ agrees with the Lagrangian intersection Floer homology of the graph of $f$ with the diagonal in $X\times X$. I think the argument is that if we choose almost complex structures for the two Floer theories in the ... | 21 | https://mathoverflow.net/users/6670 | 43357 | 27,565 |
https://mathoverflow.net/questions/43362 | 1 | Why does forcing seem to be so vacuously true?
It seems like you are just reversing the subset containment of the model of ZFC + CH to be the other way in the poset. So, why is this valid? Why are you allowed to just put the empty set at the top of the partial order\*?
\*I have just started learning set theory so I... | https://mathoverflow.net/users/nan | The consistency of ZFC + CH gives the ability to travel to a universe which models ZFC + \neg CH? | The empty set $\emptyset$ here is the forcing condition that has the least amount of information about the generic object being constructed.
Since it has the least information, you might think it should be at the bottom of the partial order. There are two replies to this:
(1) Mathematically, we are free to define... | 8 | https://mathoverflow.net/users/4600 | 43365 | 27,571 |
https://mathoverflow.net/questions/43366 | 6 | Kodaira embedding theorem says that a positive line bundle is ample, i.e. high tensor powers are holomorphically embeddable into complex projective space of high dimension.
However, ampleness is not stable under blow-ups. Usually a replacement is to consider big line bundles, which is stable under blow-ups.
Is th... | https://mathoverflow.net/users/nan | Embedding Theorem for big line bundles | If $X$ is normal, then the Iitaka fibration theorem implies that $L$ is big if and only if the rational map
$\phi\_m \colon X \dashrightarrow \mathbb{P}H^0(X, L^{\otimes m})$
is birational onto its image for some $m >0$, see [Lazarsfeld, Positivity in Algebraic Geometry I, p. 139].
I guess this is the "embedding ... | 7 | https://mathoverflow.net/users/7460 | 43368 | 27,573 |
https://mathoverflow.net/questions/43372 | 10 | This question is about p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}\_p / \mathbb{Q}\_p)$ and $(\varphi, \Gamma)$-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the category of p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}\_p / \mathbb{Q}\_p)$ is equivalent to each of ... | https://mathoverflow.net/users/2481 | Can an etale (phi, Gamma) module be an extension of non-etale ones? | In the first two cases, the slopes of $\varphi$-modules are given by the "standard" Dieudonné-Manin decomposition. In particular, subobjects of étale objects are étale.
For more info, see (for example) chapter 4.5 of Kedlaya's "Slope Filtrations Revisited".
| 9 | https://mathoverflow.net/users/5743 | 43373 | 27,577 |
https://mathoverflow.net/questions/43380 | 3 | It is well known that the upper bound on the number of quadratic residues mod n is approximately n/2 and it reaches this bound for n prime.
Is there any similar lower bound on the number of quadratic residues mod n?
Some numerical experiments indicate that it would be somewhere at n^0.65 for highly composite n with... | https://mathoverflow.net/users/10292 | Numbers with few quadratic residues | Edit: First version of this answer had a silly mistake (forgot to multiply by $n$ the first estimate). Argument is still the same but the final result changes.
If you take $n$ to be the product of the first $r$ odd primes, then the number of quadratic residues modulo $n$ is bounded below by ${(1/2)}^rn$, by the Chine... | 5 | https://mathoverflow.net/users/2290 | 43384 | 27,582 |
https://mathoverflow.net/questions/43381 | 10 | More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that
$$\left| \frac{p}{q} - r \right| < \frac{\epsilon}{q^2}.$$
I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly po... | https://mathoverflow.net/users/290 | What numbers can be approximated "pretty well" by rationals? | This is a well-studied question in diophantine approximation. You can look up Markov or Lagrange spectrum for a "description" of the numbers for which you cannot take $\epsilon$ arbitrarily small. For the answer to your last question, look up Khinchin's theorem (the answer is no, they have full measure).
| 13 | https://mathoverflow.net/users/2290 | 43385 | 27,583 |
https://mathoverflow.net/questions/16794 | 5 | In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also interested in certain simplicial complexes that are singular disks, or even more generally singular disks tiled by polygons.... | https://mathoverflow.net/users/1450 | Better term for a (simplicial) contractible plane continuum | In the end, we (Joel Kamnitzer, his student Bruce Fontaine, and I) agreed on the term "diskoid". In the abstract, the word "cactus" seemed too clever by half. When I actually wrote it into the paper, it was awkward and it didn't emphasize the main property of interest, that our shapes are meant as a mild generalization... | 1 | https://mathoverflow.net/users/1450 | 43387 | 27,585 |
https://mathoverflow.net/questions/43051 | 3 | I am trying to compare some homomorphism groups over different base rings, so given a commutative local ring $(A,\mathfrak{m})$ and a finite dimensional Azumaya algebra $R$ over $A$.
If $M$ and $N$ are two left $R$-modules which are finitely generated and torsion free over $A$, is there an $A$-isomorphism $R \otimes... | https://mathoverflow.net/users/3233 | Comparing homomorphisms over different base rings | I think this is true. There is a homomorphism $R \otimes\_A Hom\_R(M,N) \rightarrow Hom\_A(M,N)$, given from the obvious $R$-module structure of $Hom\_A(M,N)$. To check that this is an isomorphism we can make a faithfully flat extension and split $R$; so we may assume that $R$ is a matrix algebra $M\_n(A)$.
Now we us... | 3 | https://mathoverflow.net/users/4790 | 43390 | 27,587 |
https://mathoverflow.net/questions/43397 | 16 | Finding primes in signals is seen as a sign of some kind of intelligence - see e.g. the role of primes in the search for extraterrestrial life (see e.g. [here](http://www.math.dartmouth.edu/~carlp/PDF/extraterrestrial.pdf)).
This is because there are relatively few examples of numbers that appear in nature because t... | https://mathoverflow.net/users/1047 | Examples of prime numbers in nature | Somewhat longish to be a comment, so here goes:
How about examples like:
* Polygons in nature? Unfortunately, the most famous one of them is a hexagon ;-)
But I am certain there are several chemical compounds, physical structures such as crystals, and so on that exhibit polygonal structures with prime number of e... | 2 | https://mathoverflow.net/users/8430 | 43401 | 27,595 |
https://mathoverflow.net/questions/43400 | 10 | This should be really well known but I don't seem to find a statement about it nor a question in MO answering this.
Consider a Compact Hausdorff topological space $X$. The [cohomological dimension](http://en.wikipedia.org/wiki/Cohomological_dimension) of $X$ is the first natural number where the cohomology vanishes.... | https://mathoverflow.net/users/5753 | Topological dimension versus cohomological dimension | Well, I think it depends on which dimension you mean and which cohomology. The best fit I think is covering dimension and Čech cohomology. The Čech cohomological dimension is indeed bounded (more or less by definition) by the covering dimension.
**Addendum**: Just to comment on BCnrd's comment. The usual definition o... | 11 | https://mathoverflow.net/users/4008 | 43405 | 27,598 |
https://mathoverflow.net/questions/43305 | 18 | Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following statement is true:
>
> For any probability space
> $(X,\Sigma,\mu),$ for any
> measure-preserving transformation $T$
>... | https://mathoverflow.net/users/6101 | Ergodic limits along subsets of $\mathbb{N}.$ | These are called good universal sets.
Bourgain (1987) proved that sequences of the form $p(n)$, $n \in {\bf N}$, $p$ a non constant polynomial, are good.
He also proved (1988) that the set of primes is a good universal set for $L^p$ functions, $p> {(1+\sqrt{3})\over 2}$. This was later improved to $p>1$ by Wierdl, se... | 10 | https://mathoverflow.net/users/6129 | 43414 | 27,604 |
https://mathoverflow.net/questions/43336 | 12 | Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.
**Question:** Is $Y$ also Cohen-Macaulay? Are there common conditions which imply it is?
If $Y$ is not normal I know of several ways to show that the answer to the first question is no.
Th... | https://mathoverflow.net/users/3521 | Blowups of Cohen-Macaulay varieties | An example was given in Section 3 of this [paper by Cutkosky](https://doi.org/10.1007/BF01233425 "Invent Math 102, 157–177 (1990). zbMATH review at https://zbmath.org/?q=an:0718.14025"): "A new characterization of rational surface singularities." (The scheme $Z$ in the last page, which is a blow up of some $m$-primary ... | 11 | https://mathoverflow.net/users/2083 | 43415 | 27,605 |
https://mathoverflow.net/questions/43408 | 12 | In customary formulations of the Singular Value Decomposition or SVD that I have seen,
(e.g., Wikipedia or Gil Strang's textbooks) it is always stated in terms of writing an
$m \times n$ matrix $M$ (say of rank $r$) as a product $U \Lambda V$, where $U$ and $V$ are
orthogonal $m \times m$ and $n \times n$ matrices an... | https://mathoverflow.net/users/7311 | Is this formulation of the Singular Value Decomposition standard? | I just looked in Wikipedia (<http://en.wikipedia.org/wiki/Singular_value_decomposition>). There is a very thorough discussion, including a section "Geometric meaning" in which your interpretation is clearly explained. Well, there $K^n$, where $K = \mathbb R$ or $K = \mathbb C$ is used, instead of arbitrary inner produc... | 6 | https://mathoverflow.net/users/4790 | 43416 | 27,606 |
https://mathoverflow.net/questions/43411 | 5 | Is there any subset of $R^n$ homotopically equivalent to the wedge product of countable many circles.
In particular, is the union of circles in $R^2$ with center (0,n) and radius n for n=1,2 ... n
(sort of inverse Hawaiian earring) homotopically equivalent to that wedge product.
Note that the answer is no if we ask a... | https://mathoverflow.net/users/10299 | homotopical immersion of the wedge product of countable many circles in $R^n$ | You can certainly embed the real line, with perpendicular circles at the integer points, into $\mathbb{R}^3$, and this is homotopy equivalent to the wedge you want.
Let $W$ be the expanding `inverse Hawaiian earring'.
Then the obvious map $\bigvee S^1\to W$
is a homotopy equivalence. To define the inverse,
let $Z... | 5 | https://mathoverflow.net/users/3634 | 43432 | 27,616 |
https://mathoverflow.net/questions/43428 | 1 | Suppose we have two sets of discrete events, $A$ and $B$. Then I think it is true that:
$$2\sum\_{i \in A, j \in B}\Pr[i\ \textrm{AND}\ j] \leq \sum\_{i \in A}\Pr[i]+ \sum\_{j \in B}\Pr[j] +\sum\_{i, j \in A}\Pr[i\ \textrm{AND}\ j] + \sum\_{i, j \in B}\Pr[i\ \textrm{AND}\ j]$$
My intuition for why this should be tr... | https://mathoverflow.net/users/10303 | Probability space analogue of Cauchy-Schwarz inequality | yes, if $\sum\_{i,j\in A}$ means double summation (each pair $i,j$ is taken twice)
then denote $f\_i$ and $g\_i$ characteristic functions of your events from $A$ and $B$ respectively, LHS equals $2 \sum \int f\_i g\_j=\int 2(\sum f\_i)(\sum g\_j)$, RHS equals $\sum \int f\_i^2+\sum \int g\_j^2+2\sum \int f\_i f\_j+2\su... | 3 | https://mathoverflow.net/users/4312 | 43437 | 27,618 |
https://mathoverflow.net/questions/43436 | 0 | Hi Mathoverflow
I hope you bear with me that my linear algebra knowledge is a little rusty, but I have a question that might potentially very easy to answer. Nevertheless it's been bugging me for a good 3 hours now.
I'm trying to find the LU decomposition of a matrix in order to find the determinant. This would be ... | https://mathoverflow.net/users/8672 | Finding the determinant of a matrix with LU composition | The factorization you give at the end ("the true upper and lower matrices") is incorrect. To see why, just check the (1,1) element in your original matrix. Multiplying your $L$ by your $U$ gives 4 for that element, but your original matrix has a 2 there.
Meshcach's factorization is correct. The right $L$ and $U$ matr... | 2 | https://mathoverflow.net/users/9716 | 43440 | 27,620 |
https://mathoverflow.net/questions/36810 | 3 | Let $X$ be a subcomplex of a CW-complex $Y$. Is $(Y/X)^{\wedge k}$ homotopy equivalent to $Y^{\wedge k}/X^{\wedge k}$, where $\wedge k$ is the $k$-fold smash product? I know it is not true for products but am having a hard time visualizing for smash products.
| https://mathoverflow.net/users/8658 | Do Smash Products and Quotients Commute? | The easiest way I know to say what is going on is to resort to looking at
"products" of pairs:
$$
(X, A) \times (Y, B) = ( X\times Y , A\times Y \cup X\times B).
$$
The point of this notation is that the functor $(X, A) \mapsto (X/A, \*)$
carries $(X, A) \times (Y, B)$ to $X/A \wedge Y/B$. We can iterate this procedure... | 4 | https://mathoverflow.net/users/3634 | 43442 | 27,621 |
https://mathoverflow.net/questions/43446 | 4 | I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the characteristic of the field obtained by the quotient of $R$ with its unique maximal ideal $M$ and by ramification index I mean th... | https://mathoverflow.net/users/10204 | Examples of DVRs of residue char p and ramification e | The family of rings $\mathbb{Z}\_p[p^{\frac{1}{e}}]$ does what you want.
Because of your question, I gather you do not yet know the correspondence between totally ramified extensions and Eisenstein polynomials. For this see e.g. Serre's *Local Fields*, Lang's *Algebraic Number Theory*, or Section 4.3 of
[http://alp... | 6 | https://mathoverflow.net/users/1149 | 43448 | 27,623 |
https://mathoverflow.net/questions/43447 | 5 | Suppose $(C,\otimes)$ is a symmetric monoidal finitely-cocomplete category such that $\otimes$ preserves colimits. Given two morphisms $a:A\_1\to A\_2$ and $b:B\_1\to B\_2$, define $a\Box b$ to be the induced "lower right corner map" $A\_1\otimes B\_2\coprod\_{A\_1\otimes B\_1} A\_2\otimes B\_1\to A\_2\otimes B\_2$.
... | https://mathoverflow.net/users/1353 | Is the box product of morphisms associative? | Yes, that gives a monoidal product (but don't forget the monoidal unit! it's $0 \to I$ where $I$ is the monoidal unit of $\mathcal{C}$).
To simplify the description of the associativity, imagine that $\mathcal{C}$ is strict monoidal to begin with. Draw the entire cube of possible paths that lead from $A\_1 \otimes B... | 7 | https://mathoverflow.net/users/2926 | 43453 | 27,627 |
https://mathoverflow.net/questions/43434 | 10 | I'm asking the question on a bit of whim, but I do wonder what answers I would get. The Goldbach conjecture is usually discussed in the realm of the distribution of primes and/or probability. Methods I've seen in the past are mostly analytic.
Have there been methods of attack on this problems that are at their core n... | https://mathoverflow.net/users/5756 | Algebraic aspects of the Goldbach conjecture | You might be interested in this article [on Goldbach over function fields](http://arxiv.org/abs/0912.1702). The approach is rather geometric/algebraic, so it does pass your "steers away from hard analysis" test.
| 6 | https://mathoverflow.net/users/35416 | 43455 | 27,628 |
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