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https://mathoverflow.net/questions/54152 | 12 | Jon Hanke and I were just chatting and realized we didn't know the answer to the following question. If E is an elliptic curve over a number field, is there in any sense a "canonical height" on the Weil-Chatelet group H^1(G\_Q,E)? (Note that we have no clearly articulated criteria for what "canoncial" means or what kin... | https://mathoverflow.net/users/431 | Is there a canonical height on the Weil-Chatelet group? | In my opinion, instead of a "height" on the Weil-Chatelet group, one should consider a "depth", using the local duality between the points on an elliptic curve and the elements of the Weil-Chatelet group. Working over $Q$ for simplicity, there is a Pontrjagin duality between locally compact abelian groups:
$$WC\_p \tim... | 11 | https://mathoverflow.net/users/3545 | 54159 | 33,832 |
https://mathoverflow.net/questions/54161 | 6 | I'm looking for a reference for the following standard result:
Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic representation of $U$ has a fixed point.
Statements of Engel's theorem for the analogous statement about Lie algebras seem to be ... | https://mathoverflow.net/users/7868 | Reference request: representations of unipotent groups have a fixed point. | Theorem 17.5 in Humphreys's *Linear Algebraic Groups* seems to be the result you want. (Also, doesn't the result for solvable groups only imply the corresponding result for *connected* unipotent groups? The proof for unipotent groups doesn't require connectedness.)
| 6 | https://mathoverflow.net/users/396 | 54164 | 33,836 |
https://mathoverflow.net/questions/49647 | 19 | Consider the following question:
>
> 1) For a given natural number $a$, are there finitely or infinitely many natural numbers that are not of the form $anm \pm n\pm m$, where $m$ and $n$ range over positive integers? (For $a=1$ or $a=2$ you have all the natural numbers.)
>
>
>
Does this problem appear in the l... | https://mathoverflow.net/users/14726 | Is this a (well known) open problem?(infinitness and more on $anm \pm n\pm m$ ) | $\newcommand\Z{\mathbf{Z}}$
$\newcommand\Q{\mathbf{Q}}$
(Caveat: normally I wouldn't answer a question with such a limited knowledge of the general theory, but classical analytic number theory seems not so well represented by active MO members.)
Suppose that $A$ is a finite abelian group. Then I claim that given an... | 22 | https://mathoverflow.net/users/nan | 54170 | 33,842 |
https://mathoverflow.net/questions/54153 | 7 | Given a 3-manifold presented as a triangulation, a Heegaard splitting, or a Dehn surgery, what is the computational cost of converting to the other two presentations? I would like Heegaard splittings to be given as a word in the standard Dehn twist generators, and Dehn surgery as a word in the standard braid group gene... | https://mathoverflow.net/users/12695 | Computational cost of converting between 3-manifold presentations | I believe all these translations are in principle easy. The challenge is in implementing them cleanly and efficiently; the translations can be annoying and confusing.
As you describe, to go from a Heegaard splitting
to a triangulation, it's just a matter of a sequence of Pachner moves. If you allow (as is
usually sen... | 8 | https://mathoverflow.net/users/9062 | 54171 | 33,843 |
https://mathoverflow.net/questions/54175 | 23 | Every finite-dimensional vector space is isomorphic to its dual.
However for an infinite-dimensional vector space $E$ over a field $K$ this is always false since its dual $E^\ast$ is a vector space of strictly larger dimension: $dim\_KE \lt dim\_K E^\ast $ (dimensions are cardinals of course). This is a non-trivial st... | https://mathoverflow.net/users/450 | Which vector spaces are duals ? | The $\mathbf{R}$-vector space $\mathbf{R}^{(\mathbf{R})}$ has dimension $\operatorname{card} \mathbf{R}$ by definition, so it is isomorphic to $\mathbf{R}^{\mathbf{N}}$ (by the Erdős-Kaplansky theorem and because $\operatorname{card}(\mathbf{R}^{\mathbf{N}}) = \operatorname{card} \mathbf{R}$). So $\mathbf{R}^{(\mathbf{... | 15 | https://mathoverflow.net/users/6506 | 54181 | 33,849 |
https://mathoverflow.net/questions/54186 | 21 | I am starting to study noncommutative geometry and ${\rm C}^\*$ algebras so my question is:
**Does anyone know a good reference on this subject?**
I would like a basic book with intuitions for definitions and this kind of things. I come from algebraic geometry, so if the book talks a bit about the relation with alg... | https://mathoverflow.net/users/12204 | Reference: Learning noncommutative geometry and C^* algebras | First of all, let me mention that functional analysis plays a similar role in noncommutative geometry that commutative algebra plays in algebraic geometry, and it pays off to at least have a reference handy. To that end, I recommend "Banach Algebra Techniques in Operator Theory" by Ronald Douglas: it develops the essen... | 16 | https://mathoverflow.net/users/4362 | 54189 | 33,854 |
https://mathoverflow.net/questions/54180 | 1 | By a tensor category, I mean here a cocomplete $k$-linear symmetric tensor category, where $k$ is a fixed ground ring. Tensor functors are assumed to be $k$-linear and cocontinuous. I will denote the unit object as $\mathcal{O}$.
Terminology: Let $C$ be a tensor category, $\mathcal{A}$ an algebra in $C$ (i.e. an obje... | https://mathoverflow.net/users/2841 | universal property of module categories internal to a tensor category | Your functor is an equivalence. You can explicitly describe an inverse as follows: given a pair $(F, \sigma)$ as you describe, define $H: Mod\_{ \mathcal{A} } \rightarrow D$ by the formula $H(M) = F(M) \otimes\_{ F( \mathcal{A} ) } \mathcal{O}\_{D}$.
| 6 | https://mathoverflow.net/users/7721 | 54194 | 33,858 |
https://mathoverflow.net/questions/54065 | 4 | Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?
| https://mathoverflow.net/users/5292 | About injective hull | I'll follow up on what Karl said with an example closer to my own experience. Let $\mathbb{Z}$ be the ring of integers and $p$ a positive prime. Then $\mathbb{Z}/p\mathbb{Z}$ is injective as a $\mathbb{Z}/p\mathbb{Z}$ - module, being a vector space over a field, whence $\mathbb{Z}/p\mathbb{Z}$ is its own injective enve... | 2 | https://mathoverflow.net/users/8027 | 54200 | 33,861 |
https://mathoverflow.net/questions/54203 | 1 | Hi!
Perhaps it is an easy question but i don't figure out how to prove it.
Let $a\_1,...,a\_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer part of $a$) consider the matrix $A(i,j)=(a\_{i+k+l})\_{0\leq k,l\leq j}$. I denote with $D(i,j)=\det(A(i,j))$. I ha... | https://mathoverflow.net/users/4971 | Question on a relation between minors of a particular kind of matrix | This must be a consequence od the Dodgson's condensation formula. See Exercise 24 on my web [site](http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf). By the way, C. L. Dodgson was the real name of Lewis Caroll (see the MO question [link text](https://mathoverflow.net/questions/45185).
| 2 | https://mathoverflow.net/users/8799 | 54210 | 33,867 |
https://mathoverflow.net/questions/54193 | 33 | Consider graphs on $n$ nodes. I am trying to find a graph $G$ that contains all $n$-node trees as sub-graphs but contains as few edges as possible. The complete graph $K\_n$ suffices, but can we get by with fewer edges? Maybe $O(n)$ edges?
(This problem arose in the context of circuit design, where edges in $G$ corre... | https://mathoverflow.net/users/8938 | Graph containing all trees? | See Chung and Graham, [On Universal Graphs for Spanning Trees](http://www.math.ucsd.edu/~ronspubs/83_06_universal_trees.pdf). They prove that the number of edges is $\Theta(n\log n)$.
| 34 | https://mathoverflow.net/users/440 | 54211 | 33,868 |
https://mathoverflow.net/questions/54197 | 115 | The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is so r... | https://mathoverflow.net/users/12420 | Why is the Hodge Conjecture so important? | Let $K$ be one of the following fields: the complex numbers, a finite field, a number field (and we could amalgamate the last two into the more general case of a field finitely generated overs its prime subfield).
In each case we can consider the category of smooth projective varieties over $K$, with morphisms being ... | 182 | https://mathoverflow.net/users/2874 | 54218 | 33,871 |
https://mathoverflow.net/questions/54213 | 6 | If $K\_n$ is the field $\mathbb{Q}\_p(\mu\_{p^n})$, then it's easy to see that the relative different $\mathcal{D}(K\_n / K\_{n-1})$ is $(p)$ for all $n \ge 2$.
What happens if I take an arbitrary, probably totally ramified, finite extension $L/\mathbb{Q}\_p$ and look at the tower $L\_n = LK\_n$? It's clear that $\m... | https://mathoverflow.net/users/12706 | What is the different in the cyclotomic tower over a finite ramified extension of Qp? | Take $K=Q\_p$ and $L/K$ finite. It is known that the sequence $\{ p^n v\_p( \mathcal{D}(L\_n/K\_n) ) \}\_n$ is eventually constant (it's basically the valuation of the different of the extension $E\_L/E\_K$ which you get from the theory of the field of norms). This and the transitivity of the different should allow you... | 6 | https://mathoverflow.net/users/5743 | 54220 | 33,872 |
https://mathoverflow.net/questions/54216 | 0 | I have a property which is local and stable for faithfully flat base change over a base scheme $S$. So I need to prove it for $O\_{S,s}$ with $s\in S$.
Why if I can prove it for a local artinian ring then this give to me the statement for $O\_{S,s}$?
| https://mathoverflow.net/users/12198 | local statement | The implication is not true in general, if I understand you correctly. For example, consider the property "affine" for morphism of schemes. This property is fpqc locally on the target.
Say, you have a morphism of schemes $f:X \to Y$ such that for every artinian ring $A$ and morphism $\text{Spec}(A) \to Y$ the base ch... | 2 | https://mathoverflow.net/users/4101 | 54226 | 33,875 |
https://mathoverflow.net/questions/54223 | 3 | What is the source of the $k$-tuple conjecture, that every integer tuple $(k\_1,\ldots,k\_n)$ either contains all members of a congruence class mod a prime or has infinitely many primes amongst $(k\_1+c,\ldots,k\_n+c)\_{c\in\mathbb{N}}$? Of course there is also an expected density, so perhaps the forgoing is the weak f... | https://mathoverflow.net/users/6043 | Whence the k-tuple conjecture? | Quoting from "Linear Equations in Primes" by Green and Tao (see here for a preprint <http://arxiv.org/abs/math/0606088>):
"The name of Dickson is sometimes associated to this circle of ideas. In the
1904 paper [12], he noted the obvious necessary condition on the $a\_i$, $b\_i$ in order that the forms $(a\_1 n + b\_1... | 4 | https://mathoverflow.net/users/nan | 54228 | 33,877 |
https://mathoverflow.net/questions/54224 | 5 | Let $Z = \mathrm{Proj}\,k[x\_{0},x\_{1},\ldots,x\_{r}]/f$ be a closed subscheme of degree $d$, i.e., $f$ is a homogeneous polynomial of degree $d$, and $\mathcal{O}\_{Z}(1)=i^{\*}\mathcal{O}\_\mathbb{P}(1)$. Could someone please help me compute all the cohomology groups? I took the short exact sequencence given by the ... | https://mathoverflow.net/users/12468 | How to compute cohomology groups of a closed subscheme Z of projective space, defined by a homogeneous polynomial of degree d? | If you are just looking for cohomology of the structure sheaf (or of the twists $\mathcal{O}\_X(n)$), then it should be pretty easy.
You have the following short exact sequence.
$$0 \to O\_{\mathbb P} (-d + n) \to O\_{\mathbb P}(n) \to O\_Z(n) \to 0 $$
Take cohomology. Because $\mathbb{P}$ is projective space (of w... | 7 | https://mathoverflow.net/users/3521 | 54230 | 33,878 |
https://mathoverflow.net/questions/54232 | 67 | Writing a book from the beginning to the end is (so I heard) a very hard process. Planning a book is easier. This question is dual in a sense to the question "[Books you would like to read (if somebody would just write them](https://mathoverflow.net/questions/53036/books-you-would-like-to-read-if-somebody-would-just-wr... | https://mathoverflow.net/users/1532 | A book you would like to write | Gosh, what a question, Gil. What is your answer?
I have written many books in my head, but I am much too lazy actually to write a book. I guess my first choice would be
Geometric nonlinear functional analysis, volume II
and my third choice
Geometric nonlinear functional analysis, volume III
neither of which ... | 21 | https://mathoverflow.net/users/2554 | 54233 | 33,879 |
https://mathoverflow.net/questions/54221 | 16 | Let $j$ be the Klein $j$-invariant (from the theory of modular functions).
Does $j$ satisfy a differential equation of the form $j^\prime (z) = f(j(z),z)$ for
any rational function $f$?
| https://mathoverflow.net/users/12669 | does the j-invariant satisfy a rational differential equation? | No. Conceptually, the reason is that $j'(z)$ is a weakly holomorphic (= holomorphic except at the cusp at infinity, where it has a pole) modular form of weight $2$, so it cannot be expressed in terms of $j$ (weakly holomorphic modular form of weight $0$) and $z$ (not anywhere near being a modular form).
For a rigorou... | 40 | https://mathoverflow.net/users/422 | 54235 | 33,881 |
https://mathoverflow.net/questions/54237 | 36 | If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles.
For example, when $M=\mathbb{Q}$ is the trivial motive, $L(\mathbb{Q},s)$ is the Riemann Zeta function $\zeta(s)$.
There is a famous concjecture ... | https://mathoverflow.net/users/9317 | Special values of L-functions as periods | In the case $M$ is the spectrum of a number field (so that $L(M,s)$ is the Dedekind zeta function associated to the number field), it is known thanks to Borel's theorem that all non-critical values $L(M,n)$ are indeed periods. EDIT : I should add that it is very easy to prove that $\zeta(n)$ is a period for every $n \g... | 33 | https://mathoverflow.net/users/6506 | 54244 | 33,886 |
https://mathoverflow.net/questions/54239 | 41 | I want to start out by giving two examples:
1) [Graham's problem](http://en.wikipedia.org/wiki/Graham%2527s_number#Graham.27s_problem) is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a planar $K\_4$ colored with just one color. Graham's r... | https://mathoverflow.net/users/8176 | Why is "P vs. NP" necessarily relevant? | The $P \ne NP$ problem is the best way we know to formulate the belief (which was expressed even before the problem was formally stated) that certain specific algorithmic problems (such as finding a Hamiltonian cycle in a graph) requires exponential number of steps as a function of their description. The formulation is... | 26 | https://mathoverflow.net/users/1532 | 54253 | 33,893 |
https://mathoverflow.net/questions/54243 | 5 | The following graph property has come up naturally in some work I've been doing, and it seems like something that may have already been studied.
Namely, let $G$ be a graph with no loops or double edges, such that given any two edges $e, e'$, there exist triangles $T\_1, ..., T\_n$ in $G$ such that $e\in T\_1, e'\in ... | https://mathoverflow.net/users/6950 | Does this type of graph have a name? | The condition where every two vertices can be connected through a sequence of triangles sharing edges is called "strongly triangulated". The graph you ask about has a strongly triangulated line graph, which is stronger than being strongly triangulated. Another weaker property I've seen is $(3,4)$-connectivity where any... | 6 | https://mathoverflow.net/users/2384 | 54257 | 33,896 |
https://mathoverflow.net/questions/54270 | 0 | Hi folks,
I plugged the following set of numbers into Excel:
[600 470 170 430 300]
I found the mean (394), summed the numbers, divided by 5 to get variance (21704). I then did the square root to find stdev (147.32).
Using Excel or Wolfram Alpha's variance and stdev function give me differnt numbers, variance 27... | https://mathoverflow.net/users/12714 | Why does Excel/Wolfram come up with differnt stdev than I do? | The unbiased variance estimator normalizes by $n-1$, instead of $n$. This may be the difference you're seeing.
| 0 | https://mathoverflow.net/users/12710 | 54272 | 33,905 |
https://mathoverflow.net/questions/54255 | 8 | The following question came to me earlier as a "side question"; something I'd like to know, but which is not totally necessary for what I'm thinking about or doing:
Consider the genus 32 curve $X\_0(389)$, and denote its Jacobian variety as $J\_0(389)$.
I am interested in finding an upper bound for the Mordell-Weil... | https://mathoverflow.net/users/5744 | Upper bounds for ranks of modular jacobians | The rank of J0(389) over Q is 13.
The Jacobian splits into factors of dimensions 1,2,3,6,20. Combining explicit computation with Kolyvagin-Gross-zagier implies that the factors of dim 20 has rank 0. The ones of dim 2,3,6 have ranks 2,3,6. The one of dim 1 has rank 2. (this is all from memory, so....)
Similar techniques... | 17 | https://mathoverflow.net/users/8441 | 54273 | 33,906 |
https://mathoverflow.net/questions/52508 | 15 | **Background**
The *Quillen model structure* on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by the lifting property, but in the end they are those inclusions which are built up by cell attachments (or are retr... | https://mathoverflow.net/users/8032 | "Strøm-type" model structure on chain complexes? | There are several other papers, I think earlier ones, that cover this.
[32] M. Cole. The homotopy category of chain complexes is a homotopy category. Preprint (1990's)
[29] J. Daniel Christensen and Mark Hovey. Quillen model structures for relative homological
algebra. Math. Proc. Cambridge Philos. Soc., 133(2):2... | 11 | https://mathoverflow.net/users/14447 | 54279 | 33,909 |
https://mathoverflow.net/questions/51091 | 20 | Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to the way we build homotopies, lifts, etc. combinatorially in simplicial homotopy theory, but for some reason I never really... | https://mathoverflow.net/users/1353 | Computing homotopies | Sometimes easy geometric pictures have awkward seeming algebraic descriptions.
On pages 6 and 7 of *Concise*, I gave examples where I both gave a geometric picture
and explicit formulas to make the idea of such translation clear. In other cases,
(as in cofiber homotopy equivalence) I just found it quick and easy to wri... | 19 | https://mathoverflow.net/users/14447 | 54280 | 33,910 |
https://mathoverflow.net/questions/54277 | 5 | Given symmetric monoidal closed categories $C, D$, a symmetric monoidal (not closed!) functor $F:C\to D$ factors into two parts:
* the first is a symmetric monoidal **closed** functor from $C$ to a "halfway house" $C'$, followed by
* a symmetric monoidal functor from $C'$ to $D$.
To get $C'$, you do two changes of... | https://mathoverflow.net/users/756 | Do plain functors out of monoidal categories factor into a nontrivial monoidal part and a plain part? | Regarding the specific example, the construction of $C'$ can be tightened up as follows. The
first functor, from $C$ to $C'$ is not just hom-preserving (closed) but bijective on objects. The second functor, from $C'$ to $D$, has the property that it is fully faithful on maps out of the monoidal unit.
(This is enoug... | 5 | https://mathoverflow.net/users/10862 | 54285 | 33,913 |
https://mathoverflow.net/questions/54268 | 3 | * Let $X$ be some uncountable standard Borel space (e.g., the real line).
* Let $D$ be the set of Borel probability measures on $X$.
* Let $M$ be the set of signed Borel measures on $X$
* Now let $p\_1,...,p\_N$ be a finite sequence of linearly independent probability measures in $D$.
* Let $A$ be the set of all possib... | https://mathoverflow.net/users/12713 | The space of probability measures and its intersection with hyperplanes in the space of measures | Something is wrong in the statement of the question. If $X$ has four points, then a measure on $X$ is a list of four non-negative numbers that sum to 1. But if you take the plane subtended by $(1,0,1,0)$, $(1,0,0,1)$, and $(0,1,1,0)$, then it also contains $(0,1,0,1)$, and these four measures are the four extremal ones... | 6 | https://mathoverflow.net/users/1450 | 54294 | 33,916 |
https://mathoverflow.net/questions/53235 | 4 | For the purpuse of this question, a group is amenable iff there exists a Foelner sequence.
Dixmier unitarisability problem asks whether a (countable discrete) group G is amenable iff every bounded representation of G on a Hilbert space is unitarisable.
The problem is currently not solved, but at least argument "=>"... | https://mathoverflow.net/users/2631 | Is every bounded representation of Z unitarisable when all sets are measurable? | The answer is yes for separable Hilbert spaces.
If the Hilbert space is separable with basis $\lbrace e\_n \mid n \in \mathbb N\rbrace$, you only have to fix countably many inner products and define $\langle e\_n,e\_m \rangle\_{\rm new}$ for $n,m \in \mathbb N$. Assuming Folner's condition, you know that
$$\frac{1}... | 3 | https://mathoverflow.net/users/8176 | 54296 | 33,917 |
https://mathoverflow.net/questions/54299 | 1 | Let $X\subset Y$ be CW-complexes. Denote $i\colon X\to Y$ be an inclusion map.
Is it true that $i$ is deformation retract if and only if $i$ is homotopy equivalence?
When I saw some papers about h-cobordism theorem, authors usually checks inclusion map is a homotopy equivalence to prove that inclusion map is deform... | https://mathoverflow.net/users/12726 | About deformation retract | If $X$ is a subcomplex of $Y$ and the inclusion map is a homotopy equivalence, then $X$ is a deformation retract of $Y$. See for example proposition 0.16 and corollary 0.20 in [Hatcher](http://www.math.cornell.edu/~hatcher/AT/ATpage.html).
| 5 | https://mathoverflow.net/users/11771 | 54303 | 33,921 |
https://mathoverflow.net/questions/54311 | 0 | Let us consider $0\to E\to G\to H\to 0$ an exact sequence of coherent sheaves on a surface $S$ such that $rk G=rk E +1$ and the double dual of $H$ is $\mathcal{O}\_S(D)$ for some effective divisor $D$. Is it true that I have the following exact sequence:
$0\to detE\to det G\to i\_\*\mathcal{O}\_D\to 0$,
where $i$ i... | https://mathoverflow.net/users/33841 | Determinant and exact sequences of sheaves. | Provided that $S$ is smooth and $E,G$ are locally free, I find an exact sequence $0\to\det E\to\det G\to i\_\*\mathcal O\_D(\det G\vert\_D)\to 0$, as follows. Tensor the exact sequence $0\to\mathcal O\_S\to\mathcal O\_S(D)\to\mathcal O\_D(D)\to 0$ coming from the existence of a section with $\det E$ and use $\det E\oti... | 2 | https://mathoverflow.net/users/8726 | 54321 | 33,930 |
https://mathoverflow.net/questions/54315 | 5 | Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?
The recurrences $\mod p$ can be different, possibly depending on $p$ and the recurrence need not be linear, any recurrence will do.
| https://mathoverflow.net/users/12481 | Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$? | The answer to your question is NO. In fact, your hypothesis says that there is an infinite
set $\cal P$ of primes such that for each $p\in {\cal P}$ there is a function
$f\_p$ from ${\mathbb Z}\_p^r$ to ${\mathbb Z}\_p$ such that the sequence
$(a\_n)\_{n\geq 1}$ satisfies
$$ a\_{n+r+1} \equiv f\_p(a\_{n+1},a\_{n+2},a\_... | 8 | https://mathoverflow.net/users/2389 | 54323 | 33,932 |
https://mathoverflow.net/questions/54320 | 1 | Let $X$ and $Y$ be normal varieties with $D$ and $E$ Cartier divisors on $X$ and $Y$, respectively. Let $(D,E)$ denote the divisor $\pi\_X^\*(D)+\pi\_Y^\*(E)$ on the product $X\times Y$, where $\pi\_X$ and $\pi\_Y$ are the projections from $X\times Y$.
There is a natural inclusion $H^0(X,\mathcal{O}(D))\otimes H^0(Y,... | https://mathoverflow.net/users/12730 | Global sections for divisors on products of varieties | This is always true, it follows from the Künneth formula for coherent sheaves (see [Kunneth formula for sheaf cohomology of varieties](https://mathoverflow.net/questions/34673/kunneth-formula-for-sheaf-cohomology-of-varieties)).
| 5 | https://mathoverflow.net/users/4790 | 54325 | 33,934 |
https://mathoverflow.net/questions/54335 | 6 | Let $F\_2$ denote the free group of rank 2. Does anybody have a fast proof that the subgroup membership problem is undecidable for $F\_2 \times F\_2$? I saw a really fast proof last semester that started with a group with undecidable word problem and used that group to construct subgroups of $F\_2 \times F\_2$, but I'v... | https://mathoverflow.net/users/8434 | showing the subgroup membership problem is undecidable for $F_2 \times F_2$ | First, $F\_2 \times F\_2$ contains a copy of $F\_n \times F\_n$ for all $n \geq 1$, so it is enough to find such a group inside $F\_n \times F\_n$ for some $n$. Let $G$ be a finitely generated group with an unsolvable word problem and let $S$ be a generating set for $G$. Let $f : F(S) \times F(S) \rightarrow G \times G... | 12 | https://mathoverflow.net/users/317 | 54336 | 33,942 |
https://mathoverflow.net/questions/54330 | 4 | In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension restriction really necessary here? A similar restriction is used by A. Huber when she studies 'arithmetic perverse sheaves'.
| https://mathoverflow.net/users/2191 | If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one? | Gabber has announced a proof of the finiteness theorem for (direct images of constructible sheaves by) morphisms of finite type between general noetherian schemes years ago, but, being Gabber, he has not written it down. There was a seminar in the Ecole Polytechnique four years ago (I think) about his proof, and the go... | 7 | https://mathoverflow.net/users/12336 | 54339 | 33,944 |
https://mathoverflow.net/questions/54342 | 1 | Hello everyone,
I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.
In $R^2$ space, $a\_i$, $i=1,..,n>2$ are unit verctors, which are located on the unit circle uniformly. That means the angle subtended by a... | https://mathoverflow.net/users/12734 | Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I | OK, here might be an answer to the question you are meaning to ask:
Let $a\_1$, ..., $a\_n$ be unit vectors in $\mathbb{R}^d$. Let $G$ be a group acting linearly on $\mathbb{R}^d$, which permutes the $a\_i$, and such that the representation $\mathbb{R}^d$ of $G$ is irreducible. For example, maybe $d=2$ and $G$ is the... | 7 | https://mathoverflow.net/users/297 | 54345 | 33,948 |
https://mathoverflow.net/questions/54317 | 2 | Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms *from* $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective space, which may or may not be injective, so you get a map to another curve. To get a morphism *to* $C$, a particular ca... | https://mathoverflow.net/users/622 | Different ways to construct maps and the tensor products of line bundles | A possible answer is the following. Assume
$h^0(L)=n, \quad h^0(L \otimes \mu)=m$,
set
$\phi \colon \tilde{C} \to C, \quad \psi \colon C \to \bar{C} \subset \mathbb{P}^{n-1}$
and let $f=\psi \circ \phi \colon \tilde{C} \to \mathbb{P}^{n-1}$ be the composition.
Then $f^\*\mathcal{O}\_{P^n}(1)=\phi^\*L$. On th... | 4 | https://mathoverflow.net/users/7460 | 54358 | 33,958 |
https://mathoverflow.net/questions/54333 | 21 | In [G. F. Simmons' Differential Equations book (p.141)](http://books.google.com/books?id=JVWPPQAACAAJ&dq=G+F+Simmons+Differential+equations&hl=en&ei=yBZNTc3cGdH-4Abu-amoCQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ), the following claim is made:
“... As a matter of fact, there is **no known type** of second... | https://mathoverflow.net/users/5627 | Is this 1974 claim still valid? | This claim is *not* valid. The big breakthrough was the paper below:
[MR0839134](http://www.ams.org/mathscinet-getitem?mr=839134) (88c:12011)
Kovacic, Jerald J.
An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1986), no. 1, 3–43.
12H05 (34A30)
Later, M. Pet... | 29 | https://mathoverflow.net/users/11142 | 54361 | 33,961 |
https://mathoverflow.net/questions/54371 | 4 | I have a question about the symmetric group. Taking signatures of permutations defines a surjective homomorphism $S\_n \rightarrow \mathbb{Z}/2$. This is compatible with the natural inclusions $S\_n \hookrightarrow S\_{n+1}$, so we get a surjection $S\_{\infty} \rightarrow \mathbb{z}/2$. Here $S\_{\infty}$ is the direc... | https://mathoverflow.net/users/12738 | Signatures on the infinite symmetric group | The answer to the question is "no". In fact, $S'\_{\infty}$ is a perfect group, so there are no maps from it to an abelian group. Even more is true -- every element of $S'\_{\infty}$ can be expressed as a commutator! This is much stronger than simply saying that $[S'\_{\infty},S'\_{\infty}] = S'\_{\infty}$.
For these... | 16 | https://mathoverflow.net/users/317 | 54372 | 33,969 |
https://mathoverflow.net/questions/54343 | 51 | There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)!}{k!l!}Alt(\alpha\otimes\beta)$,
where $\alpha$ and $\beta$ are any forms of degree $k$ and $l$ respectively, and $Al... | https://mathoverflow.net/users/12617 | Is there a preferable convention for defining the wedge product? | I think a lot of people run into this issue. The way I think about it is the following:
Take your finite-dimensional vector space $V$ and form its tensor algebra $T(V)$. Define $\mathcal{J}$ to be the 2-sided ideal in $T(V)$ generated by elements of the form $v \otimes v$, and then define the exterior algebra to be $... | 57 | https://mathoverflow.net/users/703 | 54375 | 33,972 |
https://mathoverflow.net/questions/54376 | 7 | *Author* : Z.A. Melzak
*Book Title* : Companion to Concrete Mathematics.
*Publication* : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.
I found this book extremely nice.
To whet your appetite he talks about reification as well as some plain and less plain way to accelerate series convergence.
In... | https://mathoverflow.net/users/3005 | Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ? | According to [the Mathematics Genealogy Project](http://genealogy.math.ndsu.nodak.edu/id.php?id=17764), Zdzislaw Alexander Melzak did his graduate studies at MIT and became a professor at UBC. I'm not sure whether his name might have actually been spelled Zozislaw, though (I guess someone Polish would know which is cor... | 18 | https://mathoverflow.net/users/440 | 54378 | 33,973 |
https://mathoverflow.net/questions/54356 | 19 | What is the simplest example of a domain $R$ which is regular (in particular Noetherian) and factorial which admits a finitely generated projective module that is not free?
In fact I'll be at least somewhat happy with any example, since I can't think of one at the moment.
Some brief comments: $R$ needs to have Krul... | https://mathoverflow.net/users/1149 | Nonfree projective module over a regular UFD? | Depending on what you consider simple, let $k$ be the complex numbers, or the integers, or the field with two elements (or any other commutative ring you're fond of). Let $R=k[a,b,c,x,y,z]/(ax+by+cz-1)$. Map $R^3$ to $R$ by $(f,g,h)\mapsto xf+yg+zh$. Let $P$ be the kernel of this map.
$P$ is the universal example of ... | 16 | https://mathoverflow.net/users/10503 | 54379 | 33,974 |
https://mathoverflow.net/questions/31741 | 11 | A [Golomb ruler](http://en.wikipedia.org/wiki/Golomb_ruler) is a set of $n$ integers that determines $\binom{n}{2}$ distinct differences.
Two sets are *homometric* if they determine the same (multiset) of differences.
For example,
$$\{0,1,4,10,12,17\} \;,\; \{0,1,8,11,13,17\}$$
are a homometric pair of Golomb rulers, d... | https://mathoverflow.net/users/6094 | Largest pair of homometric Golomb rulers? | I think "[There are no further counterexamples to S. Piccard's theorem](http://dx.doi.org/10.1109/TIT.2007.899468)" by A. Bekir and S.W. Golomb is the answer. I skimmed the paper and if I understand correctly they proved that there are no homometric Golomb ruler pairs with 7 marks or more.
| 7 | https://mathoverflow.net/users/12747 | 54396 | 33,984 |
https://mathoverflow.net/questions/53884 | 1 | Can a polynomial size Context free grammar describe the finite language {$w \pi(w)$ : $\pi(w)$ is fixed string permutation, $|w|=n$ is fixed} over alphabet of {0,1}?
One case this is possible is when $\pi(w)$ is the string reverse permutation (CFG is describing palindromes).
Are there other choices for $\pi(w)$ wit... | https://mathoverflow.net/users/11847 | Can a polynomial size CFG describe the finite language \{$w \pi(w)$ : $\pi(w)$ is fixed string permutation, $|w|=n$ is fixed\} over alphabet \{0,1\}? | Quoting [chandok's answer on MO](https://math.stackexchange.com/questions/19691/can-a-polynomial-size-cfg-describe-the-finite-language-w-piw-piw-is/19727#19727) in community wiki mode.
I think a more precise question would be :
Given a permutation $\pi$ of {1...n}, what is the size of a grammar describing $L\_\pi =... | 1 | https://mathoverflow.net/users/11847 | 54409 | 33,991 |
https://mathoverflow.net/questions/54401 | 3 | Let $M$ be a complete hyperbolic manifold of dimension $n$, let $\varepsilon=\varepsilon\_n$ be the Margulis constant. Let $M\_{[\varepsilon,\infty)}$ be the thick part of $M$ with respect to $\varepsilon$. Suppose that $\pi\_1(M)$ is infinite. Is it true that $\pi\_1(M\_{[\varepsilon,\infty)})$ is also infinite. Or, i... | https://mathoverflow.net/users/10714 | Fundamental group of a thick part of hyperbolic manifold | Yes, because $\pi\_1$ of the boundary of a component of the $\epsilon$-thin part always
surjects to the fundamental group of that thin part, except in one special 2-dimensional
case: if there is a short orientation-reversing curve on a surface, then its component of
the thin part is a Moebius band, and the boundary gen... | 4 | https://mathoverflow.net/users/9062 | 54418 | 33,997 |
https://mathoverflow.net/questions/54399 | 1 | Given the number of results that are independent of ZF. It seems that once you've found a proof of a theorem that uses the axiom of choice, the odds are that it will be independent of ZF. So my question is:
-Is there any result that has a solution in ZFC which relies on AC, but has another proof that can be done only... | https://mathoverflow.net/users/10299 | If a result is apparently provable with AC, is actually independent of ZF? | Construction of the Haar integral for locally compact Hausdorff group $G$ ... as a linear functional on $C\_{00}(G)$ ... is often done using the Axiom of Choice. In the Hewitt & Ross textbook ABSTRACT HARMONIC ANALYSIS this is Theorem (15.5), and they do it without AC. They do then have an exercise (15.25) where they o... | 7 | https://mathoverflow.net/users/454 | 54424 | 34,001 |
https://mathoverflow.net/questions/54425 | 2 | In a solution to a recent post : [Fundamental group of a thick part of hyperbolic manifold](https://mathoverflow.net/questions/54401/fundamental-group-of-a-thick-part-of-hyperbolic-manifold), Igor Belegradek makes this claim that the thick part of a hyperbolic manifold is connected. To me it seems like the thick part o... | https://mathoverflow.net/users/6822 | Connectedness of the thick part of a hyperbolic manifold? | In any dimension the thin part of a hyperbolic manifold $M$ is the union of regular neigborhoods of short geodesics and of cusps. This follows from the Margulis lemma. Since removing cusps is the same as removing collars of the boundary of a compact manifold, the only way for the thin part to disconnect $M$ is if the n... | 7 | https://mathoverflow.net/users/1650 | 54427 | 34,002 |
https://mathoverflow.net/questions/54426 | 1 | It's easy to construct a sequence of rational numbers whose set of cluster (or accumulation or limit) points is of any finite cardinality, or is countably infinite, or has the cardinality of the continuum. Thus there appears no obvious barrier to forming a rational sequence with a cluster set of any cardinality up to $... | https://mathoverflow.net/users/7458 | Could the cluster set of a rational sequence be of intermediate cardinality (given not-CH)? | No, closed sets cannot have intermediate cardinality. One way to see this: suppose a
closed subset $K$ of the Cantor set $C$ has greater than countable cardinality.
It's easy to see that there must be a partition into two open and closed subsets such that
each one has greater than countably many elements of $K$, othe... | 6 | https://mathoverflow.net/users/9062 | 54429 | 34,004 |
https://mathoverflow.net/questions/17899 | 4 | Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ depending on $A$ and $K$ only such that $[K(A[\ell]):K]> C \ell^{\eta}$? What about $[K(P):K]> C' \ell^{\eta'}$, $P$ an $\ell$... | https://mathoverflow.net/users/4561 | lower bound for torsion of abelian varieties | Now let me address your last question. For the sake of simplicity, let us assume that $K$ is a global field of characteristic $p>2$ and the ring $End(A)$ of all endomorphisms of $A$ (over an algebraic closure of $K$) is the ring $Z$ of integers. Then my old results (Math. Notes: 21 (1978), 415--419 and 22 (1978), 493--... | 3 | https://mathoverflow.net/users/9658 | 54443 | 34,016 |
https://mathoverflow.net/questions/54411 | 3 | All groups here are abelian and $p$ is a prime number; I'll say $P$ is a $p$-group if every element
of $P$ has finite order which is a power of $p$.
Suppose $\mathrm{Hom}(G,P) = 0$ for every $p$-group $P$. Does it follow that
$\mathrm{map}\_\*(K(G,n), K(P,m))\sim \*$ for all $p$-groups $P$ and all
$m,n\geq 1$?
C... | https://mathoverflow.net/users/3634 | Eilenberg-Mac Lane spaces for groups that can't see $p$-groups | Yes. Your hypothesis on $G$ means that $G$ is torsion and prime to $p$: every element of $G$ has finite order prime to $p$. This in turn implies that the integral homology group $H\_mK(G,n)$ is torsion and prime to $p$, for every $m,n>0$. That in turn implies the conclusion ($H^m(K(G,n);P)=0$ for $m,n>0$ and $P$ a $p$-... | 7 | https://mathoverflow.net/users/6666 | 54453 | 34,024 |
https://mathoverflow.net/questions/54457 | 1 | Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't satisfy the unimodular column property but such that every finitely generated stably free $R$-module is free.
Thanks
... | https://mathoverflow.net/users/12532 | Unimodular column property | If "unimodular column property" means that every unimodular column can be completed to an invertible matrix, then this is equivalent to the statement that every finitely generated stably free module is free.
For finitely generated modules "Stably free implies free" is equivalent to "$P\oplus R$ free implies $P$ free... | 4 | https://mathoverflow.net/users/10503 | 54460 | 34,029 |
https://mathoverflow.net/questions/54377 | 8 | I’ve run into the following straightforward variant of lexicographic ordering, and am wondering if it has a standard name. I’ve been calling it the *flipping lexicographic* ordering, for evident reasons. I could also imagine it getting called the *parity lexicographic* ordering, but a brief search suggests that that’s ... | https://mathoverflow.net/users/2273 | Does this “flipping lexicographic” ordering have a standard name? | I [found an example](http://books.google.com/books?id=3w9eR3u8GN4C&lpg=PA61&ots=xSJTqwK-Yx&dq=%22alternating%20lexicographic%22&hl=fr&pg=PA58#v=onepage&q=%22alternating%20lexicographic%22&f=false) in the mathematical literature where the same ordering on words, and more specifically continued fractions, is called "alte... | 6 | https://mathoverflow.net/users/1450 | 54464 | 34,032 |
https://mathoverflow.net/questions/54404 | 10 | I'm doing research on the history of the Lagrange inversion theorem. The earliest predecessor I've found is the one referenced by De Morgan; viz. Jo. H. Lambert's construction in Observationes Variae in Mathesin Puram, Acta Helvetica, Vol. 3, 1758, pp. 128-168.
If anyone knows of an earlier construction I'd greatly a... | https://mathoverflow.net/users/4111 | History of the Lagrange Inversion Theorem | If you count *any* inversion of a power series as a predecessor of Lagrange
inversion, then I believe the earliest examples are Newton's inversion of the
log series to obtain the exponential series, and inversion of the inverse sine
series to obtain the sine series. The exponential series is in his *De methodis*
(1671... | 8 | https://mathoverflow.net/users/1587 | 54466 | 34,034 |
https://mathoverflow.net/questions/54468 | 0 | Hi,
How can I generate the equation of a curve that matches all arbitrarily given (x,y) pairs? I would like a polynomial of nth degree, where n does not matter, as long as the curve passes thru all the given points.
I guess this problem is NP-complete. If so, how do I find the closest matching curve in a feasible a... | https://mathoverflow.net/users/12758 | HOW TO Generate Equation of a Curve Given (x,y) pairs - algorithm? | Actually, there is an easy and standard procedure (Lagrange Interpolation) that does this. See:
<http://en.wikipedia.org/wiki/Lagrange_polynomial>
| 2 | https://mathoverflow.net/users/7311 | 54469 | 34,035 |
https://mathoverflow.net/questions/49685 | 16 | Though I'm sure it's not really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups.
I'm thinking Neumann boundary conditions to model reflection lines.
And I do realize that these spectra may vary acc... | https://mathoverflow.net/users/10909 | Hearing the 17 planar symmetry groups | I don't think that it's so hard to work out the solution non-rigorously, nor all that onerous to do it rigorously following Ian Agol's suggestion in the comments. A reference could be nice, but I don't think that it's quite necessary, because the answer itself is not all that different from its derivation.
First, in ... | 5 | https://mathoverflow.net/users/1450 | 54472 | 34,038 |
https://mathoverflow.net/questions/34914 | 6 | As we read from wiki, informally, the [reconstruction conjecture](http://en.wikipedia.org/wiki/Reconstruction_conjecture) in graph theory says that graphs are determined uniquely by their subgraphs.
1. Is there a group-theoretic formulation of this conjecture?
2. Has an analogous conjecture been made in group theory(... | https://mathoverflow.net/users/5627 | Reconstruction Conjecture: Group theoretic formulation | There are continuum 2-generated groups where all proper subgroups are cyclic of order $p$ (for the same prime $p\sim 10^{70}$), the Tarski monsters. All these groups have the same lists of proper subgroups, and the same lattice of subgroups. Also all cyclic groups of prime order have the same proper subgroups: the triv... | 8 | https://mathoverflow.net/users/nan | 54476 | 34,041 |
https://mathoverflow.net/questions/54467 | 0 | Let $k>0$ be a positive integer. Set $n=4k.$ Let $R(t)$ be a polynomial of degree $n-1$
with coefficients in $\lbrace -1,1 \rbrace$.
Consider the discrete average
$$
D(n,R) = \frac{\sum\_{j=0}^{n-1} \vert R(exp(2\pi i j/n)) \vert}{n}
$$
and the average
$$
A(n,R) = \frac{\int\_{0}^{2\pi} \vert R(exp(it) \vert dt... | https://mathoverflow.net/users/11016 | Average compared with discrete average for some $\lbrace -1,1 \rbrace$ polynomials | The claims seems to be false. Numerical integration for $k=2, (n=8)$ gives that
for $152$ out of the $256=2^8$ polynomials ($59.3\%$) you have $D(n,R) \leq A(n,R)$.
I do not see why having n a multiple of 4 should matter.
The fact that for half of polynomials the inequality holds at $n=4$ seems like a coincident. Y... | 1 | https://mathoverflow.net/users/1778 | 54489 | 34,050 |
https://mathoverflow.net/questions/54493 | 6 | Let $B \subset \mathbb{Z}^+$. Define $r\_{B,h}(n)$ to be the number of ways of writing $n$ as the sum of $h$ elements of $B$ and $R\_{B,h}(n)$ the number of ways to write $n$ as the sum of $h$ DISTINCT elements of $B$. In many applications, such as the Erdos-Tetalli theorem which finds a set $B$ such that $R\_{B,h}(n) ... | https://mathoverflow.net/users/10898 | Any rigorous way to claim that sums with repeat summands are few? | Here is a fairly crude first attempt.
First note that for any basis $B$, when $h=2$, we have
$$0\leq r\_{B,2}(n)-R\_{B,2}(n)\leq 1$$
for all n, as you note in your question. Hence assume $h\geq 3$. In this case, if $r'\_{B,h}(n)=r\_{B,h}(n)-R\_{B,h}(n)$ counts the number of representations where some elements are... | 4 | https://mathoverflow.net/users/385 | 54509 | 34,060 |
https://mathoverflow.net/questions/54491 | 6 | This is probably straightforward, but I'm having trouble writing down a precise statement. "Everyone knows" that the cobordism category $\text{2Cob}$ (all manifolds compact and oriented) is the free symmetric monoidal category on a commutative Frobenius object. What is the analogous statement for $\text{1Cob}$?
It lo... | https://mathoverflow.net/users/290 | Fill in the blanks: "1Cob is the free ____ category on a ____" | Qiaochu, yes, but you don't need to say "left and right dual" because a left dual is a right dual in a symmetric monoidal category. It would be enough to say "with a left dual", or say it as Theo did.
An equivalent way of describing it is "the free [compact closed category](http://en.wikipedia.org/wiki/Compact_close... | 5 | https://mathoverflow.net/users/2926 | 54510 | 34,061 |
https://mathoverflow.net/questions/54502 | 8 | Let $\mathbb{A}^n\_k$ be the Affine $n$-space over an algebraically closed field $k$.
Let $X$ be a variety over $k$. What would be the right definition of an "Affine bundle" i.e bundle of fiber type $\mathbb{A}^n\_k$ over $X$ (I mean local triviality in zarisky topology,or
etale .. )?. When can one get a vector bundle ... | https://mathoverflow.net/users/12202 | Affine bundles over varieties | I think the term "affine bundle" is used for at least two things: (1) A map $p:Y\to X$ such that for some open cover (in your choice of topology) there are isomorphisms $p^{-1}(U)={\Bbb A}^n \times U$ --- just like you said. (2) A torsor for a vector bundle, i.e., like (1) but with the added condition that the transiti... | 13 | https://mathoverflow.net/users/5081 | 54511 | 34,062 |
https://mathoverflow.net/questions/54513 | 70 | This question is similar to [a previous one about "urban legends"](https://mathoverflow.net/questions/53122), but not the same. It is established that Milnor proved the Fáry-Milnor theorem as an undergraduate at Princeton. For the record, Fáry was a professor in France and proved the result independently. Milnor has a ... | https://mathoverflow.net/users/1450 | The story about Milnor proving the Fáry-Milnor theorem | The story is told in some detail in Sylvia Nasar's "A Beautiful Mind", a biography of John Nash, who was a fellow student of Milnor at one point. In that version, Milnor knew that Borsuk's conjecture was an open problem; he wrote up his apparent answer not believing it to be correct, and asked Fox to look it over since... | 58 | https://mathoverflow.net/users/12767 | 54514 | 34,064 |
https://mathoverflow.net/questions/54517 | 1 | Hey guys,
The following paper uses the term `bridge' in their definition of the Tutte polynomial:
Bennett Thompson, David J. Pearce, Craig Anslow, and Gary Haggard. Visualizing the computation tree of the tutte polynomial. In Proceedings of the 4th ACM sympo- sium on Software visualization, SoftVis ’08, pages 211–2... | https://mathoverflow.net/users/10814 | The Tutte Polynomial - is a `crossing' the same as a `bridge'? | The Wikipedia page for the [Tutte polynmomial](http://en.wikipedia.org/wiki/Tutte_polynomial) doesn't use the word crossing, it also uses the word bridge. In graph theory, a bridge of a connected graph is an edge that separates the graph into two components.
However, there is a relation between the Tutte polynomial a... | 5 | https://mathoverflow.net/users/1450 | 54519 | 34,067 |
https://mathoverflow.net/questions/54003 | 25 | More specifically, letting $I=[0,1]$, do there exist $f,E$ with $E$ a (necessarily nonseparable) Banach space and $f$ a bounded Lebesgue measurable function $I\to E$ such that $f$ is not equal almost everywhere to a pointwise limit of a sequence of simple Lebesgue measurable functions? Here "simple" means having finite... | https://mathoverflow.net/users/12643 | Does there exist a measurable function which is not a.e. "strongly" measurable? | No. In fact, every Lebesgue measurable function $f\colon I\to E$ is equal almost everywhere to a limit of simple Lebesgue measurable functions. As you hint at in the question, this is easy to show in the case where $E$ is separable. The general situation reduces to the separable case due to the following result. For a ... | 37 | https://mathoverflow.net/users/1004 | 54531 | 34,074 |
https://mathoverflow.net/questions/54532 | 0 | Let $R$ be an associative ring with identity and let $x$ be an arbitrary element from the ring $R$. Could you please help me to prove that $x=ye$, where $y$ is some element in $R$ and $e$ is some primitive central idempotent in $R$. In other words, I need to prove that any element in $R$ is representable as a product o... | https://mathoverflow.net/users/12772 | I need to prove that any element in a rings is representable as a product of some element and some central idempotent | Consider the direct product $k\times k$ of two copies of some field and the element $x=(1,1)$. In this simple example you can describe all primitive central idempotents.
| 0 | https://mathoverflow.net/users/1409 | 54539 | 34,078 |
https://mathoverflow.net/questions/54516 | 2 | It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j\_{!\*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially complicated in the case when $j$ is not affine. Does an easier proof (or a plan of it:)) exist? I would like to have a proof... | https://mathoverflow.net/users/2191 | Is there an easy proof of the fact that the intermediate image functor respects weights? | The proof in BBD is not that complicated, and it doesn't matter much whether $j$ is affine or not. It uses the three following facts :
* If $f$ is a morphism of schemes, then $f\_\*$ sends a complex of weight $\geq a$ to a complex of weight $\geq a$, and $f\_!$ sends a complex of weight $\leq a$ to a complex of weigh... | 8 | https://mathoverflow.net/users/12336 | 54542 | 34,081 |
https://mathoverflow.net/questions/54537 | 2 | Hey guys,
I'm a computer science student attempting to understand a quantum algorithm that uses braid theory - something I'm completely unfamiliar. I've getting through the algorithm but I can't find any simple explanations of the following terms:
* Markov Trace
* Markov Property
For example, the following articl... | https://mathoverflow.net/users/10814 | Markov Trace and Markov Property | I think you want section 2.11 of reference [6] of the paper you are looking at, namely "A Polynomial Quantum Algorithm for Approximating the Jones Polynomial" at the [arxiv](http://arxiv.org/abs/quant-ph/0511096v2).
| 2 | https://mathoverflow.net/users/1650 | 54548 | 34,085 |
https://mathoverflow.net/questions/54547 | 1 | Let $R$ be an arbitrary associative unital ring and let $x \in R$. Can we always represent $x$ as a finite or infinite sum $x=\sum\_{i}y\_ie\_i$ where $y\_i\in R$ and each $e\_i$ is a primitive central idempotent in $R$?
P.S. Sorry if the question is stupid. Last 10 years ring theory was not my field of specializatio... | https://mathoverflow.net/users/12772 | Another question about primitive central idempotents in associative unital rings (yes, again!) | I have no idea what an infinite sum means in an arbitrary associative unital ring.
Your question is equivalent to asking whether it is possible to write 1 as a sum of primitive central idempotents. (If $1=e\_1+\ldots+e\_n$, then you can take $y\_i=x$ for all $i$.)
| 0 | https://mathoverflow.net/users/10503 | 54551 | 34,086 |
https://mathoverflow.net/questions/54550 | 24 | Serge Lang's *Differential and Riemannian Manifolds* is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers the defect of containing no exercises and few examples. This makes it difficult to learn the subject from this book, e... | https://mathoverflow.net/users/12774 | The third axiom in the definition of (infinite-dimensional) vector bundles: why? | I would refer you to the remark B in supplement 3.4A on Manifolds, Tensor Analysis and Applications of Abraham Marsden Ratiu.
It hope it could be useful, and so I quote:
>
> "The following counterexample is due to A.J. Tromba. Let $h: [0, 1]\times L^2[0, 1]\rightarrow L^2[0, 1]$ be given by $h(x,\phi)=(h'(x))(\phi)... | 7 | https://mathoverflow.net/users/12617 | 54552 | 34,087 |
https://mathoverflow.net/questions/47870 | 10 | Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains $B$) such that: there are no non-zero $C$-morphisms between $B\_1$
and $B\_2[i]$ for any $B\_1,B\_2\in Obj B$ and $i\n... | https://mathoverflow.net/users/2191 | Does a triangulated category that possesses a subcategory $B$ of generators with no extensions of non-zero degree between them have to be isomorphic to $K^b(B)$? | Mikhail, if you assume more generally that $C$ is topological (in an appropriate sense) then your claim is also true. As Matthias suggests above, this is a tilting-like theorem, Theorem 5.1.1 in '*Stable model categories are categories of modules*' by Schwede and Shipley. This assumption may comprise all examples of in... | 3 | https://mathoverflow.net/users/12166 | 54554 | 34,089 |
https://mathoverflow.net/questions/54304 | 2 | I am interested in some references on the quotient spaces obtained by quotienting G, a simple Lie group, by L, the group generated by the Levi factor of a parabolic subalgebra.
Presumably the case where L is the maximal torus is understood?
I am mostly interested in the compact case.
| https://mathoverflow.net/users/3623 | The quotient of a Lie group by the Levi factor of a parabolic subgroup | Exhausting account on homogeneous spaces of the form $G/P$ with $G$ semisimple and $P$ parabolic is given in this [book](http://www.ams.org/bookstore-getitem/item=surv-154). The relationship between $P$ and $L$ is also explained there in detail so presumably one may take these results as a starting point.
| 2 | https://mathoverflow.net/users/6818 | 54564 | 34,095 |
https://mathoverflow.net/questions/54456 | 7 | I'm working with Kahler manifolds at the moment and looking at their spin$^c$ structure. I really don't know much about spin$^c$ structures in general and don't have enough time to learn it all at present, so I was hoping someone might be able to show me a shortcut in this case.
As far as I understand, for the usual ... | https://mathoverflow.net/users/11206 | Clifford Action for Kahler Manifolds | You ask for a short-cut, and I'm going to interpret this as asking "How can I see that on a Kaehler manifold $X$ the forms $\Omega^{0,\bullet}\_X$ are a complex Clifford module, without first going through all that stuff about spin groups and their representations?"
**Warning**: My scalar factors are probably wrong.... | 5 | https://mathoverflow.net/users/2356 | 54565 | 34,096 |
https://mathoverflow.net/questions/54567 | 16 | Some experts tell me that the construction of abelian varieties from
Hilbert modular forms is an (apparently difficult) open problem. However,
in view of the construction of $l$-adic Galois representations due to Carayol for instance,
it is not clear what exactly the obstruction to the usual method of taking
the quot... | https://mathoverflow.net/users/6121 | Construction of abelian varieties from Hilbert modular forms? | There is no problem with constructing an abelian variety $A$ for *most* Hilbert modular forms of parallel weight $2$, the issue is finding such a variety for *all* $\pi$. In particular, when $d = [K:\mathbf{Q}]$ is even, there is a local obstruction to the existence of a corresponding Shimura curve which realizes the G... | 16 | https://mathoverflow.net/users/nan | 54572 | 34,099 |
https://mathoverflow.net/questions/54561 | 17 | I'm reading through the two chapters in Fulton and Harris on the representation theory of $\mathfrak{sl}(3,\mathbb{C})$, in preparation for lecturing on them this week. I'll use F&H's notation, so that $\Gamma\_{a,b}$ is the irreducible representation of highest weight $a e\_1 - b e\_3$ (with $e\_i$ the character takin... | https://mathoverflow.net/users/379 | Gap in an argument in Fulton & Harris? | I'm still on the fence as to the pedagogical value of the "lecture" approach taken by F&H, but anyway it seems essential in a formal classroom presentation to articulate clearly which features of irreducible representations are or aren't known in general at each step of the exploration of examples in low ranks. F&H def... | 16 | https://mathoverflow.net/users/4231 | 54576 | 34,102 |
https://mathoverflow.net/questions/54526 | 16 | Pythagoras and his followers believed that the Universe was made of numbers. Specifically, they thought that if you compared any magnitudes of the same kind, say the lengths of two objects, you would always get a whole number ratio. Then someone came up with a proof that the side length of a square and its diagonal are... | https://mathoverflow.net/users/5017 | Fulfilling Pythagoras' Dream using Nonstandard Models of Arithmetic and/or Surreal Numbers | First note that the axioms of [Robinson Arithmetic](http://en.wikipedia.org/wiki/Robinson_arithmetic) are usually straightforward to verify in any structure. Surely the nonnegative omnific integers satisfy these basic axioms. Note that a nonstandard model of Robinson Arithmetic cannot be wellordered because of the axio... | 12 | https://mathoverflow.net/users/2000 | 54577 | 34,103 |
https://mathoverflow.net/questions/54545 | 11 | We know that M\_g is general type for g large enough. In particular, the generic genus-g curve is not contained in a (non-isotrivial) rational family parametrized by P^1. In fact, the high-genus curves I know how to build over C(t) all have low gonality; it's easy to make a hyperelliptic curve y^2 = f(t,x), and with a ... | https://mathoverflow.net/users/431 | Can a rational family of genus-g curves have generic gonality? Can it be Brill-Noether general? | One can construct pencils of k-gonal curves of genus g by taking a K3 surface S with
Pic(S) generated by two classes: an ample class C with self-intersection 2g-2, and an elliptic curve E, so self-intersection 0, such that C.E=k. Every curve in the linear system
|C| has gonality k, and the pencil of minimal degree is ... | 12 | https://mathoverflow.net/users/12778 | 54578 | 34,104 |
https://mathoverflow.net/questions/54556 | 7 | (Related question: [What part of the fundamental group is captured by the second homology group?](https://mathoverflow.net/questions/40399/what-part-of-the-fundamental-group-is-captured-by-the-second-homology-group))
Suppose I have a path-connected space $X$ for which $\pi\_1(X)=\mathbb{Z}$. Suppose I know $\pi\_2(X)... | https://mathoverflow.net/users/8103 | Computing H_2 from pi_1=Z and pi_2 | Here's one way. Suppose first that one has a Serre fibration $F\hookrightarrow E \to S^1$ with $F$ simply connected. Then $\pi\_2(E)=\pi\_2(F)$ by the exact sequence of homotopy groups, and $\pi\_2(F)=H\_2(F)$ by Hurewicz.
The map $H\_2(F)\to H\_2(E)$ is surjective, and its kernel is the image of $\mathrm{id}-\phi$, ... | 5 | https://mathoverflow.net/users/2356 | 54582 | 34,106 |
https://mathoverflow.net/questions/52897 | 8 | Let $\mu$ be any infinite cardinal, and define a collection $N\subset[\mu]^\mu$ to be, maximal almost disjoint (MAD) over $\mu$, iff
1. $\forall\{A,B\}\in[N]^2$ $( A\cap B \in [\mu]^{<\mu})$
2. $\forall X\in[\mu]^\mu \exists A\in N$ $( X \cap A \in [\mu]^\mu)$
My questions are as follows: when $\mu$ is **singular*... | https://mathoverflow.net/users/8843 | Singular Cardinals, and A Strange Question. |
>
> **Theorem** If $0^{\sharp}$ does not exist and $\lambda$ is a singular cardinal, then any forcing adding subsets to $\lambda$ necessarily adds subsets to a cardinal below $\lambda$.
>
>
>
Proof: Let $\mathbb{P}$ be a partial order in the ground model and $G \subseteq \mathbb{P}$ be $V$-generic. Without loss ... | 6 | https://mathoverflow.net/users/11318 | 54597 | 34,112 |
https://mathoverflow.net/questions/54591 | 21 | So one of the major problems with the categories of schemes and algebraic spaces is that the "correct quotients" are oftentimes *not* schemes or algebraic spaces. The way I've seen this sort of thing rectified is either by moving a step up the categorical ladder or by defining some nonstandard quotient to "fix" things.... | https://mathoverflow.net/users/1353 | Does derived algebraic geometry allow us to take quotients with reckless abandon? | There is more than one way that derived algebraic geometry generalizes ordinary algebraic geometry. The new affines don't help you much with quotients, which are (homotopy) colimits, but they give you well-behaved intersections, which are (homotopy) limits. On the other hand, you can consider functors from affines (new... | 13 | https://mathoverflow.net/users/121 | 54599 | 34,113 |
https://mathoverflow.net/questions/54594 | 7 | Let $M$ be a compact smooth manifold without boundary. A Riemannian metric $g$ on $M$ induces a volume measure (or Lebesgue measure) $m\_g$ on $M$.
A diffeomorphism $f:M\to M$ is said to be {volume--preserving} if $f\_\*(m\_g)=m\_g$, that is, for each Borel subset $A\subset M$, $f\_\*(m\_g)(A):=m\_g(f^{-1}A)=m\_g(A)$... | https://mathoverflow.net/users/11028 | Is volume--preserving an intrinsic property? | If the volume form for $M^n$ is $\Omega$, note that the average $\Omega\_k$ of
$f^{k\*}(\Omega)$ for $0 \le k \le n$ is a volume form that has much tighter bounds for invariance by $f$ than $\Omega$, since all terms except the first and the
last are equal. You can pass to the limit of the $\Omega\_k$ as a measure,
and ... | 11 | https://mathoverflow.net/users/9062 | 54605 | 34,116 |
https://mathoverflow.net/questions/54603 | 40 | I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read to learn about the geometry that in turn inspired the ideas in Arakelov theory?
| https://mathoverflow.net/users/5309 | What should I read before reading about Arakelov theory? | To get a nice overview of how and why Arakelov theory started you could read the introduction to R. de Jong's Ph.D. thesis on
<http://www.math.leidenuniv.nl/~rdejong/publications/>
I remember that being very helpful to me.
To avoid too many complex analytic difficulties you should stick to the case of arithmetic... | 47 | https://mathoverflow.net/users/4333 | 54615 | 34,122 |
https://mathoverflow.net/questions/54612 | 30 | May I respectfully ask what the **minimal** background needed to read Wiles' proof of Fermat's Last Theorem is?
I'm not an expert on number theory, but out of curiosity I wanted to understand - at a cursory level if possible - the outline of the proof.
Thank you to all responders in advance.
My background: Junior... | https://mathoverflow.net/users/12723 | Minimal prerequisite to reading Wiles' proof of Fermat's Last Theorem | This is a very hard proof to do for an undergraduate but there are books available. Tthe book "Invitation to Fermat Wiles" ([http://www.amazon.com/Invitation-Mathematics-Fermat-Wiles-Yves-Hellegouarch/dp/0123392519](http://rads.stackoverflow.com/amzn/click/0123392519)) is an exposition on the proof written for undergra... | 24 | https://mathoverflow.net/users/12337 | 54616 | 34,123 |
https://mathoverflow.net/questions/38359 | 21 | I have often encountered definitions of the kind "stacks are equivalence classes of groupoids under Morita equivalence" in topological or differentiable context, with the notion of Morita equivalence appropriate for the context.
I have two questions:
1) can one define Deligne-Mumford or Artin stacks this way, as M... | https://mathoverflow.net/users/2234 | stacks as Morita equivalence classes | Regarding your question relating Morita equivalence as defined for internal groupoids and as defined for rings:
Given an internal groupoid $G$ (say, Lie, topological or algebraic), it defines a presheaf of groupoids on the ambient category ($Diff$, $Top$ or $Sch$). The stackification of this presheaf is the category ... | 8 | https://mathoverflow.net/users/4177 | 54617 | 34,124 |
https://mathoverflow.net/questions/54611 | 2 | hi, I'm sorry if the question is silly, but I couldn't get my head around it for a while now.
In Markov Chains (MC) proving that a state is either recurrent or transient is through Borel-Cantelli lemma (BCL): the event (state) happens infinitely often (i.o.) if the series diverges and vice versa.
The question is,... | https://mathoverflow.net/users/12418 | Borel-Cantelli Lemma on MCs (absorbing states) | Short answer:
>
> Don't.
>
>
>
In fact, one never shows that a state is absorbing through Borel-Cantelli lemma. Or that a state is recurrent, since this would mean using the part of Borel-Cantelli lemma where a series diverges, which needs independence, and the successive times of visits of a given state by a ... | 3 | https://mathoverflow.net/users/4661 | 54618 | 34,125 |
https://mathoverflow.net/questions/54619 | 9 | Let $n$ be a positive integer. How large must be a set $A\subset F\_2^n$ to ensure that if $P$ is a quadratic polynomial in $n$ variables, vanishing at all non-zero points of the sumset $2A:=\{a\_1+a\_2\colon a\_1,a\_2\in A\}$, then also $P(0)=0$?
Considering the situation where $P(x\_1,\ldots,x\_n)=\sum x\_ix\_j+\su... | https://mathoverflow.net/users/9924 | When does $P(a-b)=0$ for $a\ne b$ ensure $P(0)=0$? | Let $A$ be a maximal subset of $F^n\_2$ with $P(A+A)\equiv 0$ but $P(0)\neq 0$. By finding a maximal linearly independent subset of $A$, and applying an appropriate linear transformation, we may assume $A$ contains the basis elements $e\_1,\ldots,e\_k$ and that all other elements of $A$ are linear combinations of these... | 5 | https://mathoverflow.net/users/5513 | 54626 | 34,128 |
https://mathoverflow.net/questions/54301 | 12 | Let $X\_p$ be a projective curve over the finite field $\mathbf{F}\_p$ (i.e. a projective $\mathbf{F}\_p$-scheme pure of dimension 1) for every prime number $p$. Let $X\_\mathbf{Q}$ be a projective curve over $\mathbf{Q}$. When does there exist a flat projective integral normal scheme $X$ over Spec $\mathbf{Z}$ such th... | https://mathoverflow.net/users/4333 | Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them? | I suspect that even if you had a single curve over $\mathbb{F}\_p$, you might not find a lift
to $\mathbb{Q}$. Below I sketch an argument that works under the assumption that $\mathcal{M}\_g$ does not have Zariski dense set of points. If you believe the conjectures of Lang on rational points, this assumption should be... | 16 | https://mathoverflow.net/users/4344 | 54628 | 34,129 |
https://mathoverflow.net/questions/54632 | 6 |
>
> **Question.** Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume?
>
>
>
### Background/motivation.
A 2D version of the question was posed by Michael Goldberg in *Monthly*: find the shortest curve which divides a conv... | https://mathoverflow.net/users/5371 | Minimal surface which divides a convex body into two regions of equal volume | The conjecture as you state it is false. The variational argument (that can be interpreted in terms of fluid pressure if you like) shows that the surface has constant mean curvature and is orthogonal to the boundary. In dimension 2, this means that it is a circular arc, but in higher dimensions there are many more exam... | 19 | https://mathoverflow.net/users/4354 | 54633 | 34,131 |
https://mathoverflow.net/questions/54635 | 8 |
>
> **Possible Duplicate:**
>
> [Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?](https://mathoverflow.net/questions/18108/must-a-surface-obtained-by-exponentiating-a-plane-in-a-tangent-space-of-a-riemann)
>
>
>
The one dimensional geodesi... | https://mathoverflow.net/users/3969 | geodesic 2-dimensional submanifolds of a Riemannian manifold | Your examples (i.e. space forms) are the only manifolds with the property that the exponential map sends 2-dimensional disks to totally geodesic surfaces. One way to see this is using Jacobi vector fields.
More precisely, let $X$ and $Y$ be two orthogonal vectors in some tangent space $T\_xM$ and assume that the expo... | 6 | https://mathoverflow.net/users/10675 | 54641 | 34,136 |
https://mathoverflow.net/questions/54627 | 7 | Let $k$ be a commutative ring (with unity). Let $R$ be a $k$-algebra (with unity, but not of necessity commutative).
Let $M$ be an $\left(R,R\right)$-bimodule where $k$ acts in the same way from the left and from the right (I'd call this an $\left(R,R\right)\_k$-bimodule, but I haven't seen this notation anywhere). T... | https://mathoverflow.net/users/2530 | Hochschild H^1 (R,M) = 0 vs. H_1 (R,M) = 0 where R is a ring and M is an (R,R)-bimodule | (Let me write $A$ for your $R$, because I will mix letters up if not...)
The standard complex for $A$ over $k$ is exact, independently of the projectiveness of $A$ over $k$; call $d$ its differential. Then we have a short exact sequence
$$
0
\to
\frac{A\otimes\_kA\otimes\_kA}{d(A\otimes\_kA\otimes\_kA\otimes\_kA)}... | 6 | https://mathoverflow.net/users/1409 | 54657 | 34,146 |
https://mathoverflow.net/questions/54669 | 20 | Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many mathematical disciplines in which linear operators to which the spectral theorem applies arise. One finds quite quickly that ... | https://mathoverflow.net/users/7392 | Nice applications of the spectral theorem? | * An operator-theoretic proof that the classical [Hamburger moment problem](http://en.wikipedia.org/wiki/Hamburger_moment_problem) admits a solution (see e.g. *Methods of modern mathematical physics* by Reed and Simon, volume 2, Theorem X.4).
* Weyl's proof of the Bohr analogue of Parseval's identity for almost periodi... | 9 | https://mathoverflow.net/users/5371 | 54670 | 34,152 |
https://mathoverflow.net/questions/54634 | 8 | There are some nice families of groups as $S\_n, A\_n$, $GL(n,q)$, $SL(n,q)$, and they are useful; we know their elements, and we can get small groups as subgroups of these groups. Is it possible to get every $p$ group as a **Sylow-p subgroup** of some group in such families of groups? ( For example, the non-abelian gr... | https://mathoverflow.net/users/6761 | p-groups as Sylow subgroups | The question is somewhat loosely stated, leading to various answers and comments which are at cross-purposes. Some specific families of finite groups are mentioned, but the list seems to be left open (?) Among these families, the symmetric and alternating groups have no built-in prime $p$ to favor. Moreover, Burnside's... | 2 | https://mathoverflow.net/users/4231 | 54672 | 34,154 |
https://mathoverflow.net/questions/54677 | 14 | In how many different ways can **k** bishops be placed on an **nxn** chessboard such that no two bishops attack each other? Please try to respond with a formula and explanation.
| https://mathoverflow.net/users/12803 | How to place k bishops on an nxn chessboard | Call this number $B\_k(n)$. For fixed $k$ it is known that $B\_k(n)$ has the form
$P\_k(n)+(-1)^nQ\_k(n)$, where $P\_k$ and $Q\_k$ are polynomials. These polynomials have been computed for (at least) $k\leq 6$. We also have (unsurprisingly) the asymptotic formula
$B\_k(n)\sim n^{2k}/k!$.
For further information see <... | 23 | https://mathoverflow.net/users/2807 | 54681 | 34,160 |
https://mathoverflow.net/questions/54653 | 2 | Hello!
Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. **Is there any algorithm (more or less efficient?), that would for a given triangulation of X produce another SMALLER simplicial complex, that is also homeomorphic to X?**
My ... | https://mathoverflow.net/users/11317 | Algorithm that decreases the size of the simplicial triangulation | There are a number of papers out there for efficient computation of homology of simplicial complexes; I enjoyed reading [this paper by Dumas, Heckenbach, Saunders and Welker](http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/Publications/DHSW.pdf), but I don't know how authoritative it is. In particular, though, it seems... | 3 | https://mathoverflow.net/users/4194 | 54688 | 34,164 |
https://mathoverflow.net/questions/54685 | 1 | It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct sums of such, more precisely).
Is there a similar description for finite covers of complete regular local rings in highe... | https://mathoverflow.net/users/2234 | covers of complete regular local rings | Depending on what you mean by cover your statement isn't true in the DVR case. To make it true in that case you can throw in the condition that the cover be normal and it may also be a good idea to assume the map is flat. However, under your assumptions the complete regular local ring is a power series ring over the re... | 2 | https://mathoverflow.net/users/4008 | 54692 | 34,168 |
https://mathoverflow.net/questions/54682 | 2 | This question is related to <https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map>. If the answer to this question is yes, it would solve the abovementioned other MO question.
We equip ${\mathbb R}^3$ with the $\ell\_3$ norm
$||(x,y,z)||=(|x|^3+|y|^3+|z|^3)^{\frac{1}{3}}$. Is it true that, fo... | https://mathoverflow.net/users/2389 | Analogue of an orthogonal subspace in a noneuclidian normed space | Here is a simple proof that the property holds only for Euclidean norms, at least if the norm in question is $C^1$ smooth and strictly convex. Surely it was known way before Gromov was born.
Let $S$ denote the unit sphere of the norm. First observe that, if $v\in S$ and $w$ are such that $\|w+tv\|\ge \|tv\|$ for all ... | 4 | https://mathoverflow.net/users/4354 | 54696 | 34,171 |
https://mathoverflow.net/questions/54698 | 6 | Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S\_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines two subspaces of $V^{\otimes n}$, the subspace on which $S\_n$ acts via the trivial character and the subspace on whic... | https://mathoverflow.net/users/290 | Is there a notation for the symmetric / antisymmetric subspaces of a tensor power that distinguishes them from the symmetric / exterior power? | This only answers part of your question 0 unfortunatly. The construction is certainly functorial, but the two notions of symmetric/alternating power do not always agree. Let's write $\operatorname{Sym}^n (V)$ for the symmetric tensors, and $\operatorname{Alt} ^n (V)$ for the alternating tensors. I wish this were establ... | 7 | https://mathoverflow.net/users/6481 | 54699 | 34,173 |
https://mathoverflow.net/questions/54646 | 0 | Hello
I'm currently investigating the differential equation
$$f''=\left(\sum a\_n x^n\right)f$$
I was wondering if anyone could please provide me with a lead as to where I may find the Green's functions? Sadly I have not been able to achieve the result through my own computations
| https://mathoverflow.net/users/11439 | Green's Functions | This is indeed fairly standard. Using the two linearly independent solutions to your ODE that you already have you can just follow the procedure outlined e.g. [here](https://webspace.utexas.edu/dj955/www/notes/GreenFn.pdf) to construct the Green function.
| 2 | https://mathoverflow.net/users/2149 | 54705 | 34,176 |
https://mathoverflow.net/questions/54668 | 6 | Let $a\_n=10^n \cdot \pi$. Is the set of numbers $\{a\_n-\lfloor a\_n \rfloor : n \in \mathbb{N}\}$ dense in [0,1]?
What is the best known result near this question?
Apparently John Nash asked this on an undergraduate analysis exam (according to an anecdote told by Seymour Haber, recounted in Sylvia Nassar's biogra... | https://mathoverflow.net/users/7967 | Do the tails of the decimal expansion of pi form a dense set in [0,1]? | The comments have pretty much said all there is to be said about $\pi$. I'll just note that the fractional part of $10^n\alpha$ is known to be not just dense but uniformly distributed in $[0,1)$ for all real $\alpha$, except for a set of measure zero. There is no reason to think $\pi$ is in this exceptional set, and no... | 4 | https://mathoverflow.net/users/3684 | 54707 | 34,178 |
https://mathoverflow.net/questions/54650 | 15 | Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x\_1,x\_2,...,x\_n$ s.t. $m=(x\_1,x\_2,...,x\_n)R$. Further, the ideals $(x\_{i\_1},...,x\_{i\_j})$ with $i\_1,...,i\_j\in {1,...,n}$, are prime.
Can we make similar ... | https://mathoverflow.net/users/12800 | prime ideals in regular local rings | As Sandor pointed out, a necessary condition is that the prime ideal $P$ is a complete intersection. Here is a proof that it is also sufficient. It will suffice to prove the following:
**Claim**: Let $(R,m)$ be a Noetherian local ring and $x\in m$ a regular element on $R$. If $R/(x)$ is a domain, then so is $R$.
*... | 7 | https://mathoverflow.net/users/2083 | 54715 | 34,181 |
https://mathoverflow.net/questions/54716 | 3 | In the famous Chen's Theorem which states that every sufficiently large even positive integer $n$ can be written as $n = p + q$, where $p$ is a prime and $q$ is a product of at most two primes. This is the precise 'almost prime' definition we use, where it's either a prime or a product of two primes. Now from the prime... | https://mathoverflow.net/users/10898 | An estimate for 'almost primes'? | The number of almost-primes up to $x$ is asymptotic to $x\log\log x/\log x$. It doesn't matter whether you include the primes or not, as there are only $x/\log x$ of them. I think this gives $n\log n/\log\log n$ as a crude estimate for the $n$th almost prime.
| 5 | https://mathoverflow.net/users/3684 | 54717 | 34,182 |
https://mathoverflow.net/questions/54710 | 4 | Let $n \in \mathbb{N}, n \geq 2$. By minimal Goldbach basis $G\_{2n}$(if it is nonempty) of $2n$ , I mean the minimal set of primes such that every even number less than or equal to $2n$ can be written as a sum of two primes of that set. Minimal refers to the number of elements of the set. For instance,
$G\_{4}= \le... | https://mathoverflow.net/users/5627 | The minimal Goldbach basis | **corrected**: I'll ignore the prime 2 and sums giving 4. Then $G\_{32}=\lbrace 3, 5, 7, 13, 19, 23 \rbrace$. This is the only solution with 6 primes and there are none with 5:
A set of 5 primes could at best give 15 distinct pairwise sums. 3 is needed for 6 and 5,7 for 12. The only ways to get 32 are 3+29 and 13+19.... | 2 | https://mathoverflow.net/users/8008 | 54719 | 34,184 |
https://mathoverflow.net/questions/54722 | 5 | Given a surface M in Euclidean space, we have the generalized-Gauss-map G, i.e. map the tangent spaces into the Grassmannian G(2,n). What is the relation between DG and the second fundamental form of M, and the Gauss curvature?
| https://mathoverflow.net/users/12814 | Generalized Gauss map | Everything generalizes nicely. A nice approach to working this out is described in
Griffiths, P.
On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry.
Duke Math. J. 41 (1974), 775–814.
| 4 | https://mathoverflow.net/users/613 | 54724 | 34,186 |
https://mathoverflow.net/questions/54667 | 5 | Let $X$ be a nontrivial ringed space (i.e. all stalks are nonzero). To every locally free module $M$ on $X$ of constant rank $n$ we can associated it's determinant $\det(M)$, which is a line bundle and is defined as the $n$th exterior power of $M$. We can also define $\det(M)$ if the rank is not assumed to be constant.... | https://mathoverflow.net/users/2841 | universal property of the determinant bundle | There is a paper by Knudsen and Mumford that gives a thorough treatment of determinants of perfect complexes. Mumford has put a copy online [here](http://www.dam.brown.edu/people/mumford/Papers/DigitizedAlgGeomPapers--ForNon-CommercialUse/76b--DetDiv-Knudsen.pdf).
**Edit:** [Knudsen's 2002 paper](http://projecteuclid... | 5 | https://mathoverflow.net/users/121 | 54732 | 34,190 |
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