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https://mathoverflow.net/questions/34059 | 231 | Let $f$ be an *infinitely differentiable function* on $[0,1]$ and suppose that for each $x \in [0,1]$ there is an integer $n \in \mathbb{N}$ such that $f^{(n)}(x)=0$. Then does $f$ coincide on $[0,1]$ with some polynomial? If yes then how.
I thought of using Weierstrass approximation theorem, but couldn't succeed.
| https://mathoverflow.net/users/1483 | If $f$ is infinitely differentiable then $f$ coincides with a polynomial | The proof is by contradiction. Assume $f$ is not a polynomial.
Consider the following closed sets:
$$
S\_n = \{x: f^{(n)}(x) = 0\}
$$
and
$$
X = \{x: \forall (a,b)\ni x: f\restriction\_{(a,b)}\text{ is not a polynomial} \}.
$$
It is clear that $X$ is a non-empty closed set without isolated points. Applying Baire ... | 181 | https://mathoverflow.net/users/2029 | 34067 | 22,055 |
https://mathoverflow.net/questions/34069 | 3 | Can anyone tell me why the endomorphism ring of a finite-length module is artinian? Bonus points if you can do it without using the radical, semisimplicity, Fitting's lemma or anything fancy. If you have to or it makes the proof easier, that's OK too, but I have reason to believe that there's a simple proof (namely Den... | https://mathoverflow.net/users/3544 | endomorphism ring of a finite-length module | Never mind. I was more successful with Google this time. It turns out that the statement is simply false. In the comment above I gave an example showing that the endomorphism ring need not be left Artinian. The following paper contains a (much less trivial) example showing that the endomorphism ring also need not be ri... | 4 | https://mathoverflow.net/users/4384 | 34074 | 22,060 |
https://mathoverflow.net/questions/34058 | 3 | I've been studying some category theory lately and in particular, I became acquainted with the notions of products and coproducts, which led me to ponder the following:
Consider the category of all complex Hilbert spaces (the morphisms being linear isometries). This category has coproducts, due to the direct sum cons... | https://mathoverflow.net/users/7392 | Does the category of Hilbert spaces possess a product? | The category you specified does not have products, because it doesn't have a product of zero objects. The product of zero objects is an final object, if it exists, but final objects need to have unique maps (in this case, isometries) from all other objects. Such a final Hilbert space would need to be at least as large ... | 2 | https://mathoverflow.net/users/121 | 34075 | 22,061 |
https://mathoverflow.net/questions/34066 | 5 | Suppose $R$ is a Cohen-Macaulay ring. It is well known that if $I$ is an ideal of $R$ generated by $n$ elements, and $I$ has codimension $n$, then $R/I$ is also Cohen-Macaulay.
Now suppose that $I$ does not have codimension $n$, but (the scheme defined by) $R/I$ has several irreducible components, one of which has co... | https://mathoverflow.net/users/3077 | Irreducible components of quotients of Cohen-Macaulay rings of the "correct" dimension | Hi Alex, I think this fails for $n=2$. Start with a polynomial ring $S$ and height $2$ prime $P$ such that $S/P$ is not Cohen-Macaulay (for example let $P$ be the kernel of the map $S=k[a,b,c,d] \to k[x^4,x^3y,xy^3,y^4]$). Let $a,b$ be a regular sequence in $P$. Let $R=S[t]$ and $I=(ta,tb)$. Then $I$ has height $1$ and... | 6 | https://mathoverflow.net/users/2083 | 34085 | 22,067 |
https://mathoverflow.net/questions/34052 | 51 | How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?
| https://mathoverflow.net/users/1483 | Function satisfying $f^{-1} =f'$ | Let $a=1+p>1$ be given. We shall construct a function $f$ of the required kind with $f(a)=a$ by means of an auxiliary function $h$, defined in the neighborhood of $t=0$ and coupled to $f$ via $x=h(t)$, $f(x)=h(a t)$, $f^{-1}(x)=h(t/a)$. The condition $f'=f^{-1}$ implies that $h$ satisfies the functional equation $$(\*)... | 38 | https://mathoverflow.net/users/8050 | 34095 | 22,069 |
https://mathoverflow.net/questions/34107 | 1 | Let $\kappa$ be a cardinal, and let $P$ be a poset. Let $\mathcal{P}\_\kappa(P)$ denote the poset of $\kappa$-small subposets of $P$ and let $\mathcal{P}\_\kappa^\downarrow(P)\subseteq\mathcal{P}\_\kappa(P)$ be the subposet consisting of those subposets that are downward-closed. Then according to a reliable source, whe... | https://mathoverflow.net/users/1353 | The poset of k-small downward-closed subposets of a poset P is k-filtered when k is a regular cardinal? | A cardinal $\kappa$ is regular if (and only if) the union of fewer than $\kappa$ many sets of size less than $\kappa$ still always has size less than $\kappa$. That seems to be exactly what you have here. Also, the union of downward closed sets remains downward closed, so you don't need to take the downward closure of ... | 2 | https://mathoverflow.net/users/1946 | 34108 | 22,074 |
https://mathoverflow.net/questions/34098 | 1 | Is the isomorphism class of a fixed cardinality a set(not a proper class)? Or a fixed ordinality for that matter?
By "isomorphism" I mean just bijection for cardinals and order preserving bijection for ordinals, in the category of sets.
| https://mathoverflow.net/users/5292 | Is the isomorphism class of a fixed cardinality a set? | The isomorphism classes of the set {} and the ordinal 0 are sets, each with a single element.
let $\kappa$ be an non-zero ordinal and $\alpha$ # $\beta$ be the [natural sum](http://en.wikipedia.org/wiki/Ordinal_arithmetic#Natural_operations) of the ordinals $\alpha$ and $\beta$. Since the natural sum is the restricti... | 1 | https://mathoverflow.net/users/7727 | 34109 | 22,075 |
https://mathoverflow.net/questions/34032 | 4 | $ B = (B\_t, \mathcal{F}\_t; t\ge 0 ) $ is a 1-d Brownian family on a
measurable space $(\Omega, \mathcal{F})$ with a family of probability
measures $\{\mathbb{P}^x\}$, i.e. $\mathbb{P}^x(B\_0
= x) = 1$, and $B$ is 1-d BM starting from $x$ under $\mathbb{P}^x$.
Let $\tau$ be a given stopping time w.r.t. underlying fi... | https://mathoverflow.net/users/5656 | Continuity in intial state of Brownian Motion | Here is a simpler example that I hope convinces you
that $V$ need not be continuous, even in the one dimensional case.
Take one dimensional Brownian motion $(B\_t)$
and define the stopping time $\tau(\omega)=1\_{(B\_0(\omega)<0)}$.
Then, for any bounded measurable $f$, we have
$$V(x)=E\_x[f(B\_1)]1\_{(-\infty,0)}(x)+... | 5 | https://mathoverflow.net/users/nan | 34118 | 22,083 |
https://mathoverflow.net/questions/34120 | 5 | I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its significance, and how can it be applied to problems, or something else.
| https://mathoverflow.net/users/1483 | Sylow's theorem 3rd Proof Page 96 I.N.Herstein | This is the proof that uses the lemma that if a finite group $G$ has
a Sylow $p$-subgroup then so does each subgroup of $G$. To complete
the proof of existence of Sylow $p$-subgroups, it suffices to show
one can embed each group in a group with a Sylow $p$-subgroup.
By Cayley's theorem each finite $G$ embeds in $S\_n$ ... | 14 | https://mathoverflow.net/users/4213 | 34123 | 22,087 |
https://mathoverflow.net/questions/34088 | 42 | Let $M$ be a Riemannian manifold. There exists a unique *torsion-free* connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the Levi-Civita connection and its existence and uniqueness are usually proven by a direct calculation in coordinates. See e.g... | https://mathoverflow.net/users/2349 | A geometric interpretation of the Levi-Civita connection? | To understand the existence and uniqueness of the LC connection, it is not possible to sidestep some algebra, namely the fact (with a 1-line proof) that a tensor $a\_{ijk}$ symmetric in $i,j$ and skew in $j,k$ is necessarily zero. The geometrical interpretation is this: once one has the $O(n)$ subbundle $P$ of the fram... | 23 | https://mathoverflow.net/users/3975 | 34129 | 22,090 |
https://mathoverflow.net/questions/34142 | 17 | This should be easy to prove but I have no idea how to do it:
If $X \subseteq \mathbb{R}^2$ is borel then $f(X)$ is borel where $f(x,y) = x$
Thanks
Tobias
| https://mathoverflow.net/users/8092 | Projection of Borel set from $R^2$ to $R^1$ | This is false; take a look at <https://en.wikipedia.org/wiki/Analytic_set> for a quick introduction. For details, look at Kechris's book on **Classical Descriptive Set Theory**. There you will find also some information on the history of this result, how it was originally thought to be true, and how the discovery of co... | 22 | https://mathoverflow.net/users/6085 | 34143 | 22,099 |
https://mathoverflow.net/questions/34145 | 95 | The following question was a *research* exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum\_{1 \leq k } \left(\frac{1}{k} \times \frac{1}{k+1}\right) = \sum\_{1 \leq k } \left(\frac{1}{k} - \frac{1}{k+1}\right) = 1.$$
... | https://mathoverflow.net/users/7507 | Can we cover the unit square by these rectangles? | This problem actually goes back to Leo Moser.
The best result that I'm aware of is due to D. Jennings, who proved that all the rectangles of size $k^{-1} × (k + 1)^{-1}$, $k = 1, 2, 3 ...$, can be packed into a square of size $(133/132)^2$ ([link](https://doi.org/10.1016/0097-3165(94)90116-3)).
**Edit 1.** A web se... | 54 | https://mathoverflow.net/users/5371 | 34149 | 22,102 |
https://mathoverflow.net/questions/34087 | 4 | Suppose $(x\_\alpha)\_\alpha$ is an uncountable, linearly independent family of norm one vectors in a Banach space. Can one always select a basic sequence (or at least a minimal system) from this family? I suspect the answer is no but I cannot come up with an example.
Thank you!
| https://mathoverflow.net/users/7872 | Selecting basic sequences | Not a basic sequence. Consider $e\_0 \oplus e\_\gamma$ in $R\oplus H$ for $H$ a non separable Hilbert space, or, if you want a separable example, make $e\_\gamma$ a Hamel basis for a separable Hilbert space.
EDIT: Aug 2. Every separated sequence of unit vectors contains a minimal subsequence with bounded biorthogona... | 7 | https://mathoverflow.net/users/2554 | 34153 | 22,104 |
https://mathoverflow.net/questions/34148 | 2 | Hi there,
Sorry if this has already been asked before. I tried googling for it, but perhaps I could not find the right words to search for. My question is: Which is the fastest way to compute A\*inv(B) [edit] where A and B are matrices?
| https://mathoverflow.net/users/4430 | Efficient computation of AB^-1 for matrices | What are the sizes of $\mathbf{A}$ and $\mathbf{B}$? This information is important.
Let me assume you mean you want the efficient *numerical* computation of the matrix $ \mathbf{A} \mathbf{B}^{-1}$.
The general strategy would be to do this: Let $\mathbf{J} = \mathbf{A} \mathbf{B}^{-1}$; therefore $\mathbf{J}\mathbf... | 4 | https://mathoverflow.net/users/7851 | 34156 | 22,107 |
https://mathoverflow.net/questions/34099 | 20 | Hi!
What text on both incompleteness theorems you would recommend for beginner?
Specifically, I'm looking for the text with the following properties:
1) The proofs should be finitistic, in Godel's tradition, i. e. formalizing "I'm unprovable" (not, for instance, via formalization of halting problem);
2) The text... | https://mathoverflow.net/users/6307 | The best text to study both incompleteness theorems | First chapter of Jean-Yves Girard, "Proof Theory and Logical Complexity", Vol I, Bibliopolis, 1987
It satisfies all of your conditions, but it is not an elementary book. If I remember correctly, the authors (A.S. Troelstra and H. Schwichtenberg) of the book "Basic Proof Theory" which is published in 2001 wrote in the... | 7 | https://mathoverflow.net/users/7507 | 34163 | 22,112 |
https://mathoverflow.net/questions/33405 | 24 | ***This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't change the accepted answer, but it would be amazing to have more discussion on this question.***
I'd like a nice p... | https://mathoverflow.net/users/2467 | Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof? | There is a short conceptual proof of Gauß-Bonnet due to Chern (see also ["A panoramic view of Riemannian geometry"](http://books.google.co.uk/books?id=d_SsagQckaQC&dq=panoramic+view+berger&printsec=frontcover&source=bn&hl=en&ei=1PJVTOb8I4Xu0wTXnuT6Ag&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC8Q6AEwBA#v=onepage&q&f=... | 6 | https://mathoverflow.net/users/5371 | 34164 | 22,113 |
https://mathoverflow.net/questions/34160 | 18 | In an answer to a question [on MU](https://math.stackexchange.com/questions/540/what-is-the-riemann-zeta-function/832#832) about the Riemann zeta function, I sketched a proof that the probability distribution on $\mathbb{N}$ which assigns $n$ the probability
$$\frac{ \frac{1}{n^s} }{\zeta(s)}$$
(henceforth called t... | https://mathoverflow.net/users/290 | The Riemann zeta function and Haar measure on the profinite integers | ...because for sets with logarithmic density, the logarithmic density and the $s\to1$ zeta-measure agree. And for sets with natural density, the logarithmic and natural densities are the same.
| 7 | https://mathoverflow.net/users/935 | 34170 | 22,119 |
https://mathoverflow.net/questions/34106 | 0 | Let $(X,A)$ be a finite CW-pair $m=p^r$ for some prime $p$. Unspecified coefficient is in $\mathbb{Z}$.
From the universal coefficient theorem, We know that
$H^1(A;\mathbb{Z}\_m)=\textrm{Hom} (H\_1(A),\mathbb{Z}\_m)$ ---(1) and
$H^2(X,A;\mathbb{Z}\_m)=\textrm{Hom}(H\_2(X,A);\mathbb{Z}\_m)\bigoplus \textrm{Ext}(H\... | https://mathoverflow.net/users/7776 | About universal coefficient theorem | There is a map (i.e. a commutative diagram) from the exact sequence $$0\to Ext(H\_{n-1}(A),G)\to H^n(A;G)\to Hom(H\_n(A),G)\to 0$$ to the exact sequence $$0\to Ext(H\_n(X,A),G)\to H^{n+1}(X,A;G)\to Hom(H\_{n+1}(X,A),G)\to 0$$All the groups are functors of both the pair $(X,A)$ and the abelian group $G$. All the maps ar... | 2 | https://mathoverflow.net/users/6666 | 34177 | 22,122 |
https://mathoverflow.net/questions/34180 | 9 | We know that in a normal extension of number fields $L/K$, for any prime $P$ of $K$ and any primes $Q\_1,Q\_2$ of $L$ lying over $P$, the ramification indices and inertial degrees are the same,
$$e(Q\_1|P)=e(Q\_2|P),\quad f(Q\_1|P)=f(Q\_2|P),$$
and that this is not necessarily the case if $L/K$ is not normal. Is it nev... | https://mathoverflow.net/users/1916 | Converse to basic result on prime decomposition | Forget the ramification indices: they're nearly always 1. Let's focus on the residue field degrees. Let $E$ be the Galois closure of $L$ over $K$. For a prime $P$ in $K$, write $f\_P(E/K)$ for the common residue field degree of all primes in $E$ over $P$. Since $E$ is determined (up to isom. as an extension of $K$) by ... | 15 | https://mathoverflow.net/users/3272 | 34185 | 22,128 |
https://mathoverflow.net/questions/34186 | 36 | This question is related to [this recent but currently
unanswered MO
question](https://mathoverflow.net/questions/33879/decidability-of-matrix-algebra)
of Ricky Demer, where it arose as a comment.
Consider the structure $R^n$ consisting of $n\times n$
matrices over the reals $\mathbb{R}$, $n$-dimensional row
vectors,... | https://mathoverflow.net/users/1946 | Does the truth of any statement of real matrix algebra stabilize in sufficiently high dimensions? | One can get arithmetic progressions as *truth sets*, as in Joel's comment. Pick non-negative integers $a$ and $b$, pick a finite group $G$ which has at least one representation of degree $a$. Then there is a formula expression the statement "the vector space is a $G$-module which is a sum of irreducible representations... | 9 | https://mathoverflow.net/users/1409 | 34199 | 22,136 |
https://mathoverflow.net/questions/34188 | 12 | What can be said about the limiting distribution of the sequence of fractional parts of $\{n^{a},n>0\}$ for $a\in(1,2)$. I ran a computer experiment for $n\sqrt{n}$ and it looks like uniformly distributed. Is there a simple proof?
| https://mathoverflow.net/users/3375 | Distribution of fractional parts of n^{3/2} | Exercise 2.23 in Kuipers and Niederreiter, Uniform Distribution Of Sequences: Use Theorem 2.7 to show that the sequence $(\alpha n^{\sigma})$, $n=1,2,\dots$, $\alpha\ne0$, $1\lt\sigma\lt2$, is u.d. mod 1.
They are using $(x)$ for the fractional part. Theorem 2.7 is Let $a$ and $b$ be integers with $a\lt b$, and let ... | 8 | https://mathoverflow.net/users/3684 | 34202 | 22,139 |
https://mathoverflow.net/questions/34205 | 6 | There are short and sweet proofs of various forms of Stirling's approximation. But even the sweetest among them don't instill the same conviction in the reader as a direct bijective proof.
Computer scientists can often get away with a very weak form of Stirling's approximation:
$$(n/2)^{n/2} \leq n! \leq n^n$$
Fr... | https://mathoverflow.net/users/2036 | Bijective proof of weak form of Stirling's approximation | Think of all maps from the first $n/2$ elements of {$1,...,n$} to the last $n/2$. Say, let $a\_1< \ldots < a\_k \to z$. Make a cycle $a\_1 \to a\_2 \to \ldots \to a\_k \to z \to a\_1$. Do this for all $z$. The details are straightforward. This proves the lower bound.
| 6 | https://mathoverflow.net/users/4040 | 34227 | 22,153 |
https://mathoverflow.net/questions/34229 | 6 | In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact subsets. This made me wonder whether the following lemma holds:
**Lemma:** Let $M$ be a compact Hausdorff space, and ... | https://mathoverflow.net/users/8047 | Compact cover of a Hausdorff compact space | Note: whilst typing this, Martin posted his answer. As I come to a completely different conclusion, I'd be very interested in knowing who's right!
---
False. Let $M = [0,1]$ and $K\_i = \{0\} \cup [\frac{1}{i},1]$.
The flaw in the proof is the assumption that the $U\_j$ are increasing; ie that $U\_j \subseteq U... | 12 | https://mathoverflow.net/users/45 | 34231 | 22,155 |
https://mathoverflow.net/questions/34232 | 7 | Let $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$ be an injection for $n>m$. Can $f$ be continuous? Why?
I got this question in mind when I was trying to find a continuous map from $\mathbb{R}^{2}$ to $\mathbb{R}$.
| https://mathoverflow.net/users/1483 | Injective maps $\mathbb{R}^{n} \to \mathbb{R}^{m}$ | If $f$ is injective and continuous from $\mathbb{R}^n$ to $\mathbb{R}^m$
where $n>m$ then $f$ restricts to a continuous bijection from $S^{n-1}$,
the unit sphere in $R^n$, to a compact subset $K$ of $\mathbb{R}^m$.
Thus you can embed $S^{n-1}$, and a foriori $S^m$ in $\mathbb{R}^m$.
But there are homological obstruct... | 15 | https://mathoverflow.net/users/4213 | 34234 | 22,156 |
https://mathoverflow.net/questions/34228 | 7 | Let $R$ be a sheaf of rings on a topological space $X$. Assume $R \neq 0$. Does then $R$ have a maximal ideal? So this is a spacified analogon of the theorem, that every nontrivial ring has a maximal ideal. Currently I try to develope this sort of spacified commutative algebra and algebraic geometry. If anyone knows so... | https://mathoverflow.net/users/2841 | Does every nontrivial sheaf of rings have a maximal ideal? | Take $X=\mathbf Z$ with topology $(k,\infty)\cap\mathbf Z$ for $-\infty\leq k\leq\infty$, so that sheaves on $X$ may be identified with sequences $\dots\to F\_k\to F\_{k+1}\to\dots$, $k\in\mathbf Z$. Now take for $R$ the constant sheaf with value $\mathbf Q$. All ideals have the form $\dots\to0\_{k-1}\to0\_k\to\mathbf ... | 8 | https://mathoverflow.net/users/2035 | 34235 | 22,157 |
https://mathoverflow.net/questions/34233 | -3 | We know that by Dirichlet's formula for the Divisor function $ \displaystyle \sum\limits\_{n \leq x} d(n) = x \log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$.
What is the best approximation available till date for the given formula. I know that finding the infimum of the $\mathcal{O}'s$ is an unsolved problem, but would l... | https://mathoverflow.net/users/1483 | Dirichlet's Divisor Function | Use Wikipedia: <http://en.wikipedia.org/wiki/Dirichlet_divisor_problem> .
| 4 | https://mathoverflow.net/users/6153 | 34238 | 22,158 |
https://mathoverflow.net/questions/34241 | 5 | I'm trying to understand why the Laplacian operator is used in [blob detection](http://en.wikipedia.org/wiki/Blob_detection) in image analysis. I must admit that in trying to figure out why the Laplacian is useful in this application, I've really confused myself with the different uses of the word 'Laplace.' For instan... | https://mathoverflow.net/users/7445 | Laplacian operator and relation to the Laplace Transform | They are certainly not the same thing.
You might sometimes see them appear in the same context because transforms of Laplace-Fourier type are immensely useful for analyzing linear differential operators like the Laplacian. But the Fourier transform has better analytic properties, so that's the one you are more likely... | 5 | https://mathoverflow.net/users/2036 | 34245 | 22,162 |
https://mathoverflow.net/questions/34240 | 7 | What I mean is this. By downward Lowenheim-Skolem theorem, first-order formula Q is a always true iff it is true in every countable structure. But is there some first-order formula Q which is true in every computable structure and false in some non-computable structure? My feeling is that of course the answer should be... | https://mathoverflow.net/users/6307 | Are computable models sufficient? | There is no computable, countable nonstandard model of Peano arithmetic. This result is known as Tennenbaum's theorem after Stanley Tennenbaum. There is an online paper at [1]. So if you take any sentence $R$ in the language of PA that is not true in the standard model but is consistent with PA, it will be true in some... | 10 | https://mathoverflow.net/users/5442 | 34248 | 22,163 |
https://mathoverflow.net/questions/34237 | 28 | I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long time.
Suppose $M$ is a compact even dimensional smooth manifold with two symplectic forms $\omega\_0$ and $\omega\_1$... | https://mathoverflow.net/users/7311 | When are two symplectic forms "isotopic"? | It is known, for a long time now, that there exist examples of symplectic forms in the same cohomology class which are non-isotopic. I do not remember if there exists such example in the dimension $4$, but in dimension $6$ there are different examples. Here is an example constructed by Dusa McDuff:
Let $X$ be a produ... | 24 | https://mathoverflow.net/users/2823 | 34249 | 22,164 |
https://mathoverflow.net/questions/34246 | 4 | I would like to ask whether there is a combinatorial proof of the following recurrence relation for Catalan numbers:
$$
C\_{n+1}=\frac{4n+2}{n+2} C\_n.
$$
Thanks!~
| https://mathoverflow.net/users/6594 | Combinatorial proof of a recurrence for the Catalan numbers | What you are asking is reported as fourth proof in the wiki article for the formula of the Catalan numbers:
<http://en.wikipedia.org/wiki/Catalan_number#Fourth_proof>
| 10 | https://mathoverflow.net/users/6101 | 34259 | 22,171 |
https://mathoverflow.net/questions/34252 | 8 | If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any way I can rapidly find the eigenvalues of the matrix M=A+B?
(I would be surprised if it helps, but I actually have the... | https://mathoverflow.net/users/8105 | Eigenvalues of A+B where A is symmetric positive definite and B is diagonal | I doubt it. At least it shouldn't be easier than the case where you have the sum of two arbitrary positive definite matrices A',B' with known eigenvalues and eigenvectors. Then you could use an orthogonal basis of eigenvectors for B' and set $A=PA'P^{-1}$ and $B=PB'P^{-1}$. B would be diagonal and AB would have the sam... | 4 | https://mathoverflow.net/users/8008 | 34263 | 22,174 |
https://mathoverflow.net/questions/34269 | 9 | Dear MO Community,
I am trying to understand Mazur's 1976 notes "Rational points on Modular Curves" (which can be found in Springer Lecture Notes in Mathematics 601).
Let N be a prime number, and let $X\_0(N)$ be the usual modular curve over $\mathbb{Q}$. Say that a point on it is 'CM' if the elliptic curve corres... | https://mathoverflow.net/users/5744 | CM rational points on modular curves | Dear Barinder,
This is the subject of the arithmetic theory of complex multiplication. Given any CM
elliptic curve $E$, and any kind of level structure on $E$, this theory determines the precise
field of definition of $E$ with its given level structure. You can find a discussion of this
in Silverman II ("Advanced to... | 6 | https://mathoverflow.net/users/2874 | 34272 | 22,180 |
https://mathoverflow.net/questions/34264 | 11 | Consider a real valued function $g$ on an open interval $(a,b)$ which is the derivative of a function continuous on $[a,b]$ at each point of $(a,b)$. The function $g$ has the intermediate value property, so a monotone $g$ will have to be continuous, a general $g$ cannot have simple discontinuities, etc. With such const... | https://mathoverflow.net/users/6627 | Examples of badly behaved derivatives | Talking about how bad may be the derivative of an everywhere derivable function on the interval $[a,b]$, the natural example that occurs to my mind is: a Pompeiu derivative, that is, a derivative that vanishes in a dense set (these weird functions, however, constitue a closed linear space of $C^0[a,b]$, while it's not ... | 11 | https://mathoverflow.net/users/6101 | 34277 | 22,184 |
https://mathoverflow.net/questions/34266 | 4 | The transformation formula for a Siegel modular form can be interpreted as the statement that the modular form is a holomorphic section of a line bundle over the period domain (the quotient of the Siegel upper half-plane by a subgroup of finite index in the integral symplectic group). I've seen brief mentions of this f... | https://mathoverflow.net/users/2183 | Siegel modular forms as sections of line bundles over the period domain | Modulo torsion, the Picard group of the quotient of the Siegel upper half plane by a finite-index subgroup of $\text{Sp}\_{2g}(\mathbb{Z})$ is just $\mathbb{Z}$. This result should probably be attributed to Borel.
For a calculation of the torsion and explicit line bundles corresponding to the various pieces (plus re... | 6 | https://mathoverflow.net/users/317 | 34278 | 22,185 |
https://mathoverflow.net/questions/34281 | 4 | Is it possible to build set theory on first-order logic *without equality*?
For example, how could one show that if $x\_0=x\_1$ then $\left\{x\_0\}=\{x\_1\right\}$, where $x\_0$ and $x\_1$ are two sets? And the other way around? In my opinion, it is impossible using only the following facts:
1. axiom of extensional... | https://mathoverflow.net/users/3554 | First-order logic without equality and set theory | The converse direction fails because, in Morse-Kelley set theory plus your facts, $\{x\}$ is empty whenever $x$ is a proper class. So $\{x\_0\}=\{x\_1\}$ for any two proper classes.
The forward direction, on the other hand, seems to follow immediately from the facts you listed. If $x\_0=x\_1$ then, since facts 2, 3, ... | 5 | https://mathoverflow.net/users/6794 | 34282 | 22,186 |
https://mathoverflow.net/questions/34280 | 2 | Why is $H^p(X, \mathcal{E}xt^q(F, I))=0$ for $p>0$ and $I$ an injective sheaf and $F$ an arbitrary sheaf? I'm trying to check the hypothesis in the grothendieck spectral sequence applied to the functor $\mathcal{E}xt(F, -)$ and global sections functor, and so I'm trying to see why the sheaf $\mathcal{E}xt(F, I)$ is acy... | https://mathoverflow.net/users/8111 | Spectral sequence for Ext | $\mathcal{E}xt^q(F, I)$ is 0 for $q > 0$. On the other hand it follows easily by considering extensions by 0 that $\mathcal{H}om(F, I)$ is flabby, hence acyclic.
| 6 | https://mathoverflow.net/users/4790 | 34286 | 22,188 |
https://mathoverflow.net/questions/33711 | 17 | When $p$ is a prime $\equiv9\bmod16$, the class number, $h$, of $\mathbb Q(p^{1/4})$ is known to be even. In
[[Charles J. Parry,
A genus theory for quartic fields.
*Crelle's Journal* **314** (1980), 40--71]](http://dx.doi.org/10.1515/crll.1980.314.40)
it is shown that $h/2$ is odd when 2 is not a fourth power in $... | https://mathoverflow.net/users/6214 | If p is a prime congruent to 9 mod 16, can 4 divide the class number of Q(p^(1/4))? | Let $p \equiv 1 \bmod 8$ be a prime number, let $K = {\mathbb Q}(\sqrt[4]{p})$, and let $F$ be the quartic subfield of the field of $p$-th roots of unity. An easy exercise involving Abhyankar's Lemma shows that $FK/K$ is an unramified quadratic extension, hence the class number of $K$ is always even.
The field $KF$ h... | 16 | https://mathoverflow.net/users/3503 | 34292 | 22,190 |
https://mathoverflow.net/questions/34197 | 22 | Van der Corput [1] proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao [2] famously extended this theorem to length $k$.)
But taking this in a different direction, are *all* odd primes in a PAP-3? That is, for every prime $p>2$, is there a $k$ such that $p+k$ and ... | https://mathoverflow.net/users/6043 | Are all primes in a PAP-3? | This question is extremely close to this one
[Covering the primes by 3-term APs ?](https://mathoverflow.net/questions/2214/covering-the-primes-by-3-term-aps)
though not exactly the same.
For much the same reasons as described in the answer given there, the answer to your question is almost certainly yes, but a pr... | 26 | https://mathoverflow.net/users/5575 | 34298 | 22,195 |
https://mathoverflow.net/questions/34291 | 3 | A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't equal $f(g^\prime)$. The definition for a semigroup is analagous: just make $G$ and $G^\prime$ semigroups and make $f$ a... | https://mathoverflow.net/users/3804 | Residual finiteness of groups versus residual finiteness of semigroups | I first posted this as a comment, but I guess that this is an answer.
If a group is residually finite as a semigroup, it is residually finite as a group. This is an immediate consequence of the following easy fact: if G is a group and φ:G→S is a semigroup homomorphism, then the image φ(G) is a group and φ is a group ... | 7 | https://mathoverflow.net/users/7982 | 34299 | 22,196 |
https://mathoverflow.net/questions/34257 | 3 | We say that an object $X$ of a category $C$ is $\kappa$-compact (also $\kappa$-presentable and $\kappa$-accessible) for a cardinal $\kappa$ if $h^X(\cdot):=Hom(X,\cdot)$ commutes with all $\kappa$-filtered colimits.
In Makkai-Pare, a different but equivalent definition is given, that $X$ is $\kappa$-compact if for a... | https://mathoverflow.net/users/1353 | Equivalence of the two definitions of k-compactness/k-presentability | The definitions are equivalent as stands; no extra conditions (eg majorants for infinite sets of factorisations) are needed. This is essentially because colimits in $\mathbf{Sets}$ are computed finitarily.
One way to present the colimit $\underset{i}{\varinjlim} Hom(X,F(i))$ (see eg Mac Lane *CWM*) is as
1. the cop... | 3 | https://mathoverflow.net/users/2273 | 34305 | 22,200 |
https://mathoverflow.net/questions/34318 | 0 | Are there any examples other than using dimension for vector spaces where the easiest way to show that two objects are isomorphic is by using a classification theorem and showing that they must both be in the same class? (homeomorphisms count too)
| https://mathoverflow.net/users/nan | Isomorphism by classification | Genus for surfaces would be a simple example.
Connectedness for compact $1$-dimensional manifolds would be another!
| 3 | https://mathoverflow.net/users/1465 | 34320 | 22,211 |
https://mathoverflow.net/questions/34332 | 32 | As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1\_R)=1\_S$. But there has been a minority who do not require this, one prominent example being Herstein in *Topics in Algebra*.
What are some of the most striking consequences of not requiring ring homomorphis... | https://mathoverflow.net/users/1916 | Consequences of not requiring ring homomorphisms to be unital? | Let's suppose that our rings are commutative (which is the case that is immediately relevant to algebraic geometry).
If $\phi:A \to B$ is a (possibly non-unital) homomorphism, then $e := \phi(1\_A)$ is an
idempotent in $B$, and so we get a decomposition $B = eB \times (1-e)B,$
and the map $\phi$ factors as $A \to eB... | 36 | https://mathoverflow.net/users/2874 | 34335 | 22,217 |
https://mathoverflow.net/questions/34342 | 1 | I have the system of multi-variable polynomial (quadratic) equations with real coefficients. The number of equations is given scales as $K$ and the number of unknowns goes as $K^2$. So for for large $K$, this is an underdetermined system.
Can I conclude that I can always find $K$ large enough so that this system has ... | https://mathoverflow.net/users/8119 | Real solutions to underdetermined system of polynomial equations | However large $n$ may be, the equation
$x\_1^2 + \ldots + x\_n^2 = -1$ has no real solution.
(Similarly, the homogeneous equation $x\_1^2 + \ldots + x\_n^2 = 0$ has no nontrivial solution.)
If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial eq... | 5 | https://mathoverflow.net/users/1149 | 34344 | 22,222 |
https://mathoverflow.net/questions/34339 | 6 | Arinkin has a theorem which says that an abelian variety can be reconstructed from its derived category of coherent D-modules.
D.Orlov conjectured that this theorem is true for any variety.
My question is:
Is this conjecture proved or disproved? I wonder know the related work, examples and any other related obs... | https://mathoverflow.net/users/1851 | Reconstruction from category of D-modules on variety | As far as I understand from your statement of the conjecture, the conjecture is false, although there are similar statements that are true. If I understand correctly, a weaker question (more likely to have the answer yes) would be "can one recover a variety from its category of D-modules."
For a non-example of the we... | 6 | https://mathoverflow.net/users/2669 | 34352 | 22,228 |
https://mathoverflow.net/questions/34350 | -3 | Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{Supp}M''$. Furthermore can we obtain that $\dim M=\dim M'+\dim M''$? In fact, I want to know that given a module $M$ wi... | https://mathoverflow.net/users/5775 | An elementary question about the Krull dimension of modules | The dimension of $M$ is the maximum of the dimensions of $M'$ and $M''$, so what you are asking for cannot happen (at least not in the noetherian case). To Dylan: the dimension of a module is the dimension of its support.
| 1 | https://mathoverflow.net/users/4790 | 34357 | 22,232 |
https://mathoverflow.net/questions/34316 | 18 | OK, the title is opinionated and contentious, but I have a definite
question. I know that the title refers to the Bourbaki volume
*Groupes et Algèbres de Lie* (Chapters 4-6), published in 1968, but
>
> who said that it is the only great book that Bourbaki ever wrote?
>
>
>
The only reference I can find is the ... | https://mathoverflow.net/users/1587 | The only great book that Bourbaki ever wrote? | Google found this:
Notices of the AMS, September 1998, p. 979:
Bill Casselman's review of POLYHEDRA by Cromwell,
we find the phrase "the one great book by Bourbaki"
| 24 | https://mathoverflow.net/users/454 | 34362 | 22,236 |
https://mathoverflow.net/questions/34374 | 5 | Just like the title. I want a simple proof of the statment in the title.
$\mathbb{Q}\_p$ is the p-adic field.
I wonder which module (or vector space) will be chosen as the space for the representation. Is this statement true for arbitarily module/vector space?
Thanks!
| https://mathoverflow.net/users/8122 | Any finite dimensional admissible(smooth) irreducible representation of GL(2,Q_p) is 1-dim | What does "admissible" mean for you? Does it imply smoothness (stabilisers are open)? If not then I think the statement might be false (choose some hopelessly discontinuous injection from $\mathbf{Q}\_p$ into $\mathbf{C}$ and then consider the induced "natural" representation of $GL(2,\mathbf{Q}\_p)$ on $\mathbf{C}^2$;... | 14 | https://mathoverflow.net/users/1384 | 34381 | 22,247 |
https://mathoverflow.net/questions/34384 | 1 | It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
| https://mathoverflow.net/users/7421 | Сomplete homogeneous space which is not locally compact | A Banach space is homogeneous since the metric is arising from a norm. An infinite dimensional Banach space has the property that its unit ball is not compact; therefore the space is not locally compact.
For a concrete example, take the space of continuous real functions on an interval with the supremum norm.
| 2 | https://mathoverflow.net/users/2938 | 34386 | 22,251 |
https://mathoverflow.net/questions/34390 | 12 | It happens occasionally that one can prove that a given set is not empty by proving that it is actually large. The word "large" here may refer to different properties.
For example, one can prove that a certain set is not empty by proving that its cardinality is big, as in the proof that there exist transcendental num... | https://mathoverflow.net/users/1162 | On proving that a certain set is not empty by proving that it is actually large | Many existence proofs which exploit the idea of Baire category.
For instance, existence of a metrically transitive automorphism of the closed unit square was first obtained by the category method (see ["Measure-preserving homeomorphisms and metrical transitivity"](http://www.ams.org/mathscinet/search/publdoc.html?ar... | 16 | https://mathoverflow.net/users/5371 | 34396 | 22,257 |
https://mathoverflow.net/questions/34394 | 7 | In his book *"The geometry of geodesics"* H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
1. The space is metric
2. The space is finitely compact, i.e., a bounded infinite set has at least one accumulation point
3. [*metric convexity*] For every $x\neq z$ there exists a... | https://mathoverflow.net/users/8047 | G-spaces and manifolds | On a \**complete*\*smooth Riemannian manifold,
1. Any bounded (with respect to the distance function induced by the Riemannian metric) closed set in a manifold is compact.
2. This is telling you that there is a minimal geodesic joining $x$ to $y$ that, when extended, is also a minimal geodesic joining $x$ to $z\_1$. ... | 11 | https://mathoverflow.net/users/613 | 34398 | 22,259 |
https://mathoverflow.net/questions/34402 | 7 | Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on this space by defining $\mu(C\_{a\_i})=1/2$, where the $C\_{a\_i}=\{x\in X | x\_i=a\_i\}$ are the open subbase cylinder... | https://mathoverflow.net/users/3831 | Nice orthonormal basis for L^2(Cantor set) | Since the Cantor set with your measure is also the compact group $(\mathbb{Z}/2)^\mathbb{N}$ with Haar measure, a natural orthonormal basis is the (continuous) characters $\alpha:X\to S^1$, namely the finite products of coordinates $c\_n(x)$, $n\in\mathbb{N}$ if you view $\mathbb{Z}/2$ as `{-1,1}`. These form the discr... | 16 | https://mathoverflow.net/users/6451 | 34409 | 22,266 |
https://mathoverflow.net/questions/34358 | 4 | My knowlege in group theory is very limited and this question is out of pure curiousity:
Given a random finite group, how many conjugacy classes will it probably have?
I can try to make this question more precise:
Define $a\_n$ to be the average number of conjugacy classes in finite groups of order less or equal $n$.... | https://mathoverflow.net/users/2837 | Number of conjugacy classes in generic finite group? | In elementary terms, you have to analyze the following class equation
$ n = 1 + h\_2 + ... + h\_r $
where
* n is the order of the group G
* $h\_k$ denotes the number of elements in the k-th conjugacy class, and $ n = c\_k.h\_k$.
Dividing by n, you get
$1 = \frac{1}{n} + \frac{1}{c\_2} + ... + \frac{1}{c\_r} $ which... | 4 | https://mathoverflow.net/users/5372 | 34413 | 22,269 |
https://mathoverflow.net/questions/34364 | 8 | I have a question about graph width measures of undirected simple graphs. It is well-known that cographs (graphs which can be built by the operations of disjoint union and complementation, starting from isolated vertices) have cliquewidth at most 2. (Courcelle et al, Upper bounds to the clique width of graphs). Now con... | https://mathoverflow.net/users/5200 | Cliquewidth of Cographs + kv | I think the exponential bound is necessary. Here is why.
Consider the disconnected graph $G$ on $n$ vertices, labelled $1$ to $n$. Denote the set of vertices that new vertex $i$ is connected to by $f(i)$, a subset of $\lbrace 1,2,\ldots,n \rbrace$.
The question then becomes: is it possible to find such a function $f$... | 4 | https://mathoverflow.net/users/7252 | 34425 | 22,276 |
https://mathoverflow.net/questions/34424 | 28 | [This Wikipedia article](http://en.wikipedia.org/wiki/List_of_finite_simple_groups) states that the isomorphism type of a finite simple group is determined by its order, except that:
* L4(2) and L3(4) both have order 20160
* O2n+1(q) and S2n(q) have the same order for q odd, n > 2
I think this means that for each i... | https://mathoverflow.net/users/4947 | Number of finite simple groups of given order is at most 2 - is a classification-free proof possible? | It is usually extraordinarily difficult to prove uniqueness of a simple group given its order, or even given its order and complete character table. In particular one of the last and hardest steps in the classification of finite simple groups was proving uniqueness of the Ree groups of type $^2G\_2$ of order $q^3(q^3+1... | 53 | https://mathoverflow.net/users/51 | 34432 | 22,281 |
https://mathoverflow.net/questions/34441 | 2 | I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition.
According to wikipedia one nice thing about H-S operators is that on a separable Hilbert space $H$ the set of H-S endo... | https://mathoverflow.net/users/7 | Hilbert Schmidt operators | I think the perfect reference for you is Lars Hörmander's *The Analysis of Linear Partial Differential Operators*, vol III (the chapter on elliptic operators). There you'll find in perfect Hörmander style all you need about Fredholm, Hilbert-Schmidt, trace class operators. (As to the composition, H-S is a two-sided ide... | 3 | https://mathoverflow.net/users/6101 | 34444 | 22,287 |
https://mathoverflow.net/questions/34363 | 6 | The complete elliptic integral of the first kind
$K(m)=\int\_0^{\pi/2}\frac{\mathrm{d}t}{\sqrt{1-m\sin^2t}}$
is easily computed via the arithmetic-geometric mean iteration; to wit,
$K(m)=\frac{\pi}{2M(1,\sqrt{1-m})}$
where $M(a,b)$ is the arithmetic-geometric mean of $a$ and $b$. With a little more trickery, th... | https://mathoverflow.net/users/7934 | Simultaneously computing a complete elliptic integral and its complement | There are two possible ways to attack this problem
1. Both K and K' can be expressed in
terms of the Theta function as
described here
<http://mathworld.wolfram.com/EllipticModulus.html>.
If you compute $\Theta\_3$, you can get
both at the same time.
2. The other way is to observe that
both K and K' are expressible in... | 4 | https://mathoverflow.net/users/5372 | 34457 | 22,296 |
https://mathoverflow.net/questions/34446 | 2 | Given an undecidable collection of first-order sentences, is there necessarily a complete undecidable theory containing it? A direct attempt to prove it seems to require some control over the completions which need not exist, but I don't see a counterexample.
On a more concrete side, Macyntire proved that the theory ... | https://mathoverflow.net/users/25726 | Undecidable completion of undecidable theory, and pairs of RCF | My recollection is that Macintyre proved there are $2^{\aleph\_0}$ complete theories of pairs
$(K,L)$ where $L\subset K$ are real closed fields. This is in his thesis but I don't think
he published it anywhere else. There are later papers of Francoise Delon and Walter Bauer
that develop this further. On the other hand ... | 5 | https://mathoverflow.net/users/5849 | 34461 | 22,299 |
https://mathoverflow.net/questions/34469 | 8 | It is believed that $BPP$ has no complete problems. Even for $BPP^O$ for a suitable oracle $O$ it is believed not to have complete problems, unless P=BPP. I wonder if the class MA (the randomized version of NP) has complete problems. For example, IP has complete problems (given that it is equal to PSPACE). There is a s... | https://mathoverflow.net/users/7692 | Complete problems for randomized complexity classes | In general, for randomized classes complete problems tend to be either promise problems or approximation problems (which means they don't technically satisfy the conditions for being complete problems).
If you allow approximation problems, you can get complete problems in BPP. For example, for BPP you can ask: given... | 20 | https://mathoverflow.net/users/2294 | 34472 | 22,305 |
https://mathoverflow.net/questions/34415 | 6 | I am reading the paper *Hausdorff dimension for Horseshoes,* by McCluskey and Manning. In the following theorem
**Theorem:**
Let $\Lambda$ be a basic set for a $C^1$ axiom A diffeomorphism $f:M^2\to M^2$ with $(1,1)$ splitting
$$
T\_{\Lambda}M=E^s\oplus E^u.
$$
Define $\phi:W^u(\Lambda)\to\mathbb R$ by
$$
\phi... | https://mathoverflow.net/users/2386 | Relation between Hausdorff dimension and Bowen's equation | I think there are several different ways to make intuitive sense of this, so I'll have a bit of a go at each of them, and hope you find it helpful.
**First explanation:** *At the global level for similarity maps.*
Let $M$ be a manifold, and consider a conformal map $f\colon M\to M$. (By conformal we mean that $Df\_... | 10 | https://mathoverflow.net/users/5701 | 34475 | 22,307 |
https://mathoverflow.net/questions/34442 | 4 | I would be glad to see a reference to the following easy lemma in additive combinatorics: if $A\_1$ and $A\_2$ are two sets of remainders modulo $n^2$, each has cardinality $n > 1$ and all elements of $A\_i$ are different modulo $n$ (for $i=1,2$), then $A\_1+A\_2$ is not equal to the set of all remainders modulo $n^2$.... | https://mathoverflow.net/users/4312 | Sum of sets modulo a square | There must be an easier proof but here is a nice approach which can indeed lead to deeper results (feel free to edit for math display, I tried):
Techniques with characteristic polynomials and roots of unity can be very powerful. I like the way that the appropriate lemmas are explained in my paper with Ethan Coven "Tili... | 2 | https://mathoverflow.net/users/8008 | 34481 | 22,310 |
https://mathoverflow.net/questions/34484 | 2 | Let $X=Spec(R)$ be an affine scheme. Let $Y$ be a closed subset of $X$ and denote by $U$ its complement. Assume $U$ is quasicompact. Then $U= \cup\_{i=1}^{n} D(f\_i)$, where $f\_i \in R$.
Denote the inclusions by $j: U \to X$ and $i:Y \to X$. Let $F \in D\_{qcoh}(X)$, where $D\_{qcoh}(X)$ stands for the derived categor... | https://mathoverflow.net/users/8144 | Sheaf cohomology and torsion | The answer is yes, assuming by $(f\_1,\ldots, f\_n)$-torsion you mean that each element of each cohomology group of $F$ is killed by some power of this ideal.
There are several ways to see this. The most straightforward is probably to note that the inclusion $j\_i\colon D(f\_i) \to X$ factors via $j\colon U \to X$ so... | 3 | https://mathoverflow.net/users/310 | 34486 | 22,313 |
https://mathoverflow.net/questions/22041 | 0 | Hi Everyone
I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.
We are given a Brownian Motion $B\_t$ and we note $M\_t=\sup\_{s\le t}B\_s$.
Moreover we have a smooth real valued function $F(t,x,y)$ (for example a $C^{1,2,1... | https://mathoverflow.net/users/2642 | Looking for a version of Itô's Lemma | Let $F(t,m-b,m)=G(t,m,b)$ (this way it's easier to write). As long as $M$ has bounded variation, we can happily write (I skip arguments, which are $t,M\_t,B\_t$)
$$
dG(t,M\_t,B\_t) = G'\_t dt + G'\_m dM\_t + G'\_b dB\_t + \frac12 G''\_{bb} dt.
$$
| 2 | https://mathoverflow.net/users/8146 | 34492 | 22,318 |
https://mathoverflow.net/questions/34452 | 34 | Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals:
$
Sym\_m(E)^\*
$
and
$Sym\_m(E^\*)$ are canonically isomorphic.
This is not true in characteristic $p>0$ (one has a c... | https://mathoverflow.net/users/5771 | Symmetric powers and duals of vector bundles in char p | Here's another look at the case $m=2$. Maybe it can be used to shed more light on Torsten's nice example. $S^2E$ is part of an exact sequence $0\to \Lambda^2E\to E\otimes E\to S^2E\to 0$. $\Gamma^2E$ is part of an exact sequence $0\to \Gamma^2E\to E\otimes E\to \Lambda^2E\to 0$. The composition of $E\otimes E\to \Lambd... | 14 | https://mathoverflow.net/users/6666 | 34512 | 22,326 |
https://mathoverflow.net/questions/34510 | 1 | How to see that the cup products vanish on suspensions?
| https://mathoverflow.net/users/8152 | when cup product is a zero homomorphism | 13.66 in Switzer's Algebraic Topology: Homotopy and Homology. The idea is to use the fact that $\Sigma X$ decomposes into two copies of $CX$, say $A$ and $B$, glued along the common boundary of $X$. For any two cohomology classes $x$ and $y$ in $\tilde{E}^\* \Sigma X$, you can uniquely pull $x$ back to a class $x'$ on ... | 12 | https://mathoverflow.net/users/1094 | 34514 | 22,328 |
https://mathoverflow.net/questions/34478 | 12 | (Just to be precise, in this question, the word "knot" means "ambient isotopy class of a (EDIT: smooth) knot in $S^3$.") A knot in $S^3$ is called prime if it is not the [connected sum](http://en.wikipedia.org/wiki/Connected_sum#Connected_sum_of_knots) of two other non-trivial knots in $S^3$. Clearly any knot is a sum ... | https://mathoverflow.net/users/2669 | Is there a procedure for obtaining all knots in S^3? | If you allow yourself to consider knotted trivalent graphs, instead of just knots, then you can start with just the tetrahedron and Mobius bands, and use the operations unzip, bubbling, and connect sum to get all knotted trivalent graphs. This result is due to Dylan Thurston and Dror Bar-Natan and is written up by Dyla... | 9 | https://mathoverflow.net/users/22 | 34523 | 22,334 |
https://mathoverflow.net/questions/34524 | -3 | I'm looking for some knowledge on probability, I've scoured the net but I can't really grasp the answer.
I was having a discussion with a co-worker about roulette probability. He says that at any given spin the probability that the outcome being red or black is equal (not taking into account the 0, which is neither).... | https://mathoverflow.net/users/8154 | Roulette probability | (It's true that this question will probably be closed soon.)
Ask yourself this question: Does the roulette ball or table have a memory? If not, then past events cannot possibly affect the next probability. "No memory", or more technically independent outcomes, is a standard hypothesis in probability problems like the... | 2 | https://mathoverflow.net/users/7936 | 34531 | 22,337 |
https://mathoverflow.net/questions/34476 | 4 | Let $X$ be a scheme defined over $\mathbb{C}$ with an involution $\sigma$. How to get
a $X\_{\mathbb{R}}$ scheme defined over $\mathbb{R}$ such that $X\_{\mathbb{R}}\times\_\mathbb{R} \mathbb{C} = X$ ? How are the real algebraic bundle on $X\_{\mathbb{R}}$ and complex bundle on
$X$ related?
| https://mathoverflow.net/users/8141 | Real algebraic variety and real algebraic bundle | The paper by
Atiyah
on [K-theory and reality](http://qjmath.oxfordjournals.org/cgi/reprint/17/1/367)
Quart. J. Math. Oxford Ser. (2) 17 1966 367--386. MR0206940 discusses the topological analogue of this question, showing how to relate complex vector bundles over a space with vector bundles over the fixed points of an ... | 3 | https://mathoverflow.net/users/51 | 34534 | 22,339 |
https://mathoverflow.net/questions/34527 | 3 | This question is concerned with a possible lemma which would be very useful in one of my current research projects, but which I am currently unable to prove. The project as a whole relates to the asymptotic growth rates of certain matrix products, but once all the dynamical parts are finished with, I need to obtain som... | https://mathoverflow.net/users/1840 | A differential inclusion relating to the slope of a convex function | Unfortunately the lemma is false. Given a candidate $C$, let $\varepsilon = 1$ and $F(t) = \ln(te^C+1)$. Then the hypotheses of Lemma B hold but the conclusion fails.
| 4 | https://mathoverflow.net/users/7936 | 34537 | 22,341 |
https://mathoverflow.net/questions/34520 | -1 | let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\gamma) g +(1-\gamma)(e^{\rho s}g)^{\frac{1}{\gamma-1}}g=0$$
Find a closed form for $g$?
| https://mathoverflow.net/users/5672 | A differential equation | This seems to be a Bernoulli differential equation. Please cf. <http://en.wikipedia.org/wiki/Bernoulli_differential_equation> for the solution (in your case $n= \frac{\gamma}{\gamma-1}$).
| 0 | https://mathoverflow.net/users/6415 | 34538 | 22,342 |
https://mathoverflow.net/questions/34521 | 18 | I have been using the following result:
Given a polynomial $f(x,t)$ of degree $n$ in $\mathbb{Q}[x,t]$, if a rational specialization of $t$ results in a separable polynomial $g(x)$ of the same degree, then the Galois group of $g$ over $\mathbb{Q}$ is a subgroup of that of $f$ over $\mathbb{Q}(t)$.
However, I have b... | https://mathoverflow.net/users/4078 | Proof of the result that the Galois group of a specialization is a subgroup of the original group? | Here is a broader setup for your question. Let $A$ be a Dedekind domain with fraction field $F$, $E/F$ be a finite Galois extension, and $B$ be the integral closure of $A$ in $E$.
Pick a prime $\mathfrak p$ in $A$ and a prime $\mathfrak P$ in $B$ lying over $\mathfrak p$.
The decomposition group $D(\mathfrak P|\mathfr... | 19 | https://mathoverflow.net/users/3272 | 34558 | 22,355 |
https://mathoverflow.net/questions/31786 | 27 | For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. *bond percolation*), it is well known that there is a critical probability $0 < p\_c < 1$ depending on the lattice, such that if $ p > p\_c$ then there is almost surely an infinite component (i.e. with probability $1$), an... | https://mathoverflow.net/users/4558 | What is the right notion of self-dual (two-dimensional) percolation in R^4? | The paper "PLAQUETTES, SPHERES, AND ENTANGLEMENT" by GEOFFREY R. GRIMMETT AND ALEXANDER E. HOLROYD does not deal with the self-dual problem, but nevertheless could be of interest for you. If shows that "The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d − 1)sp... | 5 | https://mathoverflow.net/users/6415 | 34564 | 22,359 |
https://mathoverflow.net/questions/34561 | 8 | Let $M$ be a matroid admitting a coordinatization over a complex vector space. If we know that the complex coordinatization space for $M$ is connected, then may we conclude that the matroid admits a coordinatization over the real numbers?
The only examples that I am able to construct which do not have real coordinati... | https://mathoverflow.net/users/8161 | Realization space of matroids | The answer should be no. Here is the reason: It is perfectly possible to have a connected algebraic variety, defined over $\mathbb{R}$, which has no $\mathbb{R}$-points. For example, $\{ (x,y) : x^2+y^2=-1 \}$. Using [Mnev's universality theorem](http://en.wikipedia.org/wiki/Mnev%2527s_universality_theorem), you should... | 6 | https://mathoverflow.net/users/297 | 34567 | 22,362 |
https://mathoverflow.net/questions/34433 | 12 | A functor $C \to D$ between categories induces a morphism of presheaf categories $Pre(D) \to Pre(C)$. This functor has a left adjoint given by left Kan extension and I am interested in knowing when this left adjoint preserves pull-back squares.
I'm interested in any conditions that make this happen, but I am particu... | https://mathoverflow.net/users/184 | An elementary question about adjunctions between presheaf categories preserving pullbacks. | $\newcommand{\C}{\mathbf{C}} \newcommand{\D}{\mathbf{D}} \newcommand{\Lan}{\mathrm{Lan}} \newcommand{\yon}{\mathbf{y}} \newcommand{\CC}{[\C^\mathrm{op},\mathbf{Set}]} \newcommand{\DD}{[\D^\mathrm{op},\mathbf{Set}]}$
Expanding on my comment above:
Define: a category is *semi-filtered* iff every pair of arrows $x \lef... | 7 | https://mathoverflow.net/users/2273 | 34577 | 22,369 |
https://mathoverflow.net/questions/23815 | 13 | The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted as the adjacency matrix of an undirected graph $G$, and the spectral theorem gives us information about how the sequence... | https://mathoverflow.net/users/290 | Combinatorial proof of (a special case of) the spectral theorem? | I believe the answer is yes and that this might be found somewhere in the literature on the graph reconstruction problem.
Let me denote the characteristic polynomial of the graph $G$ as $\phi(G,x)$ and let its cofactors be $\phi\_{ij}(G,x)$. In the paper ["Walk Generating Functions, Christoffel-Darboux Identities and... | 6 | https://mathoverflow.net/users/2384 | 34580 | 22,370 |
https://mathoverflow.net/questions/32020 | 9 | The sequence
[A059710](http://www.research.att.com/~njas/sequences/A059710)
starts 1,0,1,1,4,10,35,...
This satisfies the polynomial recurrence relation
$$ (n+5)(n+6)a(n)=2(n-1)(2n+5)a(n-1)+(n-1)(19n+18)a(n-2)+14(n-1)(n-2)a(n-3) $$
I have a $q$-analogue of this sequence. The first few terms are:
$$1$$
$$0$$
$$1$$
$... | https://mathoverflow.net/users/3992 | Finding recurrence relation for a sequence of polynomials | Using [FriCAS](http://fricas.sourceforge.net/), one can indeed guess a q-recurrence, given the first 50 terms or so. It is not nice, though. The command issued is
`guessHolo(q)(cons(1, [qRiordan n for n in 1..60]), debug==true, safety==10)`
for the q-differential equation (a linear combination with polynomial coeff... | 5 | https://mathoverflow.net/users/3032 | 34589 | 22,376 |
https://mathoverflow.net/questions/34591 | 3 | The statement is simple:
What is the probability that a set of n-1 transpositions generates the symmetric group, $S\_n$?
The motivation is that I remembered reading that this was an open problem somewhere on the internet, and then I solved it. I'm curious to see other people's solutions, because I think it's a nice... | https://mathoverflow.net/users/5312 | Probability of generating the symmetric group | A solution (assuming that all transpositions are distinct and are choosen uniformly among all
${n\choose 2}$ possible transpositions) can be given as follows:
A set of $n-1$ transpositions $(a\_1,b\_1),\dots,(a\_{n-1},b\_{n-1})$ on the set $\lbrace 1,\dots,n\rbrace$ generates the whole symmetric group
of $\{1,\dots,n... | 11 | https://mathoverflow.net/users/4556 | 34594 | 22,379 |
https://mathoverflow.net/questions/34592 | 29 | It is well known that if $K$ is a finite index subgroup of a group $H$, then there is a finite index subgroup $N$ of $K$ which is normal in $H$. Indeed, one can observe that there are only finitely many distinct conjugates $hKh^{-1}$ of $K$ with $h \in H$, and their intersection $N := \bigcap\_{h \in H} h K h^{-1}$ wil... | https://mathoverflow.net/users/766 | Existence of simultaneously normal finite index subgroups | I think the answer to your question is no. Take $G=PSL\_d(\mathbb{Q}\_p)$. It is a simple group. Take $H\_1=PSL\_d(\mathbb{Z}\_p)$ and take $H\_2=H\_1^g$ for some $g \in G$ so that $H\_1 \ne H\_2$. Now, if I am not mistaken $H\_1$ and $H\_2$ are maximal in $G$ so together they generate $G$. Also, $G$ commensurates $H\_... | 33 | https://mathoverflow.net/users/5034 | 34603 | 22,385 |
https://mathoverflow.net/questions/33460 | 13 | Let $G$ be a torsion-free group and $C$ the ring of complex numbers. The zero divisor (idempotent, resp.) conjecture is that there is no nontrivial zerodivisor (idempotent, resp.) in $CG$.
The wiki page:
<http://en.wikipedia.org/wiki/Group_ring>
says "This conjecture (zero divisor conjecture) is equivalent to K[G] hav... | https://mathoverflow.net/users/1546 | Zero divisor conjecture and idempotent conjecture | Passman showed that whenever there are zero-divisors in a group ring one also has (non-zero) nilpotent elements. He shows that for any field $k$ and any torsionfree group $G$, the ring $kG$ is a prime ring, i.e. the zero-ideal is a prime ideal.
Now, if $a,b \in kG$ are non-zero and $ab=0$, then there exists $c \in kG... | 11 | https://mathoverflow.net/users/8176 | 34616 | 22,395 |
https://mathoverflow.net/questions/34628 | 5 | Given a linear representation $\rho$ of $SL\_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL\_n$ decomposes
into generally several Jordan blocks of size $m\_1,\dots,m\_k$.
Is it possible to describe the partition $m=m\_1+m\_2+\dots+m\_k$, say in terms
of the hig... | https://mathoverflow.net/users/4556 | Dimensions of Jordan blocks associated to representations | The answer is yes. The Jordan block decomposition of the generic nilpotent in $SL\_m$ on a representation is the same as the decomposition of any representation under the principal $SL\_2$ (which is a map of $SL\_2$ to $SL\_n$ which sends the generic nilpotent in $SL\_2$ to a generic one in $SL\_n$; people usually have... | 5 | https://mathoverflow.net/users/66 | 34633 | 22,402 |
https://mathoverflow.net/questions/1940 | 9 | Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose left and right support equals 1.
Denote by A the set of all closed unbounded operators with dense domain
affiliated w... | https://mathoverflow.net/users/402 | Maximal localizations of von Neumann algebras | I think the question is not well-posed or has a negative answer.
One first has to deal with the question whether the left-right-regular elements satisfy the Ore condition, or equivalently, we have to ask: Can we find common denominators? If one is not in the finite case, this is not possible.
For $B(H)$, let us ta... | 4 | https://mathoverflow.net/users/8176 | 34635 | 22,404 |
https://mathoverflow.net/questions/34615 | 0 | Hi
I have a time series of probabilites, vector X
I need to convert the probabilites to uniform numbers.
As I understand it if I put the series into the cdf the output is thus uniform.
The problem is I do not know what the cdf is for my series so how is this done ?
Every question/example I see seems to say ...'da... | https://mathoverflow.net/users/8178 | Transforming to uniform numbers | Replace each data point by its percentile. E.g., if $x\_{27}$ is the 45th largest of 7289 data points, let $u\_{27} = 45/7289$.
| 0 | https://mathoverflow.net/users/7651 | 34637 | 22,406 |
https://mathoverflow.net/questions/29636 | 10 | In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero.
1) Is there a standard terminology for operads with and without $O(0)$?
2) Is there standard termi... | https://mathoverflow.net/users/1465 | Operad terminology - Operads with and without O(0). | I can second Jeffrey's comment, reduced is used to say that O(1) is just the monoidal unit (it allows us to use the Boardman Vogt resolution in homotopy theory). It's my opinion that this terminology will probably stick.
I would also say that a $\mu$ in O(n) had arity n.
That the O(0) part of an operad is referred ... | 5 | https://mathoverflow.net/users/109 | 34643 | 22,411 |
https://mathoverflow.net/questions/19528 | 4 | I'm looking for adaptive controllers (adaptive in both step size and order) for stiff integrators. I have asymptotically correct error estimates for the current method and all candidate methods of order 1 higher and lower than the current method. My naive controllers have occasional problems with either oscillating bet... | https://mathoverflow.net/users/4954 | Adaptive controllers for stiff ODE and DAE integrators | Have a look at [chapter 8 of Jackiewicz's book](http://books.google.fr/books?id=SjNfr7gfUZEC&lpg=PP1&dq=General%2520Linear%2520Methods%2520for%2520Ordinary%2520Differential%2520Equations.&pg=PA417#v=onepage&q&f=false), especially section 8.10 for a general background. There's some [matlab code](http://www.math.auckland... | 2 | https://mathoverflow.net/users/469 | 34644 | 22,412 |
https://mathoverflow.net/questions/34525 | 2 | Let X be a tensored and cotensored V-category, where V is a fixed complete, cocomplete, closed symmetric monoidal category.
Define $C:=Span(X)$ to be the category of spans in X (this is the functor category $X^{Sp}$ where $Sp$ is the walking span). We notice that $C$ is automatically "tensored" over $V$ (by computin... | https://mathoverflow.net/users/1353 | A better way to compute the mapping spaces of the category of spans in an enriched tensored category? | Because of the way you've chosen to write your second description, I don't think you're going to be able to avoid using something about the fiber product in Set. But there is a general fact here: any V-functor V-category [A,X], where A and X are V-categories, inherits any V-enriched (weighted) limits that X has, constr... | 2 | https://mathoverflow.net/users/49 | 34648 | 22,414 |
https://mathoverflow.net/questions/34595 | 18 | It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k).
My question is
***How to characterize Eisenstein series among such sections using geometric datas?***
For example, we know cusp forms are just sectio... | https://mathoverflow.net/users/4245 | Eisenstein series as sections of line bundles on moduli spaces | Here is one construction:
We have the exact sequence
$$0 \to H^0(\omega^{\otimes k}(-\text{cusps})) \to H^0(\omega^{\otimes k})
\to H^0(\text{cusps}, \omega^{\otimes k}\_{| \text{cusps}}).$$
(Here I am using $\omega$ for what you called $E$; this is the traditional notation
for modular forms people.) It is easy to de... | 13 | https://mathoverflow.net/users/2874 | 34653 | 22,418 |
https://mathoverflow.net/questions/34620 | 24 | The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. But are there other examples?
**Question**: Let $X$ be a locally noetherian scheme and assume that $X$ is irreducible ... | https://mathoverflow.net/users/40 | When is an irreducible scheme quasi-compact? | There are smooth counterexamples. Let $S\_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $d > 1$, and $s\_0 \in S\_0(k)$. Blow up $s\_0$ to get another such scheme $S\_1$ with a $\mathbf{P}^{d-1}\_k$ over $s\_0$. Blow up a $k$-point $s\_1$ over $s\_0$ to get $S\_2$, and keep going. Get pair... | 21 | https://mathoverflow.net/users/3927 | 34656 | 22,421 |
https://mathoverflow.net/questions/34658 | 53 | For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can every topological manifold embed continuously into some $\mathbb R^N$, and do we get the same bound for $N$?)
| https://mathoverflow.net/users/8188 | Is there a Whitney Embedding Theorem for non-smooth manifolds? | I'm not sure about $\mathbb{R}^{2n}$, but you can embed them in $\mathbb{R}^{2n+1}$ using dimension theory. The theorem is that every compact metric space whose [covering dimension](http://en.wikipedia.org/wiki/Lebesgue_covering_dimension) is $n$ can be embedded in $\mathbb{R}^{2n+1}$. The example of non-planar graphs ... | 57 | https://mathoverflow.net/users/317 | 34659 | 22,422 |
https://mathoverflow.net/questions/34646 | 8 | Let $A$ be a $C^\*$-algebra or some norm-closed algebra of operators on a Hilbert space.
In the old paper
Hille, E. *On Roots and Logarithms of Elements of a Complex Banach Algebra*, Math. Annalen, Bd. 136, S. 46--.57 (1958)
the question is studied, which elements $x \in A$ have the property that the exponenti... | https://mathoverflow.net/users/8176 | Is the set of exponentials open? | The exponential map is not open on the von Neumann algebra of all bounded linear operators on an infinite dimensional separable complex Hilbert space. A [1987 article by Conway and Morrel](http://www.ams.org/mathscinet-getitem?mr=870760) shows that the spectrum of an element of the interior of the image of the exponent... | 5 | https://mathoverflow.net/users/1119 | 34670 | 22,428 |
https://mathoverflow.net/questions/34674 | 11 | I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an e... | https://mathoverflow.net/users/1353 | Example of a CW complex not homeomorphic to the realization of a simplicial set? | The mapping cylinder of a really messy continuous map $I\to I$
The nerve of the category in which there are two objects and each Hom set is a singleton.
| 13 | https://mathoverflow.net/users/6666 | 34677 | 22,432 |
https://mathoverflow.net/questions/34663 | 1 | Some times ago I posted [this](https://mathoverflow.net/questions/32138/reference-for-some-elementary-facts-about-principal-bundles) question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth sections over the... | https://mathoverflow.net/users/7538 | module of sections of the horizontal bundle | There is a result (for which I don't recall a reference, but you might prove it maybe with the smooth Serre-Swan theorem. See for example Nestruev, Smooth manifolds and observables), which says that for the sections of the pullback of a vector bundle $E\to M$ along a map $\phi:N\to M$ one has
$$\Gamma(\phi^\*(E))=C^\i... | 2 | https://mathoverflow.net/users/745 | 34678 | 22,433 |
https://mathoverflow.net/questions/33237 | 33 | Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one correspondence with the collection of simply laced Dynkin diagrams. The simply laced Dynkin diagrams have themselves been c... | https://mathoverflow.net/users/4362 | Is there a common genesis for ADE classifications? | I will first address the string theory part of the question.
String theory provides examples of physical systems admitting several descriptions that provide natural bridges between Kleinian singularities (and therefore Platonic solids), ALE spaces, quiver diagrams, ADE diagrams and two dimensional Conformal Field Th... | 8 | https://mathoverflow.net/users/4046 | 34680 | 22,435 |
https://mathoverflow.net/questions/34673 | 48 | What is a good reference for the following fact (the hypotheses may not be quite right):
>
> Let $X$ and $Y$ be projective varieties over a field $k$. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$ and $Y$, respectively. Let $\mathcal{F} \boxtimes \mathcal{G}$ denote $p\_1^\*(\mathcal{F}) \otimes\_{... | https://mathoverflow.net/users/5094 | Kunneth formula for sheaf cohomology of varieties | The treatment in EGA is indeed intimidating, but in fact over a field the formula is not hard to prove. You only need $X$ and $Y$ to be separated schemes over $k$, and $\mathcal{F}$ and $\mathcal{G}$ to be quasi-coherent. Then cover $X$ and $Y$ by affine open subsets $\{U\_i\}$, and $\{V\_j\}$, and write down the Čech ... | 32 | https://mathoverflow.net/users/4790 | 34683 | 22,437 |
https://mathoverflow.net/questions/34640 | 12 | Can someone please explain the difference between local rigidity and infinitesimal rigidity? Does either version of rigidity imply the other?
In particular, I'm thinking about Weil's rigidity theorem for hyperbolic metrics on manifolds of dimension $\geq 3$. I've seen it referred to as both local and infinitesimal, w... | https://mathoverflow.net/users/8183 | Local vs. infinitesimal rigidity | Infinitesimal rigidity implies local rigidity, but not conversely. Local rigidity means a representation has no deformations, whereas infinitesimal rigidity means the natural tangent space to the character variety is 0-dimensional (this tangent space is a certain cohomology group with twisted coefficients). Weil proved... | 14 | https://mathoverflow.net/users/8197 | 34684 | 22,438 |
https://mathoverflow.net/questions/34702 | 3 | $A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-sequence and an $A$-sequence. At least I know it is true when $\mbox{inj.dim }M<\infty$, from the relation $\mbox{depth }M\l... | https://mathoverflow.net/users/5292 | Depth and dimension | $A=k[[x,y]]/(x^2,xy)$ then depth$(A)=0$. Let $M=R/(x)=k[[y]]$ then $y$ is a nonzerodivisor on $M$.
| 11 | https://mathoverflow.net/users/7763 | 34708 | 22,451 |
https://mathoverflow.net/questions/34704 | 9 | $A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness) of one necessarily imply the finiteness (or infiniteness) of another?
| https://mathoverflow.net/users/5292 | Projective & injective dimensions | To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better have finite injective dimension (the converse is also quite easy).
The local rings $R$ which have finite inj. dim. over themselves are also known as [Gorenstein rings](http://arxiv.org... | 12 | https://mathoverflow.net/users/2083 | 34709 | 22,452 |
https://mathoverflow.net/questions/33831 | 9 | The following definition is from:
* Dmitry Roytenberg, "AKSZ-BV formalism and Courant algebroid-induced topological field theories", *Letters in Mathematical Physics*, 2007 vol. 79 (2) pp. 143-159, MR2301393.
>
> A *graded manifold* $M$ over base $M\_0$ is a sheaf of $\mathbb Z$-graded commutative algebras ${\rm ... | https://mathoverflow.net/users/78 | Is every graded manifold affine, and is this definition of graded manifold the right one? | First of all, you're completely right in that I didn't need to stipulate that $M\_0$ be a manifold in the definition: it follows from a graded manifold being a locally ringed space with a particular local model (it was a physically motivated survey article, so I wasn't much concerned with foundational issues). It also ... | 6 | https://mathoverflow.net/users/8203 | 34721 | 22,458 |
https://mathoverflow.net/questions/34719 | 11 | A quantitative form of the twin prime conjecture asserts that the the number of twin primes less than $n$ is asymptotically equal to $2\, C\, n/ \ln^2(n)$ where $C$ is the so-called twin prime constant. A variety of sieve methods (originating with Brun) can be used show that the number of twin primes less than $n$ is a... | https://mathoverflow.net/users/630 | What is the best known upper bound for the number of twin primes? | J Wu, Chen's double sieve, Goldbach's conjecture, and the twin prime problem, Acta Arith 114 (2004) 215-273, MR 2005e:11128, bounds the number of twin primes above by $2aCx/\log^2x$, with $C=\prod p(p-2)/(p-1)^2$, and $a=3.3996$; I don't know whether there have been any improvements.
| 19 | https://mathoverflow.net/users/3684 | 34723 | 22,460 |
https://mathoverflow.net/questions/34697 | 7 | Let $\mathcal{C}$ be a strict 2-category. A corollary of the bicategorical Yoneda lemma says that any pseudofunctor $\mathcal{C} \to \operatorname{Cat}$ is pseudonaturally equivalent to a strict 2-functor. I would like to know if the "next level" of strictification is true; namely, is it true that any pseudonatural tra... | https://mathoverflow.net/users/396 | Is a pseudonatural transformation of strict 2-functors to Cat isomorphic to a 2-natural transformation? | No, it is not true. For a counterexample, let C be the delooping of the group Z/2 (regarded as a discrete monoidal category). A strict C-module is then a category equipped with an involution, a strict C-module morphism is a functor preserving the involution strictly, and a pseudo C-module morphism preserves the involut... | 13 | https://mathoverflow.net/users/49 | 34731 | 22,464 |
https://mathoverflow.net/questions/34754 | 17 | If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H\_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal coefficient theorem for cohomology.
Now, suppose I have a fibration $F \to E \to B$ such that $F$ and $B$ have homology and ... | https://mathoverflow.net/users/798 | Are the homology and cohomology Serre spectral sequences dual to each other? | Yes, this is the case. This is easiest to see using the exact couple formalism. Suppose you have an exact couple, meaning a long exact sequence consisting of maps $i: D \to D$, $j: D \to E$, $k: E \to D$, where all the terms are (possibly graded) vector spaces over a field. Because dualization is exact (as is taking le... | 19 | https://mathoverflow.net/users/360 | 34756 | 22,479 |
https://mathoverflow.net/questions/34771 | -1 | Do Gorenstein rings necessarily have finite projective dimensions?
| https://mathoverflow.net/users/5292 | Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? | Take $A=k[x]/(x^2)$ for a field $k$. This is a self-injective $k$-algebra (that is, it is an injective module over itself), so it is Gorenstein. Yet the residue field is of infinite projective dimension.
| 2 | https://mathoverflow.net/users/1409 | 34780 | 22,495 |
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