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https://mathoverflow.net/questions/52700 | 7 | This is a simple question about terminology and provenance.
I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.
I've encountered Polignac's conjecture generalizing the twin prime conjecture by replacing pairs $(p,p+2)$ with $(p,p+k)$ for any even $k$. The even m... | https://mathoverflow.net/users/10909 | Prime constellation conjectures | From Green and Tao's "Linear Equations in Primes":
>
> We have been referring to the generalised Hardy-Littlewood conjecture because Hardy and Littlewood [28] in fact only conjectured an asymptotic for the number of $n\leq N$ for which the forms $n+b\_1,...,n+b\_t$ are all prime. If this were generalised to deal wi... | 6 | https://mathoverflow.net/users/385 | 52737 | 33,011 |
https://mathoverflow.net/questions/52734 | 6 | Part 1:
How big is the category $TVS\_{loc.conv.}$ of locally convex topological vector spaces (and continuous maps)?
In other words (and less cheekily), is there a free locally convex TVS having any given set as basis? This would imply the functor $TVS\_{loc.conv.} \to Set$ is essentially surjective and has an adj... | https://mathoverflow.net/users/4177 | On locally convex (and compactly generated) topological vector spaces | Part 1: The "cheeky" answer is: **huge**. There is a left adjoint to the forgetful functor $LCTVS \to Vect$ (in particular there is a left adjoint to the forgetful functor $LCTVS \to Sets$): Equip a vector space $V$ with the locally convex topology induced by *all* linear functionals on $V$ (or as Pietro Majer put it: ... | 4 | https://mathoverflow.net/users/11081 | 52740 | 33,014 |
https://mathoverflow.net/questions/52718 | 18 | Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in $\mathbb{Z}/n\mathbb{Z}$. My questions are:
1) Since Dedekind domains have the same factorization theorem for ideals analogous to that o... | https://mathoverflow.net/users/12312 | Generalized Euler phi function | Yes, there is a formula for $\varphi(I)$ in the case of number fields. Let $R$ be the ring of integers of a number field. As mentioned in Greg's comment, it suffices to consider the case $I=\mathfrak{p}^n$ where $\mathfrak{p}$ is a maximal ideal of $R$. Then we have a surjective ring morphism
\begin{equation}
\frac{R... | 14 | https://mathoverflow.net/users/6506 | 52741 | 33,015 |
https://mathoverflow.net/questions/52755 | 4 | Consider a dynamical system given by its [flow](http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29) $\phi(t,x)$, where $t \in R$, $x \in R^n$ and $\phi: R \times R^n \to R^n$ is (say) differentiable.
The *$\omega$-limit set*, $\omega(p)$, of a point $p \in R^n$ is the set of all $q \in R^n$ such that the... | https://mathoverflow.net/users/8460 | Omega-limit set of the omega-limit set | Stable homoclinic (or heteroclinic) loops in the plane provide a simple counterexample.
| 6 | https://mathoverflow.net/users/12120 | 52758 | 33,028 |
https://mathoverflow.net/questions/52744 | 22 | Modular forms are defined here:
<http://en.wikipedia.org/wiki/Modular_form#General_definitions>
Maass forms are defined here:
<http://en.wikipedia.org/wiki/Maass_wave_form>
I wonder if modular forms can be transfered into Maass forms.
Or they two are different categories of automorphic forms.
| https://mathoverflow.net/users/2666 | What is the relationship between modular forms and Maass forms? | In the more common terminology modular forms on the upper half-plane fall into two categories: holomorphic forms and Maass forms. In fact there is a notion of Maass forms with weight and nebentypus, which includes holomorphic forms as follows: if $f(x+iy)$ is a weight $k$ holomorphic form, then $y^{k/2}f(x+iy)$ is a we... | 39 | https://mathoverflow.net/users/11919 | 52766 | 33,033 |
https://mathoverflow.net/questions/52773 | 3 | What is the most memory efficient algorithm for calculating $A\cdot B$, where $A,B\in \mathbb{R}^{n \times n}$?
The result of this multiplication might be stored in one of the given matrices ($A$ or $B$). The 'ideal' algorithm would perform calculations with $O(1)$ additional memory and return $A\cdot B$ and $B$
| https://mathoverflow.net/users/3794 | Memory efficient matrix multiplication | [This](http://ljk.imag.fr/membres/Brice.Boyer/pdf/winoschedule.pdf) may help.
| 2 | https://mathoverflow.net/users/12138 | 52776 | 33,039 |
https://mathoverflow.net/questions/52774 | 8 | Let $K$ be a number field. I am wondering if the following exact sequence $$1 \longrightarrow[\widehat{\mathcal O}\_K^\times] \longrightarrow Gal(K^{ab}/K) \overset{\pi}{\longrightarrow} Cl\_K \longrightarrow 1$$ splits, i.e. if there is a homomorphism $s:Cl\_K \to Gal(K^{ab}/K)$ such that $s \circ \pi = id$. Here $[\w... | https://mathoverflow.net/users/5831 | On the structure of the maximal abelian Galois group of a number field | Let $K = {\mathbb Q}(\sqrt{-5})$ and let $P = (3+2\sqrt{-5})$ denote a prime ideal of norm $29$ in the ring of integers of $K$. There does not exist a quadraic extension of $K$ ramified exactly at $P$, but there is one over the Hilbert class field $K(i)$ of $K$. This means that the ray class group modulo $P$ does not s... | 8 | https://mathoverflow.net/users/3503 | 52784 | 33,044 |
https://mathoverflow.net/questions/52671 | 6 | Dear all,
I have the following problem which seems quite standard to me but nevertheless I'm stuck right now.
Given a positive integer $n$ and a multi-index $p \in \mathbb{N}\_0^n$ I want to count the number of multi-indices $k \in \mathbb{N}\_0^n$ which are dominated by $p$ (i.e. every component of $k$ is less th... | https://mathoverflow.net/users/12366 | number of muti-indices of a fixed order which are less than a given multi-index | Expanding my comment into a full answer:
By inspection, we can see that $C(n,s,p)$ is the coefficient of $x^s$ in:
$(1 + x + x^2 + \cdots + x^{p\_1}) (1 + x + x^2 + \cdots + x^{p\_2}) \cdots (1 + x + x^2 + \cdots + x^{p\_n})$
$= \Pi\_{j=1}^n \Sigma\_{i=0}^{p\_j} x^i $
$= \Pi\_{j=1}^n {{(x^{p\_j+1}-1)}\over{(x-... | 0 | https://mathoverflow.net/users/9840 | 52787 | 33,046 |
https://mathoverflow.net/questions/52688 | 52 | I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do in class or give as a project. In particular, I've found that I don't have many good examples/exercises that illustrate th... | https://mathoverflow.net/users/10206 | Interesting Calculus Questions/Exercises | You only need integration by parts to prove the irrationality of $\pi$. I'm having
my Calculus 2 students do it as a long-term group project starting Monday.
Then when you've done partial fractions, you can have them derive the quickly-converging BBP formula for $\pi$.
And you can have them do the "18th Century Sty... | 27 | https://mathoverflow.net/users/3634 | 52793 | 33,052 |
https://mathoverflow.net/questions/52785 | 1 | I am an undergraduate CS major with strong interests in applied math and theoretical computer science. In the past, I've done reasonably well grade-wise in all math-related (that is, pure math, applied or theoretical CS) classes, but I feel that I still haven't taken away as much as i could have from most.
As people ... | https://mathoverflow.net/users/942 | Best Practices for Learning Mathematics (especially in the classroom) | This is not a universal recipee for anything, rather a few random points.
1) Make sure your background matches the course expectations. If not, work on it before even thinking of taking an advanced class.
2) Read ahead, not behind. Most teachers will tell you what's coming next and if you come to the class knowing ... | 7 | https://mathoverflow.net/users/1131 | 52795 | 33,053 |
https://mathoverflow.net/questions/52797 | 6 | Hi all,
In most set-theory accounts dealing with $ 0 ^ \sharp $ and its related effects on the constructible sets, a proof of the following theorem is crucial:
(\*) Assuming there is a measurable cardinals, for uncountable limit cardinals $ \alpha , \beta $ $ L\_\alpha $ is elementary equivalent to $ L\_\beta $.
... | https://mathoverflow.net/users/10708 | Using zero-sharp to characterize L | A short proof of $(\*)$ uses the characterization of $0^\sharp$ in terms of mice. A nice account of this version is in Ernest Schimmerling's [paper](http://www.math.ucla.edu/~asl/bsl/0704/0704-002.ps) "The ABC of mice".
The idea is that a sharp is a kind of "local measurable cardinal". More formally, we can think of... | 9 | https://mathoverflow.net/users/6085 | 52800 | 33,054 |
https://mathoverflow.net/questions/52804 | 0 | I know that cartesian closed category must have finite products and exponential objects. distributive category must have finite product, finite coproduct and s.t. A\*0~0, A\*B+A\*c~A\*(B+C). I think somehow I need to use the property of expoential objects to prove the isomorphism, but not sure how to do it. I would be ... | https://mathoverflow.net/users/12394 | how to prove cartesian closed category with finite coproducts is distributive category? | If $A\times (-)$ is a left adjoint, it must commute with colimits, and in particular, $\oplus$.
That is, write $B\_1\oplus B\_2$ as $colim\_i B\_i$, then $A\times colim\_iB\_i\cong colim\_i(A\times B\_i)$, from which the result follows.
Since $A\times (-)$ must preserve colimits, also notice that the initial object... | 0 | https://mathoverflow.net/users/1353 | 52805 | 33,056 |
https://mathoverflow.net/questions/52606 | 8 | Grzegorczyk has divided the class of primitive recursive functions to Grzegorczyk-hierarchy by their rate of growth. In this hierarchy $E\_i\subset E\_{i+1}$ and the subset-relation is strict. Also $\cup\_{i}E\_i = Pr$, i.e. the union of all levels is equal to the class of primitive recursive functions.
I know that p... | https://mathoverflow.net/users/nan | Enumerating levels of Grzegorczyk-hierarchy | The usual definition of $E\_n$ is in terms of basic functions, the $n$'th generator function, closed under composition, and bounded recursion. I take it that you see how an enumeration could easily be constructed from some sort of a syntax tree, except for the difficulty that the restriction on the scheme of bounded re... | 5 | https://mathoverflow.net/users/6787 | 52806 | 33,057 |
https://mathoverflow.net/questions/52812 | 5 | Let $H$ and $N$ be two groups with $H$ cyclic. Let $f,g:H \rightarrow \mathrm{Aut}(N)$ be homomorphisms such that $N\rtimes \_f H \cong N\rtimes \_g H$. Then does that mean $f(H)$ and $g(H)$ are conjugate in $\mathrm{Aut}(N)$?
| https://mathoverflow.net/users/6761 | Do isomorphic semi-direct products correspond to conjugate automorphisms? | The answer is no. You can easily have a situation where $f$ is the trivial map, while $g$ makes $H$ act through an inner automorphism of $N$, so that in both cases $N\rtimes H\cong N\times H$. For concreteness, let $N=D\_8$ be the dihedral group of order 8, let $\sigma$ be a non-central involution in $N$, let $H=\langl... | 7 | https://mathoverflow.net/users/35416 | 52814 | 33,061 |
https://mathoverflow.net/questions/52803 | 18 | I've always been fascinated by the fact that the classical [Gauss sum](http://en.wikipedia.org/wiki/Gauss_sum) has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In particular, I have long wondered whether this apparent link between number theory ... | https://mathoverflow.net/users/3106 | Can Gauss sums derandomize any heuristic arguments? | The thing is that it is well known that for the quadratic Gauss sums, expressed as an exponential sum rather than with Legendre symbols, the path is very much not a random walk when you plot it in the complex plane. There is plenty of structure visible as approximate Cornu spirals.
Such sums are not the only Gauss s... | 8 | https://mathoverflow.net/users/6153 | 52824 | 33,068 |
https://mathoverflow.net/questions/52821 | 3 | Presentation of a semi-direct products of $N$ by $H$ can be written from presentations of $N$ and $H$. But for other extensions of $N$ by $H$ (cyclic, central etc.), which are not semi direct products, can we write presentation of the extension from presentation of $N$ and $H$?
| https://mathoverflow.net/users/6761 | Presentation of extentions of groups | To start with, following Alex's comment above, I think you need to look at some books on cohomology of groups to specify how the extension is going to be given. That means looking at the Schreier theory.
Many years ago the Schreier theory of extensions was adapted to give exactly this by Turing.(Little known paper of... | 1 | https://mathoverflow.net/users/3502 | 52833 | 33,071 |
https://mathoverflow.net/questions/52831 | 2 | Given a well ordering of a set $A$ we can define a total order $A^A$ in an obvious way (for $f \neq g$ find the least $i$ such that $f(i) \neq g(i)$ and define $f < g$ if $f(i) < g(i)$)
Does the inverse direction work? Does a total order on the powerset of $A$ give rise to a well ordering of $A$? (without choice, of ... | https://mathoverflow.net/users/4903 | Total order on the powerset | No. If this were true, then ZF would prove that "every set can be totally ordered" implies "every set can be well-ordered", which (assuming ZF is consistent) it doesn't. I can't find the original citation for this nonimplication, but it's in Howard and Rubin's "Consequences of the Axiom of Choice" for example.
| 4 | https://mathoverflow.net/users/11771 | 52836 | 33,073 |
https://mathoverflow.net/questions/52848 | 9 | Let $f(z)=\sum\_{n\geq 0}a\_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a\_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{align\*}
f(z)=\frac{P(z)}{Q(z)},
\end{align\*}
where $P(z)$ and $Q(z)$ are *coprime monic complex polynomials*. By develo... | https://mathoverflow.net/users/11765 | On rational functions with rational power series | Let there be two fields $k\subset K$, and let $f\in k[[x]]$ be a formal power series with coefficients in $k$. If $f\in K(x)$ (rational functions with coefficients in $K$) then $f\in k(x)$. A proof of this is given in J.S. Milne's notes on Etale Cohomology (lemma 27.9).
| 8 | https://mathoverflow.net/users/2384 | 52859 | 33,083 |
https://mathoverflow.net/questions/52829 | 7 | In terms of vector field analogies to closed and exact differential forms, conservative and incompressible vector fields (gradient and divergence) generalize to higher dimensions, but curl and irrotational fields do not. Why?
Cross product doesn't generalize either but one can use exterior products and hodge duals to ... | https://mathoverflow.net/users/12403 | Generalization of Curl to higher dimensions | Let ${\bf K}$ be a vector field in the neighbourhood of ${\bf p}\in{\mathbb R}^n$, and let ${\bf X}$ and ${\bf Y}$ be two tangent vectors at ${\bf p}$. These two vectors span a parallelogram $P$ with one vertex at ${\bf p}$. The "circulation" of ${\bf K}$ around $P$ computes to
$$
\int\_{\partial P}{\bf K}\cdot \mathrm... | 10 | https://mathoverflow.net/users/8050 | 52860 | 33,084 |
https://mathoverflow.net/questions/52846 | 4 | Let **C**^ = **C***op* to **Set**, for a natural transformation f:X-Y in **C**^, how to prove if f is an epimorphism, then fc is surjective for all objects c in **C**? Anyone can help me with that? Thank you in advance:)
| https://mathoverflow.net/users/12394 | prove natural transformation is epimorphism | It's a good question - possibly homework? - but good homework anyway.
One thing that makes it a good question is that if you change "epimorphism" to "monomorphism" and "surjective" to "injective" then it becomes much easier: you can solve it directly by testing at representables. (In other words, apply the definitio... | 5 | https://mathoverflow.net/users/586 | 52861 | 33,085 |
https://mathoverflow.net/questions/52856 | 0 | Guy Robin proved
$$
\frac{\sigma(n)}{n} < \exp(\gamma)ln(ln(n))+ \frac{0.6482}{ln(ln(n))},
$$
for all integers $n>2,$
where $\gamma$ is Euler's constant, and $ln$ is the logarithm to the base $exp(1).$
Question: There are analogue formulas for the sum of the $k$-th powers of the positive
divisors of $n.$
| https://mathoverflow.net/users/11016 | Analogue of Guy Robin's formula for upper bound of $\sigma(n)/n$ | Oh, yes, I gave a fairly complete answer at:
[A hierarchy of k-highly composite numbers](https://mathoverflow.net/questions/41802/a-hierarchy-of-k-highly-composite-numbers/41818#41818)
There is a procedure iniated by Ramanujan that gives a sequence of particularly large values of $$\frac{\sigma\_k(n)}{n^k},$$ essen... | 3 | https://mathoverflow.net/users/3324 | 52862 | 33,086 |
https://mathoverflow.net/questions/52850 | 0 | For any positive integer $n \in A= \mathbb{Z}$ we set
$$
\sigma(n) = \sum\_{d \mid n,\, d >0} d.
$$
J. T. B. Beard's algorithm on the title may be converted in
the following algorithm:
(i)
Take $n=n\_0$ a `well chosen` positive element of $A.$
(ii)
If $n \mid \sigma(n)$ STOP and output $n.$
(ii)
if $n \nmid \s... | https://mathoverflow.net/users/11016 | J. T. B. Beard's algorithm may catch `multi-perfect numbers` too | **Later thoughts** Up to 100,000 there are 159 starting values which lead to a multiply perfect number in 1 to 6 steps. If we take the smallest starting value which leads to each terminal value one has these 80 values: 2, 3, 5, 9, 16, 18, 30, 33, 64, 72, 98, 117, 128, 176, 234, 256, 384, 504, 512, 528, 819, 896, 1008, ... | 3 | https://mathoverflow.net/users/8008 | 52868 | 33,089 |
https://mathoverflow.net/questions/52674 | 4 | In Jones's paper "Planar Algebras I", Theorem 4.2.1 establishes that an extremal finite index subfactor admits a spherical C\*-planar algebra structure, and Theorem 4.3.1 establishes that spherical C\*-planar algebra arise from subfactors. It seems unlikely, due to the key ingredient of the proof of Theorem 4.3.1, that... | https://mathoverflow.net/users/6269 | Invertibility of the planar algebra-subfactor correspondence | Yes, strongly amenable subfactors of the hyperfinite $II\_1$-factor are completely classified by their standard invariant. The finite depth case was done by Popa's Classification of subfactors: the reduction to commuting squares (MR1055708), and the infinite depth case was finished by Popa's Classification of amenable ... | 6 | https://mathoverflow.net/users/351 | 52872 | 33,092 |
https://mathoverflow.net/questions/52864 | 3 | Let $X$ be a quasicompact quasiseparated scheme. Consider the full subcategory $\text{Qcoh}\_{fp}(X)$ of $\text{Qcoh}(X)$ which consists of the quasi-coherent modules which are locally of finite presentation.
**Question** Is every quasi-coherent module $M$ the colimit of the homomorphisms $N \to M$, where $N$ runs t... | https://mathoverflow.net/users/2841 | Colimit of locally finitely presented quasi-coherent modules | The answer is **yes**, at least if you believe Thomason-Trobaugh, *Higher algebraic $K$-theory of schemes and of derived categories*, which David Ben-Zvi already mentioned.
I quote from Appendix B.3 (p. 409f):
>
> B.3. If $X$ is a quasi-compact and quasi-separated scheme, every sheaf in $\text{Qcoh}(X)$ is a dire... | 3 | https://mathoverflow.net/users/11081 | 52876 | 33,094 |
https://mathoverflow.net/questions/52891 | 4 | Background
I'm modeling Genetic Algorithm(GA) with Markov chains and deriving the expression for the expectation of the first hittig time in the MC with 1 absorbing state and $l-1$ transient states. This results is an expression for a sum involving square of a binomial coefficient
Problem
I need to find a closed ex... | https://mathoverflow.net/users/12418 | Square of Binomial Coefficient | According to Mathematica, your sum equals:
$(1-p^2)^{l/2} \mbox{LegendreP}\left(l/2, \frac{1+p^2}{1-p^2}\right),$
or
$\, \_2F\_1\left(-\frac{l}{2},-\frac{l}{2};1;p^2\right)$
The second sum is
$n^2 \binom{2 n-2}{n-1}.$
Ain't technology grand...
**EDIT** The real question is: why do you want to know? The exp... | 5 | https://mathoverflow.net/users/11142 | 52896 | 33,104 |
https://mathoverflow.net/questions/52901 | 3 | Let $V$ be the set-theoretic universe, and suppose that $U$ is some ultrafilter over $\kappa\gt\omega.$ Then, we can go through the motions and produce the ultrapower $M = V^{\kappa}/U$.
Now, the existence of an $\omega\_1$-complete ultrafilter (and the existence of the transitive collapse of $M$) is subject to (as ... | https://mathoverflow.net/users/8843 | Ultrapowers, and Models of Set theory. | Michael,
* the existence of a well-founded $M$ and an embedding $j:V\to M$ different from the identity
is *equivalent* to
* measurability,
which is equivalent to
* the existence of an $\omega\_1$-complete non-principal ultrafilter,
which is equivalent to
* the existence of a non-principal ultrafilter ... | 6 | https://mathoverflow.net/users/6085 | 52907 | 33,109 |
https://mathoverflow.net/questions/52813 | 5 | Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic version of SAW's? We know for example that Catalan numbers count a myriad of things so perhaps SAW appear elsewhere? For ... | https://mathoverflow.net/users/934 | Self Avoiding Walk Enumerations | I strongly doubt that there exists a non-trivial bijection between self-avoiding walks on Z^d and other combinatorial objects. I don't have any intuition as to whether the generating function may have some other number theoretic interpretation, but I haven't seen or heard anything in this regard.
SAWs are sufficientl... | 6 | https://mathoverflow.net/users/6363 | 52911 | 33,113 |
https://mathoverflow.net/questions/52893 | 5 | Is there a non-finitely-generated group each of whose proper subgroups is finitely generated? If so, what form of choice (if any) is required to construct such a group?
| https://mathoverflow.net/users/11445 | Is it possible to construct (without choice, even?) a non-finitely-generated group with no proper non-finitely-generated subgroup? | (CW since this is just expanding on George Lowther’s comment to the question, which could really have been an answer in the first place; if George L wants to convert his answer to a comment himself, I can delete this one.)
For any prime $p$, the Prüfer $p$-group is as desired.
There are several constructions of thi... | 12 | https://mathoverflow.net/users/2273 | 52914 | 33,114 |
https://mathoverflow.net/questions/52922 | 9 | Suppose $X$ is a CW-complex. The monoid of homotopy self-equivalences $M = hAut(X)$ is the subspace of $Map(X,X)$ consisting of those maps with a homotopy inverse. It is a union of path components. It obviously acts on $X$, and the homotopy type only depends on the homotopy type of $X$.
It is known that we can find m... | https://mathoverflow.net/users/360 | Extreme rigidification of homotopy self-equivalences | This is a substantial revision of my original post. It shows that if we replace the "equivalence" Tyler is asking for by a "retract" then the answer is yes.
1. Given a CW space $Y$, we can take $G(Y) =$ the topological monoid of homotopy automorphisms of $Y$. The Borel construction
$$
EG(Y) \times\_{G(Y)} Y \to BG(Y)... | 4 | https://mathoverflow.net/users/8032 | 52924 | 33,120 |
https://mathoverflow.net/questions/52415 | 2 | I feel that my question is very basic, but, somewhat suprisingly, nobody was able to give me an answer so far:
Let
$\mathbf{M} := \langle \{ 0,1 \}, 0, 1, \leq, \neg \rangle$
be the structure with the obvious definitions for the relation $\leq$ and the unary mapping $\neg$.
Let us consider all elements in the ... | https://mathoverflow.net/users/8590 | Of what kind of complemented bounded poset are the structures in my quasi-variety? | Here is a suggestion which favors your characterization. Assuming you characterization is correct, the main problem is to take any appropriate self-dual bounded poset with no fixed points from the involution, and show it isomorphic to a subalgebra of a power of your structure. The idea is to use some set X such as the ... | 1 | https://mathoverflow.net/users/3402 | 52936 | 33,126 |
https://mathoverflow.net/questions/52888 | 3 | For a Riemannian manifold $(M,g)$, the geodesic flow is $\phi\_t:TM\to TM, (x,v)\mapsto (\gamma(t;x,v),\dot{\gamma}(t;x,v))$, where $\gamma(\cdot;x,v)$ is the geodesic started at $x$ with direction $v$. The related vector field of this flow can be formulated as
$X(x,v)=(v,0)$.
My question can be viewed as the sham g... | https://mathoverflow.net/users/11028 | A simple ODE on smooth manifolds | As you noticed, you need a connection $D$ on $TM$ in order to define the splitting in $T(TM)$ and the unique vector field $\xi$ with the property that $\xi$ is horizontal and satisfies $\pi\_\*(\xi\_X)=X$ for all $X\in TM$. Then the flow $\phi\_t$ of $\xi$ is the geodesic flow of the connection $D$.
To see this, rec... | 4 | https://mathoverflow.net/users/10675 | 52944 | 33,132 |
https://mathoverflow.net/questions/52935 | 13 | The recent [answer's](https://mathoverflow.net/questions/52744/what-is-the-relationship-between-modular-forms-and-maass-forms/52753#52753 "link") link to Ono's work makes me ask and wonder if his [new results](http://www.aimath.org/news/partition/folsom-kent-ono.pdf "pdf") on partition functions tell something about Ga... | https://mathoverflow.net/users/451 | partition functions and Galois representations? | Dear Thomas,
As far as I know, this work is not related directly to Galois representations, but is rather a particular calculation in the theory of $p$-adic modular forms (although it is not really described this way explicitly in the paper).
The $p$-adic theory of modular forms of half-integral weight was developed ... | 20 | https://mathoverflow.net/users/2874 | 52953 | 33,138 |
https://mathoverflow.net/questions/52934 | 5 | Let $C,C',D\in \operatorname{Cat}$, and consider the following strict pushout square
$$\begin{matrix}
C&\overset{f}{\to} &D^{op}\\
\downarrow^\pi&\swarrow&\downarrow^{\iota\_2}\\
C'&\underset{\iota\_1}{\to}&D'^{op}
\end{matrix}
$$
where $D':=C'^{op}\coprod\_{C^{op}} D$ and the 2-cell (denoted by $\swarrow$) is the ... | https://mathoverflow.net/users/1353 | Are strict pushout squares in Cat exact squares? | No, it's false for general categories; it's true for groupoids, however.
A crucial test case for Beck-Chevalley is often called (by categorical logicians especially) Frobenius reciprocity; it's the case where one forms the pullback
$$\begin{matrix}
C & \overset{\Delta}{\to} & C \times C \\
\downarrow^\Delta & & \... | 7 | https://mathoverflow.net/users/2926 | 52966 | 33,148 |
https://mathoverflow.net/questions/52970 | 6 | Let $U$ is a set. I will speak about filters on this set.
If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as
the filter whose base is $\lbrace f[A] | A \in a \rbrace$.
I will call super-embedding-1 of filter $a$ into filter $b$ a function $f$
such that $f \left[ a \right] \subseteq b$ an... | https://mathoverflow.net/users/4086 | Orderings of ultrafilters | I understand your question better now.
First, in your general context of filters the relations
$\leq\_1$ and $\leq\_2$ are not the same. To see this, let
$G=\{I\}$ be the trivial filter on a set $I$ with at
least two points, and let $\mu$ be any nonprincipal
ultrafilter on $I$. Since $G\subset \mu$, we see that
$\mu\... | 8 | https://mathoverflow.net/users/1946 | 52972 | 33,150 |
https://mathoverflow.net/questions/52958 | 2 | Let M be a smooth Riemannian manifold. Let $f:S^1\to M$ be a locally-flat injective loop. Must there exist such $\varepsilon>0$ that if we connect points $f(0),f(\varepsilon), f(2\varepsilon)... , f([1/\varepsilon]\varepsilon), f(1)=f(0) $ by shortest geodisic path then we obtain piecewise-smooth loop without self-inte... | https://mathoverflow.net/users/4298 | Uniform smoothness of locally-flat loops | I'm not sure what you intend by the terminology "locally flat injective loop". Do you intend
this in the sense of topology of manifolds, that it's a continuous injective map where
each point in the image has a neighborhood making the curve homeomorphic to a line in
$\mathbb R^n$?
If this is the correct interpretatio... | 6 | https://mathoverflow.net/users/9062 | 52973 | 33,151 |
https://mathoverflow.net/questions/52979 | 15 | I came across the problem "find all integer solutions to $y^2=x^3+17$."
I've tried several things, without any success, and I was hoping that someone could help out. (Some ideas or a reference for where to find it are both appreciated)
By numerical calculation I have found that the following integer points $(x,y)$... | https://mathoverflow.net/users/12176 | Integer Points on the Elliptic Curve $y^2=x^3+17$. | There is a standard method for computing all integral points on an elliptic curve using David's bounds and lattice reduction. The method can be found in the book:
Nigel Smart, "The Algorithmic Resolution of Diophantine Equations", Cambridge University Press.
This method is implemented in several computer algebra pack... | 22 | https://mathoverflow.net/users/4140 | 52983 | 33,158 |
https://mathoverflow.net/questions/39149 | 4 | Hello,
For an $n \times n$ real matrix, the SVD (Singular Value Decomposition) algorithm is $O(n^3)$.
I have large matrices (say $10,000 \times 10,000$) that only have elements on few diagonals, i.e. $M=(m\_{i,j})\in \mathbb{R}^{n\times n}$ such that $m\_{i,j} =0$ if $|i-j|>k$ ($k$ is set and way smaller than $n$). A... | https://mathoverflow.net/users/3958 | SVD complexity for structured sparse matrices | It depends on how small $k$ is. If $k^2 \ll n,$ the simple method of computing $M^t M$ (sparsely), then the Cholesky decomposition, then the eigenvalues, works very well. Perhaps less work is using
<http://soi.stanford.edu/~rmunk/PROPACK/>
| 1 | https://mathoverflow.net/users/11142 | 52991 | 33,162 |
https://mathoverflow.net/questions/52950 | 0 | Let $E$ be a compact subset of $\mathbb{R}^n$. Let the density function $\phi(x,y)$ be Lipschitz continuous and such that
$$
\int\limits\_E \phi(x,y)dy=1
$$
for all $x\in E$. Let us consider the non-increasing sequence of non-empty compact sets $A\_n$ such that for all $x\in A\_{n+1}$ we have
$$
\int\limits\_{A\_n} \ph... | https://mathoverflow.net/users/11768 | Convergence of sets | Anthony's interpretation of the question is that $A\_n$ is the support of $X\_n$, where $(X\_n)$ is any Markov chain of transition kernel $\phi$ such that the distribution of $X\_0$ has support $E$. If this interpretation is correct, the result holds. To see this, note that, under the assumption that $A\_1\subset E=A\_... | 1 | https://mathoverflow.net/users/4661 | 52992 | 33,163 |
https://mathoverflow.net/questions/47663 | 5 | For a hypergraph let $\chi$ be the least number of colours needed to colour the vertices, so that in each edge, each colour is used at most once (i.e., the *strong* chromatic number). Let $\Delta$ be the maximum number of hyperedges containing any vertex. Let $\omega$ be the maximum size of a *clique*, meaning a vertex... | https://mathoverflow.net/users/4020 | Hypergraph Chromatic Number vs Degree, Clique-Size | Very nice question, Dave. I had a look at it with Matej Stehlik, here in Grenoble. We found a way to give a positive answer to your question, although it uses more heavy machinery than we would like. You would expect there is an easy argument, but if it exists, we haven't found it yet.
Given a hypergraph $H$, form a ... | 6 | https://mathoverflow.net/users/2568 | 52994 | 33,164 |
https://mathoverflow.net/questions/52988 | 11 | I'd like any insight or references to the following two conjectures (see the glossary below for definitions):
Conjecture 1: For any string $x$, there exists a longest common subsequence of $x$ and its reversal $x^R$ that is a palindrome.
Conjecture 2: For any string $x$ over a two-letter alphabet, all longest commo... | https://mathoverflow.net/users/8056 | palindromic subsequences | OK. Having failed to make a counterexample, let's attempt a proof of Conjecture 1.
Suppose that $x$ and $x^R$ have a common subsequence $s$ of length $k$. This means that there are $n\_1 < n\_2 < \ldots < n\_k$ and $m\_1 > m\_2 > \ldots > m\_k$ such that $x\_{n\_i}=x\_{m\_i}$.
Now look for the place where they cross:... | 6 | https://mathoverflow.net/users/11054 | 52999 | 33,166 |
https://mathoverflow.net/questions/52996 | 29 | Modular forms of integral weight are prominent in number theory.
Furthermore, there are $\theta$-functions and the $\eta$-function, having weight 1/2,
which also have a rich theory.
But I have never seen a modular form of weight e.g. 1/3.
I have been wondering about this for a long time. Are there
examples of mod... | https://mathoverflow.net/users/3757 | Modular forms of fractional weight | I am no expert here, but I believe modular forms of fractional weight (e.g. of weight 1/3) appear more naturally as forms on metaplectic covers of GL(2) (e.g. on the cubic cover) and over fields containing the relevant roots of unity (e.g. the third roots of unity). Kubota around 1970 initiated the study of these cover... | 22 | https://mathoverflow.net/users/11919 | 53005 | 33,171 |
https://mathoverflow.net/questions/52989 | 2 | Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T-,-)\cong \mathcal C(-,\mathrm{For}-).$$
Is the same true for 2-monads on a 2-category?
| https://mathoverflow.net/users/1261 | Reference request: 2-Monads and 2-Adjunctions | As Todd says, there are several flavours of 2-monad.
If you are interested in strict 2-monads, strict algebras for these, and strict morphisms, then yes you have an adjunction (even an enriched adjunction) as usual.
If you mean something weaker, then you will have something weaker than an adjunction. In particula... | 4 | https://mathoverflow.net/users/10862 | 53007 | 33,172 |
https://mathoverflow.net/questions/53006 | 5 | I want to check that the homotopy category of cochain complexes of an idempotent splitting, preadditive category is idempotent splitting.
Let $a\xleftarrow{e}{}a$ be an idempotent chain map up to chain homotopy, $e^2\sim e$; that is, there exists maps $a\_{i-1}\xleftarrow{h\_i}a\_i$ with $e\_i^2-e\_i=h\_{i+1}d\_i+d\_... | https://mathoverflow.net/users/1068 | How do I split a homotopy idempotent? | Let me re-denote your chain complex $a$ by $C$. You can define a chain complex $D$ as the *mapping telescope* of the infinite sequence
$$
\cdots\overset e\to \quad C \quad \overset e\to \quad C \quad \overset e\to \cdots
$$
This can be constructed as follows: Form the homotopy coequalizer of the pair of maps
$$
1,S\_a:... | 6 | https://mathoverflow.net/users/8032 | 53013 | 33,176 |
https://mathoverflow.net/questions/52985 | 18 | Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of Noether says that this group is generated by linear transformations and the Cremona transformation, which is given by
$$
... | https://mathoverflow.net/users/3996 | What is known about the birational involutions of P^3? | Let $\mbox{Inv}(\mathbb P^n)$ be the
subgroup of $\mbox{Bir}(\mathbb{P}^n)$ generated by involutions and
$\mbox{PGL}(n+1, \mathbb C)$.
>
> If $n\ge 3$ then any generating set of $\mbox{Inv}(\mathbb P^n)$
> contains uncountably many involutions. More precisely
> for any $d >1$ there are
> uncountably many
> ... | 9 | https://mathoverflow.net/users/605 | 53018 | 33,178 |
https://mathoverflow.net/questions/53029 | 5 | Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of abstract groups. I think this cohomology must be well studied. Can somebody provide me some references for this cohomology. A... | https://mathoverflow.net/users/11932 | smooth cohomology of Lie groups | This is indeed very well studied. The standard references are Borel-Wallach, *Continuous cohomology, discrete subgroups and representations of reductive groups*, Annals of Math. Studies **94**, Princeton University Press (1980) and the more gentle book by A. Guichardet, *Cohomologie des groupes topologiques et des algè... | 4 | https://mathoverflow.net/users/11081 | 53034 | 33,189 |
https://mathoverflow.net/questions/34511 | 17 | It there any known way of obtaining the [Banach fixed-point theorem](http://en.wikipedia.org/wiki/Banach_fixed_point_theorem) from the [Tarski fixed-point theorem](http://en.wikipedia.org/wiki/Knaster-Tarski_theorem) or vice-versa?
| https://mathoverflow.net/users/3993 | Banach and Knaster-Tarski fixed point theorems -- are they related? | Hello,
I just found the question, so the answer might come a bit too lat, but..
Have a look at:
*Paweł Waszkiewicz, "Common patterns for metric and ordered fixed point theorems.",
In Proceedings of the 7th Workshop on Fixed Points in Computer Science (Luigi Santocanale ed.), 2010, pp. 83-87.*
I attended this talk... | 17 | https://mathoverflow.net/users/11618 | 53035 | 33,190 |
https://mathoverflow.net/questions/52997 | 5 | Where can I find a reference for the following fact:
If $C$ is a small category with finite colimits, then $\text{Ind}(C)$ is a category with all colimits, and it is the universal one in the following sense: If $D$ is a category with all colimits, then there is an equivalence of categories between finitely cocontinuo... | https://mathoverflow.net/users/2841 | Indization as adjoint from finite colimits to all colimits | Every finite loopless diagram in Ind-$C$ can be represented as a levelwise diagram, i.e. as a diagram of functors $I \to C$ for some filtered $I$. (Artin-Mazur, App. of Etale Homotopy Theory, Prop. 3.3 for example.) In particular, this holds for coequalizer diagrams. Then the colimit can be computed levelwise (Prop. 4.... | 1 | https://mathoverflow.net/users/4183 | 53039 | 33,192 |
https://mathoverflow.net/questions/53069 | 8 | Recently I was watching a talk: <http://media.cit.utexas.edu/math-grasp/Ben_Webster.html> and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation a bit):
$$ \mathfrak{Q} (\lambda, \mu) \Leftrightarrow \mathfrak{M} (\lambda, \mu)$$
Between Quiver varieties and ... | https://mathoverflow.net/users/8811 | Quiver varieties and the affine Grassmannian | Sadly, as the speaker, I would also like to know how that correspondence works. Long story short, some collaborators and I came up with a conjecture that certain pairs of symplectic varieties (algebraic varieties with an algebraic symplectic form) were in some kind of duality with each other. There is still not a good ... | 10 | https://mathoverflow.net/users/66 | 53081 | 33,211 |
https://mathoverflow.net/questions/53078 | 3 | Given $i: Z \subset X$, a closed immersion of smooth schemes over some field $k$. Is there an open subscheme $U$ of $X$ such that $Z \cap U$ is non-empty and such that the Gysin map of the Chow groups (CH is the total Chow ring here)
$$i^!: CH(U) \rightarrow CH(Z \cap U)$$
is surjective?
(Variants I would also be... | https://mathoverflow.net/users/12462 | Gysin maps between smooth schemes generically surjective? | I don't think that this is true in general. Take $X=\mathbb{P}^2$, then $CH(X)$ is finitely generated.
Moreover, finite generation holds for any open $U\subset X$ by the exact sequence
$$CH(X-U)\to CH(X)\to CH(U)\to 0$$
[Fulton, Intersection theory I, 1.8.]
On the other when $Z$ is curve of degree $3$ or more, $CH(Z\... | 3 | https://mathoverflow.net/users/4144 | 53092 | 33,216 |
https://mathoverflow.net/questions/53014 | 35 | Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}\_l \to \operatorname{Hom}\_G(T\_l A, T\_l B)$ is an isomorphism, where $A,B$ are abelian varieties over $K$, and $T\_l A$ is the Tate module of $A$. Such a statemen... | https://mathoverflow.net/users/2290 | In which ways can the isogeny theorem fail for local fields? | I believe for $l\ne p$ the theorem is 'always' false, in the sense that for every positive-dimensional abelian variety $A$ one can find many $B$s for which your map is not onto, independently of the reduction type:
Fix $A$ and take any non-constant family of abelian varieties in which $A$ is a fibre, e.g. some neighb... | 19 | https://mathoverflow.net/users/3132 | 53114 | 33,228 |
https://mathoverflow.net/questions/53116 | 4 | Let $G$ and $H$ be torsion abelian groups.
Are the following are equivalent:
1. $\mathrm{Hom}(G, H) = 0$
2. $\mathrm{map}\_\*( K(G, m), K(H,n)) \sim \*$ for all $n, m\geq 1$
?
Clearly (2) implies (1).
| https://mathoverflow.net/users/3634 | Contractible space of maps between Eilenberg-Mac Lane spaces, 2 | No, we have $Hom(\mathbb{Z}/n,\mathbb{Z})=0$ but there is a nontrivial Bockstein $K(\mathbb{Z}/n,1)\to K(\mathbb{Z},2)$.
For finite abelian groups we have (1) iff (2) iff ($|G|$ and $|H|$ are coprime).
| 4 | https://mathoverflow.net/users/10366 | 53118 | 33,230 |
https://mathoverflow.net/questions/53085 | 0 | Any reference on how to solve the problem $Ax + c = \lambda Bx$ , where $A$, $B$ are full rank matrices, $c$ and $x$ are vectors and $\lambda$ is an unknown constant. I want to solve for both $x$ and $\lambda$. Without the vector c, this is a Generalized Eigen problem and is easy to solve.
I actually don't need the ... | https://mathoverflow.net/users/12463 | Modified Eigen Problem | I think what is missing from the statement of the problem is a normalization condition on x, such as $\|x\|=1$. Problems of this kind are known as inhomogeneous eigenvalue problems, and there is some literature about them. See e.g. R.M.M. Mattheij and G. Soderlind, Linear Algebra and its Applications 88 (1987). p. 507.... | 0 | https://mathoverflow.net/users/12120 | 53123 | 33,231 |
https://mathoverflow.net/questions/53124 | 12 | In Wiki, under the item "category of groups", it states that the snake lemma fails in category of groups, however the nine lemma is valid. However, in the preface of the book " Mal'cev, protomodular, homological and semi-abelian categories ", it says "And category theory could not grasp either the conceptual foundation... | https://mathoverflow.net/users/2348 | snake lemma in category of groups | The snake lemma doesn't make sense in the category of groups in general, because maps don't always have cokernels. If you assume that the three vertical maps in your snake lemma diagram do have cokernels, that is, that their images are normal subgroups of their targets, then the snake lemma is true with the same proof ... | 13 | https://mathoverflow.net/users/12336 | 53139 | 33,242 |
https://mathoverflow.net/questions/53134 | 8 | Trivially $n^1=n^1$, and everyone knows that $3^2+4^2=5^2$. [Denis Serre](https://mathoverflow.net/questions/53048) quoted $3^3+4^3+5^3=6^3$ in a recent MathOverflow question (which prompted this one). Are any other examples known?
| https://mathoverflow.net/users/7458 | What sums of equal powers of consecutive natural numbers are powers of the same order? | There is a good discussion at <http://www.mathpages.com/home/kmath147.htm> along with some nice examples, e.g., $6^3 + 7^3 + \dots + 69^3 = 180^3$, $1134^3 + \dots + 2133^3 = 16830^3$, which apparently are part of an infinite family (starting with $3^3+4^3+5^3=6^3$). There is a table of sums of consecutive cubes equal ... | 8 | https://mathoverflow.net/users/3684 | 53150 | 33,253 |
https://mathoverflow.net/questions/53115 | 0 | **Background**
I have the following equations:
$$a+b+c=6$$
$$d+e+f=15$$
$$a+d=5$$
$$b+e=7$$
$$c+f=9$$
This is a 2x3 matrix $[a b c, d e f]$ where the marginal totals are fixed. In addition, all of the unknowns are positive.
I incorrectly told a student that he could do this analytically, and he returned with th... | https://mathoverflow.net/users/9421 | Solving 5 eqns with 6 unknowns in a 2x3 contingency matrix, is there a unique solution? | Since you have 5 linear equations but 6 unknowns so there cannot be an unique solution and there is at least one degree of freedom for the solution, for instance 5 of the unknowns depend on the 6'th.
After that you introduce one restriction: all unknowns should be positive and integer (because you define, the equati... | 1 | https://mathoverflow.net/users/7710 | 53155 | 33,258 |
https://mathoverflow.net/questions/53138 | 6 | The only examples of commutative rings of finite [global dimension](http://en.wikipedia.org/wiki/Global_dimension) I know are either:
* Dedekind domains (and fields as a degenerate special case)
* Regular local rings
* Rings constructed from the previous examples by taking direct sums, or forming the rings of polynom... | https://mathoverflow.net/users/3711 | Commutative Ring of Finite Global Dimension | At arsmath's request, I'm making this official.
(This is pretty standard commutative algebra, but I realize not everyone has gone through it.)
A commutative ring $R$ is regular if it's noetherian and its local rings are regular.
Using Serre's theorem e.g. Matsumura Commutative Ring Theory p 156, and the fact
that $Ex... | 10 | https://mathoverflow.net/users/4144 | 53158 | 33,261 |
https://mathoverflow.net/questions/53175 | 4 | The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see [this question](https://mathoverflow.net/questions/35518/upper-bound-for-tetrahedron-packing)) suggests that proving concrete bounds of this type can be quite difficult. My question i... | https://mathoverflow.net/users/12334 | Are there non-tiling polyhedra that pack arbitrarily well? | The easy topo-logic of this question depends on what you mean by a perfect tiling.
If the definition of a perfect tiling is a collection of isometric images of the polyhedron
with disjoint interiors and union equal to the whole space, it follows from compactness
in the Hausdorff topology.
Suppose you have a sequenc... | 12 | https://mathoverflow.net/users/9062 | 53183 | 33,277 |
https://mathoverflow.net/questions/53202 | 5 | Let $d\geq 1$ be a fixed integer, and $\mathbb{Z}^d$ be the lattice of all integers; consider the set $A\_d \subset [-1,1]$ defined by
$$ \alpha \in A\_d \ \iff \ \alpha=\frac{v \cdot w}{\|v\| \|w\|} \mbox{ for some } \ v,w \in \mathbb{Z}^d $$
Of course $A\_d$ is a set of quadratic irrationals and $\overline{A\_d}=[-1,... | https://mathoverflow.net/users/7979 | Angles in an integral lattice | The proof that $\pi/3$ is not a lattice angle is due to Lucas, while the proof that an $n$-gon does not embed in $\mathbb Z^d$ is due to Schoenberg, and has a nice proof by Scherrer. There is a AMM article on the topic you ask about ["Triangles with vertices on lattice points"](http://www.jstor.org/pss/2325060), which ... | 7 | https://mathoverflow.net/users/2384 | 53206 | 33,291 |
https://mathoverflow.net/questions/47258 | 18 | **Background**
notation: RV= random variable, $\mu=$ mean $m=$ median
Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV.
If $f(x)$ strictly convex:
$$\mu (f(x)) > f(\mu (x))\mathrm{\hspace{20mm}(1)}$$
Conversely if $-f(x)$ is strictly ... | https://mathoverflow.net/users/9421 | When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean? | If $f\ $ is a monotonic function then the median value of $f(X)$ is *the same* as the function applied to the median value of $X$.
If there is no monotonicity (or approximate monotonicity) you wouldn't expect them to be even close: just think of $X$ being uniformly distributed on the interval and $f(x)=|x-1/2|$ say.
... | 4 | https://mathoverflow.net/users/11054 | 53210 | 33,294 |
https://mathoverflow.net/questions/53195 | 8 | A conjecture of Galvin's is that the following is possible (which I take to mean that the consistency of the following can be proven relative to the consistency of something like ZFC, or ZFC plus some large cardinal):
>
> A partially ordered set $P$ can be decomposed into countably many chains iff the same is true ... | https://mathoverflow.net/users/7521 | Decomposing posets into countably many chains | I've investigated Galvin's Conjecture quite a bit over the past few years. Here is a [preprint](http://www-personal.umich.edu/~dorais/docs/galvin.pdf) of mine on the subject. In there, I show that Galvin's Conjecture restricted to finite-dimensional posets (posets which are subsets of some cartesian product of finitely... | 5 | https://mathoverflow.net/users/2000 | 53214 | 33,297 |
https://mathoverflow.net/questions/53219 | 1 | I am reading an article and it is written there: If A is a k-linear category (possibly without direct sums) we can embed it in the additive category A × N, where a morphism (x,m) → (y, n) is an n × m matrix with entries in A(x, y) = HomA(x, y). Of course if A is additive then A ≈ A × N. Could anyone explain it? Why is ... | https://mathoverflow.net/users/12485 | embedding a k-linear category in an additive category | It looks like if this is a wrong construction of the additivization of a $k$-linear category. Either this, or I don't understand the notation. Let me just sketch how the additivization $\operatorname{Add}(\mathcal{A})$ of a $k$-linear category $\mathcal{A}$ can be constructed:
Objects in $\operatorname{Add}(\mathcal{... | 5 | https://mathoverflow.net/users/12166 | 53224 | 33,301 |
https://mathoverflow.net/questions/53217 | 9 | Define a *Chebotarëv datum* over a number field $K$ to be a finite group $G$ together with a map $\mathfrak{p}\mapsto\gamma\_{\mathfrak{p}}$ from a cofinite set of primes of $K$ into the set of conjugacy classes of $G$ such that for every conjugacy class $c\subset G$, the proportion of $\mathfrak{p}$ with $\gamma\_{\ma... | https://mathoverflow.net/users/2821 | Chebotarëv data over number fields | I think this is false for cardinality reasons. Take some exceedingly thin, but infinite, set of primes P. Then you can take any valid Chebotarev datum and change it arbitrarily at each prime in P, and it still remains a valid datum, since P is too thin to change the densities. If the group has at least two conjugacy cl... | 9 | https://mathoverflow.net/users/2481 | 53227 | 33,302 |
https://mathoverflow.net/questions/53236 | 1 | Let $O$ be an open subset of the separable Hilbert space H and $k\geq0$ . Consider $C\_b^k(O, Sym(H))$, the space of k-times continuously differentiable maps with values in the bounded symmetric endomorphisms of $H$, bounded up to their k-th derivative. Equipped with the usual norm this space becomes a Banach space. Is... | https://mathoverflow.net/users/3509 | Separability of the space of bounded continuous maps | Say $H=L^2(R)$. Then $Sym(H)$ contains $L^\infty(R)$ isometrically (multiplication operators on $H=L^2(R)$), so that even the subspace of *constant* maps isn't separable.
ADDED : however, there seems to be a an infinite dimensional Sard theorem not requiring separability :
Hausdorff Conullity of Critical Images of... | 3 | https://mathoverflow.net/users/6451 | 53237 | 33,307 |
https://mathoverflow.net/questions/53166 | 1 | Hello all!
I recently had a question concerning algebraic dependence that has thus far gone unanswered from my professors and texts, that I hope I can phrase properly here. When answering, please reference any papers or texts that you may happen to be citing so that I can look them up later! The statement I would lik... | https://mathoverflow.net/users/12473 | Minimal Polynomials for Algebraic Dependence? | It is true if both your numbers are transcendental. It follows from the fact that height one prime ideals of $\mathbb{Q}[x,y]$ are principal. This can be found in standard algebra textbooks. Otherwise, I believe it's false. There are already counterexamples in the comments.
| 3 | https://mathoverflow.net/users/2290 | 53261 | 33,322 |
https://mathoverflow.net/questions/53274 | 23 | My understanding is if you have a homogeneous space $X = G/H$ for a reductive group $G$ and $X$ is normal and has an open $B$-orbit, for a Borel subgroup $B$ then you call $X$ spherical.
Someone asked me where 'spherical' came from and I had no idea. I asked a few more knowledgeable people and they also didn't know.... | https://mathoverflow.net/users/7 | Why are they called Spherical Varieties? | Marty is definitely correct about the origin of the terminology in the study of homogeneous spaces $G/H$ of reductive Lie groups. Due to the special example involving a quotient of special orthogonal groups the handy term \*spherical" got attached to the subgroup $H$ or the homogeneous space. In turn, this was carried ... | 14 | https://mathoverflow.net/users/4231 | 53279 | 33,333 |
https://mathoverflow.net/questions/53277 | 6 | I am having trouble making the so-called "Whitehead equivalence" explicit.
It is quite easy to draw a picture of what a Whitehead move is, given a foliation, as a kind of limit of isotopies of the foliation. What I would like to more carefully understand is precisely what is happening to the transverse measure. The ... | https://mathoverflow.net/users/10084 | Resource for Measured Foliations and Whitehead Equivalence | I don't know of a source other than FLP for understanding the Thurston compactification via measured foliations, but you might find it easier to understand this compactification using measured geodesic laminations. The nice thing about measured geodesic laminations is that you don't need clumsy operations like Whitehea... | 5 | https://mathoverflow.net/users/317 | 53283 | 33,336 |
https://mathoverflow.net/questions/53284 | 12 | The following was already known to Riemann. Suppose that one is given a connected Riemann surface $X$, a finite set $\Delta \subset X$ and a homomorphism $\phi: \pi\_1(X \backslash \Delta) \to S\_d$ where $S\_d$ is the permutation group on $d$ symbols. If this homomorphism is transitive, i.e. the image of $\phi$ acts t... | https://mathoverflow.net/users/798 | Which Riemann surfaces arise from the Riemann existence theorem? | The answer is simple: *any* complex structure arises in this way.
Indeed, any compact Riemann surface $X$ admits a holomorphic cover $\phi \colon X \to \mathbb{P}^1$.
This is straightforward in genus $0$ and $1$.
If the genus is at least $2$, then the linear system $|3K\_X|$ is very ample, so it gives an embeddin... | 13 | https://mathoverflow.net/users/7460 | 53286 | 33,337 |
https://mathoverflow.net/questions/36081 | 4 | Hi,
as the title says i'm wondering if there's a "simple" and known commutation relation between the following two differential operators. Let $E$ be a holomorphic vector bundle over a compact kahler manifold $X$, fix a Hermitian metric $h$ on $E$ and a kahler metric $g$ on $X$. Denote with $D=D^{'}+D^{''}$ the associ... | https://mathoverflow.net/users/4971 | Is there a "simple commutation" relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ the (0,1) part of the chern connection of a vector bundle and $\delta^{'}$ the adjoint of the (1,0) part? | Yes, the relation is that they anti-commute.
Let's see this very briefly (you can find it in almost all books on Kähler geometry and Hodge theory).
We want to compute $[D''\_E,\delta'\_E]$, where $[\bullet,\bullet]$ is the graded commutator and let me call $\omega$ (instead of $g$) the Kähler form. Then one has th... | 1 | https://mathoverflow.net/users/9871 | 53290 | 33,341 |
https://mathoverflow.net/questions/53240 | 2 | Hi, I know they are related questions on the board but mine is more specific. Although the answer for any non-singular matrix would be also interesting. Thanks!
UPDATE: I am sorry I though this was clear, but as I know it the $||\cdot||\_{2}$ norm is defined as follow: Let be $A\in \mathbb{R}^{m\times n};\;||A||\_{2}... | https://mathoverflow.net/users/12491 | How to compute the induced $||\cdot||_{2} $ matrix norm of an SPD matrix | To lay the question to rest, let me do two things: (i) restate it; (ii) answer it.
By $\|x\|$, we mean the Euclidean 2-norm throughout.
>
> Show that the *induced* 2-norm $$\max\_{\|x\|\not= 0} \frac{\|Ax\|}{\|x\|}$$ is given by $\sqrt{\lambda\_{\max}(A^TA)}$
>
>
>
The proof is textbook material. For the laz... | 2 | https://mathoverflow.net/users/8430 | 53291 | 33,342 |
https://mathoverflow.net/questions/53296 | 5 | Let $C$ be a continuous curve in the unit square having length $L$. Is there a lower bound on the *average distance* between the points in the unit square and $C$, as a function of $L$? Is there an asymptotic behavior that's known as $L$ gets large? (other suggestions for tags are welcome)
| https://mathoverflow.net/users/11828 | Average distance to a curve of fixed length | Here is a rough answer. I think it has to give the right order of magnitude.
$1/L$. If you draw a zigzag curve that goes up and down $L$ times it has length approximately $L$. Each point is distance no more than $1/L$ from the curve.
On the other hand if you consider a neighbourhood of a curve of width $1/(4L)$ on ... | 5 | https://mathoverflow.net/users/11054 | 53298 | 33,345 |
https://mathoverflow.net/questions/52282 | 5 | Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi\_{xy}$, where, given $e=\{x,y\}\in E$, we write $\pi\_{\{x,y\}} = \pi\_{xy}=\pi\_{yx}>0$. If $\{x,y\}\not\in E$, we set $\pi\_{xy}=0$. We write $\pi\_x$ for $\sum\_{y\sim x... | https://mathoverflow.net/users/5153 | Does generator of continuous time random walk map heat kernel from L^2 to L^2? | I was able to answer this question, although it didn't turn out to be useful in the way I thought it would be.
Let $P^\theta\_t$ be the transition operator, $(P^\theta\_tf)(x) = \sum\_{y\in G} p\_t(x,y)f(y)\theta\_y$. Then a fairly easy computation shows that $P\_t$ maps $L^2(\theta)$ to $L^2(\theta)$, and one can al... | 2 | https://mathoverflow.net/users/5153 | 53304 | 33,348 |
https://mathoverflow.net/questions/53316 | 2 | Whenever I read anything about valuations or things related to them (such as local fields) extensions always occupy a prominent position and a huge amount of effort is expended to derive results about such extensions. I have a little understanding of valuations and I think I have some understanding about how they can b... | https://mathoverflow.net/users/4692 | Why are extensions so heavily emphasized in valuation theory? | I don't find your claim that "so soon after valuations are introduced everything is ignored except for extensions" to be accurate. See for instance my notes on valuation theory (as part of a second graduate course on number theory) [here](http://alpha.math.uga.edu/%7Epete/MATH8410.html). Extensions of valuations play a... | 18 | https://mathoverflow.net/users/1149 | 53318 | 33,356 |
https://mathoverflow.net/questions/53320 | 1 | If we are given an algebraic number field L, and $ \alpha $ is an element of L whose field trace over Q is zero and whose field norm over Q belongs to Z, then does $ \alpha $ necessarily belong to the integral closure of Z in L?
The Hilbert's theorem 90 states the necessary and sufficient condition for elements to h... | https://mathoverflow.net/users/11059 | trace zero elements in algebraic number fields | The answer is no. Let $\alpha\in\mathbb{C}$ be a root of $x^3+tx+1$ for $t\in\mathbb{Q}\setminus\mathbb{Z}$, and let $L=\mathbb{Q}(\alpha)$, so that $N\_\mathbb{Q}^L(\alpha)=1$ and $tr\_{\mathbb{Q}}^L(\alpha)=0$. The element $\alpha$ cannot be integral over $\mathbb{Z}$, because its minimal polynomial over $\mathbb{Q}$... | 17 | https://mathoverflow.net/users/1916 | 53323 | 33,359 |
https://mathoverflow.net/questions/53308 | 0 | Given a sequence $a\_n$ such that $\sum\_{n\ge1} \dfrac{|a\_n|}{n^s}$ is convergent for $s>0$. Given that $\sum\_{n\ge1} \frac{|a\_n|}{n} < 1$, is it be possible to impose some sort of an upperbound for $\sum\_{n\ge1} \frac{|a\_n|}{n^s}$ for $1>s>k>0$ for some fixed $k$?
| https://mathoverflow.net/users/10046 | Bounds on Riemann Zeta type Function? | Given an integer $m$, define a sequence $a\_n$ to be $m$ at the $m$-th place and $0$ otherwise.
Then
$$\sum\_{n\ge 1} {a\_n\over n^s}=m^{1-s}$$
As any upper bound must work for all $m$, we see that no upper bound is possible for $s<1$.
| 1 | https://mathoverflow.net/users/6753 | 53325 | 33,361 |
https://mathoverflow.net/questions/53332 | 3 | Now, I am new to functional analysis. So please dont be harsh.
I was going through some papers by Balazard & Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating the characteristic function $\chi(0,1]$.
Now, what appears to me is, if we can construct a function, using functions of t... | https://mathoverflow.net/users/2865 | On the Nyman-Beurling equivalent form for RH | The Nyman-Beurling Theorem is: $\zeta(s)$ has no zeroes for $\sigma>1/p$ if and only if the subspace of functions of the form $\sum\_kc\_k\rho(\alpha\_k/x)$, for $0\le\alpha\_k\le 1$ with $\sum\_k c\_k\alpha\_k$=0, is dense in $L^p(0,1)$ (which is equivalent to the closure containing the characteristic function $\chi\_... | 3 | https://mathoverflow.net/users/6753 | 53338 | 33,368 |
https://mathoverflow.net/questions/53342 | 3 | Duro Kurepa conjectured that the function on the title is always nonzero in $\mathbb{Z}/{n \mathbb{Z}}$
provided $n>2.$ Daniel Barsky
and B\'enali Benzahgou [MR2145571 (2006a:11025)]
proved this. Thus, for all odd prime numbers $p$
$$
ku(p) = 0!+1!+ \cdots + (p-1)! \pmod{p}
$$
is an inversible element of $\mathbb{Z}/{p... | https://mathoverflow.net/users/11016 | About Duro Kurepa function: $ku(n)=0!+1!+ \cdots+(n-1)! \pmod{n}$ | Yes, if $s$ is a power of an odd prime then your function gives a unit mod $s$.
| 3 | https://mathoverflow.net/users/3684 | 53343 | 33,371 |
https://mathoverflow.net/questions/53344 | 2 | Let $X$ be a smooth projective variety of dimension $n$. Take the bundle $TX \oplus Sym^2(TX)$ over $X$ where $Sym^2(TX)$ is the second symmetric product of the tangent space.
The Grassmannian bundle $Gr(n,TX \oplus Sym^2(TX))$ has a canonical section, namely $TX$.
My question is: what is the Poincare dual of this ... | https://mathoverflow.net/users/12516 | Sections of Grassmannian bundles | Let $U$ denote the tautological subbundle. I guess that $c\_i = c\_i(U^\*)$ in your notation. Consider the composition map
$$
U \to p^\*(TX + S^2TX) \to p^\*S^2TX,
$$
where $p:Gr \to X$ is the projection. Then your section is the zero locus of this map. In other words, it is the zero locus of a global section of the ve... | 3 | https://mathoverflow.net/users/4428 | 53347 | 33,372 |
https://mathoverflow.net/questions/53346 | 2 | Let $x,y$ be vectors in $\mathbb{R}^n$ and let's use the notation $\hat x$ for the vector $x$ with its components sorted in increasing order.
The Hardy-Littlewood-Polya inequality states that
$$ x\cdot y \leq \hat x\cdot \hat y.$$
Let us also use the notation $xy\in\mathbb{R}^n$ to denote the coordinate-wise product of... | https://mathoverflow.net/users/12518 | An extension of the Hardy-Littlewood-Polya inequality? | Here is a Matlab script that will generate a quick counterexample for you:
```
function [x,y]=testIneq(n, p, q, r)
% x and y are length n vectors
% Try: [x,y]=testIneq(2,1,2,3) to get a counterexample!
flag = 1;
iter = 0;
while (flag)
iter = iter + 1;
x = randn(n,1);
y = randn(n,1);
xh = sort(x)... | 2 | https://mathoverflow.net/users/8430 | 53349 | 33,374 |
https://mathoverflow.net/questions/53354 | 7 | I was thinking about how to prove $\operatorname{Br}(K)\cong H^2(\operatorname{Gal}(\bar{K}/K),\bar{K}^\*)$ without having to introduce inductive limits and all the profinite stuff. So, I started wondering if the conditions of a direct system could be weakened for the category of abelian groups in a way that isomorphis... | https://mathoverflow.net/users/1849 | When are unions of isomorphic groups isomorphic? | The answer is no.
For a counterexample, let $G\_i=\mathbb{Z}$ be the integers and let $H\_i=\frac1i\mathbb{Z}$, for positive natural numbers $i$. The union $\bigcup\_i G\_i=\mathbb{Z}$, but $\bigcup\_i H\_i=\mathbb{Q}$.
For the revised question, where you want $G\_i$ and $H\_i$ distinct, there are still counterex... | 13 | https://mathoverflow.net/users/1946 | 53355 | 33,376 |
https://mathoverflow.net/questions/53314 | 6 | Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation group acting on $A$ has nontrivial stable subgroup. I want to see that the only flat connection in $R$ is the product con... | https://mathoverflow.net/users/12445 | Why must a reducible flat SU(2)-connection over a homology sphere be trivial? | I think the statement is that in a principal $SU(2)$-bundle $P$ over a connected manifold $M$ with $H\_1(M;\mathbb{Z})=0$, every reducible flat connection has trivial holonomy and therefore trivializes $P$.
For a Lie group $G$, gauge transformations $u$ that stabilize a $G$-connection $A$ are covariant-constant, and ... | 5 | https://mathoverflow.net/users/2356 | 53356 | 33,377 |
https://mathoverflow.net/questions/53292 | 5 | Suppose I chose two rational functions, say,
$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad
v = \frac{t^5(4+t)}{(1+4t)}.$$
Then I know that $K(X) = \mathbf{C}(u,v)$ is
the function field of the projective line
(Proof: If $K(Y) = \mathbf{C}(t)$, then
there is an inclusion $K(X) \subseteq K(Y)$ and
hence a surjection
$\math... | https://mathoverflow.net/users/12500 | Isomorphism of the function field of the projective line with $\mathbf{C}(s)$ | The usual algebraic proof of L\"uroth's theorem gives the following procedure for finding a single generator of the subfield $L= K(u\_1,\dotsc,u\_r)$ of $K(t)$: let $a$ be any non-constant coefficient of the minimal monic polynomial of $t$ over $L$. Then $K(a) = L$.
Perhaps one can concoct a fast algorithm to compute... | 3 | https://mathoverflow.net/users/2490 | 53376 | 33,389 |
https://mathoverflow.net/questions/53377 | 1 | The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. Combinatorially, this produces a copy of the Hasse diagram of Bruhat order inside $G/B$, with edges labeled by appropriate roo... | https://mathoverflow.net/users/8552 | Equivariant homology of Hilb and torus stable curves | It doesn't work as simply as it does for $G/B$, because there aren't finitely many $T$-invariant curves. Instead they will come in families whose unions are things like products of projective spaces. Although there will be a larger torus acting on each such family which has finitely many $T$-invariant curves, it won't ... | 3 | https://mathoverflow.net/users/12525 | 53383 | 33,392 |
https://mathoverflow.net/questions/53385 | 6 | Hello,
I've just learnt the notion of deficiency of a group but I don't know how work with it.
I want to construct a group with large negative deficiency ; naively I think that $(Z\_2)^n$ will work because we need $n$ generators and $n$ relations for the square of the elements being $1$, plus $n(n-1)/2$ relations of co... | https://mathoverflow.net/users/12517 | Groups with large negative deficiency | One way to bound the deficiency of a group is via the Morse inequalities. If $n$ is the number of generators and $m$ is the number of relations, then
$$1 - n+m \geq b\_0(G) - b\_1(G) + b\_2(G)$$
where $b\_i(G)$ denote the $i$-th Betti number of $G$ (with coefficients in some field $k$). This is due to the fact that... | 13 | https://mathoverflow.net/users/8176 | 53391 | 33,394 |
https://mathoverflow.net/questions/53381 | 17 | Dear all,
when reading a book of M. Berger, I learned that the injectivity radius Inj(x) on a compact Riemannian manifold depends continuously on the point x.
When the manifold is complete and non-compact, Inj may not be continuous.
For example, Inj(x) decreases to zero when x moves to the most curved point on a p... | https://mathoverflow.net/users/9915 | The continuity of Injectivity radius | The compactness is irrelevant; i.e if it is true for compact manifolds then the same is true for complete ones. (The same proof as in comact case works, but it is easier to do this way.)
If $R<\mathrm{InjRad}\_p$ then one can construct a smooth metric on a sphere with an isometric copy of $B\_R(p)$ inside.
If there... | 13 | https://mathoverflow.net/users/1441 | 53392 | 33,395 |
https://mathoverflow.net/questions/53389 | 6 | Let $G = G(n, 1/2)$ be an Erdos-Renyi graph in which each edge $e = (u,v)$ is present in the graph independently with probability $1/2$. For a subset of the vertices $S$, the cut value $c(S)$ is equal to the number of edges $(u,v)$ such that $u \in S$ and $v \not \in S$.
Clearly for any particular cut $S$, the expec... | https://mathoverflow.net/users/3027 | Max cut value in a random graph | This is addressed in:
An upper bound for the maximum cut mean value
Alberto Bertoni, Paola Campadelli and Roberto Posenato
Their bound is the same as yours; more precisely, for a random graph with $n$ vertices and $x n$ edges, for sufficiently large $x,$ they claim the size of max cut divided by $x n$ is bounded ab... | 3 | https://mathoverflow.net/users/11142 | 53393 | 33,396 |
https://mathoverflow.net/questions/53388 | 7 |
>
> **Possible Duplicate:**
>
> [What is the status of the Gauss Circle Problem?](https://mathoverflow.net/questions/19079/what-is-the-status-of-the-gauss-circle-problem)
>
>
>
The Gauss circle problem is the following: Let $N(r)$ denote the number of solutions in integer pairs $(i,j)$ to the inequality $i^2... | https://mathoverflow.net/users/10898 | Recent results on the Gauss circle problem? | The best reference on the subject is M. Huxley's monograph
MR1420620 (97g:11088)
Huxley, M. N.(4-WALC)
Area, lattice points, and exponential sums.
London Mathematical Society Monographs. New Series, 13. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xii+494 pp. ISBN: 0-19... | 6 | https://mathoverflow.net/users/11142 | 53394 | 33,397 |
https://mathoverflow.net/questions/53345 | 4 | Hello,
I want to show that the cohomological dimension (say over Z or R) of some group $K$ is 1. $K$ occurs in an exact sequence $1 \to K \to \pi\_1(X) \to \pi\_1(C) \to 1$, where $\pi\_1(X)$ has cohomological dimension 3 (in the same coefficients) and $C$ is a curve of genus greater than 2.
So I want a kind of addit... | https://mathoverflow.net/users/12517 | Cohomological dimension of a group, fibration and local coefficients | It is false. The spectral sequence shows that cohomological dimension of group extensions is subadditive. It is not additive in general as every group is resolved by free groups, eg, $F\_\infty\to F\_3\to \mathbb Z^3$.
For your hypotheses, let $G=A\*B$ be the free product of a three dimensional group $A$, say, $\math... | 6 | https://mathoverflow.net/users/4639 | 53395 | 33,398 |
https://mathoverflow.net/questions/53373 | 4 | I am interested in questions of the following form: minimize $H(f)$ given $G(f) = 0$ where $H$ and $G$ are operators of type $X \to R$ where $X = R \to R$. An example is:
Minimize $$H(f) = \int\_{-1}^1\sqrt{1+f'(x)^2}$$
Under the conditions that:
$$G(f) = \int\_{-1}^1f(x) = \pi/2 $$
$$f(-1) = f(1) = 0$$
That is... | https://mathoverflow.net/users/6210 | The Frechet derivative and Lagrange multipliers on Banach spaces | **NOTE**: this was a comment, because I thought it wasn't detailed enough for an answer; but Jules (the OP) specifically asked me to post it as an answer.
**NOTE to Jules**: However, maybe you should wait a few hours or days before accepting any answer, to give others a chance to read it (differing time zones around... | 2 | https://mathoverflow.net/users/6651 | 53397 | 33,400 |
https://mathoverflow.net/questions/53380 | 3 | So I have been mulling the following question over in my head for awhile now, and want to see if anyone else might have any ideas.
Begin with $M$ a manifold and suppose that $M$ has an antipodal map $\alpha:M\rightarrow M$, i.e. $\forall m\in M$ one has:
$\alpha(m)\neq m$
$\alpha(\alpha(m)) = m$
Let $A^n$ be... | https://mathoverflow.net/users/12301 | When does an antipodal map on a manifold extend to the antipodal map on a spheres | Let me elaborate on my comment above. Suppose $M$ is a manifold equipped with a smooth $\Bbb Z\_2$ action that is also free. Then there is an equivariant smooth embedding $M \to S^j$, for some $j$, where we give the sphere the antipodal action. If we let $N$ be the orbit space of this action on $M$, then by Whithney's ... | 5 | https://mathoverflow.net/users/8032 | 53401 | 33,403 |
https://mathoverflow.net/questions/53384 | 29 | Let $f(x)=\sum \_{n=0}^{\infty } b\_nx^n$ and $\frac{1}{f(x)}=\sum \_{n=0}^{\infty } d\_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are:
$d\_0 = \frac{1}{b\_0}$,
$d\_1 = -\frac{b\_1}{b\_0^2}$,
$d\_2 = \frac{b\_1^2-b\_0 b\_2}{b\_0^3}$
$d\_3 = -\frac{b\_1^3-... | https://mathoverflow.net/users/9404 | power series of the reciprocal... does a recursive formula exist for the coefficients | Assume $b\_0=1$ to simplify things. You want a closed formula for the recursively defined sequence $$d\_0=1$$ $$d\_n=-\sum\_{k=0}^{n-1}d\_kb\_{n-k}. $$
Let $\alpha=(\alpha\_1,\dots,\alpha\_r)\in \mathbb{N}\_ +^\omega$ be a multi-index with length $l(\alpha):=r$ and weight $|\alpha|:=\sum\_{j=1}^r\alpha\_j$. Let's denot... | 18 | https://mathoverflow.net/users/6101 | 53402 | 33,404 |
https://mathoverflow.net/questions/52816 | 2 | A little stumped! This is probably a very basic probability question, but I am lost.
At work I was asked the probability of a user hitting an outage on the website. I have some following metrics. Total system downtime = 500,000 seconds a year. Total amount of seconds a year = 31,556,926 seconds. Thus, p of system dow... | https://mathoverflow.net/users/12397 | Probability calculation, system uptime, likelihood of occurence. | First of all, this is probably best asked at [stats.stackexchange.com](https://stats.stackexchange.com/)
Aside from that, your question has a few distinct parts to it.
The calculated probability of 0.0159 is strictly the probability of there being an outage in any single random second in a year. If we make the simp... | 2 | https://mathoverflow.net/users/12309 | 53403 | 33,405 |
https://mathoverflow.net/questions/53028 | 21 | Say that $G$ is a finite group, and $V$ is an irreducible representation of $G$, over an algebraically closed field $k$. Suppose that for every $g \in G$, there is some subspace $W\_g \subset V$ which is (pointwise) fixed by $g$, such that the dimension of $W\_g$ is at least half the dimension of $V$.
If $k$ has char... | https://mathoverflow.net/users/12087 | are irreducible representations with large fixed subspaces trivial? | Unless I have misread either your question or their result, Corollary 1.2 in "Average dimension of fixed point spaces with applications" by Bob Guralnick and Attila Mar\'oti includes a positive answer to the question. You can find this on the arxiv at
<http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.3836v1.pdf>
| 15 | https://mathoverflow.net/users/36466 | 53404 | 33,406 |
https://mathoverflow.net/questions/53399 | 50 | A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See [Are there two non-homotopy equivalent spaces with equal homotopy groups?](https://mathoverflow.net/questions/3540/are-there... | https://mathoverflow.net/users/5010 | Spaces with same homotopy and homology groups that are not homotopy equivalent? | Following up on John's comment, one can consider $S^2$-fibrations over $S^2$. There are two of them since such fibrations are classified by $\pi\_1(\textrm{Diff}^{+}(S^2))=\mathbb{Z}\_2$. One of them is $S^2\times S^2$ while the other can be shown to be the connected sum of $\mathbb{CP}^2$ and $\overline{\mathbb{CP}}^2... | 37 | https://mathoverflow.net/users/1993 | 53410 | 33,410 |
https://mathoverflow.net/questions/53364 | 10 |
>
> ... "and then the
> different branches of Arithmetic--
> **Ambition, Distraction, Uglification,**
> and **Derision.**"
>
>
> (*Alice in Wonderland*, chapter IX: the Mock Turtle's story)
>
>
>
As a child I wondered for a long time what was the exact meaning of the above partition of Arithmetic quoted in the... | https://mathoverflow.net/users/6101 | The different Branches of Arithmetic | Martin Gardner's *Annotated Alice: The Definitive Edition* says only this:
>
> Needless to say, all the Mock Turtle's subjects are puns (reading, writing, addition, subtraction, multiplication, division, history, geography, drawing, sketching, painting in oils, Latin, Greek). In fact, this chapter and the one to fo... | 7 | https://mathoverflow.net/users/3106 | 53413 | 33,411 |
https://mathoverflow.net/questions/53262 | 22 | Why is it so hard to implement Haken's Algorithm for recognizing whether a knot is unknotted? (Is there a computer implementation of this algorithm?)
| https://mathoverflow.net/users/1956 | Why is it so hard to implement Haken's Algorithm for knot theory? | Regarding Haken's algorithm: It's not so hard to implement (it's essentially implemented in Regina, though at present you need to type a few lines of python to glue the bits together; a single "big red button" is on its way). However, it's hard to *run*, since the algorithm has exponential running time (and, depending ... | 29 | https://mathoverflow.net/users/12531 | 53423 | 33,416 |
https://mathoverflow.net/questions/53414 | 5 | There are three exceptional Galois groups $L\_2(5)$, $L\_2(7)$ and $L\_2(11)$ . These are cited as one of [Arnold's "trinities"](http://www.neverendingbooks.org/index.php/arnolds-trinities.html) and are connected with other trinities and the McKay Correspondence.
Ramanujan studied partition numbers and found congruen... | https://mathoverflow.net/users/12530 | Is there a connection between exceptional Galois groups and Ramanujan's partition congruences | No.
(Moderator's note: I am adding the explanation of the above answer, as provided in comments given by the poster whose account was deleted long ago, to make this a proper answer.)
Philosophically, it's a little hard to prove that something is not a coincidence, so you should take my answer to mean that I under... | 15 | https://mathoverflow.net/users/nan | 53424 | 33,417 |
https://mathoverflow.net/questions/53420 | 9 | This question came to me while reading the discussion of [magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal](https://mathoverflow.net/questions/53352/) "magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal"
That questio... | https://mathoverflow.net/users/3684 | Magic square on an infinite lattice | Here are two elementary results about those magic square arrays.
Claim: The vector subspace of collections with finitely many nonzero entries has abasis consisting of the magic squares with $4$ nonzero entries arranged as
$$\begin{array}{cc}-1 & 1 \\\1 & -1\end{array}$$.
Proof: You can always add a multiple of s... | 9 | https://mathoverflow.net/users/2954 | 53425 | 33,418 |
https://mathoverflow.net/questions/53412 | 5 | Does anyone know of a modern proof of the First Minkowski Formula for a compact embedded hypersurface $\psi \colon \mathcal{M}^n \hookrightarrow \mathbb{R}^{n+1}$ ? The integral formula is
$$ \int\_{\mathcal{M}} H \langle \psi , \nu \rangle \mathrm{d}A +A = 0$$
where $A$ is the area of $\mathcal{M}$, $H$ is the mean c... | https://mathoverflow.net/users/11266 | First Minkowski Formula | Consider the following differential $(n-1)$-form $\omega$ on $M$: for $p\in M$ and $v\_1,\dots,v\_{n-1}\in T\_pM$, define
$$
\omega(v\_1,\dots,v\_{n-1}) = [ \psi(p), \nu(p), d\psi(v\_1),\dots,d\psi(v\_{n-1})]
$$
where the square brackets denote the standard volume form in $\mathbb R^{n+1}$ (in other words, the determi... | 9 | https://mathoverflow.net/users/4354 | 53426 | 33,419 |
https://mathoverflow.net/questions/53387 | 27 | This is, in a sense, a follow up to [this question](https://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively).
[Hehl and Obukhov](http://arxiv.org/abs/0711.1535) try to give an intuitive description of torsion. I am confused about their description. I am looking at the following p... | https://mathoverflow.net/users/297 | Rolling without slipping interpretation of torsion | "Rolling without slipping" is a powerful idea, but the phrase doesn't necessarily lead one to the intended mental model. In particular,
torsion is something that is at issue only for manifolds of dimension 3 or higher. Perhaps you can imagine taking a 3-manifold, and rolling it along a hyperplane in 4-space --- but
th... | 20 | https://mathoverflow.net/users/9062 | 53433 | 33,422 |
https://mathoverflow.net/questions/53220 | 1 | Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}\_{l}$ and ${\mathbb{Q}}\_{l'}$-adic etale cohomology for two (distinct) primes $l,l'\neq p$ (i.e. the Standard Conjecture D is fulfilled).... | https://mathoverflow.net/users/2191 | Could the Kunneth decomposition of a motif depend on the choice of $l$? | Let me develop YBL's answer a bit. (I wanted to make this a comment but it was too long...)
Consider a smooth variety $U$ over $\mathbb{F}\_p$ with function field $K$ such that your motive and its two Künneth decompositions extend over U. Take a $\mathbb{F}\_q$-rational point $x$ of $U$, and look at the specializatio... | 2 | https://mathoverflow.net/users/12336 | 53435 | 33,424 |
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