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https://mathoverflow.net/questions/27657 | -4 | I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^2 + Dx + Ey + F" (see the edit below)
Experimenting with some polynomials of this sort showed me that factorization... | https://mathoverflow.net/users/5627 | Factorizing polynomials of several variables (in a different perespective) | Let K be a field. The ring $K[X\_1,...,X\_n]$ is factorial, which means that any polynomial in n variables can be factored into a product of irreducible polynomials. But of course, these polynomials are not of degree 1 in general. If a polynomial can be factored as a product of terms of degree one, then its zero set is... | 3 | https://mathoverflow.net/users/6129 | 27659 | 18,074 |
https://mathoverflow.net/questions/27658 | 7 | Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We define a new category in the following way:
An object is an object $M$ of $D^b(X)$, together with an isomorphism
$$\pi^\*M\l... | https://mathoverflow.net/users/2837 | Is this a definition of equivariant derived category? | The relation is easier to understand if $G$ is a discrete group. Then your
definition is equivalent to an action of $G$ on an object $M$ in $D^b(X)$. This
means that for each $g\in G$ we have a morphism $\phi\_g\colon M\rightarrow M$ in
$D^b(X)$ with, and this is the important part, $\phi\_{gh}=\phi\_g\phi\_h$
*in* $D^... | 12 | https://mathoverflow.net/users/4008 | 27662 | 18,076 |
https://mathoverflow.net/questions/27621 | 10 | On [this](http://www.ms.u-tokyo.ac.jp/video/vgbook/idx_vguest2006.html) page, the interview is [here](http://www.ms.u-tokyo.ac.jp/video/vgbook/2006/0602kato_L.ram). Can someone provide an English translation?
| https://mathoverflow.net/users/436 | Transcription of an interview of Kazuya Kato | I can write a short summary of the nice interview, sacrificing lots of details, if that is really what you want. But, you can probably ask the video archive stuff video@ms.u-tokyo.ac.jp? They should be happy to receive your response.
| 14 | https://mathoverflow.net/users/36665 | 27669 | 18,082 |
https://mathoverflow.net/questions/27681 | 14 | How can the liar paradox be expressed concisely in symbols? In which formal languages?
| https://mathoverflow.net/users/3441 | Formulas for the liar paradox | The [Liar](http://en.wikipedia.org/wiki/Liar_paradox) is the statement "this sentence is false." It is expressible in any language able to perform self-reference and having a truth predicate. Thus, $L$ is a statement equivalent to $\neg T(L)$.
Goedel proved that the usual formal languages of mathematics, such as the... | 23 | https://mathoverflow.net/users/1946 | 27686 | 18,095 |
https://mathoverflow.net/questions/27691 | 7 | Hanner's inequalities in the theory of $L^p$ spaces (see <http://en.wikipedia.org/wiki/Hanner>'s\_inequalities) look hard to come-up with at the first glance. Their proof (say, the one in Lieb & Loss "Analysis", Theorem 2.5.) gives no intuition (at least for me) how they come about. How does one see that these inequali... | https://mathoverflow.net/users/5498 | Hanner's inequalities: the intuition behind them | First, a probably unsatisfactory answer to your first question. I believe that Hanner proved his inequalities because he was investigating the modulus of convexity of $L^p$. So possibly he formed a guess on whatever basis, saw that guess would be correct if what have come to be known as Hanner's inequalities were true,... | 7 | https://mathoverflow.net/users/1044 | 27692 | 18,099 |
https://mathoverflow.net/questions/27693 | 56 | I hope this question is suitable; this problem always bugs me. It is an issue of mathematical orthography.
It is good praxis, recommended in various essays on mathematical writing, to capitalize theorem names when recalling them: for instance one may write "thanks to Theorem 2.4" or "using ii) from Lemma 1.2.1" and s... | https://mathoverflow.net/users/828 | Capitalization of theorem names | In English, proper nouns are capitalized. The numbered instances you mention are all usages as proper nouns, but merely refering to a lemma or corollary not by its name is not using a proper noun, and so is uncapitalized.
Thus, for example, one should write about the lemma before Theorem 1.2 having a proof similar to... | 64 | https://mathoverflow.net/users/1946 | 27695 | 18,101 |
https://mathoverflow.net/questions/27723 | 2 | Let $A\_1,A\_2,\ldots,A\_k$ be finite sets. Furthermore, for each $i\in\{1,2,\ldots,k\}$, let $B\_i$ be a set whose elements are subsets of $A\_i$.
Is there any polynomial-time algorithm that decides whether there exists a choice of precisely one element $C\_i$ of each $B\_i$ such that for all $x\in (C\_1\cup C\_2\c... | https://mathoverflow.net/users/6726 | poly-time algorithm to choose elements of sets | It seems to me that your problem is stronger than $\ell$-SAT.
In fact, let $A$ be the set of our literals. Assume that we have $p$ clauses. For each $i\in\left\lbrace 1,2,...,p\right\rbrace$, let $A\_i$ be the set of the literals occuring in the $i$-th clause, and let $B\_i$ be the set of all nonempty subsets of $A\_i$... | 3 | https://mathoverflow.net/users/2530 | 27731 | 18,122 |
https://mathoverflow.net/questions/27720 | 17 | As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions as requiring that the solution to both equations be the same element of $S$. He suggested instead replacing one of the e... | https://mathoverflow.net/users/3959 | Do these conditions on a semigroup define a group? | I am a victim of timing... I had asked this of a colleague a few days ago and had received no answers, but today at lunch he gave me a counterexample and reference (Clifford and Preston's *The Algebraic Theory of Semigroups*, volume II, pp. 82-86). The example is the Baer-Levi semigroup: the semigroup of all one-to-one... | 16 | https://mathoverflow.net/users/3959 | 27732 | 18,123 |
https://mathoverflow.net/questions/27722 | 4 | I'm wondering about a cross product for spectral sequences. I've got an idea, and I wonder if it is written up anywhere, or if it even holds water.
Let's start with three spectral sequences, $E, F$ and $G$. Assume that
$G\_1^{\*,\*} \cong E\_1^{\*,\*}\otimes F\_1^{\*,\*}$ as chain complexes.
Then the ordinary Künnet... | https://mathoverflow.net/users/3634 | Tensor product of spectral sequences? | There is no reqson for $\times\_2$ to be a chain map. Pick for example a spectral sequence such that the differential in $E\_1$ is zero, such that the one on $d\_2$ is not, and pick a any product $\times\_1$ on $E\_1$ such that it is not a chain map on $E\_2=E\_1$.
You can get such structures, though. For example, su... | 9 | https://mathoverflow.net/users/1409 | 27746 | 18,132 |
https://mathoverflow.net/questions/27729 | 14 | One of the common definitions of homology using the singular chains, i.e. maps from the simplex into your space. The free abelian group on these can be made into a chain complex and one can take the homology of this. The result is usually called singular homology.
However, one can use a smaller chain complex instead,... | https://mathoverflow.net/users/798 | What are normalized singular chains good for? | Before going to linearity, note a technical advantage of normalized geometric realization is that it preserves products. That probably has linear consequences.
The simplest technical advantage of normalized chains is the [Dold-Kan theorem](http://ncatlab.org/nlab/show/Dold-Kan+correspondence) that the normalized chai... | 10 | https://mathoverflow.net/users/4639 | 27752 | 18,136 |
https://mathoverflow.net/questions/27261 | 4 | I will begin by saying that $k=3$ might be a very specific case and this question is useless. Even if that is the case, I would like to know...
The sum of squares function $r\_k(n)$ is very famous. It counts the number of ways $n$ can be written as a sum of $k$ squares. In the case of $k=3$, when $n$ is squarefree an... | https://mathoverflow.net/users/2024 | Groups related to sum of squares function? | In my view, it depends a little what you mean by "related," but I don't see at first glance any natural group whose order is r\_k(n) for any k other than 3. Loosely speaking, representations of a form of rank m by the genus of a form of rank n are related to the set of double cosets
H(Q) \ H(A\_f) / H(Zhat)
where H... | 4 | https://mathoverflow.net/users/431 | 27759 | 18,142 |
https://mathoverflow.net/questions/27708 | 21 | When I study formal completion and formal schemes, on p.194 of Hartshorne's "Algebraic Geometry", he said "One sees easily that the stalks of the sheaf $\mathcal{O}\_{\hat{X}}$ are local rings."
Notice that here $\mathcal{O}\_{\hat{X}}$ is not the structure sheaf of X, there is a "hat" on the symbol $X$.
But I can... | https://mathoverflow.net/users/5482 | formal completion | Here is a self-contained explanation (hopefully without any blunder):
Locally, $\hat{X}$ is an affine formal scheme, so each point has a neighbourhood basis admitting of open sets $U$ admitting the following description:
there is a ring
$A$, with ideal $I$, such that the underlying topological space is $U\_0 :=$ Spe... | 30 | https://mathoverflow.net/users/2874 | 27780 | 18,158 |
https://mathoverflow.net/questions/27785 | 21 | If you have an infinite set X of cardinality k, then what is the cardinality of Sym(X) - the group of permutations of X ?
| https://mathoverflow.net/users/3537 | Cardinality of the permutations of an infinite set | $k^k$.
Easy that it's an upper bound. For lower bound split $X$ into two equinumerous
subsets; there are $\ge k^k$ permutations swapping the two subsets.
| 26 | https://mathoverflow.net/users/4213 | 27788 | 18,163 |
https://mathoverflow.net/questions/27747 | 5 | This question is motivated by some issue raised by David Speyer in [this question](https://mathoverflow.net/questions/27634/k-0r-mathbbz-but-some-f-g-projective-not-stably-free).
Let $R$ be a ring. Let $K\_0(R)$ and $G\_0(R)$ be the Grothendieck groups of f.g. projective modules and f.g. modules over $R$, respective... | https://mathoverflow.net/users/2083 | Seeking examples or proof: injectivity of Cartan homomorphism for commutative rings? | No, it is not injective in general, unless $R$ is regular notherian. There are many counterexamples; for a simple one you can take the ring $R := \mathbb C[t^2, t^3] \subseteq \mathbb C[t]$, compute that $G\_0(R) = \mathbb Z$, while $K\_0(R)$ maps onto the Picard group of $R$, which is the additive group $\mathbb C$.
... | 9 | https://mathoverflow.net/users/4790 | 27790 | 18,164 |
https://mathoverflow.net/questions/27827 | 2 | The complex projective line is isomorphic to the 2-sphere, and so, has genus $0$. Does this result for all $CP^N$, that is, is the genus of $CP^N$ equal to $0$, for all $N$?
| https://mathoverflow.net/users/1977 | Genus of complex projective space | The geometric genus (the dimension of the space of global sections of the
canonical sheaf) of projective $n$-space is zero. See Hartshorne II.8.
| 6 | https://mathoverflow.net/users/4213 | 27829 | 18,187 |
https://mathoverflow.net/questions/27839 | 2 | In light of the answers given to [this question](https://mathoverflow.net/questions/27827/genus-of-complex-projective-space), I would like to pose a more general one: Do all Grassmannian spaces have genus 0? If so, do there exist any flag manifolds with non-zero genus?
| https://mathoverflow.net/users/1977 | Genus of Grassmannians and Flag Manifolds | EDIT: My first answer was confusing and not quite accurate. Let me try again.
In arbitrary characteristic, the structure sheaf of any homogeneous space $G/P$ (for $G$ a semisimple group) has no higher cohomology. This is an instance of Kempf's vanishing theorem. The space of sections is 1-dimensional, so this implies... | 8 | https://mathoverflow.net/users/321 | 27840 | 18,193 |
https://mathoverflow.net/questions/27825 | 2 | Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product
$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^x}$
where $x > 2$ is a real number? The domain of integration is the entire complex plane. Poles are allowed in the ... | https://mathoverflow.net/users/4526 | L^2 space of holomorphic functions with given weight | Hi Daniel. As already said in my comment the space consists just of order polynomials of degree $\lfloor x - 1 \rfloor$. First, one can check that any function in the space must be holomorphic, since the weight doesn't help to integrate over poles. Then one gets from $\\| f \\| < \infty$ that $|f(x)| \leq |z|^{x-1 }$, ... | 6 | https://mathoverflow.net/users/3983 | 27844 | 18,196 |
https://mathoverflow.net/questions/27836 | 9 | Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-dimensional holomorphic representation of the complex Lie group $G(\mathbb{C})$ is actually an algebraic representation (i.e., gi... | https://mathoverflow.net/users/379 | Algebraicity of holomorphic representations of a semisimple complex linear algebraic group |
>
> "Then the claim would be immediate by
> semisimplicity if one can show that
> every irreducible representation of
> $G(\mathbb{C})$ (or perhaps of Lie
> groups in some more general class than
> these) occurs as a subquotient of a
> tensor power of a faithful one. How
> might one prove the latter? (Can one
... | 1 | https://mathoverflow.net/users/297 | 27848 | 18,199 |
https://mathoverflow.net/questions/27830 | 0 | I am working on stability of nonlinear switched systems and recently, I have proven that switched systems with homogeneous, cooperative, Irreducible and commuting vector fields , i.e., vector fields with Lie bracket equal to 0, are D-stable under some condition. I was trying to find an example for such systems but surp... | https://mathoverflow.net/users/6748 | Commuting Nonlinear Vector Fields | Any set of nonlinear co-ordinates gives you a corresponding set of commuting vector fields. So any of the co-ordinates listed in the "See also" section of
<http://en.wikipedia.org/wiki/Elliptic_coordinates>
gives an example.
| 1 | https://mathoverflow.net/users/613 | 27852 | 18,201 |
https://mathoverflow.net/questions/27854 | 13 | I feel sort of silly asking this question. Unless I'm very much mistaken the paper I'm reading assumes the following statement:
Let $G$ be a finite group. We may embed it via the Cayley embedding into an ambient permutation group $G \leq S\_{|G|}$. Then any automorphism of $G$ comes from conjugation by an element in ... | https://mathoverflow.net/users/5756 | Does every automorphism of G come from an inner automorphism of S_G? | The statement is true. Let $g \in G$ and $\pi \in Aut(G)$. Let $\lambda\_{g}$ be the corresponding left translation by $g$. Regard $\pi$ and $\lambda\_{g}$ as elements of $Sym(G)$. Then for all $x \in G$,
$(\pi \lambda\_{g} \pi^{-1})(x) = (\pi \lambda\_{g}) ( \pi^{-1}(x)) =
\pi( g \pi^{-1}(x)) = \pi(g) x = \lambda\_... | 16 | https://mathoverflow.net/users/4706 | 27856 | 18,202 |
https://mathoverflow.net/questions/27851 | 55 | Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be interested to know the status of it.
Let $f$ be a one variable complex polynomial. Supposing $f$ has a common root with ev... | https://mathoverflow.net/users/2349 | Polynomials having a common root with their derivatives | That is known as the Casas-Alvero conjecture. Check this out, for instance:
<https://arxiv.org/abs/math/0605090>
Not sure of its current status, though.
| 46 | https://mathoverflow.net/users/4290 | 27859 | 18,204 |
https://mathoverflow.net/questions/27853 | 5 | The finite dimensional irreducible unitary representations of $SU(2)$ are labelled by $j$ which needs to be half-integer, the dimension of the representation is $2j+1$. This is well-known, all is good.
If we do not require finite dimension for the representation, is it possible to make sense of representations with a... | https://mathoverflow.net/users/4526 | Infinite dimensional unitary representations of SU(2) for non-half-integer j? | I just wrote this answer on your last question. To summarize, you can't have an irreducible representation of a compact group that is infinite dimensional, unless the representation space is very exotic.
By the Peter-Weyl Theorem, all irreducible Hilbert space representations of a compact group (e.g. SU(2)) are finit... | 11 | https://mathoverflow.net/users/6753 | 27869 | 18,210 |
https://mathoverflow.net/questions/27793 | 11 | I was reading Logicomix (a fictionalised account of logic from Frege to Gödel through Russell's eyes) and there was mention about two different versions of types developed by Russell and Whitehead for Principia Mathematica, unramified (first) and ramified. I don't expect that there is too much relation to the modern ty... | https://mathoverflow.net/users/4177 | Russell and Whitehead's types: ramified and unramified | Yes, this still occurs in modern type theory; in particular, you'll find it in the [calculus of constructions](http://en.wikipedia.org/wiki/Calculus_of_constructions) employed by the [Coq](http://www.lix.polytechnique.fr/coq/) language.
Consider the type called `Prop`, whose inhabitants are logical propositions (whic... | 12 | https://mathoverflow.net/users/2361 | 27880 | 18,217 |
https://mathoverflow.net/questions/27884 | 6 | Consider the following linear Diophantine Equation::
```
ax + by + cz = d ------------ (1)
```
for all, a,b,c and d positive integers, and relatively prime, and assume a>b>c without loss of generality.
Can we find a lower bound on d which ensures at least one non-negative solution to this equation?
I know w... | https://mathoverflow.net/users/6759 | Non-negative integer solutions of a single Linear Diophantine Equation | The question of determining the lower bound on $d$ is called the Frobenius problem. For $2$ variables your bound can be improved: every integer starting from $(a-1)(b-1)$ is representable as a non-negative combination. Some general results on this problem are available in [this paper](http://www.jstor.org/pss/2371684),... | 8 | https://mathoverflow.net/users/1306 | 27886 | 18,221 |
https://mathoverflow.net/questions/27624 | 4 | I need some references for irreducible representations of the Heisenberg algebra with three generators or the category of its finite length modules.
Here, the Heisenberg algebra with three generators $x$, $y$ and $z$ is defined to be the Lie algebra whose underlying vector space is generated by $x$, $y$, and $z$, and... | https://mathoverflow.net/users/5604 | Irreducible representations of Heisenberg algebra | If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to a certain kind of equivalence (given by conjugation with algebra isomorphisms) there is a unique irreducible representation of the algebra. It is infinite dimensional, given by polynomials in one variable. If the... | 5 | https://mathoverflow.net/users/121 | 27897 | 18,229 |
https://mathoverflow.net/questions/27713 | 1 | I'm trying to solve the BVLS problem for huge (2e6x2e6) matrices which are very sparse (4 elements per row). Does anybody have a recommendation for a free solver (preferably a library of routines)?
The BVLS problem is defined as:
$\underset{l \le x \le u}{\min} \lVert Ax - b \rVert\_2^2$
| https://mathoverflow.net/users/1899 | Recommendations for a large scale bounded variable least squares (BVLS) solver for sparse matrices | This is such a well-solved problem that there are many software packages that have built in functions for this.
Here are a selection of built-in functions in different software packages that can be used:
In Matlab: lsqlin (type help lsqlin into Matlab and it tells you exactly what to type. I have just (approximat... | 2 | https://mathoverflow.net/users/2011 | 27902 | 18,233 |
https://mathoverflow.net/questions/27901 | 1 | It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X \quad x\_{n} \rightarrow p \Rightarrow f(x\_{n}) \rightarrow f(p)$$
It is also well known that if $X$ and $Y$ are met... | https://mathoverflow.net/users/4002 | Does Cauchy continuity imply uniform continuity? [No.] | No it's not true.
f(x) = x^2 on whole real line.
It maps Cauchy sequences to Cauchy sequences but it's not uniformly continuous on the whole real line.
| 12 | https://mathoverflow.net/users/3124 | 27918 | 18,247 |
https://mathoverflow.net/questions/27912 | 1 | How can one show $\displaystyle\frac{(m+n-1)!}{m ! (n-1)!}\leq \left[\frac{e (m+n-1)}{n}\right]^{n-1}$ ?
| https://mathoverflow.net/users/6766 | bound for binomial coefficients | Denote the quotient of the right and left hand sides,
$$
f(m,n)=\biggl(\frac{e(m+n-1)}n\biggr)^{n-1}\bigg/\binom{m+n-1}m.
$$
Then $f(m,1)=1$ for all $m\in\mathbb N$ and
$$
\frac{f(m,n+1)}{f(m,n)}
=\frac{e}{\biggl(1+\dfrac1n\biggr)^n}\cdot\biggl(1+\frac1{m+n-1}\biggr)^{n-1} > 1,
$$
that is,
$$
f(m,n+1)> f(m,n)>\dots> f... | 4 | https://mathoverflow.net/users/4953 | 27923 | 18,251 |
https://mathoverflow.net/questions/27896 | 14 | According to the OEIS ([A002966](http://oeis.org/A002966)) there are 294314 solutions in positive integers to the equation
$$\sum\_{i=1}^7\frac{1}{x\_i}=1$$ assuming $x\_1\leq x\_2\leq\cdots\leq x\_7$.
Similarly for 8 summands there are 159330691 solutions.
My question: What are they? Is there a way of counting th... | https://mathoverflow.net/users/6355 | Diophantine equation: Egyptian fraction representations of 1 | As far as I know, the only significant result to speed up these calculations is that $E\_2(\frac{p}{q}) = \frac{1}{2}|\lbrace d: d | q^2, d \equiv -q (mod p) \rbrace|$, where $E\_2(p/q)$ represents the number of decompositions into 2 unit fractions, and each matching $d$ represents the decomposition $\frac{p}{q} = \fra... | 12 | https://mathoverflow.net/users/6089 | 27925 | 18,252 |
https://mathoverflow.net/questions/27922 | 2 | For introduction, Ethiointegers are integers which get reversed when multiplied by another number.
For instance,
2178 \* 4 = 8712
1089 \* 9 = 9801
I couldn't find such numbers, even by another name anywhere else except in one journal which I think is not electronically accessible.
**My question**: Has number... | https://mathoverflow.net/users/5627 | Ethio Integers? | These unnamed numbers were famous enough to make it to [A Mathematician's Apology](http://en.wikipedia.org/wiki/A_Mathematician%27s_Apology). After mentioning what you wrote above Hardy writes:
>
> These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which... | 11 | https://mathoverflow.net/users/2384 | 27927 | 18,253 |
https://mathoverflow.net/questions/27931 | 38 | The $j$-function and the fact that 163 and 67 have class number 1 explain why:
$\operatorname{exp}(\pi\cdot \sqrt{163}) = 262537412640768743.99999999999925$,
$\operatorname{exp}(\pi\cdot \sqrt{67}) = 147197952743.9999987$.
But is there any explanation for these?:
$\frac{163}{\operatorname{ln}(163)} = 31.9999987... | https://mathoverflow.net/users/6769 | Why Is $\frac{163}{\operatorname{ln}(163)}$ a Near-Integer? | On the other hand, *Mathematica* gives LogIntegral[163]=43.075210908806756346563... and LogIntegral[67]=22.6520420103880266691324... so this does not appear to be connected to x/Ln[x] in the context of the Prime Number Theorem
| 4 | https://mathoverflow.net/users/6756 | 27932 | 18,255 |
https://mathoverflow.net/questions/27878 | -1 | [Akaike's information criterion](http://en.wikipedia.org/wiki/Akaike_information_criterion) is a measure of the goodness of fit of an estimated statistical model that accounts for both the fit quality and model complexity. One way to calculate AIC is as follows:
$\mathit{AIC}=2k + n[\ln(\mathit{RSS})]\,$
, where $k... | https://mathoverflow.net/users/5823 | Number of parameters in Akaike's information criterion | The question should be: how many of the parameters need to get estimated based on the data? If three of them were somehow known independently of the data, then I say for present purposes there's only one.
The nonlinearity in $x$ in the $\sin x$ term is not a concern because $f\_2(x)$ is linear in $\sin x$ and the par... | 0 | https://mathoverflow.net/users/6316 | 27966 | 18,278 |
https://mathoverflow.net/questions/27941 | 1 | The Fourier series of a function (B-spline) is given by:
$$s(x)=\sum\_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$
But the B-spline has only finite support. How can one see this using its Fourier series representation?
| https://mathoverflow.net/users/3589 | Fourier series of B-spline | A function is a piecewise polynomial if and only f it is a linear combination of functions, each of them having some derivative equal to a finite sum of Dirac measures.
The jth Fourier coefficient of the Dirac at a is $e^{2\pi ija}$, and integrating amounts to multiplying the coefficient by some power of $1\over j$.... | 4 | https://mathoverflow.net/users/6129 | 27969 | 18,280 |
https://mathoverflow.net/questions/27971 | 60 | Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a technical theorem as a glorious wellspring of intuition- something, at least from my perspective, that rings are missing... | https://mathoverflow.net/users/5869 | Why is there no Cayley's Theorem for rings? | Every (associative, unital) ring is a subring of the endomorphism ring of its underlying additive group. Rings act on abelian groups; groups act on sets. The universal action on an abelian group is its endomorphism ring; the universal action on a set is the symmetric group. Modules are rings that remember their action ... | 84 | https://mathoverflow.net/users/3710 | 27974 | 18,282 |
https://mathoverflow.net/questions/20265 | 2 | Here is my question: how to define global section functor from D-module on affine flag variety to representation of affine Lie algebra?
Let's me explain the difficulty: it seems there doesn't exist global definition of D-module on ind-scheme. For affine flag variety, it is a union of finite dimensional subvarieties, ... | https://mathoverflow.net/users/5082 | About localization theorem for affine Lie algebra? | The main problem seems to be that you think the global section functor for (twisted) D-modules on a singular variety depends on a choice of embedding into a smooth variety. This is not true - D-modules can be defined on singular spaces using the infinitesimal site, and you can define global sections without any choice ... | 2 | https://mathoverflow.net/users/121 | 27977 | 18,285 |
https://mathoverflow.net/questions/27509 | 5 | Let $\gamma$ be a simple loop in a spine of a strongly irreducible Heegaard splitting of a closed 3-manifold $M$ with torsion-free fundamental group. Does $\gamma$ necessarily correspond to a primitive element of the fundamental group of $M$, or is it possible for $\gamma$ to be a power of some other element?
I suspe... | https://mathoverflow.net/users/4325 | Is a simple loop in a spine of a strongly irreducible Heegaard splitting primitive in the fundamental group? | **New Answer:** Take a 2-bridge knot, and perform hyperbolic
Dehn filling (so that the core of the Dehn filling is geodesic),
and so that the filling slope has intersection number $>1$ with
the meridian. Then the meridian will not be primitive, since it
will be a multiple of the core of the Dehn filling. 2-bridge
knots... | 4 | https://mathoverflow.net/users/1345 | 27983 | 18,291 |
https://mathoverflow.net/questions/27984 | 5 | "Random" modules of the same size over a polynomial ring seem to always have the same Betti table. By a "random" module I mean the cokernel of a matrix whose entries are random forms of a fixed degree.
For example if we take the quotient of the polynomial ring in three variables by five random cubics:
$S = \mat... | https://mathoverflow.net/users/3293 | Does the free resolution of the cokernel of a generic matrix remain exact on a Zariski open set? | It sounds like the question you mean to ask is the following: if $X$ is an integral noetherian scheme with generic point $\eta$ and $C^{\bullet}$ is a finite complex of coherent sheaves on $X$ such that $C^{\bullet}\_ {\eta}$ is exact, then does there exist a dense open $U$ in $X$ such that $C^{\bullet}|\_U$ is exact a... | 6 | https://mathoverflow.net/users/6773 | 27986 | 18,292 |
https://mathoverflow.net/questions/27989 | 8 | We know that a countably additive translation invariant measure with $\mu([0,1]) = 1$ cannot be defined on the power set of $\mathbb R$. This is because $[0,1]$ can be partitioned into countably many congruent sets, with the help of the axiom of choice.
But I was wondering whether a *finitely additive* measure with t... | https://mathoverflow.net/users/1229 | Finitely additive translation invariant measure on $\mathcal P(\mathbb R)$ | Banach-Tarski poses a problem for existence of measures that are invariant under all rigid motions, not just translation. The existence of finitely additive translation-invariant measures that agree with Lebesgue measure on Lebesgue-measurable sets is a consequence of the Hahn-Banach theorem. This is exercise 21 in cha... | 7 | https://mathoverflow.net/users/121 | 27994 | 18,297 |
https://mathoverflow.net/questions/27965 | 1 | For me the definition of amenability of an at most countable discrete group (with counting measure) is existence of a Folner sequence. Assuming this, why is every countable discrete abelian group amenable? What is the Folner sequence that does the job?
| https://mathoverflow.net/users/5498 | Countable discrete abelian group amenable | The direct limit $G = \bigcup\_n G\_n$ of a nested sequence of countable amenable groups $G\_n$ is still amenable, since every finite set $S$ in $G$ will lie in one of the $G\_n$ and thus there must exist some finite set $F\_S$ which is not shifted very much by $S$. Since there are only a countable number of $S$, one c... | 5 | https://mathoverflow.net/users/766 | 27999 | 18,301 |
https://mathoverflow.net/questions/28000 | 13 | There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series
$$\sum\_{n=1}^\infty \frac{\chi(n)}{n^s}$$
or as an Euler product
$$\prod\_{p\mbox{ prime}} \left(1-\frac{\chi(p)}{p^s}\right)^{-1}.$$
Correspondingly, this gives two ways of restricting a D... | https://mathoverflow.net/users/6243 | What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions? | It's best to split this up into two cases.
Case 1: $\chi(a) = 1$. Then for $\Re(s) > 1$,
$$\prod\_{p \equiv a \pmod{q}} \left(1 - \frac{\chi(p)}{p^s}\right)^{-1} = \prod\_{p \equiv a \pmod{q}} \left(1 - \frac{1}{p^s}\right)^{-1} = \sum\_{n \in \left\langle \mathcal{P} \right\rangle} \frac{1}{n^s},$$
where $\left\lang... | 11 | https://mathoverflow.net/users/3803 | 28004 | 18,305 |
https://mathoverflow.net/questions/27997 | 2 | Let $X = \mathbb{R}^n$, and consider a nondegenerate representation $\rho: C\_0(X) \to B(H)$ where $B(H)$ is the algebra of bounded operators on a separable Hilbert space. The support of a vector $v \in H$ is defined to be the complement in $X$ of the union of all open sets $U$ such that $\rho(f)v = 0$ for every $f \in... | https://mathoverflow.net/users/4362 | Is this a correct interpretation of support in coarse geometry? | **Edit** I have amended the proof to cover the general case following a suggestion of Matthew Daws.
By the definition of $supp(v)$, for any $x$ in $supp(v)^c$ there exists an open set $U(x)\subset supp(v)^c$ containing $x$ such that $\rho(f)v=0$ for all $f\in C\_0(U(x)).$ If $g$ has compact support $K\subset supp(v)^... | 4 | https://mathoverflow.net/users/5740 | 28005 | 18,306 |
https://mathoverflow.net/questions/27972 | 18 | Is it true that a line bundle is relatively ample iff its restsriction to fibers is? If so, what would be the reference?
| https://mathoverflow.net/users/6772 | Relatively ample line bundles | If you admit the map to be proper and the schemes to be reasonably good it is true.
A reference I know is Lazarsfeld's book "Positivity in algebraic geometry", paragraph 1.7.
| 5 | https://mathoverflow.net/users/6430 | 28006 | 18,307 |
https://mathoverflow.net/questions/8269 | 11 | This is a question about forcing. I have seen the following fact mentioned in multiple places, but have not been able to find a proof: if a random real is added to a transitive model of ZFC, then in the generic extension the set of reals in the ground model becomes meager.
My guess is that one should be able to, in ... | https://mathoverflow.net/users/2436 | Adding a random real makes the set of ground model reals meager | The proof is based on the fact that there is a decomposition ${\bf R}=A\cup B$ of the reals such that $A$, $B$ are (very simple) Borel sets, $A$ is meager, $B$ is of measure zero, and ${\bf R}=A\cup B$ even holds if after forcing we reinterpret the sets. Nos let $s$ be a random real. If $r\in {\bf R}$ is an old real, t... | 8 | https://mathoverflow.net/users/6647 | 28024 | 18,319 |
https://mathoverflow.net/questions/28028 | 33 | It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
[the survey article](http://www.ams.org/notices/199910/fea-strichartz.pdf) by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one can write (and solve) the heat equation on the fractal.
**The ... | https://mathoverflow.net/users/5371 | How to define a differential form on a fractal? | Take a look at [Jenny Harrison, "Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes' theorem,"](http://iopscience.iop.org/0305-4470/32/28/310/) which should at least answer your question about a "correct" notion of flux and divergence for a fractal domain -- here's the abstract:
>
> By replacing the pa... | 15 | https://mathoverflow.net/users/1557 | 28029 | 18,321 |
https://mathoverflow.net/questions/28025 | 16 | I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific field right now and maybe some of you is aware of the answer.
The situation with Gödel's second incompleteness theorem is... | https://mathoverflow.net/users/6307 | Clarification of Gödel's second incompleteness theorem | The key idea Feferman is exploiting is that there can be two different enumerations of the axioms of a theory, so that the theory does not prove that the two enumerations give the same theory.
Here is an example. Let $A$ be a set of the axioms defined by some formula $\phi(n)$ (that is, $\phi(x)$ holds for exactly th... | 21 | https://mathoverflow.net/users/5442 | 28031 | 18,322 |
https://mathoverflow.net/questions/28008 | 12 | Disclaimer:
-----------
I am asking this question as an improvement to [this question](https://mathoverflow.net/questions/27881/who-is-the-last-mathematician-that-understood-all-of-mathematics), which should be community wiki. This is in line with the actions taken by Andy Putman in a similar case (cf. [meta](http://... | https://mathoverflow.net/users/1353 | At what point in history did it become impossible for a person to understand most of mathematics? | The world's output of scientific papers increased exponentially from 1700 to 1950.
One online source is [this article](http://www.its.caltech.edu/~dg/crunch_art.html) (which is concerned with what has happened since then). The author displays a graph (whose source is a 1961 book entitled "Science since Babylon" by D... | 11 | https://mathoverflow.net/users/2356 | 28033 | 18,324 |
https://mathoverflow.net/questions/25691 | 25 | G(n,p)
------
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This model is referred to as **G(n,1/2)** or more generally **G(n,p).**
Random graphs with prescribed marginal beha... | https://mathoverflow.net/users/1532 | Some models for random graphs that I am curious about | The Lovasz-Szegedy theory of *graphons* is likely to be relevant. Every measurable symmetric function $p: [0,1] \times [0,1] \to [0,1]$ (otherwise known as a graphon) determines a random graph model, in which every vertex v is assigned a colour c(v) uniformly at random from the unit interval [0,1], and then any two ver... | 16 | https://mathoverflow.net/users/766 | 28045 | 18,331 |
https://mathoverflow.net/questions/28027 | 4 | Let $L/K$ be a finite Galois extension with Galois group $G$ and $V$ a $L$-vector space, on which $G$ acts by $K$-automorphisms satisfying $g(\lambda v)=g(\lambda) g(v)$. It is known that the canonical map
$V^G \otimes\_K L \to V$
is an isomorphism. However, I can't find any short and nice proof for that. Actually ... | https://mathoverflow.net/users/2841 | Galois descent, explicit inverse map | Martin, <http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisdescent.pdf>
is a handout on this kind of stuff
and Theorem 2.14 there gives a proof of the bijection between $K$-forms of $V$ and $G$-structures on $V$. It doesn't qualify as "short", and whether it's "nice" or not is too
subjective. I wrote it f... | 6 | https://mathoverflow.net/users/3272 | 28049 | 18,333 |
https://mathoverflow.net/questions/27584 | 5 | Hi,
Having a random variable $X$ I am trying to find a stochastic process $Z\_t$ such that:
$$P[Z\_t>T] = P[X > T | X > t]$$
for all $T>t$, or a proof that such a process does not exist.
Please note that this question is not related to any homework and that I actually need this result for my research in financi... | https://mathoverflow.net/users/3160 | Process equivalent to conditional probability | Cool problem. The process you are after is certainly not unique, but here is a reasonably explicit construction of an increasing jump process $Z\_t$ with the property you want. (Under a couple of assumptions which I think are implicit in your statement).
The assumptions: 1) $X$ is a positive random variable, $P(X>0)=... | 6 | https://mathoverflow.net/users/6781 | 28050 | 18,334 |
https://mathoverflow.net/questions/28053 | 7 | This question probably has a simple and immediate answer which escapes me now. (And, I should admit, it's more my curiosity than anything else.) The only natural way to construct a group structure on the cartesian product $G\times H$ of two groups $G$ and $H$ (in particular, ``natural'' to me means that on each factor ... | https://mathoverflow.net/users/1306 | Group structures on the cartesian product of two groups | Wikipedia to the rescue!
<http://en.wikipedia.org/wiki/Zappa-Szep_product>
| 10 | https://mathoverflow.net/users/1106 | 28058 | 18,338 |
https://mathoverflow.net/questions/28056 | 31 | **Question.** Given a Turing-machine program $e$, which
is guaranteed to run in polynomial time, can we computably
find such a polynomial?
In other words, is there a
computable function $e\mapsto p\_e$, such that whenever $e$
is a Turing-machine program that runs in polynomial time,
then $p\_e$ is such a polynomial t... | https://mathoverflow.net/users/1946 | Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program? | [Edit: A bug in the original proof has been fixed, thanks to a comment by Francois Dorais.]
The answer is no. This kind of thing can be proved by what I call a "gas tank" argument. First enumerate all Turing machines in some manner $N\_1, N\_2, N\_3, \ldots$ Then construct a sequence of Turing machines $M\_1, M\_2, M... | 41 | https://mathoverflow.net/users/3106 | 28060 | 18,340 |
https://mathoverflow.net/questions/28062 | 6 | Let $M$ be a compact Riemannian manifold and let $T : M \rightarrow M$ be Anosov. I have read [here](http://books.google.com/books?id=hFN6oiecbrYC&pg=PA7) that it is an open problem to prove that $T$ is topologically mixing if $M$ is connected. [Katok and Hasselblatt](http://books.google.com/books?id=9nL7ZX8Djp4C&dq&pg... | https://mathoverflow.net/users/1847 | When is an Anosov diffeomorphism mixing? | I don't have a proper answer to your main question beyond pointing to the list of equivalent properties in Pesin's book (your first reference), which you've obviously seen already. However, I'll point out that in Ruelle's paper (your third reference), the first main theorem (on page 3), which contains a statement on ex... | 4 | https://mathoverflow.net/users/5701 | 28065 | 18,343 |
https://mathoverflow.net/questions/28063 | 14 | Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:
1. If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a [blog posting by David Speyer](http://sbseminar.wordpress.com/2008/02/14/complex-manifolds-which-are-not-algebraic/), y... | https://mathoverflow.net/users/1450 | Highly connected, compact complex manifolds | E. Calabi, B. Eckmann, *A class of compact, complex manifolds which are not algebraic.*
Ann. of Math. (2) 58, (1953). 494–500.
From Chern's MR review (MR0057539):
>
> This paper defines on the topological product $S^{2p+1} \times S^{2q+1}$ of two spheres of dimensions $2p+1$ and $2q+1$ respectively, $p$ > 0, a ... | 20 | https://mathoverflow.net/users/2356 | 28067 | 18,344 |
https://mathoverflow.net/questions/28047 | 15 | Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a result on arithmetic progressions represented by a positive binary form. So my question is whether that is the case, do w... | https://mathoverflow.net/users/3324 | The Green-Tao theorem and positive binary quadratic forms | Edit: {The answer to your question, "...do we already know that a positive binary form represents arbitrarily long arithmetic progressions?" is yes. See the second paragraph below.}
If the relative density exists, so does the Dirichlet density and they are equal. The converse is not true in general. For primes in a g... | 15 | https://mathoverflow.net/users/5513 | 28069 | 18,346 |
https://mathoverflow.net/questions/28054 | 22 | The group $Diff(S^n)$ ($C^\infty$-smooth diffeomorphisms of the $n$-sphere) has many interesting subgroups. But one question I've never seen explored is what are its "big" finite-dimensional subgroups?
For example, $Diff(S^n)$ contains a finite-dimensional Lie subgroup of dimension $n+2 \choose 2$, the subgroup of c... | https://mathoverflow.net/users/1465 | "Largest" finite-dimensional Lie subgroups of Diff(S^n), are they known? | You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint open subset. So the additive group $\mathbb R^N$ becomes a subgroup of $Diff(S^1)$ by flowing along $N$ commuting vector ... | 33 | https://mathoverflow.net/users/6666 | 28081 | 18,354 |
https://mathoverflow.net/questions/28088 | 21 | As usual I expect to be critisised for "duplicating"
[this question](https://mathoverflow.net/questions/27931/). But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my [response](https://mathoverflow.net/questions/27931/why-is-163-ln163-a-near-in... | https://mathoverflow.net/users/4953 | When is $n/\ln(n)$ close to an integer? | If $f(x)=\frac{x}{\log x}$, then $f'(x)=\frac{1}{\log x} - \frac{1}{(\log x)^2}$, which tends to zero as $x\rightarrow \infty$. Choose some large real number $x$ for which $f(x)$ is integral. Then the value of $f$ on any integer near $x$ must be very close to integral.
| 42 | https://mathoverflow.net/users/5513 | 28090 | 18,359 |
https://mathoverflow.net/questions/28103 | 2 | I have recursive polynomials
$$Q\_{n}(t)=tQ\_{n-1}(t)+\frac{1-t^{2}}{n+1}Q'\_{n-1}(t)$$
and
$$Q\_{0}(t)=1$$
Is there a theory for finding a factorisation of recursive polynomials?
It is possible to show that
$$\sum\_{s=-\infty}^{\infty}sinc(\pi(x+s))^{p+1}=Q\_{p-1}[cos(\pi x)]$$
where $$sinc(x)=sin(x)/x$$
... | https://mathoverflow.net/users/3589 | roots of recursive polynomials | I have realised that your recurrent relation is exactly the one which appears in Eq. (2.4) in [[*Izvestiya: Mathematics* **66**:3 (2002) 489--542]](http://dx.doi.org/10.1070/IM2002v066n03ABEH000387) (see
also [here](http://wain.mi.ras.ru/PS/zete_main.pdf)). The properties of the corresponding polynomials are expressed ... | 3 | https://mathoverflow.net/users/4953 | 28105 | 18,365 |
https://mathoverflow.net/questions/28093 | 12 | Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum
$$ \delta = \sum\_i f\_i\partial\_x^i$$
where there $f\_i$ are complex polynomials in x.
Let $R=\mathbb{C}[x]$, and consider the image of $\delta$ as a map on R. As a subspace of $R$, doe... | https://mathoverflow.net/users/750 | Does the image of a differential operator always contain an ideal? | No. Let $\delta=x-\partial$ and $L=Im(\delta).$ I claim that $L$ does not contain any non-zero ideal of $\mathbb{C}[x].$ Indeed, $x^k\equiv (k-1)x^{k-2}\ (\mod L)$ and, by induction,
$$x^{2n+1}\equiv (2n)!!x\equiv 0(\mod L),\ x^{2n}\equiv (2n-1)!!\ (\mod L).$$
Thus $L$ contains all odd powers of $x$ and has codime... | 14 | https://mathoverflow.net/users/5740 | 28107 | 18,367 |
https://mathoverflow.net/questions/1388 | 39 | Given a set Ω and a σ-algebra F of subsets, is there some natural way to assign something like a "uniform" measure on the space of all measurable functions on this space? (I suppose first it would be useful to know if there's even a natural σ-algebra to use on this space.)
The reason I'm asking is because I'd like to... | https://mathoverflow.net/users/445 | Is there a natural measures on the space of measurable functions? | Let I be the unit interval with the Borel $\sigma$-algebra. There is no $\sigma$-algebra on the set of measurable functions from I to I such that the evaluation functional $e:I^I\times I\to I$ given by $e(f,x)=f(x)$ is measurable, as shown by Robert Aumann [here](http://www.ma.huji.ac.il/raumann/pdf/66.pdf), so even fi... | 42 | https://mathoverflow.net/users/35357 | 28114 | 18,370 |
https://mathoverflow.net/questions/28113 | 2 | **Q** is the rational number field.
p is a prime number.
q is a prime number other than p.
$k\_{p^r}$ is a cyclotomic field.
$k\_{p^r}$=**Q**(x) where x is exp(2$\pi$i/$p^r$).
[$k\_{p^r}$:**Q**]=$p^{r-1}(p-1)$.
Question: Does q remain a prime in the integer ring of $k\_{p^r}$?
| https://mathoverflow.net/users/2666 | Cyclotomic Fields over Q and prime ideals | Theorem I.2.13 of Washington's book on cyclotomic fields says the following: $K$ is the $n$th cyclotomic field and $p\nmid n$, let $f$ be the smallest positive integer such that $p^f\equiv 1 (\mathrm{mod}~n)$. Then $p$ splits into $\phi(n)/f$ distinct primes in $K$.
| 10 | https://mathoverflow.net/users/1021 | 28121 | 18,375 |
https://mathoverflow.net/questions/28010 | 1 | Hi
Just a short question. How are the IQR of the boxplot related to the confidence interval of a sample? Is the IQR actually the 50% confidence interval?
| https://mathoverflow.net/users/5357 | Boxplot IQR and confidence interval | My answer didn't seem to score any points with anyone, and rdchat's answer is lousy, so let's look more closely.
Suppose $X\_1,\dots,X\_n$ are an i.i.d. sample from a normally distributed population with unknown mean $\mu$ and unknown variance $\sigma^2$, and we seek a confidence interval for the population mean. As ... | 1 | https://mathoverflow.net/users/6316 | 28126 | 18,376 |
https://mathoverflow.net/questions/28119 | 11 | Hello,
Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields :
*Number of roots of unity in number fields is something like the size of the constant field for function fields.*
Could an... | https://mathoverflow.net/users/2330 | Why the roots of unity are the analogs of constants ? | One answer: the roots of unity in $K$ are [precisely the elements](https://mathoverflow.net/questions/10911/english-reference-for-a-result-of-kronecker) of $K$ which have absolute value $1$ for every absolute value on $K$; the elements of the constant field have this property for function fields.
| 13 | https://mathoverflow.net/users/297 | 28129 | 18,378 |
https://mathoverflow.net/questions/26040 | 23 | Given a positive integer $a$, the *Ramsey number* $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K\_n$ are colored using only two colors, we necessarily have a copy of $K\_a$ with all its edges of the same color.
For example, $R(3)= 6$, which is usually stated by saying that in a party of... | https://mathoverflow.net/users/6085 | Ramsey multiplicity | I emailed David Conlon about this question. He agreed to let me share his answer. In short, the problem very much seems to be open (I've added the relevant tag). As Thomas mentions, the upper bound I cite is straightforward. *And nothing better is known!*
If one looks for papers on Ramsey multiplicity, a few come up,... | 12 | https://mathoverflow.net/users/6085 | 28132 | 18,381 |
https://mathoverflow.net/questions/28104 | 16 | It occured to me that the Sieve of Eratosthenes eventually generates the same prime numbers, independently of the ones chosen at the beginning. (We generally start with 2, but we could chose any finite set of integers >= 2, and it would still end up generating the same primes, after a "recalibrating" phase).
If I tak... | https://mathoverflow.net/users/2446 | Sieve of Eratosthenes - eventual independence from initial values | So, let's see if I can precisify your claim. We start with a finite "seed set" that we assert to be Ghi-Om-prime (the seed set must not contain 1). Numbers smaller than the largest seed we completely ignore. Now for numbers larger than any seed prime, we run the Seive. You claim that there is some cut-off, depending of... | 15 | https://mathoverflow.net/users/78 | 28142 | 18,389 |
https://mathoverflow.net/questions/28135 | 4 | I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
| https://mathoverflow.net/users/19029 | When is a triangular matrix totally unimodular? | [Seymour's decomposition theorem](http://en.wikipedia.org/wiki/Matroid#Regular_matroids) for regular matroids yields a polynomial-time algorithm for testing if any {0,1,-1} matrix is totally unimodular. Unfortunately, due to the sound of paint drying on this [question](https://mathoverflow.net/questions/27346/has-anyon... | 5 | https://mathoverflow.net/users/2233 | 28144 | 18,390 |
https://mathoverflow.net/questions/28092 | 5 | It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to *deciding* whether the intersection is non-empty?
| https://mathoverflow.net/users/6645 | Feasibility of linear programs | It seems this can be done in linear time. Algorithms that solve linear programs are also capable of deciding whether the LP is feasible or not and 2-d linear programs can be solved in linear time (linear in terms of the number of constraints). So to decide whether a set of n halfplanes is non-empty or not, just solve t... | 2 | https://mathoverflow.net/users/6645 | 28146 | 18,392 |
https://mathoverflow.net/questions/28147 | 150 | I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ones of $p=1$, $p=2$, and $p=\infty$. I don't know much analysis and the best thing I could think of was Littlewood's 4... | https://mathoverflow.net/users/3106 | Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$? | Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces.
* Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of *a weak* solution to the corresponding Cauchy problem with initial data from the space $L^2(\mathbb R^3)$. Unfortunately, it is still a ma... | 69 | https://mathoverflow.net/users/5371 | 28150 | 18,395 |
https://mathoverflow.net/questions/28152 | 4 | Is there a notion of fibered category with box products?
By this I roughly mean a fibration $C\rightarrow B$ where $B$ has finite products,
along with functors
$$\boxtimes: C(X)\times C(Y)\rightarrow C(X\times Y)$$
and some coherent isomorphisms, for example:
$$(f\times g)^\* (M\boxtimes N) \leftrightarrow (f^\* M) \... | https://mathoverflow.net/users/2837 | Is there a notion of "fibered category with boxproducts"? | Yes, there is. In [this paper](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html) I called it a *monoidal fibration* (with cartesian monoidal base), but I'm sure that other people had thought about it before. There are some nice things you can say especially in the case when the base is cartesian; for instance you ... | 4 | https://mathoverflow.net/users/49 | 28163 | 18,401 |
https://mathoverflow.net/questions/28168 | 0 | Let δ is a proximity.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Question: Let A and B are sets with non-empty intersection. Let both A and B are connected. Prove or give a counter-example that A∪B is also connected.
(This question arouse as a special example of a more g... | https://mathoverflow.net/users/4086 | Connectedness of a union regading a proximity | Consider $X\cap A$ and $Y\cap A$, starting from a partition $\lbrace X,Y\rbrace$ of $A\cup B$. If both intersections are nonempty we are done, as $(X\cap A)\delta(Y\cap A)$. Otherwise, $A\subseteq X$, say, but then $X\cap B$ and $Y\cap B$ are nonempty and we find $(X\cap B)\delta(Y\cap B)$.
In either case $X\delta Y$. ... | 3 | https://mathoverflow.net/users/5903 | 28173 | 18,410 |
https://mathoverflow.net/questions/28186 | 4 | Can we find a sequence $u\_n$ of positive real numbers such that
$\sum\_{n=1}^\infty u\_n^2$ is finite, yet $\sum\_{n=1}^\infty ({u\_1+u\_2+...+u\_n\over n})^2$ is infinite ?
After several attempts, I think this is not possible, but I can't prove that the finiteness of the first sum implies the finiteness of the sec... | https://mathoverflow.net/users/6129 | Convergence of the sum of squares of averages of a sequence whose sum of squares is convergent | Hardy's inequality says
$$\sum\_{n=1}^{\infty}\left(\frac{a\_1+\cdots+a\_n}{n}\right)^p\le \left(\frac{p}{p-1}\right)^{p}\sum\_{n=1}^{\infty}a\_n^p$$
for any $p>1$.
| 10 | https://mathoverflow.net/users/2384 | 28187 | 18,422 |
https://mathoverflow.net/questions/28162 | 2 | When some papers say"XXX fibration", I see it seems that it is just that the surjective map f: X ---> Y, such that the fiber is XXX,but it is really not "fibration",I didn't see it prove that it is a fibration. So could you tell me if I am wrong? Thanks!
| https://mathoverflow.net/users/2391 | Is an algebraic geometer's fibration also an algebraic topologist's fibration? | Now that the intent of the question has become clear, I'll attempt to take it out of limbo by transferring the content of the comments - my own (TP) and Boyarsky's - into a community wiki answer.
In algebraic or complex analytic geometry, a fibration is a map from a variety to a lower-dimensional variety having some ... | 5 | https://mathoverflow.net/users/2356 | 28190 | 18,424 |
https://mathoverflow.net/questions/14212 | 35 | Here's a a famous problem:
If a rectangle $R$ is tiled by rectangles $T\_i$ each of which has at least one integer side length, then the tiled rectangle $R$ has at least one integer side length.
---------------------------------------------------------------------------------------------------------------------------... | https://mathoverflow.net/users/934 | Tiling a rectangle with a hint of magic | It is not at all obvious to me that there is any deep principle at work in the double-integral proof. In my mind, the double-integral proof is really the same as the checkerboard proof. You're just trying to come up with a translation-invariant function on rectangles that is (a) additive and (b) zero if and only if the... | 15 | https://mathoverflow.net/users/3106 | 28191 | 18,425 |
https://mathoverflow.net/questions/26582 | 2 | I am looking for a reference to a proof of the following well-know fact (cited for example by
B.Farb in ``Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840); [MR1650094](https://mathscinet.ams.org/mathscinet-getitem?mr=1650094),
[DOI:10.1007/s000390050075](https://doi.org/10.1007/s0003900500... | https://mathoverflow.net/users/6206 | Reference for the geometry of horospheres | Try [Geometry of horospheres](http://www.google.com/url?sa=t&source=web&cd=4&ved=0CCEQFjAD&url=http%3A%2F%2Fwww.intlpress.com%2FJDG%2Farchive%2F1977%2F12-4-481.pdf&ei=8-EWTOknwYHyBp6y3PMI&usg=AFQjCNFd1SZWgJqlUWCA-8lYqLHdWUtP3w) by
Heintze and Im Hof.
| 6 | https://mathoverflow.net/users/1573 | 28194 | 18,428 |
https://mathoverflow.net/questions/28157 | 4 | Some days ago, I posted a question about [models of arithmetic and incompleteness](https://mathoverflow.net/questions/26676/incompleteness-and-nonstandard-models-of-arithmetic). I then made a mixture of too many scattered ideas. Thinking again about such matters, I realize that what really annoyed me was the assertion ... | https://mathoverflow.net/users/6466 | Breaking the circularity in the definition of N | Looking at the draft that was linked above, it's more clear what Kunen means. He is just saying that the informal "definition" of the natural numbers that you might think of in school is circular when examined closely. And it is, in the sense that you have to start with some undefined concept, be it "natural number", "... | 6 | https://mathoverflow.net/users/5442 | 28240 | 18,455 |
https://mathoverflow.net/questions/28237 | 17 | If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:
$$0\to Br(F)\to\oplus\_v Br(F\_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$
The last non-trivial map here is "sum", with each local $Br(F\_v)$ canonically injecting into $\mathbf{Q}/\mathbf{Z}$ by l... | https://mathoverflow.net/users/1384 | Dimension of central simple algebra over a global field "built using class field theory". | To paraphrase Igor Pak: OK, this I know. It is remarkable how difficult it is to track down a reference which gives an actual proof for this fact (moreover applicable to all global fields). The notes of Pete Clark don't give a proof or a reference for a proof, and its omission in Cassels-Frohlich is an uncorrected erro... | 12 | https://mathoverflow.net/users/6773 | 28245 | 18,458 |
https://mathoverflow.net/questions/28248 | 8 | Given an operation of say a topological group on a topological space, one can form the quotient stack X//G:
the stack associated to the action groupoid.
Does this stack satisfy some kind of universal property?
| https://mathoverflow.net/users/2837 | Universal property of X//G? | Probably the simplest example is when the space $X$ is a single point. Then $pt//G$ classifies principal $G$-bundles. Here, the action groupoid is just $G$ considered as a one-object groupoid. The one object, $pt$ becomes an atlas for the stack, so we have a representable epimorphism $pt \to pt//G$. This is in fact the... | 9 | https://mathoverflow.net/users/4528 | 28251 | 18,462 |
https://mathoverflow.net/questions/28044 | 8 | The osculating circle at a point of a smooth plane curve can be obtained by considering three points on the curve and the circle defined by them. When the three points approach $P$, the circle becomes the osculating circle at $P$.
Generalizing this method to a conic defined by five points on a curve, one obtains an ... | https://mathoverflow.net/users/6415 | Osculating conics and cubics and beyond | These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important references will be:
Topological invariants of plane curves and caustics.
Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5. American Math... | 10 | https://mathoverflow.net/users/943 | 28256 | 18,466 |
https://mathoverflow.net/questions/22263 | 4 | We are interested in the following question (definitions and references are given below):
Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there always a Lipschitz continuous function $g:R^n \rightarrow R^m$ such that $g(x) \in F(x)$ for all $x \in R^n$ ?
In general,... | https://mathoverflow.net/users/5526 | Do upper-semicontinuous polyhedral multifunctions have Lipschitz continuous selections? | Hi,
I happen to be working on semi-algebraic set-valued maps, and I might have a partial answer in [1]. I guess when you say polyhedral, you mean that the graph of the set-valued map is a union of finitely many polyhedrons. If that is the case, polyhedral set-valued maps are semi-algebraic. Semi-algebraic set-valued ... | 4 | https://mathoverflow.net/users/6821 | 28258 | 18,468 |
https://mathoverflow.net/questions/28233 | 5 | is there any algorithm known for computing (middle perversity)intersection homology of complex toric varieties based on their combinatorial data? I'm not looking for a computer program.
Regards,
Peter
| https://mathoverflow.net/users/6398 | Intersection homology for toric varieties | See
Braden, Tom and MacPherson, Robert, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), no. 3, 533--551.
| 5 | https://mathoverflow.net/users/3077 | 28260 | 18,470 |
https://mathoverflow.net/questions/28263 | 1 | This must be a well-known exercise with spectral sequences, but I don't know a reference for it. I'm trying to figure out when does $Tot$ commute with colimits.
More precisely, let $X$ be a double cochain complex of, say, $R$-modules, $R$ a commutative ring with unit, or, more generally, a double complex in an abelia... | https://mathoverflow.net/users/1246 | Tot and colimits | Imagine that all double complexes in the image of your functor X: I → C have both differentials equal to zero. Moreover, all terms of these bicomplexes outside of a fixed diagonal are also zero. Then you are asking, quite simply, whether colimits commute with countable products. If they don't, your morphism θ cannot be... | 8 | https://mathoverflow.net/users/2106 | 28273 | 18,476 |
https://mathoverflow.net/questions/28271 | 8 | Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assuming that the system has a finite set of fixed points, I am interested in knowing (or obtaining references about) what is the behaviour of its fixed point structure under perturbations of the ODEs. More specifically, i would like to k... | https://mathoverflow.net/users/6091 | Persistence of fixed points under perturbation in dynamical systems | A good reference for this sort of thing is Guckenheimer and Holmes, *Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*, Springer-Verlag, 1983, if you're not already familiar with it (and even if you are, for that matter). Chapter 3 in particular is relevant to your question.
| 3 | https://mathoverflow.net/users/5701 | 28285 | 18,483 |
https://mathoverflow.net/questions/28280 | 9 | In his answer to my question
[The Green-Tao theorem and positive binary quadratic forms](https://mathoverflow.net/questions/28047/the-green-tao-theorem-and-positive-binary-quadratic-forms)
Kevin Ventullo answers my initial question in the affirmative. What remains is the title question here, of separate interest t... | https://mathoverflow.net/users/3324 | Does a positive binary quadratic form represent a set of primes possessing a natural density | Accoring to [H. Lenstra](http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf), Chebotarev's theorem holds both for Dirichlet and
for natural density (but he doesn't give a reference in this document).
Applying Chebotarev to the extension $H/\mathbb{Q}$ where $H$ is the Hilbert
class field of $\mathbb{Q}(\... | 8 | https://mathoverflow.net/users/4213 | 28287 | 18,485 |
https://mathoverflow.net/questions/28070 | 4 | Hello all. This is probably a simple problem for you guys, but my geometry is a bit rusty and I am hoping that you can help.
I am trying to arrange an arbitrary number of objects around the circumference of an ellipse. My first stab at the problem resulted in the use of a simple rotational matrix like this (note that... | https://mathoverflow.net/users/2416 | Finding n points that are equidistant around the circumference of an ellipse | This answer assumes you are interested in finding $n$ points on an ellipse such that the arc lengths between successive points are equal.
As others mentioned, this problem involves an elliptic integral, which has no elementary expression. However, many scientific computation libraries are able to compute this functio... | 6 | https://mathoverflow.net/users/4832 | 28290 | 18,488 |
https://mathoverflow.net/questions/28275 | 3 | I'm looking for a small research problem an undergraduate would be capable of after taking just an abstract algebra course, introductory algebraic geometry (at level of Miles Reid's book and Ideals, Varieties & Algorithms), and a course in number theory. Is there a website that would have a decent listing, or possibly ... | https://mathoverflow.net/users/6824 | Looking for an undergraduate research problem in algebraic geometry or algebraic number theory | Your question is missing a crucial word in the first sentence (capable of ...). Is the missing word "understanding" or "solving"?
Anyway, here is a problem: Find the maximum number of points of a curve of genus $g$ over $\mathbb{F}\_q$, for some values of $g,q$ for which this number is not known (check for values at ... | 2 | https://mathoverflow.net/users/2290 | 28291 | 18,489 |
https://mathoverflow.net/questions/26312 | 5 | Recently Francisco Santos has announced that he has a counterexample to the Hirsch conjecture. The last I heard it was circulating among several people and there would be a public version of it available soon. I am curious how close it is to release. Also has there been any progress in the attempt to find the vertices ... | https://mathoverflow.net/users/1098 | A Counterexample to the HIrsch Conjecture | The public version is now out. It is available [here](http://arxiv.org/abs/1006.2814)
| 3 | https://mathoverflow.net/users/1098 | 28294 | 18,492 |
https://mathoverflow.net/questions/28282 | 5 | Let $G$ be a simple algebraic group over a field $k$, and let $U$ be the unipotent radical of a Borel subgroup $B$. Because $B$ normalises $U$, the group $H = B/U$ acts on the coordinate ring $\mathcal{O} = k[X]$ of the basic affine space $X = G/U$ via $(h.f)(x) = f(xh)$. We get a decomposition of $\mathcal{O}$ into a ... | https://mathoverflow.net/users/6827 | Tensor products of Weyl modules in positive characteristic | The question has an affirmative answer and a fairly long history as well, but the
proof uses some nontrivial ideas. The notation used here is nonstandard relative to that found in Jantzen's book *Representations of Algebraic Groups* (second edition, AMS, 2003). Also, a "Weyl module" (in the usual sense) of a given high... | 7 | https://mathoverflow.net/users/4231 | 28301 | 18,495 |
https://mathoverflow.net/questions/28265 | 37 | In the common Hodge theory books, the authors usually cite other sources for the theory of elliptic operators, like in the Book about Hodge Theory of Claire Voisin, where you find on page 128 the Theorem 5.22:
>
> Let P : E → F be an elliptic differential operator on a compact manifold. Assume that E and F are of t... | https://mathoverflow.net/users/956 | Proving Hodge decomposition without using the theory of elliptic operators? | The hard part of the proof of the Hodge decomposition (which is where the serious functional analysis is used) is the construction of the Green's operator. In Section 1.4 of Lange and Birkenhake's "Complex Abelian Varieties", they prove the Hodge decomposition for complex tori using an easy Fourier series argument to c... | 23 | https://mathoverflow.net/users/317 | 28307 | 18,497 |
https://mathoverflow.net/questions/28250 | 9 | I have a specific problem, but would also like to know how to tackle the general case. I will first state the genral question. Let $M$ be an embedded submanifold of $\mathbb{R}^n$ and let $F: \mathbb{R}^n \to \mathbb{R}^n $ be a smooth map. How do I go about checking whether $F(M)$ is a smooth embedded submanifold of $... | https://mathoverflow.net/users/36038 | Checking whether the image of a smooth map is a manifold | The specific $F(M)$ is not a smooth submanifold. Here is an argument.
To simplify formulas, I renormalize the sphere: let it be the set of $(z\_1,z\_2)\in\mathbb C^2$ such that $|z\_1|^2+|z\_2^2|=2$ rather than 1. Then, as Gregory Arone pointed out, $F(M)$ is the set of $(b,c)\in\mathbb C^2$ such that the roots $z\_1... | 9 | https://mathoverflow.net/users/4354 | 28308 | 18,498 |
https://mathoverflow.net/questions/28305 | 4 | I just read for the first time the definition of an internally approachable set, which says:
A set $N$ is internally approachable (I.A.) of length $\mu$ iff there is a sequence $(N\_{\alpha} : \alpha < \mu)$ for which the following holds: $N=\bigcup\_{\alpha< \mu} N\_{\alpha}$ and for all $\beta < \mu$ $( N\_{\alpha}... | https://mathoverflow.net/users/4753 | Some consequences of internally approachable structures | Yes, both (a) and (b) follow from your definition without assuming that $N$ is transitive. You say that for every $\beta\lt\mu$ the sequence $\langle N\_\alpha | \alpha\lt \beta\rangle$ is in $N$. This implies that $\beta$ is in $N$, since $\beta$ is the length of this sequence and $N$ computes this length correctly by... | 4 | https://mathoverflow.net/users/1946 | 28310 | 18,499 |
https://mathoverflow.net/questions/28112 | 6 | Let $R$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k$, and let $X$ be a proper, smooth, geometrically connected curve over $K$. Take a finite extension $L/K$ and a regular proper model $\widetilde{X}$ of $X$ over the ring of integers of $L$ whose special fi... | https://mathoverflow.net/users/5944 | Diameter of reduction graph of a curve over a complete discrete valuation ring | This is a sequel to the above comments.
Consider an elliptic curve $E$ over $K$ with additive reduction over $K$ and multiplicative reduction over some extension $L/K$. Then we can find a quadratic extension $L/K$, and the Kodaira symbole of $E$ over $K$ is $I^\*\_m$ for some $m$ (see below), and the group of compon... | 6 | https://mathoverflow.net/users/3485 | 28315 | 18,502 |
https://mathoverflow.net/questions/26313 | 1 | In page 21 of *A Problem seminar*, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the **failure probability formula**. His formula goes like this
$f\_{0}+f\_{1}+f\_{2}+\ldots$
where $f\_{n}$ is the probability that the ... | https://mathoverflow.net/users/1593 | Failure probability formula | Well, this is certainly a known idea, but I suspect it's not important enough to have its own name. For example, I believe it is used without special mention in Hammersley's "A Few Seedlings of Research" (Proc. 6th Berkeley Symp. on Math. Stat. and Prob., 1971), which has as its target audience a graduate student just ... | 3 | https://mathoverflow.net/users/4658 | 28316 | 18,503 |
https://mathoverflow.net/questions/28313 | 5 | Let $B$ be the closed unit ball in $\mathbb R^n$ and $f\colon B\to B$ a continuous map.
Consider the infinite product $B^{\mathbb Z}$ equipped with the product topology. Consider the solenoid
$$
S\_f=\{\{x\_n; n\in\mathbb Z\}: x\_{n+1}=f(x\_n)\}
$$
equipped with the induced topology.
Question: Is $S\_f$ contractibl... | https://mathoverflow.net/users/2029 | Solenoid of a continuous map of a ball, is it contractible? | No, it is not even path-connected in general, already for $n=1$.
Consider the folding map $f:[0,1]\to[0,1]$, namely $f(t)=2t$ for $t\le 1/2$ and $f(t)=2(1-t)$ for $t\ge 1/2$. There is no path connecting the orbits of the two fixed points: 0 and 2/3.
Indeed, suppose there is a continuous path $t\mapsto \{x\_n(t):n\i... | 9 | https://mathoverflow.net/users/4354 | 28318 | 18,505 |
https://mathoverflow.net/questions/28235 | 2 | Except the original Grönwall's theorem that $$\limsup\_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $$\limsup\_{\begin{smallmatrix} n\to\infty\cr n\ \text{is square free}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{6e^{\gamma}}{\pi^2}$$ and $$\limsup\_{\begin{smallma... | https://mathoverflow.net/users/2525 | Variants of Grönwall's theorem | One example possessing a limit is the colossally abundant numbers of Alaoglu and Erdos,
<http://en.wikipedia.org/wiki/Colossally_abundant_number>
where the limit of the Choie, Lichiardopol, Moree and Sole's
$$f\_1(a\_n) = \frac{\sigma(a\_n)}{a\_n \log \log a\_n}$$
is the same
$$ e^\gamma .$$
That is, the limit... | 2 | https://mathoverflow.net/users/3324 | 28328 | 18,512 |
https://mathoverflow.net/questions/28347 | 2 | Roth's theorem states that for every real algebraic $\alpha$ and $\epsilon>0$, there is a $c>0$ such that
$$|\alpha -\frac{p}{q}| > \frac{c}{q^{2+\epsilon}}.$$
Lang's conjecture strengthened this to
$$|\alpha -\frac{p}{q}| > \frac{c}{q^2 (\log q)^{1+\epsilon}}.$$
A naive further strengthening would be to ask for ... | https://mathoverflow.net/users/1894 | Is it possile for all real algebraic numbers to have continued fractions with bounded partial quotients ? | There is no example of an algebraic number of degree $> 2$ for which the boundedness or not of the entries of the continued fraction has been determined. In a 1976 Annals of Math paper, Baum & Sweet treat the analogous problem with the ring of rational integers replaced by $\mathbb{F}\_ 2[x]$ (and its infinite place). ... | 10 | https://mathoverflow.net/users/6773 | 28348 | 18,524 |
https://mathoverflow.net/questions/28345 | 6 | Iwaniec and Friedlander wrote a short survey article for the notices of the AMS, entitled "What is the Parity Phenomenon?"
<http://www.ams.org/notices/200907/rtx090700817p.pdf>
At the end of the article they refer to a young mathematician:
"Sometimes it almost seems as though there is a
ghost in the House of Prim... | https://mathoverflow.net/users/2547 | Who is the Youngster in the Automorphic Room? | They wrote a mixed technical summary and allegory, something of a prose poem. The allegorical part is concentrated in three paragraphs. These are the second paragraph, the last paragraph, and one in the middle in which people in the "Analytic Room" regard their methods as recent in that Euler is only about three hundre... | 13 | https://mathoverflow.net/users/3324 | 28352 | 18,528 |
https://mathoverflow.net/questions/28295 | 8 | I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" <http://www.math.msu.edu/~fedja/Published/paper.ps> by Nazarov and he calls it by the name of Salem Inequality (which according to him is well known, but... | https://mathoverflow.net/users/6766 | Salem Inequality | Following a cue from Wadim, this inequality is Theorem 9.1 in Chapter 5 of Zygmund's *Trigonometric series*, vol 1. Note that although the book is mostly dealing with trigonometric series, the proof is given for general lacunary $\lambda\_k.$ (Salem was a good friend of Zygmund's; see the preface to the book.)
| 8 | https://mathoverflow.net/users/5740 | 28353 | 18,529 |
https://mathoverflow.net/questions/28354 | 5 | There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated very much). Is there any extension of the Howe duality for exceptional algebras?
| https://mathoverflow.net/users/2052 | Howe duality for exceptional algebras | There is a theory of dual reductive pairs and examples for exceptional Lie algebras.
For example, in $E\_8$ we have dual reductive pairs $(A\_1,E\_7)$, $(A\_2,E\_6)$, $(G\_2,F\_4)$, $(D\_4,D\_4)$. These are used implicitly in constructions of $E\_8$; for example $(G\_2,F\_4)$
corresponds to the Freudenthal-Tits constru... | 3 | https://mathoverflow.net/users/3992 | 28359 | 18,532 |
https://mathoverflow.net/questions/28299 | 9 | Consider simple, undirected [Erdős–Rényi](http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model) graphs $G(n,p)$, where $n$ is the number of vertices and $p$ is the probability for each pair of vertices to form an edge. Many properties of these graphs are known - in particular, $G(n,p)$ is almost surely conn... | https://mathoverflow.net/users/3920 | Vertex connectivity of random graphs? | The expected connectivity cannot be higher than the expected minimal degree, which jumps to roughly $pn$ after getting into the range $p\gg\frac{\log n}{n}$. On the other hand, sloppily counting potential clusters of size $m < n/2$ that have boundaries of less than $k$ vertices gives a probability of $\binom{n}{m}\bino... | 5 | https://mathoverflow.net/users/2368 | 28366 | 18,536 |
https://mathoverflow.net/questions/28362 | 1 | The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
This is motivated by some representation theory.
The naive idea is to start with the sequence of symmetric functions $s\_... | https://mathoverflow.net/users/3992 | Hankel determinants of symmetric functions | It is unlikely to obtain for such a determinant the sum of all Schur functions indexed by partitions of $2n$ with four parts all even or all odd. Indeed, this sum is already equal to the inner product $s\_{n,n}\ast s\_{n,n}$ (see [arXiv:0809.3469](https://arxiv.org/abs/0809.3469)).
About your second question: you can... | 1 | https://mathoverflow.net/users/6768 | 28368 | 18,538 |
https://mathoverflow.net/questions/28361 | 9 | My motivation is to understand the following situation: Given absolutely and almost simple algebraic group $G$ defined over a number field $k$ and a finite valuation $v$ on $k$, when $G(k\_v)$ can be compact (with respect to the $p$-adic topology)?
I more or less understand that if $G=SL\_1(D)$ where $D$ is a divisio... | https://mathoverflow.net/users/6836 | Compact simple simply connected algebraic groups over $Q_p$ or other local non-archimedean fields | Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits" from Bull. Soc. Math. France 110 (1982), pp. 197--202, for an incredibly elegant and... | 13 | https://mathoverflow.net/users/6773 | 28377 | 18,543 |
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