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https://mathoverflow.net/questions/54727 | 1 | Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some intermediate modification.) Let $F\_C$ be a torsion free
sheaf on $C$. Pull it back: $F\_{C'}:=\nu^\*(F\_C)/Torsion$.
Suppose $F\... | https://mathoverflow.net/users/2900 | when a section descends? | First of all, I think that all of $\mathscr F\_C$'s global sections appear as global sections of $\mathscr F\_{C'}$, so the answer to your second question is that it only happens in that trivial case you're mentioning.
Observe that $\nu$ is an isomorphism outside of finitely many points, so the induced morphism $\mat... | 2 | https://mathoverflow.net/users/10076 | 54734 | 34,192 |
https://mathoverflow.net/questions/54753 | 1 | Given a centerless finite group G, with at least one automorphism which is not conjugation by an element of G. Is there any lower bound on the size of Aut(G) given in terms of G? (As big as possible, of course).
| https://mathoverflow.net/users/12822 | lower bound on Aut(G) | Only the trivial bound $2|G|$, because the alternating groups $A\_n$, $n\neq 6$ make this sharp. See e.g. the [Wikipedia article](http://en.wikipedia.org/wiki/Alternating_group#Automorphism_group).
| 8 | https://mathoverflow.net/users/35416 | 54757 | 34,206 |
https://mathoverflow.net/questions/54760 | 12 | After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity
uses this assumption, and for the only non-paracompact manifold I know, namely the long line, a similar argument seems to show the exi... | https://mathoverflow.net/users/3635 | Riemannian metrics on non-paracompact manifolds | On the contrary, the long line does not have a Riemannian metric. Every countable subset of the long line has a least upper bound, so if it were Riemannian then a geodesic ray in the long direction would have to stop short of the end. But then the Riemannian metric would break down at the "endpoint" of this geodesic.
... | 22 | https://mathoverflow.net/users/1450 | 54762 | 34,208 |
https://mathoverflow.net/questions/54743 | 5 | The following question seems pretty natural, but searching online
and looking in some obvious places didn't turn up much, so maybe
I can ask it here. (Disclaimer: I'm a newcomer to this topic, so
apologies if the question is obviously misguided.)
Suppose that $V\_1 = X\_1/\Gamma\_1$ and $V\_2= X\_2/\Gamma\_2$ are
ari... | https://mathoverflow.net/users/nan | Is the Baily--Borel compactification functorial? | A useful reference might be the article "Satake Compactification and extension of Holomorphic Mappings", Inv.Math. 16, 237-248, 1972, by Kiernan and Kobayashi. They show that if the map $V\_1 \to V\_2$ is induced from a map $X\_1 \to X\_2$ then it extends. In particular, the answer is positive for your "simplest exampl... | 3 | https://mathoverflow.net/users/519 | 54768 | 34,213 |
https://mathoverflow.net/questions/54749 | 3 | Let $X$ be a smooth projective surface over $\mathbb{C}$. How to show that for any zero-dimensional subscheme $Z$ of $X$, the Euler characteristic $\chi(\mathcal{O}\_Z,\mathcal{O}\_Z)={\sum}\_i (-1)^i \ dim \ Ext^i\_{\mathcal{O}\_X} (\mathcal{O}\_Z,\mathcal{O}\_Z)$ is the same for all $Z$ of length $n$?
(I read that... | https://mathoverflow.net/users/12395 | How to show that a form of Euler characteristic is constant for a family of schemes? | Define $\chi(\mathcal M,\mathcal N)$ for all finite length $\mathcal O\_X$-modules. It is additive in both arguments so for its computation we get
$$
\chi(\mathcal M,\mathcal N) = \sum\_{x,y}
\mathrm{lgth}\_x(\mathcal M)\mathrm{lgth}\_y(\mathcal N)\chi(k(x),k(y)).
$$
Clearly $\chi(k(x),k(y))=0$ if $x\neq y$ and the exa... | 6 | https://mathoverflow.net/users/4008 | 54770 | 34,214 |
https://mathoverflow.net/questions/53215 | 6 | When finding representations of finite groups over $\mathbb{F}\_p$, (i.e. homomorphism from $G$ to $GL(n,\mathbb{F}\_p$), it requires many times presentation of $GL(n,p)$. What is presentation of GL(n,p)?
| https://mathoverflow.net/users/12484 | Presentation of GL(n,p)? | To focus just on the question of giving a presentation of a finite general linear group over the prime field (or other finite field), leaving aside the fuzzy connection with representations of finite groups: This can be looked at profitably in the broader setting of finite groups of Lie type as studied by Steinberg in ... | 3 | https://mathoverflow.net/users/4231 | 54771 | 34,215 |
https://mathoverflow.net/questions/54623 | 1 | In her paper "[Mixed perverse sheaves for schemes over number fields](https://doi.org/10.1023/A:1000273606373 "Compositio Mathematica 108, 107–121 (1997). zbMATH review at https://zbmath.org/?q=an:0882.14006")" A. Huber defines certain weights for certain categories of $\mathbb{Q}\_l$-sheaves over a finite type $\mathb... | https://mathoverflow.net/users/2191 | Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes? | The answer to the question in your title is, I think : "in general, no". The answer to your last question is : "well, it depends how you have defined the objects, and you will have to be very careful about which morphisms of schemes you allow".
Let me be more precise. Huber wants to consider only complexes of sheaves... | 4 | https://mathoverflow.net/users/12336 | 54780 | 34,222 |
https://mathoverflow.net/questions/54752 | 9 | I know quite a bit about the abstract theory of Interpolation of Banach spaces. Today I had a colleague from Environmental sciences (who used to be in our Applied Maths department) come and ask me about (complex) interpolation of Sobolev spaces. I was, in the end, able to explain enough to give him a "black box" which ... | https://mathoverflow.net/users/406 | Interpolation of Sobolev spaces | What exactly does your colleague need interpolation for? I guess he just needs to extend some inequalities to intermediate values of the parameters. Then he can use the black box approach and his problem is reduced to computing interpolation spaces between given couples of Banach spaces. Then, two possibilities:
1) T... | 8 | https://mathoverflow.net/users/7294 | 54797 | 34,231 |
https://mathoverflow.net/questions/54720 | 14 | Hi,
I was wondering about good techniques that one can use to show that given certain coefficients, they are the Fourier coefficients of a cusp form, assuming we know the desired weight and level. I am aware of Weil's "converse theorem", but am not aware of any examples of it being used to prove something is a modular ... | https://mathoverflow.net/users/9769 | Verifying coefficients of modular forms | Maybe I can add something:
1. Wiles famous proof of Fermat's Last Theorem (and the later proof of modularity of all elliptic curves over $\mathbf{Q}$ by Breuil, Conrad, Diamond, Taylor) is a sort of ultimate example in which "given certain coefficients [associated to an elliptic curve by point count], they are [prove... | 18 | https://mathoverflow.net/users/8441 | 54805 | 34,236 |
https://mathoverflow.net/questions/54775 | 53 | The question is self-explanatory, but I want to make some remarks in order to prevent the responses from going off into undesirable directions.
It seems that every few years I hear someone ask this question; it seems to hold a perennial fascination for research mathematicians, just as quests for short proofs do. The ... | https://mathoverflow.net/users/3106 | What is the shortest Ph.D. thesis? | David Rector's thesis ("An Unstable Adams Spectral Sequence", MIT 1966) is 9 pages, according to the [record at the MIT library](https://library.mit.edu/item/000612341). I haven't seen the actual thesis for many years, but I'm pretty the actual mathematical content takes about 3 pages total, and is largely identical to... | 53 | https://mathoverflow.net/users/437 | 54810 | 34,240 |
https://mathoverflow.net/questions/54795 | 3 | I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x\_1,\dots{}x\_N\right]$. For that purpose, I evaluate Cramer-Rao bound for Gaussian i.i.d. arguments.
The function has a singularity at the point $\mathbb{x}\_0$ where $f(\mathbb{x}\_0)... | https://mathoverflow.net/users/1783 | Concentration of measure and bounds on variance | Let $X\_i$ and $X'\_i$ be i.i.d random variables. Write $f = f(X\_1, \dots, X\_n),$ and define $$f\_i = f(X\_1, \dots, X'\_i, \dots, X\_n)$$ as the same function with the $i$th input replaced by the independent copy $X'\_i$.
The [**Efron-Stein inequality**](http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/... | 2 | https://mathoverflow.net/users/238 | 54827 | 34,251 |
https://mathoverflow.net/questions/54798 | 4 | Hi,
I have an elementary question concerning an extremal problem:
Given a manifold $M$:={$(x\_1, x\_2, x\_3) $ | $x\_1^2 + x\_2^2 + x\_3^2 = c$ and $\frac{x\_1^2}{a\_1^2} + \frac{x\_2^2}{a\_2^2} + \frac{x\_3^2}{a\_3^2}= d$ and max($a\_1, a\_2, a\_3$) $> \sqrt{c} >$ min($a\_1, a\_2, a\_3$) } consider an arbitrary po... | https://mathoverflow.net/users/12826 | Minimal distance between a point and a manifold M (M is the intersection of a ball and an ellipsoid) | Lagrange method is based on a fact that extrema can occur only at points where the gradient of $f$ is a linear combination of gradients of the constraints -- i.e. at points where $\{\nabla f,\nabla g\_1, \nabla g\_2 \}$ is linearly dependent with coefficient in front of $\nabla f$ nontrivial. If the constraints are ind... | 5 | https://mathoverflow.net/users/6818 | 54834 | 34,256 |
https://mathoverflow.net/questions/54759 | 9 | There are several well-known dualization results in category theory, i.e. that such-and-such a well-known category D is isomorphic to the opposite C^{op}. Does anyone know of such a result concerning what the opposite category to Rng, rings (\*-monoid on +-group) with their homomorphisms, looks like?
I ask (naively I... | https://mathoverflow.net/users/12823 | What does Rng^{op} look like? | Here is a too-serious answer to your question, along with answers to a couple questions I think you should be asking:
The category you're interested in, as noted by others, is the category of coalgebras / corings, which is emphatically *not* the opposite category of rings --- but we're going to see exactly what's dif... | 33 | https://mathoverflow.net/users/1094 | 54835 | 34,257 |
https://mathoverflow.net/questions/42653 | 21 | This is the second [follow-up](https://mathoverflow.net/questions/42646/square-roots-of-elements-in-a-finite-group-and-representation-theory) to [this question](https://mathoverflow.net/questions/41784/roots-of-permutations) on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th... | https://mathoverflow.net/users/35416 | Number of n-th roots of elements in a finite group and higher Frobenius-Schur indicators | Here are some things you probably know. For a representation $W$ of $G$, let $\text{Inv}(W)$ denote the subspace of $G$-invariants. For an irreducible representation $V$ with character $\chi$, the F-S indicator $s\_2(\chi)$ naturally appears in the formulas
$$\dim \text{Inv}(S^2(V)) = \frac{1}{|G|} \sum\_{g \in G} \f... | 7 | https://mathoverflow.net/users/290 | 54840 | 34,261 |
https://mathoverflow.net/questions/54822 | 3 | Any closed form known for $\lim \limits\_{n \to \infty}\ \underbrace{\log\_2 \log\_2 \dots \log\_2}\_n\ \underbrace{3^{3^{\cdot^{\cdot^{3}}}}}\_n$?
Numerical evidence suggests that it is around $2.4440214614892$ and converges very fast, but I was not able to prove existence of this limit.
| https://mathoverflow.net/users/9550 | Repeated logarithm of a power tower | The answer is "no" ... also see a Usenet sci.math discussion in July, 2009.
| 2 | https://mathoverflow.net/users/454 | 54842 | 34,262 |
https://mathoverflow.net/questions/54847 | 3 | Let $A\in SL(n,\mathbb{Z})$ and $B\in\mathcal{M}\_n(\mathbb{Z})$ s.t. $\det(B)\ne 0$. Is it possible to find a power of $B^{-1}AB$ in $SL(n,\mathbb{Z})$?
| https://mathoverflow.net/users/12839 | On some type of conjugate of elements of SL(n,Z) | Yes. Write $N = det(B)$. Then, the group $SL(n, \mathbb{Z} / N\mathbb{Z})$ is finite-order, say of order $k$. If we raise $B^{-1} A B$ to the power of $k$, then it should lie in $SL(n, \mathbb{Z})$.
| 4 | https://mathoverflow.net/users/12087 | 54850 | 34,268 |
https://mathoverflow.net/questions/54806 | 0 | 1. Let $f$ be modular of level $p^nN$, $(p,n) = 1$, $p > 2$ with character $\chi\psi\eta$, where $\chi$ has conductor dividing $N$, $\psi$ conductor power of $p$ and order power of $p$, and $\eta$ conductor $p$ and order dividing $p-1$. Since $p$ is odd, one can write $\psi = \xi^{-2}$. (i) Why is the character of $f \... | https://mathoverflow.net/users/12832 | questions regarding modular forms | William Stein has answered your question (i). As for your question (ii), since $\xi$ has $p$-power order, and since any $p$-power root of unity is congruent to $1$ modulo the unique prime ideal lying over $p$ in $\mathbb Q$ adjoin the $p$-power roots of unity, we see
that $f\otimes \xi$ is congruent to $f$ modulo any p... | 4 | https://mathoverflow.net/users/2874 | 54852 | 34,269 |
https://mathoverflow.net/questions/54862 | 1 | Just browsing some old stuff in my office for other thing I found the following:
$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan.
Find three numbers such that their sum, the sum of their squares, and the sum of their cubes
, shall be a cube.
Note.--W. D. Cairns says this problem, which was prop... | https://mathoverflow.net/users/11016 | $230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan | With a quick search from the internet I was able to only find the solution $$(146, -1314, 1168)$$ by **E. T. Bell** in *The American Mathematical Monthly*, Vol. 24, No. 5 (May, 1917), p. 240. The paper can be found from <http://www.jstor.org/stable/pdfplus/2974328.pdf>
(Also, a quick computer search shows that there ... | 11 | https://mathoverflow.net/users/11716 | 54875 | 34,277 |
https://mathoverflow.net/questions/54877 | 2 | Let $\mathscr{S}$ be the set of all countable subfields of $\mathbb{C}$. Clearly, $\mathscr{S}$ is a partially ordered set under inclusion, and if $K\_1\subseteq K\_2 \subseteq \cdots$ is an ascending chain of countable subfields, then $\bigcup\_{i=1}^{\infty}K\_i$ is a countable union of countable fields, and is hence... | https://mathoverflow.net/users/6856 | Countable Fields with No Countable Extension | Every countably infinite field $F$ not only has a countable
extension $K$, but has one satisfying exactly the same
first-order truths in the language of fields. Indeed, one
can add any further structure to the language, such as
predicates for relative transcendency, and maintain this
feature. This is an immediate conse... | 11 | https://mathoverflow.net/users/1946 | 54880 | 34,280 |
https://mathoverflow.net/questions/54867 | 21 | Consider a compact connected complex manifold $X$ of dimension $n$. Siegel proved in 1955 that its field of meromorphic functions $\mathcal M (X)$ has transcendence degree over $\mathbb C$ at most $n$. Moishezon studied those complex manifolds for which the degree is $n$, and consequently these manifolds are now called... | https://mathoverflow.net/users/450 | Is a complex manifold projective just because its blow-up at a point is ? | maybe the following argument works. (It's quite possible a sign went wrong somewhere, though.)
Let $\pi: Y \rightarrow X$ be the blowup. By assumption $Y$ is projective, so it carries an ample line bundle $A$ say. Let $E$ denote the exceptional divisor of the blowup, and consider line bundles of the form $A+nE$ (for ... | 15 | https://mathoverflow.net/users/nan | 54883 | 34,282 |
https://mathoverflow.net/questions/54881 | 1 | So let $\mathbf{F}\_q$ be a finite field with $q$ elements where $q=p^m$,
$p$ a prime number and $m\in\mathbf{Z}\_{\geq 1}$. Let $f(x,y)\in \mathbf{F}\_q[x,y]$
be a smooth non-constant polynomial and let $A:=\mathbf{F}\_q[x,y]/(f)$.
Q: Does there exists an integer $N\_0$ (which depends on $A$) such that for $N\... | https://mathoverflow.net/users/11765 | About points on affine curves defined over finite fields | The answer is "yes", given what we know about the number of points on the corresponding projective curve (or more accurately a smooth model of it, in case of singularities). Since points at infinity and singular points can be bounded terms of the degree of *f*, they don't affect the asymptotics here. So the question co... | 1 | https://mathoverflow.net/users/6153 | 54884 | 34,283 |
https://mathoverflow.net/questions/54820 | 30 | Is there constructed some set of physical laws from which we can logically obtain that any function that can be implemented in some device is Turing computable?
EDIT
I believe that if we restrict ourselves to classical mechanics (I mean if we suppose that any device obey just classical mechanics laws) then CT thesi... | https://mathoverflow.net/users/8381 | Physics and Church–Turing Thesis | Just appeared on the arXiv today:
"The physical Church-Turing thesis and the principles of quantum theory,"
by Pablo Arrighi and Gilles Dowek.
<http://arxiv.org/abs/1102.1612>
**Abstract:**
>
> Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory. Yet several works... | 21 | https://mathoverflow.net/users/6094 | 54886 | 34,285 |
https://mathoverflow.net/questions/54758 | 6 | Does the following series converge? $\sum\_{n=1}^{\infty} \vert \sin n \vert ^{n}$
| https://mathoverflow.net/users/10758 | Does the following series converge? | The question has basically been answered in the comments by David Speyer and SJR. It is a theorem of Chebyshev that that for any irrational $\alpha$ and any real $\beta$, the inequality
$$|\alpha n - k - \beta| < 3/n$$
has infinitely many solutions. In particular, take $\alpha = 1/(2\pi)$ and $\beta = \frac12$. Then on... | 6 | https://mathoverflow.net/users/1450 | 54892 | 34,289 |
https://mathoverflow.net/questions/54889 | 1 | I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory.
However, I am interested in learning algebraic number theory and I recently found that the basic ideas of commutative algebra are not so familiar with me:).
Besides, the fundamental notion of algebraic numbe... | https://mathoverflow.net/users/11059 | Good Minkowski Theory and Commutative Algebra Books | You might try Pierre Samuel, "Algebraic Number Theory", for a concise introduction with basic treatments of what you are asking about.
| 4 | https://mathoverflow.net/users/6153 | 54893 | 34,290 |
https://mathoverflow.net/questions/54890 | 1 | Hi,
I'm trying to find the probability that after n trials of a multinomial rv, there have been exactly d distinct outcomes.
What I'm ultimately trying to calculate is the expected number of trials required to achieve each possible outcome.
I can see how to formulate this as a recurrence relation, but it is doubly ... | https://mathoverflow.net/users/12848 | Probability of d distinct outcomes after n trials | In the symmetric case, your second question is easy. Assume that each outcome is equally likely. You need $N\_0=1$ trial to get $1$ outcome. Once you got $k$ different outcomes, you need $N\_k$ more trials to get a new one, where $N\_k$ is geometric with parameter $1-(k/n)$, hence $E(N\_k)=n/(n-k)$. You get every possi... | 1 | https://mathoverflow.net/users/4661 | 54894 | 34,291 |
https://mathoverflow.net/questions/54864 | 4 | Consider a multiset $S$ containing $n$ positive integers $\{s\_1, s\_2, \ldots, s\_n\}$. For the purpose of this question, assume that (some of) the integers are very large with respect to $n$, i.e. much larger than $2^n$ such that the number of bits to describe them is not polynomial in $n$. I am interested in whether... | https://mathoverflow.net/users/5200 | Decreasing the size of integers in a multiset while maintaining the total order on sums of subsets | Here is a much expanded answer. the old answer is below.
There are some very interesting questions here. What I say here is far from answering them completely but does go a certain distance. The sequence [A009997](http://oeis.org/A009997) in the OEIS and its references turns out to be very relevant. I suggest looking... | 3 | https://mathoverflow.net/users/8008 | 54896 | 34,292 |
https://mathoverflow.net/questions/6974 | 5 | The Radon transform apparently was discovered around 1917 if Wikipedia is to be believed. The Cauchy-Crofton theorem is a much older theorem (mid 19th-century). But both ideas are more or less the same.
Did Radon consider his transform as a generalization of the Cauchy-Crofton theorem? Did he not know about the Cauc... | https://mathoverflow.net/users/1465 | Historical question Cauchy-Crofton theorem vs. Radon transform | I am looking at the translation of the original paper by Radon. It has only three references: To Minkowski, Funk, and von Weyl. Crofton is not mentioned in the text, and the only mention of Cauchy is irrelevant to your question. This makes it rather unlikely that he knew about it, I think. – Harald Hanche-Olsen Nov 27 ... | 1 | https://mathoverflow.net/users/1465 | 54901 | 34,296 |
https://mathoverflow.net/questions/54876 | 56 | What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various nice operations. But whereas complete intersections have an obvious geometric meaning, I don't know if this is true for C... | https://mathoverflow.net/users/2841 | Geometric meaning of Cohen-Macaulay schemes | [**EDIT:** I rewrote the first couple of paragraphs, because I realized a better way to say what I had in mind.]
There are many ways to define dimension and some of them give the same answer some of them don't.
*Depth* is a sort of dimension. Perhaps not the most obvious, but one that works well in many situation.... | 55 | https://mathoverflow.net/users/10076 | 54904 | 34,298 |
https://mathoverflow.net/questions/45393 | 12 | Does this property characterizes amenability or there are examples of non-amenable groups satisfying it?
Let $G$ be finitely generated group.
Property:
There exists $C<1$ such that for every $S\subset G$ - finite set, there exists $F \subset G$ - finite, such that
$|sF \Delta F|\leq C\cdot |F|$ for every $s\in... | https://mathoverflow.net/users/8699 | a question on Folner sets | If I am not mistaken the group should be amenable by the following arguments.
As was explained to me by Jesse Peterson, if $\pi:G\rightarrow B(H)$ is a representation of $G$ and there exists unit vector $\xi$ such that for every $g\in G$: $\|\pi(g)\xi-\xi\|\leq C$ then $\pi$ has an invariant vector.
Let $\xi\_F=\fr... | 6 | https://mathoverflow.net/users/8699 | 54911 | 34,302 |
https://mathoverflow.net/questions/54907 | 24 | This question appeared in my answer to [this](https://mathoverflow.net/questions/54838/amenability-of-groups-ii) question, but it seems to be interesting in itself. Let $G$ be an infinite finitely generated group, $\epsilon\gt 0$. Is there a finite subset $S\subset G$ such that every subset of $S$ with at least $\epsi... | https://mathoverflow.net/users/nan | A non-trivial property of all groups | This is false for the infinite dihedral group $\langle a,b\mid b^2=1, ba=a^{-1}b\rangle$. No set $S$ works for $\epsilon\le1/3$, because there is always a subset with $\lceil{\epsilon|S|}\rceil$ elements that lies entirely in $\{a^n\mid n\in\mathbb{Z}\}$, $\{a^{2n}b\mid n\in\mathbb{Z}\}$ or $\{a^{2n+1}b\mid n\in\mathbb... | 33 | https://mathoverflow.net/users/12362 | 54917 | 34,306 |
https://mathoverflow.net/questions/54923 | 15 | I'm looking for a measure-theoretic analogue to the disjoint union topology, or for work on the $\sigma$-algebra generated by canonical injections. More formally:
> For an indexed family of sets $\{A\_i\}\_{i \in I}$, define $\psi\_i : A\_i \to A$, $a \mapsto (a,i)$ (the canonical injections), where $A = \bigcup\_{i... | https://mathoverflow.net/users/10828 | Is there a "disjoint union" sigma algebra? | This does exist, and has a nice explicit description. Treating the sets $A\_i$, for convenience, as disjoint subsets of $A$, take a subset $S \subseteq A$ to be measurable exactly if $S \cap A\_i$ is a measurable subset of $A\_i$, for each $i$. The proof that this is a sigma-algebra making each $\psi\_i$ measurable, an... | 24 | https://mathoverflow.net/users/2273 | 54924 | 34,309 |
https://mathoverflow.net/questions/54895 | 8 | There is a very nice geometric proof of Deligne for the Artin Reciprocity in the geometric setting, namely for a smooth, projective, geometrically irreducible curve $C$ over a finite field $\mathbb{F}\_{q}$, with function field $K=k(C)$, and idele group $\mathbb{I}\_{K}:=\prod^{'}\_{p\in|C|}K^{\*}\_{p}$ there is a one-... | https://mathoverflow.net/users/12847 | Geometric abelian class field theory | They are different statements. What Deligne proves is the unramified case, i.e. the description of abelian extensions of $K$ unramified everywhere. If you could extend his argument to affine curves then you could possibly prove Artin reciprocity by his method. Going the other way should not be difficult. Have you looke... | 5 | https://mathoverflow.net/users/2290 | 54931 | 34,311 |
https://mathoverflow.net/questions/54925 | 2 | Let $\mathcal{A}$ be a $C^\*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$ is an injective \*-homomorphism from $\mathcal{A}$ into a subset of the bounded operators $\mathcal{B}(\mathcal{H... | https://mathoverflow.net/users/7671 | Do unitary bijections act invariantly on irreducible representations? | The answer is no.
Consider the Toeplitz algebra $\mathcal T$ with its canonical representation on $\ell^2 \mathbb N$, which is generated as a $C^\star$-algebra by the shift $S(e\_n)=e\_{n+1}$. It is well-known that the Toeplitz algebra contains all compact operators; hence the representation is irreducible.
It is w... | 4 | https://mathoverflow.net/users/8176 | 54942 | 34,317 |
https://mathoverflow.net/questions/54912 | 3 | Suppose that $R = S/I = k[x\_1, \dots, x\_n]/I$ is a (normal) domain of finite type over a field (or any semi-local ring $k$ with a dualizing complex). In this case, I can define $\omega\_R = \textrm{Ext}^{n - \dim R}\_{S}(R, S)$.
$R$ is called *quasi-Gorenstein* if $\omega\_R$ is locally free of rank 1. Presumably t... | https://mathoverflow.net/users/3521 | An easy example of a (1/quasi-)Gorenstein ring with non-trival canonical divisor class. | Karl, I think one can construct a smooth affine curve with non-zero canonical class by removing some general points from a smooth projective curve of genus $>1$. Details can be found in this [paper](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.6348) (Theorem 6), which I gave in this [answer](https://matho... | 2 | https://mathoverflow.net/users/2083 | 54948 | 34,321 |
https://mathoverflow.net/questions/54909 | 0 | It goes without saying that the name in the title tentatively refers to a series whose name one does not know yet and probably in the future I may come with a post titled "The x-sequence" or "The x-function" ,etc. I do not know whether the quotient series I am going to construct is known as a "descending quotient serie... | https://mathoverflow.net/users/5627 | The X-series (for groups) | This is called a [chief series](https://en.wikipedia.org/wiki/Chief_series) or principal series. The quotients of the terms in the series are called chief factors, and the length of the series is called the chief length. Non-isomorphic groups can have isomorphic chief factors. It depends on your definition of isomorphi... | 8 | https://mathoverflow.net/users/3710 | 54951 | 34,324 |
https://mathoverflow.net/questions/54679 | 5 | Let T be a Turing machine which, when started on an infinite blank tape extending to the left and to the right,
is programmed to print out in some recursively enumerable order, all the theorems of ZFC (one after another) on
the squares of the tape. It is also programmed to halt when and only when (if ever) it prints a ... | https://mathoverflow.net/users/4423 | A question about imaginary Turing machines. | The Turing Machine you describe here can actually be constructed (from a practical standpoint also), but it would be tedious and not of much practical use.
First note that the finite set of symbols $\{ \in, \forall, \exists, x, \prime, \land, \lor, \lnot, \rightarrow, \leftrightarrow \}$ would be more than sufficient... | 3 | https://mathoverflow.net/users/11318 | 54957 | 34,329 |
https://mathoverflow.net/questions/54927 | 2 | I am confused with morphisms of supermanifolds. Take a simple example $f:R^{0|1}\to R^{0|1}$. By (one of) definition, $f$ is a morphism of superalgebras of functions $C(R^{0|1})\to C(R^{0|1})$. Morphisms of superalgebras preserve the grading, I deduced that $f$ have the form $1\mapsto 1, \theta\mapsto x\theta$, i.e. $H... | https://mathoverflow.net/users/7341 | Morphisms between supermanifolds R^{0|1}→R^{0|1} | You are right that the set of supermanifold morphisms
$Hom(\mathbb R^{0|1},\mathbb R^{0|1})$ to itself is $\mathbb R^1$.
However, one can define for supermanifolds $X,Y$ with $\dim X=0|d$ a
supermanifold $map(X,Y)$ of morphisms from $X$ to $Y$, by $Hom(Z,map(X,Y))=Hom(Z\times X,Y)$ for all supermanifolds $Z$.
And... | 4 | https://mathoverflow.net/users/1090 | 54959 | 34,330 |
https://mathoverflow.net/questions/54936 | 1 | Last month, I asked whether there is an efficient algorithm for finding the square root modulo a prime power here: [Is there an efficient algorithm for finding a square root modulo a prime power?](https://mathoverflow.net/questions/52081/is-there-an-efficient-algorithm-for-finding-a-square-root-modulo-a-prime-power)
... | https://mathoverflow.net/users/7089 | Finding the square root modulo n, when the factors of n are known | I see my mistake now. I interpreted the paper by Manders and Adelman wrong. I thought that the theorem in their paper implied that finding a square root is NP-complete, but this not true.
| 1 | https://mathoverflow.net/users/7089 | 54960 | 34,331 |
https://mathoverflow.net/questions/54964 | 20 | $\DeclareMathOperator\nr{nr}\DeclareMathOperator\rank{rank}$This is a terminology question (I should probably know this, but I don't). Given a group $G$, consider the minimal cardinality $\nr(G)$ of a set $S \subset G$ such that $G$ is the normal closure of $S$: $G = \langle\!\langle S \rangle \!\rangle$ (nr is short f... | https://mathoverflow.net/users/1345 | How many elements does it take to normally generate a group? | According, for example, to the following paper by Gonzales-Acuna
<http://www.jstor.org/pss/1971036>
the smallest number of elements needed to normally generate a group $G$ is called the weight of $G$. This terminology is confirmed in the book
Algebraic invariants of links
by J. Hillman. I also confirm that the ... | 21 | https://mathoverflow.net/users/6206 | 54965 | 34,333 |
https://mathoverflow.net/questions/54933 | 18 | It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative.
For if a and b are elements of R, and writing + for the group operation then applying the distributive property one has
$$
\begin{align}
a+a+b+b&=a\*(1+1)+b\*(1+1)\\\\
&=(a+b)\*(1+1)\\\\
&=(a+b)\*1+(a+b)\*1... | https://mathoverflow.net/users/11629 | Superfluous definitions | A unique factorization domain is typically defined as:
a domain $D$ such that each non-zero and non-invertible element $d$ can be factored as a product of irreducible elements.
And, for any factorizations $d=u\_1 \dots u\_m$ and $d = v\_1 \dots v\_n$ with irreducibles $u\_i,v\_i$ one has
that $m=n$ and there exists... | 8 | https://mathoverflow.net/users/nan | 54968 | 34,336 |
https://mathoverflow.net/questions/54921 | 22 | Related to a question by Mark Sapir (see [here](https://mathoverflow.net/questions/54907/a-non-trivial-property-of-all-groups)) and a question by Kate Juschenko (see [here](https://mathoverflow.net/questions/54838/amenability-of-groups-ii)), let me ask the following:
>
> **Question:** Let $G$ be a finitely generate... | https://mathoverflow.net/users/8176 | Generation of finite index subgroups | (**Edited to add more detail about random walks on infinite graphs**)
Let $G$ be a group with generators $g\_1, \dots, g\_n$, and let $\epsilon > 0$ be given.
Let $K$ be a cell complex with $n$ edges corresponding to the generators with
fundamental group $G$ (this can be obtained by sewing on 2-disks for each of a poss... | 15 | https://mathoverflow.net/users/9062 | 54987 | 34,349 |
https://mathoverflow.net/questions/54974 | 12 | I feel that the following problem should be known, but I'm not sure where to look for it.
Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A\_p$ denote the set of residue classes coming from the first $\lfloor p \epsilon \rfloor$ integers. Let $B\_p$ denote the squares (modulo $p$) of ... | https://mathoverflow.net/users/nan | Arithmetic progressions modulo $p$ under the squaring map | This is a variant of a common theme. It should follow from more or less standard exponential sums estimates. The general buzzword is Erdos-Turan inequality. The answer should be yes and it might follow from the results of:
A. Granville, I. E. Shparlinski and A. Zaharescu, On the
distribution of rational functions alo... | 7 | https://mathoverflow.net/users/2290 | 54995 | 34,355 |
https://mathoverflow.net/questions/54994 | 15 | Fix an algebraically closed field $k$. Why is the general curve over $k$ of genus $g \ge 3$ automorphism-free?
I am particularly interested in seeing an argument that does not go by induction and specialization to a singular genus $g$ curve.
Let's say a curve is a smooth, projective, connected $1$-dimensional $k$-... | https://mathoverflow.net/users/5337 | Why is a general curve automorphism-free? | One way to do it is through deformation theory, provided we only consider
automorphism groups $G$ of order not divisible by the characteristic (one may of
course assume that it is cyclic of prime order). Then the
the moduli space (or just a miniversal deformation) of all curves of genus $g>1$
is smooth with tangent spa... | 14 | https://mathoverflow.net/users/4008 | 55001 | 34,361 |
https://mathoverflow.net/questions/54954 | 6 | I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."
>
> For each $\epsilon >0$ there is a number $\eta^\prime(\epsilon) > 0$ with the following property. Let $X$ be a Gaussian r.v. with values in a Banach spa... | https://mathoverflow.net/users/2586 | Why is the dimension of Gaussian variables is bounded by the dimension of the space? | None of these estimates is trivial. The upper bound follows from John's theorem that the Banach-Mazur distance between an N dimensional normed space and an N dimensional Euclidean space is at most $\sqrt N$. The lower bound is more involved and uses the Dvoretzky-Rogers lemma. A good reference for the Gaussian approach... | 5 | https://mathoverflow.net/users/6921 | 55017 | 34,370 |
https://mathoverflow.net/questions/55010 | 14 | If $n=\prod\_{i=1}^{k} p\_i^{e\_i}$ is a prime factorization of integer $n$.
> Is there a quick way to find the prime factorization of $n+1$?
Or the only way to do it is recalculating the whole factorization?
Any references and/or articles on this problem?
| https://mathoverflow.net/users/12875 | Prime factorization of n+1 | Check out the literature on Fermat numbers, $2^{2^n}+1$. If factoring $m$ helped you factor $m+1$, these numbers would be a cinch, but they're not.
| 21 | https://mathoverflow.net/users/3684 | 55019 | 34,371 |
https://mathoverflow.net/questions/54851 | 54 | For a finite group G, let |G| denote the order of G and write $D(G) = \sum\_{N \triangleleft G} |N|$, the sum of the orders of the normal subgroups. I would like to call G "perfect" if D(G) = 2|G|, since then the cyclic group of order n is perfect if and only if the number n is perfect. But the term "perfect group" is ... | https://mathoverflow.net/users/586 | Is there an odd-order group whose order is the sum of the orders of the proper normal subgroups? | I did a little computer search and I think I found an example of an odd immaculate group.
I searched for groups of the form $G=(C\_q \rtimes C\_p) \times C\_N$ with odd primes $p,q$ such that $p | q-1$ and $N$ an odd integer satisfying $(N,pq)=1$. Using Tom's notations and results, we have
\begin{equation\*}
\frac{... | 46 | https://mathoverflow.net/users/6506 | 55026 | 34,376 |
https://mathoverflow.net/questions/55027 | 9 | **Motivation:** I'm working on a computational problem at the moment, and have some very good routines for natively working with simplicial complexes and calculating homology, but the structures I'm dealing with arise naturally as cubical complexes.
**Problem:** Is there an efficient way to triangulate the n-cube, i.... | https://mathoverflow.net/users/12823 | Triangulating hypercubes | For the question as stated, it's a big open problem to triangulate the $n$-cube, and the papers that you cite are basically the state of the art. The lower bound that you give is simply a matter of comparing the volume of an $n$-cube to the volume of the largest possible $n$-simplex inside it, only with the extra idea ... | 16 | https://mathoverflow.net/users/1450 | 55030 | 34,378 |
https://mathoverflow.net/questions/55002 | 3 | Consider graph $T$ where nodes correspond to maximal cliques of some graph $G$ and two nodes can be connected if corresponding cliques intersect. Clique tree is an example when $T$ is required to be a tree and $G$ is chordal. I'm interested in graphs $T$ when tree/chordal requirements are relaxed, do they come up anywh... | https://mathoverflow.net/users/7655 | Maximal clique intersection graphs | $T$ is called the clique graph of $G$, see
<https://link.springer.com/chapter/10.1007/0-387-22444-0_5>
| 3 | https://mathoverflow.net/users/9156 | 55031 | 34,379 |
https://mathoverflow.net/questions/43430 | 3 | The following question is driving me bananas.
I am given a split extension
$0 \to \mathbb{Z}/n\mathbb{Z} \to (\mathbb{Z}/m\mathbb{Z})\ltimes (\mathbb{Z}/n\mathbb{Z})\to \mathbb{Z}/m\mathbb{Z}\to 0,$
with $m,n>2$ natural numbers, and an element $x\in \mathbb{Z}/n\mathbb{Z}$ satisfying $x^m=1$ but $x^j\neq 1$ fo... | https://mathoverflow.net/users/2051 | Order in $\mathbb{Z}/n^2\mathbb{Z}$ of an $m$th root of unity in $\mathbb{Z}/n\mathbb{Z}$ | Dror Speiser provides a counterexample in a comment:
>
> $x=2$, $n=3511$, $m= \mathrm{ord}\_n(x)=1755= \mathrm{ord}\_{n^2}(x)$. See wikipedia for more on Wiefrich primes and generalizations.
>
| 2 | https://mathoverflow.net/users/2051 | 55032 | 34,380 |
https://mathoverflow.net/questions/55037 | 3 | For a Riemannian manifold $(M,g)$, that is also a complex manifold, when does the Levi-Civita $\nabla\_g$ connection restrict to a connection on the holomorphic forms $\Omega^{(\cdot,0)}$, and when does it restrict to a connection on the anti-holomorphic forms $\Omega^{(0,\cdot)}$?
I would assume there are some suffi... | https://mathoverflow.net/users/1095 | Restriction of the Levi-Civita Connection to a Connection on the (Anti-)Holomorphic Forms | If you assume that $(M,g,J)$ is almost Hermitian, then the Levi-Civita $\nabla$ preserves $\Lambda^{0,p}$ or $\Lambda^{p,0}$ for some $p>0$ iff $J$ is $\nabla$-parallel, i.e. iff $(M,g,J)$ is K\"ahler. This is an easy exercice in Riemannian geometry.
| 4 | https://mathoverflow.net/users/10675 | 55040 | 34,383 |
https://mathoverflow.net/questions/53774 | 3 | Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly.
* I would like to know of expository references and explanations on the concept of "single/multi trace letter partition function" and how it connects to Witten Index and superconformal field th... | https://mathoverflow.net/users/2678 | Witten Index, letter partition function and superconformal representations. | I would recommend you to read [this paper](http://arxiv.org/abs/0801.1435) from 2008. It contains more review materials in it than the one you quoted.
| 3 | https://mathoverflow.net/users/5420 | 55045 | 34,385 |
https://mathoverflow.net/questions/55042 | 20 | I have a probably stupid question on schemes ...
Let $S$ be a scheme, and let $A = \mathsf{Aut}(S)$ be its automorphism group. Does $A$ carry
a scheme structure itself, that is, can one see $A$ as a group scheme ?
Thanks !
| https://mathoverflow.net/users/12884 | Automorphism group of a scheme | The answer is yes when the scheme is flat and projective over the base. This follows from the existence of the Hom scheme, which in turn is proven via the existence of the Hilbert scheme.
A readable reference is Nitin Nitsure's part of the book *Fundamental algebraic geometry*. In particular Theorem 5.23 in his note... | 31 | https://mathoverflow.net/users/1310 | 55049 | 34,388 |
https://mathoverflow.net/questions/55013 | 6 | Context: I am currently reading through the freely available lecture notes from Tristan Riviere ([here](http://www.fim.math.ethz.ch/~riviere/papers/Minicours-Vancouver0709.pdf)) on the applicability of integration by compensation in the analysis of various geometrically motivated PDEs.
I have attempted to find someth... | https://mathoverflow.net/users/11262 | Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant? | I'm too lazy to type-up the proof myself, so I'll send you to a reference.
[Chang, S.-Y. A., Wang, L. and Yang, P. C. (1999), "Regularity of harmonic maps". CPAM](http://www.ams.org/mathscinet-getitem?mr=MR1692152) has the proof in Section 3. Once you get $C^{1,\gamma}$ you immediately get RHS is in $C^\gamma$ and t... | 4 | https://mathoverflow.net/users/3948 | 55055 | 34,393 |
https://mathoverflow.net/questions/55006 | 3 | Let $F$ be the field ${\mathbb Q}(i)\subset \mathbb C$ and let $T\subset F$ be the set of all elements of complex absolute value 1.
Let $n$ be a natural number $\ge 2$ and let $\mu\_n(T)\subset\mathbb C$ be the set of all $n$-th roots of elements of $T$.
Finally, let $E=F(\mu\_n(T))$.
Question: Is the field extensio... | https://mathoverflow.net/users/nan | n-th roots of Pythagorean numbers | Tan (*The group of rational points on the unit circle*, Math. Mag. 69 (1996), 163-171)
proved that the group of rational points on the unit circle modulo torsion is isomorphic to infinitely many copies of $\mathbb Z$.
I have given a couple of references to related articles in *Kreise und Quadrate modulo $p$*, Math. ... | 4 | https://mathoverflow.net/users/3503 | 55056 | 34,394 |
https://mathoverflow.net/questions/55074 | 7 | Let $M$ be a compact manifold and let $f : M \rightarrow M$ be a homeomorphism which is isotopic to the identity. We will say that $f$ can be *fragmented* if it satisfies the following property. Let $\mathcal{U}$ be any open cover of $M$. There then exists homeomorphisms $f\_1,\ldots,f\_n$ from $M$ to itself which are ... | https://mathoverflow.net/users/317 | Fragmenting a homeomorphism of a compact manifold | If I understand you correctly, this is Corollary 1.3. of [Kirby and Edwards, Deformations of spaces of imbeddings, Ann. Math. (2) 93 1971 63–88](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=101825&vfpref=html&r=37&mx-pid=283802). (I couldn't find an online version of the journal article.)
For $k$-para... | 9 | https://mathoverflow.net/users/284 | 55078 | 34,406 |
https://mathoverflow.net/questions/55080 | 3 | I am wondering where to start with questions like:
Given a BM $dX\_t=\mu t+\sigma dB\_t$, having started at $X\_0=0$. What is the probability that $X\_t$ does not hit 0 in the time interval $[a,T]$ where $0\le a\le T$?
Here the hit level can be changed from 0 to any constant $b\gt 0$, or even to a space-time line $... | https://mathoverflow.net/users/12438 | probability question regarding brownian motion | A straightforward approach is to simply integrate the density of $X\_t$ at time $a$ (which will be normally distributed with mean $\mu$ and variance $\sigma^2 a$) against the probability of hitting 0 conditional on the value at time $a$ (which is also known in closed-form). This will give you a messy integral (with an ... | 5 | https://mathoverflow.net/users/1222 | 55081 | 34,408 |
https://mathoverflow.net/questions/55066 | 9 | The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its suspension. The H map has a nice model once one knows about James's theorem that $\Sigma \Omega \Sigma X$ splits as $\bigvee\_i ... | https://mathoverflow.net/users/4991 | Models for P map in EHP sequence | Dear Dev,
You probably know this, but the map $P$ is the Whitehead product map in the metastable range in the following sense: Suppose $X = \Sigma Y$ is a suspension. Then the generalized Whitehead product map $$P:\Sigma Y \wedge Y\to \Sigma Y$$ coincides with map
from the homotopy fiber of $$E: \Sigma Y \to \Omega ... | 5 | https://mathoverflow.net/users/8032 | 55086 | 34,411 |
https://mathoverflow.net/questions/55085 | 30 | I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical... which is Euler's proof of Euclid's theorem which asserts that there exist infinitely many primes. Here is when the factorization $\di... | https://mathoverflow.net/users/10898 | New proofs to major theorems leading to new insights and results? | Here are a few examples from the 19th century.
1. *Unsolvability of the quintic equation*. Abel (1826) proved this by algebraic
ingenuity, but without clarifying the concepts involved. Galois (1830) gave a
proof that introduced the concepts of group, normal subgroup, and solvability
(of groups), thus laying the found... | 54 | https://mathoverflow.net/users/1587 | 55088 | 34,412 |
https://mathoverflow.net/questions/54769 | 12 | The most commonly stated reason for why mathematics should be a required condition for graduating is }to improve problem-solving skills". Usually it's taken for granted that taking a mathematics course does improve one's ability to solve problems. Does anyone know of any studies that either back that up or contradict i... | https://mathoverflow.net/users/619 | Is there evidence whether undergraduate math courses improve problem-solving? | (I don't think my answer directly answers the question, but I'm hoping it would be useful.)
I assume that when you say "problem solving" you mean mathematical "problem-solving as a skill" ("being able to obtain solutions to the problems other people give you to solve," Schoenfeld, 1992).
I was unable to find any st... | 5 | https://mathoverflow.net/users/12357 | 55099 | 34,420 |
https://mathoverflow.net/questions/55092 | 12 | I am wondering, polynomials like
$S\_n^4-6n S\_n^2+3n^2+2n$ for $$S\_n=\sum\_{i=1}^n{X\_i}$$ where $$\mathbb{P}(X\_i=1)=\mathbb{P}(X\_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). While $$B\_t^4-6t B\_t^2+3t^2$$ for Brownian motion $B\_t$ is also a martingale.
Note the difference between... | https://mathoverflow.net/users/12438 | Martingales in both discrete and continuous setting | One knows that $P(S\_n,n)$ is a martingale if and only if $P(s+1,n+1)+P(s-1,n+1)=2P(s,n)$ and
that $Q(B\_t,t)$ is a martingale if and only if $2\partial\_tQ(x,t)+\partial^2\_{xx}Q(x,t)=0$.
Assume that $P(S\_n,n)$ is a martingale and, for a given $d$ and for every $h>0$, let
$$
Q\_h(x,t)=h^{d}P(x/\sqrt{h},t/h),
$$
i... | 21 | https://mathoverflow.net/users/4661 | 55101 | 34,422 |
https://mathoverflow.net/questions/55097 | 9 | Supposed I have an *n*-dimensional manifold *M* with a *k*-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a *k+1*-dimensional manifold *N* and a smooth map $N\to M$ so that the boundary maps diffeomorphically to my submanifold? Can I just take ... | https://mathoverflow.net/users/2467 | Can homologous submanifolds be connected by an immersed manifold with boundary? | As Igor has noted, an obvious obstruction is that your embedded submanifold be null-cobordant.
It seems that you are really asking about the kernel of the realization map $MO\_k(M)\to H\_k(M;\mathbb{Z}\_2)$ (assuming that you don't care about orientations). This appears as the edge homomorphism in the unoriented bord... | 5 | https://mathoverflow.net/users/8103 | 55102 | 34,423 |
https://mathoverflow.net/questions/55059 | 8 |
>
> Let $\mathfrak{p}\_1, \dotsc, \mathfrak{p}\_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x\_0, \dotsc, x\_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's terminology) tells us that there exists a nonconstant homogenous polynomial $f \not\in \cup\_i \mathfrak{p}\_i$. Is the... | https://mathoverflow.net/users/5094 | Prime avoidance in adjacent degrees | The answer is yes. The following is extracted from a preprint of Gabber-Liu-Lorenzini.
>
> Let $B=\oplus\_{n\ge 0}B(n)$ be a graded ring. Let $I=\oplus\_{n\ge 0}I(n)$ be a homogeneous ideal of $B$. Let $\mathfrak p\_1,\dots,\mathfrak p\_r$ be homogeneous prime ideals of $B$ not containing $B(1)$ and not containing... | 6 | https://mathoverflow.net/users/3485 | 55117 | 34,433 |
https://mathoverflow.net/questions/54929 | 7 | Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ or a maximal $P$). While my programs are built on sound theory, one can never be too sure. It's always good practice to c... | https://mathoverflow.net/users/12709 | Multiplication tables for H*(G/P)? | You may consult with Duan Haibao's papers on arxiv.
Also, I have a Maple realization of my own, available at
<http://www.staff.uni-mainz.de/semenov/software.html>
Unfortunately, there is no documentation, and the coding style is ugly, but examples may help.
| 1 | https://mathoverflow.net/users/5107 | 55122 | 34,437 |
https://mathoverflow.net/questions/54731 | 11 | As $\alpha$ and $\gamma$ range uniformly over $[0,1]$, what is the typical (e.g. median or root-mean-square) order of magnitude of $C\_m (\alpha,\gamma)$ := $\sum\_{1 \leq k \leq m} \left( {\rm frac}(k\alpha+\gamma) - \frac12 \right)$ where frac($x$) denotes the fractional part of $x$?
I'd settle for an answer in the... | https://mathoverflow.net/users/3621 | sums of fractional parts of linear functions of n | I think I can show that
$$\sum\_{1 \leq h,k \leq N} \frac{GCD(h,k)^2}{hk}$$
grows linearly. But I get the constant is
$$\sum\_{ GCD(i,j)=1} \frac{1}{\max(i,j) i j}$$
This constant is incredibly close to $3$. (I am omitting the $1/12$, so my $3$ is your $0.25$.) My intuition is that they can't be equal, but they agree ... | 5 | https://mathoverflow.net/users/297 | 55131 | 34,443 |
https://mathoverflow.net/questions/55132 | -1 | Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
| https://mathoverflow.net/users/8725 | eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $ | The three terms commute, whether or not $A$, $B$, and $C$ are symmetric. The eigenvalues are of the form $bc+ac+ab$, where $a$, $b$, and $c$ are eigenvalues of $A$, $B$, and $C$. You just tensor eigenvectors together and you get the answer.
| 5 | https://mathoverflow.net/users/1450 | 55134 | 34,445 |
https://mathoverflow.net/questions/55141 | 14 | Does:
$$\sum\_{1 \leq i<j} \frac{1}{i j^2} = \sum\_{1 \leq k} \frac{1}{k^3}?$$
Motivation: Call the above sum $S$, and let
$$T := \sum\_{ GCD(i,j)=1} \frac{1}{\max(i,j) i j}.$$
The sum $T$ came up in a computation on Jim Propp's question [here](https://mathoverflow.net/questions/54731/sums-of-fractional-parts-of-lin... | https://mathoverflow.net/users/297 | Does this sum equal zeta(3)? | Hi David,
This is the first example of a multiple zeta identity. Your sum S is just $\zeta(1,2)$, where the multiple zeta value is defined by:
$$\zeta(s\_1, s\_2, \ldots, s\_k) = \sum\_{0 < n\_1 < n\_2 < \cdots n\_k} \left( \prod\_{i=1}^k n\_i^{-s\_i} \right).$$
Your identity $\zeta(1,2) = \zeta(3)$ was discovered ... | 26 | https://mathoverflow.net/users/3545 | 55144 | 34,450 |
https://mathoverflow.net/questions/55154 | 1 | Let $F\_2$ denote the free group of rank two and consider the group $G=\langle a,b,c \mathbin | a^2b^2c^2=1\rangle$ which is the fundamental group of the connected sum of three projective planes. Does $G$ have $\mathbb{Z} \times F\_2$ as a subgroup? Thanks!
| https://mathoverflow.net/users/8434 | Does the group $\langle a,b,c \mathbin | a^2b^2c^2=1\rangle $ have $\mathbb{Z} \times F_2$ as a subgroup? | The answer is `no'. No hyperbolic group contains a copy of $\mathbb{Z}^2$. To give some more details, the action of $\Gamma=\pi\_1(3\mathbb{R}P^2)$ on the hyperbolic plane is free, discrete, and every element acts loxodromically. Commuting elements must have a common axis, but $\mathbb{Z}^2$ cannot act freely and discr... | 5 | https://mathoverflow.net/users/1463 | 55155 | 34,453 |
https://mathoverflow.net/questions/9063 | 15 | I've read that quotient singularities (that is, spectra of invariant subrings of finite groups acting linearly on polynomial rings) have rational singularities. Is there an elementary proof of this fact in dimension two? I have one proof, but it uses big guns like Grothendieck spectral sequences etc.
| https://mathoverflow.net/users/460 | Two-dimensional quotient singularities are rational: why? | Let me add a quick proof, also working in characteristic zero, that doesn't rely on classification (this proof is essentially originally due to S\'andor, see his Duke paper on rational singularities). It does rely on canonical modules and a certain functoriality for them. I will use the following characterization of ra... | 10 | https://mathoverflow.net/users/3521 | 55160 | 34,456 |
https://mathoverflow.net/questions/55162 | 14 | It's easy to tell when a polynomial is squarefree (or not): that's just the question of the vanishing of the discriminant, which can be dealt with as the resultant of $f$ and $f'$. However, given a polynomial of degree $2n$ $f$, when is it of the form $g^2$ for $g$ a polynomial of degree $n$?
I've been trying to work... | https://mathoverflow.net/users/622 | How can I write down polynomial relations that define when a polynomial is a square? | Say, for simplicity, you are working over $\mathbb{C}$ or in characteristic zero in general. Then you can guess one of the two values of $g(0)$ (say) and then compute the Taylor series of $\sqrt{f}$. The approach is similar to Hensel lifting: The equation for the first coefficient is non-linear; the equations for the o... | 14 | https://mathoverflow.net/users/1450 | 55163 | 34,457 |
https://mathoverflow.net/questions/55139 | 6 | Given a cubic number field and a basis $\{\gamma\_1,\gamma\_2,\gamma\_3\}$ for it over the rationals, we can write down the norm equation $N(x\_1\gamma\_1+x\_2\gamma\_2+x\_3\gamma\_3)=1$. For almost all substitutions, say $x\_1=c$, the resulting affine cubic curve is an affine part of an elliptic curve.
I was wanderi... | https://mathoverflow.net/users/2024 | When is an affine part of an elliptic curve isomorphic to an affine part of a norm equation? | If you have $C$ a curve of genus one and $P,Q,R$ points on it such that $P+Q \sim 2R, Q+R \sim 2P, P+R \sim 2Q$ (any two implies the third, btw), then $P-Q,R-P,Q-R$ have order three and, if you embed $C$ in the plane by the linear system $P+Q+R$ and choose coordinates in the affine plane such that the (inflectional) ta... | 4 | https://mathoverflow.net/users/2290 | 55170 | 34,462 |
https://mathoverflow.net/questions/55181 | 8 | A rational function is called positive if all its Taylor coefficients are positive.
Friedrichs-Lewy conjecture states the positivity of the rational function
\begin{eqnarray\*}\frac{1}{
(1-x)(1- y)+(1- y)(1-z)+(1-z)(1-x)}
= \sum\limits\_{ k,m,n\ge0}
a\_{k,m, n }x^k y^mz^n. \end{eqnarray\*}
The conjecture was first pr... | https://mathoverflow.net/users/3818 | Positivity of a rational function | The answer is yes.
This was already proved in Gabor Szegö's [original paper](http://www.springerlink.com/content/tgmq242t51414177/) from 1933:
G. Szegö, *Über gewisse Potenzreihen mit lauter positiven Koeffizienten*, Mathematische Zeitschrift, Volume 37, Number 1, 674-688, DOI: 10.1007/BF01474608
The result can b... | 9 | https://mathoverflow.net/users/8176 | 55193 | 34,472 |
https://mathoverflow.net/questions/55182 | 26 | Monads on the category **Set** of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science. We know of a good number of monads on **Set**, but they all appear (at least to me) as isolated examples (other than three big classes of them I'll ... | https://mathoverflow.net/users/2811 | What is known about the category of monads on Set? | I predict that someone such as Steve Lack or Mike Shulman will tell you about the existence of (co)limits in **Mon**, and they'll do it better than I would, so instead I'll address a question in the last paragraph: do $M(0)$, $M(1)$ and $M(0 \to 1)$ tell you much about the rest of $M$? The answer is basically no.
To... | 22 | https://mathoverflow.net/users/586 | 55197 | 34,476 |
https://mathoverflow.net/questions/55152 | 1 | Let a function $f:X \times \mathbb{R} \rightarrow \mathbb{R}$ continuous, with $X \subset \mathbb{R}$ compact, and supose that $\partial\_2 f(x,t)$ exists for all $x \in X$ and is continuous. (here $\partial\_2$ is the derivative with respect to the second coordinate)
I would like to know if $$\displaystyle \lim\_{(x... | https://mathoverflow.net/users/12223 | This limit converges to the partial derivative? | For $t\ne 0$ one has
$${f(x,t)-f(x,0) \over t}- \partial\_2 f(0,0)= \int\_0^1 (\partial\_2 f(x,\tau \thinspace \thinspace t) - \partial\_2 f(0,0))\thinspace d\tau ,$$
and here the right side is $<\epsilon$ when $(x,t)$ is in a suitable neighbourhood of $(0,0)$.
For an $f:X\times {\bf R}^n\to {\bf R}^m$ it is enough t... | 4 | https://mathoverflow.net/users/8050 | 55198 | 34,477 |
https://mathoverflow.net/questions/55180 | 3 | Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ be an intermediate extension, corresponding to the factorization of the normalization map: $Spec(\bar{R})\to Spec{R'}\t... | https://mathoverflow.net/users/2900 | on the relative conductor of curve singularity and quotient of ideals | For 1. You could try looking at various exercises in the Swanson-Huneke book on Integral Closure. There might be something there. In particular, see chapter 12 (titled, the conductor).
For 2. 3. 4. Another way to identify the conductor (or relative conductor) is to consider $\text{Hom}\_{R}(R', R)$. This module alwa... | 6 | https://mathoverflow.net/users/3521 | 55207 | 34,482 |
https://mathoverflow.net/questions/55205 | 9 | Finite simple groups (non-abelian) can generated by two elements.
Let $G=\langle x,y|x^l=y^m=(xy)^n=1,...\rangle$ be a finite simple group (non-abelian), and $\langle x,y|x^p=y^q=(xy)^r=1,...\rangle$ be another presentation of $G$. *(Here, "..." means possibly more relations).*
1) If $(1/l)+(1/m)+(1/n)<1$, then do... | https://mathoverflow.net/users/6761 | Presentations of simple groups |
>
> **No**. The simple group of order 60 is a counterexample to (1) and (3).
>
>
>
If {x,y} is a generating set of G, call its signature 1/|x| + 1/|y| + 1/|xy|. It appears that the signature of most generating sets of non-abelian simple groups are less than 1.
I'll assume that l,m,n are required to be the exac... | 7 | https://mathoverflow.net/users/3710 | 55216 | 34,487 |
https://mathoverflow.net/questions/55214 | 19 | The first place where the amenability problem for Thompson's group $F$ appears in the literature is, I believe, 1980 in a problems article by Ross Geoghegan. I have heard, however, vague comments to the effect that the problem was considered by other people before this. Does anyone have any knowledge about the existenc... | https://mathoverflow.net/users/10774 | Does the amenability problem for Thompson's group $F$ predate 1980? | Richard Thompson visited me at Princeton several times I believe in the mid '70's, gave me copies of some handwritten notes about his groups (which I shared with a few people), and raised the amenability question, which we discussed a bit. I don't think Thompson's groups were very widely known at the time.
| 24 | https://mathoverflow.net/users/9062 | 55217 | 34,488 |
https://mathoverflow.net/questions/55215 | 15 | Let $X$ be a quasi-projective variety over a field $k$. Let $D\_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D\_{perf}$ its full subcategory of perfect complexes.
This question is about Hochschild cohomology in the dg category sense. The question in the title... | https://mathoverflow.net/users/2356 | What is the Hochschild cohomology of the dg category of perfect complexes on a variety? | The answer is yes - at least if you take for the definition of $HH^\*$ the self-ext of the identity functor.
For any quasicompact quasiseparated scheme we know (thanks to Thomason-Trobaugh) that $D\_{qc}(X)$ is compactly generated by the perfect complexes. This means that $D\_{qc}(X)=Ind(D\_{perf}(X))$ -- the quasicom... | 18 | https://mathoverflow.net/users/582 | 55218 | 34,489 |
https://mathoverflow.net/questions/55111 | 8 | Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b\_1\ldots,b\_n$ let $M\_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b\_i,b\_j))$ . Given two scalar products $s,s'$ one can find by PCA a orthonormal basis $B$ of $s$ such that $M\_{s',B}$ is a diagonal... | https://mathoverflow.net/users/3969 | Is there an elementary way to show the triangular inequality for this expression ? | Does the following proof for triangle inequality for the Riemannian metric for posdef matrices help?
(I am currently travelling, so the proof is missing exact references; will add them later when I am near my books.)
The proof that recall below is one of my favorites, and I first saw it in a paper by **R. Bhatia**... | 3 | https://mathoverflow.net/users/8430 | 55220 | 34,491 |
https://mathoverflow.net/questions/45558 | 31 | Let me start with Helly's theorem: Let $A\_1$, $A\_2$, ..., $A\_{n+2}$ be $n+2$ convex subsets of $\mathbb R^n$. If any $n+1$ of these subsets intersect (this means: have nonempty intersection), the so do all $n+2$.
This assertion is, logically speaking, a definite clause: All conditions are of the form "some subsets... | https://mathoverflow.net/users/2530 | The logic of convex sets | This is an interesting question. (Even if there is no finite list of "axioms".)
For example the following is true: Suppose you have a (d+1)-dimensional polytope P. Associate a convex set in $R^d$ to every facet of P, and suppose that every non empty intersection among the facets implies a non empty intersection for ... | 10 | https://mathoverflow.net/users/1532 | 55226 | 34,496 |
https://mathoverflow.net/questions/55241 | 6 | Let $k$ be a field of characteristic zero. Wikipedia states that the natural functor from finite-dimensional formal group laws over $k$ to finite-dimensional Lie algebras over $k$ is an equivalence of categories but does not provide a reference, and I'm not sure where to find one. Does anyone know of one?
(Not really... | https://mathoverflow.net/users/290 | Reference request: equivalence of formal group laws and Lie algebras in characteristic zero | This follows from the [Baker--Campbell--Hausdorf formula](http://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula). Serre's lecture notes on Lie algebras and Lie groups should cover it.
| 10 | https://mathoverflow.net/users/2874 | 55242 | 34,503 |
https://mathoverflow.net/questions/55238 | 22 | There are the following two notions of "Gauss-Manin connection":
1. The complex-analytic one: let $f:X\to S$ be a smooth family of complex manifolds. Then we obtain a local system $R^nf\_{\ast}\mathbb{C}$ of complex vector spaces on $S$, defining a holomorphic vector bundle $\mathcal{V}=R^nf\_{\ast}\mathbb{C}\otimes\... | https://mathoverflow.net/users/12757 | analytic vs. algebraic Gauss-Manin connection | The two constructions are compatible.
Your first definition of the Gauss-Manin connexion is $ DR^{-1} (R f\_\* \mathbb{C}\_X ) $. Here $DR : D^b\_{hr}(\mathcal{D}\_X) \to D^b\_c( \mathbb{C}\_X )$ and $DR(\mathcal{M}) = \omega\_X \otimes^L\_{D\_X} \mathcal{M}$ is the analytic de Rham complex. This is an equivalence b... | 12 | https://mathoverflow.net/users/1985 | 55247 | 34,506 |
https://mathoverflow.net/questions/55237 | 3 | In the chapter on Gröbner bases from Eisenbud's "Commutative Algebra" the following statement appears as Proposition 15.15 (page 344):
*Let $F$ be a free $S$ module with basis and monomial order compatible with a given monomial order on $S$. If $M\subset F$ is any submodule and* $h\_{1},\cdots,h\_{u} \in S$ *are such... | https://mathoverflow.net/users/66825 | Is the first part of Eisenbud's Proposition 15.15's proof o.k? | Indeed, $h(f-in(f))\in M$ does not hold in general. As an example, take $S=F=k[x,y], M=\langle x+y\rangle, h=y$ and any monomial order. Then $h \cdot (x+y) \in M$, but neither $hx$ nor $hy$ is in $M$.
The proof is easily fixed, though. After ".., so by our hypothesis, $in(f) \in in(M)$" continue as follows. Let $m \i... | 5 | https://mathoverflow.net/users/3380 | 55248 | 34,507 |
https://mathoverflow.net/questions/55236 | 17 | Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H\_{\alpha}(X)$ where $\alpha$ is an ordinal? The obvious ways of trying to define an $H\_\omega(X)$ don't work out very well essentially because $S^\infty$ is... | https://mathoverflow.net/users/9417 | Ordinal-indexed homology theory? | I'm not sure whether this is what you are after, but this paper, [Semi-infinite cycles in Floer theory: Viterbo's Theorem](http://arxiv.org/pdf/0911.3714.pdf), by Max Lipyanskiy, develops a theory of ($\omega$/2+k)-dimensional cycles in an $\omega$-dimensional manifold with a choice of polarization of its tangent space... | 4 | https://mathoverflow.net/users/284 | 55253 | 34,511 |
https://mathoverflow.net/questions/55252 | 5 | Let $K$ be a field, and $A$ a finitely generated associative algebra over $K$. We suppose that $A$ has a unit and that every element $x$ of $A$ is annihilated by a non-zero polynomial $P\_x$ depending on $x$. Is $A$ a finite-dimensional vector space over $K$ ?
Under the additional assumption that there is an integer ... | https://mathoverflow.net/users/12806 | Finitely generated algebra in which every element is annihilated by a non-zero polynomial | This is the Kurosh problem, which has a negative solution. If I recall correctly, one exhibits an example using the Golod-Shafarevich lemma.
[Wikipedia](http://en.wikipedia.org/wiki/Kurosh_problem) has a page on this, in fact. The example was constructed by Golod.
| 9 | https://mathoverflow.net/users/1409 | 55264 | 34,518 |
https://mathoverflow.net/questions/55260 | 8 | Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi\_1(X) \to \pi\_1(Y)$ is an isomorphism, and $H\_n(f) : H\_n(X, G) \to H\_n(Y, G)$ is an isomorphism for all $n \ge 1$ and for any local system of coefficients $G$, then $X$ is weakly equivalent to $Y$. Does anyone have a referenc... | https://mathoverflow.net/users/4239 | Reference needed: Isomorphism on pi_1 and homology gives weak equivalence | You need to assume either that the spaces involved are simple (I believe Emmanuel Dror-Farjoun generalized that to nilpotent), or that the map f induces an isomorphism in homology with local coefficients. There is an exercise in Hatcher's book that discuss this, in Section 4.2 (Ex. 12). You should also look at Peter Ma... | 8 | https://mathoverflow.net/users/4042 | 55266 | 34,520 |
https://mathoverflow.net/questions/55256 | 6 | Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about the elementary theory of non-abelian free semigroups? I know that non-abelian free groups of finite rank have the same el... | https://mathoverflow.net/users/8434 | need references regarding the elementary theory of free semigroup and free abelian groups | Two finitely generated free semigroups of different ranks have different elementary theories. Indeed, the set of elements $x$ such that $\forall z,t \neg(x=zt)$ is exactly the set of generators of the free semigroup. So the rank of a finitely generated free semigroup is elementary definable.
For the free Abelian gro... | 6 | https://mathoverflow.net/users/nan | 55269 | 34,522 |
https://mathoverflow.net/questions/55271 | 7 | Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric realisation of this gets me the correct answer.
Now suppose I'm trying to compute homotopy colimits in some other model categor... | https://mathoverflow.net/users/1202 | When does a cosimplicial object compute homotopy colimits? | Dear Saul,
The answer to your question is the subject of chapters 16-19 of Phil Hirschhorn's book *Model Categories and their Localizations*.
To write out the answer in the general case would be prohibitively time consuming, but I'll write a little bit out.
**Definition 19.1.5** Let $M$ be a framed model categor... | 5 | https://mathoverflow.net/users/1353 | 55276 | 34,524 |
https://mathoverflow.net/questions/55244 | 40 | On the Wikipedia page1 about algebraic varieties <https://en.wikipedia.org/wiki/Algebraic_variety>, a sentence reads as follows:
[[A more significant modification is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if allowed in coordinate rings aren't seen as coordinate functions.
... | https://mathoverflow.net/users/1992 | Why must nilpotent elements be allowed in modern algebraic geometry? | I'm a bit confused by the quoted wikipedia entry, because the category of reduced rings also has coproducts (take the tensor product, and then pass to the quotient by the nilradical), and hence the category of reduced schemes, the category of varieties over a field, and so on, all admit fibre products. [Added: See Jim'... | 53 | https://mathoverflow.net/users/2874 | 55279 | 34,526 |
https://mathoverflow.net/questions/55288 | 28 | In the days before [W, TW, BCDT], how did people show that specific elliptic curves over $\mathbb{Q}$ were modular? For instance, I was reading through a paper of Buhler, Gross and Zagier from 1985 on the curve 5077a, and they say that modularity can be checked by a finite computation in the 422-dimensional space of cu... | https://mathoverflow.net/users/2698 | How to show modularity of an elliptic curve? | They explicitly computed quotients of $X\_0(N)$ and identified them with elliptic curves.
Suppose you can compute the space $S\_2(\Gamma\_0(N),\mathbf{C})$ of modular forms. An (isogeny class of) elliptic curves of conductor $N$ corresponds (by modularity) to a normalized new Hecke eigenform with coefficients in $\math... | 26 | https://mathoverflow.net/users/nan | 55292 | 34,535 |
https://mathoverflow.net/questions/55303 | 5 | The theory of sumsets $A+B$ where $A$ and $B$ are finite subsets of an additive group $Z$ is extensively studied in additive combinatorics: finding long arithmetic progressions inside them, finding lots of subsets of this form, bounding its size above and below, and so on.
A fairly natural inverse question is the fol... | https://mathoverflow.net/users/385 | How large can a non-sumset be? | As far as I know, the only result of this sort is due to Alon, found [here](http://www.tau.ac.il/~nogaa/PDFS/sumset.pdf).
| 7 | https://mathoverflow.net/users/9924 | 55305 | 34,537 |
https://mathoverflow.net/questions/55289 | 10 | Would the [Picard–Lindelöf theorem](https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem#Picard.E2.80.93Lindel.C3.B6f_theorem) still be true if the requirement that f be Lipschitz continuous in y was replaced with the requirement that f be [almost Lipschitz](https://en.wikipedia.org/wiki/Modulus_of_contin... | https://mathoverflow.net/users/nan | Existence/Uniqueness of solutions to quasi-Lipschitz ODEs | Yes. This follows from the classical uniqueness theorem due to Osgood (the original [paper](https://doi.org/10.1007/BF01707876 "Osgood, W.F. Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x,y) ohne Hinzunahme der Cauchy-Lipschitz'schen Bedingung. Monatsh. f. Mathematik und Physik 9, 331–345 (1898)... | 21 | https://mathoverflow.net/users/5371 | 55310 | 34,540 |
https://mathoverflow.net/questions/55297 | 22 | I am currently teaching an advanced undergraduate analysis class, and the following question came up.
Intuition suggests that "most" subsets of $[0,1]$ are not Lebesgue measurable. However, the power set $\mathcal{P}([0,1])$ has the same cardinality as the collection of measurable sets, so it is not clear how to make... | https://mathoverflow.net/users/6514 | Codimension of Measurable Sets | In 1917, Lusin and Sierpinski showed that the unit interval $[0,1]$ can be partitioned into $2^{\aleph\_{0}}$ many pairwise disjoint sets each having Lebesgue outer measure 1; say, $X\_{i}$, $i \in I$. Fix $i\_{0} \in I$.
For each proper subset $J$ with $i\_{0} \notin J \subset I$, let $S\_{J} = \bigcup\_{j \in J} X\_{... | 14 | https://mathoverflow.net/users/4706 | 55316 | 34,544 |
https://mathoverflow.net/questions/55262 | 28 | If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the universal properties of these three constructions, we get a universal property of $T(X)$, namely: Defining $U[\epsilon] ... | https://mathoverflow.net/users/2841 | Universal property of the tangent bundle | The definition that Martin mentions comes close to the definition of a tangent vector which I learnt as an undergraduate.
"Definition: A geometric tangent vector is an equivalence class of germs of smooth maps $\mathbb{R} \to M$ at $0$, where two germs are equivalent
if their first order jets at $0$ agree."
More g... | 20 | https://mathoverflow.net/users/9928 | 55322 | 34,549 |
https://mathoverflow.net/questions/55323 | 10 | It is well known that the plane minus $n$ points is homotopy equivalent to a wedge of circles and hence its fundamental group is free on $n$ letters.
Question: Is the plane minus an infinite sequence of points having no limit point
homotopy equivalent to an infinite wedge circles?
I'm pretty sure that this could f... | https://mathoverflow.net/users/6254 | Homotopy type of the plane minus a sequence with no limit points | Yes. More generally, if $X$ is a proper closed subset of $\mathbb{R}^2$, then every path component $M$ of $\mathbb{R}^2 \setminus X$ is homotopy equivalent to a wedge of circles (observe that $X$ might be something like a Cantor set, which makes this a little more surprising). First, $M$ is a noncompact $2$-manifold, s... | 17 | https://mathoverflow.net/users/317 | 55325 | 34,550 |
https://mathoverflow.net/questions/55321 | 3 | Let $G$ be a connected linear algebraic group over an algebraically
closed field $k$ of characteristic $p$. An element $x\in G$ is
called *regular* if its centralizer has minimal dimension among
all the elements of $G$. Suppose now that $G$ is connected simple,
let $u\in G$ be a regular unipotent element, and let $U$ b... | https://mathoverflow.net/users/2381 | Connectedness of centralizers and regular elements in unipotent groups | Suppose that $G$ is semisimple and that the characteristic is *very good* for $G$. This means that the characteristic is good, and doesn't divide $n$ if $A\_{n-1}$ is an irreducible
component of the root system of $G$. [I'll ignore the issue of whether or not "good but not very good" primes present any real issue -- an... | 6 | https://mathoverflow.net/users/4653 | 55338 | 34,559 |
https://mathoverflow.net/questions/55334 | 1 | What techniques are out there to calculate the cohomology groups of the structure sheaf $\mathcal{O}\_X$ of a smooth quasi-projective variety $X$?
For example can we conclude something from the dimension of the complement $Z = \bar{X} \X$, where $\bar{X}$ is smooth and projective. I know I could use Hodge theory to c... | https://mathoverflow.net/users/11392 | Cohomology of Structure sheaves | Not knowing anything else about your situation, the approach that comes to mind is to use
the long exact sequence:
$$\cdots \to H^i\_Z(\overline{X},\mathcal O\_{\overline{X}}) \to H^i(\overline{X},
\mathcal O\_{\overline{X}}) \to H^i(X,\mathcal O\_X) \to \cdots $$
(described somewhere in a Hartshorne exercise). This re... | 6 | https://mathoverflow.net/users/2874 | 55339 | 34,560 |
https://mathoverflow.net/questions/55333 | 6 | Take scheme morphism $f: X\to Y$ and suppose $f$ surjective. If $y \in Y$ can one find affine open $V \subset Y$ containing $y$ and affine open $U \subset X$ such $f(U) = V$ ?
Thank you.
Later: Very good answer of Kevin shows it is not true. Is there hypothese which make it true ?
For example $X$ irreducible and/or ... | https://mathoverflow.net/users/10408 | Surjective implies local affine surjective? | If $f$ is open (e.g. $f$ finite type and flat over noetherian $Y$), then your condition is trivially satisfied: let $V'$ be any affine open neighborhood of $y$ and let $U'$ be an affine open subset of $X$ such that $y\in f(U')\subseteq V'$. Take a principal open subset $V'\_h$ such that $y\in V'\_h\subseteq f(U')$, the... | 7 | https://mathoverflow.net/users/3485 | 55346 | 34,565 |
https://mathoverflow.net/questions/55341 | 4 | Hello everybody,
I would like to say, before stating the question, that I'm not a mathematician, and therefore i apologize in advance if the question is trivial, or well-known. I'll try to state it as precisely as I can.
Let us define a well founded countably branching tree, as a set
$T\subset {\mathbb{N}}^{\*}$,... | https://mathoverflow.net/users/11618 | Well ordering of countably branching well founded trees | I have several observations:
* There is a very commonly used ranking function for
well-founded trees, defined so that the rank $\rho(x)$ of
a node $x$ in the tree is the supremum of $\rho(y)+1$ for
all children $y$ of $x$. This is well-defined exactly because the tree order is well-founded. The rank of the tree itsel... | 8 | https://mathoverflow.net/users/1946 | 55349 | 34,567 |
https://mathoverflow.net/questions/55350 | 11 | Let $G$ be a finite group and $k$ a field, let us assume that char($k$) divides the group order. Let $kG$-mod denote the category of fintely generated $kG$-modules.
This category has as a tensor product $\otimes\_{k}$ with diagonal $G$-action.
Given now $M,N\in kG$-mod such that $M\otimes\_{k}N$ is projective, can we t... | https://mathoverflow.net/users/12962 | If the tensor product of two $kG$-modules is projeсtive, does either of them have to be projective? | The answer to the first question is no. Let $G$ be the group $C\_2 \times C\_2$ and let $k$ be a field of characteristic 2. Then $kG \cong k[x,y]/(x^2, y^2)$. Let $M = k[x]/(x^2)$ with $y$ acting trivially, and let $N = k[y]/(y^2)$ with $x$ acting trivially. Then neither $M$ nor $N$ is projective (for modules over a $p... | 10 | https://mathoverflow.net/users/4194 | 55352 | 34,568 |
https://mathoverflow.net/questions/55351 | 1 | Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x\_1, x\_2^\*, \ldots, x\_n^\*) = L(x\_1)$$
for some fixed values of $x^\*\_2, \ldots, x\_n^\*$?
If this seems too difficult, any insight on how t... | https://mathoverflow.net/users/nan | Extending linear operators to multi-linear ones | Following Kate's suggestion, I'm posting my previous comment as an answer. Choose any non-zero linear functional $f$ on $X$ and any vector $y\in X$ such that $f(y)=1$. Define $\mathcal L(x\_1,x\_2,\dots,x\_n)$
to be the product $L(x\_1)f(x\_2)\dots f(x\_n)$ and take all of
$x\_2^\*,\dots x\_n^\*$ to be $y$.
| 1 | https://mathoverflow.net/users/6794 | 55363 | 34,576 |
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