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https://mathoverflow.net/questions/63510 | 3 | Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r\_1, r\_2, ... r\_N) \in R$, but unknown vertex coordinates. We further specify that:
1. All vertices within a distance $d \leq T$ of one-another share an edge.
2. No vertices separated by a distance $d > T$ share an edge.
3. F... | https://mathoverflow.net/users/14324 | Is the following two-dimensional graph likely to be globally rigid? | I'm not sure how to make a real answer out of this, since you're interested in a situation where the edge lengths are given and not the positions of vertices. But -- *generic* global rigidity is a property just of the graph $G$ (generic global rigidity means global rigidity for any embedding of the graph where the posi... | 3 | https://mathoverflow.net/users/353 | 63543 | 39,280 |
https://mathoverflow.net/questions/61296 | 3 | For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,
$
\left\|u\right\|^p\_{W^\mu\_p(U)} = \left\|u\right\|^p\_{W^{\lfloor\mu\rfloor}\_p(U)} + \sum \int\_U \int\_U \frac{|D^\alpha u... | https://mathoverflow.net/users/2011 | Sobolev-Slobodeckij spaces for p=infinity | Yes, it is Hölder spaces and can be regarded as SS spaces for $p=\infty$. Actually, for natural $\mu$ more correct for functions analysis are Zygmund spaces (with differences of th second order in the definition). They are special cases of Besov spaces, which are defined for $0 < p\le \infty$, $\mu\in \mathbb R$. They ... | 2 | https://mathoverflow.net/users/14551 | 63547 | 39,281 |
https://mathoverflow.net/questions/63144 | 2 | Hello mathoverflow community !
I have a simple question that seems to have a non trivial answer.
Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar (rectangular) function $r(x)$
$$r(x)=\begin{cases}
1 & \mbox{if }0\leq x \leq 1; \\
0 & \mbox{elsewhere}
\end{cases}$$
I would... | https://mathoverflow.net/users/13373 | Decomposing a discrete signal into a sum of rectangle functions | Dear Raskolnikov, Brian and Sebastian,
thanks you very much for sharing your ideas.
I cross posted my question in two forums, so you can check the complete discussion by looking [here](https://math.stackexchange.com/questions/35388) and [here](https://mathoverflow.net/questions/63144).
From the discussion I got th... | 0 | https://mathoverflow.net/users/13373 | 63548 | 39,282 |
https://mathoverflow.net/questions/60725 | 2 | Dear community.
I would like to derive a "good" estimate on $\frac{d}{dt}f\_\epsilon(t)$, where $f\_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e.
$|f(x-\tau)+f(x+\tau)-2f(x)| \leq C |\tau|$ for all $x \in dom(f)$.
The regularization is defined as usual. We use an even function $\rho \in... | https://mathoverflow.net/users/10893 | Regularization of Zygmund functions | I don't think it is possible to make this estimate better. A proof can be done imho considering some simple function from Zygmund space, for example $f(x)=x\log|x|$, $x\in[-1,1]$. It has one point of non-smoothness and the derivative of the regularisation can be written out explicitly.
| 1 | https://mathoverflow.net/users/14551 | 63549 | 39,283 |
https://mathoverflow.net/questions/63544 | 19 | Let $M$ be the splitting field of
```
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
```
over the rationals. If I've understood some tables correctly, the splitting field is (of course) Galois over the rationals, with Galois group isomorphic to $SL(2,\mathbf{Z}/3\mathbf{Z})$.
How might I ... | https://mathoverflow.net/users/1384 | Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field. | Step I: Put the degree 24 polynomial into Magma, make it a number field, and call LSeries on it. This divides the $L$-function into a product of 7 distinct ones (Dokchitsers code, under an attribute called "prod" on the L-series object), given by Artin representations. So my plan was to compute zeros for each of these ... | 11 | https://mathoverflow.net/users/5267 | 63552 | 39,286 |
https://mathoverflow.net/questions/63545 | 8 | Is it true that an algebraically closed field $k$ definable (in the model theory sense) in a real closed field $\mathcal R$ is the algebraic closure $\mathcal{R}^{alg}=\mathcal{R}(\sqrt{-1})$?
UPDATE: if we prove that the definable set $K \subset \mathcal{R}^n$ that defines $k$ in $\mathcal R$ has the same cardinalit... | https://mathoverflow.net/users/2234 | an algebraically closed field definable in a real closed field | Suppose that $A$ is an infinite definable subset of a real closed field $R$ which is a zero-divisor-free ring under operations whose graphs are definable in $R$. Then $A$ is definably isomorphic to one of $R$, $R(\sqrt{-1})$ or the ring of quaternions over $R$. This is a special case of the main result of:
Otero, Pet... | 11 | https://mathoverflow.net/users/4706 | 63556 | 39,288 |
https://mathoverflow.net/questions/63266 | 4 | Suppose $N \subset M$ are two factors, neither of them Type I, acting on a separable Hilbert space $H$. Let $\pi\_1$ be a faithful normal representation of $N$ and $\pi\_2$ a faithful normal representation of $M'$. We can consider the von Neumann algebra $N \vee M'$. Suppose that $\pi = \pi\_1 \otimes \pi\_2$ is a fait... | https://mathoverflow.net/users/9545 | Spatial isomorphisms of tensor product of factors | Okay so I think I have an answer for existence of a spatial isomorphism in the case that the representation $\pi\_{1},\pi\_{2}$ are the identity representaion on $H.$ . Note that the condition $N\vee M'\cong N\overline{\otimes}M'$ spatially is independent of the way $M$ is represented. Indeed, suppose $M$ is represente... | 3 | https://mathoverflow.net/users/14771 | 63557 | 39,289 |
https://mathoverflow.net/questions/63553 | 2 | Hello!
I am reading an article in which there is the following statement:
Let $E\rightarrow X$ be a holomorphic vector bundle. The holomorphic sections of $E$ over a coordinate neighbourhood of $X$ are dense in the set of smooth sections of $E$.
I have some knowledge in complex geometry but I am not aware of this... | https://mathoverflow.net/users/14806 | Density of holomorphic sections | Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is false, already for the trivial bundle of rank one $E\_1=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any non-empty set $U \subset \mathbb C$ it is impossible to approximate the $C^{\infty}$ function $\bar z$ by holomorphic fu... | 5 | https://mathoverflow.net/users/450 | 63563 | 39,292 |
https://mathoverflow.net/questions/63568 | 6 | It is known that two Markov subshifts with the same entropy are "almost isomorphic" (up to a subset of measure 0) if the entropy is a logarithm of an integer (see R. L. Adler, L. W. Goodwyn, and B.Weiss. Equivalence of topological markov shifts.
Israel J. Math, 27(1):49--63, 1977). Is it true (known) if the entropy is ... | https://mathoverflow.net/users/nan | Subshifts with the same entropy | Look at Section 9 of
<http://www-users.math.umd.edu/users/mmb/papers/openfinalsub3nov2007.pdf>
| 5 | https://mathoverflow.net/users/13774 | 63571 | 39,298 |
https://mathoverflow.net/questions/63569 | 2 | Let $l$ be a prime number, $\mathbb{Z}\_l$ be the ring of $l$-adic integers, then what is the projective dimension of the ring $A:=\mathbb{Z}\_l[T,T^{-1}]$?
Is it two?
| https://mathoverflow.net/users/3848 | What is the projective dimension of the ring $\mathbb{Z}_l[T,T^{-1}]$ ? l is a prime number | $\mathbb Z\_p$ is a principal ideal domain, so it is Noetherian and its global dimension is $1$. Now, there is a general theorem that tells you that for all right Noetherian rings $R$ one has $$\operatorname{gldim}R[X,X^{-1}]=\operatorname{gldim}R+1.$$ So your answer is indeed $2$.
You'll find that theorem proved pre... | 10 | https://mathoverflow.net/users/1409 | 63575 | 39,301 |
https://mathoverflow.net/questions/63580 | 2 | Let, $G(k^{al})$ be an algebraic group, over an algebraically closed field, and $\Gamma\_{G}$ is the set of all closed subgroups of $G(k^{al})$.
Then is the map $Z\_{G}: \Gamma\_{G} \rightarrow \Gamma\_{G}$ which takes a closed subgroup to its centralizer in $G$, an involution? (probably not true)
If we now assume ... | https://mathoverflow.net/users/14812 | Is a double centralizer type theorem ( encountered in semisimple algebras) true for algebraic groups ? | When $G=\mathrm{GL}\_n$, then the centraliser $C$ of a subgroup scheme $H$ form the invertible elements of the algebra $M$ of matrices commuting with $H$. The group of such elements is Zariski dense in $M$ so $C$ and $M$ determine each other. Hence, the image of $Z\_{\mathrm{GL}\_n}$ and the question of involutivity on... | 3 | https://mathoverflow.net/users/4008 | 63584 | 39,304 |
https://mathoverflow.net/questions/63592 | 2 | Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?
| https://mathoverflow.net/users/2191 | Is every regular (excellent) scheme separated? | **-** Separated, excellent, regular: Spec$(k)$.
**-** Separated, excellent, not regular: Spec$(k[\epsilon]/\epsilon^2)$.
**-** Separated, not excellent, regular: See <http://en.wikipedia.org/wiki/Excellent_ring>
**-** Separated, not excellent, not regular: Spec$(k[\epsilon\_1,\epsilon\_2,\ldots]/\langle\epsilon\_... | 16 | https://mathoverflow.net/users/5513 | 63594 | 39,308 |
https://mathoverflow.net/questions/63595 | 2 | Let $G<\rm{GL}\_n(\mathbb{k})$ be a linear group, where $\mathbb{k}$ is an algebraically closed field. Assume that the linear action of $G$ on $\mathbb{k}^n$ is strongly-irreducible (i.e. there are no $H$-invariant proper subspaces of $\mathbb{k}^n$, except $0$, for any $H< G$ of finite index). Equip $\mathbb{k}^n$ wit... | https://mathoverflow.net/users/6227 | Existence of proper invariant subset in an irreducible action | Let $Q$ be a non-degenerate quadratic form on $\mathbb{K}^n$, and $G=O(Q)$ its orthogonal group. I think that the set $U$ of vectors $x\in\mathbb{K}^n$ with $Q(x)\neq 0$ does the job.
| 5 | https://mathoverflow.net/users/14497 | 63598 | 39,310 |
https://mathoverflow.net/questions/63529 | 19 | Is it true that the universal cover of $\mathrm{SL}\_2(\mathbb{R})$ has no non-trivial central extensions... as an *abstract* group?
(that's certainly true as a Lie group)
*Motivation:*
I have a projective action of $\mathrm{SL}\_2(\mathbb{R})$ on some Hilbert space $H$
and I'd like to know that it induces an h... | https://mathoverflow.net/users/5690 | Universal cover of SL2(R) admits no central extensions? | The answer should be negative, because the $K\_2$ of the reals is humongous.
That is, there are nontrivial central extensions. (Please do not ask a question and then explain its negation.)
Algebraic $K$ theory detects transcendentals. There is a Chern class map from $K\_2(\Bbb R)$
towards $\Omega^2\_{\Bbb R}$, where th... | 20 | https://mathoverflow.net/users/4794 | 63599 | 39,311 |
https://mathoverflow.net/questions/19577 | 14 | Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group $C^\*$ algebra $C\_r^\*(G)$ is the operator norm completion of $CG$. It's clear that $l^1(G)$ lies in $C\_r^\*(G)$.
... | https://mathoverflow.net/users/1546 | The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$ | Q1 seems to be related to symmetry of $\ell^1(G)$ (a Banach $^\*$-algebra $A$ is symmetric if, for every $a\in A$, the spectrum of $a^\*a$ is contained in $\mathbb{R}^+$). If $\ell^1(G)$ is not symmetric, then for an element $a^\*a$ with non-positive $\ell^1$-spectrum, since the $C^\*$-spectrum is clearly positive, the... | 6 | https://mathoverflow.net/users/14497 | 63622 | 39,324 |
https://mathoverflow.net/questions/63619 | 3 | Let $K$ be a finite field and $G$ be a discrete group.
>
> Is it true that for every $a=e+a\_1+\ldots+a\_n,b=e+b\_1+\ldots+b\_m\in K[G]$ with $b\_i\neq e,a\_j\neq e$ the condition $ab=0$ implies $ba=0$?
>
>
>
It is related to [this](https://mathoverflow.net/questions/58267/group-ring-and-left-zero-divisor) qu... | https://mathoverflow.net/users/8699 | Group ring and left zero divisor II | I hope I understand the question right: the $a\_i$ are to be distinct elements of $G$, as are the $b\_j$?
If so, then the answer is no. Let $K$ be $\mathbb Z/2$ and suppose that $x$ and $y$ are elements of $G$ such that $x^2=e$ and $xy$ is not equal to $yx$. Let $a=e+x$ and let $b=(e+x)(e+y)=e+x+y+xy$. Then $ab=0$ bu... | 6 | https://mathoverflow.net/users/6666 | 63623 | 39,325 |
https://mathoverflow.net/questions/63621 | 9 | Let $G$ be a semisimple linear algebraic group, $V$ a $G$-representation and $v \in V$ a vector of highest weight $\lambda$. Is it true, that for any positive root $\alpha \in R^+$ the one dimensional unipotent subgroup $U\_{-\alpha}$ acts trivially on $v$ if and only if $\langle \alpha^{\vee}, \lambda \rangle = 0$?
... | https://mathoverflow.net/users/14385 | Action on the highest weight vector of a representation of a semisimple linear algebraic group | For any irreducible representation $V$ with highest weight vector $v$ and highest weight $\lambda$, the stabilizer of $[v]\in \mathbb{P}V$ is the parabolic subgroup corresponding to those simple roots that are orthogonal to $\lambda$. By this I mean the parabolic generated by $\mathfrak{h}$, all positive root spaces, a... | 11 | https://mathoverflow.net/users/250 | 63625 | 39,327 |
https://mathoverflow.net/questions/63633 | 25 | (This question came up in a conversation with my professor last week.)
Let $\langle G,\cdot \rangle$ be a group. Let $x$ be an element of $G$.
Is there always an isomorphism $f : G \to G$ such that $f(x) = x^{-1}$ ?
What if $G$ is finite?
| https://mathoverflow.net/users/nan | element algebraically distinguishable from its inverse | The Mathieu group $M\_{11}$ does not have this property. A quote from Example 2.16 in [this paper](http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.3895v1.pdf): "Hence there is no automorphism of $M\_{11}$ that maps $x$ to $x^{−1}$."
Background how I found this quote as I am no group theorist: I used Google on "groups w... | 40 | https://mathoverflow.net/users/11919 | 63642 | 39,337 |
https://mathoverflow.net/questions/61584 | 9 | When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ is simply the isometry group for the hermitian form attached to $V$.
My question is: why does one take $F$ to be total... | https://mathoverflow.net/users/3186 | Unitary groups over number fields | If $E$ is not CM, then the action of complex conjugation on $E$ depends on how it is embedded into $\mathbb{C}$. In particular, it could have different real subfields depending on which embedding you are using. When $E$ is CM, so that it is a totally imaginary quadratic extension of the totally real subfield $F$, then ... | 13 | https://mathoverflow.net/users/434 | 63647 | 39,342 |
https://mathoverflow.net/questions/63602 | 3 | How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor of this sort 'homology' and denote it by $H\_i$, whereas a contravariant functor is called cohomology and is denoted by $H^... | https://mathoverflow.net/users/2191 | Homology or cohomology? | (CW because it's more an over-long comment than a real answer.)
I think there are too many competing normalizations to make a good choice. In lieu of sensible default, call one of them homology, and call the other cohomology and I'm sure it'll be fine. This is also probably why someone worked out the language "left-d... | 2 | https://mathoverflow.net/users/1631 | 63676 | 39,367 |
https://mathoverflow.net/questions/63677 | 2 | Maybe this is just a stupid question, but I have tried very hard and get no luck. Here is the question:
Let $S\_{k-1}$ be the unit sphere in $R^k$, i.e., the set of all $u\in R^k$ whose distance from the origin 0 is 1. So it is true that every $x\in R^k$, except for $x=0$, has a unique representation of the form $x=r... | https://mathoverflow.net/users/13905 | Polar Coordinates and Borel sets. | The point here is that $\phi^{-1}:(0,+\infty)\times S\_{k-1}\to\mathbb R^k\setminus\{0\}$
is actually continuous and thus Borel-measurable. Hence $\phi$ maps Borel sets to Borel sets.
| 4 | https://mathoverflow.net/users/7743 | 63678 | 39,368 |
https://mathoverflow.net/questions/59281 | 5 | I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy systems. I am wondering if anything similar can be found in the literature for any metric notion of dimension (Let say that by... | https://mathoverflow.net/users/4961 | Quantitative measurement of infinite dimensionality | After failing to find any evidence that the notions I asked for have been previously defined, I chose to write things down. The resulting paper is available: [A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces](http://www-fourier.ujf-grenoble.fr/~bkloeckn/papiers/largeness.pdf). Was... | 2 | https://mathoverflow.net/users/4961 | 63689 | 39,377 |
https://mathoverflow.net/questions/63690 | 6 | Let $X\subset\Pi\_1^0$ be the set of statements which are
provable in PA$+$Con(PA) but independent of PA.
Is $X$ recursively enumerable?
| https://mathoverflow.net/users/9833 | Are undecidable consequences of Con recursively enumerable? | The answer is no, and in particular, $X$ is $\Pi^0\_1$-hard. Let $\sigma(x)=\exists v\,\theta(x,v)$ be a complete $\Sigma^0\_1$-formula, where $\theta\in\Delta^0\_0$, and find a formula $\pi(x)$ such that PA proves
$$\pi(x)\leftrightarrow\forall w\,(\mathrm{Proof\_{PA}}(w,\ulcorner\pi(\dot x)\urcorner)\to\exists v\le... | 7 | https://mathoverflow.net/users/12705 | 63697 | 39,382 |
https://mathoverflow.net/questions/63499 | 3 | Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F\_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\mathscr{L}\in Pic(X)$ ${Frob\_X}\_\*\mathscr{L}\cong\bigoplus\_{s=1}^{p^{n}}\mathscr{L}\_s$, where $\mathscr{L}\_s\in Pic(... | https://mathoverflow.net/users/14252 | Ample line bundle and Frobenius morphism on smooth toric variety | I think the answer is "no". As I suggested in a now-deleted answer, we can restrict ourselves to considering the restriction to a torus invariant $\mathbb{P}^1$. A line bundle on a toric variety is determined by its restriction to each such $\mathbb{P}^1$, and is anti-ample if and only if its restriction to such a $\ma... | 3 | https://mathoverflow.net/users/297 | 63706 | 39,389 |
https://mathoverflow.net/questions/63724 | 1 | Although it's simply stated, this is neither a homework problem or trivial (I think, but I'd be happy to be proven wrong :) ).
Let $A,B$ be matrices and $x$ be a vector. Is it true that
$$ \|P\_{A+B} x\| \geq \|P\_A x\| - \|P\_B x\|, $$
where $P\_A$ is the projection onto the range space of $A$?
(or is it true if yo... | https://mathoverflow.net/users/2586 | Does this norm inequality hold for projections onto the range of a sum of matrices? | Let
$$A=\pmatrix{1&0\cr 0&0}, B=\pmatrix{0&0\cr 1&0}, x=(x\_1,x\_2).$$
Then $P\_Ax=(x\_1,0)$, $P\_Bx=(0,x\_2)$, $P\_{A+B}x=((x\_1+x\_2)/2,(x\_1+x\_2)/2)$.
Thus you are asking if
$$|(x\_1+x\_2)/\sqrt{2}|\ge |x\_1|-|x\_2|.$$
Clearly, this is false in general.
| 4 | https://mathoverflow.net/users/12120 | 63734 | 39,406 |
https://mathoverflow.net/questions/63675 | 7 | I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal subgroups. (In other words, $[H, G] \cong M\_4$.)
I have used GAP to search for such groups and, to my surprise, I could find ... | https://mathoverflow.net/users/9124 | What groups have a second maximal subgroup below exactly four maximal subgroups? | (Updated to reflect John Shareshian's excellent answer and suggestions.)
>
> **Yes**, there are infinitely many examples.
>
>
>
First off, some negative results: If $[G:H] ≤ 31$, then $G/Core(G,H)$ acting on H is one of your three examples. If H is contained in a core-free maximal subgroup M with $[G:M] ≤ 50$ ... | 7 | https://mathoverflow.net/users/3710 | 63737 | 39,408 |
https://mathoverflow.net/questions/63735 | 5 | Can all the large cardinal axioms not presently known to be
inconsistent with ZFC be arranged in a strict linear order based
on the following criterion. Let A(1) and A(2) be any distinct pair
of these axioms and let both be adjoined to ZFC. We will say that
$A(1)\lt A(2)$ if and only if the smallest cardinal number sat... | https://mathoverflow.net/users/4423 | A question about large cardinal axioms. | Garabed:
For many large cardinals, it is indeed true that the order we usually use (consistency strength over ZFC) coincides with the ordering you suggest: The least inaccessible is strictly weaker than the least Mahlo, which is strictly weaker than the least weakly compact, for example. But this ordering does not b... | 9 | https://mathoverflow.net/users/6085 | 63738 | 39,409 |
https://mathoverflow.net/questions/63745 | 14 | (partly motivated by this question, but different: [Degree of a Variety](https://mathoverflow.net/questions/63707/degree-of-a-variety))
For a hyperelliptic curve $C$ of genus $g$ (over an algebraically closed field of characteristic not two) what is the smallest $d$ for which $C$ can be embedded in some $\mathbb{P}^n... | https://mathoverflow.net/users/2290 | What is the minimal degree of a smooth projective embedding of a hyperelliptic curve? | Felipe, I believe the answer here is d=g+3. To see that you can embed your curve in this degree is straightforward - just choose a generic line bundle of degree g+3 and it will work.
In the case of hyperelliptic curves, I don't think you can do better. The key point is that any special linear series on a hyperellipti... | 12 | https://mathoverflow.net/users/14848 | 63752 | 39,416 |
https://mathoverflow.net/questions/63776 | 5 | If we assum the underlying scheme is Notherian and reduced.
| https://mathoverflow.net/users/14854 | Is the kernel of a morphism of locally free sheaves still a locally free sheaf? | Let $X$ be a smooth quasi-projective variety over a field of dimension $n$ and $Z\subseteq X$ a subvariety such that the depth of $\mathscr O\_Z$ at a fixed (closed) point of $X$ is $d$. Assume that $n-d\geq 3$, i.e., the projective dimension of $\mathscr O\_Z$ at that closed point of $X$ is at least $3$. Then consider... | 7 | https://mathoverflow.net/users/10076 | 63780 | 39,433 |
https://mathoverflow.net/questions/63775 | 6 | Let $S=Spec(R)$ where $R$ is a Henselian local ring with fraction field $K$. Let $G$ and $G'$ be finite, flat group schemes of odd order over $S$ with isomorphic generic fibers (over $Spec(K)$). Does this isomorphism extend to one over $S$?
When $R=\mathbb{Z}\_p$, the answer is yes following Fontaine's discussion in hi... | https://mathoverflow.net/users/13628 | On a Theorem of Fontaine | No. Here is a counterexample. Let $R=\mathbf{Z}\_p[\zeta\_p]$, where $p$ is a prime number and $\zeta\_p$ is a primitive $p$-th root of unity. Let $G=\mu\_p=\mathrm{Spec}(R[x]/(x^p-1))$ and let $G'$ be the constant group scheme $\mathbf{Z}/p\mathbf{Z}$. Then $G$ and $G'$ are not isomorphic because the special fiber of ... | 12 | https://mathoverflow.net/users/1114 | 63782 | 39,435 |
https://mathoverflow.net/questions/63760 | 9 | Is anything similar to the following inequality true,
>
> $\displaystyle P\{\max\_{n \leq k \leq m} |A\_k f - A\_n f| > \epsilon\} \leq C \frac{||A\_m f - A\_n f||\_1}{\epsilon}$
>
>
>
where $A\_n f = \frac{\sum\_{i=0}^{n-1} T^i \circ f}{n}$, and $T$ is a measure-preserving transformation?
My motivation for ... | https://mathoverflow.net/users/12978 | Is this ergodic inequality true? | I think the following should provide a counter example.
Take $T: [0,1] \to [0,1]$ given by $T(x) = x + \frac{p}{q} + \delta \ mod(1)$, where $\delta > 0$ is chosen small and so that $T$ is ergodic with respect to the Lebesgue measure.
Then $\|A\_q f - A\_{2 q} f\|\_1 = O(\delta)$. Furthermore
$$
\|A\_q f - A\_{q ... | 11 | https://mathoverflow.net/users/3983 | 63783 | 39,436 |
https://mathoverflow.net/questions/63784 | 1 | How can you prove the existence of a nonzero function from the subset $U= \{z| 0 \leq Re z \leq 1\}$ of $\mathbb C$ to $\mathbb C$ which is holomorphic on the interior of $U$ and vanishes on the right boundary of $U$ ?
| https://mathoverflow.net/users/3816 | Existence of a special holomorphic function | If you assume that your function is continuous on this right boundary (without that, your question should not make sense), then you can use a reflexion principle to extend its real part into a harmonic function $v$ in a neighbourhood of $z=1$. This harmonic function is the real part of a holomorphic function, thus your... | 7 | https://mathoverflow.net/users/8799 | 63785 | 39,437 |
https://mathoverflow.net/questions/62651 | 4 | The following question arose as I was playing around a little bit with pseudo-differential operators and K-theory and so on.
Let $H^s$ be the Sobolev space of s-times weakly differentiable functions $f \in L^2(R^n)$ with the usual inner product $\langle \cdot, \cdot \rangle\_s$. Note that $H^0 = L^2(R^n)$.
Let $B\_... | https://mathoverflow.net/users/13356 | Closure of all operator of order $0$ in $B(L^2(R^n))$ | I think I figured out the answer for the last question on myself. The closure of $B\_0$ can't be the whole set of all bounded operator.
This is due to the fact that the set $\Psi\_{-\infty}$ of all smoothing operator is a two-sided ideal in $B\_0$. After passing to the closures, $\bar \Psi\_{-\infty} \subset \bar B\_... | 0 | https://mathoverflow.net/users/13356 | 63790 | 39,439 |
https://mathoverflow.net/questions/63788 | 3 | Consider first-order logic with some fixed, relational vocabulary $\tau$. A sentence is a formula in this logic with no free variables.
A sentence is in [prenex normal form](http://en.wikipedia.org/wiki/Prenex_normal_form), if all quantifiers are moved to the front. For example $\exists x\exists y(P(x)\to P(y))$ is i... | https://mathoverflow.net/users/nan | Prenex normal form vs. quantifier rank | If you begin with $\psi \equiv (\exists x)R(x) \land (\exists x)(\lnot R(x)) \land (\forall y)S(y)$, the usual prenex form will have depth 3, while the original formula has depth 1. So you are asking how to find an equivalent formula to $\psi$ that only has one quantifier.
Let's assume our language only has two unar... | 12 | https://mathoverflow.net/users/5442 | 63794 | 39,442 |
https://mathoverflow.net/questions/63787 | 2 | **Simple question (but not for me):**
Does the Euler product formula diverge for any zero of the Riemann zeta function?
The reason why I ask this is that I heard we should not use the Euler product instead of the Riemann zeta function for Re(s)=<1 because it diverges on the critical strip, but I am not sure of that.
... | https://mathoverflow.net/users/14464 | Does the Euler product formula diverge for any zero of the Riemann zeta function? | I am in a hurry now but let me tell what I think. I believe that in the critical strip and off the real axis $\prod\_p (1-p^{-s})$ does not converge to any complex number (including zero). Using a similar idea as in my response to your related earlier question, this boils down to the fact that for any nonzero constants... | 7 | https://mathoverflow.net/users/11919 | 63806 | 39,447 |
https://mathoverflow.net/questions/63803 | 1 | Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond to the restriction of an analytic complex function?
| https://mathoverflow.net/users/14024 | Relation between partially computable function and complex function | Any complex-valued function on $\mathbb{N}$ can be extended to an entire function, so the answer is "yes." This follows from Theorem 15.13 of Rudin's Real and Complex Analysis, which states that for any open set $\Omega$ in $\mathbb{C}$ and subset $A$ of $\Omega$ without limit points, there exists a holomorphic functio... | 7 | https://mathoverflow.net/users/4351 | 63808 | 39,448 |
https://mathoverflow.net/questions/63809 | 0 | Is it possible to define explicitly a Lipschitz function $f:[a,b]\times[c,d]\rightarrow \mathbb{R}$ in term of $f(a,\cdot)$, $f(b,\cdot)$, $f(\cdot,c)$, $f(\cdot,d)$ if I know these functions and they are Lipschitz?
Thanks.
| https://mathoverflow.net/users/14862 | Lipschitz function | If $a=c=0$ and $b=d=1$, define $f$ by
affine interpolation between $f(x+y,0)$ and $f(0,x+y)$ if $x+y\leq 1$ respectively
$f(x+y-1,1)$ and $f(1,x+y-1)$ if $x+y\geq 1$.
The general case can be reduced to the previous one by an affine transformation.
| 4 | https://mathoverflow.net/users/4556 | 63813 | 39,450 |
https://mathoverflow.net/questions/63765 | 2 | Let $U$ be a unipotent upper triangluar group over a local field $K$ of characteristic
zero. Can we guarantee that there is a right translation invariant metric on $U$ such
that any ball of finite radius is relatively compact?
| https://mathoverflow.net/users/11056 | unipotent group and translation invariant metric | Let $G$ be a locally compact group. If $G$ is compactly generated, then word length with respect to a compact generating subset defines an invariant metric which is proper (i.e. closed balls are compact). The problem here is that a unipotent group $U$ is usually not compactly generated (if $K$ is non-archimedean). But ... | 3 | https://mathoverflow.net/users/14497 | 63817 | 39,451 |
https://mathoverflow.net/questions/63819 | 0 | Define two real numbers to be rationally equivalent provided their difference is a rational number.
from Royden *Real Analysis*
| https://mathoverflow.net/users/14864 | Find a explicit choice function of the "rationally equivalence class" | If there were a choice function for this equivalence relation, there would be a set of real numbers that is not Lebesgue measurable. But the existence of such a set is not provable in ZF (i.e., without the axiom of choice). Even with the axiom of choice, i.e., in ZFC, it's consistent that no such set is definable. (On ... | 7 | https://mathoverflow.net/users/6794 | 63820 | 39,453 |
https://mathoverflow.net/questions/63714 | 21 | It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
**My question:** Is the Euler product always divergent for
$0 < \Re(s) < 1$ ?
I thought that the absolute value of the Euler product formula is positively divergent under the above condition. Is ... | https://mathoverflow.net/users/14464 | Is the Euler product formula always divergent for 0<Re(s)<1? | Let
$$t\_P = \sum\_{p < P} \log \left| \frac{1}{1-p^{-s}} \right|$$
with $s=\sigma+it$, $\sigma \in (0,1)$ and $t$ a nonzero real.
The point of this answer is to show that the $t\_P$ jump around a great deal. Specifically, for any $M$ and $N$, there are $P$ and $Q$ with $N < P < Q$ such that $t\_Q - t\_P > M$, and oth... | 28 | https://mathoverflow.net/users/297 | 63823 | 39,454 |
https://mathoverflow.net/questions/63821 | 7 | Hello everybody,
I'm searching for references for the following independence assertions:
ZFC + $MA\_{\aleph\_{1}}$ $\not\vdash$ "Analytic determinacy"
ZFC + $MA\_{\aleph\_{1}}$ $\not\vdash$ $\neg$ ("Analytic determinacy")
i.e. $MA\_{\aleph\_{1}}$ does not settle any determinacy question. The question extends al... | https://mathoverflow.net/users/11618 | Martin's Axiom and Determinacy-axioms: independence results | ZFC plus $\text{MA}\_{\aleph\_1}$ is consistent relative to ZFC, while analytic determinacy has a little bit of large cardinal strength, namely the existence of sharps of reals. So ZFC+MA cannot prove analytic determinacy. On the other hand, analytic determinacy follows from the existence of a measurable cardinal, and ... | 15 | https://mathoverflow.net/users/6794 | 63825 | 39,456 |
https://mathoverflow.net/questions/63807 | 23 | A handle decomposition of a manifold $M$ is a useful structure to carry around. It is induced by a [Morse function](http://en.wikipedia.org/wiki/Morse_theory) $f\colon\, M\to \mathbb{R}$.
How are two handle decompositions of $M$ related? The space of Morse functions turns out not to be connected. But if one expa... | https://mathoverflow.net/users/2051 | How does the Framed Function Theorem simplify Cerf Theory? | The framed function theorem tells you that up to "contractible choice" a compact manifold admits a framed function: i.e., a function as you prescribe. Furthermore, a framed function is supposed to give you a cell structure on the manifold up to contractible choice.
**Note:** The framing data is there to give you an e... | 11 | https://mathoverflow.net/users/8032 | 63837 | 39,462 |
https://mathoverflow.net/questions/63836 | 1 | During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of you.
My question: is there someone who can point out to me a counterexample for the following implication?
This is t... | https://mathoverflow.net/users/12617 | About the geometry of completely integrable systems | ### preliminary remark
I assume that being independant for functions here means that their differentials at **any** point are linearly indenpendant (and not at **almost** any point, like it is sometimes assumed... in which case a counter-example is easy to find for $n=2$: take $f\_1$ to be the square of the norm on $... | 2 | https://mathoverflow.net/users/7031 | 63838 | 39,463 |
https://mathoverflow.net/questions/63830 | 2 | The Dedekind sum [$s(p,q)$](http://en.wikipedia.org/wiki/Dedekind_sum) can be both positive and negative. What are the known lower/upper bounds in terms of p,q? (I would prefer something that grows not faster than q)
| https://mathoverflow.net/users/2900 | On lower/upper bounds for Dedekind sum | For a fixed $q$, the maximum is $$s(1,q)=-{1\over4}+{1\over6q}+{q\over12}$$ and the minimum is $s(q-1,q)=-s(1,q)$.
| 6 | https://mathoverflow.net/users/3684 | 63866 | 39,480 |
https://mathoverflow.net/questions/63862 | 7 | A friend of mine recently explained to me a little bit about using Lie groups and symmetries to obtain solutions of PDEs. I was interested and wanted to learn a bit more about it. He's been using Olver's "Applications of Lie Groups to Differential Equations" but I found it a bit out of my reach.
I've taken a PDE cour... | https://mathoverflow.net/users/2544 | Lie Groups and PDEs | I found a solid background in PDE, together with some physics, to be a useful entry point to Olver's nice book. There's the 'Lectures on Partial Differential Equations' by V.I.Arnold which is fun to read alongside, if not before. Any solid book on mathematical methods in classical mechanics and quantum mechanics should... | 7 | https://mathoverflow.net/users/14740 | 63867 | 39,481 |
https://mathoverflow.net/questions/63882 | 4 | Hi
Both, the greedy and the LP approach for Set Cover give a O(log n) approximation. Is there some inherent difference on the two approximation approaches?
thanks
| https://mathoverflow.net/users/44243 | Set Cover:Greedy vs LP | Both greedy and LP are asymptotically optimal approximations. According to a well known result by Ran and Shmuel, getting a better than c log(n) approximation of set cover is equivalent to showing that P = NP. Here is a reference:
Ran, Shmuel (1997), "A sub-constant error-probability low-degree test, and a sub-consta... | 4 | https://mathoverflow.net/users/4642 | 63889 | 39,490 |
https://mathoverflow.net/questions/63868 | 17 | I am gathering material for an exposition and I note that some texts (e.g. Ise and Takeuchi, "Lie Groups I & II", Stillwell, "Naive Lie Theory", Hall, "Lie Groups, Lie Algebras, and Representations") define "Matrix Lie Groups" with the unwonted requirement that the group should be a closed subgroup of $GL\left(V\right)... | https://mathoverflow.net/users/14510 | Is every Lie subgroup of GL(V) isomorphic to a (maybe another) closed subgroup of GL(V)? | Any linear Lie group is Lie-isomorphic to a closed subgroup of $\operatorname{GL}(V)$: that's a result of Morikuni Goto: Faithful representations of Lie groups. II. Nagoya Math. J. 1, (1950). 91–107. From the review in MR: "A Lie group $G$ is called faithfully representable (f.r.) if there exists a topological isomorph... | 22 | https://mathoverflow.net/users/14497 | 63890 | 39,491 |
https://mathoverflow.net/questions/63898 | 3 | Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated means not belonging to a continuous family of conics in the surface.)
A description similar to the one for isolated lines... | https://mathoverflow.net/users/14639 | Isolated conics on a del Pezzo surface | While the number of lines on Del Pezzo surfaces are finite, the number of conics is infinite. More precisely, there are finitely many families $X\to P^1$ whose fibers are plane conics. Let me explain this in more detail.
As you probably know, a degree $d$ Del Pezzo surface $X$ can be realized as the blow-up of $P^2$ ... | 6 | https://mathoverflow.net/users/3996 | 63900 | 39,494 |
https://mathoverflow.net/questions/63902 | 1 | I'm sure you all are familiar with Theorem 5.3 from Sipser's TOC book:
S = "On input (M,w) where M is a TM and w is a string:
1. Construct the code of TM M2 as follows:
M2 = "On input x:
(a) If x = 0n1n for some n ≥ 0, accept.
(b) If x = 0n1n, run M on w and if M accepts w, then accept."
2. Run R on (M2).
3. If R acc... | https://mathoverflow.net/users/14878 | REGULAR TM is undecidable | First of all, the undecidability of REGULAR\_TM follows immediately from Rice's theorem.
For a direct proof, Sipser gives a reduction from the language A\_TM. He constructs a decider R for A\_TM out of a decider for REGULAR\_TM, as follows.
R inputs the pair , where M is a TM and w is a string.
It constructs the TM... | 5 | https://mathoverflow.net/users/12518 | 63904 | 39,497 |
https://mathoverflow.net/questions/63909 | 9 | Let $X$ be a smooth projective variety of dimension $\geq 2$ and $E$ a vector bundle on $X$ of rank $r\geq 2$. Is it true that, if $E$ is globally generated, then the zero locus of a general section of $E$ has codimension $\geq r$ in $X$?
| https://mathoverflow.net/users/33841 | Vanishing locus of a general section of a vector bundle. | As Donu pointed out, it may happen that the general section has no zeroes. Anyway, the the answer to your question is *yes*.
This is a special case of the following more general result "of Bertini type" about degeneracy loci of morphism of vector bundles, whose proof can be found in Ottaviani's book *Varietà proietti... | 11 | https://mathoverflow.net/users/7460 | 63911 | 39,502 |
https://mathoverflow.net/questions/63887 | 12 | It is well known how the intended model and [how the (countable) non-standard models of arithmetic look like](http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic#Structure_of_countable_non-standard_models).
It's also well known how the intended model of set theory with the axiom of infinity replaced by its ... | https://mathoverflow.net/users/2672 | Non-standard models of finite set theory | Models of ZF-Infinity that arise from models of PA via binary bits - a method first introduced by Ackermann in 1940 to interpret set theory in arithmetic- end up satisfying the statement TC := "every set has a transitive closure".
It is known that the strengthened theory ZF-Infinity+TC is bi-interpretable with PA, w... | 20 | https://mathoverflow.net/users/9269 | 63918 | 39,504 |
https://mathoverflow.net/questions/63916 | 1 | Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D\_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$
$\int\_0^T \sup\_M|D\_0^m \frac{\partial}{\partial t} g(t)|\_{g(0)} dt< \infty$, then why $g(t)$ converges in $C^{\i... | https://mathoverflow.net/users/14881 | convergence of metrics | Sounds like a home work problem?
Note that
$$g(T)=\lim\_{t\to T-}g(t)=g(0)+\int\limits\_0^T\tfrac{\partial}{\partial t}g$$
Then you get
$$|D\_0^m g(T)|\le \mathrm{Const}(m)$$
and
$$\sup\_M|D\_0^m[g(T)-g(t\_0)]|=\sup\_M\int\limits\_{t\_0}^TD\_0^m[\tfrac{\partial}{\partial t}g]\\,dt\to 0\ \ \text{as}\ \ t\_0\to T-.$$... | 2 | https://mathoverflow.net/users/1441 | 63929 | 39,509 |
https://mathoverflow.net/questions/63922 | 0 | Having had no (proper) answer to [*this question*](https://mathoverflow.net/questions/57813/stability-of-convex-sets-w-r-t-integration-over-0-1), I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be continuous. *Do there* the... | https://mathoverflow.net/users/12643 | Integral in a σ−convex set. | Counterexample. $E = L^2[0,1]$ and $\gamma \colon [0,1] \to E$ defined by $\gamma(x) = 1\_{[0,x]}$, the characteristic function of interval $[0,x]$. Then $\gamma$ is continuous, in fact $\|\gamma(x) - \gamma(y)\| = \sqrt{|y-x|}$.
Now suppose $c\_i$ and $t\_i$ are as given. Let $$u :=
E\ \text{-}\ \lim\_{\ k\to\infty\ }... | 4 | https://mathoverflow.net/users/454 | 63944 | 39,517 |
https://mathoverflow.net/questions/63943 | 15 | Vogel assigns to every simple metric Lie algebra (and more generally to every simple metric Lie algebra object in a symmetric monoidal category) a point in the orbifold $\mathbb{P}^2/S\_3$ (where $S\_3$ acts by permuting the 3 projective coordinates) based on the value of the Casimir on the various summands of the symm... | https://mathoverflow.net/users/22 | Why do sl(2) and so(3) correspond to different points on the Vogel plane? | I take it you mean $\mathfrak{sl}\_2$ is on the line $\mathfrak{sl}\_n$ and $\mathfrak{so}\_3$ is on the line $\mathfrak{so}\_n$. There is no contradiction because there is a whole line for this metric Lie algebra. The symmetric square of the adjoint decomposes as the trivial representation (which is accounted for by t... | 6 | https://mathoverflow.net/users/3992 | 63946 | 39,519 |
https://mathoverflow.net/questions/49894 | 9 | **Question.** Is there a *concrete* example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to Bernard Beauzamy ([*Introduction to Operator Theory and Invariant Subspaces*](http://books.google.com.ua/books?id=u4e... | https://mathoverflow.net/users/5371 | The Invariant Subspace Problem: examples | It seems likely that the author of the question has found the reference in the meantime. I will provide it here for the sake of completeness.
The article containing the construction of the operator described in
Bernard Beauzamy (Introduction to Operator Theory and Invariant Subspaces, Elsevier (1988), p. 345
can be ... | 5 | https://mathoverflow.net/users/14849 | 63955 | 39,523 |
https://mathoverflow.net/questions/63952 | 1 | Consider an enumeration $\{q\_1,q\_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e\_1,e\_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define
$Ae\_{2k-1}=e\_{2k-1}$ and $Ae\_{2k}=q\_ke\_{2k}$ for all $k\geq 1$.
**Question 1:** Is it possible to extend $A$ to a linear self-adjoint operator ... | https://mathoverflow.net/users/2386 | Can be this operator extended to an unbounded self-adjoint operator ? | The spectral theorem for unbounded self-adjoint operators says the following:
Up to isomorphism, any unbounded self-adjoint operator $A$ on a Hilbert space $H$ can be written in the following form:
$$H=L^2(X,\mu)$$
$$Af(x)=a(x)f(x)$$
for some measure space $(X,\mu)$ and some $\mu$-measurable real valued functio... | 4 | https://mathoverflow.net/users/5690 | 63956 | 39,524 |
https://mathoverflow.net/questions/63924 | 3 | Let $p$ and $q$ be integers. The group $K=SO(p) \times SO(q)$ can be naturally seen as a subgroup of $G=SO(p+q)$. The quotient space $G/K$ is identified with the space of oriented $p$-dimensional subspaces of $\mathbb R^{p+q}$, and carries an action of $G$ (by left multiplication).
A spherical function is a $C^\infty... | https://mathoverflow.net/users/10265 | Are there explicit formulas for spherical functions on oriented real grassmannians? | In the unoriented case, the spherical functions are computed in the following article (they turn out to be multivariate Jacobi polynomials):
A. T. James and A. G. Constantine, *Generalized Jacobi polynomials as spherical functions of the Grassmann manifold*, Proc. London Math. Soc. (3) 29 (1974), 174-192. <http://dx.... | 5 | https://mathoverflow.net/users/4720 | 63959 | 39,527 |
https://mathoverflow.net/questions/63939 | 8 | Where can i find material about the definition of the exponential morphism from the Lie algebra of an algebraic affine group to the group?
| https://mathoverflow.net/users/7614 | Lie algebras of algebraic groups | In the classical theory of Lie groups and Lie algebras, the exponential map defined in terms of the usual power series is a standard tool for passing from the Lie algebra to the group. This makes sense for matrix groups over the real and complex fields because the series converges when evaluated at a square matrix, etc... | 21 | https://mathoverflow.net/users/4231 | 63960 | 39,528 |
https://mathoverflow.net/questions/63967 | 9 | Let $f(n)=a\_3n^3+a\_2n^2+a\_1n+a\_0$, with $a\_i\in\mathbb{Z}$, $a\_3>0, a\_0\neq 0$ such that $f(n)>0$ for all positive integers $n$.
Given a prime $p$, when is $f(p)$ again prime?
For example, let $f(n)=7n^3-50n+30$. Then,
$$f(7)=2081\quad {\rm (prime)},$$
$$f(11)=19\cdot463,$$
$$f(13)=14759\quad {\rm (prime)}.$... | https://mathoverflow.net/users/4544 | Cubic polynomial mapping primes to primes | There is no non-constant polynomial sending primes to primes, aside from $f(x)=x$. Indeed, it suffices to consider the case where $f$ is irreducible, as if $f(x)$ factors as $g(x)h(x)$, $f(p)$ is clearly composite for large $p$.
Now if $f(x)\not=x$, choose some large prime $p$ such that $f$ has a non-zero root $a$ in... | 18 | https://mathoverflow.net/users/6950 | 63968 | 39,530 |
https://mathoverflow.net/questions/63974 | 13 | Let $E$ be a spectrum. For any CW complex $X$, define $h\_\*=\pi\_i(E\wedge X)$. Then we know that $h\_\*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a cofiber sequence of spaces to a long exact sequence of abelian groups, also satisfy the wedge axiom in the definition ... | https://mathoverflow.net/users/1546 | Is every homology theory given by a spectrum? | For homology theories on CW-complexes or homology theories that map weak equivalences to isomorphisms, that's Brown's representability theorem, which you can find in any textbook on stable homotopy theory. You forgot the important axiom of excision, by the way. The short answer is yes.
| 19 | https://mathoverflow.net/users/4183 | 63975 | 39,533 |
https://mathoverflow.net/questions/63950 | 15 | In section 3.2 of Kontsevich's very interesting paper ["Notes on motives in finite characteristic,"](http://arxiv.org/PS_cache/math/pdf/0702/0702206v2.pdf), he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum $(V\_1,V\_2,R)$, where $V\_1$ and $V\_2$ are vector spaces and $R$ is a linear ... | https://mathoverflow.net/users/431 | How is the Ising model an example of a lattice model as per Kontsevich? | In this section, Kontsevich is describing a pattern of index contraction of a certain tensor which reproduces a statistical mechanics sum over configurations. For his model, you have at each vertex, which I will label by the integer $k$, a copy of the same four index tensor
$$R\_{a\mu}^{b\nu}$$
where $a$ and $b$ ar... | 15 | https://mathoverflow.net/users/14689 | 63979 | 39,536 |
https://mathoverflow.net/questions/63982 | 5 | Is it true that a compact group always has a faithful, finite-dimensional unitary representation? If not, are there any reasonably simple counter-examples?
I've done some research and know that every group has *some* faithful representation, all irreducible reps of a compact group are finite, and that the irreducible... | https://mathoverflow.net/users/14891 | Finite-dimensional faithful representations of compact groups | A famous theorem is that this is true if and only if $G$ is a Lie group.
| 13 | https://mathoverflow.net/users/7392 | 63985 | 39,540 |
https://mathoverflow.net/questions/60512 | 7 | Is there any software which can automatically visualise a non-algebraic
complex curve, I mean the structure of it's ramification points and sheet?
I think a good test example would be the Lambert curve $y\exp y =x$
(what I really need is a bit more complicated family of the curves).
| https://mathoverflow.net/users/3840 | Non-algebraic curve visualisation | I have found a good and simple paper [Graphing Elementary Riemann Surfaces](http://www.apmaths.uwo.ca/~djeffrey/Offprints/riemann.pdf) by Robert M. Corless and David J. Jeffrey with an explanation how to use Maple for graphing Riemann surfaces. In particular the Lambert curve is amnong the examples they consider.
| 3 | https://mathoverflow.net/users/3840 | 63987 | 39,541 |
https://mathoverflow.net/questions/63986 | 15 | Olivier Wittenberg and I are curious about the following :
Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can $S$ contain integral curves with arbitrarily high negative self-intersection ?
| https://mathoverflow.net/users/2868 | Surfaces containing curves of arbitrarily negative self-intersection | It is a folklore conjecture that surfaces in characteristic zero has bounded negativity. For a nice account of this problem and references, see the two survey articles
[Global aspects of the geometry of surfaces](http://arxiv.org/abs/0907.4151%20) by Harbourne, and
[Recent developments and open problems in linear s... | 17 | https://mathoverflow.net/users/3996 | 63992 | 39,544 |
https://mathoverflow.net/questions/63993 | 2 | The transitive closure of a set $X$ is usually seen as a set, but it can also be seen as a graph $G(X)$ with $V(G)= TC({X})$ and $(x,y)\in E(G)$ iff $x\in y$. Such a (transitive closure) graph reveals irredundantly everything that is to know about the set ("*its hidden $\in$-structure*"). It is known that $G(X)\simeq G... | https://mathoverflow.net/users/2672 | Characterization of transitive closure graphs | There are a few problems with what you wrote.
First, you probably want $TC(\{X\})$ rather than $TC(X)$, since you want $X$ to be an element, not just a subset, since it is the node corresponding to $X$ that has no out-arrows (but I see that you have now corrected this). Otherwise, a counterexample will arise from th... | 9 | https://mathoverflow.net/users/1946 | 63995 | 39,545 |
https://mathoverflow.net/questions/63994 | 6 | In computing the etale cohomology of curves, one of the key facts one needs is the torsion in $Pic(X) = H^1(X\_{et}, \mathcal{O}\_X^\*)$ for a smooth projective curve $X$. Namely, one shows that the $n$-torsion (for $n$ prime to the characteristic, at least) is given by $(\mathbb{Z}/n\mathbb{Z})^{2g}$ where $g$ is the ... | https://mathoverflow.net/users/344 | An elementary description of the torsion in $Pic(X)$ for a smooth projective curve $X$ | As far as I know, the standard computation of the torsion in $Pic(X$) proceeds as follows:
one considers the exact sequence $0 \to Pic^0(X) \to Pic(X) \to \mathbb Z \to 0,$
the map to $\mathbb Z$ being the degree map. This shows that the torsion subgroups
of $Pic(X)$ and $Pic^0(X)$ coincide. One then shows that $Pic^0(... | 4 | https://mathoverflow.net/users/2874 | 63997 | 39,547 |
https://mathoverflow.net/questions/63931 | 2 | Hello,
consider a parabolic boundary value problem, for instance
$-\partial\_tu+\Delta u=0$, in $\Omega$,
$\partial\_\nu u=0$ on $\partial\Omega$,
in a domain $Q=(0,T)\times\Omega$, where $\Omega\subset R^n$ is bounded. Now suppose $u$ is smooth and attains its maximum in $(T,x\_0)$ for some $x\_0\in\partial\Om... | https://mathoverflow.net/users/11291 | Maximum principle corner | Pick 0< $\tau$ < T, and let
$$\phi(\tau)=\max\_{x\in\bar\Omega,0\le t\le\tau} u(x,t).$$
If this maximum is assumed at any point where $x\in\Omega$ and $t>0$, then $u$ must be constant. Suppose the maximum is at a point $(x\_0,\tau)$, where $x\_0\in\partial\Omega$, and $u$ is not constant. Then the function $\phi(\tau)$... | 1 | https://mathoverflow.net/users/12120 | 64005 | 39,551 |
https://mathoverflow.net/questions/64009 | 3 | Let $\mathcal{A}$ be a category whose collection of object forms a proper class. Then, to be able to formulate the concept of a terminal object in $\mathcal{A}$, do we have to leave ZFC? In other words, can we formulate the concept of a terminal object in $\mathcal{A}$ inside ZFC. If yes, how could we do it?
For exam... | https://mathoverflow.net/users/11398 | Universal Objects in Big Categories | In ZFC, a class is given by a class formula $\phi$, that is, a formula expressed in the language of ZFC. For example, the notion of group can be formalized by such a formula $\phi$. The notion of group homomorphism can also be so formalized, say by $\psi$. So in principle there is no problem in being able to write down... | 10 | https://mathoverflow.net/users/2926 | 64013 | 39,554 |
https://mathoverflow.net/questions/60412 | 2 | Hi everyone,
I've got a question about explicitly lifting regular sequences. Let $I$ be an ideal in a polynomial ring $S$ with some term order. We'll denote the initial ideal by $in(I)$. It is false in general that a regular sequence on $S/in(I)$ is regular on $S/I$. For example consider $I=(x+y)$, with $x>y$ Then $... | https://mathoverflow.net/users/14098 | Generic liftings of a regular sequence on the initial ideal | Hi Adam,
What you are asking is true. In fact it is actually possible to "lift" not only regular sequences but also filter regular sequences. The proof is a simple modification of the usual argument that shows the inequality between the graded Betti numbers of $I$ and the ones of $in(I).$
Say that $in(I)=in\_w(I)$ f... | 3 | https://mathoverflow.net/users/14896 | 64015 | 39,555 |
https://mathoverflow.net/questions/64017 | 9 | I am interested in the first non zero eigenvalue of the Laplace-Beltrami operator in a 2D compact manifold, and if there is a geometric characterization of its value.
I am interested in the case when you fix the volume of the manifold to some value (say $Vol = 1$), and let the other modes of the metric fluctuate. The... | https://mathoverflow.net/users/14840 | Eigenvalues of Laplacian-Beltrami operator | The first eigenvalue of a compact surface can be made arbitrarily small (even for surfaces of fixed genus); see, for example, [1], [2], [3] (and references therefrom).
However, as proved by Sarnak and Xue [4], there are arithmetic examples of (constant negative curvature) compact Riemann surfaces of arbitrarily high ... | 9 | https://mathoverflow.net/users/14564 | 64021 | 39,557 |
https://mathoverflow.net/questions/64029 | 21 | I have a topological manifold whose suspension is homeomorphic to the sphere $S^{k+1}$. Is it necessarily itself homeomorphic to $S^k$?
I know that this is not true if I replace "suspension" with "double suspension", because I found the helpfully named <http://en.wikipedia.org/wiki/Double_suspension_theorem>.
| https://mathoverflow.net/users/14901 | If a manifold suspends to a sphere... | Suppose $M$ is a closed $n$-manifold whose suspension is homeomorphic to $S^{n+1}$.
Removing the two "singular" points from the suspension gives $M\times \mathbb R$, while
removing two points from $S^{n+1}$ gives $S^n\times\mathbb R$. Thus $M\times \mathbb R$ and $S^n\times\mathbb R$ are homeomorphic, which easily impl... | 35 | https://mathoverflow.net/users/1573 | 64036 | 39,567 |
https://mathoverflow.net/questions/64016 | 3 | An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface.
Different observers in space-time have different time slices. Is there an observer that observes a *closed* trefoil, instead of an open one?
This issue has 2 subca... | https://mathoverflow.net/users/14860 | Can a closed trefoil appear as a space-time "cut" of an open trefoil? | No, a closed trefoil cannot appear, because it's not algebraically slice.
In more detail, I understand that in the usual Minkowski space, a "time slice" $T$ would be homeomorphic to $\Bbb R^3$. In case there are problems with this in a more general situation, I'm reading the question so that $T$ is implicitly assume... | 5 | https://mathoverflow.net/users/10819 | 64047 | 39,574 |
https://mathoverflow.net/questions/63826 | 4 | I am currently stuck at a problem which seems too easy to be stuck at to me...
### Summary
Let $H$ be the convex hull of the points $d\_1,\ldots, d\_n\in \mathbb{R}^d$. How can one compute
\[\min\_{x\in H}||x||^2\_2 \]
efficiently?
### Conditions one should know about
When I talk about efficiency in this quest... | https://mathoverflow.net/users/14865 | Minimum norm of convex hull | Dear all,
I think I found quite a neat solution for my problem. At first I want to give humble thanks to Roland, who greatly inspired the solution I'm going to use now. Just in case somebody else might sometimes in the future be struggling with a similar problem, I'm going to give a short outline of the ideas and a r... | 0 | https://mathoverflow.net/users/14865 | 64055 | 39,578 |
https://mathoverflow.net/questions/59857 | 1 | I would be very astonished if this algebra isn't named.
You simply have the braid AND the Temperley-Lieb generator in
the algebra. Rules are the usual Reidemeister equivalents
plus the kink and whirl move equivalents (2nd question -
I call them that way, but are there established names?):
Sn braid generator
Hn... | https://mathoverflow.net/users/11504 | Braid*Temperley-Lieb=? | This approach where you restrict your attention to algebras consisting only of the kinds of diagrams where there's the same number of strands coming in as coming out is a little bit out of fashion (its heyday was mostly the 80's and early 90's) and so people have mostly stopped naming them. The more typical thing to do... | 3 | https://mathoverflow.net/users/22 | 64062 | 39,583 |
https://mathoverflow.net/questions/63525 | 40 | I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z\_1,z\_2,…,z\_n$ be i.i.d random points on the unit circle ($|z\_i|=1$) with uniform distribution on the unit circle. Consider the random polynomial $P(z)$ given by
$$
P(z)=\prod\_{i=1}^{n}(z−z\_i)... | https://mathoverflow.net/users/13825 | Polynomials on the Unit Circle | Here is a more careful (EDIT: even more careful!) argument that gives an affirmative answer to the weaker version of the question (as stated in the edit to my previous post, I doubt that the stronger version is true).
The argument uses the following lemma, which ought to be known. If someone has a reference, please l... | 9 | https://mathoverflow.net/users/14302 | 64063 | 39,584 |
https://mathoverflow.net/questions/64064 | 9 | A $G$-structure $\pi : B\_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie algebra $\mathfrak{g} = T\_{e}G$. For finite type $G$-structures, let us call the first $k$ for which $\mathfrak{g}^{(... | https://mathoverflow.net/users/14147 | $G$-structures of finite type. | Yes, $G$-structures exist of each finite order. In other words, for every $k\ge1$, there is an $n\ge1$ and a subgroup $G\subset GL(n,\mathbb{R})$ such that its Lie algebra $\frak{g}$ satisfies ${\frak{g}}^{(k-1)}\not=0$ while ${\frak{g}}^{(k)}=0$.
There is no known classification of such algebras, but here is a simp... | 15 | https://mathoverflow.net/users/13972 | 64067 | 39,586 |
https://mathoverflow.net/questions/64050 | 1 | It is well-known that up to homeomorphism, the complete set of orientable surfaces is $\lbrace S\_g : g=0,1,\dots \rbrace$, where $S\_g$ is the sphere with $g$ handles. The complete set of non-orientable surfaces is $\lbrace N\_k : k=1,2, \dots \rbrace$, where $N\_k$ is the sphere with $k$ crosscaps.
Typically the *... | https://mathoverflow.net/users/2233 | What is the quantity 2(handles)+crosscaps called? | You could certainly call it the [Betti number](http://en.wikipedia.org/wiki/Betti_number) $b\_1$ without bringing in homology.
| 9 | https://mathoverflow.net/users/290 | 64074 | 39,589 |
https://mathoverflow.net/questions/64071 | 39 | Just exactly what the title says; often, in mathematics, particularly in the vicinity of Grothendieck, I see reference to "the yoga of...". What exactly does the term "yoga" mean in these contexts?
| https://mathoverflow.net/users/3902 | What does the term "yoga" mean in mathematics? | I've taken "yoga" to mean a part of the body of mathematics which does not consist of many actual theorems or results -- or in fact could not be formalized as just a few theorems -- but rather a collection of principles and techniques that one needs to wrap one's head around completely, after which one will be able to ... | 39 | https://mathoverflow.net/users/1310 | 64078 | 39,591 |
https://mathoverflow.net/questions/64072 | 3 | For purposes of this question "terminal symplectic variety" means a normal variety which is symplectic (in the usual sense of symplectic singularities) and whose singular locus has codimension $\geq 4$ (the equivalence of this with the usual definition of terminal is a theorem of Namikawa).
Symplectic varieties has ... | https://mathoverflow.net/users/66 | Is a terminal symplectic variety S_4? | Ben,
a terminal singularity is Cohen-Macaulay and Cohen-Macaulay implies $S\_n$ for all $n$.
By the way, I already mentioned this in my [answer](https://mathoverflow.net/questions/63257/do-some-of-the-local-cohomology-groups-of-the-structure-sheaf-on-the-singular-loc/63261#63261) to your previous question about thi... | 6 | https://mathoverflow.net/users/10076 | 64080 | 39,593 |
https://mathoverflow.net/questions/64083 | 54 | I just heard that Daniel Quillen passed on. I am not familiar with his work
and want to celebrate his life by reading some of his papers. Which one(s?)
should I read?
I am an algebraic geometer who is comfortable with cohomological methods in his field, but knows almost nothing about homotopy theory. My goal is to ga... | https://mathoverflow.net/users/5337 | Which of Quillen's Papers Should I read? | Can I be the first to recommend **Elementary proofs of some results of cobordism theory using Steenrod operations**, *Advances in Math. 7 1971 29–56 (1971)*.
From the MR review: "In this important and elegant paper the author gives new elementary proofs of the structure theorems for the unoriented cobordism ring $N^\... | 22 | https://mathoverflow.net/users/8103 | 64087 | 39,598 |
https://mathoverflow.net/questions/64052 | 2 | Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f\_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \right]$ such that $f\_{a+b} \leq f\_a \mathsf{S}^a \left( f\_b \right)$ for all integers $a, b \geq 0$, where, classica... | https://mathoverflow.net/users/14179 | Multiple ergodic averages with varying number of terms | Unless I miss something, you are overcomplicating the story. In the first place, as it has already been pointed out, you don't really need submultiplicativity as
$$
f\_n(x) \le f\_1(x) f\_1(Sx) \dots f\_1(S^{n-1} x) = F\_n(x) \;.
$$
Applying the usual ergodic theorem to the right-hand side one gets that a.e.
$$
\log F... | 0 | https://mathoverflow.net/users/8588 | 64088 | 39,599 |
https://mathoverflow.net/questions/64097 | 4 | Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||\_n : \operatorname{Ent} \to \mathbb{R}$ by $||f||\_n = \operatorname{sup}(\{|f(z)| : |z|\leq n\})$.
$\big\langle \operatorname{Ent},+,\cdot,\{||.||\_n : ... | https://mathoverflow.net/users/nan | Are all continuous linear operators on the space of entire functions "simple"? | Suppose you have a function $u(z)$ that is defined and holomorphic for $|z|>R$, with $u(z)\to 0$ as $z\to\infty$. You can then define $L\_u:\text{Ent}\to\mathbb{C}$ by $L\_u(f)=\oint\_C f(z)u(z)\\,dz$, where $C$ is a circle around the origin of radius $R'>R$. This only depends on the germ of $u$ at $\infty$. I think on... | 0 | https://mathoverflow.net/users/10366 | 64101 | 39,607 |
https://mathoverflow.net/questions/63497 | 20 | A set of reals $X$ is $\textit{strong measure zero}$ if for any sequence of real numbers $ ( \epsilon\_n ) \_{n \in \omega }$ there is a sequence of open intervals $ ( a\_n ) \_{n \in \omega }$ which covers $X$ and such that each $ a\_i $ has length less than $ \epsilon \_i $.
The Borel Conjecture (BC) is the stateme... | https://mathoverflow.net/users/14794 | Cohen reals and strong measure zero sets | (An attempt at an answer, and also my first posting here. Thanks to Andres Caicedo for the reformatting.)
I claim that a single Cohen real makes the set of old reals strong measure zero.
Reals are functions from $\omega$ to 2.
Let ${\mathbb C}$ be Cohen forcing, and let $c$ be the name of the generic real.
Let ... | 12 | https://mathoverflow.net/users/14915 | 64111 | 39,614 |
https://mathoverflow.net/questions/64110 | 3 | Let $k$ be a finite field and $G$ a connected, reductive linear algebraic group defined over $k$. It is well-known that the union of the maximal tori of $G$ is dense in $G$ (more generally, if $G$ is not reductive, the union of its Cartan subgroups is dense).
It is also true, that $G$ is generated by the maximal tori... | https://mathoverflow.net/users/14916 | Is the union of Cartan subgroups over $k$ dense? | The set of maximal tori defined over $k$ is finite: if $T$ is one of them, they correspond to the $k$-rational points of the $k$-variety $G/\text{(normalizer of }T)$. So their union is not dense, unless $G$ is a torus.
| 6 | https://mathoverflow.net/users/7666 | 64114 | 39,616 |
https://mathoverflow.net/questions/64095 | 10 | Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?
($\*$) Let $\mathcal{B}$ be a family of $\aleph\_1$-many Borel sets. If $\bigcap \mathcal{B} = \emptyset$, then, for some countable $\mathcal{C} \subseteq \mathcal{B}$, it holds that $\bigcap \mathcal{C} = \emptyset... | https://mathoverflow.net/users/13506 | A compactness property for Borel sets | It is worth noting that a construction provided by Hausdorff more than a hundred years before the result of Kubis and Vejnar also provides a counterexample. A Hausdorff gap is a family of subsets of the natural numbers $A\_\xi, B\_\xi$ for $\xi\in\omega\_1$ such that
$A\_\xi \subseteq^\* A\_\eta \subseteq^\* B\_\eta \... | 11 | https://mathoverflow.net/users/13878 | 64115 | 39,617 |
https://mathoverflow.net/questions/64112 | 2 | This question is based on Beauville's article in Szpiro's asterisque *Seminaire sur les pinceaux de courbes de genre au moins deux* from 1986.
We will work over the complex numbers $\mathbf{C}$.
Let $f:X\longrightarrow \mathbf{P}^1\_{\mathbf{C}}$ be a semi-stable curve. This means that $f$ is a projective flat morp... | https://mathoverflow.net/users/4333 | Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres | Sheng-Li Tan proved that the answer is negative (see <http://arxiv.org/pdf/alg-geom/9411002>). This had been conjectured by Beauville.
| 9 | https://mathoverflow.net/users/4790 | 64123 | 39,623 |
https://mathoverflow.net/questions/64103 | 7 | What are the crystalline realizations of Artin motives?
In a paper by Kisin and Wortmann, "A note on Artin motives" (google it and you'll find it immediately), they define a suitable category of motives incorporating crystalline realization in analogy with Deligne's absolute Hodge cycles. However, they don't say exp... | https://mathoverflow.net/users/2147 | Crystalline realizations of Artin motives | An Artin motive is just the same thing as a continuous representation of the absolute Galois group $G\_K$ (of whatever number field $K$ we are thinking about) with finite image.
For conreteness, let's write it as $G\_K \to GL(V)$, where $V$ is a finite dimensional vector space over $\mathbb Q$. (We could incorporate... | 11 | https://mathoverflow.net/users/2874 | 64124 | 39,624 |
https://mathoverflow.net/questions/64129 | 3 | I have a discrete dynamical system in $[0,1]^n$. Specifically, I am studying the dynamics of a probability distribution under certain operator $\phi$ such that $\mathbf{q}[t+1]=\phi(\mathbf{q}[t])$. The probability distribution is specified by $n$ parameters in the $n$-dimensional vector $\mathbf{q}$.
What I need to kn... | https://mathoverflow.net/users/14917 | How to study the size of basins of attraction in a discrete dynamical system? | I don't think there is a magic theorem here which works all the time.
What you want depends a lot on the specific $\phi$, ${\bf q}'$, ${\bf q}\_0$ you are working with. What you said sounds a lot like a toy model for the renormalization group, which
is a way to study central limit type theorems. For instance the classi... | 5 | https://mathoverflow.net/users/7410 | 64136 | 39,628 |
https://mathoverflow.net/questions/64116 | 11 | Consider a locally compact group $G$, considered as a measurable space with the completed Borelstructure wrt. the Haarmeasure. Consider a map $f:G \to G$, which is measurable and has an inverse, which is then also measurable. Is $f$ an homeomorphism?
What if $G$ is abelian? If not, what are necessary conditions on $G... | https://mathoverflow.net/users/10400 | Are measurable automorphism of a locally compact group topological automorphisms? | Here is a result by Adam Kleppner ([Measurable homomorphisms of locally compact groups](https://doi.org/10.1090/S0002-9939-1989-0948154-8), Proc. Amer. Math. Soc., vol. 106, no. 2, 1989, 391-395): any measurable homomorphism between locally compact groups is continuous. Actually what he really needs, for a homomorphism... | 18 | https://mathoverflow.net/users/14497 | 64140 | 39,632 |
https://mathoverflow.net/questions/64130 | 7 | This is an arithmetic follow-up to my previous question [Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres](https://mathoverflow.net/questions/64112/does-there-exist-a-non-trivial-semi-stable-curve-of-genus-1-with-only-4-singular)
Let $k$ be an algebraically closed field and le... | https://mathoverflow.net/users/4333 | The number of singular fibres of a semi-stable arithmetic surface over \Z | I have the opposite intuition -- I would think the answer would be yes for all g. In genus 1, you are asking (I think) whether there are elliptic curves with prime conductor. There are a lot of elliptic curves of prime conductor; I believe the question of whether there are infinitely many is open, and considered hard. ... | 12 | https://mathoverflow.net/users/431 | 64143 | 39,634 |
https://mathoverflow.net/questions/64144 | 0 | Hello,
I have the following function form as one of my constraints :
f(x) = MIN(0, x)
Because of the MIN, it is non-differentiable.
As I would like to use an optimizer that uses derivative based methods, I need my objective function and constraints to be differentiable.
How could I convert (or approximate) th... | https://mathoverflow.net/users/14922 | Converting (or approximating) a non-differentiable function to a differentiable function | You can try
$f(\xi) = \frac{\xi}{2}+\frac{\ln 2}{2k} + \frac{\ln(\cosh(k \xi)}{2k},$
whose derivative is
$f'(\xi) = \frac{1}{2} [\tanh(k \xi) + 1].$
This function approaches the maximum as $k\rightarrow \infty$ (change as needed for the minimum). Will this help solve your problem? Perhaps, if the lack of smooth... | 0 | https://mathoverflow.net/users/14230 | 64147 | 39,636 |
https://mathoverflow.net/questions/63978 | 7 | I am trying to understand lowest representations of loop groups as developed in Pressley and Segal's book. Specifically I want to be able to compute the weight spaces that appear in a lowest weight representation. I realize there is a formula for this -my question is along the lines of how to apply the formula correctl... | https://mathoverflow.net/users/7 | lowest weight representation of loop groups | The formula for the invariant bilinear form is given in $(4.9.3)$ on
page 64 $$\langle (x\_1,\xi\_1, y\_1),(x\_2,\xi\_2,y\_2) \rangle=\langle
\xi\_1, \xi\_2 \rangle - x\_1 y\_2-y\_1x\_2$$ As I mentioned in the
comments, $(9.3.7)$ becomes then $||\mu||^2-6m=2$. So your last
equation would be $m=\frac{1}{3}(a^2-ab+b^2)-\... | 4 | https://mathoverflow.net/users/13377 | 64153 | 39,640 |
https://mathoverflow.net/questions/64151 | 8 | I apologize in advance if this question seems too vague.
In many topology courses, concepts like the fundamental group and homology groups are introduced as a means of distinguishing non-homeomorphic spaces - for instance, $\mathbb{T}^2$ and $S^2$. Similarly, things like the rank of an abelian group, and the Krull d... | https://mathoverflow.net/users/6856 | Categorical Invariants | Many invariants are given by topological invariants composed with the nerve functor from (small) categories to topological spaces. For example, you can talk of $\pi\_0(C)$ of a category $C$ and it is exactly what you might guess, the set of connected components. But also higher homotopy groups are defined. [This](http:... | 6 | https://mathoverflow.net/users/2841 | 64155 | 39,642 |
https://mathoverflow.net/questions/64171 | 2 | **EDIT 2011.05.09**
Thanks to Junkie and Tapio Rajala for checking on me. While most of the candidates
referred to below have small factors, the "large" small factors I list below are incorrect. Also, that $2^{2557} - 2^{1278} + 1$ does not have a small factor supports my "feeling" that the candidate set should have no... | https://mathoverflow.net/users/3402 | What are the chances of finding a small factor? | 232 is fairly small, so the easiest way to determine if the number has a prime factor up to that size is probably just to trial divide. I just did that in GP:
```
test(p)=my(two=Mod(2,p));two^264097-two^132048+1;
forprime(p=2,1<<32,if(!test(p),return(p)))
```
(actually, I used a customized function rather than for... | 1 | https://mathoverflow.net/users/6043 | 64174 | 39,648 |
https://mathoverflow.net/questions/64141 | 20 | Dear community,
In light of the recent work of DeBacker/Reeder on the depth zero local Langlands correspondence, I was wondering if there is an attempt to "geometrize" the depth zero local Langlands correspondence.
In particular, in Teruyoshi Yoshida's thesis, one can see a glimpse of this for $GL(n,F)$, where $F$ ... | https://mathoverflow.net/users/8891 | Geometric construction of depth zero local Langlands correspondence | Yoshida considers the Lubin-Tate tower in his geometric realization of the depth zero supercuspidals for $GL(n)$. For unitary groups, I'm sure that the answer to your question will be found in a similar analysis for the corresponding Rapoport-Zink spaces. I don't believe this has been done yet, but it ought to be done ... | 11 | https://mathoverflow.net/users/271 | 64175 | 39,649 |
https://mathoverflow.net/questions/64120 | 2 | Hi,
(I hope this is not too basic)
I basically have a set of data from the same underlying distribution (which I would like to estimate), but I only have available the mean and N from partitions of the data. I can estimate the mean using a weighted average, and, I thought, the sample variance using weighted varianc... | https://mathoverflow.net/users/14918 | How to estimate sample mean and variance from derived data | Note: New answer.
If you don't know the variances $Var X\_1,...,VarX\_m$, you have to make parametric assumptions about the distribution of the data. For example, under the assumption of a Poisson distribution, the the maximum likelihood estimator of the variance would simply be the aggregated mean $EX$. Under the as... | 3 | https://mathoverflow.net/users/2660 | 64189 | 39,658 |
https://mathoverflow.net/questions/64107 | 2 | A similar question was already asked in [question](https://mathoverflow.net/questions/14246/spectra-of-sums-and-products-in-banach-algebras-was-spectrum-in-banach-algebr) titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".
Answer there led me to the following question.
If f... | https://mathoverflow.net/users/13481 | spectra of sums in (Banach) algebras | Here is a counterexample to $B/rad(B)$ being commutative. Let $S\colon l^{2}(\Bbb N)\to l^{2}(\Bbb N)$ be the unilateral shift. Then the spectrum of $S$ is $\overline{\Bbb D}=\{z\in \Bbb C:|z|\leq 1\}.$ Now $S^{\*}$ has the same spectrum. Consider the algebra they generate inside $B(l^{2}(\Bbb N)),$ it is a C$^{\*}$-al... | 4 | https://mathoverflow.net/users/14771 | 64194 | 39,661 |
https://mathoverflow.net/questions/64195 | 32 | I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. Please see my specific questions at the end, after "discussion".
Discussion and Background
=========================
... | https://mathoverflow.net/users/14510 | When is a finite dimensional real or complex Lie Group not a matrix group | Most of the answers can be found in Hochschild's book on the structure of Lie group.
a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)
b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear... | 25 | https://mathoverflow.net/users/14497 | 64212 | 39,670 |
https://mathoverflow.net/questions/64164 | 2 | Is there any result that relates the minimum degree of a geometric random graph to its k-vertex connectivity? I read papers where they pose the condition $$d\_{\rm min} \geq k$$
to imply that the graph is k-connected (vertex connectivity). Is that an approximation or an asymptotic result? In my case, I need, if not an ... | https://mathoverflow.net/users/13822 | $k$-connectivity of a geometric random graph | Does this help?
<http://www.maths.bath.ac.uk/~masmdp/ab/kcon.html>
| 3 | https://mathoverflow.net/users/839 | 64213 | 39,671 |
https://mathoverflow.net/questions/64210 | 5 | In the general case, I want to say that determining $|Hom(G,H)|$ is incomputable, arguing that you could use the number to test for simplicity of a presentation, but I am new to this area and I keep finding flaws with my argument.
While I believe the general case is incomputable, there are computable special cases. O... | https://mathoverflow.net/users/14869 | Computability and complexity of computing $|Hom(G,H)|$ for finitely presented groups G, H. | Take $G=\mathbb{Z}$. Then computing $|\operatorname{Hom}(G, H)|=|H|$ is the same as computing the size of a finitely presented group, and is thus wildly undecidable. This eliminates both the general case you seem to ask about, and the case of fundamental groups of surfaces (replacing $\mathbb{Z}$ with, say $\mathbb{Z}\... | 3 | https://mathoverflow.net/users/6950 | 64215 | 39,673 |
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