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https://mathoverflow.net/questions/153812 | 7 | I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions $\mathcal{D}'(\mathrm{R}^N)$ and $\mathcal{S}'(\mathrm{R}^N)$.
**Theorem 1:**
Let $X\_n$, $n\in\mathbb{N}$ and $X$ be ra... | https://mathoverflow.net/users/39261 | Generalization of Lévy's continuity theorem for nuclear spaces | There is a partial result due to Boulicaut (1973), which states
>
> **Theorem:** Let $E$ be a separable metrizable Hausdorff locally convex topological vector space. Then $E$ is nuclear if and only if for every sequence $\{\mu\_n\}$ of tight probability measures, weak convergence to a tight probability measure $\m... | 4 | https://mathoverflow.net/users/17118 | 153823 | 81,818 |
https://mathoverflow.net/questions/98406 | 6 | In a joint project, we are currently working on an online combinatorial statistic finder in which (beside other things) want to gather information about combinatorial collections and statistic, see the website <http://www.findstat.org>.
In this context, I am reading through Dominique Foata and Doron Zeilberger's pape... | https://mathoverflow.net/users/21291 | A bijective proof that the bistatistic $(\operatorname{exc},\operatorname{den})$ on permutations is Euler-Mahonian | Yes.
G.-N. Han, Distribution Euler-mahonienne : une correspondance, C. R. Acad. Sci. Paris, 310, Série I, 1990, pp. 311-314.
G.-N. Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, 310, Série I, 1990, pp. 493-496.
G.-N. Han, Une transformation fondamentale sur les réarrangements de... | 2 | https://mathoverflow.net/users/45195 | 153837 | 81,822 |
https://mathoverflow.net/questions/153827 | 1 | Say that $V$ is an affine variety defined over $k$ and that $W$ is a locally closed subset of $V$. Is there some sufficient condition for $W$ to have a canonical $k$-structure?
| https://mathoverflow.net/users/15482 | making sense of a locally closed set being defined over k | I believe that $W$ has a canonical $k$-variety structure without our imposing any additional assumptions.
The closure $\overline{W}$ has a canonical affine $k$-variety structure since it is a closed subset of $V$. By virtue of being locally closed, $W$ is open in $\overline{W}$. Therefore, $W$ has the structure of a... | 0 | https://mathoverflow.net/users/25358 | 153838 | 81,823 |
https://mathoverflow.net/questions/153830 | 5 | I'm cross-posting this question from MSE. It's the first time I do this so I'm unsure of etiquette regarding how to cross-post, if this irritates anyone please vote this down and I'll delete the post. Also if any reply appears to the [MSE post](https://math.stackexchange.com/questions/630037/a-question-about-extensions... | https://mathoverflow.net/users/42312 | A question about extensions of Markov semigroups | I think this holds in quite some generality by the following simple argument. Let $S$ be a Polish space, let's say. If $T(t)$ is a Markov-Feller semigroup on $C\_b(S)$ with kernel $p\_t(x,dy)$, then note that for $f \in C\_b(S) \cap L^p(\mu)$, $1 \leq p < \infty$,
\begin{equation}
||T(t)f||\_{L^p(\mu)}^p = \int\_S \lef... | 3 | https://mathoverflow.net/users/22157 | 153841 | 81,825 |
https://mathoverflow.net/questions/153844 | 5 | I think this is probably elementary, but some searching (and asking on the chatroom) hasn't turned up a result. Could anyone point me to a reference for (or counterexample to) the following statement?
**Given categories $\mathcal{C}, \mathcal{D}, \mathcal{E}$ and comonadic adjunctions $\mathcal{C} \rightleftarrows \... | https://mathoverflow.net/users/344 | Is a composite of (co)monadic adjunctions (co)monadic? | This is not true in general. Note that by passage to the opposite category, your question is equivalent to asking that a composite of monadic functors is monadic. A counterexample for that is the following:
The category $\mathbf{Cat}$ of small categories is monadic over the category $\mathbf{Grph}$ of (directed) grap... | 13 | https://mathoverflow.net/users/1649 | 153848 | 81,826 |
https://mathoverflow.net/questions/153815 | 5 | [A Z-number](http://en.wikipedia.org/wiki/Mahler%27s_3/2_problem)
is a (non-zero) real number $x$ such that the fractional parts
$$\left\lbrace x \left(\frac 3 2\right)^ n \right\rbrace $$
are less than $\frac12$ for all natural numbers $n$.
It is not known whether $Z$-numbers exist.
First, I am interested in f... | https://mathoverflow.net/users/12481 | Records in $Z$-numbers and a relaxation | Suppose the fractional parts of $x, \frac{3}{2} x, (\frac{3}{2})^2 x, ... (\frac{3}{2})^kx$ are under $\frac{1}{2}$, but the fractional part of $(\frac{3}{2})^{k+1} x$ is between $\frac{1}{2}$ and $1$. Then consider $y = x + 2^k$. For $n \le k, \lbrace (\frac{3}{2})^n y\rbrace = \lbrace (\frac{3}{2})^n x \rbrace$ while... | 4 | https://mathoverflow.net/users/2954 | 153853 | 81,828 |
https://mathoverflow.net/questions/153846 | 12 | For every finite dimensional Lie algebra $g$, there is a unique simply-connected Lie group $G$ whose Lie algebra is $g$. Is this true in the infinite dimensional case?
| https://mathoverflow.net/users/40291 | Are infinite dimensional Lie algebras related to unique Lie groups? | No. Van Est and Korthagen (1964) gave perhaps the first example of what they called a "[non-enlargible Lie algebra](http://www.ams.org/mathscinet-getitem?mr=160851)", having no corresponding Lie group.
Needless to say, this kind of question largely depends on the precise definitions adopted for infinite-dimensional L... | 21 | https://mathoverflow.net/users/19276 | 153864 | 81,833 |
https://mathoverflow.net/questions/153861 | 1 | Inside the moduli space of curves $\overline{\mathcal{M}}\_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}\_{0,4}$, and those of type $\overline{\mathcal{M}}\_{1,1}$. Are there divisor classes that are trivial once restricted to one or the other (... | https://mathoverflow.net/users/4096 | divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves | The boundary divisor $\Delta\_0$ has intersection number $0$ with every $F$-curve of the form $\overline{\mathcal{M}}\_{0,4}$, yet has nonzero intersection number with every $F$-curve of the form $\overline{\mathcal{M}}\_{1,1}$.
**Edit.** Also, since $\lambda$ is the pullback of a divisor class from the Satake compa... | 2 | https://mathoverflow.net/users/13265 | 153897 | 81,843 |
https://mathoverflow.net/questions/153877 | 7 | I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive integer multiplicities $m\_i$ assigned to the rows with $\sum\_{i=1}^m m\_i =n$ so $A$ is "square with multiplicity". I cons... | https://mathoverflow.net/users/19088 | Has this generalization of a determinant (assigning multiplicities to the rows) been studied? | **Summary**
There is a clear connection to the [permanent](https://en.wikipedia.org/wiki/Permanent), which, if I understand correctly, yields the same tropicalization, and a more speculative connection to the determinant. The reason is that OP's formula can be rewritten as an [immanant](https://en.wikipedia.org/wiki/... | 8 | https://mathoverflow.net/users/5740 | 153898 | 81,844 |
https://mathoverflow.net/questions/153602 | 6 | Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$.
Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over $C$, is $A$ also a stack over $C$?
"Conversely", if $A$ is a stack over $C$ and $B$ is a stack over $C$, then must $A... | https://mathoverflow.net/users/45071 | Is "stackiness" transitive? (and a couple other basic questions about stacks) | OK, I think the first and second question are not sufficiently precise. I am going to assume you mean: C is a site and B is endowed with the [topology inherited from C](http://stacks.math.columbia.edu/tag/06NV). In this case the Stacks project contains a lemma stating that the answer to your second question is "yes". S... | 10 | https://mathoverflow.net/users/44817 | 153901 | 81,846 |
https://mathoverflow.net/questions/153859 | 2 | Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$.
$(E-K)\_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two inclusion maps $E-(K \cup K') \rightarrow E-K$ and $E-(K \cup K') \rightarrow E-K'$.
Let $F$ be the functor that associa... | https://mathoverflow.net/users/12806 | Projective limit and connected components | There is a survey article (by me) that looks at the area of the homotopy theory of ends (and Proper Homotopy) that appeared in the Handbook of Algebraic Topology (1995). If the space concerned is reasonably nice (e.g. an infinite simplicial complex or similar) then the space of ends has a neat interpretation as rays go... | 2 | https://mathoverflow.net/users/3502 | 153912 | 81,848 |
https://mathoverflow.net/questions/153867 | 5 | Let $A\_2(N)$ denote the moduli space of principally polarized abelian surfaces with level $N$ stucture. The absolute Igusa invariants $i\_1$ $i\_2$ and $i\_3$ give three different maps from $A\_2(1)$ to $\mathbb P^1$. Now what I want to do is construct a family of rational maps $f\_N : A\_2(N) \to C(N)$ where all the ... | https://mathoverflow.net/users/23501 | Maps from the moduli space of abelian surfaces with level stucture to curves | The answer to the first question is negative. It follows from an old result of Matsushima that $b\_1(\mathcal{A}\_g(n))=0$ for $g\geq 2$ (Annals of Math. 75 (1962), 312-330). If there is a dominant map $\mathcal{A}\_g(n)\rightarrow C$, $H^1(C,\mathbb{Q})$ injects into $H^1(\mathcal{A}\_g(n),\mathbb{Q})$, which implies ... | 6 | https://mathoverflow.net/users/40297 | 153916 | 81,850 |
https://mathoverflow.net/questions/153850 | 0 | Let $M$ is a smooth n-manifold and $g$ is a $Z\_2$-graded Lie algebra, we denote the algebra of smooth $g$-valued function on $M$ by $C^{\infty} (M,g)$. I wanna find all graded derivation of $C^{\infty} (M,g)$ and their relation to algebra of smooth function and smooth vector field on $M$. In the case of space-time man... | https://mathoverflow.net/users/40291 | Derivations of algebra of smooth $g$-valued function? | If $g$ is finite-dimensional, the algeba in question is isomorphic to $g \otimes C^\infty(M,\mathbb R)$. There is a full description of derivations of Lie algebras of the form $L \otimes A$ in terms of $L$ and $A$ (see, for example, arXiv:math/0302334, Corollary 2.2 and references therein). For example, if $L$ is perfe... | 2 | https://mathoverflow.net/users/1223 | 153924 | 81,854 |
https://mathoverflow.net/questions/153876 | 3 | Say that an autonomous system $\dot{u} = f(u)$ in $\mathbb{R}^{m}$ has the property that for any two solutions $x(t), y(t)$ corresponding to initial conditions $x(0)$ and $y(0)$ the trajectories are converging towards ech other:
$d(x(t),y(t)) \leq e^{-\alpha t}d(x(0),y(0))$
for some $\alpha > 0$. Must there be... | https://mathoverflow.net/users/45212 | Convergence of trajectories and asymptotic stability | As observed by P. Grover, the flow of the system $g^t:\mathbb{R}^m\to\mathbb{R}^m$ is a contraction for any $t>0$. In particular, for any $k\in\mathbb{N}\_ +$ there is a fixed point of $g^{1/k}$, which is, of course, a fixed point of the $k$ th compositional iterate, $g^1$. By uniqueness, these fixed points coincides, ... | 4 | https://mathoverflow.net/users/6101 | 153925 | 81,855 |
https://mathoverflow.net/questions/153936 | 3 | We say that a Turing degree $a$ is a strong minimal cover of $b$ if $a$ is strictly above $b$ and if any $c$ strictly below $a$ is (not necessarily strictly) below $b$. It is known that some degrees do have strong minimal covers and some other don't.
It is known that 0 has strong minimal covers. Those strong minimal ... | https://mathoverflow.net/users/14490 | Is there a Turing degree which is a strong minimal cover and does not have itself a strong minimal cover? | Not known, I think. Traditional ways to produce a SMC $ a $ by completely controlling the structure of $[0, a] $ (starting with Spector 1956, see Lerman's 1983 book) led to c.e. traceable degrees, but those all have SMCs as shown by Ishmukhametov, 1999. Nontraditional way would be to follow Kumabe's construction of a m... | 5 | https://mathoverflow.net/users/4600 | 153955 | 81,864 |
https://mathoverflow.net/questions/153940 | 13 | A classical theorem of Erdos shows that the density of numbers n with $\gcd(n,\varphi(n))=1$ is zero. Actually, something stronger is proven:
the number of integers $n\leq N$ with $\gcd(n,\varphi(n))=1$ is $(1+o(1))\frac{e^{-\gamma}N}{\log(\log(\log(N)))}$.
Now I was wondering what can be said on the number (say $t(N... | https://mathoverflow.net/users/45242 | Proportion of square-free integers $n$ with $\gcd(n,\varphi(n))$ a prime | Peter Mueller has given a full answer to your question already. But let me say a word about what's going on "behind the scenes." The key observation, which goes back at least to Erdos but is possibly older, is that for any fixed integer $M$, the set of $n$ with $\phi(n)$ divisible by $M$ has asymptotic density $1$. Thi... | 10 | https://mathoverflow.net/users/16510 | 153969 | 81,870 |
https://mathoverflow.net/questions/153970 | 1 | I'm doing a project involving tilings of Minkowski space. For instance in 2d I have rectangular tiles determined by a spacelike line segment: the rectangle is the region caused by the line segment. Four of these fit together at a vertex and there is a condition on the lengths of the four spacelike lines that determines... | https://mathoverflow.net/users/7227 | manifolds whose charts are maps to Minkowski space | These are the $(G,X)$-structures (see a [survey](http://www.math.sunysb.edu/~mlyubich/Archive/Geometry/Hyperbolic%20Geometry/Goldman.pdf?origin=publication_detail) of Goldman), for $G$ the group of isometries of Minkowski space and $X$ Minkowski space. Or you could just say that they are the flat Lorentzian manifolds.
... | 5 | https://mathoverflow.net/users/13268 | 153974 | 81,873 |
https://mathoverflow.net/questions/141408 | 7 | It's a duplicate of [this](https://math.stackexchange.com/questions/484892/closed-subspaces-of-locally-convex-inductive-limits) question, since I really want to get an explanation.
Let $\left(V\_{n},\phi\_{n,n+1}\right)\_{n\in\mathbb{N}}$ be an inductive sequence of LCTV spaces. A locally convex inductive limit of $\... | https://mathoverflow.net/users/35953 | closed subspaces of locally convex inductive limits | Here is my favorite example which is very natural and simple: Let $\mathscr E (\Omega)$ be the Frechet space of smooth functions on an open set in $\Omega\subseteq\mathbb R^d$ so that it dual $\mathscr E'(\Omega)$ is the space of compactly supported distributions. This is a nice inductive limit of Banach spaces. Hence ... | 10 | https://mathoverflow.net/users/21051 | 153975 | 81,874 |
https://mathoverflow.net/questions/153967 | 8 | The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few lower bounds on the smallest eigenvalue are known in the literature, but I could not find any *upper* bound. Hence my quest... | https://mathoverflow.net/users/26039 | Spectrum of an adjacency matrix | Since the eigenvalues are real, and since their sum is the trace of $A$, which is zero, we see that either all eigenvalues are zero, or there are both positive and negative eigenvalues. So no non-empty graph has a positive semidefinite adjacency matrix.
I do not think there is much in the way of upper bounds on the l... | 9 | https://mathoverflow.net/users/1266 | 153980 | 81,875 |
https://mathoverflow.net/questions/153979 | 1 | Is there a classification of groups having the property that any set of $d$ elements (say including the identity) is contained in a proper subgroup?
It is appealing to call the maximum such integer (when finite) some sort of "dimension" or measure of being "not cyclic". As one example, we have elementary abelian grou... | https://mathoverflow.net/users/45255 | Groups with no small generating set | Check out ["Rank of a group"](http://en.wikipedia.org/wiki/Rank_of_a_group) on wikipedia.
| 7 | https://mathoverflow.net/users/20787 | 153981 | 81,876 |
https://mathoverflow.net/questions/153953 | 9 | Let $\Sigma(p,q,r)$ be the Brieskorn homology 3-sphere with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (so not the 3-sphere or the Poincare sphere). The fundamental group is given by $$ \pi\_1(\Sigma(p,q,r)=\langle a,b,c\, |\, a^p=b^q=c^r=abc \rangle $$, the centrally extended triangle group for $(p,q,r)$.
The isotopy gro... | https://mathoverflow.net/users/7608 | What is the order of the isotopy group of the Brieskorn homology 3-sphere? | In fact, much more is known: $Diff(\Sigma(p,q,r))\simeq Isom(\Sigma(p,q,r))$ when $\frac1p+\frac1q+\frac1r <1$ by a result of [McCullough and Soma](http://projecteuclid.org/euclid.jdg/1361800869). The metric is a homogeneous metric on $\Sigma(p,q,r)$ modeled on the homogeneous space $\widetilde{SL\_2(\mathbb{R})}$. In ... | 10 | https://mathoverflow.net/users/1345 | 153982 | 81,877 |
https://mathoverflow.net/questions/133707 | 4 | This question is motivated by a computational issue. Suppose $R$ is a product of orders in numberfields such that there is no ring homomorphism $R \to \mathbb Z$, then can one write an algorithm that actually proves that there is no ring homomorphism $R \to \mathbb Z$ if one is only able to compute $R/pR$ for primes $p... | https://mathoverflow.net/users/23501 | Does there exist an order in a number field of deg>1 with a map to F_p for all p? | Since the answer is already in the comments I'm just putting it here as community wiki.
As Noam Elkies noted:
>
> Yes, a proper subgroup of a finite group G cannot intersect every conjugacy class. This has appeared here at least once before: [mathoverflow.net/questions/2697](https://mathoverflow.net/questions/269... | 1 | https://mathoverflow.net/users/23501 | 153983 | 81,878 |
https://mathoverflow.net/questions/153977 | 12 | I have the following question. I guess it's quite simple for experts.
Unfortunately, I could not come up with an answer yet.
Let $X$ be a Banach space which contains no copy of $c\_0$.
Does it impply that $X''$ (the bidual of $X$) contains no copy of $c\_0$?
| https://mathoverflow.net/users/24193 | Containment of $c_0$ | No. For complicated and important examples, consider any $\mathcal{L}\_\infty$ space that does not contain a subspace isomorphic to $c\_0$. The first such examples were constructed by Bourgain and Delbaen in the early 1980s. Some had their duals isomorphic to $\ell\_1$. The constructions there were subsequently used by... | 16 | https://mathoverflow.net/users/2554 | 153984 | 81,879 |
https://mathoverflow.net/questions/153996 | 5 | Let $X$ be a complex surface and $X^{[n]}$ be the Hilbert scheme of finite analytic subspaces $Z$ for which $dimH^0(Z,\mathcal{O}\_Z)=n$. I have trouble understanding $X^{[n]}$. That's what i've worked out:
If we take $n$ distinct points $p\_1,p\_2,\cdots,p\_n$ there is nothing to see ($I\_Z=\{f\in\mathcal{O}\_X|f(p\... | https://mathoverflow.net/users/45265 | Hilbert scheme of points on a surface | Let $H\_{\vec n}$ be the subfamily where the $n$ points cluster in groups of size $n\_1,\ldots,n\_k$ (adding up to $n$). Then it's obvious that $\dim H\_{\vec n} = \sum\_i \dim H\_{n\_i}$, so we can study just the case that all $n$ lie in the same spot, and add $2$ for moving the spot. The dimension of that "punctual" ... | 6 | https://mathoverflow.net/users/391 | 153999 | 81,883 |
https://mathoverflow.net/questions/153993 | 2 | Suppose $f: \mathbb{R} \to \mathbb{R} $ is a function. Is it equivalent that:
1) $f$ is measurable
2) the area under $f$ (i.e $\{ (x,y)\ | \ f(x)\leq y\} $) is measurable in the product measure of the Borel measure and Lebesgue measure.
We think the answer is yes, gave a proof, cannot find anything wrong with it.... | https://mathoverflow.net/users/45264 | Equivalent Definition of Measurable Function | * $1 \Rightarrow 2$ by the measurability of $\leq$.
* $2 \Rightarrow 1$ by noting that the inverse image of $\{ f(x) \leq y \}$ under the measurable function $x \mapsto (x, c)$ is $\{ f(x) \leq c \}$. The latter must be measurable, and this sufficient for measurability of $f$.
I don't know why this wouldn't be mentio... | 2 | https://mathoverflow.net/users/3676 | 154002 | 81,885 |
https://mathoverflow.net/questions/153998 | 7 | Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms $n\cdot2^{n+a}+b$.
It seems evident that the same result would hold without the $n$ out front, that is, for $a,b,c$ with $a>0$ and $b... | https://mathoverflow.net/users/6043 | Are primes of density 0 in $a\cdot b^n+c$? | Unfortunately, proving such a result for a sequence like $2^n-3$ (say) seems very difficult. The reason one can handle $n\cdot2^n+1$ by sieve methods is because that sequence is equidistributed modulo each odd number $m$. (This equidistribution statement is maybe due to Rieger; I think Hooley uses a weaker result.)
... | 13 | https://mathoverflow.net/users/16510 | 154003 | 81,886 |
https://mathoverflow.net/questions/153991 | 5 | I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation.
My own calculations are quite slow, even for $n = 8$. I suppose my question is how large could one expect to scale this computation, if one desires an explicit... | https://mathoverflow.net/users/45255 | Decomposing the conjugacy representation of Sym$(n)$ for small $n$ | The multiplicity of the irreducible character of $S\_n$ indexed by the partition $\lambda$ of $n$ in the action of $S\_n$ on itself by conjugation is the coefficient of the Schur function $s\_\lambda$ in the Schur function expansion of $1/(1-p\_1)(1-p\_2)(1-p\_3)\cdots$, where $p\_i$ is a power sum symmetric function. ... | 13 | https://mathoverflow.net/users/2807 | 154015 | 81,893 |
https://mathoverflow.net/questions/153633 | 1 | Let $N$ a large number and $P=P(N)$. We know that the "tail" of singular series of Goldbach problem is $$ \underset{q>P}{\sum}\,\frac{\mu(q)^{2}}{\phi(q)^{2}}\overset{q}{\underset{a=1}{\sum}^{\*}}e\left(-N\frac{a}{q}\right).$$
(the symbol \* indicates the condition $(a,q)=1$ ).
What are the best estimates for it (condi... | https://mathoverflow.net/users/41635 | Tail of singular series of Goldbach problem | Here is an unconditional error:
Restricting to summation over $q$ with $gcd(q,N)=n$
and using the formula for Ramanujan sums
implies that the quantity you stated
equals
$$\sum\_{n|N}\frac{\mu^2(n)}{\phi(n)} \sum\_{m>P/n}\frac{\mu(m)}{\phi^2(m)},$$
where the summation is restricted to $m$ with $$\gcd(m,n)=\gcd(m,N/n)=... | 4 | https://mathoverflow.net/users/9232 | 154026 | 81,900 |
https://mathoverflow.net/questions/154005 | 4 | Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A\_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted to zero or one), such that for all $n$, the matrix product $A\_n A\_{n-1}$ makes sense. Suppose in addition that the f... | https://mathoverflow.net/users/42278 | Weak ergodicity of nonhomogenous products of 0-1 matrices | I think there's a counterexample to the statement in the generality given here (with 6\*6 matrices).
There are 2 matrix types in the counterexample:
$$
C=\begin{pmatrix}
1&1&0&0&0&0\\
1&1&0&0&0&0\\
0&0&1&1&0&0\\
0&0&1&1&0&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\end{pmatrix}
\text{ and }
D=\begin{pmatrix}
1&1&0&0&1&0\\
1&1&0... | 2 | https://mathoverflow.net/users/11054 | 154039 | 81,906 |
https://mathoverflow.net/questions/154047 | 0 | Let $X$ and $Y$ be Noetherian schemes over $\mathbb{C}$ and suppose there is a $1-1$-correspondence between closed points of $X$ and that of $Y$. Does this imply that the dimension of $X$ is the same as the dimension of $Y$?
| https://mathoverflow.net/users/43198 | Topological properties of schemes | No. For instance, $X=\textrm{Spec }\mathbb C[[t]]$ and $Y=\textrm{Spec }\mathbb C$ both have one closed point, but $\dim X=1$ and $\dim Y=0$.
| 6 | https://mathoverflow.net/users/30827 | 154048 | 81,912 |
https://mathoverflow.net/questions/154031 | 1 | Let $\mathbb{Z}\_p$ denotes the $p$-adic integers for a prime $p$. Suppose $M$ is a finitely generated torsion $\mathbb{Z}\_p[[T]]$-module such that $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion points of $M$) are both finite dimensional $\mathbb{F\_p}$-vector spaces. How to prove $$\lambda(M) = \dim\_{\mathbb{F}\_p} ... | https://mathoverflow.net/users/33900 | Iwasawa algebra | For such an $M$, set $\lambda^{\prime}(M) := \dim\_{\mathbf{F}\_p} M/pM - \dim\_{\mathbf{F}\_p} M[p]$; since $\mu(M) = 0$, the dimensions are actually finite---this can be extracted from the argument below. Also, if $M$ is itself finite, then $\lambda^{\prime}(M) = 0$.
Since $\mu(M) = 0$, due to the structure theorem... | 2 | https://mathoverflow.net/users/5498 | 154049 | 81,913 |
https://mathoverflow.net/questions/153942 | 7 | I am attempting to translate Borel's "Cohomologie de $\text{SL}\_{n}$ et valeurs de fonctions zeta aux points entiers" paper into English. Since I know no French, this is a rather crude process heavily involving Google Translate (and also a little common sense). However, I am unable to interpret the phrase "variété à c... | https://mathoverflow.net/users/45241 | What does "variété à coins" translate to in English? | The translation is "manifold with corners".
| 9 | https://mathoverflow.net/users/13552 | 154065 | 81,918 |
https://mathoverflow.net/questions/154034 | 3 | This is probably quite naïve, maybe even stackexchange-worthy.
Consider a quadratic form such as $Q(x,y) = 3x^2+y^2$. We know that, for primes $p \equiv 1 \pmod{3}$, there exist integer solutions to $Q(x,y)=p$.
There are nontrivial algorithms to find such solutions $x,y$. But I am wondering: is there a "concise" f... | https://mathoverflow.net/users/45255 | Representing primes explicitly with binary quadratic forms | The following result of Jacobi seems concise enough to qualify, although I share your pessimism about the application.
It is known that for odd primes $p\equiv 1\pmod{3}$, there is a unique way of writing $4p = L^2 + 27M^2$, where $L\equiv 1\pmod{3}$. Jacobi showed that $L$ is the least absolute residue of $-\binom{... | 7 | https://mathoverflow.net/users/16510 | 154087 | 81,927 |
https://mathoverflow.net/questions/154072 | 12 | While studying universal constructions on principal bundles, I've stuck on a quite a basic question, namely:
Given a Lie group $G$, does there exist a principal $G$-bundle $\pi \colon P \to B$, for *some* base $B$, that admits a connection $\theta$ whose holonomy group is the full group $G$?
I suppose the answer is... | https://mathoverflow.net/users/7519 | Are there principal $G$-bundles whose holonomy group is $G$? | As I mentioned in my comment, when $G$ is connected, you can do this with $B=\mathbb{R}^2$ and $P$ being the trivial bundle $P = G\times\mathbb{R}^2$.
Here is one construction: Let $\frak{g}$ be the Lie algebra of $G$ and let $\gamma:TG\to\frak{g}$ be the canonical left invariant, $\frak{g}$-valued $1$-form on $G$. ... | 16 | https://mathoverflow.net/users/13972 | 154089 | 81,928 |
https://mathoverflow.net/questions/154076 | 17 | In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in algebraic geometry.
My question is that do these concepts actually coincide? I mean, for example, a proper morphism of sch... | https://mathoverflow.net/users/36961 | Analogy between topology and algebraic geometry | I suppose it's true that this is an aspect that deserves to receive more attention.
One place where algebraic geometry is systematically done via the topos theory of the étale toposes of the given spaces is Jacob Lurie's "[Structured Spaces](http://ncatlab.org/nlab/show/Structured+Spaces)" and generally the
"[E-∞ G... | 8 | https://mathoverflow.net/users/381 | 154091 | 81,929 |
https://mathoverflow.net/questions/154068 | 1 | From Grothendieck's lemma, we know that all holomorphic vector bundles over the complex projective line are direct sums of line bundles, and so, are $SU(2)$-equivariant.
I wonder, do there exist non-equivariant vector bundles over complex projective $N$-space, viewed as an $SU(N)$ or $U(N)$ homogeneous space? If so,... | https://mathoverflow.net/users/42100 | Non-equivariant vector bundles over complex projective $N$-space | In some sense, for every $n>1$, most holomorphic bundles on $\mathbb{C}P^n$ are non-equivariant. For instance, let $Z\subset \mathbb{C}P^2$ be a non-empty, zero dimensional closed subscheme that is locally a complete intersection, e.g., a collection of $m>0$ distinct, reduced points. Denote by $\mathcal{I}\_Z$ the (coh... | 6 | https://mathoverflow.net/users/13265 | 154108 | 81,932 |
https://mathoverflow.net/questions/154106 | 5 | Is there an algebra $A$ (for example a group) such that $Th(A)$ is logically equivalent to $id(A)$? In other words, is there an algebra $A$ such that
$$
Mod(Th(A))=Var(A)?
$$
Clearly finite algebras do not have this property. It seems that such an algebra should be relatively free.
This question is related to my prev... | https://mathoverflow.net/users/44949 | The existence of an algebra whose set of identities and first order theory are equivalent | I imagine that definitions of $Mod, Th, Var$ and so on have not changed since I saw them decades ago. The trivial one element algebra in any finite type (and likely any infinite type) is an easy example which satisfies $Mod(Th(\textbf{A})) \approx Var(\textbf{A})$. Since it is expected that $Th(\textbf{A})$ is strong e... | 5 | https://mathoverflow.net/users/3206 | 154110 | 81,933 |
https://mathoverflow.net/questions/154118 | 3 | Currently I am writing a paper with several collaborators; although I am the primary author to this (I have done a large (>85%) majority of the work and have actually written the paper) my last name begins with W. I feel obligated to include their names on the paper, however, I fear that doing so will degrade my owners... | https://mathoverflow.net/users/45326 | Authorship, and order of authors | If they are co-authors on the paper then add them alphabetically as is customary for math journals. Otherwise if they are not co-authors then just mention them in acknowledgements. If you want to record the author contributions, you can add this as a separate section after acknowledgements. You can also list yourself a... | 3 | https://mathoverflow.net/users/39754 | 154119 | 81,937 |
https://mathoverflow.net/questions/154050 | 5 | Let $d>1$ be an integer. If $n\geq 0$ is an integer we have a notion of $d$-dimensional *partitions* of $n$; the number of these, denoted $p\_d(n)$, is the number of ways we can stack $n$ ($d$-dimensional) boxes in a corner of a $d$-dimensional "room". No closed formula is known for $p\_d$, for *any* $d>1$. As far as I... | https://mathoverflow.net/users/30827 | On the generating functions for Euler characteristic of Hilbert schemes of points | Yes.
Write $\mathcal P\_d= 1 + p\_d$, so $\mathcal P\_d^{\chi(X)}= \sum\_{k=0}^{\infty} \left( \begin{array}{c} \chi( X) \\ k \end{array}\right) p\_d^k$.
I will show that $\left( \begin{array}{c} \chi( X) \\ k \end{array}\right) p\_d^k$ is the generating function for the stratum of $Hilb^n X$ consisting of subsche... | 2 | https://mathoverflow.net/users/18060 | 154125 | 81,941 |
https://mathoverflow.net/questions/154033 | 7 | Theorem 5.10.3 from *Introduction to dynamical systems*, by Brin & Stuck:
Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent:
1. $NW(f)=M$,
2. every unstable manifold is dense in $M$,
3. every stable manifold is dense in $M$
4. $f$ is topologically transitive,
5. $f$ is topologica... | https://mathoverflow.net/users/43741 | Is there a similar theorem in the partially hyperbolic case? | I think this runs into trouble at condition 1. For example, let $f : M \to M$ be an Anosov diffeomorphism that satisfies these conditions. The map $g: M \times S^1 \to M \times S^1$ defined by $g = f \times identity$ is partially hyperbolic (center direction is along the $S^1$ fibers) and the non-wandering set is the w... | 4 | https://mathoverflow.net/users/1227 | 154126 | 81,942 |
https://mathoverflow.net/questions/154131 | 0 | Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e.
$$ A(\theta) = \left\{ y \in \mathbb{R}^n \; : \sum\_{k=1}^n y\_k^2 = 1 \; \text{and} \; y\_1 > \cos(\theta) \right\} . $$
I'd like to know the area o... | https://mathoverflow.net/users/32723 | What is the area of the piece of an $n$-sphere within a given angle of a vector? | This is
>
> [the volume of a ball in spherical geometry](http://en.wikipedia.org/wiki/Spherical_cap#Hyperspherical_cap)
>
>
>
| 4 | https://mathoverflow.net/users/11142 | 154132 | 81,945 |
https://mathoverflow.net/questions/154157 | 3 | I am aware that the following question is a very basic one and therefore I would not be at all offended if it were to be closed. Moreover, I am not familiar at all with category theory.
Let $\mathcal{C}$ be a concrete category and $X$ be a free object of $\mathcal{C}$.
>
> If $Y\_1$ and $Y\_2$ are both free subo... | https://mathoverflow.net/users/15551 | Intersection of free objects | It's not in general true for the category of modules for a ring.
For example, let $R=\mathbb{C}[x]/(x^2)$, let $X=R\oplus R$ be the free module on two generators, and let $Y\_1$ and $Y\_2$ be the submodules of $X$ generated by $(1,0)$ and $(1,x)$ respectively. Then $Y\_1$ and $Y\_2$ are both free modules on one gener... | 6 | https://mathoverflow.net/users/22989 | 154164 | 81,953 |
https://mathoverflow.net/questions/154154 | 5 | You have an anistropic semisimple algebraic group $G$ defined over a non-archimedean local field $k$. When can you say that the $k$-rational part of the conjugacy class of a $k$-rational point is compact?
| https://mathoverflow.net/users/15482 | conjugacy classes in anisotropic semisimple groups | The answer is always. If $G$ is an anisotropic group over a non-archimedean local field $k$, then $G(k)$ is compact. The elements of $G(k)$ are semi-simple (if $char k =0$) and hence their conjugacy classes are closed in the Zariski topology and therefore also in the $k$-topology.
[Edit] To round it off, the questio... | 7 | https://mathoverflow.net/users/23291 | 154165 | 81,954 |
https://mathoverflow.net/questions/153668 | 3 | In 1990, Hofer proved that the displacement energy of a standard ball in $C^{n}$ equals it's Gromov area.
Here is the baby case: Consider a smooth bounded function $f:R^{2}\rightarrow R$. Consider the vector field $(-\frac{\partial f}{\partial y},\frac{\partial f}{\partial x})$, which is the rotation of the gradient fi... | https://mathoverflow.net/users/44651 | Fundamental proof of the baby case of Hofer's theorem about displacement energy | (To match the standard definitions, you want $f$ or at least $df$ to have compact support. This doesn't really matter for this simple computation. I just need the Hamiltonian flow to be complete.)
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be a smooth path. Then, define a map
$\bar \gamma \colon [0,1]\times[0,1] \to... | 2 | https://mathoverflow.net/users/477 | 154186 | 81,960 |
https://mathoverflow.net/questions/153819 | 7 | Let $X$ be an alphabet and denote by $X^{\omega}$ the set of all infinite sequences (i.e. words) in $X$. A subset $L \subseteq X^{\omega}$ is called [$\omega$-regular](http://en.wikipedia.org/wiki/Omega-regular_language) if it is acceptable by some Büchi-Automaton, equivalently if it is of the form
$$
L = \bigcup\_i^... | https://mathoverflow.net/users/37580 | Generalising the adherence operator and its closure properties with regard to regular (rational) languages | Then answer is yes. Since $F(L)$ is a regular language, it suffices to prove the following result:
>
> If $K$ is a regular language, then $R(K) = \{u \in A^\omega \mid F(u) \subseteq K \}$ is $\omega$-regular.
>
>
>
Since regular languages are closed under complement, it suffices to show that $R(K^c)$
is $\ome... | 4 | https://mathoverflow.net/users/38236 | 154187 | 81,961 |
https://mathoverflow.net/questions/154184 | 6 | **Definitions:**
A set $X$ is *of PA degree relative to a set $Y$* if every infinite $Y$-computable binary tree has an infinite $X$-computable path.
A set $X$ is *low* if $X'$ is computable from $\emptyset'$.
**Easy facts:**
By the relativized Low basis theorem and using the fact that a low relative to a low is s... | https://mathoverflow.net/users/8833 | Is 0' of PA degree relative to a non-low set? | No, by the Arslanov completeness criterion $0'$ is only DNC (Diagonally non-computable) relative to low sets. And PA implies DNC.
| 6 | https://mathoverflow.net/users/4600 | 154190 | 81,964 |
https://mathoverflow.net/questions/55599 | 10 | By results of Størmer and Woronowicz, every positive map $\Phi \colon \mathcal{M}\_{d \times d} \rightarrow \mathcal{M}\_{d' \times d'}$ for $dd' \leq 6$ can be decomposed as a convex combination
$$\Phi = p \phi + (1-p) ~ T \circ \psi$$
where $\phi$, $\psi$ are *completely* positive maps and $T$ is the transpositio... | https://mathoverflow.net/users/98 | Decomposability of positive maps | I'm quite late answering this question, but I figured it still deserves an answer. The answer to your question is "no": when $d,d^\prime \geq 3$ there does not exist such a finite (or even countable) set of positive maps.
In [arXiv:1209.0437](http://arxiv.org/abs/1209.0437), we considered the following problem: given... | 10 | https://mathoverflow.net/users/11236 | 154191 | 81,965 |
https://mathoverflow.net/questions/154200 | 1 | ${\bf Question:}$
We can write $U(N)\approx \prod\_{k=1}^ N S^{2N-1} $ where "$\approx$" means "locally equal to", $S^m$ is $m-$ sphere and $U(N)$ is the unitary group of dimension $N^2$. I want to check whether we can use this to show that
$$Gr(2,5)\approx S^7\times S^5$$
where $Gr(2,5)$ being a complex Grassma... | https://mathoverflow.net/users/35615 | On the Grassmannian Gr(2,5) and spheres | That is definitely not true. In particular, $H^2(\text{Gr}(k,\mathbb{C}^n),\mathbb{Z})$ is isomorphic to $\mathbb{Z}$ for every pair of integers $n>1$ and $1<k<n$. Yet, by Künneth, $H^2(S^7\times S^5,\mathbb{Z})$ vanishes.
**Edit.**Your second guess is still false, for almost precisely the same reason as the first. I... | 7 | https://mathoverflow.net/users/13265 | 154201 | 81,970 |
https://mathoverflow.net/questions/154194 | 1 | I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and smaller than $1$, while the functions $a\left(x\right)$ and $f\left(x\right)$ are $\mathbb{R}\rightarrow\mathbb{R}$. Mor... | https://mathoverflow.net/users/44965 | A linear algebraic q-difference equation [SOLVED] | Putting for simplicity $x=e^t, q=e^{-b}, F(t)=f(e^t), A(t)=a(e^t)$, we obtain
$$A(t)F(t)=F(t-b).$$
Now you can assign $F$ arbitrarily on any interval of length $b$, for example on $(0,b)$,
and this formula defines you a solution everywhere left of this interval.
If you want a continuous function, you want $A(b)F(b)=F(0... | 3 | https://mathoverflow.net/users/25510 | 154207 | 81,973 |
https://mathoverflow.net/questions/154161 | 3 | Maybe it's a very simple question, but I have a problem with the following series
$$\sum\limits\_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$
where $p, q \in \mathbb{R}$. I know about five ways how to calculate the series
$$\sum\limits\_{n=1}^{\infty}\frac{1}{n^2+r^2}, \,\, r>0.$$
The most "beautiful" way for me is a Poisson su... | https://mathoverflow.net/users/19829 | Series of the inverse quadratic trinomial | This can be expressed in terms of elementary functions, if $p/2$ is an integer. Suppose, for example that it is a positive integer. Then your sum is
$$S:=\sum\_{m=p/2+1}^\infty\frac{1}{m^2+c}=\frac{1}{2}\sum\_{|m|>p/2}\frac{1}{m^2+c},\quad c=q-p^2/4,$$
where summation in the last sum is over positive and negative integ... | 4 | https://mathoverflow.net/users/25510 | 154210 | 81,974 |
https://mathoverflow.net/questions/154209 | 3 | Consider the problem of coloring each point of $S^2$ with one of two colors (say "black" or "white") so that among any three points of $S^2$ which are the vertices of an equilateral spherical triangle with 90 degree sides, one and only one of these three points will be colored "white". This problem comes from Quantum M... | https://mathoverflow.net/users/4423 | Is there a simple topological proof for a topological theorem about $S^2$? | If I understand the problem correctly, one wants a coloring of the points of the unit sphere $S^2$ so that for every triangle with edge lengths $\pi/2$, precisely two vertices are colored black and one is colored white?
If that's correct, then I think there might be a topological proof. If one takes a black point $p... | 8 | https://mathoverflow.net/users/1345 | 154217 | 81,976 |
https://mathoverflow.net/questions/141370 | 3 | Why is the dimension of the cartan subalgebra $2n-\text{rank}(A)$ in the defintion from Kumar's book. From a few examples I can see why the defintion is the way it is, but, I would like a better understanding of why the dimension of the Cartan subalgebra is $2n-\text{rank}(A)$ and not $n$ or something else. I
To be ... | https://mathoverflow.net/users/17260 | Kac Moody algebra defintion | I'll just elaborate on my comment from last year.
A Kac-Moody algebra is defined by generators-and-relations, with starting data given by an $n \times n$ matrix $A$. It can be written as $L \rtimes D$, where $D$ is a $n - rank(A)$-dimensional commutative Lie algebra that acts on $L$ by derivations. The standard examp... | 4 | https://mathoverflow.net/users/121 | 154228 | 81,980 |
https://mathoverflow.net/questions/154208 | 7 | I'm reading [*Distance Regular Graphs*](http://rads.stackoverflow.com/amzn/click/3642743439) by Brouwer, Cohen, and Neumaier. In section 1.8, they explained [Hadamard graphs](http://mathworld.wolfram.com/HadamardGraph.html).
Conversion from a Hadamard Matrix into a Hadamard Graph
-------------------------------------... | https://mathoverflow.net/users/11361 | Equivalence of Hadamard Graph and Hadamard Matrix | First, the book by Brouwer, Cohen and Neumaier is known for a degree of terseness;
any one who uses it will have struggled with it at some point.
You describe a construction that, from an $n\times n$ Hadamard matrix, produces a
bipartite graph on $4n$ vertices that is regular of degree $n$.
This graph has diameter fo... | 7 | https://mathoverflow.net/users/1266 | 154237 | 81,984 |
https://mathoverflow.net/questions/154234 | 3 | I am teaching a course on linear algebra and came to this theorem: every $m \times n$ matrix $A$ with rank $r$ admits a factorization $A = CR$ where $C$ is an $m \times r$ matrix and $R$ is an $r \times n$ matrix.
Then some students raised this question: are there any applications of such factorization?
As I am not... | https://mathoverflow.net/users/15424 | Applications of rank factorization or full rank decomposition | This decomposition can be used to give a short proof of Finsler's lemma, as can be seen in the appendix of (1) (Fortunately, the page that has the appendix is part of the preview offered by [google books](http://books.google.com.br/books?id=mz2r1XJqtPAC&pg=PA256 "this page")).
Finsler's lemma itself find many applica... | 2 | https://mathoverflow.net/users/22389 | 154239 | 81,986 |
https://mathoverflow.net/questions/154218 | 2 | For brevity, let $LG=\mathbb{T}\ltimes \tilde{L}G$, the affine loop group and let $G$ be a simple simply conneceted Lie group. I have a map $\phi:L\mathfrak{g} \to L\mathfrak{g}$ that is equivariant. Is it possible to lift this map to an equivariant map on the level of groups? How is it done?
| https://mathoverflow.net/users/17260 | Is is possible to lift an equivariant map of Loop lie algebras to an equivariant map of Loop groups? | Use theorem 40.3 of
* Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997.[(pdf)](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)
You will have to handle some fundamental group obstructions... | 2 | https://mathoverflow.net/users/26935 | 154246 | 81,990 |
https://mathoverflow.net/questions/154193 | 0 | During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have in my mind:
Let $\mathcal{L}$ be an algebraic language. A negated identity in $\mathcal{L}$ is a formula of the form
$$
... | https://mathoverflow.net/users/44949 | Negated varieties and their relatively free algebras | This refers to the PS in the question (I cannot make comments): the term algebra $T\_L(X)$ seems to satisfy the sentence $\forall x\forall y: x^2+y^2\ne -1$ but $T\_L(X)$ does not satisfy any non-trivial identity. It follows that no non-trivial identity is a consequence of that sentence.
| 1 | https://mathoverflow.net/users/32831 | 154251 | 81,993 |
https://mathoverflow.net/questions/154222 | 3 | Let $E$ be a smooth elliptic curve over an algebraically closed field of characteristic zero. Let $\mathcal{L}$ be a line bundle of degree $3$. Heisenberg group $H\_3$ acts on global sections of $\mathcal{L}$. Let me choose basis $z\_1, z\_2, z\_3$ in $H^0(E, \mathcal{L})$ such that generators of of order three points ... | https://mathoverflow.net/users/21029 | Hesse pencil and Schrodinger representation of Heisenberg group | The answers are **yes** and **no**.
**Yes.** Denote by $g$ the first of mentioned generators of order $3$. Consider the embedding $E\subset{\mathbb P}\_k^2$ given by $\mathcal L$, pick as $0\in E$ one of the inflection points, and denote by $\oplus$ the operation on $E$. Let $p\_1+p\_2+p\_3$ be the divisor of the int... | 3 | https://mathoverflow.net/users/40352 | 154252 | 81,994 |
https://mathoverflow.net/questions/154255 | 9 | Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ whose image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?
| https://mathoverflow.net/users/36688 | A lagrangian version of the Withney theorem | No: being a Lagrangian submanifold of $\mathbb{R}^{2n}$ imposes strong conditions on $M$. If $M$ is embedded in $\mathbb{R}^{2n}$, the bundle $T\_M\oplus N$ (normal bundle) is trivial; if $M$ is Lagrangian, the symplectic form induces an isomorphism $N\cong T\_M^\*$. Thus $T\_M\oplus T\_M$ is trivial; this implies for ... | 12 | https://mathoverflow.net/users/40297 | 154258 | 81,996 |
https://mathoverflow.net/questions/153939 | 16 | My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space but are just told which sets are null - what can we say about the measure space? Note that, for example, membership in $... | https://mathoverflow.net/users/33647 | Which sigma-ideals in a sigma-algebra are ideals of null sets? | First of all, one should mention that not every triple (X,B,μ) (i.e., what is often called a *measure space*)
satisfies the property that its C\*-algebra of bounded functions is a von Neumann algebra (= W\*-algebra) or that the map L^∞→(L\_1)\* is an isomorphism.
One has to impose additional conditions to ensure that t... | 12 | https://mathoverflow.net/users/402 | 154260 | 81,997 |
https://mathoverflow.net/questions/154253 | 0 | Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \mathbb{R}$ is defined:
$$\int\_{\partial\Omega} fdS = ?$$
without the use of a transformation of coordinates in the sense th... | https://mathoverflow.net/users/45379 | Defining surface integral on boundary of $C^1$-domain | Take any piecewise continuous function $f$ defined on the boundary $\partial \Omega$, and extend $f$ to be constant along all lines perpendicular to the boundary, at least until those lines collide with one another. If $\partial \Omega$ is compact, this extends $f$ to some $\varepsilon$-neighborhood, as long as $\parti... | 1 | https://mathoverflow.net/users/13268 | 154279 | 82,005 |
https://mathoverflow.net/questions/154308 | 1 | Let $[n]$ denote the set of integers $\{1,2,\ldots,n\}$. A subset of $2^{[n]}$ is **partition-free** if it does not contain a partition of $[n]$.
>
> What is the maximum size of a partition-free subset of $2^{[n]}$?
>
>
>
Note that it is easy to get such a subset of size $2^{n-1}$: for some choice $x\in[n]$, w... | https://mathoverflow.net/users/556 | Partition-free subsets of $2^{[n]}$ | Given a set $S$ and its complement, you can use at most one, so you can't do better than $2^{n-1}$.
| 2 | https://mathoverflow.net/users/3684 | 154310 | 82,014 |
https://mathoverflow.net/questions/154309 | 6 | From the introduction of Ribet-Stein:
>
> Shimura showed that if we start with the elliptic curve $E$ defined by the equation $y^2 +y = x^3 −x^2$ then for “most” $n$ the image of $\rho$ is all of $\mathrm{GL}\_2(\mathbf{Z}/n\mathbf{Z})$.
>
>
>
Here $\rho$ is the representation of $\mathrm{Gal}(\overline{\mathb... | https://mathoverflow.net/users/45408 | What did Shimura say about $y^2 + y = x^3 - x$? | Goro Shimura, A reciprocity law in non-solvable extensions. J. Reine Angew. Math. 221 1966 209--220.
| 10 | https://mathoverflow.net/users/36469 | 154311 | 82,015 |
https://mathoverflow.net/questions/154305 | 2 | Let $f\in S\_k(\Gamma\_0(N))$, $(p,N)=1$ and $T\_r:S\_k(\Gamma\_0(Np))\longrightarrow S\_k(\Gamma\_0(N))$ the trace map. Does then $T\_r(f\mid V\_p) = f\mid T\_p$ hold?
| https://mathoverflow.net/users/45300 | Relation between Hecke Operator and Trace map | You don't define your notations, as Olivier points out, but I'm going to assume that $V\_p$ is the map $S\_k(\Gamma\_0(N)) \to S\_k(\Gamma\_0(Np))$ given by $f(z) \mapsto f(pz)$.
Then the answer is "yes" (at least up to a constant factor, something like $p^{k-1}$, that depends on your choice of conventions); in fact,... | 2 | https://mathoverflow.net/users/2481 | 154319 | 82,020 |
https://mathoverflow.net/questions/154321 | 5 | Let $\left[ n \right]=\{{1,2,\cdots,n\}}$ and call a family $\mathcal{F} \subset 2^{\left[n\right]}$ **partition-free** if it does not contain any partition of $\left[n\right]$. A [recent question](https://mathoverflow.net/questions/154308/partition-free-subsets-of-2n) asked for the maximal size of such a set. The answ... | https://mathoverflow.net/users/8008 | Can a partition free family in $2^{[n]}$ always be enlarged to one of size $2^{n-1}$? | Yes. Suppose $\mathcal{F}$ is partition-free of smaller size. Then there is some $A$ for which $\mathcal{F}$ contains neither $A$ nor its complement. It must be possible to add either $A$ or its complement to $\mathcal{F}$ to get a larger partition-free family, as otherwise $\mathcal{F}$ would already have to contain p... | 14 | https://mathoverflow.net/users/22989 | 154322 | 82,021 |
https://mathoverflow.net/questions/153389 | 0 | Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
1. $P$ is Lagrangian
2. P involutive
3. dim$P\cap\bar P \cap TM$ is constant
This definition shows that every complex polarization induces a real isotrpic distr... | https://mathoverflow.net/users/nan | An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable | The standard example is the case when $M=T^\*Q$ is a cotangent bundle over a manifold $Q$ with its usual symplectic form, and $P$ is the vertical polarization. Then the isotropic and coisotropic polarizations agree (both equaling the vertical polarization), and their integral manifolds are the fibers of the bundle proj... | 1 | https://mathoverflow.net/users/17945 | 154354 | 82,031 |
https://mathoverflow.net/questions/154352 | 4 | I am interested in knowing how we can use the concepts of Limits and Colimits in modeling problems in every day life? Could anyone provide (Software) engineering examples, perhaps? Or describe intuitively in general for what sorts of modeling problems we can use these concepts? (Providing even few examples will help in... | https://mathoverflow.net/users/45438 | What are the uses of Limits and Colimits of Category Theory in every day problems? | Here are some resources, not necessarily limited to limits and colimits:
1. Since you specifically ask about software engineering, perhaps you can loko at Steve Easterbrooks's slides "[An Introduction to Category Theory for Software Engineers](http://www.cs.toronto.edu/~sme/presentations/cat101.pdf)" (which I found b... | 11 | https://mathoverflow.net/users/1176 | 154355 | 82,032 |
https://mathoverflow.net/questions/154356 | 3 | This is a problem I asked in SE, but it seems the question is more suitable for MO.
Consider a ring $R$ (not necessary with identity or commutative) such that for any proper two-sided ideal $I$, $R\cong\frac RI$ (as rings), e.g., $\Bbb{Z}\_{2^\infty}$ with zero product. Is it true that the set of two-sided ideals of... | https://mathoverflow.net/users/44785 | On rings $R$ for which $R \cong \frac RI$ for any proper two-sided ideal $I$ | If there are ideals $I\_1,I\_2$ not forming a chain, we can assume that $I\_1,I\_2\ne0=I\_1\cap I\_2$ just passing to $R\simeq R/(I\_1\cap I\_2)$. Pick any $0\ne r\in R$. By the Zorn lemma, there is a maximal ideal $I$ in the set of the ideals with the property $r\notin I$. Passing to $R\simeq R/I$, we can assume that ... | 7 | https://mathoverflow.net/users/40352 | 154362 | 82,033 |
https://mathoverflow.net/questions/154360 | 1 | What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to
$Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1.
This is a theorem from Friedman's book on pdes. While i can plough through the proofs which were full of estimates, i don't ... | https://mathoverflow.net/users/32325 | What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''? | Change the variables: define
$$
u(t,s,x)=v(\underbrace{\frac{t+s}{2}}\_{\tau},\underbrace{{t-s}}\_{\sigma},x).
$$
You get
$
\partial\_t u+\partial\_s u=\partial\_{\tau} v
$
and the equation becomes
$$
\partial\_{\tau} v+F(x,v,D\_x
v,D\_x^2 v)=0,$$
where the function $v$ depends on the real parameter $\sigma$.
| 2 | https://mathoverflow.net/users/21907 | 154370 | 82,037 |
https://mathoverflow.net/questions/154215 | 8 | I am reading the paper "chain independence and common information" (<http://ttic.uchicago.edu/~yury/papers/independ.pdf>). In this paper, an inequality is used several times (without proof) which looks interesting to me. I would appreciate if anybody can help me prove it.
The inequality is as follows: For random variab... | https://mathoverflow.net/users/41666 | Inequality in information theory | The answer is as follows. We need to show that
$$I(Z; X)-I(X; Y)\leq H(Z|Y).$$ The left hand side of this is simplified as $H(X|Y)-H(X|Z)$, so we need to show that $H(X|Y)-H(X|Z)\leq H(Z|Y)$. Since conditioning reduces the entropy we have
$$H(X|Y)-H(X|Z)\leq H(X|Y)-H(X|Y,Z)=I(X;Z|Y)\leq H(Z|Y).$$
| 7 | https://mathoverflow.net/users/41666 | 154374 | 82,039 |
https://mathoverflow.net/questions/154331 | 21 | Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential operators play a major role in the K-theoretic *proof* of the index theorem, but it seems to me that they are of no use... | https://mathoverflow.net/users/13356 | Applications of Atiyah-Singer using pseudodifferential operators | Boundary problems for elliptic differential equations are often studied by reducing to equations on the boundary. These equations are, as a rule, pseudo-differential but not differential. If the boundary problem is elliptic then the pseudo-differential operator is elliptic, thus Fredholm, and its index is of interest f... | 16 | https://mathoverflow.net/users/nan | 154375 | 82,040 |
https://mathoverflow.net/questions/154340 | 3 | I would like prove that, under the conditions described below, no non-trivial variety exists.
>
> Let $\mathcal{V}$ be a variety of algebras e.g. rings, semigroups, semilattices.
>
>
>
Further suppose that:
>
> 1. The empty algebra exists i.e. $\mathcal{V}$ has no constants.
> 2. The dual condition also ho... | https://mathoverflow.net/users/5152 | A variety of algebras satisfying some dual conditions | It's false. Take the theory generated by unary operations $\zeta$ and $\iota$ and a binary operation $\cdot$, subject to the equations
$$
\zeta(x) = \zeta(y),
\qquad
\iota(x) = \iota(y),
\qquad
\zeta(x) \cdot y = \zeta(x),
\qquad
\iota(x) \cdot y = y.
$$
An example of an algebra is any ring, with $\zeta(x) = 0$, $\iota... | 6 | https://mathoverflow.net/users/586 | 154376 | 82,041 |
https://mathoverflow.net/questions/154386 | 10 | A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such shapes are known as [Steinmetz Solids](http://en.wikipedia.org/wiki/Steinmetz_solid), and for two and three cylinders we have... | https://mathoverflow.net/users/12176 | The intersection of $n$ cylinders in $3$-dimensional space | This is a heuristic, suggesting $f(n)=O( 1/n^2)$.
Consider the sphere of radius $r$. Each cylinder intersects this sphere in a great circle. The great circles divide the sphere into a number of regions. By induction we can see that this is at most $n(n-1)+2$. So if the sphere is relatively evenly divided into regions... | 4 | https://mathoverflow.net/users/18060 | 154390 | 82,045 |
https://mathoverflow.net/questions/154379 | 18 | In topology, it is common to use the [compact-open topology](http://en.wikipedia.org/wiki/Compact-open_topology) on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear maps from $H$ to itself. In functional analysis, there are [many t... | https://mathoverflow.net/users/5690 | compact-open topology on $B(H)$ | It's easy to see that the compact-open topology agrees with the strong operator topology on norm-bounded subsets of $B(H)$. Bill Johnson mentioned this in a comment. I think this shows that of all the "usual" topologies on $B(H)$ the only candidates for agreeing with the compact-open topology are the strong and the ult... | 6 | https://mathoverflow.net/users/23141 | 154398 | 82,049 |
https://mathoverflow.net/questions/154399 | 8 | Let $q$ be a prime power. It is well known that all Singer subgroups (subgroups of order $q^n-1$) in $GL(n,q)$ are conjugate. My question is: If $H$ is a cyclic subgroup of order $m$ in $GL(n,q)$, $m\mid (q^n-1)$ and $m$ is quite large (for example, $m= (q^n-1)/2$ for $q$ odd), is it necessary that $H$ must be containe... | https://mathoverflow.net/users/41560 | Cyclic subgroups of GL(n,q) | Yes. Put the generator in rational canonical form. Because its order is prime to $q$, it is semisimple, so this puts it in block diagonal form where each block is a companion matrix of an irreducible.
It is contained in a Singer subgroup if and only if it has a single $n \times n$ block.
Suppose it is not in a Sin... | 11 | https://mathoverflow.net/users/18060 | 154400 | 82,050 |
https://mathoverflow.net/questions/154368 | 2 | In one paper about number theory author stated 2 lemmas
**Lemma 1.** If $p$ is a prime $\equiv3(mod $ $4)$ then $x^2+y^2-pz^2$ represents a non-zero rational number $m$ if and only if $m$ is not of the form $kps^2$ with $\left(\frac{k}{p}\right)=1$ or $ks^2$ with $k\equiv p(mod$ $8)$.
**Lemma 2.** If $p$ and $q$ ar... | https://mathoverflow.net/users/39304 | Representation of rationals by quadratic form | Lemma B (for binary) (completing the square and a few cases to check): Given integers, $f(x,y) = a x^2 + b x y + c y^2 ,$ with discriminant $\Delta = b^2 - 4 a c $ not a square. Given a (always positive) prime $r$ with Legendre $(\Delta|r) = -1,$ so that $\Delta \neq 0 \pmod r$ in particular. IF $f(x,y) \equiv 0 \pmod ... | 2 | https://mathoverflow.net/users/3324 | 154441 | 82,060 |
https://mathoverflow.net/questions/154433 | 6 | The classification of [doubly transitive](http://en.wikipedia.org/wiki/2-transitive_group) groups with simple socle is
known. A good account of such classification can be found for example
in [this](http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1112/blms/13.1.1) paper:
Cameron, Peter J. Finite permutation gr... | https://mathoverflow.net/users/17845 | Doubly primitive groups with simple socle | None of these groups are $2$-primitive except for ${\rm Sp}(2d,2)$.
For ${\rm PSL}(d,q)$ with $d>2$, the $2$-point stabilizer fixes two projective points, say $\langle v\_1 \rangle$ and $\langle v\_2 \rangle$, so it also fixes other points, such as $\langle v\_1+v\_2 \rangle$. These fixed points, other than $\langle ... | 8 | https://mathoverflow.net/users/35840 | 154456 | 82,069 |
https://mathoverflow.net/questions/154449 | 0 | By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?
**EDIT** By torsion sheaf I mean the stalk at the generic point is torsion. The curve is not necessarily reduced.
| https://mathoverflow.net/users/32151 | Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve | Just making the above comment an answer.
Take $C:=\text{Proj}\ \big(k[x,y,z]/z^2\big)$. Then $C\_{\text{red}}={\mathbb P}\_k^1$ and the coherent sheaf $F:={\mathcal O}\_{{\mathbb P}\_k^1}(-2)$ over ${\mathbb P}\_k^1$ is known to satisfy
$H^1\big({\mathbb P}\_k^1,{\mathcal O}\_{{\mathbb P}\_k^1}(-2)\big)=k$. Since $F$... | 3 | https://mathoverflow.net/users/40352 | 154466 | 82,075 |
https://mathoverflow.net/questions/154461 | 14 | I am not exactly a group theorist, so this may be well-known.
Let $G$ be a finitely generated group such that the cardinality of minimal generating sets of $G$ is bounded above.
Does it follow that $G$ is finite?
This is true if $G$ is abelian, but I have no idea about the general case.
| https://mathoverflow.net/users/44860 | Minimal generating sets of groups | Just to expand my comment, a [Tarski Monster](http://en.wikipedia.org/wiki/Tarski_monster_group) is an infinite group in which, for some fixed prime $p$, all proper nontrivial subgroups have order $p$. It was proved by Olshanskii in 1979 that they exist for all primes $p>10^{75}$.
A set of $p+1$ distinct elements of ... | 20 | https://mathoverflow.net/users/35840 | 154467 | 82,076 |
https://mathoverflow.net/questions/154478 | 0 | Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by $p$ and $\bar p$ marginals of $P$ on $X$ and $\bar X$ respectively. Does there exist a universally measurable map $f:X\... | https://mathoverflow.net/users/11768 | Existence of a map connecting two marginals of a product measure | No. The measures $p, \bar{p}$ could be almost anything; there's nothing in the given conditions that forces $\bar{p}$ to be a possible pushforward of $p$.
As an extreme example, take $X = \{0\}$ to be a singleton, $\bar{X} = [0,1]$, $A = X \times \bar{X}$, $P= \delta\_0 \times m$ where $m$ is Lebesgue measure on $[0,... | 2 | https://mathoverflow.net/users/4832 | 154479 | 82,080 |
https://mathoverflow.net/questions/154358 | 7 | If you toss a coin $2\ell-1$ times you get a sequence of outcomes, say, $HTHTHTH$ for $\ell = 4$. I am trying to work out the probability that there are at least two runs (in other words contiguous subsequences of outcomes) of length $\ell$ that are identical. In this case $HTHT$ occurs twice, starting at the first and... | https://mathoverflow.net/users/45564 | Probability two matching runs of coin tosses | Inspecting the bound of Bjorn Kjoss-Hanssen's answer with the numerical results in the question, it seems that the upper bound is about twice as large as the true value. I give a refined upper bound that saves this factor of $2$, and which I expect is close to the correct answer. Numerically it seems that $\binom{\ell}... | 6 | https://mathoverflow.net/users/38624 | 154481 | 82,081 |
https://mathoverflow.net/questions/154474 | 4 | Good evening. Is there a reasonable notion of being self-adjoint for the adjoint operator on Banach Spaces? Kind regards, Alex
| https://mathoverflow.net/users/45494 | Self-Adjointness for Banach Spaces | Among the first hits in Google are
>
> [The theory of self-adjoint operators in banach spaces with a Hermitian form](http://link.springer.com/article/10.1007/BF01875301), V. A. Shtraus,
> Siberian Mathematical Journal, 1978, Volume 19, Issue 3, pp 483-489
>
>
>
and
>
> [Self-adjoint operators on real Ban... | 4 | https://mathoverflow.net/users/9652 | 154493 | 82,087 |
https://mathoverflow.net/questions/154468 | 2 | Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A\_k^n$, and that variety inherits the structure of $k$-manifold.
However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim\_{\Bb... | https://mathoverflow.net/users/38448 | What turns $k$-variety into $k$-manifold? | About quaternionic analysis:
The naive approach to manifolds needs the implicit function theorem.
If you try to mimic complex analysis and define a mapping between quaternionic right vector spaces as quaternionically differentiable if the real derivative is right quaternonically linear, then you end up with quaterni... | 4 | https://mathoverflow.net/users/26935 | 154500 | 82,090 |
https://mathoverflow.net/questions/154490 | 16 | In the literature on representation theory of $GL\_2(\Bbb F\_p)$ and $GL\_2(\Bbb Q\_p)$, the irreducible representations with trivial Jacquet module are often called "cuspidal" or "supercuspidal". Why are these representations called cuspidal and who started calling them that? Also, is there any difference between cusp... | https://mathoverflow.net/users/45510 | Etymology of cuspidal representations | Cuspidal modular forms vanish at the cusps of $SL\_2(\mathbb{Z}) \backslash \mathbb{H}$.
Via Strong approximation you can lift the to functions on $SL\_2(\mathbb{Q}) \backslash SL\_2(\mathbb{A})$ because
$$ SL\_2(\mathbb{Z}) \backslash \mathbb{H} \cong SL\_2(\mathbb{Q}) \backslash SL\_2(\mathbb{A}) / \prod\limits\_p ... | 12 | https://mathoverflow.net/users/10400 | 154502 | 82,092 |
https://mathoverflow.net/questions/154498 | 32 | The formula $V-E+F=2$ is so simple that I can't believe that it was really Euler (or perhaps Descartes) who first observed it (I mean the formula itself in some generality, not necessarily a valid proof). To have a concrete question: Is there any reference to this formula in ancient mathematics?
| https://mathoverflow.net/users/21051 | Did ancient mathematicians know Euler's characteristic for convex polyhedra? | there is no doubt the answer to your question is "no"; for a wonderful and scholarly recent book on the whole story, see [Euler's Gem: The Polyhedron Formula and the Birth of Topology](http://books.google.com/books/p/princeton?id=KUYLhOVkaV4C&printsec=frontcover&source=gbs_ViewAPI&hl=en) by David Richeson.
>
> They... | 37 | https://mathoverflow.net/users/11260 | 154505 | 82,093 |
https://mathoverflow.net/questions/145875 | 5 | I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field.
Given a periodic graph (actually a physical lattice or crystal structure), we want to examine some periodic coloring ( or ordered but not periodic coloring) of the g... | https://mathoverflow.net/users/40780 | Crystal structure, lattice, Graph and coloring | There is a theory of color symmetry in mathematical crystallography and in particular, it is applied to lattices. The basic idea is that each coset of a sublattice is assigned a unique color, and what one usually aims for is a symmetrically colored lattice.
Some introductory papers on the topics are: 1.[Color groups ... | 5 | https://mathoverflow.net/users/20072 | 154507 | 82,095 |
https://mathoverflow.net/questions/154510 | 0 | I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important theorem, proved by Eisenbud and Schreyer, in their paper "Betti Numbers of Graded Modules and Cohomology of Vector Bundles" in... | https://mathoverflow.net/users/nan | The Existence of Pure Resolutions, Given a Degree Sequence? | It looks like your problem is having very little knowledge about Koszul complexes. I'm not very good at references, but Eisenbud definitely covers this topic in his CA-AG book.
As for your questions:
1. This is nothing but the Koszul complex associated to the corresponding sections of $\mathcal{O}\_{\mathbb{P}}(1,1... | 1 | https://mathoverflow.net/users/10941 | 154525 | 82,101 |
https://mathoverflow.net/questions/144200 | 7 | I'm looking for a reference and/or the original source for the following fact:
An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does not contain a subgroup isomorphic to a finite index subgroup of $SL(3,{\Bbb Z})$ or $SO(2,3)\_{\Bbb Z}$.
| https://mathoverflow.net/users/39082 | Characterisation of Q-rank 1 | The proof of Kazhdan's property (T) for real simple Lie groups of real rank at least two as given in the old Bourbaki talk of Kirilllov and Delaroche involves showing (property (T) for $H=SL\_3({\mathbb R}), Sp\_2({\mathbb R})$ and then showing) that any such $G$ contains a subgroup locally isomorphic to $H$.
Exactl... | 6 | https://mathoverflow.net/users/23291 | 154526 | 82,102 |
https://mathoverflow.net/questions/154203 | 2 | Assume $x\in X=\mathbb{P}^1\_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}\_p$ and denote by $I\_x$ the ideal sheaf of $x$.
Can we write down a basis for the free $\mathbb{Z}$-module $M(n)=H^0(X,I\_x\otimes\mathcal{O}\_X(n))$?
I think... | https://mathoverflow.net/users/43247 | Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf | For the sake of simplicity I will tackle the case that $x$ is the closed point $V\_+(p,x\_0)$ and compute $H^0(X,\mathscr{I}\_x(n))$ using \v{C}ech cohomology. In our case, the relevant complex looks like
$$ 0 \to (x\_0,p)\_{x\_0}(n) \times (x\_0,p)\_{x\_1}(n) \stackrel{d}{\to} (x\_0,p)\_{x\_0x\_1}(n) \to 0$$
where... | 3 | https://mathoverflow.net/users/21278 | 154527 | 82,103 |
https://mathoverflow.net/questions/154426 | 19 | I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in that case), the alternating group $\mathfrak{A}\_n$ has a nontrivial central extension by $\mathbb{Z}/3$. The construction... | https://mathoverflow.net/users/40297 | Reference for the triple covering of A_6 | The oval construction for $3\cdot A\_6$ can be found on p.110 of
>
> *Symmetric Generation of Groups With Applications to Many of the Sporadic Finite
> Simple Groups* by Robert Curtis.
>
>
>
An e-version is [here](http://f3.tiera.ru/2/M_Mathematics/MA_Algebra/MAtg_Group%20theory/Curtis%20R.T.%20Symmetric%20ge... | 17 | https://mathoverflow.net/users/801 | 154536 | 82,105 |
https://mathoverflow.net/questions/154550 | 3 | Is there a known example of an algebra $(A, +, \cdot)$ with two binary commutative (see P.S below) and idempotent operations $+$ and $\cdot$ satisfying the identity $(a+b)(c+d)=ac+bd$?
---
Actually I need to know is there any algebra $A$ having signature $\mathcal{L}$ such that for any $n$-ary $f\in \mathcal{L}$... | https://mathoverflow.net/users/44949 | Examples of algebras satisfying (a+b)(c+d)=ac+bd | Assuming the properties of idempotency and commutativity, $a+b=(a+b)(a+b)=(a+b)(b+a)=ab+ba=ab+ab=ab$. So you will have essentially identical operations for + and \*. I don't know what happens if you drop one or more of idempotency or commutativity.
| 5 | https://mathoverflow.net/users/35626 | 154557 | 82,112 |
https://mathoverflow.net/questions/9147 | 3 | It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum\_i^r\frac{a\_i-1}{a\_i})$, where $r$ is the number of cones with stabilizer of orders $a\_1,...,a\_r$ respectively.
Now, I don't know... | https://mathoverflow.net/users/2196 | Branched coverings over orbifolds with reflector lines | A detailed write up of what you are looking for is also available in Chapter 13 of [Thurston's notes](http://library.msri.org/nonmsri/gt3m/). Specifically, formula 13.3.4 on page 331, has the desired formula
$$\chi(O) = \chi(X\_O) -\frac{1}{2} \sum\_{i=1} ^N (1-\frac{1}{n\_i}) -\sum\_{j=1}^M(1-\frac{1}{m\_j}),$$
where... | 4 | https://mathoverflow.net/users/27453 | 154559 | 82,114 |
https://mathoverflow.net/questions/154554 | 26 | It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases where we force below some sort of master condition, for example, but this basically amounts to asking my question for the c... | https://mathoverflow.net/users/1058 | When does the choice of the generic matter? | Let me give a couple of examples to show at least that being forcing agnostic is not equivalent to being weakly homogeneous.
First, start in any model of set theory $V$, and then add a Cohen real $c$. Work in $V[c]$. Let $\mathbb{P}=\mathbb{1}\oplus\text{Add}(\omega,1)$, the lottery sum of trivial forcing and the fo... | 14 | https://mathoverflow.net/users/1946 | 154561 | 82,115 |
https://mathoverflow.net/questions/154450 | 1 | Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int\_{\partial\Omega} \frac{1}{|y|^{n-2}} dS(y) < \infty$$
where $dS$ is the surface measure.
My problem is that there is no easy way (for me) to conver... | https://mathoverflow.net/users/45379 | How to show this integral on boundary of Lipschitz domain is finite? | The integral over the complement of any nbd of the origin is finite. You can bound the contribution to the integral in a nbd of the origin, changing variable with a bi-lipschitz local chart: This reduces to the case of a flat boundary, that gives a finite value. Note that the area formula is not even needed, but only t... | 0 | https://mathoverflow.net/users/6101 | 154569 | 82,118 |
https://mathoverflow.net/questions/154580 | 1 | Let $X$ be a closed convex subset of a Banach space, and let $A\subseteq X$ be a Borel set. Denote
$$
B\_r(A) :=\{y\in X:\exists x\in A \text{ such that }\|x-y\|\leq r\}
$$
and by $H(A)$ the convex hull of $A$. Is that true that
$$
B\_r(H(A)\cap X)\cap X = H(B\_r(A)\cap X)\cap X?
$$
Perhaps, there are some additiona... | https://mathoverflow.net/users/11768 | Commutativity of convex hulls and closed balls | This seems kind of trivially true (even without the assumption of $A$ Borel), unless I'm mistaken. First, let's see that $H(B\_r(A)) \subset B\_r(H(A))$. Any $y \in H(B\_r(A))$ can be written as $\sum \lambda\_i y\_i$, with $y\_1, \dots, y\_m \in B\_r(A)$, and $\lambda\_1, \dots, \lambda\_m$ non-negative reals adding t... | 6 | https://mathoverflow.net/users/2698 | 154584 | 82,122 |
https://mathoverflow.net/questions/154587 | 1 | Dirichlet's theorem shows that, for any fixed prime integer a,
"big prime numbers mod a" are uniformly distributed between
1 and a-1. If we similarly pick different prime integers
b,c,..., are these uniform distributions independent of each
other?
| https://mathoverflow.net/users/20757 | Joint Modular Distribution of Primes | Yes (asymptotically of course). The prime number theorem for arithmetic progressions tells us that primes are (asymptotically) uniformly distributed in the $\phi(m)$ reduced residue classes modulo $m$ for any integer $m$, even composite $m$. You can quickly convince yourself that, if $a$ and $b$ are primes, the indepen... | 1 | https://mathoverflow.net/users/5091 | 154588 | 82,123 |
https://mathoverflow.net/questions/154592 | 0 | My question is, that given a vector field only numerically discrete in space, is there a way to estimate its vector potential?
Theoretically, I see [this](http://en.wikipedia.org/wiki/Vector_potential#Theorem) which requires the vector field over all of $\mathbb{R}^3$. Further, I also see [Will Jagy's answer using Po... | https://mathoverflow.net/users/36263 | Estimating the vector potential | Let's put the origin in your corner of ${\mathbb R}^3$, and say you have measurements of the field $\vec{F}$ at lattice points $\epsilon [i,j,k]$ for integers $0 \le i \le m$, $0 \le j \le n$, $0 \le k \le p$. One version of the
vector potential at $(x,y,z)$ is
$$ {\vec V}(x,y,z) = \left[ \int\_0^z F\_2(x,y,t)\ dt, \... | 2 | https://mathoverflow.net/users/13650 | 154593 | 82,125 |
https://mathoverflow.net/questions/154465 | 7 | Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by $\lambda\in X\_{\*}(T)^{+}$.
Let $Gr\_{\lambda}$ such a strata and $\overline{Gr}\_{\lambda}$ the closure in $Gr$.
Then, $\overl... | https://mathoverflow.net/users/27398 | Whitney stratification and affine grassmanian | The point is following: $\overline{Gr^\lambda}$ is a finite dimensional variety acted upon by the pro-algebraic group $G(\mathbb{C}[[t]])$.
This action factors through the action of some finite dimensional algebraic group called $G(\mathcal{J}^l)$. The strata you describe are just the orbits of this group action and ... | 5 | https://mathoverflow.net/users/32972 | 154594 | 82,126 |
https://mathoverflow.net/questions/154601 | 4 | In ergodic theory is common to use the decay of correlation property to deduce
properties analogues to those of i.i.d. random variables.
Call $X\doteq [0,1].$
Examples of decay of correlation properties in dynamics:
1) For $f:X\to X,$ $f(x)=Nx$ mod1, where $N\in \{2,3,4,\ldots,\}$ is fixed, the Lebesgue measure... | https://mathoverflow.net/users/39115 | Decay of Correlation, references for a non-standard way | As you are perhaps aware, the standard method for investigating decay of correlations of expanding maps is using operator theory, either directly or via an induced dynamical system. The decay of correlations can be deduced from the spectral properties of the Ruelle transfer operator acting on the space of Hoelder funct... | 6 | https://mathoverflow.net/users/1840 | 154606 | 82,129 |
https://mathoverflow.net/questions/154595 | 3 | Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., the topology such that the dual space is the space of distributions.
Question: Is $\mathcal D(\mathbb{R}^n)$ separable?... | https://mathoverflow.net/users/45560 | Is the space of test functions separable? | I suppose, you mean the usual topology on $D({\mathbb R}^n)$ defined for example in [Rudin's book](http://rads.stackoverflow.com/amzn/click/0070542368). Take $D\_N=\{\varphi\in D({\mathbb R}^n):\ {\rm supp}\varphi\subseteq\{x\in{\mathbb R}^n:\ |x|\le N\} \}$, where $N\in{\mathbb N}$. Each $D\_N$ can be considered as a ... | 6 | https://mathoverflow.net/users/18943 | 154611 | 82,130 |
https://mathoverflow.net/questions/154610 | 3 | Recall a graph is *chordal* if it contains no induced cycle of length 4 or more, and *outerplanar* if it has a crossing-free embedding in the plane such that all vertices are on the same face. While studying a certain problem, I came across *chordal outerplanar graphs*, which are graphs that are both chordal and outerp... | https://mathoverflow.net/users/39475 | Have chordal outerplanar graphs been studied before? | Yes, every inner face must be a triangle, as chords drawn outside a cycle prevent a drawing from being outerplanar. Drawings of chordal graphs are then just triangulations of polygons; conversely, every triangulated polygon is outerplanar (certainly) and chordal (by induction, as cycles are sub-[triangulated polygons])... | 5 | https://mathoverflow.net/users/25485 | 154613 | 82,131 |
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