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https://mathoverflow.net/questions/70416 | 3 | Let $A$ and $B$ be finite abelian groups with coprime order, and let $G=A\rtimes{}B$ be a semidirect product, via any action. Let $C\subseteq{}A$ be the subgroup of the elements of $A$ which are fixed by the action of $B$, so that $C=Z(G)\cap{}A$. Then we have
$$
A = C \oplus G'.
$$
Is there a quick reference for th... | https://mathoverflow.net/users/3680 | Reference for decomposition in invariants and derived subgroup in a semidirect product of abelian groups | Theorem 2.3 in Chapter 5 of the book "Finite Groups" by Daniel Gorenstein states that if $A$ is a $p'$-group of automorphisms of an abelian $p$-group $P$, then $P = C\_P(A) \times [P,A]$ (all groups here are assumed to be finite). You can deduce your result easily from this.
| 3 | https://mathoverflow.net/users/35840 | 70448 | 43,154 |
https://mathoverflow.net/questions/70446 | 10 | Consider $S\_{n}$ the symmetric group and for each $\sigma\in S\_{n}$ let $U\_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the average
$$
\mathbb{E}(A):=\sum\_{\sigma \in S\_{n}}{w(\sigma) U\_{\sigma} A U\_{\sigma}^{\*}}
$$
where the $w(\sig... | https://mathoverflow.net/users/13825 | Average over Random Permutations | You can calculate this directly. All you need to know is the probability that $(\sigma(i),\sigma(j))=(k,l)$ for each pair $(k,l)$. Write $e\_i$ for the $i$th basis vector, $v=\sum\_i e\_i$. Let $E = \mathbb{E}(A)$. Then:
$$ e\_i^T E e\_i = \frac{\theta-1}{\theta+n-1} e\_i^TAe\_i + \frac{1}{\theta+n-1}\mathrm{Tr} A$$
... | 4 | https://mathoverflow.net/users/327 | 70465 | 43,162 |
https://mathoverflow.net/questions/70424 | 3 | Let $k,p$ be positive integers. Is there a closed form for the sums
$$\sum\_{i=0}^{p} \binom{k}{i} \binom{k+p-i}{p-i}\text{, or}$$
$$\sum\_{i=0}^{p} \binom{k-1}{i} \binom{k+p-i}{p-i}\text{?}$$
(where 'closed form' should be interpreted as usual, i.e. meaning free of sums and hypergeometric functions).
We know ... | https://mathoverflow.net/users/16479 | Yet another sum involving binomial coefficients | Your first sum is the Delannoy number $D(k,p)$. See [OEIS sequence A008288](http://oeis.org/A008288)
| 8 | https://mathoverflow.net/users/13650 | 70474 | 43,168 |
https://mathoverflow.net/questions/70481 | 4 | Is there an example of a Cohen-Macaulay local domain $R$ of characteristic $p>0$ for which the Hilbert-Kunz multiplicity $e\_{HK}(R)$ is *not* equal to its Hilbert-Samuel multiplicity $e(R)$? If no example, a result that states they may not in general be equal would also be helpful, of course.
| https://mathoverflow.net/users/16046 | Hilbert-Kunz multiplicity of Cohen-Macaulay local domains | In general $e\_{HK}(R) \leq e(R)$. Most of the time the inequality is strict.
The case $e(R)=2$ and $\dim R=2$ is studied carefully in the paper by Yoshida-Watanabe "Hilbert-Kunz multiplicity of two-dimensional local rings" , [available here](https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-162/i... | 5 | https://mathoverflow.net/users/2083 | 70482 | 43,169 |
https://mathoverflow.net/questions/70414 | 3 | I have a rather abstract paper on triangulated categories; I would say that it is of average size and quality. I want to find an appropriate journal to publish it; I would like it to be accepted in two months or so. Which journals of high enough reputation would be appropriate here?
Being more precise: this is the pa... | https://mathoverflow.net/users/2191 | Where could I publish an average paper on triangulated categories? | As this may be of use to others as well, I will try to provide some general points on journal selection for an `average' paper.
One obvious general approach to take is (i) to look where the references of your paper were published (and I note that there are lots still unpublished so here that may raise a problem... so... | 11 | https://mathoverflow.net/users/3502 | 70487 | 43,172 |
https://mathoverflow.net/questions/70489 | 3 | Let $G/k$ be a finite group scheme over a field $k$ and $X$ be $k$-scheme of finite type. An action of $G$ on $X$ is a $k$-morphism $\mu : G \times\_k X \rightarrow X$ satisfying the usual conditions. In SGA3-V-4 and 5, it states that the quotient $X/G$ exists if $\mu$ is a finite flat morphism with other conditions. B... | https://mathoverflow.net/users/5482 | what´s a finite group scheme action on a variety? | If $G$ is finite over $k$, then $\mu$ is automatically finite and flat. Indeed the morphism $G\to\mathrm{Spec}\;k$ is finite and flat and hence so is the projection $p:G\times\_k X\to X$. But $\mu$ only differs from $p$ by an automorphism of $G\times\_k X$ and hence it is also finite and flat. Namely $\mu=p\circ \alpha... | 11 | https://mathoverflow.net/users/2308 | 70495 | 43,176 |
https://mathoverflow.net/questions/70479 | 1 | I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:
(1) - I am only allowed to use a construction block, $v\_i$, consisting of a single vertex with $N$ edges of uniform length $L$.
(2) - The edges must exhibit rotational symmetry around any $v\_i$ sim... | https://mathoverflow.net/users/16497 | Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges | I think that your constraints are very restrictive, and that the only possibilities are the 5 Platonic solids: tetrahedron, octahedron, cube, icosahedron, and dodecahedron.
| 2 | https://mathoverflow.net/users/5690 | 70497 | 43,177 |
https://mathoverflow.net/questions/63879 | 22 | There are two explanations in Silverman ( Arithmetic of Elliptic Curves), one in exercises developing the Weil reciprocity law ( for algebraic curves) and then generalizing, and then there is a different, somewhat computational (in my opinion) proof in one of the chapters.
[I should also point out that in case of el... | https://mathoverflow.net/users/14812 | Conceptualizing Weil Pairing for elliptic curves ( and number fields) | Not sure if it will be helpful, but I wrote a survey article whose title could have been "Where do pairings really come from, anyway?" It was for a cryptography conference on pairings. I tried to explain, from a functorial point of view, the origins and relationships of the various pairings on abelian varieties associa... | 13 | https://mathoverflow.net/users/11926 | 70498 | 43,178 |
https://mathoverflow.net/questions/70510 | 4 | Consider the definition of a $\mathcal{C}^{\infty}$-scheme given in Dominique Joyce's "Algebraic geometry over C-infty rings". As far as I uderstand (not being an expert either in C-infty rings nor in logic, nor in categories) the setting is the following. You have the Lawvere theory $\mathrm{Euc}$ given by objects $X^... | https://mathoverflow.net/users/4721 | Can ordinary schemes be described as sheves of algebras/models for a certain Lawvere theory? | Your proposed Lawvere theory seems to me the usual one associated to commutative algebras over your base ring. Your category $\mathcal T$ is the opposite of the category of polynomial rings in finitely many variables. This in turn is the opposite of the category of free commutative algebras over your base ring. This is... | 5 | https://mathoverflow.net/users/15934 | 70512 | 43,185 |
https://mathoverflow.net/questions/70515 | 1 | Are there an results on functions annihilated by the n-times iterated Cauchy-Riemann operator ${\partial\over\partial\bar z}$, aka functions $f$ that for some $n\in\mathbb{N}$ satisfy the following equation?
$${\partial^n f\over\partial\bar z^n}=0$$
EDIT: I have posted the same questions only a very short period of tim... | https://mathoverflow.net/users/16504 | n-times iterated Cauchy-Riemann operator | You can prove by induction that, if $f$ satisfies your equation, then there exist holomorphic functions $f\_0,\ldots,f\_{n-1}$ on the domain of $f$ such that
$$
f = f\_0(z) + f\_1(z)\ \bar z + \cdots + f\_{n-1}(z)\ {\bar z}^{n-1}.
$$
Is this the kind of answer you had in mind?
| 10 | https://mathoverflow.net/users/13972 | 70516 | 43,187 |
https://mathoverflow.net/questions/56762 | 3 | I am interested in the question: Does there are exist concept of support in representation theory?
When I say support I mean number of non-zero values of $f \in C[G]$.
Do you know theorems which talks about the action of elements of $C[G]$ with small support in different representations?
The only example I know abo... | https://mathoverflow.net/users/4246 | Representations and support. | Although perhaps the question was directed more at finite groups: for reductive or semi-simple real Lie groups, (serious) results of Harish-Chandra (starting in the early 1950s) show that among regular semi-simple elements of the group, the supports of characters of principal series are the smallest, being confined to ... | 2 | https://mathoverflow.net/users/15629 | 70519 | 43,188 |
https://mathoverflow.net/questions/70440 | 7 | The answer to this question should be well known, but it's a hard question to search for online.
Suppose we want to approximate the function $x^n$ by a polynomial of degree $d$ in the $L\_\infty$ norm on $[-1,1]$. What is a good estimate of the error of the best approximator, in terms of $n$ and $d$?
I know this qu... | https://mathoverflow.net/users/658 | Uniform approximation of $x^n$ by a degree $d$ polynomial: estimating the error | For large $n$ and fixed $\epsilon > 0$ there is a polynomial of degree $d = O\_\epsilon(\sqrt{n})$ that uniformly approximates $x^n$ to within $\epsilon$ on all of $[-1,+1]$. The polynomial can be taken to be the truncated Čebyšev expansion of $x^n$, as the original proposer (OP) suggested. As $\epsilon \rightarrow 0$,... | 8 | https://mathoverflow.net/users/14830 | 70527 | 43,192 |
https://mathoverflow.net/questions/70520 | 5 | Let $\mathcal{M}$ be an algebraic stack, and let $X$ be its coarse moduli space (assume it exists as a scheme).
We know that $h\_X(Spec(k))=\mathcal{M}(Spec(k))$ if $k$ is algebraically closed. Is there anything intelligent we can say about $h\_X(U)$ for a general scheme $U$?
For example, can you come up with an al... | https://mathoverflow.net/users/5756 | If X is the coarse moduli space of the algebraic stack M, is there a nice description of Hom(_,X)? | This is more an answer to the comment of *unknowngoogle*.
• The object *U* ↦ π0(ℳ(*U*)) that you described is the initial **presheaf** to which the stack ℳ maps.
• One can also consider the sheafification of *U* ↦ π0(ℳ(*U*)),
which is the initial **sheaf** to which ℳ maps.
• Finally, there is the (possibly non... | 8 | https://mathoverflow.net/users/5690 | 70529 | 43,193 |
https://mathoverflow.net/questions/69813 | 2 | I place two spherical particles, $P\_1$ and $P\_2$ (with radii $r\_1$ & $r\_2$), into a cylindrical container of radius $r\_c$ ($r\_1$ & $r\_2$ $\leq \frac{1}{2}r\_c$) and height $h$. While $P\_1$ is immobilized at the centerpoint of the cylindrical container, $P\_2$ has a coefficient of diffusion $D$, and can freely d... | https://mathoverflow.net/users/3248 | Residency time of a spherical Brownian particle in a cylindrical container with another spherical particle at a fixed position | You can use the [Feynman-Kac](http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) formula to get the [Moment Generating Function](http://en.wikipedia.org/wiki/Moment-generating_function) of the time it takes the particle to leave.
I will consider the case where you fix $P\_1$ and let $P\_2$ move. Whatever the g... | 4 | https://mathoverflow.net/users/7949 | 70537 | 43,199 |
https://mathoverflow.net/questions/70542 | 1 | Let $P$ be the set of all odd prime numbers. I am looking for all $s\in(1,\infty)$ for them
$
A=\prod\_{p\in P} (1+\frac{1}{(p-1)^s})^{p-1}
$
exists (i.e. is finite). I know that it should be somehow related to Riemann zeta function but I was not sure how can I pursue the calculations.
If I use natural logarithm ... | https://mathoverflow.net/users/13736 | Some infinite products related to prime numbers. | The observation regarding the logarithm shows that the product exists if $s >2$, since
$\ln(1+x) < x$ for $x >0$, so that the expression for $\ln(A)$ is less than
$\sum\_{p \in P} (p-1)^{1-s}$, which converges. However, for $1 < s \leq 2$, the product diverges,
since, for a given $p$, the contribution to the product fr... | 7 | https://mathoverflow.net/users/14450 | 70544 | 43,202 |
https://mathoverflow.net/questions/70535 | 1 | Has any work been done on generalizing statistical computations to arbitrary structures? I was wondering what would be necessary for them to be meaningful. For example, the mean of a set is the "sum" of the set (probably a commutative monoid, since finite multisets are "free commutative monoids"?) but then how would on... | https://mathoverflow.net/users/202 | Statistical calculations over algebraic structures | There is a whole research area called Algebraic Statistics, although its boundaries are pretty blurry in my opinion. But you could do worse than to start with Seth Sullivant's web page for some idea of what it is all about:
<http://www4.ncsu.edu/~smsulli2/Pubs/publications.html>
Titles like "Algebraic factor analy... | 5 | https://mathoverflow.net/users/8719 | 70546 | 43,203 |
https://mathoverflow.net/questions/38579 | 5 | To find a counterexample disproving a generalization of a theorem in the theory of scale functions on locally compact totally disconnected groups, initiated by George Willis, I am looking for a group with certain properties. Alternatively I am looking for arguments why such a group cannot exist. The precise question is... | https://mathoverflow.net/users/43085 | Examples of certain locally compact totally disconnected groups | This question just got bumped to the front page by the Mathoverflow bot. But the question was already answered in the comments by BCnrd. Following advice from [this](http://mathoverflow.tqft.net/discussion/328/should-we-do-anything-if-a-question-is-answered-well-in-the-comments/) Meta article, I'm going to copy BCnrd's... | 2 | https://mathoverflow.net/users/11540 | 70550 | 43,204 |
https://mathoverflow.net/questions/70547 | 19 | Haar measure is a measure on locally compact abelian groups which is invariants to translations. For example, the Lebesgue measure on the reals is such measure.
It can be shown without the use of the axiom of choice that the Haar measure exists and it is unique up to a scalar, that is if we want the measure of the un... | https://mathoverflow.net/users/7206 | Haar measures in Solovay's model | If $(X,\mathcal{S})$ is standard Borel space and $\mu$ a continuous measure on $(X,\mathcal{S})$ then there is a Borel isomorphism $F:X\to [0,1]$ that sends $\mu$ to Lebesgue measure on $[0,1]$. (See Kechris 17.41.) Since the isomorphism preserves measure this shows that any measurable subset of $[0,1]$ has measurable ... | 9 | https://mathoverflow.net/users/13878 | 70552 | 43,206 |
https://mathoverflow.net/questions/70523 | 24 | This is motivated by the discussion [here](https://mathoverflow.net/questions/13176/is-every-flat-unramified-cover-of-quasi-projective-curves-profinite). The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any category ... | https://mathoverflow.net/users/344 | Grothendieck's Galois theory without finiteness hypotheses | Check out Section 2 of Noohi's paper *Fundamental groups of topological stacks with slice property*, Algebr. Geom. Topol. **8** (2008) pp 1333–1370, doi:[10.2140/agt.2008.8.1333](http://doi.org/10.2140/agt.2008.8.1333), arXiv:[0710.2615](https://arxiv.org/abs/0710.2615).
| 9 | https://mathoverflow.net/users/4790 | 70556 | 43,208 |
https://mathoverflow.net/questions/70555 | 6 | Let $X\rightarrow Y$ be a finite etale map. Let $R$ be a strict henselian ring with residue field $k$. Say that we have a map $Spec(k)\rightarrow X$ and so also $Spec(k)\rightarrow Y$. Assume that the map $Spec(k)\rightarrow Y$ factors thusly: $Spec(k)\rightarrow Spec(R)\rightarrow Y$. Then the question is: would there... | https://mathoverflow.net/users/5756 | Do etale maps satisfy the following? | I think this is true. ~~Namely, we can lift each $\mathrm{Spec}(R/\mathfrak{m}^n)\to Y$ to $X$ (because formal etaleness allows the lift to exist for any nilpotent thickening) successively so as to be compatible with all the previous ones (in fact, it has to be, by uniqueness in the lifting property). This successive l... | 10 | https://mathoverflow.net/users/344 | 70557 | 43,209 |
https://mathoverflow.net/questions/70543 | 5 | I have a graph of Points G(V,E) and I want to find the shortest path covering all the edge, I want the minimum number of edge repetitions .
Which is the best way to reduce this problem to well know problems like TSP , Hamiltonian circuit , Hamltonian completion ?
Thank you.
| https://mathoverflow.net/users/16515 | Edge Covering Shortest path | That is Chinese Postman Path. Search for Chinese Postman Problem...
E.g., this section from some book looks comprehensive: <http://ie454.cankaya.edu.tr/uploads/files/Chp-03%20044-064.pdf>
| 5 | https://mathoverflow.net/users/7076 | 70566 | 43,216 |
https://mathoverflow.net/questions/70578 | 6 | Having read this [link](https://math.stackexchange.com/questions/49763/relationship-between-brownian-motion-and-1-2-delta) on math stackexchange, I would like to submit to your wisdom the following questions.
Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian motion?
More speci... | https://mathoverflow.net/users/19072 | Fractional Brownian motion and Laplacian | Your construction has been carried out for much more general situations. See
Baudoin, F., Coutin, L. Operators associated with a stochastic differential equation driven
by fractional Brownian motions, Stoch. Proc. Appl. 117, 5, 550–574, 2007. ArXiv: math/0509511.
| 5 | https://mathoverflow.net/users/12205 | 70592 | 43,224 |
https://mathoverflow.net/questions/70493 | 3 | This is shamelessly close to my other question: [A Question on Koszul duality and $B(\infty)$ structures on $HH^\*$](https://mathoverflow.net/questions/70151/a-question-on-koszul-duality-and-b-infty-structures-on-hh). Maybe this one will get a better response. Rather than rewrite that one, I am going to ask about a spe... | https://mathoverflow.net/users/6986 | dg-lie structure on $HH^*$ and Koszul duality | Hi,
Your Question:When are the dg-Lie algebra structures on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1] quasi-isomorphic ?
this is always true.
Step 1:
From my paper with Felix and Thomas, looking at the proof, you can see that
dg-Lie algebra structures on Hochschild cochains: $HC... | 5 | https://mathoverflow.net/users/16526 | 70594 | 43,225 |
https://mathoverflow.net/questions/69800 | 11 | Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^\*(C)$
$$
HH^\*(C) \otimes HH^\*(C) \to HH^\*(C)
$$
$$
a \otimes b \mapsto ab
$$
A degree-0 action of Hochschild cohomology on the Hochschild hom... | https://mathoverflow.net/users/284 | Relation between Gerstenhaber bracket and Connes differential | Hi,
Your formula is due (without the signs!) due to Ginzburg Calabi-Yau algebras (9.3.2)
as explained in Lemma 15 of my paper, Batalin-Vilkovisky algebra structures on Hochschild Cohomology, Bull. Soc. Math. France 137 (2009), no 2, 277-295
(sorry for quoting myself!)
Here is Lemma 15
Lemma 15 [17, formula (9.3.2... | 6 | https://mathoverflow.net/users/16526 | 70597 | 43,227 |
https://mathoverflow.net/questions/70503 | 8 | What can I say about it？
Can I say the stalks equal the tensor products of the corresponding factors stalks？
Thanks！
| https://mathoverflow.net/users/3525 | Stalks of structure sheaf of fibre product？ | Let $X,Y$ be $S$-schemes. Then a point of $X \times\_S Y$ corresponds to a pair of points $x \in X, y \in Y$ lying over the same $s \in S$ together with a prime ideal $\mathfrak{p} \subseteq \mathcal{O}\_{X,x} \otimes\_{\mathcal{O}\_{S,s}} \mathcal{O}\_{Y,y}$ which restricts to the maximal ideals in $\mathcal{O}\_{X,x}... | 11 | https://mathoverflow.net/users/2841 | 70598 | 43,228 |
https://mathoverflow.net/questions/70599 | 12 | Let $M$ be a smooth connected oriented without boundary non-compact manifold
of dimension n.
Let $k$ be a principal ideal, e. g. the integers $Z$
Let $H\_n(M)$ and $H^n(M)$ be the homology and cohomology in degree n of $M$
with coefficients in $k$.
It is well-known that $H\_n(M)=0$.
Is $H^n(M)$ also trivial ?
... | https://mathoverflow.net/users/16526 | n-th integral cohomology of a non-compact manifold of dimension n | By a version of Poincaré duality, $\mathrm H^n(M, \mathbb Z)$ is isomorphic to the $0$-th Borel-Moore homology group $\mathrm H\_0^{\mathrm BM}(M, \mathbb Z)$, (see, for example, IX-4 in Iversen's book on cohomology of sheaves). On the other hand $\mathrm H\_0^{\mathrm BM}(M, \mathbb Z) = 0$; this can easily be seen co... | 9 | https://mathoverflow.net/users/4790 | 70601 | 43,230 |
https://mathoverflow.net/questions/70602 | 6 | Is Vladimir Voevodskys ICM lecture available in videotaped format somewhere?
Strangely it is not at the IMU homepage (but Lafforgues is) <http://www.mathunion.org/Videos/ICM2002/>
Was it not taped (Why not?)?
If not is at least a transcript available somewhere?
Is there a least video of the opening ceremony, and the ... | https://mathoverflow.net/users/14802 | Vladimir Voevodskys 2002 ICM Lecture. | Some of Voevodsky videos are here:
<http://video.ias.edu/taxonomy/term/42>
<http://www.mathnet.ru/PresentFiles/425/425.flv>
<http://www.mathunion.org/Videos/ICM98/ICMs/vladimir_voevodsky.html>
<http://claymath.msri.org/voevodsky2002.mov>
The last two talks, *Algebraic Cycles and Motives* and *An Intuitive Int... | 11 | https://mathoverflow.net/users/2149 | 70604 | 43,232 |
https://mathoverflow.net/questions/70605 | 41 | Let $C$ be a nonsingular projective curve of genus $g \geq 0$ over a finite field $\mathbb{F}\_q$ with $q$ elements. From this curve, we define the zeta function
$$Z\_{C/{\mathbb{F}}\_q}(u) = \exp\left(\sum^{\infty}\_{n = 1}{\frac{\# C(\mathbb{F}\_{q^n})}{n} u^n}\right),$$
valid for all $|u| < q^{-1}$. This zeta functi... | https://mathoverflow.net/users/3803 | From Zeta Functions to Curves | To determine which potential zeta functions are actual zeta functions of curves is *very* difficult. The zeta function of a curve is determined by the zeta function of its Jacobian so one could instead ask which potential zeta functions are zeta functions of abelian varieties. This problem is solved (by Tate though as ... | 42 | https://mathoverflow.net/users/4008 | 70606 | 43,233 |
https://mathoverflow.net/questions/70609 | 2 | Is there any necessary and sufficient condition for a complex polynomial to have a real root?
A complex polynomial has a real root if and only if...?
| https://mathoverflow.net/users/15620 | Real roots for polynomials | Yes. If your polynomial is not yet real, replace $P$ by $P\bar P$ ($\bar P$ has complex conjugated coefficients). Therefore we may suppose that $P\in{\mathbb R}[X]$.Using the Euclid algorithm, you may find the g.c.d of $P$ and $P'$. Dividing $P$ by this g.c.d, your are left with the case where $P$ is real and has simpl... | 11 | https://mathoverflow.net/users/8799 | 70614 | 43,235 |
https://mathoverflow.net/questions/70616 | 1 | Consider the set $S\_n$ of all strings of length $n$ ($n$ integer, $n \geq 3$)
representing an expression in RPN
( <http://en.wikipedia.org/wiki/Reverse_Polish_notation>. )
Assumptions (to simplify):
* A string represents always a "valid" expression, where, by "valid", here it is merely meant that when evaluate... | https://mathoverflow.net/users/16343 | Operator probability in a RPN string | I'm going to assume that "valid" expressions are well-formed in the usual sense. This means that not only are there (as you required) no operators lacking operands but also you do not end up with two or more operands on the stack and no operation to combine them. Such an expression necessarily has $n$ operators and $n+... | 2 | https://mathoverflow.net/users/6794 | 70621 | 43,238 |
https://mathoverflow.net/questions/70624 | 2 | I was asking myself if there exists a sort of canonical relation between the standard contact structure on $J^1 N$ and $J^1 M$, for an arbitrary submanifold $N$ of $M$.
My starting point is that, constructing the spaces of $k$-jets and their structures of smooth manifolds, it is remarked that, for any smooth manifold... | https://mathoverflow.net/users/12617 | if $N$ is a submanifold of $M$, then what is the relation between $J^1 N$ and $J^1 M$? | Well, the short answer is that, when $N$ is a submanifold of $M$, the contact manifold $J^1(N)$ is a subquotient of $J^1(M)$. The point is that restriction of germs of functions to submanifolds implies that, if $S\subset J^1(M)$ is the submanifold consisting of those $1$-jets with source a point of $N$, then the canoni... | 5 | https://mathoverflow.net/users/13972 | 70629 | 43,242 |
https://mathoverflow.net/questions/70654 | 3 | The answer to the following question is probably known since long ago, although unknown to me, since I am not a differential geometer.
Let $X$ and $Y$ be 2-dimensional, smooth manifolds and let $Z$ be an open piece of a hypersurface in $X\times Y$ near a point $(x\_0, y\_0)$ with the properties that both projections... | https://mathoverflow.net/users/16546 | Invariants for subspaces of product manifolds | There is a theory due to Tresse, which is unfortunately fairly complicated. It is explained, at least partly, in the book of Arnol'd, Geometric Methods in the Theory of Ordinary Differential Equations. Elie Cartan wrote a difficult paper on it, and this paper was explained more clearly in a paper of Bryant, Griffiths a... | 5 | https://mathoverflow.net/users/13268 | 70656 | 43,254 |
https://mathoverflow.net/questions/70619 | 3 | ([I asked this ten days ago](https://math.stackexchange.com/questions/50051/stricter-permutation-patterns) on math.SE, but I received no reply, so I'm trying again here.)
A lot of work has been done on [patterns in permutations](http://en.wikipedia.org/wiki/Permutation_pattern), where a permutation is said to match a... | https://mathoverflow.net/users/3356 | Stricter permutation patterns | Yes, these things have been studied extensively in two ways:
1) As bivincular patterns, introduced in a 2010 paper by Bousquet-Mèlou, Claesson, Dukes and Kitaev.
2) If you require the occurrence in the permutation to be completely consecutive in values (like 6354) then the inverse of your pattern is called a consec... | 6 | https://mathoverflow.net/users/340 | 70662 | 43,257 |
https://mathoverflow.net/questions/70640 | 8 | In "Au-dessous de $\text{Spec}(\mathbb{Z})$", Toen and Vaquié define schemes relative to a complete, cocomplete symmetric monoidal category $C$ using a functorial approach.
In the introduction the authors mention that there should be a description of the underlying topological space $|\text{Spec}(A)|$ of an affine sc... | https://mathoverflow.net/users/2841 | Spectrum of an algebra object and Reconstruction of Schemes | Florian Marty studied this question in his thesis. The relevant chapter is available as [arXiv:0712.3676](http://arxiv.org/abs/0712.3676) (otherwise, the thesis is available [here](http://thesesups.ups-tlse.fr/540/)). He describes the space $|\mathrm{Spec}(A)|$ as the set of prime ideals endowed with the Zariski topolo... | 9 | https://mathoverflow.net/users/1017 | 70681 | 43,263 |
https://mathoverflow.net/questions/70689 | 5 | Let $ \begin{bmatrix}
A& B \\\\ B^\* &C
\end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$.
Are the following plausible inequalities true? I have seen a lot of similar results, but for the following inequalities, I cannot locate them in the literature or find that they have been pointed out to be... | https://mathoverflow.net/users/3818 | Ask some matrix eigenvalue inequalities. | **Item 1** is true. This is part of Problem 22 (b) in Section 3.5 of Horn and Johnson [HJ94], which states that for [Ky Fan norm](http://en.wikipedia.org/wiki/Singular_value_decomposition#Ky_Fan_norms) ||⋅|| (and in fact for any unitarily invariant norm) and a positive semidefinite block matrix $\begin{pmatrix}A & B \\... | 17 | https://mathoverflow.net/users/7982 | 70702 | 43,275 |
https://mathoverflow.net/questions/70676 | 10 | I am currently reading an article in which the author goes to certain lengths which could be avoided if the following result were true:
>
>
> >
> > *Lemma (proposed)*: Let $T$ be an ergodic measure-preserving transformation of a probability space $(X,\mathcal{F},\mu)$, and let $(f\_n)$ be a sequence of integrable... | https://mathoverflow.net/users/1840 | Non-oscillatory behaviour in the subadditive ergodic theorem | I believe the Lemma you propose is true, via a relatively straightforward adaptation of the proof given for the additive case in Giles Atkinson, *Recurrence of co-cycles and random walks*, J. Lond. Math. Soc. (2) **13** (1976), 486-488. (I assume this is the result you were referencing.) This may already be known and w... | 3 | https://mathoverflow.net/users/5701 | 70704 | 43,277 |
https://mathoverflow.net/questions/70661 | 8 | Let $M$ be a finitely generated graded module over a graded ring $R$. Let $\mathcal{F}$ be the corresponding coherent sheaf on $\operatorname{Proj} R$. There is a natural map of graded $R$-modules
$$\phi \colon M \to \Gamma^\*(\mathcal{F}) := \bigoplus\_{n} \Gamma(\operatorname{Proj} R, \mathcal{F}(n)).$$
If I recall R... | https://mathoverflow.net/users/5094 | A name for "not quite saturated" graded modules | To elaborate on Karl's comment:
Let $m$ be the irrelevant ideal of $R$, then there is a short exact sequence:
$$0 \to H\_m^0(M) \to M \to \Gamma^\*(\mathcal{F}) \to H\_m^1(M) \to 0$$
(see Eisenbud's book, Theorem A4.1, p. 693). Here $H\_m^i(M)$ denote the local cohomology modules. So the map is injective precisel... | 6 | https://mathoverflow.net/users/2083 | 70709 | 43,280 |
https://mathoverflow.net/questions/70647 | 10 | Artin conjectured that if $a$ is an integer which is not a square and not $-1$ then $a$ is a primitive root for infinitely many primes. This conjecture has not been resolved, but partial results are known: Heath-Brown showed that there are at most two prime numbers $a$ for which the conjecture fails.
I'd like to know... | https://mathoverflow.net/users/16160 | Approximate primitive roots mod p | A result of Erdos and Murty asserts that if $\epsilon(p)$ is any decreasing function tending to zero, then $I(p) \leq p^{1/2-\epsilon(p)}$ for almost all primes $p$ (i.e., all but $o(\pi(x))$ primes $p \leq x$).
Kurlberg and Pomerance (see Lemma 20 in the paper mentioned below) show that for a positive proportion of... | 8 | https://mathoverflow.net/users/16510 | 70711 | 43,281 |
https://mathoverflow.net/questions/70713 | 6 | I have read that the Riemann Hypothesis is equivalent to
$\pi(x)=\text{Li}(x)+O(\sqrt{x}\log x)$
Is there an analogous statement saying the Riemann Hypothesis is equivalent to
$\pi(x)=\frac{x}{\log x}+ O(f(x))\quad$ for some $f$
or
$\pi(x)=\frac{x}{\log x}+ g(x) + O(h(x))\quad$ for some elementary function $g... | https://mathoverflow.net/users/16557 | Error term of the Prime Number Theorem and the Riemann Hypothesis | It is not hard to show that
$$\mathrm{Li}(x) = \frac{x}{\log x} \sum\_{k=0}^{m - 1}{\frac{k!}{(\log x)^k}} + O\left(\frac{x}{(\log x)^{m + 1}}\right)$$
for any $m \geq 0$ (just use the definition of $\mathrm{Li}(x)$ and repeated integration by parts). Thus
$$\pi(x) = \frac{x}{\log x} \sum\_{k=0}^{m - 1}{\frac{k!}{(\log... | 20 | https://mathoverflow.net/users/3803 | 70717 | 43,284 |
https://mathoverflow.net/questions/70714 | 55 | In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its expected topology where "expected" makes sense because there are sensible measures on the space of hypersurfaces. See Welschin... | https://mathoverflow.net/users/10839 | Random manifolds | To define a random $n$-manifold you typically need to define a *complexity* on the set $\mathcal M\_n$ of all $n$-manifolds you want to consider, which satisfies a finiteness property: for every $k$, there are only a finite number of manifolds having complexity at most $k$. There are various ways to do this in a combin... | 26 | https://mathoverflow.net/users/6205 | 70718 | 43,285 |
https://mathoverflow.net/questions/70668 | 10 | An affine manifold is a topological manifold which admits a system of charts such that the coordinate changes are (restrictions of) affine transformations. Let $M$ be a compact affine manifold. Let $G$ be the fundamental group of $M$ and $\tilde M$ be its universal cover. One can show that each $n$-dimensional affine m... | https://mathoverflow.net/users/8176 | Affine manifolds | There is a conjecture due to Markus which states that any compact affine manifold has parallel volume (*i.e.* the linear part of $\varphi$ lies in $\mathrm{SL}(n;\mathbb{R})$) if and only if it is complete. To the best of my knowledge, this conjecture is still open, which goes towards saying that there should be no eas... | 9 | https://mathoverflow.net/users/13022 | 70721 | 43,287 |
https://mathoverflow.net/questions/70716 | 7 | Let $X$ be a scheme and $\mathcal{F}$ be quasi-coherent module on $X$. It is clear that if $\mathcal{F}$ is locally free of rank $n$, then $\det(\mathcal{F}) := \wedge^n \mathcal{F}$ is invertible, i.e. locally free of rank $1$. But what about the converse?
**Question.** Assume $\wedge^n \mathcal{F}$ is invertible. D... | https://mathoverflow.net/users/2841 | Characterization of locally free modules via exterior powers | Here's an argument for $n=2$. After further localisation if necessary, we may assume that $X=\text{spec}(k)$ and that we have an isomorphism $\alpha:\Lambda^2(F)\simeq k$. As this is surjective we see that $1$ can be written as a sum of terms $\alpha(u\wedge v)$; after yet more localisation we may assume that some such... | 5 | https://mathoverflow.net/users/10366 | 70723 | 43,288 |
https://mathoverflow.net/questions/70690 | 7 | The non-endpoint Strichartz estimates for the (linear) Schrödinger equation:
$$
\|e^{i t \Delta/2} u\_0 \|\_{L^q\_t L^r\_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u\_0\|\_{L^2\_x(\mathbb{R}^d)}
$$
$$
2 \leq q,r \leq \infty,\;\frac{2}{q}+\frac{d}{r} = \frac{d}{2},\; (q,r,d) \neq (2,\infty,2),\; q\neq 2
$$
are easily... | https://mathoverflow.net/users/13127 | Endpoint Strichartz Estimates for the Schrödinger Equation | From my less-than-expert (where's Terry when you need him?) point of view, a possible reason seems to be the following (I wouldn't call it something going wrong or even a difficulty):
The *statement* of restriction estimates only give you estimates where the left hand side is an *isotropic Lebesgue space*, in the sen... | 4 | https://mathoverflow.net/users/3948 | 70726 | 43,289 |
https://mathoverflow.net/questions/70652 | 5 | It is with some sort of reverential fear that I've come here to write. I've been reading you for a long time, but writing is another story... In any case, I suppose it is too late now to back out!
Then, I am looking for (as many as possible) references to known "different" proofs of the classical spectral theorem for... | https://mathoverflow.net/users/16537 | References for "different" proofs of the spectral theorem for compact operators | Let $T$ be a compact operator on the Banach space $X$ and $\lambda$ a non zero point in the spectrum $\sigma(T)$ of $T$. Then $\lambda$ is in the boundary of $\sigma(T)$ since $T$ is compact and hence is an approximate eigenvalue of $T$. Take a net $x\_a$ of norm one vectors in $X$ s.t. $Tx\_a$ converges and $\lambda x... | 2 | https://mathoverflow.net/users/2554 | 70738 | 43,297 |
https://mathoverflow.net/questions/70674 | 5 | Assume that $X$ is a smooth 3-fold and let $C\subseteq X$ a curve with a unique singular point of multiplicity $2$. Does there exist a smooth surface $S$ inside $X$ which contain $C$ ?
Clearly if the multiplicity of $C$ was at least 3 then it would be very easy to find counter-examples. On the other hand, if the mult... | https://mathoverflow.net/users/15642 | Singular curves in a 3-fold? | Here is a quick proof that any complete local Cohan-Macaulay ring of dimension $1$ and multiplicity $2$ is a *hypersurface*, so the answer to your last question is always yes.
Call such ring $R$ with maximal ideal $m$. Since $e(R)=2$ and $R$ is CM, there is a regular element $x\in m$ such that $$length(R/xR) = e(R)=2... | 4 | https://mathoverflow.net/users/2083 | 70742 | 43,298 |
https://mathoverflow.net/questions/70666 | 6 | Suppose we take the "even" indefinite lattice from page 50 in Serre *A Course in Arithmetic* (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
called $H$ in pages 189-191 of Larry J. Gerstein *Basic Quadratic Forms*.
What I cannot find in any detail is a proof of this arithme... | https://mathoverflow.net/users/3324 | Lorentzian characterization of genus | A good reference for this assertion is Cassels's "Rational Quadratic Forms", though you have to dig a bit. Let me see if I can outline the proof. First, I think Conway and Sloane assume $f$ and $g$ are classical integral (i.e. correspond to even lattices). In my copy of SPLAG, at the end of subsection 2.1 of that chapt... | 7 | https://mathoverflow.net/users/2698 | 70748 | 43,302 |
https://mathoverflow.net/questions/70683 | 4 | Kleene's O is a $\Pi\_1^1$ complete set that decides every hyperarithmetic statement. A Turing Machine that uses this set as an oracle to decide a hyperarithmetic question can only look at a finite segment of the oracle before making a decision. The possible questions are all of the form $n$ is or is not a notation for... | https://mathoverflow.net/users/16554 | Hyperarithemtic statements decidable by induction up to a recursive ordinal | Every $\Pi^1\_1$ set is many-one reducible to Kleene's $\mathcal{O}$. In particular, the universal $\Pi^1\_1$ set is many-one reducible to Kleene's $\mathcal{O}$. Therefore, every $\Pi^1\_1$ sentence (and in particular hyperarithmetical sentences) can be decided by making a single query to Kleene's $\mathcal{O}$.
| 1 | https://mathoverflow.net/users/2000 | 70754 | 43,307 |
https://mathoverflow.net/questions/70679 | 3 | This question is elementary. Let $G$ be a simple algebraic group over $\mathbb{C}$, and let $B$ be a choice of Borel subgroup, with unipotent radical $U$ with Lie algebra $\mathfrak{n}$. Then the Springer resolution of the nilpotent cone is $Z = G \times\_B \mathfrak{n}$; it is identified with the cotangent bundle of $... | https://mathoverflow.net/users/1594 | Cohomology of Springer resolution | The reason your argument doesn't work is because it's not true that $\text{Sym}^l \mathfrak n^\vee$ has a filtration with 1-dimensional graded pieces where $T$ acts with anti-dominant weights. In fact, this is false even in the case $l = 1$. I'll assume, as you do, that $B$ corresponds to the positive roots. Then the w... | 9 | https://mathoverflow.net/users/1528 | 70764 | 43,309 |
https://mathoverflow.net/questions/70740 | 21 | It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^p(\partial\Omega) $$
by $Tu(y) = u(y)$ for $y\in\partial\Omega)$ can be extended continuously to a linear map on Sobo... | https://mathoverflow.net/users/3948 | Image of the trace operator | The image you are looking for equals the Besov space $B\_{p,p}^{1-\frac1p} (\partial \Omega )$. See
H. Triebel. Interpolation theory, function spaces, differential operators. Leipzig, 1995
(in fact, I used the earlier Russian edition, Moscow, 1980).
| 14 | https://mathoverflow.net/users/12205 | 70766 | 43,310 |
https://mathoverflow.net/questions/70765 | 2 | I've been thinking recently about moduli spaces defined over $\mathbb{Z}$, and this led me to the following question:
### Question
Riemann existence says that if we have a variety over $\mathbb{C}$, $X\_{\mathbb{C}}$, then $\widehat{\pi\_1^{top}(X\_{\mathbb{C}}(\mathbb{C}))}\cong\pi\_1^{et}(X\_{\mathbb{C}})$. For w... | https://mathoverflow.net/users/5309 | In Riemann Existence, what is the interpretation of the space of complex-geometric points? | The set of points should be given by the second choice, i.e., the set of $\operatorname{Spec} \mathbb{C}$-points, over $\operatorname{Spec} \mathbb{C}$. However, there is an additional step you need to do before defining $\pi\_1$ (besides choosing a basepoint), which is applying an analytification functor to endow the ... | 2 | https://mathoverflow.net/users/121 | 70769 | 43,311 |
https://mathoverflow.net/questions/70761 | 8 | In [this answer](https://mathoverflow.net/questions/70547/haar-measures-in-solovays-model/70564#70564), Gerald Edgar mentions that Haar measure is naturally defined on the $\sigma$-algebra of [Baire sets](http://en.wikipedia.org/wiki/Baire_set) (the smallest $\sigma$-algebra that contains all the compact $G\_\delta$ se... | https://mathoverflow.net/users/2000 | Haar measure for large locally compact groups | Let's try this example. Let $\mathbb R$ be the additive group of reals with the usual topology, and let $\mathbb R\_\mathrm{d}$ be the additive group of reals with the discrete topology. Our group is $G = \mathbb R \times \mathbb R\_\mathrm{d}$. A **big** group. Any compact set in $G$ has finite projection on the $y$-a... | 16 | https://mathoverflow.net/users/454 | 70772 | 43,314 |
https://mathoverflow.net/questions/70460 | 3 | The Zeta-function can be written as the following infinite Hadamard product of its non-trivial zeroes:
$\zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod\_\rho \left(1- \frac{s}{\rho} \right)}{2(s-1)\Gamma(1+\frac{s}{2})}$
this also implies that:
$\zeta(1-s) = \pi^{\frac{(1-s)}{2}} \dfrac{\prod\_\rho \left(1- \frac{(1-s... | https://mathoverflow.net/users/12489 | Non trivial zeros of the Zeta function | Using the notation $s=u+1/2$ your conjecture can be reformulated and generalized as follows.
**Proposition.** Let $v\_1,v\_2,\dots,v\_N$ be arbitrary positive numbers, then all solutions of the equation
$$ \prod\_{n=1}^N \frac{v\_ni-u}{v\_ni+u} = 1 $$
are real.
**Proof.** The degree of the polynomial $\prod\_{n=1}... | 12 | https://mathoverflow.net/users/11919 | 70774 | 43,315 |
https://mathoverflow.net/questions/70773 | 4 | I've seen, on several occasions, papers whose purpose it is to construct a moduli space over $\mathbb{Z}$ for a moduli problem for which a moduli space over $\mathbb{C}$ was already constructed. Let's give things names:
Let $X$ be a (coarse should be enough) moduli space over $\mathbb{Z}$ for some moduli problem. Let... | https://mathoverflow.net/users/5309 | A question about moduli spaces over $\mathbb{Z}$ | In general, maps over $W$ from $W$ to $X \times W$ are the same as maps from $W$ to $X$, by the universal property of products. Here, $W = \operatorname{Spec} \mathbb{C}$. I think one possible reason for the apparent incongruity is that lots of complex points of the base-changed space end up more generic after dropping... | 4 | https://mathoverflow.net/users/121 | 70775 | 43,316 |
https://mathoverflow.net/questions/70747 | 3 | It's well-known that the exceptional locus of a birational morphism is covered by rational curves, in various degrees of generality. The best result I know in this direction is the following:
**Theorem (Hacon–McKernan):** Let (X,Δ) be a dlt pair, and f: Y -> X a birational morphism. Then the fibres of f are rationall... | https://mathoverflow.net/users/nan | Exceptional loci are covered by rational curves: easy case | Artie, here is a sketch of an argument that I think should work.
*Definition* (just in case someone needs this): A threefold singularity is called *comopound Du Val* or *cDV* if a general hyperplane section through the singular point has Du Val singularities (a.k.a. rational double point).
*Remark* Since a rational... | 3 | https://mathoverflow.net/users/10076 | 70777 | 43,317 |
https://mathoverflow.net/questions/70784 | 7 | Suppose $V$ is a model of ZF. Within $V$ we have $L$ which is a model of ZFC, furthermore $L[A]$ is a model of choice for every $A\in V$.
Suppose $A=\emptyset$ then clearly $L[A]=L$, furthermore if $A\in L$ then $A\cap L\in L$, therefore $L[A]=L$. Recall also that if $A' = L[A]\cap A$ then $L[A] = L[A']$.
On the ot... | https://mathoverflow.net/users/7206 | For models of ZF, if for some $A$ we have $L[A] = L$, what can we deduce about $A$? | Regarding your last question, it is easy to see that if
$A\cap L=\varnothing$, then $L[A]=L$, because at every stage, if we
have agreement $L\_\alpha[A]=L\_\alpha$ so far, then having
$A$ as a predicate doesn't help us to define any new sets
beyond the empty predicate (since the answer for whether a
set in $L\_\alpha$ ... | 7 | https://mathoverflow.net/users/1946 | 70786 | 43,324 |
https://mathoverflow.net/questions/63229 | 3 | Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z\_T:=T\cap Z$. Is there a formula relating $mult\_p(Z)$ and $mult\_p(Z\_T)$?
| https://mathoverflow.net/users/2348 | Multiplicity of a singular point | The answer is yes (auniket's comment meant probably $T$ is generic and not $p$). This is a classical result : if $A$ is a noetherian local ring of dimension $d>0$, with **infinite** residue field, then there exists a system of parameters $(f\_1,...,f\_d)$ such that $\mathrm{mult}(A)=\mathrm{mult}(A/(f\_1,...,f\_{d-1}))... | 2 | https://mathoverflow.net/users/3485 | 70787 | 43,325 |
https://mathoverflow.net/questions/42685 | 10 | EDIT: After talking to some experts on the subject, I have concluded that a) the answer is not obvious or well-known for locally compact groups in general, b) the answer should be 'no' and I have some idea how to construct examples, but would rather try to write them up properly somewhere. Perhaps this question should ... | https://mathoverflow.net/users/4053 | pro-discrete = locally compact and open normal subgroups have trivial intersection? | Let $K$ be an infinite profinite group and let $K\_n < K$ be a decreasing family of open subgroups with trivial intersection. Thus, $K$ acts continuously on the discrete space $X = \sqcup\_n K/K\_n$ and this in turn gives rise to a continuous action of $K$ on the free abelian group $\mathbb Z[X]$. Define $G$ to be the ... | 2 | https://mathoverflow.net/users/6460 | 70798 | 43,331 |
https://mathoverflow.net/questions/70796 | 3 | Hello,
I began to read a few weeks ago an article about automorphic L-functions in which a formula like $L(s,\pi\times\pi')=L(s,\pi)L(s,\pi')$ appeared. Unfortunately, I can't find it back. Could someone give me some reference?
Thank you in advance.
| https://mathoverflow.net/users/13625 | Product of automorphic L-functions | As David Hansen says, Langlands proved that there is an automorphic representation (consisting of specific Eisenstein series) whose $L$-function is your right hand side. This representation is denoted by $\pi \boxplus \pi'$ and is called the isobaric sum of $\pi$ and $\pi'$, it mimics the direct sum of Galois represent... | 9 | https://mathoverflow.net/users/11919 | 70802 | 43,333 |
https://mathoverflow.net/questions/70795 | 2 | Let $G$ be a finite group, for each irreducible character $\chi$, we define ${\bf Z}(\chi)$ to be the set of all $x\in G$ such that $|\chi(x)|=\chi(e)$ when $e$ is the identity of the group.
For every irreducible charcter we know that
$\frac{|G|}{\chi(e)^2} \leq \frac{1}{\chi(e)} \sum\_{x\in G} |\chi(x)| \leq \frac{... | https://mathoverflow.net/users/13736 | Some special characters of finite groups | The rightmost inequality can be improved for all non-linear irreducible characters, and if $Z(\chi) = 1$ and $G$ is non-trivial, then $\chi$ must certainly be non-linear. Using Cauchy-Schwarz, we have $$\left(\sum\_{x \in G} |\chi(x)|\right)^{2} \leq m(\chi) \left(\sum\_{x \in G} |\chi(x)|^{2}\right),$$ where $m(\chi)$... | 6 | https://mathoverflow.net/users/14450 | 70804 | 43,334 |
https://mathoverflow.net/questions/70790 | 32 | By Matiyasevich, for every recursively enumerable set $A$ of natural numbers there exists a polynomial $f(x\_1,...,x\_n)$ with integer coefficients such that for every $p\ge 0$, $f(x\_1,...,x\_n)=p$ has integer solutions if and only if $p\in A$.
Now suppose that $A$ is a set of natural numbers with membership proble... | https://mathoverflow.net/users/nan | The NP version of Matiyasevich's theorem | I don’t know about the particular form of the polynomial you are using, but in general, it is a well-known open problem whether every NP set can be represented by a Diophantine equation with a polynomial bound on the length of the solutions. [Adleman and Manders](http://dx.doi.org/10.1016/0022-0000%2878%2990044-2) prov... | 22 | https://mathoverflow.net/users/12705 | 70811 | 43,339 |
https://mathoverflow.net/questions/70813 | 4 | Let $G$ be a finite group acting on a finite set $\Omega$. A general question is to determine the sequence $o\_k(\Omega)$, where $o\_k(\Omega)$ is the number of orbits on $G$ for the natural action of $G$ on the set of $k$-subsets of $\Omega$. It's well-known that if $G=S\_n$ and the action on $\Omega =[n] := \{1, \ldo... | https://mathoverflow.net/users/2784 | The number of orbits of a permutation action | For $r=2$ a $k$-subset can be thought of as a graph with vertices $[n]$ and $k$ edges. Hence the number of orbits is equal to the number of isomorphism classes of graphs on $n$ vertices and $k$ edges. Counting them seems like a fairly intractable problem.
| 5 | https://mathoverflow.net/users/4008 | 70817 | 43,342 |
https://mathoverflow.net/questions/70801 | 7 | The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right regular representation on $L^2(G(F))$ or perhaps better on $L^2(G(F)/Z(F))$ or $L^2(G(F)^1)$ decompose, where $Z$ is the cen... | https://mathoverflow.net/users/10400 | How does the right regular of GL(n, R) and GL(n,Qp) decompose? | For the Lie (a.k.a. "archimedean") case, interpreting the $L^2$ question as asking for Plancherel measure, Harish-Chandra in-principle did this for a large class of reductive groups. The early non-compact example was $GL\_n(\mathbb C)$ treated by Gelfand and Naimark, where the orbital-integral idea already appeared, in... | 10 | https://mathoverflow.net/users/15629 | 70818 | 43,343 |
https://mathoverflow.net/questions/70767 | 2 | Profunctors from a category to itself seem like they'd be useful in representing the result of a program analysis; I can imagine a profunctor that given some information about a function it tells you what information you can derive about the composition of that function with something else.
Please post references bel... | https://mathoverflow.net/users/756 | References to using profunctors in program analysis? | I don't really understand what you're after, it'd be great if you could expand a little bit; anyway, some pointers to the use of profunctors in related matters (possibly you already know about all this things):
Hughes' arrows
---------------
This is what I think is more related to your question; *arrows* (these ar... | 5 | https://mathoverflow.net/users/4315 | 70820 | 43,345 |
https://mathoverflow.net/questions/70692 | 21 | Why is the Chebyshev function
$\theta(x) = \sum\_{p\le x}\log p$
useful in the proof of the prime number theorem. Does anyone have a conceptual argument to motivate why looking at $\sum\_{p\le x} \log p$ is relevant and say something random like $\sum\_{p\le x}\log\log p$ is not useful or for that matter any other ... | https://mathoverflow.net/users/16557 | Why is the Chebyshev function relevant to the Prime Number Theorem | There are several ideas here, some mentioned in the other answers:
**One:** When Gauss was a boy (by the dates found on his notes he was approximately 16) he noticed that the primes appear with density $ \frac{1}{\log x}$ around $x$. Then, instead of counting primes and looking at the function $\pi (x)$, lets weight ... | 25 | https://mathoverflow.net/users/12176 | 70835 | 43,351 |
https://mathoverflow.net/questions/70822 | 5 | One can define the [linking number](http://en.wikipedia.org/wiki/Linking_number) of disjointly embedded curves $K,L\subset S^{3}$ in a variety of ways, as is discussed in Chapter 5.D of Rolfsen's "Knots and Links". One way is the Gauss Integral
$$\mathrm{lk}(K,L) = \frac{1}{4\pi}\int\_{K\times L}\dfrac{\mathbf{x}-\math... | https://mathoverflow.net/users/8103 | Generalised linking numbers (where they shouldn't be) | Sort of obvious, but: in general you get lots of linking numbers. If spaces $K$ and $L$ are mapped disjointly into $\mathbb R^n$ then for every $a\in H\_i(K)$ and $b\in H\_j(L)$ with $i+j=n-1$ you get a number. It can be obtained by pulling back a generator of $H^{n-1}(S^{n-1})$ via the evident map $K\times L\to S^{n-1... | 9 | https://mathoverflow.net/users/6666 | 70837 | 43,352 |
https://mathoverflow.net/questions/70838 | 8 | Let $a, b\in A\_+$ be positive elements of some *C*\*-algebra $A$.
Assume furthermore that $a$ is invertible.
Is it true that
$$
\exists! x\in A\_+\quad:\quad xax=b\quad ?
$$
Already in the case $A=M\_2(\mathbb C)$, I don't know how to solve this.
| https://mathoverflow.net/users/5690 | Solving the equation $xax=b$ in a C*-algebra. | I do not know about general C\*-algebras, but the statement is true for complex matrices.
**Uniqueness:** Assume *b* = *x**a**x*. Then *a*1/2*b**a*1/2 = *a*1/2*x**a**x**a*1/2 = (*a*1/2*x**a*1/2)2, which implies that *a*1/2*x**a*1/2 = (*a*1/2*b**a*1/2)1/2. Since *a* is invertible, *x* must be *a*−1/2(*a*1/2*b**a*1/2)1... | 13 | https://mathoverflow.net/users/7982 | 70843 | 43,354 |
https://mathoverflow.net/questions/70840 | 1 | I'm hoping someone can help me out with finding an example of the following:
a nontrivial fiber bundle $Y \hookrightarrow Z \rightarrow X$ where $X,Y,$ and $Z$ are all compact even dimensional spin manifolds with first Pontryagin classes satisfying $p\_1(Z)=0$ and $p\_1(X)\neq 0$. I'd also like dim $Y\geq8$.
Thank... | https://mathoverflow.net/users/13377 | Example of a nontrivial fiber bundle with total space compact, spin, and $p_1=0$ | $X=\mathbb CP^3$, $Z=S^7\times S^1$ mapping to $X$ by product projection on $S^7$ followed by the usual circle-bundle $S^7\to\mathbb CP^3$. So $Y=S^1\times S^1$. Oh, you wanted $dim(Y)$ to be at least $8$, so cross it with six more circles.
| 8 | https://mathoverflow.net/users/6666 | 70844 | 43,355 |
https://mathoverflow.net/questions/70663 | 6 | General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n pairs of independent draws from F(x)? Less technically, what is the distribution of the maximum associated with the kth gr... | https://mathoverflow.net/users/16548 | Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums? | This question has been answered by Bogdan Lataianu at this link:
<https://stats.stackexchange.com/questions/13259/what-is-the-distribution-of-maximum-of-a-pair-of-iid-draws-where-the-minimum-is>
| 3 | https://mathoverflow.net/users/16548 | 70846 | 43,356 |
https://mathoverflow.net/questions/70735 | 3 | Background
----------
I've met this problem when I was trying to convert a elliptic PDE problem
into the corresponding variational problem in order to apply finite element method.
The PDE is an elliptic PDE with non-zero Dirichlet boundary condition:
Denote
$$
Lu=-\nabla\cdot(a\nabla u)+bu
$$
Then the equation i... | https://mathoverflow.net/users/14941 | Finding an $H^1$ function given its values on $\partial\Omega$ | Here is an explicit construction not requiring local straightening of the boundary.
I think it got to do the trick but I didn't check the details. Let $d(x)$ be the distance function from the point $x$ to the boundary $\partial\Omega$, denote by $K(a,t)=(4\pi t)^{-(n-1)/2}e^{-a^2/4t}$ a kernel corresponding to the fun... | 3 | https://mathoverflow.net/users/14551 | 70847 | 43,357 |
https://mathoverflow.net/questions/70848 | 0 | Hi there,
I am struggling with a theorem about truncated Dirichlet series. I am trying to prove the following theorem:
Let $(a\_n)\_n \subset \mathbb{C}$ and $N \in \mathbb{N}$. Then $\sup\_{t \in \mathbb{R}} \vert \sum\_{n=1}^N a\_n n^{-it} \vert = \sup\_{\Re s \ge 0} \vert \sum\_{n=1}^N a\_n n^{-s} \vert$.
I thin... | https://mathoverflow.net/users/16609 | Truncated Dirichlet series take their supremum on the imaginary axis | Observe that $\vert \sum\_{n=1}^N a\_n n^{-s} \vert$ is bounded in the half-plane $\Re(s)\geq 0$, and it is very small for $\Re(s)$ large. As a result, there is a strip $0\leq\Re(s)\leq b$ such that the supremum here is the same as in the half-plane, but the supremum on the right edge $\Re(s)=b$ is much smaller. Now yo... | 2 | https://mathoverflow.net/users/11919 | 70856 | 43,360 |
https://mathoverflow.net/questions/69900 | 4 | Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p\_1 + \ldots + p\_k \leq n \quad (1) $$ where $k$ is arbitrary and $p\_1 \leq \ldots \leq p\_k$ are odd prime numbers. I have edited the answer and gave three attempts I tried to use in order to find... | https://mathoverflow.net/users/1737 | Asymptotics for the number of ways to sum primes such that the sum is <= n | The question has been answered on math.stackexchange, the answer here is just for the sake of completeness.
From <https://math.stackexchange.com/questions/52737/estimating-an-integral> we see that an asymptotically equivalent estimate is $2\sqrt{n\log{n}}\;e^{\sqrt{n/\log{n}}}.$
| 2 | https://mathoverflow.net/users/1737 | 70859 | 43,362 |
https://mathoverflow.net/questions/70851 | 5 | Given a family of Boolean algebras $\mathcal{B}=\{B\_i\colon i\in I\}$ with respective Stone spaces $S\_i$, the algebra of clopen (both closed and open) subsets of the product space $\textstyle\prod\_{i\in I}S\_i$ is called the *free product* of $\mathcal B$. This algebra is typically denoted by $\textstyle\bigotimes\_... | https://mathoverflow.net/users/15129 | Free product of Boolean algebras | Translating this to Boolean spaces, you are looking for a Boolean space X which is not second countable, but cannot be written as a product of two factors of the same type (i.e., not second countable).
Have you considered the compact space $[0,\omega\_1]$? It is certainly not the product of two uncountable spaces, a... | 11 | https://mathoverflow.net/users/14915 | 70860 | 43,363 |
https://mathoverflow.net/questions/69914 | 6 | When trying to prove something about a program, the known techniques are Hoare logic and temporal logics.
An alternative is to transform a program in a mathematical (logical) expression. So, rather that mathematics is used to prove some properties of the program, the program itself is a piece of mathematics.
Loops ... | https://mathoverflow.net/users/5917 | Program transformation as alternative for Hoare logic or temporal logic | It appears to me that the gist of your suggestion is to translate a program into a relation and reason about the transitive closure of that relation. It is orthogonal that this relation is definable in first order arithmetic.
The idea of translating a program into a relation is rather old and I doubt there is a uniq... | 3 | https://mathoverflow.net/users/13475 | 70869 | 43,366 |
https://mathoverflow.net/questions/58988 | 7 | Are there non-compact complex manifolds that
a) Don't embed in C^n (holomorphically)
and
b) Cannot be covered by a finite number of coordinate open sets?
If b) can be satisfied, then I think so can a) be by taking a product with a compact complex manifold. If one takes a Riemann surface of infinite genus, one does not ... | https://mathoverflow.net/users/3709 | An example of a complex manifold without a finite open cover | Fornaess and Stout proved that EVERY complex manifold (connected and second countable) can be covered by finitely many open subsets biholomorphic to a polydisc (Lemma II.1 in MR0470251). They even have an explicit bound on the size of the cover in terms of the dimension of the manifold. Further results of a similar fla... | 26 | https://mathoverflow.net/users/16620 | 70877 | 43,372 |
https://mathoverflow.net/questions/70881 | 5 | In the [n-lab entry about shape theory](http://ncatlab.org/nlab/show/shape+theory) one can read that
>
> Strong Shape Theory [...] has, especially
> in the approach pioneered by Edwards
> and Hastings, strong links to proper
> homotopy theory. The links are a form
> of duality related to some of the more
> ge... | https://mathoverflow.net/users/7031 | Duality between proper homotopy theory and strong shape theory | A quick summary of the story is told in section 7 of S. Mardešić's "Shape Theory" from the ICM proceedings (1978) (find [here](http://www.mathunion.org/ICM/ICM1978.1/)). Strong shape theory was introduced by Edwards and Hastings keeping in mind this duality, which was inspired by Chapman's complement theorem (mentioned... | 2 | https://mathoverflow.net/users/2384 | 70884 | 43,375 |
https://mathoverflow.net/questions/70897 | 8 | Given two (simple, undirected, finite) graphs $G\_1 = (V\_1, E\_1)$ and $G\_2 = (V\_2, E\_2)$, let their automorphism groups be $Aut(G\_1)$ and $Aut(G\_2)$.
I'll recall that the cartesian product $G\_1 \times G\_2$ has vertex set $V\_1 \times V\_2$ , and two vertices $(a,b) , (x,y) \in V\_1 \times V\_2$ are adjacent ... | https://mathoverflow.net/users/16629 | Automorphism group of the cartesian product of two graphs. | All you need is in W. Imrich, S. Klavzar: "Product graphs: structure and recognition".
John Wiley & Sons, New York, USA, 2000. See also: Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. "Graphs and their Cartesian Products". A. K. Peters (2008).
One issue is that the automorphism group of the Cartesian product of $... | 13 | https://mathoverflow.net/users/1266 | 70898 | 43,382 |
https://mathoverflow.net/questions/70865 | 6 | I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}\_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$.
The groups $\operatorname{Kh}\_{\operatorname{Lee}}^\ast(L)$ for a link $L$ are very simple: there is an isomorphism:
$$\bigoplus\_{\text{orientations of... | https://mathoverflow.net/users/35353 | Is the complete functorial structure for Khovanov--Lee homology known? | As the title suggests, the paper [Fixing the functoriality of Khovanov homology](http://arxiv.org/abs/math/0701339) by Clark, Morrison and Walker fixes the functoriality in Khovanov homology. The same techniques (disoriented surfaces) fix the functoriality in Lee homology. In fact, the paper is written in such a way th... | 8 | https://mathoverflow.net/users/284 | 70902 | 43,383 |
https://mathoverflow.net/questions/70887 | 8 | Let $S$ be the spectrum of $\mathbf{Z}$ or the spectrum of an algebraically closed field. (Actually, one can take $S$ to be any noetherian integral regular scheme.)
Let $f:X\longrightarrow Y$ be a finite morphism of integral normal projective flat $S$-schemes which is etale above the complement of $B$, where $B\subse... | https://mathoverflow.net/users/16625 | Ramification divisor associated to a cover of a regular scheme | Your hypothesis imply that $\omega\_{Y/S}$ is an invertible sheaf (because $Y\to S$ is locally complete intersection).
(**EDIT**) As $f$ is flat at points of codimension $1$ ($Y$ is normal) and we are only interested on codimension 1 cycles, we can restrict $Y$ and suppose that $f$ is flat.
Then the dualizing sheaf... | 3 | https://mathoverflow.net/users/3485 | 70903 | 43,384 |
https://mathoverflow.net/questions/60944 | 8 | Let $\pi:Y\longrightarrow \mathbf{P}^1\_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch locus $D$. Let us suppose that $\pi$ is tamely ramified.
**Question 1.** Does this mean that for every prime number $p$ s... | https://mathoverflow.net/users/4333 | Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings | **Question 1**: it is not exactly the meaning of tameness (ramification index prime to $p$ and separable residue extension). But if $p>\deg \pi$, then $\pi$ is tame at $p$.
**Question 2**: The strict transform of $D$ is horizontal because it is finite birational to $D$, but the preimage of $D$ by $f$ is not horizont... | 4 | https://mathoverflow.net/users/3485 | 70907 | 43,385 |
https://mathoverflow.net/questions/70905 | 4 | Suppose *n* is a natural number, at least 3. Is the following true?
*Any two subgroups of the alternating group $A\_n$ that are conjugate inside the symmetric group $S\_n$ (With the natural embedding of $A\_n$ in $S\_n$) are also conjugate inside $A\_n$?*
Background: Elements of $A\_n$ that are conjugate in $S\_n$ ... | https://mathoverflow.net/users/3040 | Subgroups of alternating group conjugate in symmetric group conjugate in alternating group? | What you need is an example of a subgroup $G\subset A\_n$ such that the normalizer of $G$ in $S\_n$ is contained in $A\_n$. If every automorphism of $G$ is inner, then it will be enough if the centralizer of $G$ in $S\_n$ is contained in $A\_n$. How about $n=8$ and $G$ the diagonal copy of $S\_4$ in $S\_4\times S\_4\su... | 7 | https://mathoverflow.net/users/6666 | 70912 | 43,388 |
https://mathoverflow.net/questions/70608 | 10 | In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a bit the regularity of a function, given the knowledge that such function already satisfies local bounds via another more i... | https://mathoverflow.net/users/5628 | What would the best treatment of Gehring's lemma look like? | I'm not sure about "the larger context", but let me try to dissect the proof a bit so that there will be no mystery left there.
It runs upon 3 main ideas:
1) The "lack of concentration implies better summability" principle. In the nutshell, it is the following. Assume that we have some positive integrable function ... | 13 | https://mathoverflow.net/users/1131 | 70914 | 43,389 |
https://mathoverflow.net/questions/70916 | 3 | If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does **not** contain a closed (and hence discrete in the subspace topology) infinite cyclic subgroup?
An easy necessary condition comes from the usual decomposition theorem t... | https://mathoverflow.net/users/16107 | When does a LCA group not contain a (closed) infinite cyclic subgroup? | In general, you have for a compactly generated group $G = \mathbb{R}^n \times \mathbb{Z}^n\times K$, with $K$ compact. And there is no way to embed $\mathbb{Z}$ discretely in something compact, see Deitmar-Echterhoff Principles of harmonic Analysis on page 96. These are a reasonably nice family of groups, because the H... | 3 | https://mathoverflow.net/users/10400 | 70919 | 43,392 |
https://mathoverflow.net/questions/70554 | 2 | I am interested in singularity theory by topology.
I want to understand following results.
$f$ is a smooth map of a closed surface $M$ which has only
fold points and cusps as its singularities.
Suppose that a closed curve $c$ in $M$ intersects
a singular set $S(f)$ transversely at a finite number of points.
Then th... | https://mathoverflow.net/users/16516 | Thom's result and Poincaré duality | Let me attempt to answer your question as I understand it.
Let $x\in H^1(M)$ be the Poincaré dual of $[c]\in H\_1(M)$ (all (co)homology groups are with $\mathbb{Z}\_2$ coefficients). The result of Thom you state is that $w\_1(M)\in H^1(M)$ is the Poincaré dual of $[S(f)]\in H\_1(M)$.
It is well known that the cup p... | 3 | https://mathoverflow.net/users/8103 | 70924 | 43,395 |
https://mathoverflow.net/questions/70917 | 14 | Let $\mu$ be a probability measure on a set of $n$ elements and let $p\_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by
$$
E(\mu)=-\sum\_{i=1}^np\_i\log(p\_i)
$$
with the usual convention that $0\cdot(-\infty)=0$.
The following are two fundamental properties:
>
> **Property 1:** $E(... | https://mathoverflow.net/users/13809 | Entropy of a measure | I don't know if someone has already defined such entropy and I am not an expert on these things, but (depending what is wanted) I would start with something like
$$E(\mu)=\sup\left\{-\sum\_{i=1}^n\mu(A\_i)\log(\mu(A\_i)) : \mathbb{N} = \bigcup\_{i=1}^n A\_i, A\_i \text{ pairwise disjoint}\right\}.$$
Some properties t... | 6 | https://mathoverflow.net/users/11716 | 70932 | 43,401 |
https://mathoverflow.net/questions/70934 | 7 | Hello,
I looked through MathOverflow's existing entries but couldn't find a satisfactory answer to the following question:
What is the relationship between **No**, Conway's class of surreal numbers, and **V**, the Von Neumann set-theoretical universe?
In particular, does **V** contain all the surreal numbers? If ... | https://mathoverflow.net/users/7154 | Surreal Numbers and Set Theory | Yes, every surreal number is also an element of $V$ (at least, once you choose some method of encoding ordered pairs of sets as sets). The (highly recursive) characterization of which elements of $V$ are surreal numbers is precisely the definition of the surreal numbers: an element of $V$ is a surreal number just in ca... | 14 | https://mathoverflow.net/users/3902 | 70935 | 43,402 |
https://mathoverflow.net/questions/70922 | 10 | Is it true that if the average of a continuous function $f:\mathbb{R}^2\rightarrow[0,1]$ over a unit circle centered around $(x,y)$ is $f(x,y)$ for all $(x,y)\in\mathbb{R}^2$, then $f$ is necessarily constant?
| https://mathoverflow.net/users/16641 | On the average of continuous functions $f:\mathbb{R}^2\rightarrow[0,1]$ | Yes, any such $f$ is constant. In fact, if we relax the condition so that $f$ is only required to be bounded below, but not above, then it is still true that $f$ is constant. This can be proven by martingale theory, as can the statement that harmonic functions bounded below are constant ([Liouville's theorem](http://en... | 16 | https://mathoverflow.net/users/1004 | 70940 | 43,404 |
https://mathoverflow.net/questions/70949 | 0 | What is the conditional probability or probability of classes of languages?
Let $E,C,S,F,R $ be the class of computably enumerable languages,computable languagesl,context-sensitive anguages,context-free languages and regular languages respectively. $E$ is class of all computably enumerable languages and it's subset o... | https://mathoverflow.net/users/14024 | What is the conditional probability or probablity of classes of languages? | The probability measure you're asking about is nonatomic, but your events of interest are countable. Therefore the unconditional probability of each is zero, and the conditional probabilities are undefined.
| 3 | https://mathoverflow.net/users/5963 | 70950 | 43,408 |
https://mathoverflow.net/questions/70946 | 5 | I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iterati... | https://mathoverflow.net/users/112114 | Periodic sequences in symbolic dynamics | Intuitively this should happen for a large class of dynamical systems, but I don't know the right necessary and sufficient conditions.
A class of examples satisfying this is given by polyhedral billiards, where you assign a symbol to each face and correspond orbits to sequences in the obvious manner. It is a result ... | 4 | https://mathoverflow.net/users/2384 | 70955 | 43,411 |
https://mathoverflow.net/questions/66834 | 1 | In mathematics, a [Green's function](http://en.wikipedia.org/wiki/Green%27s_function) is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A [fundamental solution](http://en.wikipedia.org/wiki/Fundamental_solution) for a linear partial d... | https://mathoverflow.net/users/nan | Green's functions of Stokes flow? | I moved this question to [math.SE](https://math.stackexchange.com/q/42714/9464) a month ago. This is indeed the problem I got from the research, though it may not very appropriate here.
@Willie Wong gave a very nice [answer](https://math.stackexchange.com/questions/42714/greens-functions-of-stokes-flow/50398#50398) ... | 1 | https://mathoverflow.net/users/nan | 70960 | 43,415 |
https://mathoverflow.net/questions/70965 | 13 | Simplicial homology require that we cover the space X with a simplicial complex. But singular homology relaxes the requirement of such a discretization by considering all possible simplices in X. Although the later is not computationally favorable, it is helpful in proving many things.
Cellular homology, like simplic... | https://mathoverflow.net/users/13883 | Singular analog of cellular homology | Just thinking on my feet: From what I can tell, the trouble with setting up a "singular cell" version of homology is that simplices, whether part of a simplicial complex or on their own, have boundaries that are themselves simplices. So the boundary of a singular simplex is a singular complex simply by restricting. On ... | 8 | https://mathoverflow.net/users/6646 | 70967 | 43,420 |
https://mathoverflow.net/questions/70969 | 30 | Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int\_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert Schmidt iff $k \in L^2(X \times X, \mu \otimes\mu)$!
Q1:The main point of this questions, what are necessary and suffic... | https://mathoverflow.net/users/10400 | When is an integral transform trace class? | There are many results of the kind you ask about in the book
I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators. Providence, RI: American Mathematical Society, 1969.
It contains both necessary and sufficient conditions, and counter-examples.
| 17 | https://mathoverflow.net/users/12205 | 70972 | 43,423 |
https://mathoverflow.net/questions/70952 | 2 | Let $x\_n$ be a increasing sequence of negative real numbers that converge to $0$. Let $f$ be a function defined on $x\_n$ such that $f(x\_n)$ is a increasing sequence of negative numbers that converge to zero. Can I find a $C^\infty$ or real-analytic function $g$ defined on a ball around zero such that $g(x\_n)=f(x\_n... | https://mathoverflow.net/users/15856 | smooth/real analytic interpolation of monotonic sequence | As fedja says in the comments, the answer is "No" in general. However, there is a known condition on when it is possible to interpolate a sequence by a smooth function. This can be found in Section 2.8 of Chapter I of Kriegl and Michor's [The Convenient Setting for Global Analysis](http://www.ams.org/online_bks/surv53/... | 3 | https://mathoverflow.net/users/45 | 70974 | 43,425 |
https://mathoverflow.net/questions/70880 | 2 | There is an equation
$$
w(x) = g(x)+\int\limits\_0^M w(y)f(x-y)\,dy
$$
where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits\_{-\infty}^\infty f(t)\,dt\leq 1$. With regards to $g$ we know that $0\leq g(t)\leq 1$. This equation should be solved for $w(x)$ on $[0,M]$. Functions $g,... | https://mathoverflow.net/users/11768 | Integral Fredholm equation of the second type | Just in the case someone will be interested in a problem of such a kind. Very nice methods are developed by Prof. Kendall E. Atkinson. I read some of his papers and also used his toolbox for MATLAB which solves these problems very precise. One can find the description of a toolbox [here](http://homepage.math.uiowa.edu/... | 1 | https://mathoverflow.net/users/11768 | 70977 | 43,427 |
https://mathoverflow.net/questions/70942 | 9 | If I have a smooth compact algebraic scheme of dimention $2$ over $Spec(\mathbb{Z})$ whose generic fiber is a surface in minimal model (say of general type). Then:
(a) Is it true that the special fibers are in minimal model as well? (I would guess the answer is no in general but at least in an open subscheme of the b... | https://mathoverflow.net/users/4685 | minimal model of a "surface" over $Spec(\mathbb{Z})$ | The question is local on the base, so you can replace $\mathbb Z$ with a discrete valuation ring $R$. Moreover, the Kodaira dimension and the minimality of surfaces are stable by field extension, so one can suppose the residue field of $R$ is algebraically closed.
Then the positive answer is given in Katsura and Uen... | 9 | https://mathoverflow.net/users/3485 | 70978 | 43,428 |
https://mathoverflow.net/questions/70954 | 7 | Define a complete embedding of Boolean algebra as an homomorphism of Boolean algebras which preserves also the sup and inf operations. Notice that if $\mathbb{B}$ and $\mathbb{D}$ are complete boolean algebras, $i:\mathbb{B}\to\mathbb{D}$ is a complete embedding and
$G$ is $V$-generic for $\mathbb{D}$, then $H=i^{-1}[... | https://mathoverflow.net/users/16645 | complete embeddings of boolean algebras and preservation of stationarity | Your situation can happen.
Let $\mathbb{B}=\text{Add}(\omega\_1,1)$ be the forcing to
add a Cohen subset $S\subset \omega\_1$, and let
$\mathbb{D}$ be the forcing that first adds such a set $S$,
and then shoots a club through it $C\subset S$. Note that
$\mathbb{B}$ is countably closed in $V$ and therefore
stationary-... | 6 | https://mathoverflow.net/users/1946 | 70983 | 43,431 |
https://mathoverflow.net/questions/70987 | 3 | Are there *practical* ways to construct the [Szemerédi partitions](http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma) of a given graph (on a computer)? I found [this algorithmic version](http://en.wikipedia.org/wiki/Algorithmic_version_for_Szemer%C3%A9di_regularity_partition) of the lemma (also see the refer... | https://mathoverflow.net/users/8776 | Constructing Szemerédi partitions on a computer | Most of this information is apparently available through links from Wikipedia, so my apologies if this is not helpful (or useful enough to warrant being an answer).
I highly doubt this can be done in a practical way. The algorithm of Frieze and Kannan requires $O\left(\varepsilon^{-45}\right)$ steps, most of which re... | 4 | https://mathoverflow.net/users/6461 | 70989 | 43,434 |
https://mathoverflow.net/questions/70896 | 2 | I am calculating numerical solutions of time-dependent Schroedinger equation
$\frac{d\Psi}{dt} = - i H \Psi$
where $\Psi$ is an $N$-element complex vector and $H$ is an $N \times N$ complex matrix, which is ``almost normal''. That is, $H = H\_0 + i D$, where $H\_0$ is normal (and often Hermitian), $D$ is Hermitian ... | https://mathoverflow.net/users/1580 | Numerical solution of linear Schroedinger ODE with almost-normal Hamiltonian matrix | This should really be a comment, but it is too long.
Since Euler is out of question on such a long interval, my next suggestion would be to try to run the 3rd order Runge-Kutta about which I am pretty certain that it is not screwed up anywhere. The recursion step is
$$
\begin{aligned}
x(t+\tau)&=
\cr
&x(t)
\cr
+&\f... | 4 | https://mathoverflow.net/users/1131 | 70995 | 43,437 |
https://mathoverflow.net/questions/70990 | 8 | A triangle group has a presentation of the form,
$G=\langle a, b; a^{\alpha}, b^{\beta}, c^{\gamma}, abc\rangle, \alpha, \beta, \gamma \geq 2$
(I believe that these are also called von Dyke groups, or "ordinary" triangle groups, with triangle groups being something slightly different, but names are beside the point... | https://mathoverflow.net/users/6503 | Torsion in triangle groups | According to the survey article [H. Zieschang, On Triangle Groups, Russian Mathematical Surveys (October 1976), 31 (5), pg. 226-233](http://iopscience.iop.org/0036-0279/31/5/R18), this fact is proved in the Russian paper,
[H. Zieschang, “Discrete groups of plane motions and plane group images”, Uspekhi Mat. Nauk, 21:3(... | 7 | https://mathoverflow.net/users/1392 | 70997 | 43,438 |
https://mathoverflow.net/questions/70979 | 4 | Suppose $\mathcal C$ is a preadditive, Karoubi category with a zero object. What further assumptions on $\mathcal C$ are required to ensure that the endomorphism ring of an indecomposable object is a local?
By an object $X$ being indecomposable, I mean that in any biproduct decomposition $X \cong X\_1 \oplus X\_2$, o... | https://mathoverflow.net/users/16654 | endomorphism rings of indecomposable objects | Hi benjamin.
You're asking if $\mathcal{C}$ is a so called Krull-Schmidt-category. There are several sufficient conditions know from ring theory if $\mathcal{C}$ is a module category. For example the argument George mentioned gives an affirmative answer if $\mathcal{C}$ is the category of finitely generated (left)mod... | 5 | https://mathoverflow.net/users/3041 | 71009 | 43,443 |
https://mathoverflow.net/questions/71015 | 2 | If I had access to potentially unlimited CPUs and wanted to quickly check 100 million digit numbers for primality using a map-reduce architecture, how many CPUs would be necessary? Each of the mapped compute instances would perform efficient checks against the number in question with an assigned range of divisors (e.g.... | https://mathoverflow.net/users/16665 | How many cpus needed to check a 100 million digit prime number efficiently? | This question is a bit unclear, still I will try to give some sort of answer.
Some intial remarks:
First, at the moment noone suceed in proving primality for a 100 million (decimal) digit number. The current record is (I believe) close to 13 million digits (in binary this would still not be 100 million).
Second... | 3 | https://mathoverflow.net/users/nan | 71019 | 43,447 |
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