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https://mathoverflow.net/questions/75925 | 3 | I hope this question is focused enough – it's not about real problem I have, but to find out if anyone knows about a similar thing.
You probably know the [Heisenberg uncertainty principle](https://en.wikipedia.org/wiki/Uncertainty_principle): For any function $g\in L^2(\mathbb{R})$ for which the respective expression... | https://mathoverflow.net/users/9652 | Do you know this form of an uncertainty principle? | There exists a plethora of inequalities relating weighted $L^p$ norms of a function and its derivatives. For instance you have the Caffarelli-Kohn-Nirenberg family of inequalities
$$\| |x|^{-\gamma}u\|\_ {L^{r}}\le C \||x|^{-\alpha}\nabla u\|^{a}\_ {L^{p}}\||x|^{-\beta}u\|^{1-a}\_ {L^{q}}
$$
which hold for a quite larg... | 4 | https://mathoverflow.net/users/7294 | 75930 | 46,050 |
https://mathoverflow.net/questions/75933 | 4 | Let $T(n)$ denote the number of transitive binary relations on an $n$-element set. So T(1) = 2 and T(2) = 13, for of the 16 possible relations on a 2-element set {a,b}, the only three which are not transitive are
(i) {(a,b), (b,a)},
(ii) {(a,a), (a,b), (b,a)},
(iii) {(b,b), (a,b), (b,a)}.
There is some literature o... | https://mathoverflow.net/users/17637 | Number of transitive relations on a set | If $P(n)$ is the number of partial orders, then $\log\_2 P(n) = n^2/4 + o(n^2)$, an old result of Kleitman. Look in MathSciNet for many different sharpenings. Now if $T(n)$ is the number of transitive relations, then Klaska proved that $T(n)$ and $2^n P(n)$ are asymptotically equal. Therefore, $\log\_2 T(n) = n^2/4 + o... | 5 | https://mathoverflow.net/users/9025 | 75936 | 46,054 |
https://mathoverflow.net/questions/75896 | 12 | The so-called "compact AR Problem" reads:
>
> Is every compact convex set in a metrizable topological vector space an absolute retract?
>
>
>
It is open according to the chapter by T. Banakh, R. Cauty and M. Zarichnyi in ["Open Problems in Topology II"](http://books.google.com/books?id=XdXnQCV5K08C&lpg=PA609&... | https://mathoverflow.net/users/10819 | Status of the compact AR problem? | So if I understand right, you haven't checked MathSciNet? Well, it's not much use:
* Park's paper is <http://www.ams.org/mathscinet-getitem?mr=2144066> but there is no review.
* Park's survey paper is <http://www.ams.org/mathscinet-getitem?mr=2188170> but there is only a "summary".
* Those are the only things you are... | 0 | https://mathoverflow.net/users/406 | 75944 | 46,056 |
https://mathoverflow.net/questions/75938 | 3 | Definition of Condition Expectation( of random variable) can be seen here:
<http://en.wikipedia.org/wiki/Conditional_expectation>
My silly question is:
Intuitively, conditional expectation seems to be a restriction of a measurable function on the sub-$\sigma$-algebra. Why don't people call it a restriction instead of... | https://mathoverflow.net/users/17708 | Why isn't conditional expectation called restricted random variable? | Here's an instructive example. Let $\sigma\_1 = \{ \emptyset ,[0,1/2),[1/2,1),[0,1)\}$, and $\sigma\_2 = \{\emptyset, [0,1)\}$, both sigma algebras. Let $X(\omega) = 4$ if $\omega\in[0,1/2)$ and $X(\omega)=5$ if $\omega\in[1/2,1)$. Now, $X$ is $\sigma\_1$-measurable, and $E(X \mid \sigma\_2) = 4.5$ for $\omega \in[0,1)... | 6 | https://mathoverflow.net/users/935 | 75945 | 46,057 |
https://mathoverflow.net/questions/75923 | 7 | Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations.
First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of another field $k$. We only consider the case $k=\mathbb{Q}$ or $\mathbb{C}$. By definition, A *model* of $K$ is a variet... | https://mathoverflow.net/users/17294 | The space of valuations of a function field | Simply put (more or less already said above). Let $K=C(x,y)$ and let $xi(t)$ be a generalized power series (a power series with well-ordered exponents) where the exponents are non-negative real numbers. Assume (this is important) that $P(t,\xi(t))\neq 0$ for any $P\in K$. Then the map: $\nu(P(x,y))=ord\_{t}(t,\xi(t))$ ... | 3 | https://mathoverflow.net/users/17956 | 75952 | 46,060 |
https://mathoverflow.net/questions/75956 | 5 | Would you please give some references concerning the number of indecomposable modules over preprojective algebras of type $A\_n$?
More precisely, I need references about the following claim: The number of indecomposable modules over the preprojective algebra of type $A\_n$ is:
1) only one for $n=1$,
2) four for ... | https://mathoverflow.net/users/17845 | Indecomposable modules over preprojective algebras | This is stated in [Rigid modules over preprojective algebras](http://www.math.uni-bonn.de/~schroer/preprints/rigid.pdf) by Geiss, Leclerc and Schröer, Sections 8.1-8.3. As noted in my comment a proof of the $n=5$ statement can be found in [The module theoretic approach to quasi-hereditary algebras](http://www.mathemati... | 7 | https://mathoverflow.net/users/15887 | 75963 | 46,064 |
https://mathoverflow.net/questions/75964 | 2 | Let's say that $X$ is an integral scheme of finite type over a field and $Y \subset X$ is a closed subscheme. Given a vector bundle $E$ on $Y$, is $E$ the restriction to $Y$ of a vector bundle on a neighborhood $U$ of $Y$ in $X$?
| https://mathoverflow.net/users/132 | Extending a vector bundle to a torsion free sheaf | No, this is not even true for line bundles.
For an example, let $X = \mathbb{P}^2$ and $Y$ a smooth curve in $X$ of genus $>0$ (over an algebraically closed field). Since $X$ is smooth, any line bundle on an open set $U$ extends to a line bundle on $X$ so the map $\operatorname{Pic}(X) \to \operatorname{Pic}(U)$ is s... | 10 | https://mathoverflow.net/users/519 | 75969 | 46,065 |
https://mathoverflow.net/questions/75943 | 1 | Let $\mathcal{M}\_g$ be the moduli space of smooth curves of genus $g$. Let $Z$ be the closure in $\mathcal{M}\_g$ of the set of smooth curves of genus $g$ which are a cyclic cover of the projective line.
**Question.** Is $Z$ irreducible?
**Question.** What is the dimension of $Z$? Do we have non-trivial bounds?
... | https://mathoverflow.net/users/4333 | The locus of cyclic covers in the moduli space of curves | The following paper:
M. Cornalba, On the locus of curves with automorphisms, Annali di Matematica pura ed applicata (4) 149 (1987), 135-151. Erratum, Annali di Matematica pura ed applicata (4) 187 (2008), 185-186.
(A revised version incorporating the changes described in the Erratum is available on the author's web ... | 5 | https://mathoverflow.net/users/10610 | 75972 | 46,068 |
https://mathoverflow.net/questions/75978 | 1 | Let $M$ and $N$ be two real square matrices of size $p+q$. The matrix $M$ is nonsingular. The matrix $N$ has the following block structure, where $A$ is a $q{\times}p$ matrix. $N = \left(\begin{array}{cc} \text{O}\_{p,p} & \text{O}\_{p,q} \\ A & \text{Id}\_{q} \end{array}\right)$. I would like to conclude that the alge... | https://mathoverflow.net/users/10271 | Preserving the algebraic multiplicity of the zero eigenvalue | Take $p=q=1$, let $M = \begin{pmatrix} 0 & 1 \newline 1 & 0 \end{pmatrix}$ and let $N = \begin{pmatrix} 0 & 0 \newline 0 & 1 \end{pmatrix}$. Then the algebraic multiplicity of the zero eigenvalue in $MN = \begin{pmatrix} 0 & 1 \newline 0 & 0 \end{pmatrix}$ is $2$.
| 3 | https://mathoverflow.net/users/7709 | 75984 | 46,071 |
https://mathoverflow.net/questions/75987 | 2 | Let $n$ be a positive integer, and let $\mathbf{v}$ be a non-zero vector in $\mathbb{R}^n$.
Let $\; p : [0,1] \to \mathbb{R}^n \;$ be injective and $C^1$ and such that $p'$ is nowhere zero.
Does there always exist $\; f : ([0,1] \times \mathbb{R}^n) \to \mathbb{R}^n \;$ such that $f\hspace{0.01 in}$ is $C^1$ an... | https://mathoverflow.net/users/nan | Are all $C^1$ arcs tame? | If $p$ is continuously differentiable up to and including the endpoints, then it has a $C^1$ extension to $(-\varepsilon,1+\varepsilon)$. Then it has a tubular neighborhood $U$ parametrized by a $C^1$ diffeomorphism $i:(-\varepsilon,1+\varepsilon)\times D^{n-1}\to U\subset\mathbb R^n$,
where $i(t,0)=p(t)$. In the cylin... | 3 | https://mathoverflow.net/users/4354 | 75988 | 46,073 |
https://mathoverflow.net/questions/75982 | 3 | **Background:**
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ denote the positive roots in a root system of a torus of $G$. Then we have the hyperalgebra $\bar U(N)$ of $N$ which ... | https://mathoverflow.net/users/1528 | Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra | I don't have a full answer, but the following calculation may be useful.
Consider the example $G = SL\_3$ so that $N$ is the Heisenberg group on three generators and in characteristic zero, $\bar{U}(N)$ is the enveloping algebra of the three-dimensional Heisenberg Lie algebra. Thus
$\bar{U}(N) = k[x,z][y ; z \frac... | 2 | https://mathoverflow.net/users/6827 | 75992 | 46,074 |
https://mathoverflow.net/questions/75986 | 7 | Let $H$ be a group, $\phi$ an automorphism of $H$ of order n and fix $h\_0 \in H$. I wonder, what the restrictions are, such that
$$G:= \lt H,g \mid g^n=h\_0,\quad \forall h \in H: ghg^{-1}=\phi(h) \gt$$
defines a group which has $H$ as normal subgroup such that $G/H$ is cyclic of order n.
There are two obvious res... | https://mathoverflow.net/users/17734 | Restrictions for presenting groups with cyclic quotient | Your conditions are sufficient. Indeed, consider the $H$-by-cyclic group $G\_0=\langle H, g\mid ghg^{-1}=\phi(h), h\in H\rangle$. By (1) and (2), both $g^n$ and $h\_0$ are central in that group. Hence the element $u=h\_0^{-1}g^n$ is central. Now factor out the central subgroup $\langle u\rangle$: $G=G\_0/\langle u\rang... | 3 | https://mathoverflow.net/users/nan | 76001 | 46,077 |
https://mathoverflow.net/questions/75994 | 5 | Let $G$ be a real reductive Lie group, $P=MN$ its parabolic subgroup with Levi decomposition. Suppose $\sigma$ is a smooth admissible irreducible representation of $M$, extend this to $P$ by letting $N$ act trivially. Form the unitarily induced representation $Ind\_P^G(\sigma)$.
My question is what is the contragredi... | https://mathoverflow.net/users/1832 | what's the contragredient of induced representation | Yes. The point is that $Ind\_P^G(\sigma)$ is by definition equal to the space of
sections of a certain $G$-equivariant vector bundle $E\_{\sigma}$ on $G/P$ and $Ind\_P^G(\sigma')$
is equal to the sections of the corresponding bundle $E\_{\sigma'}$. Now the point
is that because you are using unitary induction there is ... | 8 | https://mathoverflow.net/users/3891 | 76003 | 46,079 |
https://mathoverflow.net/questions/75976 | 11 | Hello,
I am trying to understand the calculus of pseudodifferential operators on manifolds. All the textbooks I could put my hand on define the principal symbol of a pseudodifferential operator locally, then prove that it transforms well, hence becomes a "global" object. Is there any good way to define the the princi... | https://mathoverflow.net/users/17965 | Symbol of pseudodiff operator | There is an invariant way of defining pseudodifferential operators, and a (much simpler and quite classical) invariant way of defining symbols.
The latter appears already in the old Atiyah-Singer volume from the early '60's. Choose any point $(x\_0, \xi\_0)$ in the cotangent bundle. Choose a function $\phi \in \mathc... | 14 | https://mathoverflow.net/users/17969 | 76004 | 46,080 |
https://mathoverflow.net/questions/75990 | 4 | Let $X$ be a compact metric space, and
let $C(X)$ be the Banach space of continuous real-valued function on $X$, with
the maximum norm.
Suppose $S\subset C(X)$ is a set of functions with the following property:
For every ball $B(a,r)\subset X$ and for every $\epsilon>0$, there exists a function
$f\in S$ such that... | https://mathoverflow.net/users/17970 | Dense sets in the space of continuous functions | No, $S$ does not have to span $C(X)$.
Taking the case with $X=[0,1]$, let $\mu$ be any atomless finite signed measure whose positive and negative parts $\mu^+$,$\mu^-$ have full support, so that $\mu^+(U) > 0$ and $\mu^-(U) > 0$ for any nonempty open $U$. Then, the set $S=\{f\in C(X)\colon\int f\,d\mu=0\}$ satisfies ... | 12 | https://mathoverflow.net/users/1004 | 76005 | 46,081 |
https://mathoverflow.net/questions/75975 | 3 | It is my understanding that a connection form on a principal G-bundle over a manifold X is defined to be a Lie algebra-valued 1-form $\alpha$ which reproduces the Lie algebra generators of the fundamental vector fields at every point. That is, if at $p$ a vector field $v$ takes the value $\frac{d}{dt}(exp(tg)p) |\_{t=0... | https://mathoverflow.net/users/17913 | Principal bundle connections | First off, Prof. Figueroa-O'Farrill is correct in noting that you omitted the $G$-equivariance condition for the connection, which reduces to $G$-invariance in the case where $G$ is abelian (in particular, when $G=\mathbb{C}^\*$).
1. The choice of normalization conventions really just comes down to how you identify t... | 4 | https://mathoverflow.net/users/17945 | 76008 | 46,083 |
https://mathoverflow.net/questions/76006 | 2 | In Chriss and Ginzburg's "Representation Theory and Complex Geometry", they describe a geometric construction of representations of the affine Hecke algebra, using the Borel-Moore homology of generalized Springer fibers.
Briefly, let $G$ be a sufficiently nice algebraic group, and choose a semisimple $s \in G$ and s... | https://mathoverflow.net/users/12694 | Action of centralizer on Borel-Moore homology of Springer Fibers for Affine Hecke Algebra | Borel-Moore homology isn't a homotopy invariant, but it is an isotopy invariant; if two homeomorphisms are homotopic through a homotopy which is a homeomorphism over any point in [0,1], they induce the same map on Borel-Moore homology.
One can see this instantly by writing Borel-Moore homology as homology of the 1-po... | 2 | https://mathoverflow.net/users/66 | 76010 | 46,084 |
https://mathoverflow.net/questions/76013 | 9 | Is there a conceptual way to understand where the Fast Fourier Transform is avoiding redundant computation and thus achieving $O(n\log n)$ instead of $O(n^2)$.
Consider a standard example of the FFT to multiply two polynomials faster. Its not obvious to me conceptually why the FFT should yield a faster way to multipl... | https://mathoverflow.net/users/16557 | Why is the Fast Fourier Transform efficient? | Conceptually the FFT takes advantage of a shortcut similar to the distributive law for multiplication. To compute $$(x\_1 + x\_2)(x\_3 + x\_4)$$ on could either add first (twice) and then multiply (once), or one could expand $$sx\_1x\_3 + x\_1x\_4 + x\_2x\_3 + x\_2x\_4$$ and multiply (four times) and then add (three ti... | 12 | https://mathoverflow.net/users/8719 | 76017 | 46,089 |
https://mathoverflow.net/questions/75998 | 7 | By the Bruhat decomposition of $GL(n, \mathbb{F}\_q) / B\_n$ we know that $$[n]! = \sum\_{ \sigma \in S(n)} q^{l(\sigma)}$$ where $[n]! = \prod\_{j=1}^n (1+q + \cdots + q^{j-1})$ and $l(\sigma)$ is the length of the permutation $\sigma \in S(n)$ (also known as the number of involutions of $\sigma$).
We also know that... | https://mathoverflow.net/users/6862 | Two ways of generalizing factorials via symmetric groups | It seems like there is no hope for a nice closed form for $F(q,\theta)=\sum\_{\sigma\in S\_n}q^{l(\sigma)}\theta^{[\sigma]}$. When the sum is restricted to $\sigma\in S\_n$ which are involutions, the computation can be found in I. Gessel's paper ["A q-analog of the exponential formula"](http://www.sciencedirect.com/sci... | 8 | https://mathoverflow.net/users/2384 | 76022 | 46,093 |
https://mathoverflow.net/questions/76029 | 3 | Let $S\_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$.
The cone $S\_+$ seems exceptionally symmetrical, and I am curious to know its symmetry groups. Let me be precise. An $R\... | https://mathoverflow.net/users/nan | Linear and Isometric Automorphism Groups of the PSD Cone | The general linear group $GL(n, R)$ acts on $S+$ by $g(x) = g x g^t,$ and this is the full linear automorphism group.
<http://www.math.umbc.edu/~gowda/tech-reports/trGOW11-03.pdf>
and references therein for more details and related results. I am pretty sure the result goes back to at least Minkowski (for $n=2$ this... | 3 | https://mathoverflow.net/users/11142 | 76039 | 46,101 |
https://mathoverflow.net/questions/72955 | 11 | The context
-----------
In a [beautiful paper](http://arxiv.org/abs/math/9803041), Malikov-Schechtman-Vaintrob defined a canonical sheaf of vertex algebras equipped with a differential on any manifold $X$ (either in the $C^\infty$, complex analytic or algebraic context). They called it the **chiral de Rham complex** ... | https://mathoverflow.net/users/7031 | different N=2 SUSY structures on the chiral de Rham complex of a Calabi-Yau manifold? | Ooops, I should start reading this site. In some sense all of these N=2 structures are the same. Unfortunately we now understand the situation well better than we did then.
To simplify the answer you may think of the $C^\infty$ Chiral de Rham (CDR) as the tensor product of a holomorphic CDR with a anti-holomorphic on... | 4 | https://mathoverflow.net/users/17980 | 76048 | 46,104 |
https://mathoverflow.net/questions/76037 | 25 | I wanted to put this originally on math.stackexchange, since I considered it to be a straightforward question and probably a fairly known fact. After I failed to solve the problem, I browsed through literature and what a surprise - two books claim it is primitive recursive, one resource claims it isn't, and neither one... | https://mathoverflow.net/users/4925 | Inverse Ackermann - primitive recursive or not? | The inverse Ackermann function is primitive recursive.
One way to see this is to use the fact that a function $f$ is primitive recursive when and only when
1. the graph of $f$ is primitive recursive, and
2. $f$ is bounded above by some primitive recursive function.
The graph of the Ackermann function is primitiv... | 32 | https://mathoverflow.net/users/2000 | 76054 | 46,106 |
https://mathoverflow.net/questions/76044 | 7 | Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and Abramovich-Vistoli (algebraically). Additively, this is nothing but the ordinary cohomology of the inertia stack, but it ... | https://mathoverflow.net/users/1310 | Intuition behind the age grading in quantum cohomology of orbifolds | It is my understanding that the original motivation did come from GW theory: Chen and Ruan came across the orbifold cohomology ring, and its product, as a byproduct and first step of defining the orbifold quantum cohomology. The strongest motivation I have for the grading also comes from this perspective, and maybe we ... | 5 | https://mathoverflow.net/users/1102 | 76055 | 46,107 |
https://mathoverflow.net/questions/75995 | 9 | Euclid's proof of the infinitude of primes requires me to take an arbitrary finite set of primes and multiply them together. If I want to formalize this proof in Peano Arithmetic, I need to know that any finite set of numbers has a product. Carl Mummert's comment on Russell O'Connor's answer to [this question](https://... | https://mathoverflow.net/users/10503 | Formalizing Euclid's proof of the infinitude of primes | The definition of iterated product is the easy part of the proof, the bottleneck is elsewhere. Even quite weak theories of bounded arithmetic, such as PV (this is an open theory with function symbols for polynomial-time functions), can prove that the product of a sequence of natural numbers exists (and is divisible by ... | 14 | https://mathoverflow.net/users/12705 | 76058 | 46,109 |
https://mathoverflow.net/questions/76063 | 2 | Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where
* $f$ is real-analytic on the open interval $(0,1)$
* $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-C\leq f(x)\leq C$ for $x\in[0,1]$).
Is it true that there is a real-analytic continuation of $f$ to the interval... | https://mathoverflow.net/users/17984 | Does a bounded real function have an analytic continuation | $f(x) = \sqrt{1-x^2}$ is real analytic on $(-1,1)$, bounded and continuous on $[-1,1]$, but of course not even one-sided differentiable at the endpoints. But then Igor's example is one-sided differentiable of all orders at the endpoint $0$, but still not real analytic in any neighborhood of $0$.
| 4 | https://mathoverflow.net/users/454 | 76066 | 46,114 |
https://mathoverflow.net/questions/76069 | 2 | Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):
Proposition: Let $F$ a constructible sheaf on $X\_{et}$, the small étale site of $X$, $X$ proper over a field $k$. Then $H^{i}(X,F)$ is finite for $i\geq0$.
(false?)
He deduces it via Hochschild-Serre from the statement, t... | https://mathoverflow.net/users/17987 | Finiteness of étale Cohomology Groups | (This was going to be a comment, but it's too long.)
I don't see how the big étale site appears, even in the proof of cor IV.2.8 of Milne. Seems like he's just base changing to the integral closure of the field but using small étale sites all time. (Though I'm not that familiar with Milne, as I use SGA 4 and 4 1/2 as... | 3 | https://mathoverflow.net/users/12336 | 76070 | 46,115 |
https://mathoverflow.net/questions/76052 | 4 | Let $f,g \in \mathbf {C}[x\_1, \ldots, x\_n]\subseteq \mathbf {C}[x\_1, \ldots, x\_{n+1}]$ be two polynomials with complex coefficients and suppose that there exists $h\_1 \in \operatorname{GL}\_{n+1}(\mathbf C)$ such that $h\_1 \cdot f=g$. Does there exist $h\_2 \in \operatorname{GL}\_n(\mathbf C)$ such that $h\_2\cdo... | https://mathoverflow.net/users/3380 | GL(n+1)- vs. GL(n)-orbits | Yes, this is true. Let me consider the case when there is no linear change of coordinates $(x\_1,...,x\_n)$ such that $g=g(x\_1,...,x\_{n-1})$. In this case it is clear that the vector $v=h\_1(0,...,1)$ does not belong to the plane $(x\_{n+1}=0)\subset \mathbb C^{n+1}$. Hence we can find a linear transformation $h\_3$ ... | 3 | https://mathoverflow.net/users/943 | 76076 | 46,118 |
https://mathoverflow.net/questions/76038 | 10 | Dear all,
a question came to me when I read the paper "Complete three dimensional manifolds with positive Ricci curvature and scalar curvature/ R. Schoen and S.-T. Yau, 1982".
The question is as follows.
suppose we have a sequence of complete non-compact smooth submanifolds $\Sigma\_k\subset M$ and each $\Sigma\_k$... | https://mathoverflow.net/users/9915 | Smooth convergence of minimizing varifolds | The reason for this is Allard's regularity theorem.
Roughly speaking Allard's theorem says that if near a point of the support of a stationary varifold the varifold has unit density and area close to that of the ball of the appropriate dimension (for a 2-varifold it would be area of a disk) then the support of the va... | 14 | https://mathoverflow.net/users/26801 | 76080 | 46,121 |
https://mathoverflow.net/questions/76083 | 9 | Hi,
I received an answer to a question a while back. The question was about how we can present a category as a collection of arrows and a large list of algebraic relations between them. One of the answers I got was about Freyd's "Categories, Alegories", and here it is:
[products in a category without reference to o... | https://mathoverflow.net/users/10007 | Categories presented with Arrows only, no objects: partial monoids | Of course you can define a (just-arrow) category $\mathcal C$ like a partial algebra which consist of:
a set $\mathcal C$ (namely the set of arrows of your category), a set $D\_\mathcal{C} \subseteq \mathcal C \times \mathcal C$ (the set of pair of composable arrows) and
a map $\circ \colon D\_\mathcal{C} \to \mathc... | 13 | https://mathoverflow.net/users/14969 | 76089 | 46,127 |
https://mathoverflow.net/questions/76082 | 2 | Does there exist a holomorphic map from a neighborhood of $\mathbb C$ to $S^3 \subseteq \mathbb C^2$?
| https://mathoverflow.net/users/15197 | Holomorphic map from a neighborhood in $\mathbb C$ to S^3 | Suppose such a map exists and is non-constant. Let $f,g$ be its components. By composing the map with a holomorphic function on the right and with an element of $U(2)$ on the left, we can assume that in a neighborhood of 0 we have $f(z)=z$ (i.e., we can take $f$ as a local coordinate) and $g(z)=1+az+\cdots$ with $a\neq... | 4 | https://mathoverflow.net/users/2349 | 76090 | 46,128 |
https://mathoverflow.net/questions/76071 | 6 | So I'm reading the part in Ana Cannas da Silva's book "Lectures on Symplectic Geometry" available (on her website) about hamiltonian group actions on a symplectic manifold. She starts by defining $\mathbb R$-actions and $\mathbb S^1$ actions by saying that the vector field on $M$ that they generate must be hamiltonian.... | https://mathoverflow.net/users/14359 | Question about the definition of hamiltonian group action. | Just so you're aware, not every author insists that a momentum map be infinitesimally equivariant (Prof. Figueroa-O'Farrill's condition 2), although it is part of da Silva's definition (edit: actually, on checking, da Silva requires the slightly stronger condition of equivariance, i.e. $\mu(g\cdot p)=\mu(p)\circ\mathrm... | 3 | https://mathoverflow.net/users/17945 | 76092 | 46,129 |
https://mathoverflow.net/questions/76097 | 1 | What kind of conditions we need to make morphisms of schemes quasi-projective?
I am really interested in the following case:
If $f : X \to Y$ is an etale, of finite type and separated morphism of schemes, then is it quasi-projective?
If so, which conditions we use?
If necessary, please assume that the scheme $Y... | https://mathoverflow.net/users/39742 | What kind of conditions we need to make morphisms of schemes quasi-projective? | The answer is yes, if you assume that $Y$ is quasi-compact, and $f:X\to Y$ is of finite type and separated.
Every etale morphism is unramified, which implies it is quasi-finite (Milne, Prop 3.2). This in turn implies by Zariski's main theorem (Milne, Thm 1.8), that $f$ factors as an open immersion followed by a fini... | 6 | https://mathoverflow.net/users/4709 | 76100 | 46,133 |
https://mathoverflow.net/questions/75999 | 2 | Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
EDIT: Would you use 'map of finite length'? (because $S/f(\mathfrak{m})S$ has finite length over $S$?) Assume $R$ and $S... | https://mathoverflow.net/users/16046 | Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary? | One can notice that your hypothesis is equivalent to say that the closed fiber of $\mathrm{Spec}(S)\to \mathrm{Spec}(R)$ has dimension $0$ because this closed fibers is equal to $V(\varphi(\mathfrak m)S)$, and $V(I)$ of an ideal $I$ in $S$ is reduced to the closed point if and only the nilradical of $I$ is equal to $\m... | 4 | https://mathoverflow.net/users/3485 | 76107 | 46,136 |
https://mathoverflow.net/questions/76102 | 5 | Let $f\colon X\to Y$ be a finite morphism between two smooth irreducible varieties over an algebraically closed field $k$. Let $x\in X$ be a (closed) point, and $y = f(x)$. Define the multiplicity of $f$ at $x$ to be $m\_f(x) := \dim\_k\mathcal{O}\_{X,x}/M\_y\mathcal{O}\_{X,x}$ where here $\mathcal{O}\_{X,x}$ is the lo... | https://mathoverflow.net/users/17995 | Upper semicontinuity of multiplicities for finite morphisms between varieties | M. Lejeune-Jalabert and B. Teissier. *Normal cones and sheaves of relative jets*. Compositio
Math., 28:305–331, 1974
Actually the result is much more general, not just for finite morphisms.
| 3 | https://mathoverflow.net/users/1939 | 76110 | 46,137 |
https://mathoverflow.net/questions/76116 | 1 | Let $P$ be an interval in $\mathbf{R}$, $n \in \mathbf{N}$. Assume that a function $f: P \rightarrow \mathbf{R}$ satisfies $\Delta^{n+1}\_h f(x)=0$ for every $x \in P$ and every $h>0$ such that $x+(n+1)h \in P$. I want to know whether there exists a function $F: \mathbf{R} \rightarrow \mathbf{R}$ such that $\Delta^{n+1... | https://mathoverflow.net/users/17110 | Extension of polynomial functions | Yes, such an extension does exist.
Let $\{ e\_i \}\_{i \in I}$ be a basis of $\mathbb{R}$ over $\mathbb{Q}$. Pick any finite subset $e\_1, ..., e\_k$ of the $e\_i$s, and look at the restriction of $f$ to $P' = (\mathbb{Q}e\_1 \oplus \cdots \oplus \mathbb{Q}e\_k) \cap P$. It's easy to see that there is a unique extens... | 1 | https://mathoverflow.net/users/2363 | 76120 | 46,142 |
https://mathoverflow.net/questions/76118 | 7 | In his paper "Spin structures and quadratic forms on surfaces", Johnson constructs a bijection between the set of spin strucutres on a **smooth closed orientable surface** $S$ and the set of quadratic forms on $H\_1(S,\mathbb{Z}\_2)$.
It seems to me that this result (and all the proofs) extends to the case of a **no... | https://mathoverflow.net/users/17998 | Spin structures and quadratic forms on surfaces | It is not quite true. There is an algebraic gadget one can produce from a Spin structure on a surface with boundary (where we *fix* the Spin structure along the boundary), and one gets a bijection from Spin structures to such gadgets, but the process is horribly non-canonical. It does however let you prove that there a... | 12 | https://mathoverflow.net/users/318 | 76123 | 46,144 |
https://mathoverflow.net/questions/76126 | 3 | Assume that X and Y are abelian schemes (or even abelian varieties) over a base T. If $\ S --> T$ is a PD nilpotent thickening (i.e. the ideal of $\ S$ in $ \ T$ is a nilpotent divided power ideal) and S is of characteristic $\ p>0$ . If $\ X\_{0}$ and $ \ Y\_{0}$ are reductions of $\ X$ and $\ Y$ to $\ S$. If $\ f\_{0... | https://mathoverflow.net/users/18000 | lifting the isomorphisms between abelian schemes over PD thickenings | No, this is very far from being true.
For a counterexample, let $S = Spec(\mathbb{Z}/p)$ and $T = Spec(\mathbb{Z}/p^2)$. The versal deformation space of an elliptic curve $E$ over $S$ is isomorphic to $Spec(\mathbb{Z}\_p[[x]])$ so lifts of $E$ to $T$ are parametrized by the set of homomorphisms of local algebras $Hom... | 1 | https://mathoverflow.net/users/519 | 76132 | 46,147 |
https://mathoverflow.net/questions/76134 | 39 | What is a purely topological characterisation of the real line( standard topology)?
| https://mathoverflow.net/users/18005 | Topological Characterisation of the real line. | Here are a few examples that came up in a first search. Ward in "The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198 proved the characterization of the real line as a connected, locally connected separable metric space, such that every point is a strong cut point ... | 39 | https://mathoverflow.net/users/2384 | 76139 | 46,149 |
https://mathoverflow.net/questions/76133 | 6 | Hello,
I'm searching for a good subject of Dynamical Systems theory in which I can propose a theme for a undergraduate research opportunity program. As I'm a undergraduate student, I have had only be exposed to subjects such as linear and abstract algebra, real analysis (calculus, basic topology of $\mathbb{R}^{n}$, ... | https://mathoverflow.net/users/17760 | Dynamical Systems for undergraduate students | I would recommend reading through [*Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering*](http://www.google.com/products/catalog?q=nonlinear+dynamics+and+chaos&oe=utf-8&rls=org.mozilla%3Aen-US%3Aofficial&client=firefox-a&um=1&ie=UTF-8&tbm=shop&cid=3730006166953307001&sa=X&ei=... | 8 | https://mathoverflow.net/users/934 | 76142 | 46,152 |
https://mathoverflow.net/questions/76143 | 6 | For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}\_X$ or $-K\_X$ is ample. (I prefer the second notation, although it is not very correct),
In characteristic 0, as far as I know, there is a classification. $X$ ha... | https://mathoverflow.net/users/1887 | Del pezzo surfaces in positive characteristic | The classical classification of (smooth) Del Pezzo surfaces as blow-ups relies on the Kodaira vanishing theorem in characteristic zero, but is actually true over any algebraically closed field. See for example Kollar's *Rational curves on algebraic varieties* book, section III.3. ([This paper by Xie](http://arxiv.org/a... | 10 | https://mathoverflow.net/users/3996 | 76149 | 46,157 |
https://mathoverflow.net/questions/76152 | 2 | I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter.
To clarify, suppose I have a 1-parameter family $T\_h$ of self-adjoint operators in $L(H)$, $h \in I$ open and suppose that every $T\_h$ has a discrete and well-ordered spectr... | https://mathoverflow.net/users/16702 | Smooth dependence of the spectrum on the operator | Look at Kato's book on Perturbation Theory for Linear Operators.
| 1 | https://mathoverflow.net/users/12120 | 76157 | 46,161 |
https://mathoverflow.net/questions/76153 | 10 | $N$ points are generated randomly within a unit square, with a uniform distribution.
What is the probability that the points form a connected graph, given that two points are connected if the distance between them is less than or equal to $d$?
(this should obviously be some function of $N$ and $d$).
If you don't kno... | https://mathoverflow.net/users/18009 | Probability of Generating a Connected Graph | Just a few more comments to the answers and references already posted. I will denote your graph by $G(n,d(n))$. I'm not sure if this is satisfactory enough, but with fairly standard methods one can prove that if $\mu=ne^{-\pi nd(n)^2}\to 0$ as $n\to \infty$ then the graph is aas connected. In general the probability th... | 10 | https://mathoverflow.net/users/2384 | 76164 | 46,163 |
https://mathoverflow.net/questions/76158 | 0 | [Question cross posted on stack-exchange]
I'm slowly working through Part III of the book, and I'm scratching my head a bit while reading the proof of Lemma 3.2 (here reproduced):
Let $X, A$ be a pair of CW-spectra, and $Y, B$ a pair of spectra such that $\pi\_\*(Y, B) = 0$. Suppose given a map $f: X \to Y$ and a h... | https://mathoverflow.net/users/15331 | Understanding a proof in Adams' Stable Homotopy and Gen. Coh | Well perhaps Adams did things a little out of order, but this should work:
We know that $h \circ i\_0 = f \circ j$ where $j: A \rightarrow X$ is the inclusion map. Now choose representatives *of the compositions*, $k$ and $k'$, so that $k = k'$ on some cofinal spectrum in $A$. Next choose representatives for $h, f, i... | 2 | https://mathoverflow.net/users/6936 | 76169 | 46,165 |
https://mathoverflow.net/questions/76065 | 1 | OK, the heading was a bit tersely formulated...
If you have a quantum group and an irrep, you theoretically know the
R matrix (mathematicians are a notoriously idle lot, they give the
general formula and thus the problem is solved :-) - and the
characteristic equation of the R matrix is a valid skein equation.
No... | https://mathoverflow.net/users/11504 | Knot polynomials: Skein>Matrix>Group? | The Kauffman 2-variable knot polynomial probably can't be obtained from a quantum group if by this you mean the usual q-deformed universal enveloping algebras. If your two variables are $(r,q)$ and $r=\pm q^n$ then it can be obtained from quantum groups of type B,C or D (depending on $n$), see Wenzl's paper Comm. Math.... | 3 | https://mathoverflow.net/users/6355 | 76173 | 46,167 |
https://mathoverflow.net/questions/76174 | 1 | Why are two curves over a field k homeomorphic?
I have been able to prove that any variety of positive dimension over a field k has the same cardinality as k.
| https://mathoverflow.net/users/18013 | Any two curves over k homeomorphic | Alright I'll just answer so that I can get enough "reputation" to be able to comment and ask for a question to be closed because it belongs in other sites (see the faq).
A bijection between two topological spaces $f:X \rightarrow Y$ where open sets are complements of finite sets is a homeomorphism.
| 20 | https://mathoverflow.net/users/17980 | 76180 | 46,170 |
https://mathoverflow.net/questions/76191 | 3 | A cone is a $R\_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R\_+$-linear map. That is a map $f:C\to C$ such that $f(\alpha x +\beta y)=\alpha f(x) + \beta f(y)$ for all $\alpha,\beta \ge 0 $ and $x,y\in C$.
Supp... | https://mathoverflow.net/users/nan | Does the automorphism group of a cone determine the cone? | No. Consider solid angles in $\mathbb R^2$. They all (except the half-plane) are isomorphic, yet one may be strictly included in another.
Furthermore, a generic cone in $\mathbb R^d$, $d\ge 3$, has a trivial automorphism group, so you may have $Aut(C)=Aut(D)$ in the strongest possible sense but $C\ne D$.
| 9 | https://mathoverflow.net/users/4354 | 76196 | 46,176 |
https://mathoverflow.net/questions/9177 | 24 | Let $(V,A)$ be a [tournament](http://en.wikipedia.org/wiki/Tournament_%28graph_theory%29). A subset of vertices $V'\subseteq V$ is stable if
there exists no $v\in V\setminus V'$ such that $V'\cup${$v$} contains an inclusion-maximal transitive subtournament with source $v$.
(In other words, $V'$ is stable if for ever... | https://mathoverflow.net/users/2647 | Disjoint stable sets in tournaments | The statement is false. The existence of a counter-example using a probabilistic argument was shown by Chudnovsky, Kim, Liu, Norin, Scott, Seymour, and Thomasse.
| 8 | https://mathoverflow.net/users/2647 | 76199 | 46,178 |
https://mathoverflow.net/questions/76193 | 4 | Let $f \in \mathbb{Z}[x]$ be monic, irreducible and hyperbolic (no roots of absolute value $1$), and such that $f(0)= \pm 1$.
Denoting as $c\_{p}(x)$ the cyclotomic polynomial $$c\_{p}(x)=1+x+\cdots +x^{p-1},$$
my question is: how can be characterized the (certainly finite?) set of primes $p$ for which $f$ and $c\_{p}$... | https://mathoverflow.net/users/7456 | Cyclotomic polynomials coprime to a fixed polynomial | Not an answer, but the set of $p$ that you define is quite likely infinite (Oops, see David's comment). Here is an argument. Let $K$ be the splitting field of the polynomial $f(x^2)$ and $\ell$ a prime that splits completely in $K$. There are lots of those primes and is a reasonable assumption (?) that infinitely many ... | 3 | https://mathoverflow.net/users/2290 | 76200 | 46,179 |
https://mathoverflow.net/questions/76189 | 16 | Please allow me to ask a potentially dumb question (or maybe more precisely, a question floating on clouds of ignorance):
>
> Why is a max-plus algebra called a *tropical algebra*?
>
>
>
| https://mathoverflow.net/users/8430 | What's tropical about tropical algebra? | A lot of sources mention that the adjective "tropical" is given in honor of Imre Simon, but it seems hard to find who precisely coined the term. I found some sources which attribute this to some French mathematicians. [Here](http://bit-player.org/2009/treats-tropiques) is what Bryan Hayes writes on the topic:
>
> F... | 20 | https://mathoverflow.net/users/2384 | 76202 | 46,180 |
https://mathoverflow.net/questions/76186 | 5 | I would like to apply the usual 'functoriality properties' of the perverse $t$-structure to torsion (constructible complexes of) sheaves (I am in the algebraic setting, so these are etale sheaves, but probably the difference from the 'topological case' is not very large here) i.e. I want to use the corresponding left a... | https://mathoverflow.net/users/2191 | Functoriality properties of the perverse $t$-structure for torsion (constructible complexes of) sheaves | If you want only $\mathbb{Z}/\ell\mathbb{Z}$ coefficients (not general $\mathbb{Z}/\ell^m\mathbb{Z}$), then there is only one middle perverse t-structure, which is good. The way the exactness properties of the 4 operations is proved in BBD is to reduce to $\mathbb{Z}/\ell/\mathbb{Z}$ coefficients (see 4.0), so the answ... | 5 | https://mathoverflow.net/users/12336 | 76213 | 46,186 |
https://mathoverflow.net/questions/76212 | 1 | So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$
be 3 monic polynomials such that $f=gh$. So I would like to have a simple example
of a ring $R$ for which one has that $f,g\in R[x]$ but $h\notin R[x]$.
P.S. May be working with an non-maximal order of a Dedekind ring is good enough. Ne... | https://mathoverflow.net/users/11765 | quotient of integral polynomials not being integral | We show that $h$ must be in $R[x]$. Suppose that we have polynomials
$$g = x^n + a\_1x^{n-1} + ... + a\_n \in R[x]$$
$$h = x^m +b\_1x^{m-1} + ... + b\_m \in K[x]$$
such that $f = gh \in R[x]$ but $h \notin R[x]$. Let $r:= \min \{i \mid b\_i \notin R\}$. Since $f \in R[x]$ we have $b\_r + a\_1b\_{r-1} + ... + a\_{r-1... | 8 | https://mathoverflow.net/users/17901 | 76219 | 46,188 |
https://mathoverflow.net/questions/76216 | 10 | My question is about the theory of complexity, but let me first explain my motivation, which comes from number theory or more precisely from trying to understand a question/conjecture of Serre and a remark he made about it. Yesterday at Harvard, Jean-Pierre Serre gave a wonderful colloquium on "Variation with $p$ of th... | https://mathoverflow.net/users/9317 | Polynomial-time complexity and a question and a remark of Serre | Yes, if $1\le r< s$, there exists a set of integers computable in time $O((\log x)^s)$, but not in time $O((\log x)^r)$, where $x$ is the input number written in binary. This is an instance of the [time hierarchy theorem](http://en.wikipedia.org/wiki/Time_hierarchy_theorem). However, the only known such problems are ob... | 12 | https://mathoverflow.net/users/12705 | 76221 | 46,189 |
https://mathoverflow.net/questions/76194 | 1 | Let $G$ be a compact (connected) Lie group. Suppose that a $G$-principal bundle $\pi:P\rightarrow Q$ is given.
Is it always possible to equip $P$ and $Q$ with Riemannian metrics, s.t. $\pi$ is totally geodesic? Notice that $g\_P,$ the metric on $P,$ does not have to be $G$-invariant and $\pi$ does not have to be a Ri... | https://mathoverflow.net/users/3509 | Is it possible to make the principal bundle projection map totally geodesic? | I think the (negative) answer follows from Vims' paper [Totally Geodesic Maps](http://projecteuclid.org/euclid.jdg/1214429276).
Indeed, assume for simplicity that the manifolds $P$ and $B$ are compact. Vilms proves that any totally geodesic map factors as a totally geodesic Riemannian submersion followed by an immer... | 6 | https://mathoverflow.net/users/1573 | 76222 | 46,190 |
https://mathoverflow.net/questions/76198 | 7 | It is known (by Gauss) that for a prime $p \equiv 1 \pmod 3$ there is a "unique" writing of $4p=A^2+27B^2$ where $A=1+p-M\_p$ and $M\_p$ is the number of solutions of $X^3+Y^3+Z^3=0$ in the projective plane $\mathbb{P^2(\mathbb F\_p)}$.
It's also known that the points on the cubic $y^3=x^3+1$ are in 1-1 correspondenc... | https://mathoverflow.net/users/18019 | Gauss Theorem and Weil Conjectures for elliptic curves | I am going to interpret "the Weil conjectures" as a synecdoche for "thinking about counting points in terms of eigenvalues of Frobenius operators." In this case, the answer is a resounding yes! However, the specific facts about eigenvalues of Frobenius which which are being used here are easier than the ones discussed ... | 8 | https://mathoverflow.net/users/297 | 76223 | 46,191 |
https://mathoverflow.net/questions/76226 | 4 | For 3-dim Poincare Conjecture, the assumption is 'simply connected'.
I am wondering whether simply connectedness assumption in 3-dim implies the same homotopy groups as the 3-sphere?
or If we switch the assumption of 'simply connected' to 'homotopy 3-sphere', would it be easier to proof Poincare Conjecuture.
| https://mathoverflow.net/users/16750 | "homotopy sphere" assumption in Poincare Conjecture | See the fifth paragraph of
<http://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf>
| 10 | https://mathoverflow.net/users/11142 | 76229 | 46,192 |
https://mathoverflow.net/questions/76220 | 2 | How can I evaluate the sum $$\sum\_{a+b=1 ;\ a,b \in \mathbb F\_p}\left(\frac{a}{p}\right) \chi(b)$$
where $\chi$ is a multiplicative character of $\mathbb F\_p^{\*}$ of order three?
Thanks!
| https://mathoverflow.net/users/18019 | How can I evaluate the sum $\sum_{a+b=1;\ a,b \in \mathbb F_p} \left(\frac{a}{p}\right) \chi(b)$ where $\chi$ is a multiplicative character of $\mathbb F_p^{*}$ of order three? | If I've understood the question correctly, this is just a Jacobi sum. The question only seems to make sense for $p$ a prime congruent to 1 mod 3. In this case, the order 3 character and the quadratic character are distinct, and not inverses of each other either, so the absolute value of the Jacobi sum is going to be $\... | 4 | https://mathoverflow.net/users/1384 | 76234 | 46,197 |
https://mathoverflow.net/questions/76236 | 5 | Let X be a compact Kähler manifold, $M\subseteq X$ be a complex submanifold, is there a biholomorphism between a neighborhood of the zero section of the normal bundle $NM$ of $M$ and a neighborhood $U(M)\subseteq X$?
| https://mathoverflow.net/users/4971 | Biholomorphism between neighborhood of a complex submanifold and a neighborhood of zero section of its normal bundle | The existence of such a biholomorphism is rather rare. For example, as a simple exercise you can check that already for a conic in $\mathbb CP^2$ such a biholomorphism does not exist. At the same time, in the case $M$ is "exceptional" in $X$, for example, $X$ is obtained by a blow up from $X'$ at a point $x'\in X'$ and... | 8 | https://mathoverflow.net/users/943 | 76238 | 46,199 |
https://mathoverflow.net/questions/76228 | 13 | Let $r,s,t>1$ be positive integers. Must there exist a finite group $G$ with elements $x$ and $y$ such that $ord(x)=r$, $ord(y)=s$, and $ord(xy)=t$?
The answer is probably "yes." Is there a nice description of such a $G$?
| https://mathoverflow.net/users/18027 | Is $ord(xy)$ independent of $ord(x)$ and $ord(y)$ in a finite group? | Let $a$ and $b$ be elements of a group $G$. If $a$ has order $m$ and $b$ has
order $n$, what can we say about the order of $ab$? The next theorem shows
that we can say nothing at all.
THEOREM: For any integers $m,n,r>1$, there exists a finite group $G$ with
elements $a$ and $b$ such that $a$ has order $m$, $b$ has or... | 17 | https://mathoverflow.net/users/18030 | 76240 | 46,200 |
https://mathoverflow.net/questions/76245 | 1 | Hi All,
Let us consider a P x Q real matrix (P >= Q). It can be thought of as an element of $\mathbb{R}^{PQ}$. We are considering Lebesgue measure over that space. My question is whether the subspace of matrices with repeated singular values are of measure 0 or not.
Any suggestions would be welcome.
Thanks
Ashin... | https://mathoverflow.net/users/18033 | Measure of Subspace of Matrices with repeated Singular Values | The eigenvalues of
$$
\widehat M = \begin{pmatrix}0&M\\ M^T&0\end{pmatrix}
$$
are the squares of the singular values of $M$.
A symmetric $n\times n$ matrix $N$ has a repeated eigenvalue if and only if the rank of
$$
N\otimes I - I\otimes N
$$
is less than $n^2-n$. So the set of matrices $\widehat M$ with a repeated ... | 3 | https://mathoverflow.net/users/1266 | 76249 | 46,205 |
https://mathoverflow.net/questions/76188 | 13 | It is probably a trivial question. But I don't see the answer.
Is there any Hurewicz fibration $\mathbb{R}^n\to \mathbb{S}^n$ ?
Is there any fibration $X\to \mathbb{S}^n$, when $X\subset \mathbb{R}^n $?
I appreciate any help. Thank you very much!
| https://mathoverflow.net/users/18017 | is there any fibration $\mathbb{R}^n\to \mathbb{S}^n$? | Edit: The following simplifies the original answer (which unnecessarily used singular cohomology).
If $f:\Bbb R^n\to S^n$ is a fibration, then as Mark noted, a fiber $F$ of $f$ is *weak* homotopy equivalent to $\Omega S^n$ (using the 5-lemma, see Prop. 4.66 in Hatcher). I claim that $F$ is in fact homotopy equivalent... | 12 | https://mathoverflow.net/users/10819 | 76251 | 46,207 |
https://mathoverflow.net/questions/76255 | 10 | Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x\_1,x\_2,..,x\_N)$ along with linear constraints $x\_n \ge 0, \forall n$
$x\_n \le c\_n$
and $\sum\_{n=1}^N x\_n = 1$ find an optimal (local) solution by randomly sampling the feasible region. $c\_n$ are just constants.
The issue I am ... | https://mathoverflow.net/users/18035 | Random Sampling a linearly constrained region in n-dimensions... | Your constraints $x\_n \geq 0$, $\sum\_{n=1}^N x\_n = 1$, are those for the [standard simplex](http://en.wikipedia.org/wiki/Simplex#The_standard_simplex). You could try [uniform sampling from the standard simplex](http://www.cs.cmu.edu/~nasmith/papers/smith+tromble.tr04.pdf), and then reject any sample that doesn't als... | 6 | https://mathoverflow.net/users/9716 | 76258 | 46,210 |
https://mathoverflow.net/questions/30653 | 24 | Does there exist an analog of the HNN Embedding Theorem for the class of countable amenable groups? In other words, is it true that every countable amenable group embeds into a 2-generator amenable group? Perhaps easier, is it true that every countable amenable group embeds into a finitely generated amenable group?
| https://mathoverflow.net/users/4706 | HNN Embedding Theorem for Amenable Groups? | If I am not mistaken, the answer is "yes".
**Theorem.** *Every countable amenable (respectively, elementary amenable) group embeds into a $2$-generated amenable (respectively, elementary amenable) group.*
The proof is based on the following lemma, which admits a quite elementary proof using wreath products (see [P.... | 18 | https://mathoverflow.net/users/10251 | 76261 | 46,212 |
https://mathoverflow.net/questions/76235 | 4 | Given any collection $\mathcal{C} = \{E\_1, E\_2, ..., E\_m\}$ of finite and nonempty discrete sets, is there a set $I$ such that
$$ \forall E\_k \in \mathcal{C}, \; E\_k \cap I \neq \emptyset,$$
and
$$ \forall i \in I, \; \exists E\_k \in \mathcal{C}, E\_{k} \cap I = \{i\} $$
?
For example, for $E\_1 = \{1,2\}$, $E... | https://mathoverflow.net/users/18028 | Hitting set problem variant | Let $I$ be a minimal set that intersects each $E\_j$, where minimal means that no point can be removed from $I$ without it no longer intersecting each $E\_j$. Take any $i\in I$. We know $i$ lies in some $E\_j$, otherwise $I$ was not minimal. If every $E\_j$ that contains $i$ also contains another element of $I$, then w... | 4 | https://mathoverflow.net/users/9025 | 76267 | 46,216 |
https://mathoverflow.net/questions/76274 | 3 | Below is actually a statement in textbook. But I don't have a good intuition of it.
If we want a stochastic process $W\_t$ to satisfy
i). $s\neq t$ implies $W\_s$ and $W\_t$ are independent,
ii). $\{W\_t\}$ is stationary,
iii). $E[W\_t]=0$ for all t,
then $W\_t$ cannot have continuous paths.
I hope someone can poin... | https://mathoverflow.net/users/17708 | Non-existence of such a continuous stochastic process | Something much stronger holds. One can actually show that no nontrivial such process has measurable sample parths. No assumption on the mean and no stationarity assumption is necessary. This is Proposition 1.1. in Y. Sun, [The almost equivalence of pairwise and mutual
independence and the duality with exchangeability](... | 5 | https://mathoverflow.net/users/35357 | 76277 | 46,224 |
https://mathoverflow.net/questions/76205 | 8 | This is a question about the open problem [Fibonacci divisibility](http://garden.irmacs.sfu.ca/?q=op/fibonacci_divisibility) from the Open Problem Garden.
The problem, originally stated in 1960 by D.D. Wall, has several equivalent formulations one of which is:
Find a prime $p$ with $p^2|a\_{p-\left(\frac{p}{5}\righ... | https://mathoverflow.net/users/17879 | Is the Crandall, Dilcher and Pomerance heuristic concerning Wall-Sun-Sun primes still state of the art? | As quid mentioned, Klyve and I have done some computational investigations on Fibonacci-Wieferich/Wall-Sun-Sun primes. In particular, we collected all primes $p < 9.7\times10^{14}$ such that $F\_{p-(p/5)} \equiv Ap \pmod{p^2}$ with $|A| < 2\times10^6$. I've just crunched our data for primes in the range from $6.5\times... | 9 | https://mathoverflow.net/users/2000 | 76288 | 46,228 |
https://mathoverflow.net/questions/76227 | 10 | Consider the following situation: I have a set $A$ of $n$ vertices and a set $B$ of $N = n^2$vertices. I consider the bipartite graph $(A, B)$ and put at random $M = n^{1 + \varepsilon}$ edges (or I could put each edge independently with probability $p$ such that $pnN = M$, this shouldn't make a big difference). Then I... | https://mathoverflow.net/users/2192 | Random bipartite graphs | Take the case of choosing edges independently with probability $p=n^{-2+\epsilon}$. As you say, it won't make much difference compared to choosing $n^{1+\epsilon}$ edges. Assume $\epsilon<\frac12$.
Consider a particular vertex on the left. The number of possible 2-edge paths from that vertex to any other vertex on th... | 9 | https://mathoverflow.net/users/9025 | 76289 | 46,229 |
https://mathoverflow.net/questions/76282 | 5 | Suppose I have a generic 3-dimensional affine linear subspace of the 6-dimensional space of symmetric $3 \times 3$ matrices. Does such a space necessarily intersect the set of definite (either positive or negative definite) symmetric matrices?
Genericity is important here; this is certainly not necessarily true for a... | https://mathoverflow.net/users/18048 | Existence of definite symmetric matrices satisfying affine linear constraints | No. Consider the 3-dimensional affine subspace consisting of all matrices having $1,1,-1$ on the diagonal (non-diagonal entries are arbitrary). A small perturbation of this subspace cannot intersect the cone of definite matrices. Indeed, a matrix from a perturbed subspace either has diagonal entries close to $1,1,-1$, ... | 7 | https://mathoverflow.net/users/4354 | 76292 | 46,231 |
https://mathoverflow.net/questions/76286 | 11 | This question arises from [HNN Embedding Theorem for Amenable Groups?](https://mathoverflow.net/questions/30653/hnn-embedding-theorem-for-amenable-groups/76261#76261)
Recall that a group $G$ is called *SQ-universal* if every countable group is isomorphic to a subgroup of a quotient of $G$. The first non-trivial examp... | https://mathoverflow.net/users/10251 | SQ-universality in the class of amenable groups | Here's what you were looking for:
MR2254627 (2007k:20086)
Erschler, Anna(F-PARIS11-M)
Piecewise automatic groups. (English summary)
Duke Math. J. 134 (2006), no. 3, 591–613.
20F65 (20F69 43A07 57M07)
"The main result of the paper under review is stated as follows: For any function f:N→N there exists a finitely ... | 10 | https://mathoverflow.net/users/4706 | 76305 | 46,234 |
https://mathoverflow.net/questions/70108 | 4 | Does there exist a polynomial Hamiltonian function $H$ on some $\mathbb{R}^{2n}$ such that
1. Any polynomial function $P$ such that $\{P,H\}=0$ is of the form $p(H)$ for some polynomial $p$ in one variable;
2. There exists a smooth function $F$ such that $\{F,H\}=0$, and yet $F$ is *not* of the form $f(H)$ for any s... | https://mathoverflow.net/users/8203 | Non-polynomial integrals of motion for polynomial dynamical systems | Here is a tentative Hamiltonian answer having 2 degrees of freedom.
Take $H = (1/2)(1 + a x^2 + bxy + cy^2) (p\_x ^2 + p\_y ^2)$.
for essentially any parameters $a, b, c$ for which
$ax^2 + bxy + cy^2$ is positive definite ($a x^2 + bxy + c y^2) > \epsilon (x^2 + y^2)$)
but NOT a multiple of $x^2 + y^2$. This $H$ is... | 2 | https://mathoverflow.net/users/2906 | 76308 | 46,235 |
https://mathoverflow.net/questions/37036 | 8 | Hope, MO is the right place for this question (if not so: where would you pose it?).
Consider a two-body system in classical mechanics. As long as the interaction depends only on the distance of the two bodies, the two-body problem is integrable/solvable. Now consider the two bodies in a fixed external field. (This i... | https://mathoverflow.net/users/2672 | Two interacting bodies in an external field | I seriously doubt there is any general criteria. However there are
more than one beautiful explicit examples of an external field which
lead to an integrable problem. The simplest and probably best known is
that of a constant field.
An absolutely beautiful description of this and its solution can be found in the book
`... | 8 | https://mathoverflow.net/users/2906 | 76309 | 46,236 |
https://mathoverflow.net/questions/76291 | 10 | I'd like to prove the non-existence of a real analytic function, injective, non-surjective
that sends rationals to rationals.
Is it a classical result ? If not, any hints on how to prove it ?
Thanks in advance for you help.
| https://mathoverflow.net/users/18049 | Real analytic function, injective, non surjective and preserving the rationals ? | The statement in the question is not true. Given any two enumerable and dense sets in open intervals of the reals, there is a (complex) analytic1 function giving a bijection between them. See the following paper: [Analytic Transformations of Everywhere Dense Point Sets](http://www.jstor.org/stable/1989166), Philip Fran... | 15 | https://mathoverflow.net/users/1004 | 76310 | 46,237 |
https://mathoverflow.net/questions/76307 | 12 | Hirzebruch, in the paper 'Arrangements of Lines and Algebraic Surfaces'
constructs a special $K3$ surface out of a 'complete quadrilateral' in
$CP^2$. A complete quadritlateral consists of
4 points in general position and the $6$ lines joining them.
Over each line Hirzebruch forms the local 2:1 fold cover to get... | https://mathoverflow.net/users/2906 | A K3 over $P^1$ with six singular $A_1$- fibers? | [**EDITED** to give likely identification of the OP's surface at the end]
An elliptic K3 surface has discriminant $\Delta$ of degree $24$, and $\Delta$ has valuation 2 or 3 at an $A\_1$ fiber depending on whether the Kodaira type is I$\_2$ or III, so if that's the only kind of singular fiber there must be at least $2... | 10 | https://mathoverflow.net/users/14830 | 76311 | 46,238 |
https://mathoverflow.net/questions/76314 | 11 | Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)\_{\mathbb Q} = CH(X)\otimes\_{\mathbb Z} \mathbb Q$. My question is
>
> Suppose $CH(X)\_{\mathbb Q}$ is finite-dimensional as a $\math... | https://mathoverflow.net/users/2083 | Genus of smooth varieties with small Chow group | As you suspect, the answer to your first question is yes.
A. Roĭtman in
"Rational equivalence of zero-dimensional cycles". Math. USSR-Sb. 18 (1974), 571--588, generalised Mumford's theorem to show that
if $h^q(X, \mathcal{O}\_X) > 0$ for any $q>0$ then $CH\_0(X)$ is infinite dimensional. If $q > 1$
then there is no ... | 12 | https://mathoverflow.net/users/519 | 76316 | 46,241 |
https://mathoverflow.net/questions/76295 | 19 | What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it can be thought of as a measure over a uniform distribution of phases on a circle, i.e. a matrix representing $M \in U(... | https://mathoverflow.net/users/1674 | Intuition for Haar measure of random matrix | You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices.
To form your intuition, consider $N=1$. You then have $U=e^{i\phi}$, with $0<\phi\leq 2\pi$ and $d\mu(U)=d\phi$ measures the perimeter of the unit circle. This is a uniform measure, b... | 22 | https://mathoverflow.net/users/11260 | 76322 | 46,244 |
https://mathoverflow.net/questions/76318 | 1 | I am reading a paper on Riemann surfaces and faced a problem about one of the refernces the author gave in the exlaination of one of the results.
Here is a summary of what I am reading:
Let $X\_1$ and $X\_2$ be two surfaces. Let $\pi\_1:\widetilde{X\_1}\longrightarrow X\_1$, $\pi\_2:\widetilde{X\_2}\longrightarrow ... | https://mathoverflow.net/users/18054 | A theorem by Hopf on surfaces | Since the surfaces of genus $\ge 1$ are aspherical, the homotopy types of maps are classified by their induced homomorphisms of fundamental groups. This is essentially what the final result says. (But, since the fundamental group depends on a marked point in a nontrivial way, and homotopies do not respect marked points... | 4 | https://mathoverflow.net/users/4354 | 76323 | 46,245 |
https://mathoverflow.net/questions/76327 | 6 | The following question has been on [math.SE](https://math.stackexchange.com/q/66394/9464) for several days. Without having a satisfying answer, I'd like to ask the experts here.
In mathematics, the [big $O$ notation](http://en.wikipedia.org/wiki/Big_O_notation) is used to describe the *limiting behavior* of a functi... | https://mathoverflow.net/users/nan | Is $O(10^{-6})$ an acceptable notation in numerical analysis? | I have never seen this in civilized literature. Your professor is probably a visigoth.
| 7 | https://mathoverflow.net/users/11142 | 76332 | 46,247 |
https://mathoverflow.net/questions/76328 | 5 | In perfect elastostatics, the unknown is the displacement $x\mapsto y$, where $x\in\Omega\subset{\mathbb R}^3$ is the reference configuration, and $y\in{\mathbb R}^3$. It obeys to an 2nd-order PDEs. When we rewrite the PDE as a 1st-order system, the unknown becomes the deformation gradient $F(x):=\nabla y$.
A fundam... | https://mathoverflow.net/users/8799 | Elastostatics and homotopy type | Complementing Dmitri's answer: one can see $SO(3)\cong \mathbb{P^3}(\mathbb{R})$ as the 3-skeleton of $K(\mathbb{Z}/2,1)\cong \mathbb{P^\infty}(\mathbb{R})$. Assuming $\Omega$ and its boundary are non-pathological, they will have the homotopy type of 2-dimensional $CW$-complexes. If $X$ is a 2-dimensional $CW$-complex,... | 5 | https://mathoverflow.net/users/2349 | 76338 | 46,251 |
https://mathoverflow.net/questions/76325 | 6 | The Navier-Stokes equations can be written on a Riemannian manifold as:
$$\dot{u}+\nabla\_u u+ \Delta u=(df)^\* $$
$$d^\* u=0$$
where $\nabla$ is the Levi-Civita connection, $u$ is a vector field, $\Delta$ is the Laplacian, $df$ is the differential of $f$, $(df)^\* $ is the dual of $df$ via the metric, and $d^\*u$ is t... | https://mathoverflow.net/users/12806 | Navier-Stokes equations in Riemannian geometry | The answer and comments about Arnold and Marsden papers are a little off side. They concern the equation of inviscid fluids, called Euler equation. This differs from Navier-Stokes by the highest-order derivatives $\Delta u$. This changes completely the functional analysis background. Also, Euler equation has a geometri... | 10 | https://mathoverflow.net/users/8799 | 76346 | 46,255 |
https://mathoverflow.net/questions/75308 | 7 | Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one requirement. For example the uniform polyhedra have all vertices and edges congruent, but not all faces, and their duals ... | https://mathoverflow.net/users/15516 | Not quite regular polyhedra | It turns out that these polyhedra that have congruent vertices and faces have a name. They are the *Noble Polyhedra*. If one insists that they also be convex the Noble polyhedra are the regular polyhedra plus the [disphenoids](http://en.wikipedia.org/wiki/Disphenoid) mentioned in Douglas Zare's answer.
When one allo... | 11 | https://mathoverflow.net/users/15516 | 76355 | 46,258 |
https://mathoverflow.net/questions/76347 | 4 | Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a\_1,a\_2,\dots,a\_n\)}$, and then we can get the difference between two consecutive numbers and get the gap list:
$\{(a\_1, a\_2-a\_1,\dots ,a\_n-a\_{n-1})\}$
... | https://mathoverflow.net/users/8379 | Distribution of the biggest gap | This is the answer to a slightly modified version of
the problem. I hope that it would also lead to a solution
of the original version.
As I point out in my answer to [Math StackExchange question 66430](https://math.stackexchange.com/questions/66430/)
("What is the distribution of gaps?"),
if, in addition to the gap... | 3 | https://mathoverflow.net/users/nan | 76361 | 46,262 |
https://mathoverflow.net/questions/76352 | 12 | Orthogonal group of the quadratic form over fields, somehow, is well-studied. Indeed
E. Cartan has proved for quadratic forms over the reals or complexes that any
orthogonal transformation is a product of at most $n$ symmetries, where $n$ is
the dimensionality of the underlying vector space. This result was generalize... | https://mathoverflow.net/users/8419 | Orthogonal group of quadratic form | Yes, these groups are arithmetic lattices, and are therefore finitely generated.
I believe Selberg showed that they are cofinite volume (with respect to the
discrete action on the appropriate symmetric space).
When the form is definite, it is a finite group. When the form is Lorentzian,
the group may be shown to have... | 14 | https://mathoverflow.net/users/1345 | 76363 | 46,263 |
https://mathoverflow.net/questions/76349 | 9 | I was woolgathering about the notion of a scheme, and it occurred to me that I know of no non-affine scheme $S$ that is the union of $Spec(O\_K)$'s of some number field $K$ (I allow $K$ to vary - so that $S$ might be $Spec(O\_K)\cup Spec(O\_L)$ for example).
It would an interesting notion if one could patch rings of ... | https://mathoverflow.net/users/5756 | Is there a connected non-affine scheme $S$ such that it is the union of rings of integers of number fields? | If $i: \mathrm{Spec}(O\_K)\to S$ is an open immersion into a connected separate scheme $S$, then $i$ is an isomorphism. Indeed, the canonical morphism $\pi : \mathrm{Spec}(O\_K)\to \mathrm{Spec}(\mathbb Z)$ is finite (hence proper) and can be decomposed into $i$ followed by the canonical morphism $S\to \mathrm{Spec}(\m... | 28 | https://mathoverflow.net/users/3485 | 76368 | 46,266 |
https://mathoverflow.net/questions/76371 | 3 | Hi,
Fix $N > 3$ and consider the modular curve $X(N)$ parametrizing elliptic curves with
full level N structures. Let $\pi : E(N)\to X(N)$ be the universal elliptic curve. Then
$V=R^1\pi\_\*\mathbf Q$ defines a local system on $X(N)$. Define $H^1\_c = H^1\_c(X(N),Sym^kV)$ and $H^1=H^1(X(N),Sym^kV)$ for $k>=0$. Parabo... | https://mathoverflow.net/users/36285 | parabolic-Eisenstein decomposition of cohomology of modular curve | The fact that $H^1\_p$ is a direct summand in $H^1$ is obvious if we consider
just those objects as $\mathbb{Q}$-vector space (since a sub vector space always has a suplementary).
What you mean probably is "why is $H^1\_p$ a direct summand of $H^1$ as a module over the Hecke operators?". Then the answer is still yes, b... | 6 | https://mathoverflow.net/users/9317 | 76375 | 46,268 |
https://mathoverflow.net/questions/76374 | 2 | 1. This is surely well known. Let $X$ and $Y$ be smooth, projective, connected complex varieties. Then
$$H^2(X\times Y,Z/n)=H^2(X,Z/n)\oplus H^2(Y,Z/n) \oplus (H^1(X,Z/n)\otimes H^1(Y,Z/n))$$
for any $n>1$. I think I can prove this using a counting argument and the K\"unneth formula with coefficients in $Z$, but it se... | https://mathoverflow.net/users/18063 | Second cohomology group with finite coefficients of the product of two varieties | The identity map of $H^2(X,\mathbb{Z}/n)\oplus H^2(Y,\mathbb{Z}/n)$ can be decomposed as $$H^2(X,\mathbb{Z}/n)\oplus H^2(Y,\mathbb{Z}/n)\to H^2(X\times Y,\mathbb{Z}/n)\to H^2(X,\mathbb{Z}/n)\oplus H^2(Y,\mathbb{Z}/n)$$ where the first arrow is the sum of the pullbacks and the second arrow is the sum of the maps induced... | 1 | https://mathoverflow.net/users/2349 | 76382 | 46,270 |
https://mathoverflow.net/questions/76386 | 35 | It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace class.
The following attractive statement, however, is false:
**Non-theorem**:
*Let $a$ and $b$ be bounded operators on... | https://mathoverflow.net/users/5690 | tr(ab) = tr(ba)? | **EDIT:** Bill Johnson has pointed out a gap in my initial answer. It seems that bridging this gap is just as difficult as proving that $tr(AB)=tr(BA)$. Below I give two other proofs of this equality. The flawed proof is also reproduced at the end.
**Proof 1** (alluded to by Bill in his comment to Gjergji's answer): ... | 28 | https://mathoverflow.net/users/430 | 76389 | 46,275 |
https://mathoverflow.net/questions/41129 | 5 | I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of this subspace is $o(n)$.
It is not difficult to show that a symmetric function does not satisfy this property
by consid... | https://mathoverflow.net/users/9808 | A Boolean function that is not constant on affine subspaces of large enough dimension | Unless I am misreading it, the paper [Affine dispersers from subspace polynomials](http://web.mit.edu/swastik/www/affine-disperser.pdf) by Ben-Sasson and Kopparty gives an explicit construction which is nonconstant on any affine subspace of dimension less than $6 n^{4/5}$.
| 3 | https://mathoverflow.net/users/658 | 76401 | 46,282 |
https://mathoverflow.net/questions/76293 | 0 | I'm studying intersection of curves with a fixed plane cubic, the first case I consider is of course lines, in particular lines intersecting the cubic at only one point. The problem is quite easy and I've solved it in terms of basic intersection theory on $\mathbb{P}^2$. Now I'm considering a new way of attacking the p... | https://mathoverflow.net/users/14339 | sequence of sheaves for studying intersection | 1. is ok. For 2: I don't think you can prove the existence of flexes this way, but assuming P is a flex, then unicity of the line does follow from the iso. BTW, the unique "conic" that intersects B in 6P (P a flex) is the tangent line, doubled.
| 1 | https://mathoverflow.net/users/1939 | 76403 | 46,283 |
https://mathoverflow.net/questions/76326 | 7 | Consider the following game between *Alice* and *Bob*.
$\Sigma$ is a finite nonempty alphabet, $\Delta \notin \Sigma$ denotes
a special symbol, and $k > 0$ is a positive integer constant representing
the length of contexts.
1. *Alice* gets a message $w \in \Sigma^{\omega}$.
She then finds an appropriate index $i > k... | https://mathoverflow.net/users/18057 | Can you hide a letter without losing information? | We say that Alice catches the word if she can make the desired move. We prove that a protocol exists by the induction on $d=|\Sigma|$.
Your example states the base for $d=2$. Assume that we know Alice's strategy for $d-1$ letters; let $k'$ be the length of the words in the catching triples. Note that in fact Alice ca... | 5 | https://mathoverflow.net/users/17581 | 76418 | 46,291 |
https://mathoverflow.net/questions/76402 | 14 | Observe that we have $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$.
More generally, assume that $K$ is a finite extension of Q. Is there any $\alpha \in K$ such that $K=Q(\alpha^n)$ for every $n \in N$?
| https://mathoverflow.net/users/13947 | $Q(\sqrt{2})=Q((\sqrt{2}+1)^n)$ | In a more general setting, the following is true : let $K$ be an infinite field and $L/K$ be a finite separable extension whose Galois closure $M$ contains only finitely many roots of unity (this assumption is true for number fields). Then there exists $\alpha \in L$ such that $L=K(\alpha^n)$ for every $n \geq 1$.
Pr... | 21 | https://mathoverflow.net/users/6506 | 76424 | 46,293 |
https://mathoverflow.net/questions/73945 | 32 | Given a finitely generated $\def\CC{\mathbb C}\CC$-algebra $R$ and a $\CC$-point (maximal ideal) $p\in Spec(R)$, I define the *singularity type* of $p\in Spec(R)$ to be the isomorphism class of the completed local ring $\hat R\_p$, as a $\CC$-algebra.
Do there exist non-algebraic singularity types? That is, does ther... | https://mathoverflow.net/users/1 | Wanted: example of a non-algebraic singularity | I got this example from Frank Loray. I'll explain the analytic version, but the formal variant works just as well.
Let $U\subset \mathbb{C}$ be open. Choose two holomorphic functions $f$, $g$ which are algebraically independent over $\mathbb{C}$ (e.g. $f(z)=z$, $g(z)=e^z$). For simplicity, assume that $f$, $g$, $0$,... | 35 | https://mathoverflow.net/users/7666 | 76442 | 46,305 |
https://mathoverflow.net/questions/76429 | 3 | Hello! I have a problem with the following Lemma, which is mentioned in Serre's book "Trees" on page 60. In the book it is the Example 6.3.4.:
*Lemma*: Let $G$ be a group acting (without inversion) on a tree $X$. Let $X^G$ be the set of fixed points of $G$ in $X$ ($X^G$ is a subgraph of $X$). Let $G'$ be a subgroup o... | https://mathoverflow.net/users/17255 | Fixed points of a group-operation on a tree, Serre's book "Trees" 6.3.4. and Prop 27 | The first two questions have been answered. The third question is also easy. If $G/H$ is cyclic and we assume that $H$ has fixed points, let $T$ be the subtree of fixed points of $H$. Then $G/H$ acts on that tree. Since $G/H$ is cyclic and the action is without inversions, it either has a fixed point, whence $G$ has a ... | 3 | https://mathoverflow.net/users/nan | 76453 | 46,312 |
https://mathoverflow.net/questions/76380 | 6 | I am reading Bjorn Poonen's very nice survey on Hilbert's Tenth problem
([http://www-math.mit.edu/~poonen/papers/uniform.pdf](http://www-math.mit.edu/%7Epoonen/papers/uniform.pdf)), and while I believe I understand the mathematics well, I have widespread difficulties with the meta-mathematics of these questions. To ill... | https://mathoverflow.net/users/9317 | A meta-mathematical question related to Hilbert tenth problem | There's a general "trick" for handling all issues of this sort. Take any mathematical theorem that a platonist regards as meaningful. Formalize it as a formal theorem T in ZFC. The formalist will now accept the sentence, "ZFC proves T."
Here, the only potentially confusing concept is that of truth. But to say that so... | 13 | https://mathoverflow.net/users/3106 | 76454 | 46,313 |
https://mathoverflow.net/questions/76377 | 1 | Hi all,
I am encountering a problem in calculating the sum of multinomial coefficients. The original problem is about a signal source with $k$ symbols under uniform distribution, i.e.
$p\_0=p\_1=\cdots=p\_{k-1}= \dfrac{1}{k}$.
My problem is to find an appropriate string length $N$ with two concerns:
* The pr... | https://mathoverflow.net/users/18064 | Restriction sum of multinomial coefficients | The probability that a sequence does not contain all symbols is bounded above by the expected number of symbols omitted, $k (\frac{k-1}{k})^N \approx k \exp (\frac{-N}{k})$. You can ensure this is small by choosing $N$ to be large relative to $k \log k$.
You can get an exact probability easily using inclusion-exclusi... | 1 | https://mathoverflow.net/users/2954 | 76460 | 46,316 |
https://mathoverflow.net/questions/36693 | 68 | At the end of the paper [*Division by three*](http://arxiv.org/abs/math/0605779v1) by Peter G. Doyle and John H. Conway, the authors say:
*Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat d... | https://mathoverflow.net/users/8176 | Nelson's program to show inconsistency of ZF | Nelson claimed to have succeeded just now.
<http://www.math.princeton.edu/~nelson/papers/outline.pdf>
I hope consensus about this forms soon, so I can know what to do with the rest of my life. If only I had been born a few years later, I wouldn't be put into the position of worrying that my chosen career path is do... | 25 | https://mathoverflow.net/users/11145 | 76462 | 46,317 |
https://mathoverflow.net/questions/76413 | 16 | Here is the short version:
>
> Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E\_d(\mathbb{Q})\_{tors}$ vary, as $E\_d$ runs through the quadratic twists of $E$?
>
>
>
Here is the longer version:
I have been playing with SAGE this morning. I inserted the elliptic curve ('11a1') $$E : y... | https://mathoverflow.net/users/13741 | Torsion subgroups in families of twists of elliptic curves | **Theorem** (originally due to Setzer?): Fix $E/\mathbb{Q}$ with $j(E)$ not 0 or 1728. Then for all but finitely many inequivalent twists $E\_d$, the torsion subgroup $E\_d(\mathbb{Q})\_{tors}$ is isomorphic to $E[2](\mathbb{Q})$, so in particular $E\_d(\mathbb{Q})\_{tors}$ has order 1, 2, or 4. (Probably he also prove... | 18 | https://mathoverflow.net/users/11926 | 76464 | 46,319 |
https://mathoverflow.net/questions/76480 | 6 | Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ($n$ is prime to the residue field characteristic) is non-zero?
| https://mathoverflow.net/users/2191 | An example of an affine variety with non-zero Chow groups | I think there will be many such varieties.
For example, let $Q$ be a smooth $4$-dimensional projective quadric, $Q'$ a hyperplane section of $Q$ and $A = Q \backslash Q'$. For any $i$, we have the localisation sequence
$$ CH^{i-1}(Q') \to CH^i(Q) \to CH^i(A) \to 0 \ . $$
Since $CH^1(Q') = \mathbb{Z}/n\mathbb{Z}$... | 5 | https://mathoverflow.net/users/519 | 76482 | 46,322 |
https://mathoverflow.net/questions/76477 | 6 | Is it true that groups $\langle a,b \mid a^n b^k=b^ka^{n+1}, b^la^s=a^sb^{l+1}\rangle$ are non-trivial for almost all (in any sense:))) $n,k,l,s\in\mathbb N$?
| https://mathoverflow.net/users/4298 | G = [G,G] with two generators | The group is always trivial. Indeed, $a^{-ns}b^{l^n}a^{ns}=b^{(l+1)^n}$. On the other hand,
$a^{ns}=b^ka^{(n+1)s}b^{-k}$. Substitute $a^{ns}$ from the second equality to the first. You will get that $a^{(n+1)s}$ conjugates $b^{l^n}$ to $b^{(l+1)^n}$. Since $a^{ns}$ does the same, you get that $a^s$ commutes with $b^{l... | 16 | https://mathoverflow.net/users/nan | 76489 | 46,327 |
https://mathoverflow.net/questions/76495 | 8 | Is it known to be consistent with ZF that there is no non-principal ultrafilter on any infinite set? (Feel free to use your favorite interpretation of "infinite" in this context.
If infinite just meant infinite ordinals, that would be fine, too. You may use all kinds of large cardinals.)
| https://mathoverflow.net/users/7743 | Existence of non-principal ultrafilters on sets | Yes, it is consistent with ZF that every ultrafilter is principal. This is a result of Andreas Blass, *A model without ultrafilters*, Bull. Acad. Polon. Sci. 25 (1977), 329–331.
| 13 | https://mathoverflow.net/users/2000 | 76499 | 46,330 |
https://mathoverflow.net/questions/76509 | 49 | How to think about coalgebras? Are there geometric interpretations of coalgebras?
If I think of algebras and modules as spaces and vectorbundles, what are coalgebras and comodules? What basic examples of coalgebras should one keep in mind?
Anything that helps to think about coalgebras without headache is welcome ;) ... | https://mathoverflow.net/users/2837 | What is a coalgebra intuitively? | It seems like there are two basic sources of examples:
1.
The basic structure you have on a space (set, scheme...) is the diagonal morphism $\Delta :X \to X\times X$. Functions on spaces are contravariant, which is why functions on a space form an algebra: $f.g = \Delta ^\ast (f \boxtimes g)$. We also have the mo... | 39 | https://mathoverflow.net/users/7762 | 76518 | 46,337 |
https://mathoverflow.net/questions/76515 | 7 | Let $R$ be the hyperfinite $II\_1$-factor. It is well-known that it is the smallest $II\_1$-factor, in the sense that every $II\_1$-factor contains a copy of $R$.
Now, let $\omega$ be a free ultrafilter on the natural numbers and construct the tracial ultrapower $R^\omega$. It is well-known, that this is a quite big... | https://mathoverflow.net/users/13809 | How big are the ultrapowers of the hyperfinite $II_1$-factor? | This is not true. If we denote by $\mathbb F\_{\mathbb R}$ the free group with generators indexed by $\mathbb R$ then any separable von Neumann subalgebra $N \subset L\mathbb F\_{\mathbb R}$ must necessarily be contained in the von Neumann subalgebra $L\mathbb F\_I$ for some countably infinite subset $I \subset \mathbb... | 9 | https://mathoverflow.net/users/6460 | 76524 | 46,340 |
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