parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/12452 | 7 | [John's Theorem](http://en.wikipedia.org/wiki/John_ellipsoid) can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ellipsoid. By using this theorem, one can prove that the ellipsoid of maximal... | https://mathoverflow.net/users/2043 | Maximal Ellipsoid | Here's an attempt at a low-brow proof. Take a max area ellipse. Apply an affine transform to
make it a circle; then the problem becomes to show that a minimal area parallelogram containing a circle is a square. It is easy to see that both the height of the parallelogram
and its base are at least the diameter. Q.E.D.
| 12 | https://mathoverflow.net/users/2653 | 12459 | 8,435 |
https://mathoverflow.net/questions/12461 | 7 | Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ under both metrics.
**My question:** Is there a coordinate transformation taking one metric to the other, such that t... | https://mathoverflow.net/users/238 | Changing coordinates so that one Riemannian metric matches another, up to second derivatives | The answer is yes. Just use geodesic normal (also known as exponential) co-ordinates. If you have a book or two on Riemannian geometry, just look for that or a discussion of the exponential map.
[ADDITIONAL COMMENT]
For a 2-dimensional metric, it's a nice exercise to figure all of this out using Jacobi fields. In fac... | 11 | https://mathoverflow.net/users/613 | 12466 | 8,439 |
https://mathoverflow.net/questions/12442 | 1 | **Background**
Ray tracing is very common in computational geometry and the problem is then to find the point of intersection between the equation of a **line** and the equation of a **plane** in **3D**.
The parametric form of the line is given by
$\mathbf{p}\_\mathrm{line}=\mathbf{p}\_\mathrm{a} + \xi (\mathbf{p... | https://mathoverflow.net/users/3414 | Unusual ray tracing | The sets you mention (plane, ellipse) can be expressed as the zero sets of certain polynomial functions; you are asking about the set on which both of these polynomials vanish simultaneously -- this is a basic question in algebraic geometry. One common computational solution is to apply Buchberger's algorithm for compu... | 2 | https://mathoverflow.net/users/1557 | 12470 | 8,441 |
https://mathoverflow.net/questions/12473 | 0 | I'm in the middle of trying to prove something at the moment and am looking for a decomposition of the Lie algebra $\mathfrak{su}(3)$ into a tensor product of some algebra $A$, and another $B$ containing $\mathfrak{su}(2)$, or some such result. Does anyone know of anything?
| https://mathoverflow.net/users/2612 | Does there exist an $A$ and $\mathfrak{su}(2) \subset B$ such that $\mathfrak{su}(3) \simeq A \otimes B$ | $\mathfrak{su}(3)$ cannot be decomposed as a tensor product, since there are no nonabelian simple Lie algebras of dimension 4, 2 or 1 (thus, any 8-d Lie algebra which is a tensor product has a proper ideal, which $\mathfrak{su}(3)$ doesn't).
| 8 | https://mathoverflow.net/users/66 | 12476 | 8,445 |
https://mathoverflow.net/questions/12451 | 5 | There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this literature at <http://www.math.ucr.edu/~jbergner/OneInfty.pdf>.
Does anyone know if similar comparisons have been made for... | https://mathoverflow.net/users/3413 | references for models of stable infinity categories | There seems to be a little confusion in your question about what stable ($\infty$,1)-categories are, so I want to address that first.
The definition of a stable infinity category that I'm familiar with is Jacob Lurie's notion (definition 29 [DAGI:Stable Infty-Categories](http://www.math.harvard.edu/~lurie/papers/DAG... | 6 | https://mathoverflow.net/users/184 | 12480 | 8,448 |
https://mathoverflow.net/questions/12483 | 4 | Suppose I have two $n$-sided polygons A and B. Is there a non-trivial upper bound on the number of parameters (eg. area, perimeter, etc) of the two polygons, that need to be the same, for A and B to be identical?
(For example, if we consider a simpler case to the problem when say A is constrained to lie inside B, then... | https://mathoverflow.net/users/3327 | Uniqueness of a polygon | I guess you are looking for a nice answer, but here is a stupid one.
On the other hand I'm sure that there is no "nice answer".
A polygon can is uniquely determined by length of sides $\ell\_i$ and angles $\alpha\_i$.
Thus we have to find a complete set of invariants for sequence $(\alpha\_1,\ell\_1,\alpha\_2,\ell\_2... | 5 | https://mathoverflow.net/users/1441 | 12487 | 8,450 |
https://mathoverflow.net/questions/12463 | 10 | What can be said about extensions à la $\mathbb{Q}\_p(\sqrt[n]{a})/\mathbb{Q}\_p$? Ramification behaviour, valuation ring, ...?
I find it hard to say anything general - for example, as a function of the $p$-adic valuation of $n$ and/or $a$. Of course some special cases are rather easy to handle, and I understand wha... | https://mathoverflow.net/users/1107 | adding an n-th root to Q_p | If $n$ is prime to $p$, then ${\mathbb Q}\\_p(a^{1/n})$ is unramified if $n | v\_p(a)$,
and is tamely ramified otherwise.
To see this, we note that we may first of all divide $a$ by powers of $p^n$, and so assume
that $0 \leq v\_p(a) < n.$
If in fact $v\_p(a)=0$, i.e. $a$ is a unit, then the extension is unramified,... | 11 | https://mathoverflow.net/users/2874 | 12488 | 8,451 |
https://mathoverflow.net/questions/12472 | 7 | Let X be a DM stack over a field k. We follow the definition in Laumon and Moret-Bailly's book, so that its diagonal is quasi-compact (and hence diagonal is of finite type). Then is the diagonal necessarily finite?
**Edit:** I meant to ask if the inertia $I\to X$ is finite. Recall that the inertia stack $I$ is define... | https://mathoverflow.net/users/370 | Is the inertia stack of a Deligne-Mumford stack always finite? | I'm not sure about the suggested equivalence in the last two sentences of your question, but at least the statement about etale group schemes has a negative answer.
That is, it is possible to have an etale group scheme $G \rightarrow S$,
with $G$ and $S$ both finite type over a field $k$, but $G$ not finite over $S$.... | 6 | https://mathoverflow.net/users/2874 | 12490 | 8,452 |
https://mathoverflow.net/questions/12462 | 8 | how does limsup and liminf for a sequence of sets, apply to probability theory. any real world examples would be much appreciated
| https://mathoverflow.net/users/3421 | limsup and liminf for a sequence of sets | For a sequence of subsets $A\_n$ of a set $X$, the $\limsup A\_n$ $= \cap\_{N=1}^\infty ( \cup\_{n\ge N} A\_n )$ and $\liminf A\_n$ $= \cup\_{N=1}^\infty (\cap\_{n \ge N} A\_n)$.
If $ x \in \limsup A\_n$ then $x$ is in all of the $\cup\_{n\ge N} A\_n$, which means no matter how large you pick $N$ you will find an $A\... | 23 | https://mathoverflow.net/users/3370 | 12497 | 8,459 |
https://mathoverflow.net/questions/12500 | 8 | I think it should be a standard procedure to construct such things, can anyone give a reference or give a hint? Can this be done over any base scheme?
| https://mathoverflow.net/users/1238 | (nontrivial) isotrivial family of elliptic curves | Hint: use quadratic twists.
**Edit**: So as not to drag things out, I hope it's okay if I just give you a standard example. Let
$E\_0: y^2 = x^3 + Ax + B$
be your favorite elliptic curve over $\mathbb{Q}$ (i.e., any will do). Consider the
elliptic curve
$E: t y^2 = x^3 + Ax + B$
over the rational function fie... | 3 | https://mathoverflow.net/users/1149 | 12501 | 8,460 |
https://mathoverflow.net/questions/12503 | 5 | Dear group theorists,
Let $n \geq 1$ and $I$ be an infinite set (you may assume $I$ to be countable). Is the abelian group $(\mathbb{Z}/n)^{(I)}$ (direct sum of copies $\mathbb{Z}/n$) a direct summand of $(\mathbb{Z}/n)^I$ (direct product of copies $\mathbb{Z}/n$)? This question is motivated by [that one](https://mat... | https://mathoverflow.net/users/2841 | (Z/n)^(I) is a direct summand of (Z/n)^I | Yes. The ring ${\mathbb Z}/n {\mathbb Z}$ is injective over itself, and over a Noetherian
ring, direct limits of injectives are again injective; thus ${\mathbb Z}/n{\mathbb Z}^{(I)}$
is injective over ${\mathbb Z}/n{\mathbb Z}$. Finally, any embedding of an injective splits,
as follows directly from the property of bei... | 12 | https://mathoverflow.net/users/2874 | 12504 | 8,461 |
https://mathoverflow.net/questions/12469 | 16 | I was inspired by the following algebraic topology orals question:
"Is $S^1$ the loop space of another space?"
This is easy to see if you recognize that $S^1$ is a $K(\mathbb{Z},1)$, and the loop space of any $K(G,n)$ is a $K(G,n-1)$.
I then also remembered that the loop space functor is a functor from pointed t... | https://mathoverflow.net/users/1622 | Group Structure on CP^infinty | Here's a few thoughts on your questions.
1. See algori's answer. (Incidentally, the "necessity is clear" step is because if $G$ is a topological group then it has a classifying space and then $G \simeq \Omega B G$, hence is homotopy equivalent to a loop space.)
2. For $CP^\infty$, here's a construction that makes it ... | 12 | https://mathoverflow.net/users/45 | 12517 | 8,471 |
https://mathoverflow.net/questions/12423 | 0 | Is there a univariate probability distribution $p\_{\lambda,\alpha}(\beta)$ over the reals, parameterized by $\lambda > 0$ and $1 >= \alpha >= 0$, such that $p\_{\lambda,\alpha} \propto \exp(-\lambda(\alpha \beta^{2} + (1 - \alpha)|\beta|)$? If so, is it a proper distribution (integrates to 1 over the real line)? Does ... | https://mathoverflow.net/users/3407 | univariate prior corresponding to weighted sum of L1 and L2 penalties? | I don't know if that particular density has a name, but it is a valid exponential family density. The normalisation constant does not have a closed form expression, but with a bit of work you can write as a function of the cumulative Gaussian density, by noting that, since the density is symmetrical around 0:
$\int{\ex... | 1 | https://mathoverflow.net/users/3358 | 12523 | 8,476 |
https://mathoverflow.net/questions/12178 | 8 | Given 100 boxes. Each contains arbitrary number of red, blue and green balls, i.e., 100 non-negative integer triples $(r\_i,b\_i,g\_i)$.
Prove it's always possible to find 51 boxes so that the total number of balls of each color in these boxes is no less than the ones from the rest 49 boxes.
For n boxes, replace 51... | https://mathoverflow.net/users/3350 | Balls in boxes (partition) | This proof uses a combinatorial equivalent of the Borsuk-Ulam theorem. I think that the proof is a little more complicated than the average proofs here, so please check my related [paper](http://www.cs.elte.hu/~dom/cikkek/necklace.pdf) if you have difficulties to understand.
Octahedral Tucker's lemma. If for any set-... | 3 | https://mathoverflow.net/users/955 | 12524 | 8,477 |
https://mathoverflow.net/questions/12502 | 1 | With my personal interest and hobby I started this ..
Given a sequence of numbers 1,2,3 .... N
where N is the highest among the sequence and length of the sequence as well ..
I tried my best to bring up a relationship where y=f(n) .. so that .. y (equal or not-equal to n) is an unique value for each value o... | https://mathoverflow.net/users/2972 | Seeking for a formula or an expression to generate non-repeatative random number .. | When we hear random permutations, we bring in our intuition about permutations, and try to give a method which could generate a complicated permutation. Thus, I think we didn't pay enough attention to your examples like n\*3 mod N, which for most situations would not be an acceptable way of generating random numbers. T... | 2 | https://mathoverflow.net/users/2954 | 12543 | 8,489 |
https://mathoverflow.net/questions/12549 | 10 | Let $\pi:P \rightarrow M$ be a principal $G$-bundle, and let $A \in \mathfrak{g}$, where the Lie algebra of $G$ is indicated. The *fundamental field* $A$# used to define connections is given by
$A$#$(p) := \frac{d}{dt}(\exp(At)p)|\_{t=0}$.
$A$# is well defined since $e^{At}p$ can be regarded as a vector in $\pi^{-1... | https://mathoverflow.net/users/1847 | What is a good way to think about a fundamental field on a principal G-bundle? | If you have a homogeneous space $X$ with structure group $G$ (in your case, the fiber passing through $p$) then left multiplication give you a nice action $L: X \times G \to X$. Then for a fixed $p \in X$, $L(p) : G \to X$. The differential of this guy is a map $L(p)\_\* : TG \to TX$ which takes an element of $T\_gG$ t... | 10 | https://mathoverflow.net/users/2510 | 12552 | 8,493 |
https://mathoverflow.net/questions/12538 | 4 | EDIT: Proved it on my own. It easily follows from the Witt integrality theorem. Sorry for posting.
Let $P\in\mathbb{Z}\left[\Xi\right]$ be a polynomial (where $\Xi$ is a family of symbols that we use as indeterminates, for instance $\Xi=\left(X\_1,X\_2,X\_3,...\right)$). Let $n\in\mathbb{N}$.
Prove or disprove that... | https://mathoverflow.net/users/2530 | Do n-th Witt polynomials generate {P | P' is divisible by n} ? | Solved. The key is that the set of polynomials $P$ for which there exist polynomials $P\_d\in\mathbb{Z}\left[\Xi\right]$ for all divisors $d$ of $n$ such that $\displaystyle P=\sum\_{d\mid n}dP\_d^{n/d}$ is a subring of $\mathbb{Z}\left[\Xi\right]$ (by the Witt integrality theorem, which is 9.73 in [Hazewinkel's Witt v... | 5 | https://mathoverflow.net/users/2530 | 12555 | 8,495 |
https://mathoverflow.net/questions/12489 | -4 | Is the given expression, monotonically increasing or decreasing with increasing x?
$\frac{1}{x \log(x)} \sum\_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$
EDIT: This is the derivative of the prime counting function $\pi(x)$ w.r.t. x, ie.,
$\frac{d\pi(x)}{dx} \sim \frac{1}{x \log(x)} \sum\_{i=1}^{\infty} \frac{\mu(i)}{i... | https://mathoverflow.net/users/2865 | Is the given expression, monotonically increasing or decreasing with increasing x? | I'm not sure I should bother answering this question, because it seems like the original poster may not have asked the right question. However, it is a nice exercise in basic asymptotics.
---
For $x$ sufficiently large, the sum in question is decreasing.
First, note that this sum is equal to
$$\sum\_{k \geq 1}... | 8 | https://mathoverflow.net/users/297 | 12557 | 8,497 |
https://mathoverflow.net/questions/12531 | 4 | Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on $X$. By analogy with what happens for finite groups acting on vector spaces, one is tempted to study a sheaf written $\m... | https://mathoverflow.net/users/nan | Non-commutative versions of X/G | Noncommutative versions of sheaves and holomorphic functions are not very well understood. Better understood are noncommutative versions of measurable, continuous, or smooth functions. I generally work with the continuous functions, i.e. $C^\* $-algebras, or various subalgebras that deserve to be called smooth. I'll de... | 5 | https://mathoverflow.net/users/703 | 12560 | 8,499 |
https://mathoverflow.net/questions/12566 | 4 | I'm not sure if my question make sense, but it would also be interesting to know if it didn't, so I will ask anyway:
There exist a countable model of ZF. This means (if I understand it correctly) that we can find a model of ZF in ZF, such that the set of elements in the model is countable seen from the outer ZF.
*... | https://mathoverflow.net/users/2097 | Is every model of ZF countable "seen from the outside"? | Technically, a model of ZF consists of a set with some relation, representing "being an element of". So the literal answer is no. An uncountable model is simply uncountable. Your argument that one can see every model as a model in a model which is countable doesn't work. Cardinality in a model depends on whether there ... | 8 | https://mathoverflow.net/users/35357 | 12568 | 8,503 |
https://mathoverflow.net/questions/12079 | 6 | We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient identities have generalizations to this setting? Just as two simple examples, we have
$\sum\_{k=0}^n \binom{n}{k} = 2^... | https://mathoverflow.net/users/1916 | Gamma function versions of combinatorial identites? | Chapter 5.5 of *Concrete Mathematics* discusses generalizing binomial coefficient identities to the Gamma function. It doesn't discuss the two integrals you mention, though.
Doing a bit of thinking on my own, if $n$ is a positive integer then
$$\int\_{z=0}^n \binom{n}{z} dz = \int\_{z=0}^n \frac{n! dz}{\Gamma(1+z) \G... | 6 | https://mathoverflow.net/users/297 | 12573 | 8,506 |
https://mathoverflow.net/questions/12579 | 2 | I understand that every finite set is recursively enumerable, as I see that one could just encode each element of some finite set on a Turing Machines tape, and then have the machine check each member against any input to determine set membership....
...However, it isn't clear to me how there is an analog to this met... | https://mathoverflow.net/users/3453 | Why is every finite set Diophantine? | Or, very simply stated, given the finite set $S = \{a\_1, \dots , a\_k\}$, consider the diophantine equation: $$(n-a\_1)\dots(n-a\_k)=0.$$
EDIT: Then we can write S as $\{ \ n \ | \ \exists x : (n-a\_1)\dots(n-a\_k)=0\ \}$. (Thanks David)
| 9 | https://mathoverflow.net/users/2693 | 12582 | 8,512 |
https://mathoverflow.net/questions/12527 | 1 | Let's assume we have a [simple linear regression](http://en.wikipedia.org/wiki/Simple_linear_regression) y(x) = a + bx. Is there a way to obtain a probability density function of y for any given x? Would the concept of confidence regions be useful here in any way?
I'm not a big expert in statistics and your help will... | https://mathoverflow.net/users/3432 | Distribution function of dependent variable, confidence regions | You need to make your answer more precise. First, the regression should be
y = a + bx + e
where e has some distribution, let's say described by a density g(e)
Now, do you mean:
1) Obtaining the density f(y|x) theoretically?
This one is easy. By the transformation of densities
f(y|x) = 1/(a+bx)\*g(y/(a+b... | 1 | https://mathoverflow.net/users/3458 | 12595 | 8,520 |
https://mathoverflow.net/questions/12530 | 16 | Let $f = a\_0 + a\_1 x + \ldots + a\_n x^n$ ($f \ne 0$), where $a\_i \in \{-1, 0, 1\}$. Let $p(f)$ be the largest number such that $f(x)$ is divisible by $y$ for any integer $x$ and for any $1 \leq y \leq p(f)$. Let $g(n)=max\_f\; p(f)$. Is it true that $g(n) = o(n)$? What is the best upper or lower bound on $g(n)$ can... | https://mathoverflow.net/users/3448 | Question about polynomials with coefficients in Z | I'll prove the upper bound $g(n) = O(n^{1/2+o(1)})$, which is essentially best possible if Kevin Costello's heuristics are correct.
Suppose that $q$ is a prime with $q > n^{1/2}+1$. Reducing $f(x)$ modulo $x^{q-1}-1$ in $\mathbb{Z}[x]$ amounts to reducing the *exponents* modulo $q-1$, so the result is a polynomial $h... | 20 | https://mathoverflow.net/users/2757 | 12596 | 8,521 |
https://mathoverflow.net/questions/12584 | 22 | In ordinary membership-based set theory, the **axiom schema of replacement** states that if $\phi$ is a first-order formula, and $A$ is a set such that for any $x\in A$ there exists a unique $y$ such that $\phi(x,y)$, then there exists a set $B$ such that $y\in B$ if and only if $\phi(x,y)$ for some $x\in A$. That is, ... | https://mathoverflow.net/users/49 | When does collection imply replacement? | The answer is no, if you allow me to adopt some weak-but-equivalent forms of the other axioms. And the reason is interesting:
* A shocking number of the axioms of set theory are true in the non-negative real line R+, with the usual order < being used to interpret set membership. (!)
Let's just check. For example, ... | 34 | https://mathoverflow.net/users/1946 | 12597 | 8,522 |
https://mathoverflow.net/questions/12601 | 18 | This question originated from a conversation with Dmitry that took place here
[Is there a complex structure on the 6-sphere?](https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere)
The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of a holomorphic vector bundle on... | https://mathoverflow.net/users/2349 | A topological consequence of Riemann-Roch in the almost complex case | I believe that the displayed equation is valid for almost complex manifolds. This is closely related to a computation I talked about [here](https://mathoverflow.net/questions/10630/why-do-todd-classes-appear-in-grothendieck-riemann-roch-formula).
Let $r\_1$, $r\_2$, ..., $r\_n$ be the chern roots of the tangent bundl... | 15 | https://mathoverflow.net/users/297 | 12604 | 8,526 |
https://mathoverflow.net/questions/12072 | 14 | Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring $\mathcal{C}(k)$ of all Cauchy sequences in $k$ -- viewed as a subring of $k^{\infty} = \prod\_{i=1}^{\infty} k$ -- by the max... | https://mathoverflow.net/users/1149 | What is the prime spectrum of a Cauchy series ring? | In this answer I will treat the case in which $|\text{ }|$ is not discrete.
I first claim that $\mathfrak m\_0$ is not the restriction of any proper ideal in
$k^{\infty}.$ Indeed, choose $x \in k$ such that $0 < |x| < 1$. Then $(x^i)$
is an element of $\mathfrak m\_0$ which is invertible in $k^{\infty}$ (with
inverse... | 5 | https://mathoverflow.net/users/2874 | 12607 | 8,528 |
https://mathoverflow.net/questions/12559 | -2 |
```
Cardinal Equivalence Theorem
```
For each boolean formula, |quantifications| = |assignments|.
The set of valid quantifications has some cardinality, call that |Q(B)|.
The set of satisfying assignments has some cardinality, call that |P(B)|.
Those two numbers are equal, |Q(B)| = |P(B)|, range from 0 through 2... | https://mathoverflow.net/users/3446 | cardinal equivalence: for each boolean formula, |quantifications| = |assignments|. | I think Daniel might be asking about the following proposition:
>
> Let $f:\{0,1\}^n\to\{0,1\}$ be any function (i. e., an "n-ary boolean function"). The number of true formulas $$ Q\\_1 x\\_1 \ldots Q\\_n x\\_n : f(x\\_1,\ldots,x\\_n) = 1,$$ where each $Q\\_i$ is a quantifier $\forall$ or $\exists$, is equal to t... | 11 | https://mathoverflow.net/users/302 | 12608 | 8,529 |
https://mathoverflow.net/questions/12606 | 19 | Today I was studying for a qualifying exam, and I came up with the following question;
>
> Is there a simple description in terms of the subspaces universal covers for the universal cover of a wedge product?
>
>
>
This question came about after calculating universal covers of the wedge of spheres ($\mathbb{S}^... | https://mathoverflow.net/users/348 | Universal Covering Space of Wedge Products | If $X$ and $Y$ are two reasonable spaces with universal covers $\tilde{X}$ and $\tilde{Y}$, there is a nice picture of the universal cover $\widetilde{X \vee Y}$ which has the combinatorial pattern of an infinite tree. The tree is bipartite with vertices labeled by the symbols $X$ and $Y$. The edges from an $X$ vertex ... | 23 | https://mathoverflow.net/users/1450 | 12611 | 8,532 |
https://mathoverflow.net/questions/12569 | 15 | Hi everyone,
I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the moment, but I would like to have a more detailed source that goes through or gives an overview of the constructions of clas... | https://mathoverflow.net/users/259 | References for logarithmic geometry | I put up some old notes by Illusie [here](http://math.harvard.edu/~tdp/Illusie-Log.geometry.lecture.notes-single-page.pdf) for you; they're very detailed and treat log smoothness, the log de Rham complex, and other topics in their second exposé. They're my favourite first reference.
There is also Ogus's book, the lat... | 13 | https://mathoverflow.net/users/307 | 12612 | 8,533 |
https://mathoverflow.net/questions/3030 | 1 | The famous Black-Scholes framework is usually derived using a hedging approach where a self-financing portfolio is constructed and the resulting stochastic differential equation is being solved under some conditions.
The self-financing portfolio is basically a dynamic trading strategy where according to the actual pr... | https://mathoverflow.net/users/1047 | Matching dynamic trading strategies with derivatives | At last I found two papers as a starting point that do exactly that:
* Asset allocation and derivatives by Martin B Haugh and Andrew W Lo
MIT Sloan School of Management and Operations Research Center, Cambridge, USA, 14 November 2000
<http://alo.mit.edu/wp-content/uploads/2015/08/AssetAllocationDerivatives2001.pdf> <... | 2 | https://mathoverflow.net/users/1047 | 12615 | 8,535 |
https://mathoverflow.net/questions/12621 | 3 | In "Rings, Modules, and Algebras in Stable Homotopy Theory" Elmendorf, Kriz, Mandell and May introduce a notation of spectra indexed over an universe $\mathcal{U}$ as a collection of pointed topological spaces index by finite subspaces of the universe. Has anyone seen a definition using pointed simplicial sets instead?... | https://mathoverflow.net/users/2146 | Are there universe-indexed spectra over simplicial sets? | Yes to both interpretations of your question. It is not clear to me where you want to put pointed simplicial sets.
One interpretation of your question is that you want to replace pointed topological spaces with pointed simplicial giving the notion of a spectrum as a functor from supspaces of U to pointed sSet. This ... | 8 | https://mathoverflow.net/users/184 | 12625 | 8,540 |
https://mathoverflow.net/questions/12629 | 3 | Is the category of groups with group-homomorphisms the same as the category of models of group theory with elementary maps?
If not so: why?
| https://mathoverflow.net/users/2672 | Category of groups = Category of models of group theory? | The Categories will be fundamentally different. The category of groups with group homomorphisms, (even with monomorphisms) enjoys a directedness property: any two groups can map monomorphically into their direct sum.
But in the category of models of group theory under elementary maps, the finite groups map elementar... | 3 | https://mathoverflow.net/users/1946 | 12635 | 8,542 |
https://mathoverflow.net/questions/12657 | 12 | This is an elementary question, but a little subtle so I hope it is suitable for MO.
Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$.
The characteristic polynomial $T - \lambda I$ splits into linear factors like $T - \lambda\_iI$, and we have the Jordan canonical form:
$$ J = \begin{bmatrix} J\_1 \\\ ... | https://mathoverflow.net/users/2938 | Proving "almost all matrices over C are diagonalizable". | The discriminant of the characteristic polynomial of a matrix depends polynomially on the coefficients of the matrix, and its vanishing detects precisely the existence of multiple eigenvalues. Therefore the set where the discriminant does not vanish is contained in the set of diagonalizable matrices.
Now the set wher... | 36 | https://mathoverflow.net/users/1409 | 12659 | 8,559 |
https://mathoverflow.net/questions/786 | 9 | Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. There are plenty of other connections to homological algebra. If we use cyclic homology, there are connections to geometry a... | https://mathoverflow.net/users/351 | Hochschild/cyclic homology of von Neumann algebras: useless? | The question is not well-posed. There are various versions of cyclic theory (for instance) which differ according to continuity conditions that are assumed. In Connes' original IHES papers he deals with both discrete (useful for arbitrary rings) and topological (useful in the $C^\infty $-setting.)
The basic problem i... | 10 | https://mathoverflow.net/users/3468 | 12664 | 8,562 |
https://mathoverflow.net/questions/12638 | 48 | Do you find it a good idea to take lecture notes (even detailed lecture notes) in mathematical lectures?
Related question: [Digital Pen for Math: Your Experiences?](https://mathoverflow.net/questions/12898/digital-pen-for-math-your-experiences)
| https://mathoverflow.net/users/1532 | Taking lecture notes in lectures | I usually bring a pad of paper with me to talks, but don't take notes. I do write down things I want to revisit later, and sometimes it turns into full blown note-taking. Taking notes reduces my ability to concentrate on the lecture, so if it's a really difficult lecture or if the material is totally new to me, taking ... | 65 | https://mathoverflow.net/users/1 | 12673 | 8,570 |
https://mathoverflow.net/questions/12652 | 26 | There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will probably have the best chance of an answer. Roughly the free loop space is a space which is supposed to be the space of ma... | https://mathoverflow.net/users/184 | Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds? | They are the same.
If you want the full gory details, read "A Convenient Setting of Global Analysis" by Kriegl and Michor (available as a free PDF via the AMS bookstore). For a gentler approach, read my online seminar notes "The Differential Topology of Loop Spaces". Michor has also written a fair bit on manifolds of... | 16 | https://mathoverflow.net/users/45 | 12685 | 8,578 |
https://mathoverflow.net/questions/12688 | 21 | I always had trouble remembering this. Is it true that a curve over a non-algebraically-closed field is normal implies that it's non-singular? How about a 1 dimensional scheme? How about dimension 2? I think I heard once that surfaces over a non-algebraically closed field is normal does imply that it's non-singular. Is... | https://mathoverflow.net/users/3238 | Nonsingular/Normal Schemes | For curves over a field $k$, normal implies regular. (The point is that a normal Noetherian local ring
of dimension one is automatically regular, i.e. a DVR.) If $k$ is not
perfect, then it might not be smooth over $k$.
The reason is that in this case it is possible to have a regular local $k$-algebra of dimension o... | 40 | https://mathoverflow.net/users/2874 | 12690 | 8,582 |
https://mathoverflow.net/questions/11603 | 18 | Finite projective planes are fascinating objects from many perspectives. In addition to the geometric view, they can be viewed as combinatorial block designs.
From the geometric perspective, there are two very important structural properties for projective planes: the [Theorem of Desargues](http://en.wikipedia.org/wi... | https://mathoverflow.net/users/2000 | Are there analogues of Desargues and Pappus for block designs? | I passed on your question to [John H. Conway](http://en.wikipedia.org/wiki/John_Horton_Conway). Here is his response: (**NB**. Everything following this line is from Conway and is written from his point of view. Of course, in the comments and elsewhere on the site, I am not Conway.)
I think it's wrong to focus on blo... | 10 | https://mathoverflow.net/users/1079 | 12720 | 8,606 |
https://mathoverflow.net/questions/12679 | 2 | What does a coequalizer in the category of primitive recursive functions look like? I know that in Set, a coequalizer is a *minimum* congruence but...what is it in particular in the category of primitive recursive functions (which is a subcategory of Set)?
| https://mathoverflow.net/users/3338 | Coequalizer in the category of primitive recursive functions | For those readers unfamiliar with the class of [primitive recursive functions](http://en.wikipedia.org/wiki/Primitive_recursive_function), let me say that you may simplify things somewhat by fruitfully thinking of them as poor cousins of the computable functions. They were introduced, before Turing, as a natural class ... | 8 | https://mathoverflow.net/users/1946 | 12721 | 8,607 |
https://mathoverflow.net/questions/12717 | 11 | In the familiar case of (smooth projective) curves over an algebraically closed fields, (closed) points correspond to DVR's.
What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points? Is there a cha... | https://mathoverflow.net/users/3238 | Points and DVR's |
>
> What if we have a non-singular projective curve over a non-algebraically closed field? The closed points will certainly induce DVR's, but would all DVR's come from closed points?
>
>
>
Yes: let $L/k$ be a function field in one variable, so it can be given as a finite separable extension of $K = k(t)$. Then t... | 11 | https://mathoverflow.net/users/1149 | 12723 | 8,609 |
https://mathoverflow.net/questions/12716 | 1 | Is it possible to express the mean curvature of a surface of revolution in terms of the first derivative of the polar tangential angle?
To be specific: Let $r=u(\theta)$ be a polar curve in the first quadrant of the xy-plane. The surface of revolution is generated by rotating this curve around the y axis. Let $\psi$ ... | https://mathoverflow.net/users/3477 | mean curvature and polar tangential angle | Yes, it is true $H=F(u,\theta,\psi,\psi')$.
You ask for suggestion: *Calculate both principle curvatures,*
$k\_1=k\_1(u,\theta,\psi)$ *and* $k\_2=k\_2(u,\theta,\psi,\psi')$.
Here $k\_2$ is the curvature of original curve $r=u(\theta)$
and $k\_1$ is principle curvature in the normal direction.
| 2 | https://mathoverflow.net/users/1441 | 12727 | 8,613 |
https://mathoverflow.net/questions/12633 | 1 | Hi,
As part of one of my courses I need to simulate Gaussian, Student-T and Clayton copulas. The only way to do it that I am aware of uses the conditional copulas, so
$$C\_{1|2}(u, v) = P[X = F\_1^{-1}(u) | Y = F\_2^{-1}(v)]$$
What are the formulae for conditional copulas for Gaussian, Student-T and Clayton?
| https://mathoverflow.net/users/3160 | Conditional copulas | I understand that you want to generate and plot samples of Gaussian, t and Clayton copulas.
The method of sampling is different for the first two cases (elliptical copula) and the
third case (Archimedean copula).
Here is a brief description of the sampling. A reference for both cases is given:
In the first two case... | 4 | https://mathoverflow.net/users/1059 | 12728 | 8,614 |
https://mathoverflow.net/questions/9474 | 31 | Reading [Serre's letter to Gray](http://books.google.de/books?id=4Vm4Lq1WSHgC&pg=RA1-PA722&lpg=RA1-PA722&dq=Jean-Pierre+Serre+%22Extensions+icosaedriques%22&source=bl&ots=CTXk-_vVqx&sig=kFq8FuyhYs7sGPOukgPe8kbQcBY&hl=de&ei=vXwvS8jxApjknAOiwY3PBA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CA0Q6AEwAA#v=onepage&q=&f=fals... | https://mathoverflow.net/users/451 | Do there exist modern expositions of Klein's Icosahedron? | "Geometry of the Quintic" is available for free at my website.
Jerry Shurman
| 37 | https://mathoverflow.net/users/3479 | 12729 | 8,615 |
https://mathoverflow.net/questions/12658 | 2 | Suppose you have a finite dimensional real Hilbert space $V$ and you form the tensor algebra
$$T(V) = \mathbb{R} \oplus V \oplus (V\otimes V) \oplus (V\otimes V \otimes V) \oplus \cdots$$
where the multiplication is given by the tensor product. Now imagine you have a finite set of vectors $S$ which you use to form an i... | https://mathoverflow.net/users/3467 | Norm on quotient algebra of a tensor algebra |
>
> Intuitively, I would expect Q in this case to be a two dimensional vector space with basis 1,[a]. Is this correct? Projecting onto the orthogonal complement seems to give zero however.
>
>
>
The first part is correct. This may be easiest to see by considering the isomorphism, in the present case where $V$ is... | 3 | https://mathoverflow.net/users/1119 | 12736 | 8,620 |
https://mathoverflow.net/questions/11811 | 4 | ### Question
Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What are some of the available methods for showing that this map has good reduction, or integral reduction, at some s∈S? By this I mean: what are some popular conditions that imply this? What are the tricks people usually use?
### C... | https://mathoverflow.net/users/2665 | Methods of showing a map has integral or good reduction | You might have a look [here](http://www.math.u-bordeaux1.fr/~liu/articles/tokyo.ps).
| 7 | https://mathoverflow.net/users/3485 | 12739 | 8,622 |
https://mathoverflow.net/questions/12676 | 9 | Is there a relative version of sheaf cohomology?
EDIT: I rather mean the cohomology of pairs.
| https://mathoverflow.net/users/3470 | Relative version of sheaf cohomology? | It turns out that my previous answer dealt with the wrong question. The answer to the
new question is also yes:
local cohomology $H\\_Z(X,\mathcal F)$ corresponds to cohomology of the pair $(X,X\setminus Z)$.
| 7 | https://mathoverflow.net/users/2874 | 12743 | 8,624 |
https://mathoverflow.net/questions/12745 | 5 | Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on
* tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential map sends a vector to a point on a curve naturally defined in terms of the vector;
* unital Banach algebras?
(NB. I a... | https://mathoverflow.net/users/1847 | What is an exponential? | I think the exponential function for unital Banach algebras is a special case of the exponential functions for Lie groups modeled on topological vector spaces: the set of invertible elements in an unital Banach algebra naturally is a Banach Lie group, and the exponential function of this Lie group is the "classical" ex... | 6 | https://mathoverflow.net/users/3108 | 12747 | 8,626 |
https://mathoverflow.net/questions/12742 | 7 | By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions
in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$
the distance between $x$ and the nearest integer, for any real number $x$
(thus $\rho(3)=0,\rho(3.2)=0.2,\rho(3.5)=0.5, \rho(3.9)=0.1$ etc).
An "approximate" version... | https://mathoverflow.net/users/2389 | Variant of Fermat's last theorem | The condition you want is very weak, and there are clearly many accidental solutions.
You can add severe restrictions and still find many solutions. For example, (as Steven Sivek pointed out) you can force $u\_n = v\_n$ and then by the theory of simple continued fractions, there are infinitely many $p\_n/q\_n$ so tha... | 8 | https://mathoverflow.net/users/2954 | 12752 | 8,629 |
https://mathoverflow.net/questions/12758 | 18 | **Question.**
Let $C\_1,\dots,C\_k$ be conjugacy classes in the symmetric group $S\_n$. (More explicitly,
each $C\_i$ is given by a partition of $n$; $C\_i$ consists of permutations whose cycles
have the length prescribed by the partition.) Give a necessary and sufficient condition on
$C\_i$ that would ensure that ther... | https://mathoverflow.net/users/2653 | Deligne-Simpson problem in the symmetric group | This question is a more than 100 years old problem and it is called in the litearture "Hurwitz exitence problem". This is an open problem. Though many partial cases are solved. For example you can check the following article
On the existence of branched coverings between surfaces with prescribed branch data, I
Ekater... | 9 | https://mathoverflow.net/users/943 | 12759 | 8,633 |
https://mathoverflow.net/questions/12732 | 28 | There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the following question: **is the theory of categories decidable?**
More specifically, I was wondering whether or not statem... | https://mathoverflow.net/users/1079 | Is the theory of categories decidable? | Thanks for clarifying your question. The formulation that
you and Dorais give seems perfectly reasonable. You have a
first order language for category theory, where you can
quantify over objects and morphisms, you can compose
morphisms appropriately and you can express that a given
object is the initial or terminal obj... | 37 | https://mathoverflow.net/users/1946 | 12760 | 8,634 |
https://mathoverflow.net/questions/12746 | 5 | Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition.
What's a reference for that?
| https://mathoverflow.net/users/381 | transfinite composition of weak equivalences in sSet | I don't have a complete reference (and like Tyler, I don't know exactly what result you want). But here are some observations:
* there is a functor $\mathrm{Ex}^\infty$, which replaces a simplicial set with a weakly equivalent fibrant replacement, and which commutes with filtered colimits. (See ch. 3 of Goerss-Jardin... | 7 | https://mathoverflow.net/users/437 | 12766 | 8,637 |
https://mathoverflow.net/questions/12769 | 3 | Given a partial order $R\_{\leq}$ over a set $D$, the set of upper bounds under $R$ of a subset $S$ of $D$ is commonly defined as $\{ y \in D | \ \forall x\in S, x R y \}$.
(The set of lower bounds of $S$ may be defined as the set of upper bounds of $S$ under the converse relation $R^{-1}$)
Is there a common name f... | https://mathoverflow.net/users/3492 | Name for "lower/upper bounds" of arbitrary relations? | If your relation is at all order-like, then I would recommend just staying with the upper/lower bound terminology. And unless I misunderstand you, the example you describe is actually a (strict) partial order, no? If the relation only goes from disjoint sets D to D', then this is (vacuously) transitive, irreflexive and... | 2 | https://mathoverflow.net/users/1946 | 12778 | 8,643 |
https://mathoverflow.net/questions/12767 | 9 | I have a little problem. I'm probably being just so careless..... Here k-varieties are all integral separated k-schemes of finite type over k, where k is a field.
Suppose $X, Y$ are $k$-varieties, and let $f :X \to Y$ be a morphism of $k$-varieties that is one to one and onto. Then, when can we say this $f$ is an iso... | https://mathoverflow.net/users/3168 | When two k-varieties with the same underlying topological spaces isomorphic? | The condition you are looking for is **seminormality**. A variety (or a reduced scheme) $Y$ is *seminormal* if any proper bijective morphism $f:X\to Y$, with $X$ reduced, *inducing isomorphisms on residue fields* $k(y)=k(x)$ for points $x\in X$, $y=f(x)\in Y$, is an isomorphism. A basic fact is that any variety has a u... | 12 | https://mathoverflow.net/users/1784 | 12795 | 8,656 |
https://mathoverflow.net/questions/12763 | 2 | Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of t?
The transformation of the triangle from V -> V1 over time means the vertices Va,Vb,Vc each move linearly and indep... | https://mathoverflow.net/users/3491 | find the collision of a particle with a swept triangle. | This type of problem is usually called "continuous collision detection". There is a substantial literature on this subject as you'll discover if you try a [google search](http://www.google.com/search?q=%22continuous+collision+detection%22) on those key words. It's rare that someone wants to do this for one triangle at ... | 7 | https://mathoverflow.net/users/1233 | 12805 | 8,664 |
https://mathoverflow.net/questions/12804 | 24 | A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of cardinality $<\lambda$) and b. for all cardinals $\mu<\lambda$ we have $\mu^+<\lambda$ where $\mu^+$ is the successor of $\mu$. Str... | https://mathoverflow.net/users/2349 | Large cardinal axioms and Grothendieck universes | A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equivalent to the existence of one inaccessible cardinal. These cardinals and the corresponding universes have been studied in... | 31 | https://mathoverflow.net/users/1946 | 12809 | 8,668 |
https://mathoverflow.net/questions/12814 | 18 | I know this should be pretty simple, but right now the only way I can see how to prove it is to sit down and write out explicit formulae for the group law, and see that everything works out. What's the geometric or abstract-nonsense reason why the abelian group structure of elliptic curves behaves nicely under homomorp... | https://mathoverflow.net/users/382 | Why does the group law commute with morphisms of elliptic curves? | This follows by a rigidity property of certain morphisms. It is important to note that elliptic curves are complete, that is, proper and integral schemes. Then we have the following "Rigidity Lemma" (see Mumford's Abelian Varieties, the beginning of chapter II (page 43 of the old edition), for example):
Let $X$ be a ... | 33 | https://mathoverflow.net/users/88 | 12818 | 8,673 |
https://mathoverflow.net/questions/12819 | 9 |
>
> A graph is Hamiltonian if and only if
> its closure is Hamiltonian.
>
>
>
I am looking for a simple (i.e. short) proof of the theorem, that I can use as part of an article on topological sorting.
I've not been able to find one in the literature I have access to. Any help would be appreciated.
| https://mathoverflow.net/users/3499 | Proof of Bondy and Chvátal Theorem | Let $G=G\_0, G\_1, G\_2$, etc. be a sequence of graphs where each $G\_i$ is formed by performing a single closure step to $G\_{i-1}$ — that is, add an edge $uv$ to $G\_i$ when $u$ and $v$ together have at least $n$ neighbors. If any graph in this sequence is Hamiltonian, let $k$ be the minimum $k$ such that $G\_k$ is H... | 12 | https://mathoverflow.net/users/440 | 12821 | 8,675 |
https://mathoverflow.net/questions/12482 | 3 | I'm trying to understand the topology behind a certain group which fits into a truncated crossed complex, so I've been trying to understand Brown's construction of the classifying space of a crossed complex. I asked a similar question [here](https://mathoverflow.net/questions/2734/classifying-space-of-a-crossed-complex... | https://mathoverflow.net/users/343 | Explicit classifying spaces for crossed complexes | It is not clear to me what you need / want. The classifying space of a cyclic group is constructed using a presentation and then killing off higher identities that may be around (there aren't any!). From that viewpoint the question you seem to ask is related to the combinatorial group theory of the group in question (o... | 2 | https://mathoverflow.net/users/3502 | 12850 | 8,695 |
https://mathoverflow.net/questions/12861 | 13 | Let G be a finite group and let F be an algebraically closed field. If the characteristic of F is 0, then the number of irreducible F-representations of G is given by the number of conjugacy classes of elements of G. A paper I'm reading says that if the characteristic of F is p>0, then the number of F-irreps of G is th... | https://mathoverflow.net/users/493 | Representations in characteristic p | Yes, you are correct. The point is that a $p$-group acting in char. $p$ always has
a fixed point (and so acts trivially on an irrep.). So *every* irrep. of $G$
in char. $p$ factors through $G'$, as you anticipated.
The proof of the claim about $p$-groups is not hard. One approach (in general,
even when $P$ is not nec... | 19 | https://mathoverflow.net/users/2874 | 12862 | 8,703 |
https://mathoverflow.net/questions/12765 | 30 | I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and flat ring maps). However, I've never seriously studied Algebraic geomtry. Can anyone recommend a book that builds stacks d... | https://mathoverflow.net/users/1353 | Algebraic stacks from scratch | Another good place to look are the notes of [Master's course on stacks by Betrand Toen](https://ncatlab.org/nlab/show/Master+course+on+algebraic+stacks). I think they pretty much do exactly what you are looking for.
Here's the quick summary: You will want to read section 1 of Cours 2 where the term geometric context ... | 17 | https://mathoverflow.net/users/473 | 12866 | 8,705 |
https://mathoverflow.net/questions/12865 | 26 | As everybody knows, the ZFC axioms may serve as a foundation for (almost)
all of contemporary mathematics, and it is also well-known that several results
are "indecidable" in ZFC, which means that they cannot be proved or disproved within
ZFC.
It is therefore natural to look for "new axioms" to add to ZFC and make i... | https://mathoverflow.net/users/2389 | Using consistency to create new axioms in set theory | Such constructions are interesting! However, they are often done with PA instead of ZFC (see note). For an interesting discussion, I recommend Torkel Franzén's book *Inexhaustibility: a non-exhaustive treatment* (Lecture Notes in Logic 16, ASL, 2004). You can also read this excellent [blog article](http://xorshammer.co... | 30 | https://mathoverflow.net/users/2000 | 12867 | 8,706 |
https://mathoverflow.net/questions/12847 | 15 | Suppose a compact Lie group $G$ acts on a manifold $M$ with only one orbit type $G/H$ ($H$ denotes the stabiliser group). Then the manifold $M$ becomes a fibre bundle over the quotient manifold $X:=M/G$ with typical fibre $G/H$ and structure group $G$.
On the one hand one could look at the cotangent bundle $T^\* X$ ... | https://mathoverflow.net/users/3509 | cotangent bundle symplectic reduction and fibre bundles | These two symplectic manifolds are canonically symplectomorphic.
Notice first, that the map $\mu$ vanishes on the sub-bundle of $T^\* M$ of 1-forms vanishing on the fibers of the fibration $M\to X$. Let us call this sub-bundle by $T\_h ^\* M$ (h- for horizontal).
To construct the symplectomorphism notice that ther... | 13 | https://mathoverflow.net/users/943 | 12868 | 8,707 |
https://mathoverflow.net/questions/12871 | 7 | Suppose $G$ is a finite group, and $l$ is a prime, with $l$ coprime to the order $|G|$. (Thus we have complete reducibility for $G$ representations.) Is there a straightforward condition on $l$ which ensures that every irreducible representation of $G$ is liftable to a characteristic zero representation? (For instance,... | https://mathoverflow.net/users/3513 | When are all characteristic l representations liftable | Yes, $\operatorname{gcd}(l,|G|) = 1$ is sufficient. This is an easy consequence of Brauer's modular representation theory. See Serre's *Linear Representations of Finite Groups*, especially Chapter 18.
| 10 | https://mathoverflow.net/users/1149 | 12872 | 8,709 |
https://mathoverflow.net/questions/12889 | 6 | Let $h:M\to M$ be a homeomorphism of a compact manifold. Let $p:\tilde M\to M$ be a covering. 1) Is it always possible to lift $h$ to $H:\tilde M\to \tilde M$ so that everything fits into the commutative diagram?
2) Given such a diagram assume additionally that p is a self-covering. Is it true that $H$ is necessarily h... | https://mathoverflow.net/users/3375 | Lifting a homeomorphism, always possible? | No. Take any homeomorphism that doesn't preserve the subgroup of $\pi\_1$ that lift to closed paths in the covering. For example, take the 2:1 covering $S^1\to S^1$ take the product with the identity map on $S^1$. Let $h$ be the homeomorphism switching the factors.
In general, I believe a homeomorphism will lift if a... | 9 | https://mathoverflow.net/users/66 | 12890 | 8,718 |
https://mathoverflow.net/questions/12892 | 5 | I'm posting my answer to [this question](https://mathoverflow.net/questions/5146/algebraic-geometry-versus-complex-geometry) as its own question:
Let $V$ be an irreducible projective variety over $\mathbb{C}$. Let $U$ be a Zariski open set in $V$. I'll use $V(\mathbb{C})$ and $U(\mathbb{C})$ to mean $V$ and $U$ equip... | https://mathoverflow.net/users/1335 | Why can't subvarieties separate? | [This has been completely rewritten at the request of Autumn Kent.]
Let $X$ be an irreducible topological space and $U$ a non-empty open subset of $X$. Then $U$ is also irreducible -- see e.g. Proposition 141 on page 88 [here](http://alpha.math.uga.edu/%7Epete/integral.pdf). (Surely it's also in Hartshorne and lots o... | 10 | https://mathoverflow.net/users/1149 | 12896 | 8,721 |
https://mathoverflow.net/questions/12894 | 40 | I am writing a short paper in the area of combinatorics.
When the paper is complete, I would like to be able to submit it to arXiv.
The reasons that I would like to submit to arXiv are:
1. To obtain a date and time stamp from a central authority so that I can prove the work is mine.
2. To promote access to the pa... | https://mathoverflow.net/users/126024 | Submitting to arXiv when unaffiliated | I think you need to be endorsed first. See this link: <http://arxiv.org/help/endorsement>
| 17 | https://mathoverflow.net/users/2264 | 12897 | 8,722 |
https://mathoverflow.net/questions/12906 | 1 | The collection of all groups is a proper class, since every set gives rise to a group. But what about the collection of all isomorphism classes of groups? By which argument do I see, that it is a set or a proper class?
| https://mathoverflow.net/users/2672 | Is the collection of isomorphism classes of groups a proper class? | Since free groups are isomorphic if and only if they have a basis of the same cardinality (probably assuming some choice axiom here), the collection of all isomorphism classes of groups has at least the size of the collection of all isomorphism classes of sets, hence is not a set.
| 10 | https://mathoverflow.net/users/45 | 12909 | 8,729 |
https://mathoverflow.net/questions/12902 | 4 | This is a spin-off question from [How to select a journal?](https://mathoverflow.net/questions/7284/how-to-select-a-journal). Is there is any data available regarding processing time (acceptance time, time from submission to publication, or similar) specifically for combinatorics journals? Ideally, this data would be u... | https://mathoverflow.net/users/2264 | Combinatorics journals processing time | One possibility is the "backlog" published annually in the AMS Notices. The latest version is [here (as PDF)](http://www.ams.org/notices/200910/rtx091001316p.pdf).
| 8 | https://mathoverflow.net/users/45 | 12917 | 8,734 |
https://mathoverflow.net/questions/12782 | 15 | **Situation**
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by $\underline{G\text{-mod}}$ the associated stable category with loop functor $\Omega$.
For any Frobenius category $({\mathcal A},{\mathcal... | https://mathoverflow.net/users/3108 | Tate Cohomology via stable categories | To address Hanno's question about checking that composition gives a graded-commutative ring structure on $End^{\*}(\mathbb{Z}) = \oplus\_i [\mathbb{Z}, \Omega^{-i} \mathbb{Z}]$ suppose first that
$a \stackrel{f}{\to} b \stackrel{g}{\to} c \stackrel{h}{\to} \Omega^{-1} a$
is a distinguished triangle in the stable ... | 4 | https://mathoverflow.net/users/310 | 12924 | 8,737 |
https://mathoverflow.net/questions/12920 | 24 | Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The *Stokes theorem* tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any differentiable $n$-form $\omega$ on $M$, we have $\int\_{\partial M} \omega = \int\_M d\omega $.
But Stokes theorem... | https://mathoverflow.net/users/1246 | Stokes theorem for manifolds with corners? | The most general form of Stokes' theorem I know of is proved in the book
Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.
The aim in the book is to provide a version of the divergence theorem which holds also in cases where the boundary has certain singularities (as y... | 22 | https://mathoverflow.net/users/3509 | 12927 | 8,740 |
https://mathoverflow.net/questions/12935 | 7 | It is well known that for any set A in R^d there exists a measurable set E such that E contains A and m\*(A)=m\*(E). Is it possible to go the other direction?
In other words, is it true that for any measurable set E (such that m(E)>0) there is a non-measurable subset A such that m\*(A)=m\*(E)?
| https://mathoverflow.net/users/3532 | Non Lebesgue measurable subsets with "large" outer measure | A set $E$ with positive Lebesgue measure can be decomposed as a union $E = A \cup B$ where each of $A$ and $B$ have zero inner measure, and therefore each of $A$ and $B$ are nonmeasurable with $m^\*(A) = m^\*(B) = m(E)$.
An example for this construction is a [Bernstein set](http://en.wikipedia.org/wiki/Bernstein_set... | 15 | https://mathoverflow.net/users/454 | 12938 | 8,750 |
https://mathoverflow.net/questions/12943 | 56 | For any set X, let SX be the symmetric group on
X, the group of permutations of X.
My question is: Can there be two nonempty sets X and Y with
different cardinalities, but for which SX is
isomorphic to SY?
Certainly there are no *finite* examples, since the symmetric
group on n elements has n! many elements, so the... | https://mathoverflow.net/users/1946 | Can the symmetric groups on sets of different cardinalities be isomorphic? | According to the Schreier–Ulam–Baer theorem, the nontrivial normal subgroups of $S(X)$ are *(i)* the subgroup $S\_\mathrm{fin}(X)$ of permutations of $X$ of finite support, *(ii)* the subgroup $A\_\mathrm{fin}(X)$ of $S\_\mathrm{fin}(X)$ of even permutations, and *(iii)* for each cardinal $\kappa$ the subgroups $S\_{<\... | 53 | https://mathoverflow.net/users/1409 | 12946 | 8,756 |
https://mathoverflow.net/questions/6834 | 11 | I was wondering when the Kunneth formula holds for motivic cohomology:
$$
H^p(X,A(\alpha)) = \bigoplus\_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma))
$$
where $H^p(X,A(\alpha))$ is defined as you wish: by higher Chow groups, Hom groups in $DM(X)$, etc... The case I'm most interested in is $A=... | https://mathoverflow.net/users/1985 | Kunneth formula for motivic cohomology | I now remember a nice argument, why there's no Kunneth formula for Chow groups of $X \times X$ unless $X$ has a Tate motive. Let $X$ be smooth projective of dimension $d$.
We start with a decomposition of a diagonal:
$$
[\Delta] = \sum\_{i,j} \alpha^i\_j \beta^{d-i}\_j \in \oplus\_i CH^i(X) \otimes CH^{d-i}(X)
$$
We c... | 11 | https://mathoverflow.net/users/2260 | 12948 | 8,757 |
https://mathoverflow.net/questions/12873 | 6 | Let $K$ be a field, $K^\times$ its multiplicative group and $I$ an infinite set. Is then $(K^\times)^{(I)} \subseteq (K^\times)^I$ a direct summand? If not, is it possible to characterize the fields for which this is true?
In any case, it's a pure subgroup. If $K$ is finite, the answer is [yes](https://mathoverflow.n... | https://mathoverflow.net/users/2841 | Split powers of the multiplicative group of a field | Re: "I don't know an example of an abelian group $G$ such that $G^{(I)}$ is not a direct summand of $G^I$, but I'm pretty sure that there is one."
Let $G$ be the the integers, and $I$ a countable indexing set. If $G^{(I)}$ were a direct summand, let $P$ be a complement summand.
We arrive at a contradiction as foll... | 5 | https://mathoverflow.net/users/3199 | 12950 | 8,758 |
https://mathoverflow.net/questions/12957 | 4 | ### Question
Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$?
### Motivation
Studying abelian varieties is awesome. Studying objects in long exact sequences is awesome. How do (somewhat forceful... | https://mathoverflow.net/users/2024 | Can an abelian variety be represented as the cohomology of some other object? | Let $\mathcal{A}/S$ be an Abelian scheme. Then the dual Abelian scheme is given by $R^1\pi\_\*\mathcal{A}$, if I remember correctly. Also, $\mathcal{A}^\vee(V) = \mathrm{Ext}^1\_V(\mathcal{A},\mathbf{G}\_m)$.
| 2 | https://mathoverflow.net/users/nan | 12958 | 8,762 |
https://mathoverflow.net/questions/12956 | 6 | Is there a standard name for taking a homomorphism from the free object over an algebraic structure? Roughly speaking, this should amount to *evaluation* of any element of the free object under the homomorphism.
For example, take the free commutative monoid over the set of natural numbers; elements of this monoid are... | https://mathoverflow.net/users/3539 | Terminology: Name for a homomorphism from the free object? | The nLab has proposed the name "adjunct", which seems OK to me. So, for example, you could say something like "let $F(S)$ be the free group on the set $S$, let $G$ be a group, let $f:S\to G$ be a set map, and let $g:F(S)\to G$ denote the adjunct of $f$." I think it would be even a bit better to say "left adjunct" inste... | 3 | https://mathoverflow.net/users/1114 | 12964 | 8,766 |
https://mathoverflow.net/questions/12914 | 3 | How to define Kahler differential in an abelian category or more general category? Say exact category?
Is there any interesting example?
| https://mathoverflow.net/users/3156 | How to introduce Kahler differential in category? | It turns out that the whole complex of concepts
* [module](http://ncatlab.org/nlab/show/module)
* [derivation](http://ncatlab.org/nlab/show/derivation)
* [Kähler differentials](http://ncatlab.org/nlab/show/K%C3%A4hler+differential)
* [cotangent complex](http://ncatlab.org/nlab/show/cotangent+complex)
* [quasicoherent... | 7 | https://mathoverflow.net/users/381 | 12974 | 8,770 |
https://mathoverflow.net/questions/12966 | 3 | (I'm happy to work over an algebraically closed field....)
Let $\mathcal{C} \rightarrow Spec (R)$ be a (flat) family of proper, prestable curves where $R$ is a DVR. Suppose the generic fiber is smooth and the special fiber, $C\_0$, is reduced but may be reducible.
Given a finite map of curves $f\_0: D\_0 \rightarr... | https://mathoverflow.net/users/397 | Extending maps of curves | Stated as such, the question is really too vague.
For example, if $C\_0, D\_0$ are smooth of genus bigger than 1, and $f\_0$ is purely inseparable, then $f\_0$ can not lift to $R$ (even after extention) if $R$ has characteristic zero. Actually, as the generic fibers of $\mathcal C$ and $\mathcal D$ are equal to $g(C\_... | 6 | https://mathoverflow.net/users/3485 | 12975 | 8,771 |
https://mathoverflow.net/questions/12969 | 14 | Does anyone know of such a domain?
| https://mathoverflow.net/users/3186 | non-Dedekind Domain in which every ideal is generated by at most two elements | You may find Matlis' paper [The Two-Generator Problem for Ideals](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-17/issue-3/The-two-generator-problem-for-ideals/10.1307/mmj/1029000474.full) to be interesting, as its main theorem concerns the class of integral domains in which every ideal is gen... | 11 | https://mathoverflow.net/users/nan | 12978 | 8,773 |
https://mathoverflow.net/questions/12980 | 0 | Let $A$ be an algebra, $H$ a Hopf algebra, and
$$
\beta\_A: A \to A \otimes H, ~~~~~ a \mapsto a^{(1)} \otimes a^{(2)}
$$
a right $H$-coaction. This induces a right $H$-coaction on $A \otimes A$ defined by
$$
\beta\_{A \otimes A}: a \otimes b \mapsto a^{(1)} \otimes b^{(1)} \otimes a^{(2)}b^{(2)}.
$$
My question is: D... | https://mathoverflow.net/users/1867 | Coaction on the Universal Calculus | If $A$ is an $H$-comodule algebra (that is, if the multiplication map $\mu$ is a map of comodules), then the answer is yes (trivially, because the category of $H$-comodules has kernels). If it isn't, then probably not, as then you have no compatibility between the algebra structure on $A$ and the comodule structure.
| 2 | https://mathoverflow.net/users/1409 | 12981 | 8,775 |
https://mathoverflow.net/questions/12962 | 2 | The blowdown of the zero section of the canonical bundle of the first del Pezzo surface $dP\_{1}$, the blowup of $CP^{2}$ at one point, is a Calabi-Yau cone. I was just wondering if this cone admitted a smoothing. Is the same thing true for the blowdown of the zero section of the canonical bundle of the second del Pezz... | https://mathoverflow.net/users/3566 | Existence of smoothing of Calabi-Yau cones over $dP_{1}$ and $dP_{2}$ | The answer to this question is contained in the article of Mark Gross, page 33
Deforming Calabi-Yau Threefolds
<http://arxiv.org/abs/alg-geom/9506022>
The first cone can not be smoothed the second one can be smoothed
(in the terminology of Gross, which is standard, $dP\_1$ is a del-Pezzo of degree 8, $dP\_2$ is... | 3 | https://mathoverflow.net/users/943 | 12985 | 8,779 |
https://mathoverflow.net/questions/12973 | 109 | It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary operation of *symmetric difference* forms a group, and in ZFC there is a bijection between $S$ and the set of finite subs... | https://mathoverflow.net/users/932 | Does every non-empty set admit a group structure (in ZF)? | In ZF, the following are equivalent:
(a) For every nonempty set there is a binary operation making it a group
(b) Axiom of choice
Non trivial direction [(a) $\to$ (b)]:
The trick is Hartogs' construction which gives for every set $X$ an ordinal $\aleph(X)$ such that there is no injection from $\aleph(X)$ into $... | 174 | https://mathoverflow.net/users/2689 | 12988 | 8,782 |
https://mathoverflow.net/questions/12949 | 19 | Let $\kappa$ be an infinite cardinal. Then there exists at least one [real-closed field](http://en.wikipedia.org/wiki/Real-closed_field) of cardinality $\kappa$ (e.g. [Lowenheim-Skolem](http://en.wikipedia.org/wiki/Lowenheim-Skolem); or, start with a function field over $\mathbb{Q}$ in $\kappa$ indeterminates, choose a... | https://mathoverflow.net/users/1149 | Are there as many real-closed fields of a given cardinality as I think there are? | Hi Pete!
There's been a lot of study of this and similar problems. I believe that Shelah's theorem, from his 1971 paper "The number of non-isomorphic models of an unstable first-order theory" (Israel J. of Math) answers your question about real closed fields in the positive.
The best big result on such questions th... | 17 | https://mathoverflow.net/users/3545 | 12991 | 8,784 |
https://mathoverflow.net/questions/12992 | 11 | Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the fibers of $F$ vanish.
(I think I am using standard terms: the tensor product over $O\_X$ with the local ring at $x$ is th... | https://mathoverflow.net/users/1048 | When does a quasicoherent sheaf vanish? | If the scheme is locally noetherian, this is true and can be proved by noetherian induction. In fact, you can even replace $M$ with an object of bounded derived quasi-coherent category, if you are interested in such things.
The proof is relatively straightforward: For a complex of modules $M$ over the ring $R$, we ma... | 8 | https://mathoverflow.net/users/2653 | 12996 | 8,788 |
https://mathoverflow.net/questions/12968 | 5 | I am looking for a power series $\displaystyle f(z) = \sum\_{n=0}^{+\infty} a\_n z^n$ that converge uniformly on $\mathcal{D} = \Big\{ z \in \mathbb{C} \ / \ \vert z \vert \leq 1 \Big\}$ but not normally. Of course, a proof that this is impossible would be even better. It seems close to [this question](https://mathover... | https://mathoverflow.net/users/3543 | Mode of convergence of a power series | An example, due to Fejér, appears in Hille's *Analytic function theory* on [page 122](http://books.google.com/books?id=4Qu41N_OT7gC&lpg=PA122&ots=RibiKvgZOJ&dq=%22uniformly%20but%20not%20absolutely%22&pg=PA122#v=onepage&q=&f=false) of the second edition, volume 1.
Erdős wrote a [paper](http://www.renyi.hu/~p_erdos/19... | 8 | https://mathoverflow.net/users/1119 | 13003 | 8,791 |
https://mathoverflow.net/questions/13001 | 2 | I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).
He makes the sparsity assumption on $\theta \in \mathbb{R}^m$ that for some $0<p<2$ and $R>0$ we have $\|\theta\|\_p\leq R$. Then if $\theta\_N$ denotes $\theta$ with everythin... | https://mathoverflow.net/users/3547 | l^p space inequality related to compressed sensing | Suppose that $\|\theta\|\_p=1$. Then, as you correctly observed, if we drop $N$ largest coefficients, we'll have $|\theta\_i|\le N^{-1/p}$ for all the remaining ones, whence $|\theta\_i|^2\le N^{-(2-p)/p}|\theta\_i|^p$. Now just add those inequalities up, use the fact that $\sum\_i|\theta\_i|^p\le\|\theta\|\_p^p=1$ and... | 5 | https://mathoverflow.net/users/1131 | 13004 | 8,792 |
https://mathoverflow.net/questions/12990 | 2 | so, you have, for any two members of the algebraic structure A and B and any nonnegative real values a, b:
two operations: \* and +, such that
a\*A + b\*A = (a+b)\*A is in the structure
A + B = B + A is in the structure
0\*A + B = B
but there is no guarantee that X s.t.
X + A = B
is in the structure.
... | https://mathoverflow.net/users/634 | What is it called if a vector space doesn't have an additive inverse? | If the structure in question is a subset of a vector space, like it is in your example, I would call it a [convex cone](http://en.wikipedia.org/wiki/Convex_cone).
| 12 | https://mathoverflow.net/users/1096 | 13007 | 8,793 |
https://mathoverflow.net/questions/13002 | 2 | Consider the following pair of principal bundle descriptions of $\mathbb{CP}^2$:
$$
\mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1).
$$
If I have a principal $U(2)$-bundle connection for $\mathbb{CP}^2$, will that correspond to a principal $U(1)$-bundle connection? (Where by correspond I suppose I mean give the same co... | https://mathoverflow.net/users/1977 | Principal Bundle Connection Correspondence for two descriptions of the $\mathbb{CP}^2$ | The answer is no. You can see that for example from the connection one forms which are Lie algebra valued. In the first case they are u(2) valued and in the second case they are u(1) valued. However, in the case of CP2 (which also generalizes to CPn), the U(1) and SU(2) factors of the isotropy group U(2) commute, this ... | 1 | https://mathoverflow.net/users/1059 | 13009 | 8,794 |
https://mathoverflow.net/questions/13005 | 30 | The key step in Kontsevich's proof of deformation quantization of Poisson manifolds is the so-called formality theorem where 'a formal complex' means that it admits a certain condition. I wonder why it is called 'formal'. I only found the definition of Sullivan in Wikipedia: 'formal manifold is one whose real homotopy ... | https://mathoverflow.net/users/3478 | What is 'formal' ? | I would guess that the terminology goes back to the work of Sullivan and Quillen on rational homotopy theory. You should probably also look at the paper of [Deligne-Griffiths-Morgan-Sullivan](https://doi.org/10.1007/BF01389853) on the real homotopy theory of Kähler manifolds. Actually, I think that at least some famili... | 28 | https://mathoverflow.net/users/83 | 13019 | 8,800 |
https://mathoverflow.net/questions/13017 | 16 | Inspired by [a recent question](https://mathoverflow.net/questions/12873/split-powers-of-the-multiplicative-group-of-a-field) on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is cyclic. Is this sufficient? ... | https://mathoverflow.net/users/290 | Which commutative groups are the group of units of some field? | The following paper claims an answer to this question:
>
> Dicker, R. M.
> A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field.
> Proc. London Math. Soc. (3) 18 1968 114--124.
>
>
>
You can find it here:
[http://alpha.math.uga.edu/~pete/Dicker1966.pdf... | 18 | https://mathoverflow.net/users/1149 | 13024 | 8,803 |
https://mathoverflow.net/questions/13027 | 17 | Freyd's [Abelian Categories](http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf) is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that an *object* is what arrows with the same *domain* (another deri... | https://mathoverflow.net/users/2672 | Primacy of arcs/arrows over vertices/objects | Qu 1: At about the time of Freyd's book there were two approaches to defining categories. One came from algebraic topology and homological algebra, thus from Eilenberg and MacLane and used the objects and arrows definition, the other was motivated by differential geometry and used the arrows only formalism. This second... | 15 | https://mathoverflow.net/users/3502 | 13033 | 8,808 |
https://mathoverflow.net/questions/12970 | 12 | Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\sigma\in K$ exists such that $f(\sigma)=\sigma$. This will be called the fixed simplex property (FSP).
One can give e... | https://mathoverflow.net/users/2578 | Does a triangulation without fixed simplex property always exist? | EDITED. The arugment related to Mostov rigidity is completed according to a nice suggestion of Tom Church
The answer to the first question is no. There exsit manifolds of dimension 3 such that every simlicial map of the manfiold to itself (for any simplicial decomposition) has a fixed point (and hence a fixed simplex... | 11 | https://mathoverflow.net/users/943 | 13034 | 8,809 |
https://mathoverflow.net/questions/13020 | 5 | Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional
$$E(f) = \int\_{\mathcal{M}} X\_p(f)^2 dV$$
where $X\_p(f)$ is the directional (Lie) derivative of $f$ along $X$ at the point $p$ and $dV$ is a volume form on $\mathcal{M}$ -- this funct... | https://mathoverflow.net/users/1557 | Is there a name for this differential operator and/or its corresponding spectrum? | There is an simple explicit formula for your operator in terms of known operators. To see this, note that $\delta E\_f(g)$ (the differential of $E$ at the function $f$ in the direction $g$) is equal to
$$
2 \int X(f) X(g) dV = 2\int \left[L\_X(g X(f) dV )-g\left(X(X(f)) dV+X(f)L\_X(dV)\right)\right]
$$
where $L\_X$ is ... | 5 | https://mathoverflow.net/users/380 | 13036 | 8,810 |
https://mathoverflow.net/questions/13008 | 20 | What is the expected value of the determinant over the uniform distribution of all possible 1-0 NxN matrices? What does this expected value tend to as the matrix size N approaches infinity?
| https://mathoverflow.net/users/3551 | Expected determinant of a random NxN matrix | As everyone above has pointed out, the expected value is $0$.
I expect that the original poster might have wanted to know about how big the determinant is. A good way to approach this is to compute $\sqrt{E((\det A)^2)}$, so there will be no cancellation.
Now, $(\det A)^2$ is the sum over all pairs $v$ and $w$ of p... | 33 | https://mathoverflow.net/users/297 | 13040 | 8,813 |
https://mathoverflow.net/questions/13038 | 9 | This question was prompted by my interpretation of [a question](http://letterstonature.wordpress.com/2010/01/26/genus-analogues/) by cosmologist Berian James.
**Background**
Some cosmologists have suggested using the cosmological dark matter density, which defines a function $f:M\to \mathbb{R}$ with $M$ the spatial... | https://mathoverflow.net/users/394 | Is there a Morse theory for sections of bundles or more generally for maps? | You are asking a very classical question. It seems to me, that general answer is No. But there were some interesting attempts (could not find referencies..). In some cases one can prove that there is no "Morse theory" for a class of maps, since they satisfy h-principle and one can eliminate (almost all) singularities.
... | 11 | https://mathoverflow.net/users/2823 | 13041 | 8,814 |
https://mathoverflow.net/questions/13021 | 8 | The following graph theoretic notion appeared in an [economics paper entitled: "Prize competition under limited comparability,](http://www.homepages.ucl.ac.uk/~uctprsp/framing2.pdf) by Michele Piccione and Ran Spiegler which studies models of economics were the firms are rational but the consumers are not.
A graph i... | https://mathoverflow.net/users/1532 | Weighted Regular Graphs | The examples you gave can be extended. If a graph has a transitive group then it is weighted regular. Also if the graph such that one point is connected to all other points and if that point is removed the graph remaining has a transitive group then it is weighted regular.
Also there exists graphs that are not weigh... | 3 | https://mathoverflow.net/users/1098 | 13065 | 8,831 |
https://mathoverflow.net/questions/13016 | 3 | This is prop 2.6b on p.28 of Silverman's the Arithmetic of Elliptic curves.
It says that let $\phi: C\_1 \rightarrow C\_2$ be a non-constant map of projective non-singular irreducible curve. (probably over an algebraically closed field, but I am not too sure) Then for all but finitely many $Q \in C\_2$, #$\phi^{-1} (... | https://mathoverflow.net/users/nan | Generic fiber of morphism between non-singular curves | Here is a complete proof: as remarked in the answer by Norondion, we can reduce to
the case when $C\\_1 \rightarrow C\\_2$ is generically separable, i.e. $k(C\\_1)$ is separable
over $k(C\\_2)$. Let $A \subset k(C\\_1)$ be a finite type $k$-algebra consisting of the regular
functions on some non-empty affine open subse... | 5 | https://mathoverflow.net/users/2874 | 13067 | 8,832 |
https://mathoverflow.net/questions/13058 | 7 | I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism between schemes obtained from f by base change should be proper.
Now, first I guess that "proper" here actually means "c... | https://mathoverflow.net/users/828 | What is a proper stack? | As requested, an answer on terminology
My favorite reference on basics for DM stacks is [Edidin's paper](http://arxiv.org/abs/math/9805101), which I find much easier to read than Laumon & Moret-Bailly (who of course deal with Artin stacks).
Short summary. Suppose $P$ is a property of morphisms $f:X\to Y$ in the categ... | 9 | https://mathoverflow.net/users/2653 | 13076 | 8,836 |
https://mathoverflow.net/questions/13032 | 13 | In the symplectic topology view on Gromov-Witten-Invariants some authors use what they call a Kuranishi structure on the moduli of stable maps. These were introduced by Fukaya and Ono and are also used in their big book on Fukaya categories.They are also used a lot in recent papers by Joyce.
The key feature of the K... | https://mathoverflow.net/users/473 | Obstruction bundle for spaces with Kuranishi structure | Here's a view of the symplectic side of the bridge.
The Kuranishi model (see Donaldson-Kronheimer, *The geometry of four-manifolds*, ch. 4) goes like this. You're interested in a (moduli) space $M$ cut out as $\psi^{-1}(0)$, for some smooth but nonlinear map of Banach spaces, $\psi \colon (E,0) \to (F,0)$ such that $... | 13 | https://mathoverflow.net/users/2356 | 13078 | 8,838 |
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