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https://mathoverflow.net/questions/32082 | 6 | I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that $\gamma$ is contractible. Then for any contraction of this geodesic at some point its length will be equal to $2\pi$.
It w... | https://mathoverflow.net/users/943 | Contracting a geodesic on a space of curvature less than 1 | There is an article by B. Bowditch, Notes on locally CAT(1) spaces, in Geometric Group Theory, R.Charney, M.Davis, and M.Shapiro eds., de Gruyter (1995) that will likely be of help. The article is about curve shortening in locally CAT(1) spaces.
| 8 | https://mathoverflow.net/users/7021 | 32086 | 20,841 |
https://mathoverflow.net/questions/32063 | 6 | Following the notation of [Etingof-Nikshych-Ostrik](http://arxiv.org/abs/0909.3140) what is Out(G-mod) for a finite group G?
That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the property that they're just G-mod as a left (resp. right) G-module up to equivalence o... | https://mathoverflow.net/users/22 | What is Out(G-mod) for a finite group G? | The group $\operatorname{Out}(G\operatorname{-Mod})$ is equivalent to the group $\operatorname{Aut}\_{\textbf{Tw}}(k[G])$ of gauge equivalence classes of twisted automorphisms of the Hopf algebra $k[G]$ defined by Davydov in [*Twisted automorphisms of Hopf algebras*](https://arxiv.org/abs/0708.2757). In Davydov's other... | 4 | https://mathoverflow.net/users/396 | 32091 | 20,844 |
https://mathoverflow.net/questions/32083 | 5 | For a finite type morphism $f:X\to S$, $X$ is a regular scheme, should there always exist an open (dense) subscheme $U\subset S$ such that the fibre of $f$ at each Zariski point of $U$ is regular? All schemes are excellent.
If the answer is 'yes', then: could one choose such an $U$ such that the preimage of any regul... | https://mathoverflow.net/users/2191 | For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular | Over a field of characteristic zero, your result is true. This is Corollary III.10.7 in Hartshorne.
---
In characteristic $p$ no. The simplest example is to take $k$ an algebraically closed field and map $\mathbb{A}^1\_k$ to itself by $x \mapsto x^p$. For every $t \in k$, the fiber above $t$ is $\mathrm{Spec}\ k... | 14 | https://mathoverflow.net/users/297 | 32096 | 20,849 |
https://mathoverflow.net/questions/32051 | 3 | First, let me explain what I mean by "synthetic" in the title, which is a proof that reasons purely axiomatically and does not explicitly invoke local coordinate charts (either via concrete expansions or Penrose-style abstract tensor notation). For example in Euclidean geometry, one can either prove statements using Eu... | https://mathoverflow.net/users/4642 | "Synthetic" proof of geodesic flow equation? | If $(s,t) \mapsto \Gamma(s,t)$ is a family of curves in Riemannian manifold $M$, where $s \in [0,1]$ is the curve parameter and $t \in (-\delta,\delta)$ is the variation parameter, let $S = \partial\_s\Gamma, T = \partial\_t\Gamma \in T\_{\Gamma(s,t)}M$. Note that $[S,T] = 0$. Then the derivative of the energy function... | 7 | https://mathoverflow.net/users/613 | 32101 | 20,851 |
https://mathoverflow.net/questions/19946 | 46 | Short version of the question: Presumably, it's poly$(t)$. But what polynomial, and could you provide a reference?
Long version of the question:
I'm sort of surprised to be asking this, because it's such an extremely basic sounding question. Here are some variants on it:
1. How much time does it take to compute $\... | https://mathoverflow.net/users/658 | What is the time complexity of computing sin(x) to t bits of precision? | For state-of-the-art arithmetic algorithms, I'd recommend this book (a work in progress), available online, from Brent and Zimmermann:
<http://www.loria.fr/~zimmerma/mca/pub226.html>
See chapter 4.
As Steve points out, log, exp, and trig functions are $O(M(n) \log n)$ (in fact they're all calculated from log), where ... | 19 | https://mathoverflow.net/users/7106 | 32114 | 20,856 |
https://mathoverflow.net/questions/32109 | 3 | Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with the walls, when it is subjected to an elastic response (i.e., its direction vector undergoes a reflection in the facet t... | https://mathoverflow.net/users/1104 | Connections between a polytope's symmetry group and the existence of periodic orbits | Dear Zach Conn,
I think your problem is very interesting but quite difficult to approach even in dimension 2: for instance, it is a well-known open problem to decide whether every irrational triangular billiard has a periodic orbit. As far as I know, the conjectural answer is yes, but, besides the case of acute trian... | 2 | https://mathoverflow.net/users/1568 | 32115 | 20,857 |
https://mathoverflow.net/questions/32117 | 13 | Does there exists a subset E of R with positive measure and without containing any midpoints (i.e. x,y distinct in E, (x+y)/2 not in E)?
| https://mathoverflow.net/users/7663 | Set of real numbers with positive measure containing no midpoints | According to James Foran, Non-averaging sets, dimension, and porosity, Canad Math Bull 29 (1986) 60-63, "It follows from the Lebesgue Density Theorem that a measurable, non-averaging subset (of
$(0,1]$) cannot have positive measure."
| 11 | https://mathoverflow.net/users/3684 | 32124 | 20,861 |
https://mathoverflow.net/questions/32126 | 24 | There is an example of a function that is unbounded on every open set. Just take $f(n/m) = m$ for coprime $n$ and $m$ and $f(irrational) = 0$.
I want to generalize this in a way to get a function which is not just unbounded on every open set, but whose range is equal to $\mathbb{R}$ on every open set. The latter cons... | https://mathoverflow.net/users/7079 | Function with range equal to whole reals on every open set | See [Conway's base 13 function](http://en.wikipedia.org/wiki/Conway_base_13_function).
| 43 | https://mathoverflow.net/users/4213 | 32127 | 20,862 |
https://mathoverflow.net/questions/31350 | 10 | At the time of writing, the most recent blog post over at *What's new* by Terrence Tao is [Cayley graphs and the geometry of groups](http://terrytao.wordpress.com/2010/07/10/cayley-graphs-and-the-geometry-of-groups/), and that (excellent, as with most of Tao's writing) post most immediately inspired this question. Also... | https://mathoverflow.net/users/78 | Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"? | Several people have been working on what might be described as geometric group theory for compactly generated, totally disconnected topological groups.
The main definitions as well as several nice results are in this paper:
Krön, Bernhard; Möller, Rögnvaldur G.
Analogues of Cayley graphs for topological groups.
M... | 10 | https://mathoverflow.net/users/3955 | 32139 | 20,867 |
https://mathoverflow.net/questions/32145 | 1 | I have encountered a few problems regarding the minimal subgroups of a finite group $G$. Any references and/or answers regarding the following questions will be very welcome.
1)If $G$ is a finite group, what can be said about its minimal normal subgroups (under inclusion)?
2) Is there a criterion to decide when an ... | https://mathoverflow.net/users/7670 | Minimal normal subgroups of a finite group | Try reading the [planetmath.org article](http://planetmath.org/groupsocle) on the socle of a group.
| 6 | https://mathoverflow.net/users/6153 | 32148 | 20,872 |
https://mathoverflow.net/questions/32116 | 3 | It is well known that every group of exponent $n=2$ is abelian. I remember having seen that this is also the case for $n=3$. (can someone give a proof). How does this generalize to any $n$ or to any prime $p$.
| https://mathoverflow.net/users/7489 | Exponent of a group | The group defined by $\langle x,y,z; x^3 = y^3 = z^3 = 1, yz = zyx, xy = yx, xz = zx\rangle$ has order 27, exponent 3 and is non-abelian.
(Checking exponent 3 basically comes down to ensuring that $(yz)^3 = (y^2z)^3 = (yx^2)^3 = 1$. Or by using Gap.)
| 16 | https://mathoverflow.net/users/6503 | 32155 | 20,877 |
https://mathoverflow.net/questions/32149 | 1 | I have been studying measure theory of late, and i was stuck up in these two things.
BOUNDED CONVERGENCE THEOREM: It states that $f\_n$ is a sequence of measurable functions, defined on a measurable set $E$ and if $f\_{n} \to f$ pointwise on $E$,and is $f\_{n}$ is uniformly bounded, that is if $|f\_{n}(x)| \leq M$ fo... | https://mathoverflow.net/users/1483 | Fatou's Lemma and the bounded convergence theorem. | I am not sure if the question fits MO standards as it is an elementary measure theory question (and if it's not, expect to be tazered by the MO police). Here goes an answer, anyway.
To expand on Robin Chapman's comment, first, the theorem as stated is false withouth the assumption that $E$ has finite measure. The cor... | 6 | https://mathoverflow.net/users/2562 | 32156 | 20,878 |
https://mathoverflow.net/questions/32150 | 3 | Let $i$, $k$ be integers such that $2 \leq i \leq k$. I would like to show that the sum
$$
\sum\_{j=1}^{i-1} \frac{(-1)^{j-1}(i-j)^k}{(i-j)! (j-1)!}
$$
is positive. I have carried out extensive numerical experiments to check this for small values of $k$. In fact, much more should be true. Define polynomials
$$
U(x)=(x+... | https://mathoverflow.net/users/4140 | Positivity of a finite sum | These are [Stirling numbers of the second kind](http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind). More precisely
your sum is S(k,i-1) where $S$ denotes Stirling number of the second kind.
| 11 | https://mathoverflow.net/users/4213 | 32157 | 20,879 |
https://mathoverflow.net/questions/31936 | 8 | I recently heard of a game between two players "Line" and "Point" and wanted to look for more information on it. However, without knowing the name of it (if it has one) finding more information is hard, has anyone heard of it? Is there a winning strategy for one of the players?
The game is as follows, it is played on... | https://mathoverflow.net/users/3121 | Choosing lines and points in D^2 | Line actually has a winning strategy: it can force a convergent sequence. The problem was posed and solved in the following paper:
J. Maly and M. Zeleny (2006), A note on Buczolich's solution of the Weil gradient problem: a construction based on an infinite game, [*Acta Mathematica Hungarica*, Vol. 113, pp. 145-158.]... | 5 | https://mathoverflow.net/users/3376 | 32159 | 20,880 |
https://mathoverflow.net/questions/32069 | 23 | Are deformations of Nakajima quiver varieties also Nakajima quiver varieties ?
In case the answer to this is (don't k)no(w), here are some simpler things to ask for.
1. (If you're a differential geometer) Is any hyperkahler rotation / twistor deformation of a Nakajima quiver variety also a Nakajima quiver variety ?... | https://mathoverflow.net/users/7653 | Deformations of Nakajima quiver varieties | I do not know a general statement. I just want to give a comment:
>
> Now if I take dimension vector $(2,2)$ I can presumably get $Hilb^2$ of these surfaces, for an appropriate stability condition.
>
>
>
No. You only get the symmetric product of $T^\*P^1$ if you work on quiver varieties with the dimension vec... | 17 | https://mathoverflow.net/users/3837 | 32165 | 20,883 |
https://mathoverflow.net/questions/32135 | 4 | If $f:X\to S$ is a universal homeomorphism, is $f':X\times\_S X\to X$ always a nil-immersion? This seems to be easy, yet possibly I miss something. Should I give references to this fact in a paper?
| https://mathoverflow.net/users/2191 | If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ a nil-immersion? | In general, the answer is no. As Brian Conrad points out in the comments above, purely inseparable field extensions do not have this property. However (and maybe this is what you were getting at?), it is true that if $X\to S$ is a universal homeomorphism, then the diagonal $X\to X\times\_SX$ *is* a nilimmersion.
| 5 | https://mathoverflow.net/users/3049 | 32166 | 20,884 |
https://mathoverflow.net/questions/32163 | 7 | What are the curves of contact on a convex body $B$ rolling down an inclined plane?
Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic to a straight line on a plane by rolling $B$ so that the geodesic
is the point of contact, but it doesn't appear that th... | https://mathoverflow.net/users/6094 | Rolling a convex body: Geodesics vs. rolling curves | The rolling motion of a convex symmetric body on a *horizontal* plane is a classical problem. In the symmetric case, Chaplygin was the first who showed that the full equations of motion can be reduced to a linear integrable system of two ODEs. A modern exposition of Chaplygin's results can be found in the very recent [... | 5 | https://mathoverflow.net/users/5371 | 32170 | 20,886 |
https://mathoverflow.net/questions/32169 | 22 | On a Riemannian manifold, a coordinate system is called "isothermal" if the Riemannian metric in those coordinates is conformal to the Euclidean metric:
$$g\_{ij} = e^{f} \delta\_{ij}$$
My question is: Why are such coordinate systems called "isothermal"? It must have something to do with classical thermal physics. ... | https://mathoverflow.net/users/6871 | Why are they called isothermal coordinates? | Isothermal coordinates are harmonic. In other words, it solves $\triangle\_g u = 0$. So locally it is a stationary solution of the heat equation. In physics, for a steady state distribution of temperatures, each level set is called an isotherm.
| 24 | https://mathoverflow.net/users/3948 | 32172 | 20,888 |
https://mathoverflow.net/questions/32158 | 3 | Hello all,
I need some theoretical pointers (formulas, articles, online links) on how to merge Singular Value Decompositions (SVD) of two matrices (two different sets of observations over the same set of features).
That is, I have two SVDs: $A=U\_A\*S\_A\*V^T\_A$ and $B=U\_B\*S\_B\*V^T\_B$ and want to know SVD $A|B... | https://mathoverflow.net/users/7672 | distributed incremental SVD | The people aspiring for the [Netflix prize](http://en.wikipedia.org/wiki/Netflix_Prize) like incremental SVDs. See
1. <https://issues.apache.org/jira/browse/MAHOUT-371>
2. B.M. Sarwar, G.Karypis, J.A. Konstan, and J. Reidl. Incremental singular value deocmposition algorithms for highly scalable recommender systems. ... | 4 | https://mathoverflow.net/users/5372 | 32177 | 20,891 |
https://mathoverflow.net/questions/32179 | 5 | In a paper I read recently, the authors use the fact that if two groups G and H have no nontrivial common quotient, then neither do GxG and HxH. It's unclear from the context whether this is true for all groups, or just groups of the type that are important for this paper, and they don't prove the claim.
I've been tr... | https://mathoverflow.net/users/913 | Common quotients of direct products | If $Q$ is a quotient of $G\times G$, then it has a normal subgroup $A$ (the image of $G\times 1$) such that both $A$ and $Q/A$ are quotients of $G$. If it is also a quotient of $H\times H$, then it has a normal subgroup $B$ such that both $B$ and $Q/B$ are quotients of $H$. Now assume that every common quotient of $G$ ... | 13 | https://mathoverflow.net/users/6666 | 32185 | 20,895 |
https://mathoverflow.net/questions/32160 | 8 | Let $g$ be a Lie algebra over $\mathbb{C}$. Then the equivalence between the derived category of modules over $U(g)$ and the coderived category of co-modules over it's Chevalley complex $C\_\*(g)$ in which $M\rightarrow C\_\*(g,M)$ is a classical example of Koszul duality. In the case when $g$ is a finite dimensional L... | https://mathoverflow.net/users/6986 | Koszul duality and modules over the Chevalley complex | The derived category of finite-dimensional $g$-modules is not a full subcategory of the derived category of arbitrary $g$-modules for a finite-dimensional Lie algebra $g$, in general (e.g., for a semi-simple Lie algebra $g$). However, one can consider the full subcategory of the derived category of $g$-modules consisti... | 11 | https://mathoverflow.net/users/2106 | 32194 | 20,902 |
https://mathoverflow.net/questions/30000 | 18 | The aim of this question is to understand SYZ conjecture ("Mirror symmetry is T-Duality").
I don't expect a full and quick answer but to find a better picture from answers and comments.
The whole idea is to construct the Mirror C.Y $Y$ from $X$ intrinsically as follows.
One considers the moduli of special Lagrangian ... | https://mathoverflow.net/users/5259 | Do you understand SYZ conjecture | Hi-
Just saw this thread. Maybe I should comment. The conjecture
can be viewed from the perspective of various categories:
geometric, symplectic, topological. Since the argument is
physical, it was written in the most structured (geometric)
context -- but it has realizations in the other categories
too.
Geometric: ... | 18 | https://mathoverflow.net/users/1186 | 32197 | 20,904 |
https://mathoverflow.net/questions/32121 | 0 | I'm trying to write a simple recommendation system. I have a set of products that exist in a set of categories and I know whether a given customer liked a subset of the items. From this I can deduce an "affinity" from each customer to each category (a number 0..1 defining the fraction of the products from that category... | https://mathoverflow.net/users/7664 | Clustering sets of sparse vectors with high dimensionality | Let affinity range between -1 and 1, with zero representing indifference/no opinion/haven't tried it. Customers have a right to hate.
Storage is much easier: You need only store the non-zero affinities. e.g. A.J. = [MathOverFlow : 1, Coffee : 1, Decaf: -1]
My consulting fee is $5. I accept cash or check.
| 1 | https://mathoverflow.net/users/35508 | 32205 | 20,910 |
https://mathoverflow.net/questions/32108 | 1 | Let p=(v1,…,vn) be a self-avoiding walk in a graph G. Let d(p) be the number of unique i, 1≤i<n such that there's a self-avoiding walk q that starts at vn and ends at vi without visiting any other vertices in p. Let d(G) be maxp d(p) with maximum taken over all self-avoiding walks in G. Is d(G) related to treewidth of ... | https://mathoverflow.net/users/7655 | Is this measure related to treewidth? | If a graph has bounded treewidth, it has $O(n)$ edges. And if it has $O(n)$ edges, it has $2^{O(n)}$ self-avoiding walks, since a self-avoiding walk can be specified as a set of edges. This is in contrast with arbitrary graphs where the number of self-avoiding walks can be exponential in $n\log n$. But it's not really ... | 1 | https://mathoverflow.net/users/440 | 32214 | 20,917 |
https://mathoverflow.net/questions/32193 | 19 | These days, lots of people are excited by Frobenius algebras because commutative Frobenius algebras are the same thing as 2D topological quantum field theories.
...but this seems like teaching an old dog new tricks. Can anyone sum up (using only diet representation theory :-P), why Frobenius algebras were invented an... | https://mathoverflow.net/users/800 | Why did people originally like Frobenius algebras? | Frobenius's original turn-of-the-century perspective was the nonvanishing of a determinant. Brauer–Nesbitt–Nakayama studied some equivalent definitions in the late 30s and early 40s. For instance, an equivalence between the left and right regular representations is a rare and beautiful thing; this gives an equivalence ... | 10 | https://mathoverflow.net/users/3710 | 32216 | 20,918 |
https://mathoverflow.net/questions/32217 | 2 | Suppose $R$ is a commutative ring, and $S \subset R^{n\times n}$ is an $R$-module. We are given $H\_0,\dots,H\_n \in R^{n\times n}$, and we know that for all $r \in R$,
$$H\_0 + r H\_1 + \dots r^n H\_n \in S$$
The question is: when can we conclude that $H\_i \in S$ for all $i=0,1,\dots,n$ ?
Clearly this is true when ... | https://mathoverflow.net/users/7667 | For which rings does there exist an invertible Vandermonde matrix? | The fact that the $H\_i$ are matrices is irrelevant; the question is whether a module generated by elements $H\_0\dots,H\_n$ is necessarily generated by all the elements of the form $\Sigma\_{0\le i\le n}r^iH\_i$. If this is true in modules of matrices then it's true in all free modules, and if so then it's true when t... | 15 | https://mathoverflow.net/users/6666 | 32227 | 20,923 |
https://mathoverflow.net/questions/32221 | 3 | I have a way of generating random parameterized maps from $S^1 \to \mathbb{R}^3$. This method can create very simple knots, such as ellipses, but can also create knots with more crossings than I can count. The images can be knotted as I have seen the method generate a figure eight knot. I know that with a parameterizat... | https://mathoverflow.net/users/7681 | What can one do with randomly generated parameterized knots in 3 space? | Not very likely. Any physical questions about random knotting will come with all sorts of requirements about the random process that generates the knots, and so it's unlikely that whatever method you have in mind would be helpful in any given situation.
Also, it's very easy to produce random knots, and without specif... | 1 | https://mathoverflow.net/users/3 | 32230 | 20,926 |
https://mathoverflow.net/questions/32231 | 8 | Let $A\_0, \dots, A\_{n-1}$ be upper triangular matrices with ones on the diagonal. Let $B\_{n-1}, \dots, B\_0$ be of the same form.
I am interested in bounding
$$|| A\_0 \dots A\_{n-1} B\_{n-1}^{-1} \dots B\_0^{-1}||$$
and in particular showing that this product is close to the identity when $A\_i$ and $B\_i$ are c... | https://mathoverflow.net/users/344 | Techniques to bound products of upper triangular matrices and their inverses | The standard trick to quantify the joint continuity of a product operation such as $(A\_1,\ldots,A\_k) \to A\_1 \ldots A\_k$ (which is essentially what you are trying to do here) is to split a difference such as
$$ A\_1 \ldots A\_k - B\_1 \ldots B\_k $$
as the telescoping sum of $k$ expressions of the form
$$ A\_... | 10 | https://mathoverflow.net/users/766 | 32233 | 20,928 |
https://mathoverflow.net/questions/32196 | 5 | Let all schemes below be excellent.
Let $X\_0$ be a regular (not necessarily smooth, projective) non-empty scheme of finite type over the generic point $\eta$ of a regular connected scheme $S$. As the answers to my question
[For a morphism f from a regular scheme, should there exist an open subscheme U of the target s... | https://mathoverflow.net/users/2191 | Model of a scheme regular over the generic point | To synthersize a bit: let $S$ be an excellent (quasi-excellent is enough) integral scheme, let $X\_0$ be a regular scheme of finite type over $K=K(S)$. I will suppose $X\_0$ separated to avoid possible pathologies.
1. There exists a separated integral noetherian scheme $X$ over $S$ with generic fiber isomorphic to $... | 6 | https://mathoverflow.net/users/3485 | 32234 | 20,929 |
https://mathoverflow.net/questions/32147 | 10 | Let $G$ be a locally compact group and let $A(G)$ be the http://eom.springer.de/f/f120080.htm>Fourier algebra of $G$, which we view as the predual of the group von Neumann algebra $\mathcal M(G)$. Let $MA(G)$ be the space of multipliers of $A(G)$, i.e., $\varphi \in MA(G)$ if and only if $\varphi \psi \in A(G)$ for all... | https://mathoverflow.net/users/6460 | Examples of groups without the n-positive approximation property | Haagerup actually proved (lattices of) higher rank Lie groups do not have $1$-positive approximation property, nor bounded approximation property, i.e., there is no uniformly bounded sequence of compactly supported multipliers that converges pointwise to $1$. (It's still open whether the reduced group C$^\*$-algebra of... | 13 | https://mathoverflow.net/users/7591 | 32247 | 20,939 |
https://mathoverflow.net/questions/32137 | 10 | If R is a commutative noetherian ring, M and N are modules with M finite. It is well known in commutative algebra that $AssHom\_R(M,N)=Supp(M)\cap Ass(N)$. But I want to know whether there is a formular for $AssExt\_R^i(M,N) ?$ Thanks!
| https://mathoverflow.net/users/5775 | The associated prime ideals of $Ext^i_R(M,N)$ | The short answer is no, as hinted at in the comments by Karl and Graham. I would argue that even the question of understanding the minimal primes of $\text{Ext}^i(M,N)$ (which is the minimal set of the associated primes, hence an easier question) is intractable. Let's assume that $R$ is Noetherian and $M,N$ are finitel... | 5 | https://mathoverflow.net/users/2083 | 32249 | 20,940 |
https://mathoverflow.net/questions/32245 | 9 | One of my favorite theorems is that of Fáry-Milnor, stating that the total curvature of a knot in $\mathbb R^3$ which is not an unknot (an ununknot) is at least $4\pi$.
Can one quantify the way in which knottedness forces an increase in total curvature?
| https://mathoverflow.net/users/1409 | The total curvature of very knotty knots | Fields medalist Michael Freedman got involved with knots using a very simple technique, assign a sort of energy integral that becomes infinite if there is a genuine self-crossing, that is if you try to force the curve to change isotopy class. The results relate, at least, to Ryan's comment "the figure 8 knot is twice a... | 10 | https://mathoverflow.net/users/3324 | 32250 | 20,941 |
https://mathoverflow.net/questions/32255 | 7 | In a joint paper that I am working on, we are interested in taking the intersection product $[X] \cap [Y]$ of the fundamental classes of two compact, oriented pseudomanifolds $X$ and $Y$ in a compact, oriented manifold $M$. Now, the usual intersection product takes values in $H\_\*(M)$, but I need an intersection produ... | https://mathoverflow.net/users/1450 | Intersection product in a manifold, taking values in one factor | I don't think you have to get involved with strata or transversality at all. Poincare duality (Edit: and cap products) will do all the work. Let's assume:
$M$ is a compact oriented $n$-manifold.
$\xi$ is a $p$-dimensional homology class of $M$. (We don't care if it comes as the fundamental class $[X]$ of some pseu... | 14 | https://mathoverflow.net/users/6666 | 32258 | 20,946 |
https://mathoverflow.net/questions/32269 | 11 | Consider a game where one player picks an integer number between 1 and 1000 and
other has to guess it asking yes/no questions.
If the second player always gives correct answers than it's clear that in worst
case it's enought to ask 10 questions. And 10 is the smallest such number.
What if the second player is allow... | https://mathoverflow.net/users/7694 | Guess a number with at most one wrong answer | Yes, there is a way to guess a number asking **14** questions in worst case. To do it you
need a linear code with length 14, dimension 10 and distance at least 3. One such code can be built
based on Hamming code (see <http://en.wikipedia.org/wiki/Hamming_code>).
Here is the strategy.
Let us denote bits of first pla... | 24 | https://mathoverflow.net/users/7079 | 32270 | 20,954 |
https://mathoverflow.net/questions/32268 | 9 | Does anyone know of a monograph/survey on the modern history of (basic or elliptic) hypergeometric functions and their applications?
I haven't had much time to search the literature, and because it is summer it is hard to reach professors or specialists, which is why I am asking the question here. It is also likely t... | https://mathoverflow.net/users/2384 | Recent work on hypergeometric functions | Gjergji, there are remarkable articles by Richard Askey:
(1) "Ramanujan and hypergeometric and basic hypergeometric series"
in
[*Russian
Math. Surveys* **45**:1 (1990) 37--86](http://dx.doi.org/10.1070/RM1990v045n01ABEH002325);
reprinted in *Ramanujan: essays and surveys*, Hist. Math. **22**
Amer. Math. Soc., Pro... | 8 | https://mathoverflow.net/users/4953 | 32275 | 20,959 |
https://mathoverflow.net/questions/32276 | 4 | Let $Q$ be a smooth conic (the zero set of a homogeneous degree 2 polynomial)
in $\mathbb{P}^2(\mathbb{C})$ and let $j:Q\rightarrow\mathbb{P}^2(\mathbb{C})$ be the
immersion in the projective plane. How can I compute $j\_\*H\_2(Q,\mathbb{Z})\subseteq H\_2(\mathbb{P}^2(\mathbb{C}),\mathbb{Z})$? (in $H\_i(X,\mathbb{Z})$... | https://mathoverflow.net/users/4971 | Homology of a complex projective conic | The subgroup $j\_\* H\_2(Q)$ must be generated by twice the generator of
$H\_2(P^2(\mathbb{C}))$
(I'm dropping the coefficient group from my notation).
To see this, your map $\psi$ decomposes as the embedding from $P^1$ into
$P^2$ (which induces isomorphism on $H\_2$) composed with the map $[x,y]\to[x^2,y^2]$
on $P^1$.... | 4 | https://mathoverflow.net/users/4213 | 32278 | 20,961 |
https://mathoverflow.net/questions/32282 | 4 | Let $\mathfrak{F}(0)$ be the set of all bijections $\mathbb{N}\mapsto\mathbb{N}$, and let $\mathfrak{F}(n+1)$ be the set of all bijections $\mathfrak{F}(n)\mapsto\mathfrak{F}(n)$. Given $\alpha\in\mathfrak{F}(n)$, is there a lower bound on the cardinality of $C\_{\mathfrak{F}(n)}(\alpha)$, the centralizer of $\alpha$? ... | https://mathoverflow.net/users/6856 | What is the cardinality of the centralizer of a bijection? | As far as I understand, the answer is negative (assuming $n>0$). It is easy to prove the following:
**Claim.** Let $X$ be an uncountable set. Let $G$ be the group of all bijections $X\to X$. Then for any
$g\in G$, its centralizer $Z(g)\subset G$ satisfies $|Z(g)|=|G|$. (This implies the claim, since all of $\mathfrak... | 4 | https://mathoverflow.net/users/2653 | 32284 | 20,963 |
https://mathoverflow.net/questions/32287 | 13 | Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a space $f: Y \to X$ we associate the abelian group $H^n (Y, f^{\*}F)$ (Sheaf cohomology).
If $X$ is a point, Then this ... | https://mathoverflow.net/users/2095 | Representing cohomology of a sheaf à la Eilenberg-Maclane | Sheaf cohomology over $X$ is representable in the homotopy category of oo-stacks over X / spaces over X, yes.
Details, links and references are at <http://ncatlab.org/nlab/show/cohomology>
| 8 | https://mathoverflow.net/users/381 | 32291 | 20,966 |
https://mathoverflow.net/questions/32290 | 2 | I ran into the following algorithmic problem while experimenting with classification algorithms. Elements are classified into a polyhierarchy, what I understand to be a poset with a single root ("largest" element), please correct me if I am mistaken. I have to solve the following problem, which looks a lot like the [se... | https://mathoverflow.net/users/7702 | Selecting k sub-posets | You seem to rely on a notion of a vertex preceding another (you use the terms "lattice" and "polyhierarchy", and refer to the direction "down"). So the edge relation $E$ appears to be transitive, forming a strict partial order.
To show why the $k$ parameter is important, you suggest an example where the target set $G... | 2 | https://mathoverflow.net/users/7252 | 32300 | 20,969 |
https://mathoverflow.net/questions/32133 | 28 | Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p\_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p\_0 + p\_1x + \dots + p\_n x^n$.
I am looking for a proof that
$-\operatorname{adj}(A) = p\_1 I + p\_2 A + \dots + p\_n A^{n-1}$.
In the case wher... | https://mathoverflow.net/users/7667 | Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$? | Here is a direct proof along the lines of the standard proof of the Cayley–Hamilton theorem. [*This works universally, i.e. over the commutative ring $R=\mathbb{Z}[a\_{ij}]$ generated by the entries of a generic matrix $A$.*]
The following lemma combining Abel's summation and Bezout's polynomial remainder theorem is ... | 25 | https://mathoverflow.net/users/5740 | 32303 | 20,971 |
https://mathoverflow.net/questions/32262 | 4 | The endomorphisms of an abelian group form a ring under pointwise group operation and composition. Every ring is isomorphic to a subring of the endomorphism ring of some abelian group (left module over itself).
Is every ring isomorphic to the endomorphism ring of some abelian group? (not just a subring)
| https://mathoverflow.net/users/nan | Representation of rings | Generally it's difficult to characterize rings isomorphic to an endomorphism ring of an abelian group. Interest in such problems was sparked by a problem given by Fuchs in his widely-read monograph *Abelian Groups*, cf. the excerpt below from the introduction to the [paper [1]](http://dx.doi.org/10.1016/j.jalgebra.2006... | 2 | https://mathoverflow.net/users/6716 | 32305 | 20,972 |
https://mathoverflow.net/questions/32296 | 18 | It is well-known that topological K-theory is blessed with the Bott periodicity theorem, which specifies an isomorphism between $K^2(X)$ and $K^0(X)$ (where $K^n$ is defined from $K^0$ by taking suspensions). I am wondering if other generalized cohomology theories have their own periodicity theorems, and if there is a ... | https://mathoverflow.net/users/4362 | Periodicity theorems in (generalized) cohomology theories | In the spirit of first approach, there is a conjecture for a Clifford-algebra type proof of the 576-fold periodicity of TMF. This is a generalized cohomology theory constructed by piecing together all the elliptic cohomology theories together in a suitable way. I heard about this conjecture from Andre Henriques, who is... | 14 | https://mathoverflow.net/users/798 | 32314 | 20,977 |
https://mathoverflow.net/questions/32311 | 16 | The representation theory of finite groups over the complex numbers is classical und it is usually quite easy to compute the set of isomorphism classes of irreducible representations, at least for small examples. Now, sometimes one is not content with the representation theory over the complex numbers or even over any ... | https://mathoverflow.net/users/2039 | Explicit integral representation theory | You might be asking about four separate types of modules:
* irreducible Z[G] modules,
* Z-forms of irreducible Q[G] modules,
* indecomposable Z[G] modules, or
* indecomposable Z[G] modules that are finitely generated and free as Z-modules.
I'll assume the last is the main concern.
The irreducible modules of ZS3 ... | 25 | https://mathoverflow.net/users/3710 | 32321 | 20,980 |
https://mathoverflow.net/questions/32316 | 2 | This is again a question about forcing. Start in $L$, the constructible universe. CH holds. Let $\lambda$ be an inaccessible cardinal, also let $\lambda$ > $\aleph\_0$. For each $\alpha < \lambda$, let $P\_\alpha$ be the set of all functions such that $dom(p\_\alpha) \subset \aleph\_0$, $|dom(p\_\alpha)|<\aleph\_0$ and... | https://mathoverflow.net/users/3859 | Collapsing cardinals before the first inaccessible | There are several small mistakes in your question, and one big mistake, leading to your confusion.
First, the small mistakes:
* You say that $\lambda$ is inaccessible and also $\lambda>\aleph\_0$. This is redundant, since the usual definition has that $\lambda$ is an *inaccessible cardinal* if and only if it is an ... | 6 | https://mathoverflow.net/users/1946 | 32326 | 20,983 |
https://mathoverflow.net/questions/32308 | 4 | Many books describe how one can construct "by hand" a table of ordinals $1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +2,\ \ldots,\ \omega\cdot 2,\ \omega\cdot 2 +1,\ \ldots,\ \omega^{2},\ \ldots,\ \omega^{3},\ \ldots\ \omega^{\omega},\ \ldots,\ \omega^{\omega^{\omega}},\ \ldots, \epsilon\_{0},\ \ldots$.
But does this ... | https://mathoverflow.net/users/5292 | Can one really construct an "ordinal table"? | Since ordinal numbers have a unique division, logarithm and subtraction properties, when given an ordinal $\alpha$ you can write any other ordinal as a finite polynomial in $\alpha$, when $\alpha = \omega$ you get what's known as "Cantor normal form of $\gamma$ for the base $\omega$".
I.e., any ordinal $\gamma$ can b... | 6 | https://mathoverflow.net/users/7206 | 32330 | 20,986 |
https://mathoverflow.net/questions/32313 | 0 | Non-uniform circuits, according to my understanding, are those which have different circuit depending on the input size. Constant depth circuit are those whose depth is constant in the input size. So if for example we considered an instance *k* of the complexity class *TC0* which is a non-uniform constant-depth circuit... | https://mathoverflow.net/users/7685 | Non-uniform constant-depth circuits | Recall "non-uniform circuits" are represented as an infinite sequence {$C\_n$}, where $C\_n$ is the circuit that will handle $n$-bit inputs. The usual definition of "constant depth" means that the depth may vary from circuit $C\_n$ to circuit $C\_{n'}$, but the depth of $C\_n$ is never larger than a fixed number $d$. T... | 7 | https://mathoverflow.net/users/2618 | 32333 | 20,989 |
https://mathoverflow.net/questions/32320 | 8 | Consider the problem of deciding a language $L$; for concreteness, say that this is the [graph isomorphism problem](http://en.wikipedia.org/wiki/Graph_isomorphism_problem). That is, $L$ consists of pairs of graphs $(G, H)$ such that $G\simeq H$. Now the time complexity of deciding this problem as stated depends on how ... | https://mathoverflow.net/users/6950 | What is the relationship between "translation" and time complexity? | When it comes to the time complexity of problems, the encoding of the problem can be totally crucial. In general, the encoding of the problem cannot be separated from the complexity of the problem itself.
The first canonical example of this (as mentioned before in [answering another question](https://mathoverflow.ne... | 9 | https://mathoverflow.net/users/2618 | 32336 | 20,991 |
https://mathoverflow.net/questions/32071 | 17 | It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are all sorts of good reasons for this (Yoneda's lemma being a big one, and the fact that this is a topos). There are other v... | https://mathoverflow.net/users/6936 | Is the dual notion of a presheaf useful? | Let me try and take a stab at this question. I will give not so much as an answer to your question, but more of a rambling collection of remarks. The post turned out to be *much longer* than I wanted, because as B. Pascal once complained, I did not have the time to write a shorter one. Expect some inconsistencies, a lo... | 14 | https://mathoverflow.net/users/2562 | 32358 | 21,003 |
https://mathoverflow.net/questions/32353 | 4 | In Qing Liu's book *Algebraic geometry and arithmetic curves* I came across several confusing definitions. Several times he defines a notion only for a subclass of schemes/morphisms but later he is never *explicitly* mentioning these extra conditions again. Here are some examples:
* Let $X$ be a locally Noetherian sc... | https://mathoverflow.net/users/717 | Confusing definitions in Liu's Algebraic geometry and arithmetic curves? | Dear Arminius, I'm certainly not going to answer your questions "why doesn't he say...?":
Qing is a frequent and friendly contributor to MO and he will answer himself if he wants to.
Here is what I think is the consensus about your questions.
1) For a scheme *regular* definitely implies locally noetherian: De Jong ... | 6 | https://mathoverflow.net/users/450 | 32363 | 21,006 |
https://mathoverflow.net/questions/32370 | 4 | How do I see that there is a group of an arbitrary cardinality? Is this also true for abelian groups? Also, given a commutative ring $R\neq 0$ how do I see that there is an $R$-module of arbitrary cardinality?
I'm sure I saw this result somewhere but I can't seem to find it anywhere (books, google,...) Thanks!
| https://mathoverflow.net/users/5292 | Group & modules of arbitrary cardinality | For any algebraic theory that is expressible in first order logic in a countable language, and this includes groups, rings, fields, partial orders, lattices, etc. etc., then the basic fact is expressed by the [Lowenheim-Skolem theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem), which asserts t... | 9 | https://mathoverflow.net/users/1946 | 32371 | 21,009 |
https://mathoverflow.net/questions/32369 | 2 | I would like to ask if there are any set of functions $u\_n(x)$ which is orthogonal to $x^n$?
i.e.:
$\int\_0^1 x^n u\_m(x) dx = \delta\_{n,m}$
Edit: For clarification, this question asked for all non-negative integer m and n.
| https://mathoverflow.net/users/5217 | Functions orthogonal to x^n | If $f \in L^2([0, 1])$ and $\int\_0^1 x^n f(x)\, dx=0$ for all $n\ge N$ where $N$ is a nonnegative integer then $f$ is zero almost everywhere. To see this note that $x\mapsto x^N f(x)$ is an $L^2$ function orthogonal to all polynomials, and the polynomials are dense in $L^2([0,1])$. So the answer to your question is "n... | 9 | https://mathoverflow.net/users/4213 | 32372 | 21,010 |
https://mathoverflow.net/questions/32351 | 9 | I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?
| https://mathoverflow.net/users/4246 | Doubly-transitive groups | In Section 7.7 "The Finite 2-transitive Groups" of the book *Permutation groups* by John D. Dixon and Brian Mortimer, the authors describe the complete list of finite 2-transitive groups without proofs but with references.
They list eight infinite families: the alternating, symmetric, affine and projective groups in ... | 10 | https://mathoverflow.net/users/970 | 32384 | 21,018 |
https://mathoverflow.net/questions/32396 | 7 | Let $X$ be a projective variety over $\mathbb{Z}$, and suppose that $X$ has everywhere good reduction. Let $Y$ be the blow-up of $X$ at an integral point.
Then is it the case that $Y$ also has everywhere good reduction?
The example situation that I have in mind is the following (my main motivation is del Pezzo surf... | https://mathoverflow.net/users/5101 | Good reduction and blow-ups | It is true if the integral point $T$ is actually a section (as in your example), because you then blow-up a smooth scheme $X\to {\rm Spec}(\mathbb Z)$ along a smooth center $T\simeq {\rm Spec}(\mathbb Z)$. In general, as $T$ is flat over $\mathbb Z$, the fiber $Y\_p$ of $Y$ at a prime $p$ is the blow-up of $X\_p$ along... | 8 | https://mathoverflow.net/users/3485 | 32417 | 21,034 |
https://mathoverflow.net/questions/32413 | 3 | Let $E$ be a Banach space and $f:E\to E$ be a continuous map. By $f^n$ we denote the $n$-th iterate of $f$, i.e. $f^n:=\underbrace{f\circ f\circ\cdots \circ f}\_{\text{n times}}$. Let $x\_0$ denote a fixed point of $f^n$.
1) Is it true that the question as to when $x\_0$ is a fixed point of $f$ has been resolved in ... | https://mathoverflow.net/users/3014 | When is a fixed point of f^n a fixed point of f? | Actually I can't see the role of reflexivity for an answer to the question as it is, unless further properties on $f$ are assumed.
In general, to start with the obvious case: if $f$ is a contraction, it has a fixed point, which is also the unique fixed point of the contraction $f^n$, therefore it is $x\_0$, so $x\_0... | 7 | https://mathoverflow.net/users/6101 | 32419 | 21,035 |
https://mathoverflow.net/questions/32386 | 4 | Let $k$ be a field which I don't suppose to be algebraically closed. Then, the endomorphism ring of any irreducible representation of a finite group over $k$ is a division ring.
What is known about these division rings? When are they fields? When they are, are these fields Galois over $k$? At least, normal? When not,... | https://mathoverflow.net/users/2530 | Galois theory of endomorphism rings of irreducible representations | Concerning
>
> are the centers of the division rings Galois over $k$
>
>
>
A finite dim'l $k$-algebra $A$ is
*split* provided $\operatorname{End}\_A(S) = k$
for every simple $A$-module $S$.
[This terminology is consistent with that used
in other contexts -- $A$ is split
just in case the reductive quotient... | 3 | https://mathoverflow.net/users/4653 | 32420 | 21,036 |
https://mathoverflow.net/questions/32422 | 2 | $(A\_{\alpha})\_{\alpha\in B}$ a family of sets indexed by a set $B$. Is $\Pi \_ {\alpha\in B} \ A\_{\alpha}$ a set? I can see that it is a set if $A\_{\alpha}=A \ \ \forall\alpha$ because in that case the product is a subset of the power set ${\cal P}\ (A\times B)$. As far as I understand, the axiom of choice only say... | https://mathoverflow.net/users/5292 | Is an arbitrary product of sets a set? | The standard definition of $(A\_\alpha)\_{\alpha \in B}$ a family of sets indexed by a set $B$ is that one is given a function $A$ with domain $B$ so that for $\alpha \in B$ we understand $A\_\alpha$ as $A(\alpha)$. Then, yes, the product is a set.
The function $A$ itself is an element of the powerset of $B \times \... | 13 | https://mathoverflow.net/users/5147 | 32424 | 21,038 |
https://mathoverflow.net/questions/7493 | 16 | I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces some of the entries by zeros.
Show that there still remains a non-empty subset of rows that add up to the all zero vector.... | https://mathoverflow.net/users/2229 | A riddle about zeros, ones and minus-ones | Another answer (I guess they must be equivalent):
* Write each *original* line as a difference of two 0/1 vectors.
* Adapt this representation to the modified lines by changing *only the subtrahends*.
* You now have a function from {0,1}^n to {0,1}^n. Find a cycle.
| 13 | https://mathoverflow.net/users/7732 | 32425 | 21,039 |
https://mathoverflow.net/questions/32236 | 4 | Where can I find a complete classification of the $l$-adic Weil-Deligne representations for a local field $F$ of residual characteristic $p$ (with $p$ different from $l$)? Thanks.
| https://mathoverflow.net/users/7686 | Weil-Deligne representations | Here are the references:
D. Rohrlich. Elliptic curves and the Weil-Deligne group. Elliptic curves and related topics, 125{157, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994.
or
J. Tate, Number theoretic background.
| 4 | https://mathoverflow.net/users/1816 | 32453 | 21,055 |
https://mathoverflow.net/questions/32450 | 15 | The [wikipedia article](http://en.wikipedia.org/wiki/Chevalley%E2%80%93Shephard%E2%80%93Todd_theorem#Statement_of_the_theorem/%22wikipedia%20article%22) claims that the theorem "was first proved by G. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwards gave a uniform... | https://mathoverflow.net/users/6772 | Chevalley–Shephard–Todd theorem | There are indeed many presentations (if I remember correctly Bourbaki has it)
but the proof is very elegant and short so that I find it hard to refrain from giving
it. Let $H$ be the normal subgroup of the finite $G\subset \mathrm{GL}\_n$
generated by the pseudo-reflections. By the other direction $X:=\mathbb{A}^n/H$
i... | 21 | https://mathoverflow.net/users/4008 | 32456 | 21,058 |
https://mathoverflow.net/questions/32318 | 11 | Modern statements of Gödel's incompleteness theorems are usually in terms of first-order predicate logic. However, I've often read the claim that they extend to arbitrary formal systems that can prove basic propositions about numbers. Indeed, according to Wikipedia, the original theorems referred to the type theory of ... | https://mathoverflow.net/users/6904 | Most general formulation of Gödel's incompleteness theorems | Raymond Smullyan gave a very general formulation in terms of representation systems. They appear in his "Theory of Formal Systems", and in the first and last chapters of "Godel's Incompleteness Theorems". They generalise first- and higher-order systems of logic, type theories, and Post production systems.
A represent... | 14 | https://mathoverflow.net/users/7247 | 32463 | 21,065 |
https://mathoverflow.net/questions/32469 | 0 | I need the following sum ( in the sense of principal value):
$$\sum\_{s=-\infty}^{\infty}\frac{(-1)^{s}e^{-2\pi isy}}{x+s}$$
It is possible to show that
$$\sum\_{s=-\infty}^{\infty}\frac{e^{-2\pi isy}}{x+s}=\pi\frac{e^{\pi ix(2FractionalPart[y]-1)}}{\sin(\pi x)}$$
Hence the sum I need is
$$f(y)=\sum\_{s=-\infty}^{... | https://mathoverflow.net/users/3589 | series Sum[(-1)^n/(x+n)] | This is in a typical complex analysis text
$$
\frac{1}{x} + \sum\_{s = 1}^{\infty}\left( \frac{1}{x + s} + \frac{1}{x - s}\right) = \pi \cot (\pi x)
$$
When grouped this way, it converges...
Maple says
$$
\frac{1}{x} + \sum\_{s = 1}^{\infty} \left(\frac{\operatorname{e} ^{(-2 i\pi s y)}}{x + s} + \frac{\operatorname{... | 3 | https://mathoverflow.net/users/454 | 32475 | 21,073 |
https://mathoverflow.net/questions/32477 | 16 | On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this is a open problem?
When I asked this problem few years ago, someone told me it's open. I don't know if this is still o... | https://mathoverflow.net/users/6886 | Smallest area shape that covers all unit length curve | Whereas I don't know of any recent progress in this problem, let me mention one result for
*closed* curves.
>
> **Theorem.** A closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$.
>
>
>
This was proved in 1974 by H.H. Johnson ([link 1]... | 16 | https://mathoverflow.net/users/5371 | 32483 | 21,077 |
https://mathoverflow.net/questions/32495 | 1 |
>
> **Possible Duplicate:**
>
> [Cardinality of the permutations of an infinite set](https://mathoverflow.net/questions/27785/cardinality-of-the-permutations-of-an-infinite-set)
>
>
>
Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?
| https://mathoverflow.net/users/5292 | Cardinality of symmetric group | I assume you mean an infinite set $X$. You need to use the Axiom of Choice to prove this fact, but I'm not sure to what extent it is necessary.
Since $|X \times X| = |X|$ (uses AC) and $Sym(X) \subseteq \mathcal{P}(X \times X)$, it is clear that $|Sym(X)| \leq 2^{|X \times X|} = 2^{|X|}$.
Since $|X \times 2| = |X|$... | 4 | https://mathoverflow.net/users/2000 | 32499 | 21,089 |
https://mathoverflow.net/questions/32427 | 5 | I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S\_n$, and I don't want these to get lost. I hope this one is not quite as vague as the last one.
This here is an attempt to generalize Exercise I.21 in ... | https://mathoverflow.net/users/2530 | Left U_n-invariants of SL_n - an exercise in Kraft-Procesi | No, they don't. $U\_n(K)$ is performing upward row operations, so any $m\times m$ minor that uses the last $m$ rows will be $U\_n(K)$-invariant, e.g. any single bottom entry. You won't be able to generate those linear functions using your higher-degree functions. (Victor's disproof is nice too!)
What *is* true is tha... | 5 | https://mathoverflow.net/users/391 | 32500 | 21,090 |
https://mathoverflow.net/questions/32401 | 6 | For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite nonabelian group $G$ such that $S^{\lambda}(V)$ is still irreducible?
This is not obvious to me even for the symmetric and exterior powers (alt... | https://mathoverflow.net/users/290 | Can the image of a Schur functor always be made an irreducible representation? | Since the Guralnick and Tiep paper is very long, I thought I would summarize my understanding of it. Note that I only learned about this result from moonface last night, so I am hardly an expert. This answer is community wiki, in case anyone can improve on my summary.
We want to establish the following result: Let $G... | 3 | https://mathoverflow.net/users/297 | 32503 | 21,091 |
https://mathoverflow.net/questions/32502 | 2 | Suppose a bounded sequence $(x\_n)$ converges to $x$ in the Cesaro sense (i.e., $\frac{1}{n}(x\_1 + x\_2 + \dots + x\_n)\rightarrow x$) in a separable Hilbert space $H$. How to prove that some subsequence $(x\_{n\_k})$ converges weakly to $x$?
| https://mathoverflow.net/users/5498 | Cesaro convergence implies weak convergence of a subsequence | If we take $x\_n = (-1)^n x$ then $x\_n$ converges to $0$ in Cesaro sence. But no subsequence of $x\_n$ converges weakly to $0$. $x\_n$ is also a bounded sequence.
Hence your statements seems wrong.
| 8 | https://mathoverflow.net/users/7079 | 32508 | 21,095 |
https://mathoverflow.net/questions/32515 | 2 | Matlab has a set of dot operators, such as .\*, ./, .^. Each of these operators consists of a dot and a normal algebraic operator. They perform element-wise algebraic operations on a matrix. For example, consider the following codes
```
A = [1 2 3; 3 2 1];
x = [1 2 4];
B = A.^2
y = 1./x
```
The result is
```
B... | https://mathoverflow.net/users/7595 | How to write Matlab's dot operators in mathematical expressions? | Your matrix $B$ is the Hadamard product of $A$ and $A$ which uses the notation $B = A \circ A$. However I don't know of any others, particularly for expressing $y$.
See: <http://en.wikipedia.org/wiki/Matrix_multiplication#Hadamard_product>
| 4 | https://mathoverflow.net/users/3121 | 32522 | 21,105 |
https://mathoverflow.net/questions/32527 | 4 | I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot. I am primarily interested in surfaces topologically
equivalent to a sphere in $\mathbb{R}^3$, so measures that rely
on the genus are not useful.
Ideally the measure would achieve its min... | https://mathoverflow.net/users/6094 | Measures of the complexity of a metric | I think you would be pretty happy with the Willmore functional for, well, compact orientable
$C^\infty$ surfaces in $\mathbb R^3.$ It is just the integral of the square of the *mean* curvature or
$$ \frac{1}{2 \pi} \int\_{M^2} \; \; H^2 \; dS $$
This quantity is at least 2, and is only equal to 2 for a round sphere. Th... | 8 | https://mathoverflow.net/users/3324 | 32529 | 21,108 |
https://mathoverflow.net/questions/32349 | 1 | Suppose given a d-dimensional Brownian motion $B\_t$ starting from the origin and a centered ball with radius 1. Define T as the first hitting time of the sphere (boundary of the ball). How can one prove that T and $B\_T$ are independent?
| https://mathoverflow.net/users/7713 | Independence of conditional hitting distribution and hitting time | The short, and somewhat heuristic, answer is that rotational invariance implies that given $T$ the distribution of $B\_T$ is uniform on the sphere $S^{d-1}$. Since the conditional distribution does not depend on $T$, this is independence of the two variables.
In more detail, and with more rigor, let $\Omega$ denote t... | 5 | https://mathoverflow.net/users/6781 | 32532 | 21,110 |
https://mathoverflow.net/questions/32486 | 9 | First let me give a precise formulation of the question; I'll give some background/motivation at the end.
If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently high multiple of every Weil divisor is Cartier), then a *small Q-factorial modification* or *SQM* of X means a biratio... | https://mathoverflow.net/users/nan | How much can small modifications change the nef cone? | In the world of Calabi-Yau (as opposed to Fano) varieties, one does not expect miracles of Mori dream space type. A specific example which gives a positive answer to your first question is described in the paper <http://xxx.lanl.gov/abs/math/0102055> of Michael Fryers: a Calabi-Yau threefold (a degenerate quintic) whic... | 8 | https://mathoverflow.net/users/6107 | 32536 | 21,113 |
https://mathoverflow.net/questions/32540 | 4 | If {c(n)} is an arbitrary sequence of irrational numbers converging to 0 then Q + c(n), the set obtained by adding c(n) to the set of rational numbers Q, is clearly disjoint from Q for each n.
Is there an uncountable dense set of the real numbers, say D, for which a sequence {c(n)} converging to 0 exists such that D... | https://mathoverflow.net/users/6627 | disjoint translates of a dense uncountable set | Choose your favorite c(n), and define $D$ as any complementary $\mathbb{Q}$-vector space of $\mathrm{span}(\{c(n): n\in\mathbb{N}\})$ in $\mathbb{R}$ as $\mathbb{Q}$ vector space. This $D$ is not measurable, of course (countably many translates of it cover $\mathbb{R}$, so it can't be of measure zero, and on the other ... | 3 | https://mathoverflow.net/users/6101 | 32541 | 21,115 |
https://mathoverflow.net/questions/32528 | 4 | See my previous question [What is the product in the 2-category of spans?](https://mathoverflow.net/questions/32526/what-is-the-product-in-the-2-category-of-spans) for notation. In brief, $\mathcal S$ is a category with finite limits, $\operatorname{Span}(\mathcal S)$ is the 2-category whose 1-morphisms are diagrams in... | https://mathoverflow.net/users/78 | Do all equivalences in the 2-category of spans come from isomorphisms? | Yes. Suppose we have spans $X\_1 \stackrel{f\_1}{\leftarrow} Y \stackrel{f\_2}{\to} X\_2$ and $X\_2 \stackrel{g\_2}{\leftarrow} Z \stackrel{g\_1}{\to} X\_1$ with an isomorphism $Y \times\_{X\_2} Z \stackrel{\omega}{\to} X\_1$ such that $\omega = f\_1 \circ \pi\_1 = g\_1 \circ \pi\_2$, where $\pi\_1$ and $\pi\_2$ are th... | 1 | https://mathoverflow.net/users/396 | 32544 | 21,118 |
https://mathoverflow.net/questions/32533 | 49 | Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic programming is almost as easy, and there's a good deal of semi-definite, second-order cone and even integer programming methods th... | https://mathoverflow.net/users/942 | Is all non-convex optimization heuristic? | If the question is "Are there non-convex global search algorithms with provably nice properties?" then the answer is "Yes, lots." The algorithms I'm familiar with use interval analysis. Here's a seminal paper from 1979: [Global Optimization Using Interval Analysis](http://old.ict.nsc.ru/interval/Library/Thematic/GlobOp... | 37 | https://mathoverflow.net/users/7759 | 32550 | 21,124 |
https://mathoverflow.net/questions/32498 | 4 | Can anyone provide a reference to proofs of statements of the following type: The higher algebric group cohomology of a reductive group $G$ over $\mathbb{C}$ vanishes.
I am interested not just in finite dimensional modules but also "rational representations" for instance the functions on a vector space $\mathbb{C}^{n... | https://mathoverflow.net/users/3396 | Group Cohomology for Reductive Groups | Rational representations are directed unions of finite-dimensional ones, on which all linear representations of $G$ are completely reducible (either by an ad hoc definition of "reductive group" or a theorem applied to a good definition). So the functor of $G$-invariants on the category of rational representations is ex... | 9 | https://mathoverflow.net/users/3927 | 32551 | 21,125 |
https://mathoverflow.net/questions/32512 | 5 | Assume that M=G/K is a non-compact Hermitian symmetric space, for G the real points of a semisimple (or even simple) algerbaic group and K a maximal compact subgroup. M admits the structure of a complex manifold. Now, if X is a complex subvariety of M then its pre-image in G is a real analytic subvariety of G. My quest... | https://mathoverflow.net/users/7753 | Complex subvarieties of hermitian symmetric spaces | I will explain how to check it in any particular case, but I am not sure whether there is a good classification available (perhaps, it is even easy and I did not think enough about it). Helgason's book and papers of Joe Wolf and Alan Huckleberry may contain some relevant information.
Since the complex structure on $M... | 2 | https://mathoverflow.net/users/5740 | 32561 | 21,131 |
https://mathoverflow.net/questions/32526 | 9 | Let $\mathcal S$ be a category with finite limits. The 2-category $\operatorname{Span}(\mathcal S)$ has the same objects as are in $\mathcal S$. For objects $X,Y$, the hom category in $\operatorname{Span}(\mathcal S)$ between $X$ and $Y$ is the category of diagrams in $\mathcal S$ of the form $X \leftarrow \bullet \rig... | https://mathoverflow.net/users/78 | What is the product in the 2-category of spans? | If $\mathcal S = Set$, it looks to me like the product in $Span(\mathcal S)$ is given by disjoint union.
Define functors $F\_Z: Span(Z,X\sqcup Y) \to Span(Z,X)\times Span(Z,Y)$
$$
F\_Z(Z \leftarrow A \rightarrow X \sqcup Y) = ((Z \leftarrow A \times \_{X\sqcup Y}X \rightarrow X), (Z \leftarrow A \times \_{X\sqcup Y}Y... | 7 | https://mathoverflow.net/users/7762 | 32567 | 21,136 |
https://mathoverflow.net/questions/32570 | 9 | Suppose we have a recursively enumerable set of polynomials $\mathcal{P}=\{ p\_1({\bf x}), p\_2({\bf x}), \ldots\}, p\_i \in \mathbb{Z}[{\bf x}], {\bf x} = (x\_1, \ldots, x\_n)$. Let $V(\mathcal{P})$ denote the affine variety in $\mathbb{C}^n$ defined by $\mathcal{P}$. Is there an algorithm to compute $V(\mathcal{P})$?... | https://mathoverflow.net/users/1345 | Variety defined by a recursively enumerable set of polynomials | There is no such algorithm, even if n is zero. Take p\_i to be 0 unless you can find a counterexample of length i to your favorite unsolved math problem (such as "is 2i+4 the sum of 2 primes?"), in which case p\_i is 1.
| 6 | https://mathoverflow.net/users/51 | 32571 | 21,138 |
https://mathoverflow.net/questions/32568 | 25 | Let $(T,X)$ be a discrete dynamical system. By this I mean that $X$ is a compact Hausdorff space and $T: X \to X$ a homeomorphism.
For example, take $X$ to be the sequence space $2^{\mathbb{Z}}$ and $T$ the Bernoulli shift. Then there is a dense set of periodic points, and there is another (disjoint) dense set of poi... | https://mathoverflow.net/users/344 | Is there a dynamical system such that every orbit is either periodic or dense? | I believe you will find such examples for $X=\mathbb{C}$ and $T$ a rational map in
[Mary Rees](http://www.liv.ac.uk/~maryrees/maryrees.homepage.html), **Ergodic rational maps with dense critical point forward orbit**, *Ergodic Theory and Dynamical Systems* 4 (1984), 311-322. [official version](http://journals.cambrid... | 13 | https://mathoverflow.net/users/3993 | 32574 | 21,141 |
https://mathoverflow.net/questions/32554 | 59 | I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to make everything seem like it has a point (going back to the basic goals of "what is an algebraic group" and "what does t... | https://mathoverflow.net/users/6545 | Why the Killing form? | Hi Ryan,
I presume given your description of the students that they know finite groups pretty well, and have seen the averaging idempotent $e=\frac{1}{|G|}\sum\_{g\in G} g$, and how this can be used to construct an invariant inner product on any representation of a finite group. Perhaps you can convince them that com... | 38 | https://mathoverflow.net/users/1040 | 32576 | 21,143 |
https://mathoverflow.net/questions/32552 | 4 | Suppose a simple graph has $n$ vertices and $m$ edges. If the vertices are labelled, then each edge then corresponds to a transposition in a natural way. A theorem in Godsil and Royle's Algebraic Graph Theory, section 3.10, asserts the following:
If the graph is not a tree, then some product of the $m$ distinct trans... | https://mathoverflow.net/users/7760 | characterization of trees in terms of products of transpositions | Notation: αβ means apply α then β.
If the graph is not a tree, then either it contains a cycle or it contains less than n−1 edges. In the latter case, we get a contradiction since less than n−1 transpositions cannot multiply to a big cycle.
So suppose that the graph contains a cycle (12...k), and assume that for ea... | 4 | https://mathoverflow.net/users/7732 | 32578 | 21,145 |
https://mathoverflow.net/questions/32597 | 31 | John Hubbard recently told me that he has been asking people if there are compact surfaces of negative curvature in $\mathbb{R}^4$ without getting any definite answers. I had assumed it was possible, but couldn't come up with an easy example off the top of my head.
In $\mathbb{R}^3$ it is easy to show that surfaces o... | https://mathoverflow.net/users/2510 | Compact surfaces of negative curvature | You will find examples (topologically, spheres with seven handles) in section 5.5 of *Surfaces of Negative Curvature* by E. R. Rozendorn, in *Geometry III: Theory of surfaces*, Yu. D. Burago VI A. Zalgaller (Eds.) EMS 48.
Rozendorn tells us that «from the visual point of view, their construction seems fairly simple.»... | 27 | https://mathoverflow.net/users/1409 | 32598 | 21,161 |
https://mathoverflow.net/questions/32607 | 6 | Shimura (Crelle 221, 1966) considers the elliptic curve $E:y^2+y=x^3-x^2$ (although he doesn't use this equation) of conductor $11$ whose associated modular form is
$$
q\prod\_{k=1}^{+\infty}(1-q^k)^2(1-q^{11k})^2=\sum\_{n=1}^{+\infty}c\_nq^n
$$
where $q=e^{2i\pi\tau}$ and $\tau$ is in the upper half of $\bf C$. For a ... | https://mathoverflow.net/users/2821 | Reciprocity law for number fields defined by torsion points of modular elliptic curves | The first question is answered in Serre's
[[Propriétés galoisiennes des points d'ordre fini des courbes elliptiques,
*Invent. Math.* **15**:4 (1972) 259--331]](http://dx.doi.org10.1007/BF01405086), on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the... | 5 | https://mathoverflow.net/users/5015 | 32616 | 21,173 |
https://mathoverflow.net/questions/32437 | 5 | ### Universal codings of integers
A (binary) *coding of the integers* is a prefix-free code of the natural numbers, whose codewords are non-decreasing in size. A coding is *universal* if it is short enough (log n + o(log n)), but that's not important.
Some examples:
* The unary coding 0, 10, 110, ...; code length... | https://mathoverflow.net/users/7732 | When do cofinal chains of universal codings of the integers exist? | I believe I can answer your first question. But the answer involves forcing, which I cannot explain here (see Kunen's Set Theory. An introduction to independence proofs).
Assuming CH, there is a scale of codes. Why? Enumerates all codes as
$(c\_\alpha)\_{\alpha<\omega\_1}$. Construct a sequence $(a\_\alpha)\_{\alpha<\o... | 4 | https://mathoverflow.net/users/7743 | 32622 | 21,177 |
https://mathoverflow.net/questions/32626 | 15 | This question is mainly a curiosity, but comes from a practical experience (all players of *Race for the galaxy*, for example, must have ask themselves the question).
Assume I have a deck of cards that I would like to shuffle. Unfortunately, the deck is so big that I cannot hold it entirely in my hands. Let's say tha... | https://mathoverflow.net/users/4961 | How to shuffle a deck by parts? | A truly uniform distribution, no. (Well, your question is not completely well posed, but I will argue this for most ways of making it so.) There are $(kn)!$ factorial ways to shuffle a deck of $kn$ cards. So you want each permutation to occur with probability $1/(kn)!$. In particular, for every prime $p \leq kn$, you w... | 14 | https://mathoverflow.net/users/297 | 32631 | 21,180 |
https://mathoverflow.net/questions/32624 | 21 | Dirichlet's theorem states that for any coprime $k$ and $m$ there exists infinitely many primes $p$ such that $p \equiv k \pmod m$.
Some special cases of this theorem are easy to prove without any analytic methods. Those cases include, for example, $m=4, k=1$ and $m=4, k=3$.
Both cases could be proved by considerin... | https://mathoverflow.net/users/7079 | Special cases of Dirichlet's theorem | There is a simple non-analytic proof for $p\equiv 1 \bmod n$; see e.g. Proposition $3$ in [this note](https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf68b26a49636398ee2dce/1472161971095/primes1mod4.pdf). The proof gives a (Euclidean) argument that infinitely many primes divide the values of an inte... | 21 | https://mathoverflow.net/users/6950 | 32635 | 21,184 |
https://mathoverflow.net/questions/32623 | 2 | if $P\_{1}$ and $P\_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P\_{1})=P\_{2}$. then, does $\deg (P\_{1}\cap K(x))=\deg (P\_{2}\cap K(x))$, where K(x) is the rational function field?
in particular, is this true over the hermitian functio... | https://mathoverflow.net/users/7773 | behavior of places of a function field under automorphism | No, not in general, that is not without particular requirements for $x$:
take $F=\mathbb{R}(y)$, the rational function field in one variable over the reals.
Then the equation $\sigma (y)=y+1$ determines an automorphism of $F/\mathbb{R}$.
Let $P\_1$ be the place associated to the polynomial $y^2+1$; then $\deg (P\_1... | 3 | https://mathoverflow.net/users/3556 | 32636 | 21,185 |
https://mathoverflow.net/questions/32620 | 13 | Lehmer's conjecture for Ramanujan's tau function,
$$
\Delta(q)=q\prod\_{n=1}^\infty(1-q^n)^{24}=\sum\_{m=1}^\infty\tau(m)q^m,
$$
asserts that $\tau(m)$ never vanishes for $m=1,2,\dots$.
In the [recent question](https://mathoverflow.net/questions/31058/)
it was asked why it is important to have the nonvanishing.
I am ... | https://mathoverflow.net/users/4953 | Lehmer's conjecture for Ramanujan's tau function | One of the canonical references for questions like this is Serre's "Quelques applications du theoreme de densite de Chebotarev", Publ. Math. IHES 54. He proves, for example, that the number of primes $0\leq p \leq X$ with $\tau(p)=0$ is $\ll X (\log{X})^{-3/2}$ unconditionally, and is $\ll X^{\frac{3}{4}}$ under GRH.
... | 5 | https://mathoverflow.net/users/1464 | 32643 | 21,190 |
https://mathoverflow.net/questions/32641 | 17 | Robin Chapman [introduced me](https://mathoverflow.net/questions/32126/function-with-range-equal-to-whole-reals-on-every-open-set/32127#32127) to [Conway's Base 13 Function](http://en.wikipedia.org/wiki/Conway_base_13_function). Now, my real analysis is a tiny bit rusty, so maybe my question has a really simple and qui... | https://mathoverflow.net/users/3948 | Is Conway's base-13 function measurable? | Call the support set $S$. The answer is yes it is Lebesgue measurable and no, it has zero measure. It is even Borel measurable, which would take a tiny bit more effort to prove.
Note that $S$ is included in the set $T$ of numbers in which the "digits" '+','-', and '.' appear finitely many times in the "base 13 expans... | 16 | https://mathoverflow.net/users/5963 | 32647 | 21,193 |
https://mathoverflow.net/questions/32652 | 3 | I am a student almost without background on algebraic geometry (but I do know basic graduate algebra and topology). Now I am trying to understand something about algebraic stacks.
I want to start with this short AMS article first: <http://www.ams.org/notices/200304/what-is.pdf>
This article tries to define algebrai... | https://mathoverflow.net/users/7780 | Elliptic curves and algebraic stacks | You should really learn algebraic geometry first. After you're done with that, try reading Mumford's paper "Picard Groups of Moduli Problems", which is a fascinating window into the mind of one of the people who later invented algebraic stacks as he was himself figuring out what you are also trying to figure out. From ... | 8 | https://mathoverflow.net/users/6545 | 32653 | 21,196 |
https://mathoverflow.net/questions/24545 | 10 | Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma\_1, \ldots, \gamma\_n$ corresponding to geometric cycles $Z\_i$ on a target space $X$, pull them back and then the integral
$$\langle \gamma\_1 \cdots \gamma\_n\rangle=\int\_{\overline{\mathcal{M}}\_{g... | https://mathoverflow.net/users/1703 | What is the geometry behind psi classes in Gromov-Witten theory? | The following answer is unfortunately not quite correct, but it may be useful anyway. I will of course be ignoring any virtual fundamental class issues.
Imagine that you are computing a Gromov-Witten invariant where you require the i-th marked point to land at a specific point (i.e. your i-th insertion γi is the clas... | 13 | https://mathoverflow.net/users/7437 | 32670 | 21,208 |
https://mathoverflow.net/questions/32637 | 28 | I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring $R$ lives in the category Ab of Abelian groups as an internal monoid $(\mu\_R, \eta\_R)$. A module is then just an Abel... | https://mathoverflow.net/users/800 | When are modules and representations not the same thing? | Here is my representation theorist's perspective: the key difference between representations and modules is that representations are "non-linear", whereas modules are "linear". I'll concentrate on the case of groups as the most familiar, but this applies more generally.
As Greg has already mentioned, in the most gen... | 15 | https://mathoverflow.net/users/5740 | 32677 | 21,215 |
https://mathoverflow.net/questions/32666 | 29 | Hello,
recently, I've been reading some algebra and sometimes I stumble up on the concept of something "being too big" to be a set. An example, is given in (<http://www.dpmms.cam.ac.uk/~wtg10/tensors3.html>) , where he writes, "Let B be the set of all bilinear maps defined on VxW. (That's the naughtiness - B is too big... | https://mathoverflow.net/users/7607 | When is something too big to be a set? | I just want to give a refinement of other answers so far, as well as a different point of view (namely, that of a person who knows little about set theory but who also encounters these kinds of issues).
As others have mentioned, the root cause of the problem is that there are big logical problems with considering "th... | 30 | https://mathoverflow.net/users/4362 | 32681 | 21,219 |
https://mathoverflow.net/questions/32690 | 17 | This question is influenced by the following riddle:
>
> You are given a rectangular set in the plane with a rectangular hole cut out (in any orientation). How do you cut the region into two sets of equal area?
>
>
>
*SPOILER ALERT!!* - The answer is that you can cut through the center of both rectangles, and ... | https://mathoverflow.net/users/1540 | Planar sets where any line through the center of mass divides the set into two regions of equal area. | Assume that $A$ is compact and convex. If there is a point $P$ such that any line through it is a bisector of $A$ then $A$ has to be centrally symmetric. In fact a stronger result is known (see the [paper](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-16/issue-1/A-theorem-on-partitions-of-mas... | 19 | https://mathoverflow.net/users/5371 | 32696 | 21,229 |
https://mathoverflow.net/questions/32705 | 35 | If $f: [a,b] \to V$ is a (nice) function taking values in a vector space, one can define the definite integral $\int\_a^b f(t)\ dt \in V$ as the limit of Riemann sums $\sum\_{i=1}^n f(t\_i^\*) dt\_i$, or as the final value $F(b)$ of the solution $F: [a,b] \to V$ to the ODE problem $F'(t) = f(t); F(a) = 0$.
In a simil... | https://mathoverflow.net/users/766 | What is the standard notation for a multiplicative integral? | This type of construction also arises in topology and algebraic geometry as "iterated integrals" or "Chen's iterated integrals". There are many sources of which a famous one by Chen himself is: [Link](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-83/issue-5/Iterated-... | 13 | https://mathoverflow.net/users/6579 | 32713 | 21,241 |
https://mathoverflow.net/questions/32478 | 14 | I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
| https://mathoverflow.net/users/2191 | A nice explanation of what is a smooth (l-adic) sheaf? | I'll give an answer, only because I'm interested in chasing down these references
myself. But all I'm doing is assembling references. I assume that
BCnrd will keep me honest. [July 21: I've added some remarks about constructibility, to
makes this more useful (at least to me).]
Since I'm a complex geometer rather an a... | 27 | https://mathoverflow.net/users/4144 | 32717 | 21,243 |
https://mathoverflow.net/questions/32725 | 9 | What are the websites for general position in applied or industrial mathematics(or financial mathematics) related jobs (that is if we have to find a non academic job temporarily) ?Thanks!
| https://mathoverflow.net/users/2391 | website for jobs in applied or industrial mathematics (or financial math) | For finance, I recommend wilmott.com and gloriamundi.org. There's also www.mathjobs.org.
| 6 | https://mathoverflow.net/users/613 | 32727 | 21,250 |
https://mathoverflow.net/questions/32699 | 5 | Problem
-------
Consider two *d* x *d* complex matrices, *R* and *S*, whose entries lie in the unit disk:
$\quad |R\_{i,j}|<1 \quad$ and $\quad |S\_{i,j}|<1 $.
Say that *R* is constructed by randomly choosing complex numbers from the unit disk, but *S* is constructed as
$\quad S\_{i,j} = f(i/d,j/d)$
where $f(... | https://mathoverflow.net/users/5789 | Dependence of trace norm on matrix size for smooth vs. random matrices. | To flesh out Helge's answer a bit before I go to bed:
Assume that $f(x,y)$ is a smooth function on the unit square.
Define the functions $f\_n(x,y) = f(\frac{\lfloor nx \rfloor}{n}, \frac{\lfloor ny\rfloor}{n})$. This is a piecewise step function. Observe that the operator $S$ of dimension $d$ is the same if you def... | 5 | https://mathoverflow.net/users/3948 | 32728 | 21,251 |
https://mathoverflow.net/questions/32734 | 7 | Does there exist a notion of Jordan curve homotopy?
In particular, suppose we have two Jordan curves $C\_0 : S^1 \rightarrow \mathbb{R}^2$ and $C\_1 : S^1 \rightarrow \mathbb{R}^2$. When does there exist a continuous function $f: S^1 \times [0,1] \rightarrow \mathbb{R}^2$ such that:
$f(x,0) = C\_0(x)$, $f(x,1) = C... | https://mathoverflow.net/users/7726 | Jordan Curve Homotopy | Homotopies through embeddings are usually called isotopies.
There is a subtlety called local flatness that comes up in higher dimensions. Let $E$ be any embedding of $\mathbb R$ in $\mathbb R^3$ such that $E(s)=(s,0,0)$ when $s<-1$ or $s>1$. Define a homotopy $H\_t$ with $E\_0(s)=(s,0,0)$ for all $s\in \mathbb R$ an... | 10 | https://mathoverflow.net/users/6666 | 32742 | 21,259 |
https://mathoverflow.net/questions/32730 | 4 | Given some number $n$ and a seed number $s$<$n$, I want a random number generator (RNG) that will go through all integers `<$n$ before coming back to $s$. The resulting random number must be roughly uniformly distributed (which of course it will be if you go through the entire sequence, so I mean for any "large" subseq... | https://mathoverflow.net/users/6898 | Does an "efficient" random number generator exist? | Your requirements aren't rigorously stated, so it's hard to say what you can prove exists or doesn't exist. In the strictest sense, a pseudo-random number generator cannot possibly be "roughly uniformly distributed". Every PRNG is an expansion of entropy from its settings to the set of possible sequences, and it is eas... | 7 | https://mathoverflow.net/users/1450 | 32743 | 21,260 |
https://mathoverflow.net/questions/32720 | 72 | This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of choice, and examples of non Lebesgue measurable set would be [Vitali sets](http://en.wikipedia.org/wiki/Vitali_set), whic... | https://mathoverflow.net/users/2938 | Non-Borel sets without axiom of choice | No, it is not possible. It is consistent with ZF without choice that
>
> the reals are the countable union of countable sets. (\*)
>
>
>
From this it follows that all sets of reals are Borel. Of course, the "axiom" (\*) makes it impossible to do any analysis. As soon as one allows the bit of choice that it is... | 73 | https://mathoverflow.net/users/6085 | 32746 | 21,261 |
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