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https://mathoverflow.net/questions/30647 | 4 | **Background**
Let $m,n$ be positive integers and consider the cyclic group $\mathbb{Z}\_{mn}$.
We have a natural epimorphism $\mathbb{Z}\_{mn} \to \mathbb{Z}\_n$ coming from the exact sequence
$$ 0 \to \mathbb{Z}\_m \to \mathbb{Z}\_{mn} \to \mathbb{Z}\_n \to 0 $$
where the homomorphism $\mathbb{Z}\_m \to \mathbb... | https://mathoverflow.net/users/394 | Fibered products of cyclic groups | The group $G=\mathbb{Z}\_{\ell n} \times\_{(\mathbb{Z}\_n,r)} \mathbb{Z}\_{m n}$
is isomorphic to $\mathbb{Z}\_{an} \times\mathbb{Z}\_b$
where $a$ is the least common multiple of $\ell$ and $m$ and $b$ is
their highest common factor.
There is $r'\in\mathbb{Z}\_{\ell n}^\times$ with $r'\equiv r$ (mod $n$).
Then $g=(r'... | 6 | https://mathoverflow.net/users/4213 | 30656 | 19,955 |
https://mathoverflow.net/questions/30664 | 5 | Let $\phi\in C^\infty\_c(\mathbb R)$ be a smooth function with *compact support*.
For $h>0$ define the difference quotient $\phi\_h\in C^\infty\_c(\mathbb R)$ by $\phi\_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$.
By definition, for fixed $t\in\mathbb R$, we have $\phi\_h(t)\to\phi'(t)$ as $h\to 0$.
>
> Question: Can we... | https://mathoverflow.net/users/1291 | Uniform convergence of difference quotient | By Taylor's theorem
$$\phi(t+h)=\phi(t)+h\phi'(t)+h^2\phi''(t+u(h,t)h)/2$$
where $0\le u(h,t)\le1$. So
$$\phi\_h(t)=\phi'(t)+h\phi''(t+u(h,t)h)/2.$$
As $\phi''$ is in $C^\infty\_c$ it's pretty clear that $\phi\_h\to\phi'$
uniformly.
| 4 | https://mathoverflow.net/users/4213 | 30667 | 19,962 |
https://mathoverflow.net/questions/30655 | 9 | There is a conjecture by Birkhoff which claims that for a simple closed $C^2$ plane curve $C$, if the billiard ball map is integrable then the curve is an ellipse.
Integrability here might be formulated as follows: there exists a neighbourhood of $C$ in the interior $Int(C)$ that is foliated by caustics (caustics be... | https://mathoverflow.net/users/7031 | Birkhoff conjecture about integrable billiards | I haven't heard of any recent breakthroughs. The strongest result that I know is due to [Misha Bialy](https://doi.org/10.1007/BF02572397):
>
> **Theorem.** If almost every phase point of the billiard ball map in a strictly convex billiard table belongs to an invariant circle, then the billiard table is a disc.
>
>... | 5 | https://mathoverflow.net/users/5371 | 30668 | 19,963 |
https://mathoverflow.net/questions/30635 | 6 | All rings below are commutative with $1$.
Suppose $A\subset B$ is a subring and that $A\rightarrow A'$ is a faithfully flat ring homomorphism. [You may assume the rings are actually ${\mathbb C}$-algebras if it helps.] Suppose that $B' = A'\otimes\_A B$ is a localization of $A'$, i.e. there is a multiplicatively clo... | https://mathoverflow.net/users/2628 | Checking locally whether a homomorphism is a localization | Let $A$ be the coordinate ring of a smooth affine curve $X$ over $\mathbb C$, and let $p$ be a point of infinite order in the class group of $A$. Let $B$ be the coordinate ring of $X \smallsetminus \{p\}$, and let $C$ be the coordinate ring of an open subscheme $U$ of $X$ containing $p$ such that $p$ is principal in $U... | 5 | https://mathoverflow.net/users/4790 | 30670 | 19,964 |
https://mathoverflow.net/questions/30669 | 21 | Sometimes, dealing with the concrete and familiar Banach spaces of everyday life in maths, I happen nevertheless to ask myself about the generality of certain constructions. But, as I try to abstract such constructions up to the level of general Banach spaces, I immediately feel the cold air of wilderness coming from t... | https://mathoverflow.net/users/6101 | Banach spaces with few linear operators ? | Examples were constructed (about two years ago?) by [Argyros and Haydon](http://arxiv.org/abs/0903.3921). See this [blog post](http://gowers.wordpress.com/2009/02/07/a-remarkable-recent-result-in-banach-space-theory/) for some non-technical discussion. It seems worth noting, as one is almost obliged to, that the space ... | 17 | https://mathoverflow.net/users/763 | 30671 | 19,965 |
https://mathoverflow.net/questions/30646 | 10 | If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in
Proceedings of the International Congress of Mathematics (1950), 729–730,
where R.M. Robinson proved that Gödel Incompleteness Theorem still applies to Peano Axioms if we drop the induction schema (hence showin... | https://mathoverflow.net/users/1234 | How to locate the paper that established Robinson Arithmetic? | Hi Jose,
it's in the British library collection:
<http://snurl.com/z16ud>
Haven't checked what the fees are, but you could order it from there.
Alternatively, you could try the LMS:
<http://www.lms.ac.uk>
A good chance they will have the procs in their library, and you can get photocopies for a nominal fee.
Several... | 3 | https://mathoverflow.net/users/7248 | 30686 | 19,976 |
https://mathoverflow.net/questions/30661 | 66 | What are nice examples of topological spaces $X$ and $Y$ such that $X$ and $Y$ are **not** homeomorphic but there do exist continuous bijections $f: X \to Y$ and $g: Y \to X$?
| https://mathoverflow.net/users/2060 | Non-homeomorphic spaces that have continuous bijections between them | Recycling an old (ca. 1998) sci.math post:
" Anyone know an example of two topological spaces $X$ and $Y$
with continuous bijections $f:X\to Y$ and $g:Y\to X$ such that
$f$ and $g$ are not homeomorphisms?
Let $X = Y = Z \times \{0,1\}$ as sets, where $Z$ is the set of integers.
We declare that the following s... | 31 | https://mathoverflow.net/users/3528 | 30695 | 19,982 |
https://mathoverflow.net/questions/30659 | 18 | I am a 19 yr old student new to all these ideas. I made the transformation $X(z)=\sum\_{n=1}^\infty z^n/n^2$. Therefore $X(1)=\pi^2/6$ as we all know (it is $\zeta(2)$). To calculate $X(1)$, I integrated
$$\frac{Y(z)}{z}=\sum\_{n=1}^\infty \frac{z^{n-1}}{n} $$
between 0 to 1;
We all know $Y(z)=-\log(1-z)$; so doing i... | https://mathoverflow.net/users/7346 | Establishing zeta(3) as a definite integral and its computation. | Dear Vamsi,
Unlike the special values $\zeta(2n)$ (for $n \geq 1$), which are known to be simple algebraic expressions in $\pi$ (in fact just rational multiples of $\pi^{2n}$), it is conjectured (but not known) that the values $\zeta(2n+1)$ are genuinely new irrationalities (and that in fact
each is genuinely differe... | 52 | https://mathoverflow.net/users/2874 | 30698 | 19,984 |
https://mathoverflow.net/questions/30662 | 9 | I am looking for a (Hausdorff or better) space $X$ and a subset $A$ of $X$ that is relatively countably compact (every sequence from $A$ has an accumulation point in $X$) such that the closure of $A$ is not countably compact.
It is known that in many "nice" spaces such examples do not exist (a classical case being norm... | https://mathoverflow.net/users/2060 | Relatively countably compact subsets without countably compact closure. | Let $X$ be the space
$(\omega+1)\times(\omega\_1+1)-\{(\omega,\omega\_1)\}$, putting the transfinite order topology on each coordinate and the product topology on the whole space.
```
(0,omega_1) (1,omega_1) (2,omega_1) ---> O
: : :
: : ... | 9 | https://mathoverflow.net/users/1946 | 30701 | 19,987 |
https://mathoverflow.net/questions/30611 | 3 | Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same as the sheaf cohomology $H^i(E,\mathbb{Z})$ for the constant sheaf $\mathbb{Z}$, by taking the group cohomology of $L$ wi... | https://mathoverflow.net/users/1355 | Relation between sheaf and group cohomology | I doubt that in general one can construct a reasonable sheaf on $U$ with the required properties. To see what kind of bad things can happen, let us try to understand why this works for $X$ an elliptic curve and the sheaf $\cal{O}^{\times}$ on it.
We have the derived global sections functor from the $D^b$ of sheaves o... | 4 | https://mathoverflow.net/users/2349 | 30704 | 19,988 |
https://mathoverflow.net/questions/30696 | 5 | How can we prove that the moduli space,$M\_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M\_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface with $n$ punctures? Thanks! (It is very intuitive, but it seems that I can't make it)
| https://mathoverflow.net/users/2391 | How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent? | A compact Riemann surface of genus $g$ with $n$ boundary components has a unique realization as a hyperbolic surface with geodesic boundary. One may see this by reflecting through the boundary and uniformizing. The uniqueness of the uniformization implies it is invariant under reflection, and therefore the fixed point ... | 8 | https://mathoverflow.net/users/1345 | 30706 | 19,990 |
https://mathoverflow.net/questions/29880 | 1 | I would like to draw a curve between two points that minimizes the square of the second derivative integrated along the curve.
$J(y) = \int\_{1}^{0} {y}''^2 dx $
The first derivative for the start and end point are known and must be preserved, and all values on the curve between the start and end point must fall w... | https://mathoverflow.net/users/7168 | Minimizing functionals constrained in a box | Just to elaborate a bit on what Rahul and I mentioned in the comments.
Take the action functional to be $\int\_0^1 (y'')^2 dt$, with prescribed boundary conditions $y(0) = a$, $y(1) = b$, $y'(0) = c$, $y'(1) = d$. For finding the free evolution, take the variation of the function relative to $y$ and set it to zero. ... | 0 | https://mathoverflow.net/users/3948 | 30713 | 19,996 |
https://mathoverflow.net/questions/25062 | 10 | So I understand in theory the definition of Ext and Tor, but when it comes to actually computing them, I'm stuck. For example, could someone show me how to compute $\text{Ext}(\mathbb{Z}/m\mathbb{Z}, \mathbb{Z}/n\mathbb{Z})$? I tried this by taking an injective resolution ($0 \to \mathbb{Z}/m\mathbb{Z} \to \mathbb{Q}/\... | https://mathoverflow.net/users/2503 | Examples of computing Ext and Tor functors? | $\mathbb{Q}/\mathbb{Z}$ is a pretty terrible abelian group, or a rather hard one, there may be better injective resolutions to work with. It would certainly be easier to do the projective resolution, use $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/n \to 0$. this will surely be easier to work through than the one in... | 11 | https://mathoverflow.net/users/3901 | 30714 | 19,997 |
https://mathoverflow.net/questions/30709 | 4 | Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements of the positive cone.
Assume now that we can show $$k\Vert x\Vert\le \Vert Tx\Vert$$ for some
$k>0$ and every $x\in ... | https://mathoverflow.net/users/nan | When can closedness of the range of an operator be checked on a positive cone? | I'm reluctant to say no when the question is "Are there *any* natural, general conditions ..." but I'll take my chances. The following seems to be a counterexample with every nice property I can think of. Let X and Y both be the Hilbert space $\ell^2$, and label an orthonormal basis as $\{e\_n,f\_n:n\in\mathbf N\}$. Le... | 2 | https://mathoverflow.net/users/6794 | 30717 | 19,998 |
https://mathoverflow.net/questions/30716 | 2 | I am interested in the possibility of generating probability distributions using inequality constraints. For instance assume that we have three urns with total of a 10 balls. Thus,
$a + b + c = 10$
Additionally it has the constraint
$a \geq b \geq c$
The allowable triples will correspond to the integer partitio... | https://mathoverflow.net/users/7168 | Inequality constraints, probability distributions, and integer partitions | Whether this is a valid approach for calculating the probability will depend on what assumptions you are making on the probabilities of the various partitions. In particular, this calculation seems to require for its validity that all partitions be equally likely. Do you actually want (10,0,0) to be as likely as (4,3,3... | 3 | https://mathoverflow.net/users/3684 | 30722 | 20,001 |
https://mathoverflow.net/questions/30725 | 0 | I think it is possible to use only cosine function, but why the formula is used with sine?
I am trying to understand but don't know what to do after Fourier transform with imaginary part and real part.
| https://mathoverflow.net/users/3195 | Why do we use sine and cosine in fourier transform together? | I'm not sure which Fourier transform you mean, but I have only seen Fourier series written where the phase, e.g. $e^{2\pi i n x}$ for some integer n, is decomposed into its real and imaginary parts $cos(2\pi n x) +i sin(2\pi n x)$. Since sine and cosine are related to each other by translation, $sin (\pi/2 - x) = cos (... | 4 | https://mathoverflow.net/users/7361 | 30728 | 20,004 |
https://mathoverflow.net/questions/30723 | 6 | In some of my classes (e.g. graph theory, mechanics), the professors encourage the students to visualize solutions to problems; I do well in these classes. In other classes (e.g. linear algebra), we are encouraged instead to reason about abstract concepts; I usually do worse in these classes (relative to the same set o... | https://mathoverflow.net/users/4135 | What subfields of mathematics better lend themselves to visualization? | The classes that lend themselves to visualization certainly include graph theory
and mechanics, and I expect you would do well in classical geometry too. In
other cases, it may depend on which book you use. For example, I am sure
you would enjoy the approach to complex analysis in Tristan Needham's book
*Visual Complex... | 9 | https://mathoverflow.net/users/1587 | 30733 | 20,009 |
https://mathoverflow.net/questions/30749 | 4 | Suppose I have a square n\*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be satisfied (the other requirements follow from the definition).
Is there some standard measure that says to what extent thi... | https://mathoverflow.net/users/7368 | Is there a standard measure for how close a matrix is to being a distance metric ? | There are a couple of plausible measures you could employ. One would be to minimize the Frobenius distance between the given matrix (call it $D$) and the target matrix $X$ . Since the space of all distance matrices that satisfy triangle inequality can be expressed using linear constraints, you end up with a least-squar... | 6 | https://mathoverflow.net/users/972 | 30754 | 20,023 |
https://mathoverflow.net/questions/30750 | 7 | This may turn out to be a bit embarassing, but here it is. It is probably not true that the ascending and descending central series (\*) of a nilpotent group have the same terms. (Or at least one of MacLane-Birkhoff, Rotman and Jacobson would have mentioned it.) However, I have been unable to find an example where they... | https://mathoverflow.net/users/4367 | Nilpotent group with ascending and descending central series different? | It's false even if $m=2$. Try the product of a group of order $2$ and a dihedral group of order $8$. The center has order $4$ and the quotient by the center is abelian. The commutator subgroup has order $2$.
| 15 | https://mathoverflow.net/users/6666 | 30755 | 20,024 |
https://mathoverflow.net/questions/30643 | 12 | We are talking about ordinary reals in constructive mathematics.
1. Let represent each real number by infinite converging series:
$$r = [\;(a\_0,b\_0),(a\_1,b\_1),...,(a\_i,b\_i),...\;]$$
$$where\quad a\_i \leq b\_i\quad and \quad a\_i \leq a\_{i+1} \; and \; b\_{i+1} \leq b\_i$$
And interval $(a\_i,b\_i)$ converge... | https://mathoverflow.net/users/7257 | Are real numbers countable in constructive mathematics? | I am going to attempt another answer which directly addresses what you wrote.
It **cannot** be shown constructively that every infinite subset of $\mathbb{N}$ is in bijective correspondence with $\mathbb{N}$. Thus your reasoning has a flaw when going from step 4 to step 5. By "infinite set" here I mean that there is ... | 9 | https://mathoverflow.net/users/1176 | 30757 | 20,026 |
https://mathoverflow.net/questions/30742 | 5 | You start with a bag of N recognizable balls. You pick them one by one and replace them until they have all been picked up at least once. So when you stop the ball you pick has not been picked before but all the others have been picked once or more.
Let $P\_N$ be the probability that all the others were actually pick... | https://mathoverflow.net/users/7238 | Limit probability | One way to think about this sort of problem is to embed in continuous time. Take $N$ independent Poisson processes of rate 1. (Think of $N$ independent Geiger counters, each going off at rate 1, if you like). A point in the $i$th process corresponds to picking the $i$th ball. Since the processes are independent and all... | 12 | https://mathoverflow.net/users/5784 | 30760 | 20,028 |
https://mathoverflow.net/questions/30748 | 0 | Given this modified Dirichlet function: $f(x) = 0$ if $x$ is in $\mathbb{Q}$, else $f(x) = x$. I am wondering if this function is Darboux integrable on the interval $[0, 2]$.
I managed to show that every lower Darboux sum is equal to zero, therefore the lower Darboux integral is 0. My intuition tells me that the uppe... | https://mathoverflow.net/users/7369 | Modified Dirichlet function Darboux integrable on $[0,2]$? | The function is not integrable by Darboux (and equivalently by Riemann) as any point in (0,2] is a discontinuity point. (take a rational sequence approaching $x$, the values of $f(x\_n)$ are constantly $0$. If $x$ is irrational, then $\lim f(x\_n) \neq f(x)$)
If you want to stick to the Darboux integral definition, s... | 2 | https://mathoverflow.net/users/7206 | 30764 | 20,032 |
https://mathoverflow.net/questions/30759 | 1 | Given a graph with a list of edges, is it possible to always construct a set of cycle bases for those edges, such that each and every edge is shared by at most 2 cycle bases?
The above question assumes that each and every edge must somehow belong to at least one cycle. IN other words, there is no vertex that is conne... | https://mathoverflow.net/users/807 | In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? | Consider the complete graph on 7 vertices. It has 21 edges, so any set of cycles that utilizes each edge at most twice has size at most 42/3=14. But the cycle space of the graph has dimension 21-7+1=15, so you cannot have a basis with the requested property.
| 5 | https://mathoverflow.net/users/2368 | 30767 | 20,033 |
https://mathoverflow.net/questions/30765 | 16 | Is there a notion of an octonion prime?
A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime.
A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is prime.
I know there is an eight-square identity that underlies the octonions.
Is there a parallel statement, something lik... | https://mathoverflow.net/users/6094 | Gaussian primes, quaternion primes, ... octonions? | You should probably read "On Quaternions and Octonions" by J.H. Conway and D.A. Smith
P.S. It's "octonion" not "octonian"
Edit: The first thing you will find is a discussion of integral numbers. For the complex numbers you have $\mathbb{Z}[i]$ (aka Gaussian integers) which is an $A\_1\times A\_1$ lattice. You also ... | 16 | https://mathoverflow.net/users/3992 | 30768 | 20,034 |
https://mathoverflow.net/questions/30771 | 1 | For all $n$, I need to find examples of rings $A\subset B$ such that:
i) $\dim A-\dim B\gt n$
ii) $\dim B-\dim A\gt n$
(where $\dim$ is the *Krull dimension*)
| https://mathoverflow.net/users/7332 | Krull Dimension | $\mathbb{Q} \subset \mathbb{Q}[x\_0, \dots, x\_n] \subset \mathbb{Q}(x\_0, \dots, x\_n)$.
| 14 | https://mathoverflow.net/users/460 | 30776 | 20,037 |
https://mathoverflow.net/questions/30481 | 9 | If I define an additive functor to be a functor on abelian categories such that the action of $F$ on ${\rm Hom}(A,B)$ is a group homomorphism, do I necessarily have that $F(\text{zero object}) = \text{zero object}$?
| https://mathoverflow.net/users/7313 | F(0) = 0? F: additive functor | Since the OP asked for a detailed answer:
Let $A$ be an object of an abelian (or additive) category. Then $A$ is a zero object if and only if the zero endomorphism is the identity endomorphism (and then $Hom(A,A)$ is the zero ring). If $F$ is any functor, it sends the identity morphism of $A$ to the identity morphism... | 18 | https://mathoverflow.net/users/2653 | 30779 | 20,040 |
https://mathoverflow.net/questions/30081 | 18 | I've never seen an authoritative explanation for the choice of the lower case letter $\ell$ or $l$ to denote an arbitrary prime different from a given prime $p$. This now has its own LaTeX command \ell, but has been in use at least since the old work of Taniyama and Weil involving *L* functions. That use of the upper c... | https://mathoverflow.net/users/4231 | Origin of symbol *l* for a prime different from a fixed prime? | This elaborates quim's answer. Kummer did indeed use $\lambda$ for denoting primes (in connection with cyclotomic fields); he borrowed the notation from Jacobi's articles on cyclotomy as well as from his notes of the number theory lectures in 1836/37. When Hilbert rewrote Kummer's contributions in his Zahlbericht, he s... | 10 | https://mathoverflow.net/users/3503 | 30799 | 20,053 |
https://mathoverflow.net/questions/30798 | 7 | i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general characterizations of "approximate" linearity? for example, what can be said if $|f(x-y)-f(x)+f(y)-f(0)|$ is bounded by some sma... | https://mathoverflow.net/users/7373 | approximately linear functions | Let $E$ and $E'$ be Banach spaces. Mappings $f:E\to E'$, which satisfy the inequality
$$\|f(x + y) − f(x) − f(y)\| \leq\epsilon$$
for all $x, y \in E$, are called $\epsilon$-additive (or approximately additive). The main result concerning approximately additive functions is due to D. Hyers
([link](http://www.ams.org/ma... | 14 | https://mathoverflow.net/users/5371 | 30801 | 20,054 |
https://mathoverflow.net/questions/30800 | 2 | Say we have a branched $G$-Galois covering $X \rightarrow Y$ of surfaces, and assume that the branch locus (in $Y$) is a divisor with normal crossings. I will always assume that the ramification is tame, and you can pretend we're doing this with varieties over $\mathbb{C}$ if you prefer. This implies that the inertia g... | https://mathoverflow.net/users/5309 | What happens to inertia groups after blow ups? | I'll do this over $\mathbb{C}$, since you'll let me get away with it.
The geometric fact you need to know is the following: If $D \subset Y$ is a branch divisor for the $G$ covering $X \to Y$, then the inertia group of $D$ is generated by the monodromy around a small loop $\gamma$ encircling $D$. In other words, nea... | 3 | https://mathoverflow.net/users/297 | 30803 | 20,055 |
https://mathoverflow.net/questions/30795 | 5 | Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional hyperbolic mapping tori are unknotted and unlinked with respect to the fiber, where "short" depends only on the genus of the... | https://mathoverflow.net/users/4325 | Rotation part of short geodesics in hyperbolic mapping tori | This should follow from Minsky's paper *[The classification of Kleinian surface groups, I: Models and bounds](https://arxiv.org/abs/math/0302208)* on a priori bounds for surface groups, which is used in the proof of the ending lamination conjecture.
The punctured torus case (*[The classification of punctured-torus grou... | 8 | https://mathoverflow.net/users/1345 | 30816 | 20,060 |
https://mathoverflow.net/questions/30818 | 9 | The sum $\sum\_{n=1}^{\infty} 1/n^{s}$ is convergent for all real $s>1$ and diverges for all real $s \le 1$. The same holds for the sum $\sum\_{p \ prime} 1/p^{s}$. Thus, for the functions $f(n)= 1/n^s, s \in \mathbb{R}$ the sum $\sum\_{n=1}^{\infty}f(n)$ shows the same convergence behaviour as the sum $\sum\_{p \ prim... | https://mathoverflow.net/users/6415 | Sum f(p) over all primes convergent with sum f(n) over all natural numbers divergent? | I think, you are mistaken, sum $\sum 1/(p\log p)$ converges, since $p\_n\log p\_n$ behaves like $n(\log n)^2$
| 11 | https://mathoverflow.net/users/4312 | 30820 | 20,063 |
https://mathoverflow.net/questions/30828 | 8 | Given a natural number k, are there only finitely many finite simple groups with the property that all elements have order at most k?
This holds if I only look at the finite simple groups I understand
(e.g. alternating groups and SL(k,finite field)), but it's not clear to me whether this holds for all finite simple ... | https://mathoverflow.net/users/2985 | Finite simple groups with upper bound on order of elements | The classification of the finite simple groups implies that there are only finitely many finite simple groups of a given exponent $k$. To see this, first note that we can ignore the sporadic groups, as well as the cyclic groups of prime order. It is also also clear that there are only finitely many alternating groups o... | 14 | https://mathoverflow.net/users/4706 | 30832 | 20,070 |
https://mathoverflow.net/questions/29087 | 25 | For a given $n$, is there any characterization for the commutative subalgebras of $M\_n(\Bbb{C})$? I would like to know how many commutative subalgebras there are for each possible dimension.
In view of Chapman's answer, I am refining my previous question:
Given $k\leq n$, is there any way of describing the commuta... | https://mathoverflow.net/users/6985 | Commutative subalgebras of M_n | If you are only concerned about commutative subalgebras of $M\_n(\mathbb{C})$ then there is a fairly easy characterization. So any self-adjoint abelian algebra is generated by a single self adjoint element (spectral theorem). Call this element T. Then T is diagonalizable and so the algebra it form will be the algebra o... | 7 | https://mathoverflow.net/users/5732 | 30849 | 20,081 |
https://mathoverflow.net/questions/30463 | 2 | Suppose we have two finite groups given by presentations
$G=F/N\_1 , H=F/N\_2$
where $N\_1 \subset N\_2$ and $F$ is a free group of finite rank.
The canonical map $\pi: G \rightarrow H $ induces the inflation map between Schur Multipliers,
$Inf: M(H) \rightarrow M(G)$. (Take a cocyle in M(H) and compose with $\pi ... | https://mathoverflow.net/users/7307 | Hopf's formula and inflation map | I am very far from an expert, but I suspect you have written the map you want going in the wrong direction.
Have a look, for example, at the early part of the book review by Van der Kallen in The Bulletin of the AMS, Vol 10, Number 2 (1984), pages 330-333
He explains that "There is now some confusion as to what the... | 3 | https://mathoverflow.net/users/4648 | 30865 | 20,091 |
https://mathoverflow.net/questions/30868 | 18 | Is every finite codimensional subspace of a Banach space closed? Is it also complemented? I know how to answer the same questions for finite dimensional subspaces, but couldn't figure out the finite codimension case.
| https://mathoverflow.net/users/5498 | Subspaces of finite codimension in Banach spaces | It's a standard result that a linear functional from a Banach space to the underlying field (real or complex numbers) is continuous if and only if the its kernel is closed. Notice that its kernel is of codimension one. So, use the axiom of choice to find a discontinuous linear functional, and you have found a codimensi... | 14 | https://mathoverflow.net/users/406 | 30870 | 20,094 |
https://mathoverflow.net/questions/30788 | 15 | Inspired by [this](https://mathoverflow.net/questions/30345/what-is-an-example-of-a-compact-smooth-manifold-whose-k-theory-and-cech-cohomolog) question, I wonder if anyone can provide an example of a finite CW complex X for which the order of the torsion subgroup of $H^{even} (X; \mathbb{Z}) = \bigoplus\_{k=0}^\infty... | https://mathoverflow.net/users/4042 | Torsion in K-theory versus torsion in cohomology | [This paper](http://arxiv.org/pdf/hep-th/0005103) by Volker Braun shows that the orientable 8-manifold $X=\mathbb{RP}^3\times \mathbb{RP}^5$ gives an example. One has $$K^0(X) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/4\oplus (\mathbb{Z}/2)^2$$
and
$$H^{ev}(X) \cong \mathbb{Z}^2 \oplus (\mathbb{Z}/2)^5.$$
Braun does the c... | 13 | https://mathoverflow.net/users/2356 | 30893 | 20,110 |
https://mathoverflow.net/questions/30907 | 11 | One way to define toric varieties is as quotients of affine $n-$space by the action of some torus. However, this is not strictly true as we need to throw away "bad points" which ruin this construction.
For example consider the construction of projective space as a toric variety. Let $\mathbb{G}\_m$ act on $\mathbb{A}... | https://mathoverflow.net/users/5101 | Toric varieties as quotients of affine space | You want to read Section 2 of *[The homogenous ring of a toric variety](https://arxiv.org/abs/alg-geom/9210008)*, by David Cox. Nick Proudfoot has written an expository note, *[Geometric invariant theory and projective toric varieties](https://arxiv.org/abs/math/0502366)* on the projective case, which you might find he... | 7 | https://mathoverflow.net/users/297 | 30908 | 20,119 |
https://mathoverflow.net/questions/30910 | 5 | Has the Robinson-Schensted correspondence, as explained by [Wikipedia](https://en.wikipedia.org/wiki/Robinson-Schensted_algorithm) or [Richard Stanley](http://math.mit.edu/%7Emusiker/rstan7-8.pdf), been implemented in any of the standard programming languages. I'm using Python, but I'm open to Java, C++, Mathematica, M... | https://mathoverflow.net/users/1358 | Implementation of the Robinson-Schensted correspondence | It doesn't require linked lists, just arrays that can grow.
There's a [Java applet online](http://www.math.uconn.edu/~troby/Goggin/BumpingAlg.html) that implements it.
I'm sure there are other implementations online, but since I couldn't find any, as a start, here's a simple Python implementation. [Though it feels... | 12 | https://mathoverflow.net/users/111 | 30914 | 20,122 |
https://mathoverflow.net/questions/30911 | 10 | Hartshorne's "Algebraic geometry" begins with the definition of (quasi-)affine and (quasi-)projective varieties over some fixed algebraically closed field. At a first glance, these seem to be quite different, so that I would have expected that one would pose questions *either* on quasi-affine *or* on quasi-projective v... | https://mathoverflow.net/users/1291 | Is Hartshorne's definition of the category of varieties natural? | A quasiaffine variety IS quasi projective. Indeed it is an open set in an affine variety, which in turn is open in its projective closure. So one only considers quasiprojective varieties.
| 19 | https://mathoverflow.net/users/828 | 30916 | 20,124 |
https://mathoverflow.net/questions/30894 | 12 | I've been thinking about the [equiangular lines (or SIC-POVM) conjecture](http://en.wikipedia.org/wiki/SIC-POVM), and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking specifically of geometric fixed point theorems, like Brouwer's. So my (rather vague) ques... | https://mathoverflow.net/users/2294 | Fixed point theorems and equiangular lines | The book "Fixed point theory" by Dugundji and Granas is a nice reference. The headers of the sections in the book give some kind of classification of fixed point theorems.
* results based on compactness
* order theoretic results
* results based on convexity
* Borsuk theorem and topological transitivity
* homology and... | 7 | https://mathoverflow.net/users/6129 | 30933 | 20,135 |
https://mathoverflow.net/questions/30935 | 2 | If A and B are disjoint subsets of real numbers, and one of them is measurable can we say
m\*(A U B)=m\*(A)+m\*(B)?
I am unable to find counter example. I feel this is not true.
| https://mathoverflow.net/users/7401 | about measure theory | This is true. If for example A is measurable it is measurable in the sense of Caratheodory so that
For every set C we will have
$m\*(C) = m\*(C\cap A) + m\*(C \setminus A)$.
This with $C=A \cup B$ is your assertion
| 6 | https://mathoverflow.net/users/7402 | 30940 | 20,140 |
https://mathoverflow.net/questions/30938 | 6 | A lot is known about geometric and topological properties of diffeomorphism groups of surfaces (here, I am mainly thinking about the work of Smale and Eells-Elworthy). Is there anything known for orbisurfaces ? My first guess would be that many of these groups must have contractible components since singular points imp... | https://mathoverflow.net/users/7325 | Diffeomorphism groups of orbifolds | The result you want can be found in the following paper:
MR0955816 (89h:30028)
Earle, Clifford J.(1-CRNL); McMullen, Curt(1-MSRI)
Quasiconformal isotopies. Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 143--154,
Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
What they prove is actually ... | 7 | https://mathoverflow.net/users/317 | 30941 | 20,141 |
https://mathoverflow.net/questions/30715 | 12 | It is well known as Cohen's theorem that a commutative ring is Noetherian if all its
prime ideals are finitely generated. Is this statement true or false when prime ideals are replaced by maximal ideals?
| https://mathoverflow.net/users/5775 | A remark on Cohen's theorem | See the following paper (and search for its citations for related work)
Gilmer, R; Heinzer W.
A non-Noetherian two-dimensional Hilbert domain with principal maximal ideals,
Michigan J. Math. 23 (1976), 353-362
[Link](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-23/issue-4/A-non-Noe... | 7 | https://mathoverflow.net/users/6716 | 30942 | 20,142 |
https://mathoverflow.net/questions/29061 | 6 | C. Pugh and M. Shub showed in 1971 that, given an ergodic action of $G=\mathbb{R}^k$ on some separable finite measure space $(X,\mu)$, then all elements of $G$ , off a countable family of hyperplanes, are ergodic.
Is there an analogous statement in the topological setting, with ergodicity replaced by minimality (i.e.... | https://mathoverflow.net/users/6129 | Minimal elements of minimal R^k actions | A colleague pointed out the following counterexample.
Let $h\_t$ be the horocyclic flow on a negatively curved compact surface S. This R action is known to be minimal. Now Consider the $R^2$ action on $S\times S$ given by $(s,t)\rightarrow (h\_s,h\_t)$. This action is again minimal.
The action of the diagonal $\{(s,... | 4 | https://mathoverflow.net/users/6129 | 30945 | 20,144 |
https://mathoverflow.net/questions/30925 | 8 | Famous Robinson Schensted Knuth correspondence gives a correspondence between the matrices with non-negative integer entries and pair of semi standard tableaux. The proof that I have seen is highly combinatorial e.g. in Knuth's paper [Permutations, matrices, and generalized young tableaux]. Does there exist a geometric... | https://mathoverflow.net/users/7386 | Geometric proof of Robinson-Schensted-Knuth correspondence? | This depends on the meaning of the word "geometric". If you are thinking of RSK and want the geometry in the way the algorithm is *presented* (in the case of permutations only), Viennot's paper [Une forme geometrique de la correspondance de Robinson–Schensted](https://doi.org/10.1007/BFb0090011) [mentioned](https://mat... | 18 | https://mathoverflow.net/users/4040 | 30950 | 20,146 |
https://mathoverflow.net/questions/30948 | 8 | I'm trying to read [a paper](http://arxiv.org/abs/0808.0350) of Boris Kalinin on the cohomology of dynamical systems for a project. The material is geared towards topologically transitive Anosov diffeomorphisms (which is how the initial (abelian) results were proved by Livsic). However, he axiomatizes things for homeom... | https://mathoverflow.net/users/344 | Kalinin's formulation of the Anosov closing lemma | The closing lemma as stated by Kalinin can be found in many textbooks e.g. Katok-Hasselblatt "Introduction to the modern theory of dynamical systems", corollary 6.4.17.
The closing lemma really gives a periodic point close to x,
with iterates also close to the iterates of x until the orbit of x returns. That's not ju... | 3 | https://mathoverflow.net/users/6129 | 30954 | 20,149 |
https://mathoverflow.net/questions/30917 | 5 | What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?
| https://mathoverflow.net/users/nan | Relation between motivic cohomology and Quillen K-theory | You should look at Marc Levine's preprint "K-theory and motivic cohomology of schemes, I". The version on the UIUC K-theory server seems to be older than the version on his [website](http://www.uni-due.de/~bm0032/publ/Publ.html) .
Roughly speaking, the motivic spectral sequence starts from motivic cohomology and con... | 7 | https://mathoverflow.net/users/4042 | 30955 | 20,150 |
https://mathoverflow.net/questions/30501 | 8 | Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$ as $c\_0+c\_1x+c\_2x^2+\dots$ with each $c\_n$ in $\mathbb Z/2\mathbb Z$.
**Question.** Is it true that when $n$ is ... | https://mathoverflow.net/users/6214 | Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$ | (Part 1)--My argument uses the following curious fact about ideals in $Z[i]$ and $Z[\sqrt{-2}].$
Suppose $n=8m+1$. Let $I=I(n)$ and $J=J(n)$ be the number of ideals of norm $n$ in $Z[i]$ and $Z[\sqrt{-2}]$. Then $I\equiv J$ (4) except when $m$ is odd triangular, in which case $I\equiv J+2$ (4).
As a corollary we find... | 1 | https://mathoverflow.net/users/6214 | 30958 | 20,153 |
https://mathoverflow.net/questions/30891 | 6 | A binary quartic form
$aX^4+bX^3Y+cX^2Y^2+dXY^3+Y^4$
decomposes as a product of linear factors $Y-t\_jX$, $j=1,...,4$.
I would like to have an explicit formula for symmetrization of the crossratio of $t\_j$.
| https://mathoverflow.net/users/7393 | explicit formula for the j-invariant of binary quartic form | The $j$ invariant is
$j=\frac{S^3}{S^3-27T^2}$
where
$S=a-\frac{bd}{4}+\frac{c^2}{12}$
and
$T=\frac{ac}{6}+\frac{bcd}{48}-\frac{c^3}{216}-\frac{ad^2}{16}-\frac{b^2}{16}$
for more details see my article ["A computational solution to a question by Beauville on the invariants of the binary quintic"](https://ww... | 11 | https://mathoverflow.net/users/7410 | 30967 | 20,159 |
https://mathoverflow.net/questions/30890 | 6 | Hello,
Maybe it is too vague a question, but I would like to ask if anybody could say some explanatory words about the importance (for infinity category study) of studying all the kinds of fibrations there are for simplicial sets (there are more than eight: right, left, mid, Kan, trivial, and so on..). In general, it... | https://mathoverflow.net/users/2095 | Fibrations of Simplicial sets | I'm not an expert, but, here is my understanding. Right-fibrations are important because they are the infinity-version of a category fibered in groupoids (that is an infinity-category fibered in infinity-groupoids). In particular, given an infinity-category $C$,there is a model structure on $sSet/C$, called the contrav... | 8 | https://mathoverflow.net/users/4528 | 30984 | 20,171 |
https://mathoverflow.net/questions/30904 | 7 | It is known that to prove completeness of first-order logic for countable languages WKL0 is enough. But, is it the weakest subsystem where one can prove it?
What about the incompleteness theorems? Is it known which are the weakest subsystems of second order arithmetic where one would be able to prove each of them?
| https://mathoverflow.net/users/6466 | Weakest subsystems of second order arithmetic for mathematical logic | In fact, the incompleteness and completeness theorems can be proven in subsystems of second-order arithmetic weaker than RCA-0: incompleteness can be proven in EFA (first-order elementary arithmetic), which proves exponentiation total, but cannot prove iterated exponentiation to be total. In fact, systems much weaker t... | 7 | https://mathoverflow.net/users/3154 | 31008 | 20,182 |
https://mathoverflow.net/questions/30998 | 9 | While giving the first of eight lectures on introductory model theory and its applications yesterday, I stated Hilbert's 17th problem (or rather, Artin's Theorem): if $f \in \mathbb{R}[t\_1,\ldots,t\_n]$ is positive semidefinite -- i.e., non-negative when evaluated at every $x = (x\_1,\ldots,x\_n) \in \mathbb{R}^n$ -- ... | https://mathoverflow.net/users/1149 | Sums of two squares in (certain) integral domains | Let $K$ be a complex quadratic number field such that $K(i)$ has class number $1$. If $K$ has class number $\ne 1$, then $K(i)$ must be the Hilbert class field of $K$, which, in this case, coincides with the genus class field of $K$. By genus theory, the discriminant of $K$ must have the form $d = -4p$ for a prime numb... | 9 | https://mathoverflow.net/users/3503 | 31010 | 20,183 |
https://mathoverflow.net/questions/31009 | 6 | Hi guys,
I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want to use a similar procedure to prove that a manifold whose isometries act transitively is complete, i.e there is always ... | https://mathoverflow.net/users/4890 | Action of the group of isometries on a manifold | By the Hopf-Rinow theorem, you only have to prove that the manifold is a complete metric space. By homogeneity, the injectivity radius is bounded from below by a uniform positive constant. Using this and the compacity of balls whose radius is smaller than the injectivity radius of their center it is easy to check the c... | 14 | https://mathoverflow.net/users/4961 | 31012 | 20,184 |
https://mathoverflow.net/questions/31006 | 1 | Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet map. Now take a $G$-linearised coherent sheaf $(\mathcal{F}, \lambda)$, one can construct the sheaf of invariants of $\m... | https://mathoverflow.net/users/6949 | The fiber of the sheaf of invariants | It all works out as well as you could want in every possible sense because of the freeness of the action. As you know, you can make a cover by $G$-stable affine opens, so the real work is in that case. So we focus on the affine case, and then all hypotheses on the affine can be removed: let $A$ be any ring whatsoever a... | 3 | https://mathoverflow.net/users/6773 | 31029 | 20,188 |
https://mathoverflow.net/questions/31004 | 80 | At various times I've heard the statement that computing the group structure of $\pi\_k S^n$ is *algorithmic*. But I've never come across a reference claiming this.
Is there a precise algorithm written down anywhere in the literature? Is there one in folklore, and if so what are the run-time estimates? Presumably th... | https://mathoverflow.net/users/1465 | Computational complexity of computing homotopy groups of spheres | [Francis Sergeraert](http://www-fourier.ujf-grenoble.fr/~sergerar/) and his coworkers have implemented his effective algebraic topology theory in a program named Kenzo. It seems capable of computing any $\pi\_n(S^k)$ (in fact homotopy groups of any simply connected finite CW complex), although I don't know how far it i... | 44 | https://mathoverflow.net/users/6451 | 31042 | 20,192 |
https://mathoverflow.net/questions/30989 | 4 | Some NP-complete optimization problems, like the knapsack problem, have a solution reachable in polynomial time that is guaranteed to be within arbitrary ε of the optimum answer. (aka PTAS - polynomial time approximation scheme)
Some decision problems, like testing primes, have probabilistic solutions (like Rabin's) ... | https://mathoverflow.net/users/7418 | Does NP = "epsilon-P" (PTAS / BPP)? | The answer to this question is essentially given in previous answers, but I'll try to state it more completely. It really depends on the problem. All NP-complete problems are equivalent in how hard it is to find their exact solution, but they vary widely in how hard it is to approximate them. Many of them can be shown ... | 10 | https://mathoverflow.net/users/2294 | 31046 | 20,195 |
https://mathoverflow.net/questions/31035 | 16 | I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at Hatcher, I still feel quite confused. I remember one friend told me combinatorically one can do this from posets back to poset... | https://mathoverflow.net/users/5175 | Deeper meanings of barycentric subdivision | There can be many reasons for subdividing simplices, barycentrically or otherwise.
For a simplicial complex (triangulated space) there are the simplicial homology groups. These are known to be isomorphic to the singular homology groups, therefore (1) invariant under homeomorphism, and in particular (2) invariant und... | 17 | https://mathoverflow.net/users/6666 | 31048 | 20,196 |
https://mathoverflow.net/questions/31020 | 1 | Which method would you recommend for error estimation of the following approximation?
$$\frac{1}{K} \sum\_{j=0}^{K-1}\frac{cos(2\pi\frac{j}{K}u)}{P\_{n}(\cos[\pi\frac{j}{K}])}\approx\int\_{0}^{1}\frac{cos(2\pi xu)}{P\_{n}(\cos[\pi x])}dx$$
Here $P\_{n}$ some polynomial
$u=1,2...K/2$
$\frac{1}{12k^2}f''(\psi)$ is a ve... | https://mathoverflow.net/users/3589 | trapezoidal rule error approximation. What if f''(x)/12n^2 doesn't work? | My first guess would be to use the Euler-Maclaurin summation formula ([Wikipedia article](http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_summation_formula)). This proves, amongst other things, that the error goes down exponentially if the integrand is a periodic function on [0,1].
**Added:** After thinking abou... | 3 | https://mathoverflow.net/users/2610 | 31053 | 20,199 |
https://mathoverflow.net/questions/31050 | 8 |
>
> Let $G$ be the group of orientation-preserving homeomorphisms (or, if you prefer, diffeomorphisms) of the real line. Does there exist a natural way to associate, to each function $f \in G$, a homomorphism $\phi\_f \colon \mathbb{R} \to G$ such that $\phi\_f(1) = f$?
>
>
>
Motivation: Many of us, in high scho... | https://mathoverflow.net/users/5094 | Homomorphisms from $\mathbb{R}$ to $\mathrm{Homeo}^+(\mathbb{R})$, or "fractional iterations" | This has some relation with [this question](https://mathoverflow.net/questions/26462/noninteger-iterates-of-functions-how-to-get-ode-from-flow-at-a-given-time/26472#26472), but it is obviously different.
In general, a homeomorphism of $\mathbb{R}$ which preserves the orientation may have or not fixed points.
If it... | 8 | https://mathoverflow.net/users/5753 | 31056 | 20,202 |
https://mathoverflow.net/questions/31001 | 5 | In fact, it is a simple problem. I just want to know whether there are some interesting proof.
$Z[x\_1, x\_2, ......, x\_{n^2-1}]$ and $Z[y\_{11}, ......, y\_{1n}, y\_{21}, ......, y\_{nn}]/(det(y\_{ij})-1))$, where $Z$ is integer.
One way to prove is select a prime number,say $p=2$,then localize these two rings,... | https://mathoverflow.net/users/1851 | How to prove these two rings are not isomorphic | Does your critic dislike that the argument seems not applicable over general rings? But it is: if there's an isomorphism over some ring $R$ then we can descend to a finitely generated subring and pass to the quotient by a maximal ideal to get such an isomorphism over a finite field, and then count points.
Or maybe y... | 7 | https://mathoverflow.net/users/3927 | 31057 | 20,203 |
https://mathoverflow.net/questions/31058 | 16 | This is a problem I had a look at some years ago but always had the feeling that I was missing something behind its motivation.
D.H. Lehmer says in his 1947 paper, “The Vanishing of Ramanujan's Function τ(n),” that it is natural to ask whether τ(n)=0 for any n>0.
My question is: Why is it natural to wonder whether ... | https://mathoverflow.net/users/7330 | The vanishing of Ramanujan's Function tau(n) | The key to your question is [lacunarity](https://en.wikipedia.org/wiki/Lacunary_function) in modular functions.
The tau function, as we know, occurs as the coefficient of the [Discriminant function](https://planetmath.org/ModularDiscriminant), which in turn is the 24th power of the [Eta function](https://mathworld.wo... | 10 | https://mathoverflow.net/users/5372 | 31070 | 20,209 |
https://mathoverflow.net/questions/30975 | 14 | Let $F(s)=\sum\_{n\geq 1}\frac{a\_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma\_a$. It can be shown that $\forall \sigma >\sigma\_a:$
$$\lim\_{T\to\infty}\frac{1}{2T}\int\_{-T}^{T}F(\sigma+ it)n^{it}\mathrm{d}t=\frac{a\_n}{n^{\sigma}}.$$
The natural question arises, given some ... | https://mathoverflow.net/users/1849 | Dirichlet series expansion of an analytic function | A.F. Leont'ev continued to work on [general Dirichlet series](http://en.wikipedia.org/wiki/General_Dirichlet_series) well into 1980s (until his death in 1987). Actually, he published three monographs on the subject from 1976 to 1983! He made a short summary of his earlier results for the 1974 ICM in Vancouver (a free p... | 8 | https://mathoverflow.net/users/5371 | 31075 | 20,211 |
https://mathoverflow.net/questions/31077 | 1 | Define
g(x) = (1+x) ln(1+x) - x.
One can check that g is strictly monotonically increasing for x>=0 by checking its derivative is ln(1+x). So g is invertible and its inverse is also strictly monotically increasing.
Is there an explicit closed form for its inverse?
With a page of calculations I can prove that
(1/2... | https://mathoverflow.net/users/7438 | Inverse of (1+x) ln(1+x) - x | Just to expand on Qiaochu's comment: let $W$ stand for the [Lambert W-function](http://en.wikipedia.org/wiki/Lambert_W_function), then if $g(x)=z$, we readily find that
$$
x=\exp\big(W((z-1)/e)+1\big)-1.
$$
| 6 | https://mathoverflow.net/users/2149 | 31080 | 20,213 |
https://mathoverflow.net/questions/30589 | 3 | Let $G=GL(2,p)$ be the group of linear transformations. It acts on the set $X=F\_p^2$.
Then $C^X$ is a linear representation of $G$. What is the decomposition of this representation into irreducible representations?
*Note*: Let $d\_i$ denote the number of isomorphic irreducible representations. Then $\sum d\_i^2$ is ... | https://mathoverflow.net/users/4246 | Decomposition of GL(2,p) into irreducible representations | **Note:** *This was written up concurrently with David's answer, but wasn't proofread and didn't get past the captcha stage due to technical problems.*
It is more common to consider the representation of $G=GL(2,\mathbb{F}\_p)$ on $\mathbb{C}^Y$, where $Y=X\setminus 0$. This is a direct sum over all multiplicative ... | 3 | https://mathoverflow.net/users/5740 | 31083 | 20,216 |
https://mathoverflow.net/questions/31100 | 1 | In need for something equivalent to the continuity-definition of real functions I use the following definition of "coarse-continuity" for sequences. Has it been known already? Has it even got a name?
Definition: A function $f(x)$ with $x \in \mathbb{N}$ is called coarsely continuous if and only if there exists a fix... | https://mathoverflow.net/users/7441 | Has coarse continuity been known already? | Functions like this are called [Lipschitz](http://en.wikipedia.org/wiki/Lipschitz_continuity). The definition works for maps between any two metric spaces. There is also the notion of being coarse lipschitz:
If you have a function $f : X \to Y$ between two metric spaces, and constants $K \geq 1$ and $C \geq 0$, then ... | 6 | https://mathoverflow.net/users/1335 | 31101 | 20,228 |
https://mathoverflow.net/questions/14918 | 17 | Are there examples of sets of natural numbers that are proven to be decidable but by non-constructive proofs only?
| https://mathoverflow.net/users/2672 | Non-constructive proofs of decidability? | When I teach computability, I usually use the following example to illustrate the point.
Let $f(n)=1$, if there are $n$ consecutive $1$s somewhere in the decimal expansion of $\pi$, and $f(n)=0$ otherwise. Is this a computable function?
Some students might try naively to compute it like this: on input $n$, start ... | 22 | https://mathoverflow.net/users/1946 | 31111 | 20,235 |
https://mathoverflow.net/questions/31038 | 5 | Let A and B be C\*-algebras, and let $\phi:A\rightarrow B$ be a *surjective* \*-homomorphism. Then $\phi$ is non-degenerate, and so we can extend it to \*-homomorphism between the multiplier algebras: $\tilde\phi: M(A)\rightarrow M(B)$. It's rather tempting to believe that then, surely, $\tilde\phi$ is also surjective.... | https://mathoverflow.net/users/406 | Surjective *-homs between multiplier algebras | This is true if $A$ is $\sigma$-unital, and is sometimes called the "noncommutative Tietze extension theorem". A good reference is Proposition 6.8 in Lance's *Hilbert $C^\*$-modules*. Proposition 3.12.10 in Pedersen's *$C^\*$-algebras and their automorphism groups* covers the separable case, which was first proved by A... | 6 | https://mathoverflow.net/users/1119 | 31117 | 20,239 |
https://mathoverflow.net/questions/31113 | 113 | Zagier has a very short proof ([MR1041893](https://mathscinet.ams.org/mathscinet-getitem?mr=1041893), [JSTOR](https://www.jstor.org/stable/2323918)) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \... | https://mathoverflow.net/users/3635 | Zagier's one-sentence proof of a theorem of Fermat | [This paper](http://www.math.tugraz.at/~elsholtz/WWW/papers/zagierenglish9thjuly2002.ps) by Christian Elsholtz seems to be exactly what you're looking for. It motivates the Zagier/Liouville/Heath-Brown proof and uses the method to prove some other similar statements. Here is a [German version](http://www.math.tugraz.at... | 95 | https://mathoverflow.net/users/6950 | 31121 | 20,242 |
https://mathoverflow.net/questions/31118 | 5 | Is there a way to determine a formula giving all *integer* values of $x$ for which the value of a polynomial $P(x)$ with *integer coefficients* is a square?
That is, is there a closed formula for:
$X = \{ x \in \mathbb{N} : \exists \ n \in \mathbb{N} : P(x) = n^2 \}$ ?
I'm interested in particular in $P'(x) = 8x^... | https://mathoverflow.net/users/7258 | Integer polynomials taking square values | There's a fairly detailed explanation of the solution to a similar equation [here](https://web.archive.org/web/20150908090650/http://mathforum.org/library/drmath/view/73118.html). See also [this page](https://www.alpertron.com.ar/QUAD.HTM), which can give you an automated step-by-step solution to such quadratic diophan... | 5 | https://mathoverflow.net/users/353 | 31125 | 20,244 |
https://mathoverflow.net/questions/31109 | 38 | I am interested in the differences between algebras and coalgebras. Naively, it does not seem as though there is much difference: after all, all you have done is to reverse the arrows in the definitions. There are some simple differences:
The dual of a coalgebra is naturally an algebra but the dual of an algebra nee... | https://mathoverflow.net/users/3992 | Is there an explicit construction of a free coalgebra? | There cannot be a "free coalgebra" functor, at least in what I think is the standard usage. Namely, suppose that "orange" is a type of algebraic object, for which there is a natural "forgetful" functor from "orange" objects to "blue" objects. Then the "free orange" functor from Blue to Orange is the *left* adjoint, if ... | 25 | https://mathoverflow.net/users/78 | 31126 | 20,245 |
https://mathoverflow.net/questions/30898 | 8 | It seems that there are three basic ways to prove an inequality eg $x>0$.
1. Show that x is a sum of squares.
2. Use an entropy argument. (Entropy always increases)
3. Convexity.
Are there other means?
Edit: I was looking for something fundamental. For instance Lagrange multipliers reduce to convexity. I have not... | https://mathoverflow.net/users/nan | Ways to prove an inequality | I don't think your question is a mathematical one, for the question about what do all inequalities eventually reduce to has a simple answer: axioms. I interpret it as a metamathematical question and still I believe the closest answer is the suggestion above about using everything you know.
An inequality is a fairly g... | 9 | https://mathoverflow.net/users/2384 | 31135 | 20,250 |
https://mathoverflow.net/questions/31072 | 2 | As a phd-student I've wandered into a question of colourings of graphs and wondered what was known about them.
Given a finite graph G, where the maximum degree of a vertex is d, I'm interested in colourings where not only are no adjacent vertices the same colour, but also that no vertex has two neighbours of the same... | https://mathoverflow.net/users/5965 | Colourings of Graphs with extra conditions | Qiaochu Yuan commented that your problem is equivalent to coloring what is known as the **square** $G^{2}$ of the graph $G$. For more details on coloring the square of a graph, see "The chromatic number of graph powers", N. Alon and B. Mohar, Combinatorics, Probability and Computing (1993) 11, 1-10. On-line at <http://... | 2 | https://mathoverflow.net/users/5883 | 31139 | 20,253 |
https://mathoverflow.net/questions/31147 | 15 | I'm looking for a good reference that has a detailed treatment of obstruction theory in the case where the target space is not simple. The specific situation I am interested in involves lifting a map of 3-skeletons from a $K(G, 1)$ to an arbitrary homotopy 3-type $X$ to the 4-skeletons (and hence to a true map between ... | https://mathoverflow.net/users/396 | Obstruction theory for non-simple spaces | Paul Olum developed some obstruction theory for maps into non-simple spaces back in the 1940-ies and 50-ies. You may want to check out his paper "Obstructions to extensions and homotopies", Annals of Mathematics, Vol 52, 1950, pp 1-50, if you have not looked at it yet.
| 14 | https://mathoverflow.net/users/6668 | 31149 | 20,257 |
https://mathoverflow.net/questions/31150 | 4 | The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\varpi\right)$ is the Legendre symbol.
Is there a good way to calculate this? The product converges very slowly, and none... | https://mathoverflow.net/users/6043 | Calculating the infinite product from the Hardy-Littlewood Conjecture F | This problem was studied by a few, and the ideas involve too much latex to write here. Mainly there are ideas of transforming to crazy weighted sums and then use ERH to bound errors from crazier integrals. It suffices to say that the culmination of this research is the freely available paper:
>
> [New Quadratic Pol... | 2 | https://mathoverflow.net/users/2024 | 31161 | 20,264 |
https://mathoverflow.net/questions/31153 | 10 | I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.
What are some research problems with the following properties(Experimental mathematics is a field containing problems with the criteria below):
1. Accessible to undergraduates
2. There can be many re... | https://mathoverflow.net/users/6886 | Problem suggestions for polymath for undergraduates research | Pick any of the problems in the archives of [Al Zimmermann's Programming Contests](http://www.azspcs.net/), and make progress either on the theoretic side (tighter upper bounds / lower bounds / asymptotics) or the computational side.
A specific nice example could be [Point Packing](http://www.azspcs.net/Contest/Point... | 2 | https://mathoverflow.net/users/25 | 31162 | 20,265 |
https://mathoverflow.net/questions/31163 | 15 | Let $l^2$ be a Hilbert space of infinite sequences $(z\_0, z\_1, \cdots)$ with finite $\sum\_{i=0}^{\infty} |z\_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(T)=l^2$?
| https://mathoverflow.net/users/7079 | Unbounded linear operator defined on $l^2$ | No there aren't any simple, or even any constructive, examples of everywhere defined unbounded operators. The only way to obtain such a thing is to use Zorn's Lemma to extend a densely defined unbounded operator. Densely defined unbounded operators are easy to find.
Zorn's lemma is applied as follows. Let $A$ be an o... | 15 | https://mathoverflow.net/users/6781 | 31166 | 20,267 |
https://mathoverflow.net/questions/31165 | 6 | Let $K$ be a algebraic number field of degree $n$ over $\mathbb{Q}$, and $O$ its ring of integers. Let $P$ be a prime ideal of $O$ and $(p)=P \cap \mathbb{Z}$.
Is it true that the localization $O\_{P}$ is a rank $n$ free module over $\mathbb{Z}\_{(p)}$ (the localization of $\mathbb{Z}$ at $(p)$) if and only if $P$ is t... | https://mathoverflow.net/users/7456 | Localizations as free, finite rank modules | Well, if $P$ is not the only prime above $p$, then $O\_P$ cannot be a finitely-generated $\mathbb{Z}\_{(p)}$-module for the following reason. Suppose $Q$ is another prime ideal above $p$ and select $\beta\in Q\setminus P$. Then $\beta^{-1}\in O\_P$. If $O\_P$ were finitely-generated as a module over $\mathbb{Z}\_{(p)}$... | 6 | https://mathoverflow.net/users/4351 | 31182 | 20,276 |
https://mathoverflow.net/questions/31154 | 24 | Recall that a **$(k,k+1,\dots,k+n)$-TQFT** is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some target category, usually your favorite version of $n$-Vect. When $k=0$, a full "classification" of TQFTs with a give... | https://mathoverflow.net/users/78 | What's the current state of the classification of not-fully-extended TQFTs? | Moore and Seiberg's result (Phys. Lett. **212B** (1988) p.451) on classifying modular functors can be thought of as classification of (1,2,3) theories. (M&S only do the 1 and 2 of (1,2,3), but it's not hard to extend to 3 as well; see "On Witten's 3-manifold Invariants" [here](http://canyon23.net/math/).)
My guess is... | 21 | https://mathoverflow.net/users/284 | 31184 | 20,277 |
https://mathoverflow.net/questions/31179 | 3 | Let $S$ be an irregular surface of general type over $\mathbb{C}$ and $a \colon S \to A:=\textrm{Alb}(S)$ be its Albanese map. Let Def($S$) and Def($A$) be the bases of the Kuranishi family of $S$ and $A$, respectively. Then $a$ induces a map $f \colon \textrm{Def}(S) \to \textrm{Def}(A)$, whose differential is $f\_\* ... | https://mathoverflow.net/users/7460 | Variation of the Albanese map | Yes, this is true. The point is that the Albanese is well defined in families, as a family of abelian varieties; that is, given a flat family of smooth projective varieties, there is a projective smooth family of Albanese varieties over the same base. This family is the dual of the family of Pic^0, which is well known ... | 4 | https://mathoverflow.net/users/4790 | 31190 | 20,281 |
https://mathoverflow.net/questions/31091 | 0 | In Lambek and Scott's "Introduction to higher order categorical logic" (1988), they state that every Heyting Algebra can be understood as a bicartesian closed category.
On the other hand, fixing a bicartesian closed category, and using $A \cong B$ to denote that morphisms1 exists between $A$ and $B$, we can see that ... | https://mathoverflow.net/users/7348 | Bicartesian closed categories and Heyting algebras | As Andreas Blass observed, those identities do not hold in all bicartesian closed categories. However, they are true if "isomorphism" is replaced by "equimorphism." In a poset category, equimorphism and isomorphism are the same and thus these equations do verify that a bicartesian closed poset category is a Heyting alg... | 2 | https://mathoverflow.net/users/2000 | 31192 | 20,282 |
https://mathoverflow.net/questions/30168 | 4 | Suppose a $\mathcal{H}^{1}$ measurable set $A\subset \mathbb{R}^{n}$ has positive Hausdorff density $\Theta^{1}(\mathcal{H}^{1},A,x)=c>0$ in a point $x\in A$.
If we have a decomposition $A=B\cup C$ with disjoint $B$ and $C$, can it happen that $\Theta^{1}(\mathcal{H}^{1},B,x)$ and $\Theta^{1}(\mathcal{H}^{1},C,x)$ d... | https://mathoverflow.net/users/1272 | Does positive density imply existence of the density for some part of a decomposition? | Today at our problem coffee someone (I don't know if he wants his name mentioned) showed me a counterexample to the first part of my question, in case anyone else is interested.
We take $X:= [0,1]\subset \mathbb{R}^{2}$ - for $\mathbb{R}^{n}$ the same argument should work, only with slightly more complicated notation... | 1 | https://mathoverflow.net/users/1272 | 31196 | 20,283 |
https://mathoverflow.net/questions/31198 | 20 | Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar curvature R.
>
> **Question** [Edited] Is there a nice formula which expresses the scalar curvature at a point of ... | https://mathoverflow.net/users/7466 | Curvature of a Lie group | See Exercice 1 in Chapter 4 of Do Carmo's "Riemannian Geometry".
The formula is $R(X,Y)Z = \frac 1 4 [[X,Y], Z]$.
In particular, if $X$ and $Y$ are orthonormal, the sectional curvature of the generated plane is
$K(\sigma)= \frac 1 4 \|[X,Y]\|^2$
Which is always $\geq 0$.
EDIT: In view of the comments, it ... | 23 | https://mathoverflow.net/users/5753 | 31200 | 20,285 |
https://mathoverflow.net/questions/26839 | 26 | There is a unique nonempty set $B$ of nonnegative integers such that
every positive integer can be written in the form
$$b + s^2, b\in B, s\ge0$$
in an even number of ways.
$B = \{0, 1, 2, 3, 5, 7, 8, 9, 13, 17, 18, 23, 27, 29, 31, 32, 35, 37,$
$ 39, 41, 45, 47, 49, 50, 53, 55, 59, 61, 63, 71, 72,$
$ 73, 79, 81, 83, ... | https://mathoverflow.net/users/935 | How thick is the reciprocal of the squares | In a related question, (Why are there usually..), O'Bryant characterized the elements of B that = 3 mod 8, and asked why the number of such that are at most X appears to be small; my
answer to his question showed that the number is O(Xloglog(X)/log(X)). In this answer I'll
sketch a proof that the same result holds for ... | 6 | https://mathoverflow.net/users/6214 | 31204 | 20,289 |
https://mathoverflow.net/questions/31173 | 3 | This can be seen as a follow up my question here:
[Is there a notion of "fibered category with boxproducts"?](https://mathoverflow.net/questions/28152/is-there-a-notion-of-fibered-category-with-boxproducts)
Given a monoidal fibration $f:E\rightarrow B$
(i.e. a strict monoidal functor between monoidal categories wh... | https://mathoverflow.net/users/2837 | Notion of stack fibered in monoidal categories? | I would take the view point that a monoidal category is a bicategory with one object. (Then a category fibered in monoidal categories should be the same thing as a weak functor into bicategories that "only hits monoidal categories".) In other words, what you should have is that this fibration is a 2-stack when viewed a... | 4 | https://mathoverflow.net/users/4528 | 31206 | 20,291 |
https://mathoverflow.net/questions/31212 | 5 | Let $g, r, a, b$ be positive integers. In Friedman's urn model we have an urn with $r$ red and $g$ green balls in it. In each step we take one ball out of urn, register its color and return it to the urn. Additionally, we put $a$ more balls of this color and $b$ more balls of the other color.
Let $X\_n$ be the relati... | https://mathoverflow.net/users/6159 | Intuitive "proof" or explanation of a result in Friedman's urn | You might be interested in [the article](http://www.jstor.org/stable/2238205?seq=2) by David A. Freedman on Friedman's urn. He reports a simple and intuitive proof due to Ornstein, which only uses the strong law
of large numbers.
In his notation the urn contains $W\_n$ white balls and $B\_n$ black balls at time $n$,... | 4 | https://mathoverflow.net/users/5371 | 31218 | 20,295 |
https://mathoverflow.net/questions/30238 | 13 | Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold.
When starting to write this answer I found this related [MO question](https://mathoverflow.net/questions/15411/finite-generated-group-realized-as-fundamental-group-of-manifolds/15421#... | https://mathoverflow.net/users/5753 | Constructing 4-manifolds with fundamental group with a given presentation. | Related to question (1), suppose you wanted to get a 4-manifold with boundary as a submanifold of $\mathbb{R}^4$ with fundamental group a finitely presented group, with presentation complex $K$. It is a [result of Curtis](http://www.ams.org/mathscinet-getitem?mr=140114) that any 2-complex $K$ is
homotopy equivalent to ... | 7 | https://mathoverflow.net/users/1345 | 31226 | 20,299 |
https://mathoverflow.net/questions/31223 | 7 | In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}\_l}$-)sheaves over a variety $X\_0$ defined over a finite field $F$. Weights start to behave better when one extends coefficients from $F$ to its algebraic closure i.e. passes from $X\_0$ to $X$.
Now, BBD was published in 1982. Are... | https://mathoverflow.net/users/2191 | In what setting does one usually define mixed sheaves and weights for them? | Admittedly, I'm almost completely ignorant about the $\ell$-adic setting. But, in case the following at least gets at the spirit of your question: in the de Rham setting, I found an old (1990s) preprint of Saito ("On the formalism of mixed sheaves," now TeXed up and available [on the arxiv](http://arxiv.org/abs/math/06... | 7 | https://mathoverflow.net/users/2628 | 31229 | 20,301 |
https://mathoverflow.net/questions/31178 | 12 | I've asked a question like this before, but now I'm more interested in counting the number of covers.
We suppose given the following data.
1. A positive integer $d$
2. A finite set of closed points $B= ( b\_1,\ldots,b\_n )$ in $\mathbf{P}^1\_\mathbf{C}$
3. Branch types $T\_1,\ldots, T\_n$.
**Question.** How many ... | https://mathoverflow.net/users/4333 | Counting branched covers of the projective line and Spec Z | One thing to keep in mind is that the analogue of Spec Z is really P^1 over a finite field k, not P^1/C. And here already one does *not* have a simple "Hurwitz-type formula" for the number of G-covers with given branching which are defined over k.
Just to give an example which may be illustrative; suppose that G = S\... | 9 | https://mathoverflow.net/users/431 | 31233 | 20,303 |
https://mathoverflow.net/questions/31243 | 5 | What is the entropy of a normal distribution with mean 0 and variance \sigma?
Thanks!
| https://mathoverflow.net/users/7475 | entropy of normal distribution | I found [here](http://www.cis.hut.fi/ahonkela/dippa/node94.html) that "the negative differential entropy of the normal distribution"
(which may not be what you are asking for?) is:
$$-\frac{1}{2} [ \log (2 \pi \sigma^2 ) + 1 ] ,$$
independent of $\mu$.
| 6 | https://mathoverflow.net/users/6094 | 31245 | 20,309 |
https://mathoverflow.net/questions/31208 | 6 | Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors
$X= \{x\_i\}$ and $Y=\{y\_j\}$ satisfying $x\_i\cdot y\_j=f(i,j)$, in such a way that
the vectors in $X$ and in $Y$ are as small as possible. More precisely,
set $m= \max\... | https://mathoverflow.net/users/6442 | Inner products and norms | Such questions have been dealt with. Note first that your $m$ is just the norm of the matrix $F$ (see Robin's comment) as an operator from $\ell\_1^n$ to $\ell\_\infty^n$. $M$ also has a name, it is the $\gamma\_2$ norm of this operator (This is the minimal product of
$$\|Y\|\_{1\to 2}\|X\|\_{2\to\infty}$$ over all fac... | 10 | https://mathoverflow.net/users/6921 | 31246 | 20,310 |
https://mathoverflow.net/questions/31250 | 53 | A naive and idle number theory question from a topologist (but not a knot theorist):
I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ and the $3$-sphere. I gather that from an etale point of view the former is $3$-dimensional and simply connected; from t... | https://mathoverflow.net/users/6666 | Prime numbers as knots: Alexander polynomial | [This article](http://arxiv.org/abs/0904.3399v1) appears to discuss the relationship between Alexander polynomials in knot theory and Iwasawa polynomials in number theory, although I haven't looked at it in detail. I discovered this paper in [This Week's Finds 257](http://math.ucr.edu/home/baez/week257.html), which giv... | 20 | https://mathoverflow.net/users/396 | 31253 | 20,312 |
https://mathoverflow.net/questions/31131 | 11 | Given a CM field we can use its maximal order (and a choice of CM type) to construct an abelian variety $\mathbb{C}^g/\Lambda$ with complex multiplication by the maximal order.
How do I (or where can I find information on) explicitly write down equations for a projective embedding of this variety, and the action of t... | https://mathoverflow.net/users/2024 | CM field to Torus to Abelian Variety? | This is not the answer. I am adding some relevant papers (some possessing good examples) which won't fit in the comments section.
I have been wanting to know the answer to this question as well. It seems one has to find an ample line bundle, and then calculate the Riemann theta relations which define the projective ... | 3 | https://mathoverflow.net/users/5372 | 31255 | 20,313 |
https://mathoverflow.net/questions/31248 | 20 | I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
* objects are groupoids;
* 1-morphisms are (left-principal?) bibundles;
* 2-morphisms are bibundle homomorphisms.
On the right, I have the 2-category ALG, whose:
* objects are algebras (over $\mathbb C$, say);
* 1-mor... | https://mathoverflow.net/users/78 | What is the precise relationship between groupoid language and noncommutative algebra language? | I think the problem is that the left hand side is part of commutative geometry, while the right hand side is part of noncommutative geometry (regardless of whatever vague claims I may have made in my previous response). Groupoids (or stacks) certainly have interesting noncommutative aspects that are captured by the con... | 11 | https://mathoverflow.net/users/582 | 31256 | 20,314 |
https://mathoverflow.net/questions/31254 | 0 | Any hints how to compute this sum
$$\sum\_{i=1}^{N-1}\left[i\frac{K}{N}\right]^{p}?$$
where K < N , $\left[\cdot\right]$ denotes fractional part,
$p\in N$
| https://mathoverflow.net/users/3589 | sum of fractional parts | The article [On Certain Sums of Fractional Parts](https://doi.org/10.1007/BF01238638 "Arch. Math 25, 41–44 (1974). zbMATH review at https://zbmath.org/?q=an:0277.10005") by Gandhi and Williams answers your question for $p=1$; it's likely that since 1974 this result has been generalized, but I wasn't able to find a refe... | 3 | https://mathoverflow.net/users/35336 | 31258 | 20,316 |
https://mathoverflow.net/questions/31278 | 7 | Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}\_1$ and let $\mathbb{F}\_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}\_n$ on $\mathcal{R}$.
Does the von Neumann crossed product $\mathcal{R}\rtimes\_{\alpha}\mathbb{F}\_n$ have the QWEP?
Remarks: Since $\mathbb{F}\_n$ ... | https://mathoverflow.net/users/5210 | Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP? | Yes. If $a$ and $b$ are generators of $\mathbb F\_2$ then $\mathcal R \rtimes\_\alpha \mathbb F\_2$ decomposes as an amalgamated free product of $(\mathcal R \rtimes\_\alpha \langle a \rangle)$ and $(\mathcal R \rtimes\_\alpha \langle b \rangle)$ over $\mathcal R$, where each of these are hyperfinite. Brown, Dykema, an... | 9 | https://mathoverflow.net/users/6460 | 31282 | 20,329 |
https://mathoverflow.net/questions/31270 | 19 | Hello, I would like to ask you if there is a mathematical theory, that is complete (in the sense of Goedel's theorem) but practically applicable. I know about Robinson arithmetic that is very limited but incomplete already. So, I would like to know if there is some mathematics that could be practically used (expressive... | https://mathoverflow.net/users/6702 | Complete mathematics | You probably intended to restrict the question to effectively axiomatizable theories. Otherwise, for example, the first-order theory of the standard model of arithmetic is a complete theory, as is the theory of the standard model of ZFC.
Gödel's incompleteness theorem establishes some limitations on which effective ... | 27 | https://mathoverflow.net/users/5442 | 31286 | 20,331 |
https://mathoverflow.net/questions/31275 | 28 | Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a *Smith normal form*. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we additionally suppose that $R$ is an integral domain?
What can we say if we impose the additional condition that the diagon... | https://mathoverflow.net/users/nan | Does Smith normal form imply PID? | The implication is false without the assumption that R is Noetherian, because finite matrices don't detect enough information about infinitely generated ideals.
For example, let R be the ring
$$
\bigcup\_{n \geq 0} k[[t^{1/n}]]
$$
where $k$ is a field (an indiscrete valuation ring). Any finite matrix with coefficient... | 33 | https://mathoverflow.net/users/360 | 31287 | 20,332 |
https://mathoverflow.net/questions/31288 | 0 |
>
> **Possible Duplicate:**
>
> [AC in group isomorphism between R and R^2](https://mathoverflow.net/questions/25375/ac-in-group-isomorphism-between-r-and-r2)
>
>
>
Somewhere, I recall being told that there is an isomorphism between $\mathbb{R}$ and $\mathbb{C}$ under addition. However, despite a rather leng... | https://mathoverflow.net/users/6856 | Is there an Isomorphism between R and C under addition? | As vector spaces over the rationals, they have the same dimension, so the only tricky part is the difficulty in finding a basis.
| 5 | https://mathoverflow.net/users/3684 | 31289 | 20,333 |
https://mathoverflow.net/questions/26079 | -2 | Given a directed acyclic graph `G` and a path made up from its set of nodes `N`, what is the closest approximate match to N, equipped with an intuitive notion of distance?
A path in a directed acyclic graph is essentially a string (that uses the set of nodes as the alphabet), so when comparing paths one can make use ... | https://mathoverflow.net/users/6321 | Algorithm for determining if a path exists in a graph or if not, the closest edit distance. | If graph is acyclic you can use some sort of dynamic programming.
Let $a\_{u,k}$ be the best distance you can get if you start from vertex $u \in G$ and consider only $k$ last vertices of your given path.
It's quite straightforward how to calculate all $a\_{u,k}$ based on all values of $a\_{u',k'}$ with $u'$ "after" ... | 1 | https://mathoverflow.net/users/7079 | 31296 | 20,337 |
https://mathoverflow.net/questions/31295 | 17 | Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (*Questions with distributions etc. are possible also but I want to keep things simple here*).
### FINAL CLARIFICATION/REWRITE!!
Thanks to all who have commented so far. I will need some more time to d... | https://mathoverflow.net/users/6651 | Let a function f have all moments zero. What conditions force f to be identically zero? | The answer to the first question is no. The following example is standard in probability theory, see e.g. Billingsley "probability and measure", example 30.2.
$$f(x) = {1\over \sqrt{2\pi}\ x}\ e^{-{(\ln x)^2\over 2}}\ \sin(2\pi \ln x) \ {\bf 1}\_{[0,\infty[}(x)$$
You can check that all the moments are zero using th... | 19 | https://mathoverflow.net/users/6129 | 31299 | 20,339 |
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