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https://mathoverflow.net/questions/93659 | 2 | You have three points. A,B and C.
They define a circle segment that starts at A, goes through B and ends at C.
Find the smallest bounding box that encompases the circle segment.
Here is a picture:
<https://docs.google.com/drawings/d/14YwCO0UeMzu-rTLmqULg7HDU5XPYZvSBQdyfu0l71Fs/edit>
I started creating thee equatios... | https://mathoverflow.net/users/22802 | Find the bounding box of a circle segment. | I managed to do it. The first phase is to calculate the circle centre and the angles, just as Joseph said. Then you have to calculate the angles between the vector (1,0) and the (x\_n-cx, y\_n-cy) where n is [0,1,2].
The first and last angle together with the circle define a sector.
Now, calculate the four compass poin... | 0 | https://mathoverflow.net/users/22802 | 93934 | 55,140 |
https://mathoverflow.net/questions/93935 | 7 | Let $f\colon\thinspace M\to N$ be a map of closed smooth manifolds, with $\dim M > \dim N$. Recall that a *submersion* is a smooth map whose differential is surjective at every point in the domain.
>
> Can one give conditions which guarantee that $f$ is homotopic to an submersion?
>
>
>
These conditions would ... | https://mathoverflow.net/users/8103 | Submersions of closed manifolds | Let $F$ be the homotopy fibre of $f$ (ie the space of pairs $(x,u)$, where $x\in M$ and $u$ is a path from $f(x)$ to a specified baspoint in $N$). If $f$ is homotopic to a submersion $f'$, then (using Ehresmann) $F$ will be homotopy equivalent to a closed manifold $(f')^{-1}\{\text{point}\}$, and in particular, it will... | 6 | https://mathoverflow.net/users/10366 | 93940 | 55,144 |
https://mathoverflow.net/questions/93873 | 3 | Let $G$ be a locally compact Hausdorff group.
It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \in W^{\ast}(G)$ is a function assigning to each representation $\pi$ a bounded operator $T(\pi) \in B(H\_{\pi})$. This $T... | https://mathoverflow.net/users/22789 | Universal $W^*$-algebras of locally compact groups: where is the error in this argument? | A link to the literature: I think of $C^\*(G)^\*$ as being $B(G)$, the Fourier-Stieltjes algebra, realised as a (non-closed) algebra of continuous functions on $G$. Any member of $C^\*(G)^\*$ can be realised as the composition of a representation $\pi$ on $H$ with a vector functional $\omega\_{\xi,\eta}$ on $H$. The re... | 6 | https://mathoverflow.net/users/406 | 93941 | 55,145 |
https://mathoverflow.net/questions/93864 | 8 | If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful properties (see the third part of Cox's Primes of the Form for a great introductory reference).
My question is: does ... | https://mathoverflow.net/users/21857 | j-invariant fixed point? | As Kevin Buzzard pointed out in the comments, the $j$-invariant has many possible normalizations. Automorphisms of the affine line naturally act on the target space, so any $aj+b$ with invertible $a$ is a decent replacement.
That said, any normalization has infinitely many fixed points. I'll give an argument for the ... | 9 | https://mathoverflow.net/users/121 | 93943 | 55,146 |
https://mathoverflow.net/questions/93942 | 11 | I've heard this result from my differential manifold class, and I don't know how to prove it.
Does anyone know how to construct such diffeomorphism? Please tell me, thanks a lot.
Any comments are welcome.
| https://mathoverflow.net/users/15289 | Why S^3-K and SL(2,R)/SL(2,Z) are diffeomorphic? Here K is a trefoil in S^3. | Here is a sketch:
$SL(2,\mathbb R)/SL(2,\mathbb Z)$ can be identified with the space of lattices of covolume 1 in $\mathbb R^2$. Indeed, $SL(2, \mathbb R)$ can be identified with a couple of vectors, and multiplying this couple by an element of $SL(2, \mathbb Z)$ does not alter the lattice they generate.
Let $\Lam... | 15 | https://mathoverflow.net/users/9248 | 93946 | 55,148 |
https://mathoverflow.net/questions/93947 | 2 | I work now a lot with strings of characters and other finite sequences and found that I need many times a good notation for "cutting the end" a string. If $a$ is a finite sequence and $a'$ is its right end, then there is a sequence $x$ such that $a = x a'$ (where the product is defined by concatenation). What I would l... | https://mathoverflow.net/users/nan | Notation for ends of a string | A good reference is "G. Rozenberg, A. Salomaa Eds., Handbook of Formal Languages - Vol 1 Word Language Grammar, Springer, 1997" in Chapter 6: Combinatorics of Words:
Let $\sum$ be a finite alphabet.
If $u$ is a word then $u=a\_{1}\ldots a\_{n}$, with
$a\_{i}\in\sum$.
For a pair $\left(u,v\right)$ of words we define... | 6 | https://mathoverflow.net/users/20989 | 93952 | 55,149 |
https://mathoverflow.net/questions/93948 | 13 | Hello !
If I a have a grothendieck Site (C,J), I can consider :
* The Stacks on (C,J) : category fibered in groupoid over C which statisfy suitable descent condition with respect to the covering sieve of J...
* Internal Groupoid in the topos Sh(C,J).
Clearly those two notions are very close. but not exactly equiv... | https://mathoverflow.net/users/22131 | Link between internal groupoids and stacks on a topos ? | The $2$-category of internal groupoids in $Sh(C,J)$ is equivalent to the $2$-category of sheaves of groupoids on $(C,J)$. To see this, given a groupoid $G$ in sheaves,the strict $2$-functor $$C \mapsto \left(G\_0\left(C\right) \rightrightarrows G\_1\left(C\right)\right)$$ is a sheaf of groupoids. So, there is a $2$-fun... | 12 | https://mathoverflow.net/users/4528 | 93953 | 55,150 |
https://mathoverflow.net/questions/93956 | 2 | On a compact Riemannian maniflod $(M,g)$, for an elliptic
complex
$\mathcal{C}\_0\overset{L\_1}{\longrightarrow}\mathcal{C}\_1\overset{L\_2}{\longrightarrow}\mathcal{C}\_2$ where $L\_1$ and $L\_2$ are partial differential operators of order
$l\_1$, $l\_2$ respectively, and $\mathcal{C}\_\*$ are vector bundles
over $M$.... | https://mathoverflow.net/users/22880 | A priori estimate of elliptic complex | Yes. Check Sections 6.1 and 6.2 of
>
> Charles B. Morrey: Multiple Integrals in the Calculus of Variations, Springer Verlag.
>
>
>
| 3 | https://mathoverflow.net/users/20302 | 93959 | 55,153 |
https://mathoverflow.net/questions/93844 | 3 | The Zariski-Fujita theorem says that on a projective variety $X$, if a Cartier divisor $D$ is ample on its base locus, then some positive multiple $mD$ is base point free. I'm wondering if the following related statement is true.
In the same setting, suppose the base locus $Bs(|D|)= \bigcup\_{i=1}^{N} C\_i$ is a unio... | https://mathoverflow.net/users/11661 | Curves contained in the stable base locus | I don't think this is true. Let $X$ be obtained by blowing up $\mathbb P^2$ at one point, and then at a point on the exceptional curve. Call the exceptional divisors $E$ and $F$ respectively. Let $D = E+3F$. Then $D \cdot E = 1$, but $E$ is in the stable base locus.
| 4 | https://mathoverflow.net/users/22892 | 93983 | 55,165 |
https://mathoverflow.net/questions/93992 | 4 | A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional.
Q.1. Is every closed meagre subset of an $n$-dimensional locally compact Hausdorff space of dimension $\le n-1$ (for $n\ge 1$)?
Q.2. If so, is there anything unusual about meagre sets in $0$-di... | https://mathoverflow.net/users/22260 | closed meagre sets | For Q3, the answer is yes. Think of the Cantor set as consisting of the points that have a base-3 expansion containing only 0's and 2's. Now take the subset of those points where the 2's occur only in even-numbered positions.
Also, in the first sentence, "either countable or homeomorphic to the Cantor set" isn't rig... | 4 | https://mathoverflow.net/users/6794 | 93993 | 55,170 |
https://mathoverflow.net/questions/93998 | 6 | Suppose we are given a diagram of topological spaces. We can restrict ourselves to the diagrams over finite partially ordered sets and let all spaces be good enough (e.g. CW-complexes). One can take the colimit and homotopy colimit of this diagram and there is a set of rules to manipulate these objects, I mean first of... | https://mathoverflow.net/users/22897 | Equivariant colimits and homotopy colimits | Shulman's paper ``Homotopy limits and colimits and enriched homotopy theory''
(on arXiv) gives a thorough study of homotopy colimits in enriched contexts,
and his enriching category $V$ can be taken to be the category of $G$-spaces
for a topological group $G$. It specializes to answer your questions, in greater
general... | 12 | https://mathoverflow.net/users/14447 | 94007 | 55,174 |
https://mathoverflow.net/questions/94008 | 2 | If a topological space $X$ enjoys the property that every ultrafilter $U$ on $X$ has a single limit, must $X$ be a Hausdorff space?
(Ultrafilters here consist of arbitrary subsets (so not necessarily, for example, $z$-sets or closed sets) but the limit of such a $U$ means the intersection of the closures of all its s... | https://mathoverflow.net/users/22161 | Does every ultrafilter has single limit imply Hausdorff separation | An ultrafilter $U$ on the space $X$ is said to converge to the point $p$ if every neighborhood of $p$ belongs to $U$. (This is equivalent to saying that the complement of an element of $U$ is never a neighborhood of $p$. Thus it is also equivalent to saying that $p$ belongs to the closure of every element of $U$.)
If... | 9 | https://mathoverflow.net/users/6666 | 94011 | 55,176 |
https://mathoverflow.net/questions/94022 | 4 | Hi,
I have a question related to Serre Duality.
First, I would like to recall some preliminar definitions.
**Definition 1.** A variety $X$ is $\mathbb{Q}$-Gorenstein if it is Cohen-Macaulay and $\omega\_X^{\otimes m}$ is a line bundle for some $m>0$, where $\omega\_X$ is the dualizing sheaf of $X.$ In particular... | https://mathoverflow.net/users/nan | Ext groups and Serre duality for proper $\mathbb{Q}$-Gorenstein varieties. | **EDIT:** After some clarification from the questioner with respect to the $Q$-Gorenstein definition, let me point out that additional hypotheses are needed I think.
I don't think there's any hope that this can be true without additional hypotheses. I'll start by claiming that I can pick $\mathcal{E, F}$ such that $\... | 7 | https://mathoverflow.net/users/3521 | 94024 | 55,183 |
https://mathoverflow.net/questions/94027 | 3 | This is a part of exercise E4 from Chapter VII of Kunen's Set Theory.
The hint (courtesy of A. Miller) goes like this: let ${\bf P} = Fn(I,2)$, $(I \geq \omega\_{1})^M$. Let G be ${\bf P}$-generic over M, N = ${\bf L}(\mathcal P \left({\omega}\right))^{M[G]}$, and assume AC in N such that $(\kappa = |\mathcal P \lef... | https://mathoverflow.net/users/22905 | Cohen forcing: why is the cardinality of $\mathcal P \left({\omega}\right)$ in ${\bf L}(\mathcal P \left({\omega}\right))$ independent of G? | I'm a little confused by your question, since if we add uncountably many Cohen reals, which is what your forcing does, then the inner model $L(\mathbb{R})$, which is the same as your $L(P(\omega))$, does not actually satisfy the axiom of choice, so your assumption that $N\models\text{AC}$ is not warranted. In particula... | 4 | https://mathoverflow.net/users/1946 | 94035 | 55,186 |
https://mathoverflow.net/questions/94017 | 21 | Is there a direct way to seeing that $B{\mathbb{N}}\simeq S^1$, i.e. the classifying space of the monoid of natural numbers is homotopy equivalent to the circle?
Here, since the natural numbers ${\mathbb{N}}$ is not a group, some care is needed to define the classifying space $B{\mathbb{N}}$ properly. One way to do ... | https://mathoverflow.net/users/6301 | Why is the classifying space of the natural numbers homotopy equivalent to the circle? | Method I: Symmetric products.
Contained inside the simplicial set $N\mathbb{N}$ is a copy of the simplicial circle $S^1$, generated by the zero-simplex and the 1-simplex $[1]$. This consists of all simplices of the form $e\_i = (0,\ldots,0,1,0,\ldots,0)$, together with the basepoint $(0,\cdots,0)$, in the simplicial ... | 18 | https://mathoverflow.net/users/360 | 94047 | 55,191 |
https://mathoverflow.net/questions/94030 | 1 | Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0 $ is exact with $E ,F$ rank two bundles on $X$ and $K$ a line bundle on a (smooth?) divisor $D \subset X$. Can one give information on the cokernel of the injection $Sym^n(E) \to... | https://mathoverflow.net/users/5100 | cokernel of the symmetric product of an injection. | Hi Adam,
Say $n=2$. Let $EF\subset Sym^2(F)$ be the subsheaf generated locally by products of sections of $E$ and $F$. Then a diagram chase should give an extension
$$0\to (EF)/Sym^2(E)\to Sym^2(F)/Sym^2(E)\to Sym^2(F)/EF\to 0$$
of sheaves supported on $D$.
To get a better sense of this, say $E=O\_X\oplus O\_X$,
$F... | 1 | https://mathoverflow.net/users/4144 | 94049 | 55,193 |
https://mathoverflow.net/questions/94052 | 16 | As an undergraduate I learned point-set topology from Munkres's book, as did many others.
One topic that gets a lot of attention is the separation axioms. For example, a space $X$ is normal if any two closed, disjoint subsets of $X$ can be separated by open neighborhoods.
Some of the axioms (e.g. Hausdorff) turn up... | https://mathoverflow.net/users/1050 | Where else do the (topology) separation axioms turn up? | These separation axioms play an important role in determing duals of spaces of continuous functions (i.e. in functional analysis). Examples are:
* If $X$ is normal, then the dual of the space of bounded real-valued continuous functions $C\_b(X)$ is the space of regular bounded finitely additive measures on $X$.
* If... | 12 | https://mathoverflow.net/users/10194 | 94053 | 55,195 |
https://mathoverflow.net/questions/94038 | 30 | Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup\_{n\in \mathbb{Z}^+} (a\_n,b\_n)$ where $(a\_n,b\_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in \mathbb{Z}^+$ ,$f$ coincides with a polynomial on $(a\_n,b\_n)$.
Is it necessary that $f$ coincide with a polynomia... | https://mathoverflow.net/users/22907 | is f a polynomial provided that it is "partially" smooth? | I believe the answer is "no". The key lemma is:
**Lemma.** Let $f: [c,d] \to {\bf R}$ be smooth, let $I$ be a compact subinterval of $(c,d)$, $q$ be an interior point of $I$, let $n \geq 1$, and let $\varepsilon > 0$. Then there exists a smooth perturbation $g: [c,d] \to {\bf R}$ of $f$ which agrees with $f$ outside ... | 21 | https://mathoverflow.net/users/766 | 94077 | 55,207 |
https://mathoverflow.net/questions/94066 | 13 | **Can the Yang-Mills or Chern-Simons action functionals be considered as [possibly *perfect*] Morse functions?** I assume we would be in an equivariant scenario due to considering the configuration spaces with gauge-groups/transformations. Or at least how far away are they from a Morse-Bott function (and from being per... | https://mathoverflow.net/users/12310 | Yang-Mills and Chern-Simons functionals as Morse functions | The problem with the CS functional is that the Morse indices of its critical points are infinite. In particular, this functional cannot be perfect. The Floer complex does not compute the homology of any particular space (though it might compute the homology of a certain spectrum).
On 4-manifolds the YM functional has... | 11 | https://mathoverflow.net/users/20302 | 94082 | 55,211 |
https://mathoverflow.net/questions/94084 | 6 | Is there a name for the class of groups in the title, and any sort of characterization? Free groups and surface groups are in the class, but presumably there are many more...
| https://mathoverflow.net/users/11142 | Groups where every two generator subgroup is free | There is no name, there are lots of examples. Guba gave many non-trivial examples in Guba, V. S. Conditions under which 2-generated subgroups in small cancellation groups are free.
Izv. Vyssh. Uchebn. Zaved. Mat. 1986, no. 7, 12–19 and here: Guba, V. S.
A finitely generated simple group with free 2-generated subgroups.... | 16 | https://mathoverflow.net/users/nan | 94085 | 55,212 |
https://mathoverflow.net/questions/94086 | 2 | [This question](https://mathoverflow.net/questions/94038/is-f-a-polynomial-provided-that-it-is-partially-smooth) asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" of an open interval... | https://mathoverflow.net/users/20729 | Partitions of an interval | Unless I'm misunderstanding your defintion of having an 'next' interval to the left or right, what about a Cantor set like construction: divide (0,1) into three equal intervals. Let $(\frac{1}{3},\frac{2}{3})$ be included in your decomposition, then divide $(0,\frac{1}{3})$ and $(\frac{2}{3},1)$ into three equal interv... | 5 | https://mathoverflow.net/users/12301 | 94088 | 55,214 |
https://mathoverflow.net/questions/93463 | 19 | The structure of finite simple groups of Lie type of arbitrary rank can be described well via BN-pairs. BN-pairs basically generalize the Bruhat decomposition of matrices into monomial $N$ and triangular $B$ matrices and come with a "Weyl group" $N/(B\cap N)$, that has to be a Coxeter group.
Conversely, Tits showed i... | https://mathoverflow.net/users/22709 | (weak?) BN-Pair / Tits System for Sporadic Groups | There has been a lot of work done on various generalizations of the concept of the building, to apply them to sporadic groups. These generalizations are variously known as diagram geometries, chamber systems, etc.
Names like G.Stroth, S.Smith, M.Ronan, A. Delgado, D. Goldschmidt, B. Stellmacher, etc. spring to mind. Th... | 11 | https://mathoverflow.net/users/11100 | 94089 | 55,215 |
https://mathoverflow.net/questions/94033 | 4 | Hi!
Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:
1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geod... | https://mathoverflow.net/users/19689 | Pleated surfaces do not curl up too much | This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d\_N(g(x), g(y)) \leq d\_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting... | 1 | https://mathoverflow.net/users/1650 | 94095 | 55,220 |
https://mathoverflow.net/questions/94076 | 1 | Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?
<http://www.jstor.org/discover/10.2307/1970887?uid=3739256&uid=2&uid=4&sid=56038494493>
| https://mathoverflow.net/users/22786 | Intertwining Integral defined on a Weyl group? | It is not true that the intertwiner depends only upon the Weyl element but also upon the parabolic subgroup $P$. Moreover, it is better to consider the intertwiner to depend upon a set of complex parameters. Its analytic properties give alot of information about the irreducibility, growth of matrix coeffecients, and th... | 2 | https://mathoverflow.net/users/10400 | 94096 | 55,221 |
https://mathoverflow.net/questions/94036 | 6 | Let $C$ be a quasi-category. Then there is an imbedding
$$ C^{op} \times C \to \mathrm{Kan}$$
where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's book by choosing a model for $C$ as the nerve of a fibrant simplicial category $\mathfrak{C}$ and then taking the nerve of... | https://mathoverflow.net/users/344 | $(\infty, 1)$-Yoneda embedding via the Grothendieck construction | Let $\mathcal{M}$ be the simplicial set defined by the formula $Hom( \Delta^{J}, \mathcal{M} ) =Hom( \Delta^{J^{op} } \star \Delta^{J}, \mathcal{C} )$, so that an $n$-simplex of $\mathcal{M}$ is a $(2n+1)$-simplex of
$\mathcal{C}$. The inclusions of $\Delta^{J^{op} }$ and $\Delta^{J}$ into $\Delta^{J^{op} } \star \Delt... | 11 | https://mathoverflow.net/users/7721 | 94102 | 55,224 |
https://mathoverflow.net/questions/94109 | 1 | I think I could write down a projective resolution, tensor with the twisted coefficients and find the first cohomology of the standard torus.
BUT, I was wondering if there is an easier way to understand the first (co)homology groups. (Just checking, by Poincaré duality they should be isomorphic, right?)
Is there so... | https://mathoverflow.net/users/11084 | Twisted cohomology of torus | Let me offer you a projective resolution of $\mathbb{Z}$ as a module over $\pi\_1(T)=\mathbb{Z}^2$, obtained from the cellular chain complex of $\mathbb{R}^2$, the universal cover of $T$. The cell structure on $\mathbb{R}^2$ is induced by the usual cell structure on $T$ with one vertex, two edges and one $2$-cell.
Th... | 3 | https://mathoverflow.net/users/12166 | 94118 | 55,228 |
https://mathoverflow.net/questions/94119 | 5 | Let $f(x)=\sum\_{n\geq 0}\frac{1}{n!n!}x^{n}$, is there an explicit formula for $f(x)$?
| https://mathoverflow.net/users/22925 | Is there an explicit formula for $\sum x^n/(n!)^2$? | This is, for $x \ge 0$, a special case of the modified Bessel function of the first kind. Have a look here: <http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html>
$$
\sum\_{n = 0}^{\infty} \frac{x^{n}}{(n!)^{2}} = \mathrm{I}\_0 \left(2 \sqrt{x}\right)
$$
| 15 | https://mathoverflow.net/users/6698 | 94123 | 55,231 |
https://mathoverflow.net/questions/94014 | 29 | Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as Quantum Electrodynamics).
EDIT: the reason for asking "essentially different" is the following. It is possible to intuiti... | https://mathoverflow.net/users/22789 | Mathematical foundations of Quantum Field Theory | If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somewhere has, but that's certainly not mainstream thought in QFT research, either on the mathematics or theoretical physics ... | 13 | https://mathoverflow.net/users/2622 | 94124 | 55,232 |
https://mathoverflow.net/questions/94116 | 2 | For my thesis I've been doing a lot of research concerning upwards closed sets and anti chains. A while ago while searching I thought I stumbled across a proof that gave an upper bound on the number of anti-chains/uppersets in a partially ordered set.
Now I'm not 100% sure someone has proven a fair upper bound (ofc n... | https://mathoverflow.net/users/22390 | Is there an upper bound on the number of uppersets/antichains in a partially ordered set | If the poset is a power set, then the answer is $2^{{n \choose n/2}(1-o(1))}$.
A brief history of the problem:
<http://mathworld.wolfram.com/DedekindsProblem.html>
<http://oeis.org/A014466>
<http://www.ams.org/journals/proc/1969-021-03/S0002-9939-1969-0241334-6/S0002-9939-1969-0241334-6.pdf>
| 3 | https://mathoverflow.net/users/955 | 94128 | 55,234 |
https://mathoverflow.net/questions/94137 | 3 | Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a canonical fashion. An [Ehresmann connection](http://en.wikipedia.org/wiki/Ehresmann_connection#Formal_definition) is a choi... | https://mathoverflow.net/users/238 | How do we use an Ehresmann connection to define a semispray? | Isn't the spray associated to the connection just the geodesic differential equation? Let $\pi\_M : TM \to M$ be bundle projection, $\pi\_{TM} : TTM \to TM$ bundle projection.
A double tangent vector $w \in TTM$ represents parallel transport of $\pi\_{TM}(w)$ if $v(w) = 0$. i.e. if $w$ is horizontal. You can identif... | 3 | https://mathoverflow.net/users/1465 | 94139 | 55,238 |
https://mathoverflow.net/questions/94120 | 2 | I will be grateful for any ideas to solve the series
$$\sum^\infty\_{k=0}\frac{x^k z^k}{k!} \frac{\Gamma(1+a+2k)}{\Gamma(2+k)}{}\_2F\_1(1,1+a+2k;2+k;z)$$
$a$ is a nonegative integer, $z$ and $x$ are real numbers.
| https://mathoverflow.net/users/19493 | Infinite Series of 2F1 | Your expression simplifies to $s(k)=(xz)^k/k!\*\sum\_{n=0}^\infty z^n\*\frac{(2k+a+n)!}{(k+n+1)!}$ where $s(k)$ is the sequence you sum from $k=0$ to infinity. Since $a$ is non-negative, $\frac{(2k+a+n)!}{(k+n+1)!k!}>\binom{2k+a+n}{k}$ for $a>1$ and the limit of $\frac{(2k+a+n)!}{(k+n+1)!k!}$ for large $k$ or $n$ goes ... | 3 | https://mathoverflow.net/users/22917 | 94146 | 55,240 |
https://mathoverflow.net/questions/93615 | 5 | If $f: \mathbb{R}^n \to \mathbb{R}$ is a Lipschitz function and $X$ is a standard $n$-dimensional Gaussian vector with $\mathbb{E} f(X) = 0$, then $f(X)$ is subgaussian (in a way that does not depend on $n$). If $f$ is $\mathcal{C}^1$, this is equivalent to saying that $|\nabla f|$ bounded implies $f(X)$ is subgaussian... | https://mathoverflow.net/users/21652 | Concentration of Gaussian vectors | To answer my own question, this follows from a more general result that is mentioned in ["On measure concentration of vector valued maps"](http://webmail.impan.gov.pl/cgi-bin/ba/pdf?ba55-3-07) by Ledoux and Oleszkiewicz, Theorem 4: for any convex function $\Psi: \mathbb{R}^k \to \mathbb{R}$,
$$
\mathbb{E} \Psi(f(X)) \l... | 3 | https://mathoverflow.net/users/21652 | 94161 | 55,247 |
https://mathoverflow.net/questions/94083 | 4 | Consider a random variable $X = \sum\_{i=1}^{m} X\_i$, where each $X\_i$ is an indicator
random variable that is $1$ with probability $k/m$ and $0$ otherwise. We are interested in the quantity $S\_X(m) = E[\min(X,k)]$. The motivation is that we have a bin of capacity $k$. At each step, a ball is thrown into the bin wi... | https://mathoverflow.net/users/11247 | Capped binomial random variables | The answer to your question is **positive** and, for example, follows immediately from Corollary 4 of
>
> C. A. León and F. Perron (2003), [Extremal properties of sums of Bernoulli random variables](http://dx.doi.org/10.1016/S0167-7152%2803%2900037-3), *Statistics and Probability Letters*, vol. 62, 345–354.
>
>
>... | 7 | https://mathoverflow.net/users/17114 | 94164 | 55,249 |
https://mathoverflow.net/questions/94133 | 5 | I have a simple application of [Polya's enumeration theorem](http://en.wikipedia.org/wiki/Polya_enumeration_theorem) to counting nonisomorphic pentago boards with a given number of stones. In terms of the theorem, the color generating function is $f(x) = 1+x+y$ corresponding to empty, black, or white, there are $6 \tim... | https://mathoverflow.net/users/22930 | Using Polya's enumeration theorem for explicit generation of instances | There are two very general methods for this type of problem, called orderly generation, and generation by canonical construction path. One place that describes these is the book Classification Algorithms for Codes and Designs (Springer 2006) by Kaski and Östergård (where they call the second method "canonical augmentat... | 3 | https://mathoverflow.net/users/9025 | 94173 | 55,254 |
https://mathoverflow.net/questions/94001 | 4 | Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation is taken to be semisimple... Are they considering its semisimplification?
Sorry for the simple question.
Thanks
| https://mathoverflow.net/users/36285 | semisimplicity of automorphic Galois representations | Do you mean the local Galois representations or the local Galois representations ?
The global Galois representations they are constructing correspond to cuspidal automorphic representations of GL(n). They are expected to be always irreducible, though I'm not sure when this is known exactly. But it is known in the cas... | 4 | https://mathoverflow.net/users/12336 | 94176 | 55,255 |
https://mathoverflow.net/questions/94138 | 1 | $A$ is symmetric positive definite matrix and $S$ is such that $A=SS^{T}$. Further
$y=Sz$
Does there exist a simple ( or any verifiable) relation exist only involving $A$,$y$ and $z$ ?
Thanks
| https://mathoverflow.net/users/22429 | Matrix elimination | Your last comment confirms the tentative answer I gave in the comment above. All you need to check is the scalar relation $y^T A^{-1} y = z^T z$.
| 0 | https://mathoverflow.net/users/2622 | 94183 | 55,257 |
https://mathoverflow.net/questions/94182 | 7 | Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a pure numerical coincidence?
[It's possible that my question amounts to groundless mysticism, but I had to wonder.]
On... | https://mathoverflow.net/users/22431 | Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions? | The cobordism groups can be calculated using the Adams spectral sequence, which is based on the homological algebra of modules over the Steenrod algebra. This works most nicely for the unoriented cobordism groups, which form a polynomial algebra over $\mathbb{Z}/2$ with one generator in each degree not of the form $2^j... | 13 | https://mathoverflow.net/users/10366 | 94185 | 55,258 |
https://mathoverflow.net/questions/94179 | 9 | Prikry-Silver forcing $\mathbb{V}$ (sometimes just Silver forcing) is the forcing notion consisting of all partial functions $p:\omega\rightarrow 2$ with co-infinite domain. In "Combinatorics on ideals and forcing with trees" Marcia Groszek mentions (without proof) that a Prikry-Silver real has minimal real degree, but... | https://mathoverflow.net/users/2436 | Intermediate extension of a Prikry-Silver extension? | I don't know a reference for this, but the $A$ that you are looking for is the collection of
all domains of conditions in the Silver generic filter.
Equivalently, you can consider the set of all complements of domains of conditions in the filter. This is a non-principal ultrafilter on $\omega$.
Silver forcing can ... | 11 | https://mathoverflow.net/users/7743 | 94186 | 55,259 |
https://mathoverflow.net/questions/94165 | 16 | In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that certain sums converge uniformly, then for all ideles $a$, we have
$\frac{1}{|a|}\displaystyle\sum\_{\xi\in k}\hat{f}(\x... | https://mathoverflow.net/users/7998 | Analogues of the Riemann-Roch Theorem | Perhaps you might want to look in the book project of Frankenhuijsen <http://research.uvu.edu/Math/machiel/RH.pdf> and search for Riemann Roch theorem.
>
> Quote from page 3: The function $ζ\_C$ (zeta function of a curve) satisfies the functional equation $ζ\_C (1 − s) = ζ\_C (s)$. This functional equation can be pr... | 6 | https://mathoverflow.net/users/10400 | 94192 | 55,264 |
https://mathoverflow.net/questions/94180 | 2 | Let $X$ be a compact Kahler manifold of complex dimension $n$. The Aubin--Calabi--Yau theorem says that if we fix a smooth form $\rho$ in the Chern class $c\_1(X)$, then every Kahler class on $X$ contains a unique Kahler metric $\omega$ whose Ricci-form is $\rho$. Alternatively, one may fix a volume form $dV$ on $X$, t... | https://mathoverflow.net/users/4054 | Are there hermitian metrics with the volume form of a Kahler metric? | A property of a hermitian metric to be conformally Kähler is a very restrictive property and a generic hermitian metric is not conformally Kähler.
The property of the volume form to be equal to some fixed volume form is senseless if we allow to multiply the metric by a conformal coefficient. Combining this, we see tha... | 3 | https://mathoverflow.net/users/14515 | 94199 | 55,267 |
https://mathoverflow.net/questions/94150 | 1 | Let $A$ be an abelian variety defined over $\mathbf{C}$ (of dimension $>1$) and let $\Theta\_A$ be the holomorphic tangent sheaf of $A$.
>
> **Question.** How does one compute $H^1(A,\Theta\_A)$ ?
>
>
>
If $A$ has dimension $1$ then using Serre's duality one finds that
$H^1(A,\Theta\_A)\simeq H^0(A,\omega\_A^... | https://mathoverflow.net/users/11765 | complex deformations of abelian varieties | Hugo,
Although this was already discussed in the comments, perhaps I can write few more details
here. The material can be found in many books such as Mumford's Abelian Varieties or
the book on the same by Birkenhake and Lange.
Claim $\dim H^1(A,\Theta)= g^2$.
The first thing to observe is that $A$ is a group, so ... | 7 | https://mathoverflow.net/users/4144 | 94208 | 55,272 |
https://mathoverflow.net/questions/94195 | 4 | Suppose we have a group $G$ which can be generated by two elements $x$, $y$. Call $H$, $K$, $L$ the subgroups of $G$ generated by $x$, $y$ and $y^{-1}x^{-1}$, respectively.
With these data, we can build up a graph $\mathcal{G}$, by declaring the vertices of $\mathcal{G}$ to be the right-cosets of $H$, $K$ and $L$, an... | https://mathoverflow.net/users/22606 | Representing groups with two generators as graph automorphisms | I think what you are trying to remember is the triangle group $\langle x,y\mid x^k=y^l=(xy)^m=1\rangle$ (see [Wiki](http://en.wikipedia.org/wiki/Triangle_group)). Depending on whether $1/k+1/l+1/m$ is less than, equal to or greater than 1, the group corresponds to a tessellation of a hyperbolic plane, Euclidean plane o... | 4 | https://mathoverflow.net/users/nan | 94210 | 55,274 |
https://mathoverflow.net/questions/94151 | 2 | Alright, here I go again, don't know if I'm missing something here but let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I want to compute the cohomology of this complex. The hypercohomology of this complex is the cohomology of the complex
tot$(C^\bullet(F^\bullet)(X))$
of global... | https://mathoverflow.net/users/22191 | Spectral sequences in Hypercohomology of sheaves | You seem to be concluding that the hypercohomology of **any** cochain complex $F^\bullet$ must vanish (except, perhaps, in degree zero)?
To see where you've gone wrong, start with your favorite sheaf ${\cal O}$, and let $F^\bullet$ be an injective resolution of ${\cal O}$. Then (pretty much directly from the definiti... | 3 | https://mathoverflow.net/users/10503 | 94215 | 55,277 |
https://mathoverflow.net/questions/93871 | 8 | Let $(M, \xi)$ be a closed contact manifold with co-oriented contact structure $\xi = \ker \alpha$. Let $\mathrm{Cont}(M, \alpha)$ be the group of diffeomorphisms that preserve the contact form $\alpha$, and let $\mathrm{Cont}^+(M, \xi)$ be the diffeomorphisms that preserve $\xi$ with its co-orientation (i.e. that pull... | https://mathoverflow.net/users/477 | strong contactomorphism group inside contactomorphism group | One of the problems with this question is that $\text{Cont}(M,\alpha)$ depends heavily on the choice of contact form $\alpha$, so while the previous answer shows that there is a choice of contact form for which $\text{Cont}(M,\alpha)$ and $\text{Cont}^+(M,\xi)$ can't be homotopy equivalent, it's not immediately clear i... | 9 | https://mathoverflow.net/users/22948 | 94217 | 55,279 |
https://mathoverflow.net/questions/94194 | 13 | Given a presentation $ < X ; R >$ of a group $G$. Suppose we know for some reason that $G$ is the fundamental group of a three-dimensional finite volume manifold.
Then there is a injective group homomorphism $G\rightarrow Isom(\mathbb{H}^3)$. Mostows rigidity theorem then tells us that it is unique up to composition... | https://mathoverflow.net/users/3969 | Finding hyperbolic metrics by approximation | To my knowledge this hasn't been done in theory (although see Harriet Moser's thesis <http://www.math.columbia.edu/~moser/>). But it certainly has been done in practice by Jeff Weeks' program SnapPea. Note that $\mbox{Isom}(\mathbb{H}^3) = \mbox{PSL}(2, \mathbb{C}) = \Gamma$. So your source group $G = \pi\_1(M^3)$ alre... | 12 | https://mathoverflow.net/users/1650 | 94219 | 55,280 |
https://mathoverflow.net/questions/94213 | 7 | If $G = \langle X | R \rangle$ is a $\delta$-hyperbolic group presentation, then Dehn's algorithm provides a linear time solution to the word problem, but the linear constant is horribly exponential in $\delta$ since the Dehn presentation consists of all words of length $8 \delta$ equal to the identity.
Are there an... | https://mathoverflow.net/users/14163 | Asymptotics of the number of required Dehn relators in hyperbolic groups | Here is an idea of an example (just for a start). Take the finite group $A\_n$ (it is hyperbolic). It has a short presentation, see [this paper.](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=0CDYQFjAC&url=http%3A%2F%2Fwww.ma.huji.ac.il%2F~alexlub%2FPAPERS%2FPresentation%2520of%2520Finite%2520Simpl... | 2 | https://mathoverflow.net/users/nan | 94222 | 55,282 |
https://mathoverflow.net/questions/94227 | 2 | I have two "rookie questions" about elliptic surfaces:
1. Let $S$ be an elliptic surface over $\mathbb{C}$, i.e. a smooth, projective algebraic surface equipped with a morphism $f: S \to C$ to a curve $C$ such that the generic fibre is an elliptic curve over $\mathbb{C}(C)$. The Kodaira dimension of an elliptic surfa... | https://mathoverflow.net/users/1107 | Elliptic surface with $\kappa = 1$ | If $g(C) \geq 2$ then $\textrm{kod}(S)=1$.
The same holds also if $g(C)=1$ and $f$ is not locally trivial.
See [Barth-Hulek-Peters-Van de Ven, Compact Complex Surfaces], Proposition 12.5 page 215 (Chapter V).
| 4 | https://mathoverflow.net/users/7460 | 94228 | 55,283 |
https://mathoverflow.net/questions/94025 | 14 | Is there a "local" algorithm which takes as its input a parking function and returns the non-crossing partition labelled by that sequence?
Background: A parking function is a sequence of positive integers which is a permutation of a sequence $a\_1\leq \ldots \leq a\_n$ satisfying $a\_k \leq k$ for all $k$. In a 1996 ... | https://mathoverflow.net/users/15365 | Parking functions to non-crossing partitions | I still hope there is a complete (and easy) answer to the question, but as mentioned in my comment above, and since no one else answered so far, I give a description of the inverse map that is not completely local, but still "better" than first labelling all chains in the noncrossing partition lattice. (I just worked i... | 8 | https://mathoverflow.net/users/21291 | 94241 | 55,287 |
https://mathoverflow.net/questions/94239 | 4 | The $2$-fiber product of two functors $f : A \to C$ and $g : B \to C$ is given by the category of triples $(a,b,\theta)$, where $a \in A, b \in B, \theta : f(a) \cong g(b)$. This is the correct notion of fiber product when we think of $\mathrm{Cat}$ as a $2$-category. When we just regard it as a $1$-category, we consid... | https://mathoverflow.net/users/2841 | Pullbacks of (Waldhausen) categories | Nice observation. A sufficient condition so that the $2$-categoricall pull-back coincides with the strict one is that either $f\colon A\rightarrow C$ or $g\colon A\rightarrow C$ is a fibration, in the sense that it satisfies the isomorphism lifting property. These are fibrations for a model category structure on small ... | 6 | https://mathoverflow.net/users/12166 | 94247 | 55,293 |
https://mathoverflow.net/questions/94252 | 4 | A good treatment have been given to the quaternion equations. Indeed, Ivan Niven in his paper Equations in Quaternion given in this link <http://jones.math.unibas.ch/~massierer/algebra-hs11/niven(quaternions).pdf>
gave an algorithm on how to find all the solution of a given quaternion polynomial equation. Actually, in... | https://mathoverflow.net/users/19510 | The octonion equations | The paper
On Octonionic Polynomials
Rogério Serôdio (in advances in applied clifford algebras -- who knew such a journal existed?) seems to address precisely the questions you ask.
| 2 | https://mathoverflow.net/users/11142 | 94257 | 55,297 |
https://mathoverflow.net/questions/94226 | 22 | The following problem has bothered me for a long time.
Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly in the following way. For every step of the "walk", it will choose a real number $\Delta x$ uniformly from the interval $[-1,1]$, ... | https://mathoverflow.net/users/22954 | A random walk with uniformly distributed steps | Here is an argument that proves the conjecture. More generally, it shows that if $X\_1,\dots,X\_n$ is a "generic" sequence of positive real numbers, and we form a sum by permuting the terms randomly and putting random signs on the terms (uniform distribution over all possibilities), then the probability that all partia... | 29 | https://mathoverflow.net/users/14302 | 94262 | 55,299 |
https://mathoverflow.net/questions/94275 | 1 | Does the following hold true $\forall T>0,a>0,c>0$ (in particular for c arbitrarily small):
$P\_0(\int\_0^T e^{-aB\_s}ds<{c})>0$?
This is a minor result which will improve several steps in a couple of proofs in my thesis but although it seems intuitive to me that it is a correct statement, I'm having trouble sho... | https://mathoverflow.net/users/22968 | distribution of specific exponential functional of brownian motion | Yes. Let $M$ be defined by $e^{-aM}=c/(2T)$. Now if $(B\_s)$ satisfies $B\_s>-1$ for all $s\in [0,c/(2e^a)]$ and $B\_s>M$ for all $s\in[c/(2e^a),T]$, then $\int\_0^{c/(2e^a)} e^{-aB\_s}\,ds\le e^a\cdot c/(2e^a)=c/2$.
Similarly $ \int\_{c/(2e^a)}^Te^{-aB\_s} \, ds < Te^{-aM} < c/2$. Combining the two, assuming that $... | 1 | https://mathoverflow.net/users/11054 | 94276 | 55,304 |
https://mathoverflow.net/questions/94267 | 9 | On page 58 of Mark Hovey's book *Model Categories*, he states the following definitions:
>
> "A subset $U$ of a space $X$ is
> compactly open if for every continuous
> $f:K\rightarrow X$ where $K$ is
> compact Hausdorff, $f^{-1}(U)$ is open
> in $K$... A space $X$ is called a
> $k$-space, ***or Kelley space***... | https://mathoverflow.net/users/11540 | Are k-spaces named for Kelley? | Engelking cites [this paper](http://www.ams.org/journals/proc/1950-001-03/S0002-9939-1950-0036503-X/home.html) as the place where $k$-spaces were introduced, though the author, David Gale, says the notion was first defined by Hurewicz. The $k$ probably refers to the German `kompakt'.
| 12 | https://mathoverflow.net/users/5903 | 94284 | 55,310 |
https://mathoverflow.net/questions/94236 | 2 | considering the Hamitlonian for the Selberg Operator $ y^{2} ( \partial \_{x}^{2}+ \partial \_{y}^{2}) $ given in the Hamiltonian form
$ H=g\_{ab}p^{a}p^{b} $ with $ ds^{2} = \frac{dx^{2}+dy^{2}}{y^{2}}$ being the metric
can we obtain from this H the functional determinant expression for the selberg zeta ??
can w... | https://mathoverflow.net/users/21933 | selberg trace from classical physics | This is not really an answer, but somewhat long comment.
Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects.
Let me suggest some steps towards physical point of view (which are standa... | 3 | https://mathoverflow.net/users/10446 | 94289 | 55,314 |
https://mathoverflow.net/questions/94259 | 1 | Let $X$ be an algebraic variety (over any field). The definition of the Lie bracket of two vector fields on $X$ (i.e. sections of the tangent sheaf) which I know characterizes vector fields as derivations of the structure sheaf, and then one checks that the commutator of two derivations is a derivation. My first questi... | https://mathoverflow.net/users/3544 | Lie bracket of algebraic vector fields | In [an old blog post](http://qchu.wordpress.com/2011/02/26/the-quaternions-and-lie-algebras-i/) I give a definition of the Lie bracket of two derivations (of a ring, not necessarily commutative, in which $2$ is invertible) that doesn't require limits and doesn't require that you verify that the commutator of two deriva... | 1 | https://mathoverflow.net/users/290 | 94298 | 55,317 |
https://mathoverflow.net/questions/94296 | 12 | By accident I came across the following,
$$\lim\_{n\to\infty}\frac{1}{n}\sum\_{r=1}^n\frac{n\ (\mathrm{mod}\ r)}{r}=0.4227\ldots\approx 1-\gamma,$$
where the numerator is the remainder of $n$ divided by $r$. Is it known whether we have equality in the above expression or is it just a numerical coincidence? Has this... | https://mathoverflow.net/users/22978 | Limit of $\frac{1}{n}\sum_{r=1}^n\frac{n\ (\mathrm{mod}\ r)}{r}$ | This follows by elementary computation: we have
\begin{align}
\sum\_{r=1}^n\frac{n\bmod r}r&=\sum\_{r\le n}\frac nr-\sum\_{r\le n}\left\lfloor\frac nr\right\rfloor\\\\
&=nH\_n-|\{(r,s)\in\mathbb N^2:1\le rs\le n\}|\\\\
&=nH\_n-2\sum\_{r\le\sqrt n}\left\lfloor\frac nr\right\rfloor+\lfloor\sqrt n\rfloor^2\\\\
&=nH\_n-2nH... | 19 | https://mathoverflow.net/users/12705 | 94299 | 55,318 |
https://mathoverflow.net/questions/94303 | 8 | Let $G\subseteq\mathrm{Gl}\_n(\mathbb{C})$ be a subgroup of the general linear group and assume that $\rho:G\to\mathrm{Gl}(V)$ is a representation. Understand the complex vector space $V$ as an affine algebraic variety. Then, it appears to be well-known that the quotient $V/G$ has the structure of an algebraic variety ... | https://mathoverflow.net/users/9947 | Why can I divide an affine variety by the action of the general linear group? | If $G$ is reductive, try looking at [Fogarty, Kirwan, Mumford, Geometric Invariant Theory, p. 27](http://books.google.ie/books?id=dFlv3zn_2-gC&pg=PR3&lpg=PR3&dq=fogarty+kirwan&source=bl&ots=nR5HzpIUeY&sig=8Uxe2guLjSavpLeBdA1swNQctL8&hl=en&sa=X&ei=LJ-NT6SzL8fW0QX7k4XsDA&ved=0CCAQ6AEwAA#v=onepage&q=fogarty%2520kirwan&f=f... | 10 | https://mathoverflow.net/users/13268 | 94305 | 55,321 |
https://mathoverflow.net/questions/94307 | 3 | I'm putting the finishing touches on my masters thesis and need a reference for the following fact (which my advisor told me):
>
> Let $G$ be the $n\times n$ grid and identify the sides to make it a torus. A simple random walk on $G$ is expected to take $O(n^4)$ time before it hits every vertex.
>
>
>
I've bee... | https://mathoverflow.net/users/11540 | Reference Request: Cover time for simple random walk on $n \times n$ torus | Markov Chains and Mixing Times by Levin, Peres and Wilmer. Section 11.3.2.
<http://research.microsoft.com/en-us/um/people/peres/markovmixing.pdf>
The expected cover time is of order $n^2(\log n)^2$.
| 6 | https://mathoverflow.net/users/12105 | 94309 | 55,323 |
https://mathoverflow.net/questions/94294 | 6 | Let $X\_t$ be a family of algebraic varieties (my interest is Calabi-Yau varieties, but I don't think that's important) over $\mathbb{C}$, smooth for $t \neq 0$, on which a group $G$ acts fibre-wise. Suppose further that $X\_0$ admits at least one crepant resolution. Does there always exist an equivariant crepant resol... | https://mathoverflow.net/users/22975 | Do equivariant crepant resolutions always exist? | Consider $\mathbb Z/2$ acting on $\{xy-zw=t\}$ by $x\leftrightarrow y$. This swaps the two small resolutions of the central fibre (the 3-fold ordinary double point $xy=zw$ in $\mathbb C^4$). So there can't be an equivariant small resolution.
(A formal proof might go along these lines: $\mathbb Z/2$ does act on the bl... | 6 | https://mathoverflow.net/users/7653 | 94324 | 55,329 |
https://mathoverflow.net/questions/94272 | 4 | Consider the following attempt at a ``thought-free'' proof of Morley's
Theorem.
Let $(x\_1,y\_1)$, $(x\_2,y\_2)$ and $(x\_3,y\_3)$ denote three vertices of
a generic triangle.
Let $(a\_1,b\_1)$, $(a\_2,b\_2)$ and $(a\_3,b\_3)$ denote three more points
in the plane.
For all $\lbrace i,j,k\rbrace=\lbrace 1,2,3\rbra... | https://mathoverflow.net/users/10909 | Morley's Theorem and real algebraic geometry | This isn't a matter of real versus complex geometry -- the equations you give aren't enough to encode Morley's theorem over the reals. Let $z\_1 z\_2 z\_3$ be our original triangle; I will use indices that are cyclic modulo $3$. Let $\ell^0\_i$ be the trisector of the angle at $z\_i$ which is closer to $z\_{i-1}$ and l... | 3 | https://mathoverflow.net/users/297 | 94326 | 55,331 |
https://mathoverflow.net/questions/94316 | 5 | There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand operator spaces which are not c.b. In particular, I failed this task for $\mathcal{K}(H)$ and $\mathcal{B}(H)$, where $H$ is a... | https://mathoverflow.net/users/22469 | Projections which are not completely bounded | Symmetrisation (or anti-symmetrisation).
That is: let $T:K(H)\to K(H)$ be the transpose map and let $P=({\rm id}+T)/2$. Then $P:K(H)\to K(H)$ is a projection onto the subspace of symmetric [NOT self-adjoint] compact operators. Since $T$ is not completely bounded, $P$ is not completely bounded.
This argument also sh... | 7 | https://mathoverflow.net/users/763 | 94329 | 55,333 |
https://mathoverflow.net/questions/91877 | 0 | I'm trying to sort out two different uses of the term "compound distribution" and figure out the relationship.
The Wikipedia article on [compound distribution](http://en.wikipedia.org/wiki/Compound_distribution) -- which I wrote -- defines a compound distribution as an infinite mixture, i.e. if $p(x|a)$ is a distribu... | https://mathoverflow.net/users/22313 | compound distribution in Bayesian sense vs. compound distribution as random sum? | I can't say in general, but in the actuarial literature, the random sum of random variables is called an *aggregate distribution*, as in aggregate insured losses. Your definition of compound distribution is the one used in insurance.
| 0 | https://mathoverflow.net/users/nan | 94335 | 55,335 |
https://mathoverflow.net/questions/94308 | 5 | This is a refined/sheafified version of this [previos question](https://mathoverflow.net/questions/67121/sections-of-2-vector-bundles) of mine.
Let $(X,\mathcal{O}\_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}\_X$ is a sheaf of $\mathbb{K}$-algebras for some field $\... | https://mathoverflow.net/users/8320 | Pushforwards of stacks of algebras? | Yes, there are several different formalisms that achieve this - for example it's treated in Lurie's DAG XI or Toen-Vezzosi [Caractères de Chern, traces équivariantes et géométrie algébrique dérivée](http://arxiv.org/abs/0903.3292) in the derived context, and in many places (eg Gaitsgory [The notion of category over an ... | 8 | https://mathoverflow.net/users/582 | 94336 | 55,336 |
https://mathoverflow.net/questions/94334 | 3 | Let $Q=(Q\_{0},Q\_{1},h,t)$ be a finite quiver where $Q\_{0}$ are the vertices, $Q\_{1}$ the arrows and we have two maps $h: Q\_{1} \rightarrow Q\_{0}$ (head) and $t: Q\_{1} \rightarrow Q\_{0}$ (tail). Fix a field $K$ and associative to $Q$ two vector spaces $R=K^{Q\_{0}}$ and $A=K^{Q\_{1}}$ i.e vector spaces consistin... | https://mathoverflow.net/users/22669 | Quiver on tensor product | There is a basis of the tensor product $A^{\otimes n}$ given by $\gamma\_1\otimes \cdots \otimes \gamma\_n$ where $\gamma\_1,\dots, \gamma\_n$ are a list of $n$ elements of $Q\_1$ that are composable. That is, where concatenating $\gamma\_1\cdots \gamma\_n$ gives a path of length $n$. So, just as $A$ has a basis given ... | 3 | https://mathoverflow.net/users/66 | 94340 | 55,339 |
https://mathoverflow.net/questions/94331 | 9 | The notion of orbifold is quite well established by now. I would like to ask how one should call a point of an orbifold with non-trivial stabilizer? Should one call this *a singular point*? Of something else?
For some reason, I was not able to find any text that fixes this innocent bit of terminology concerning orbi... | https://mathoverflow.net/users/13441 | A terminological question concerning orbifolds. | Take a look at Thurston's notes (Chapter 13 of his "The geometry and topology of 3-manifolds", available at the MSRI's site: <http://library.msri.org/books/gt3m/>). Thurston calls these points *singular points*. (Same as Satake in his original paper "The Gauss-Bonnet Theorem for V-manifolds" from 1957 where he introduc... | 12 | https://mathoverflow.net/users/21684 | 94342 | 55,340 |
https://mathoverflow.net/questions/94310 | 11 | Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
| https://mathoverflow.net/users/3960 | Symmetric functions on three parameters being perfect squares | Yes. By straightforward search the smallest example is
$\lbrace x,y,z \rbrace = \lbrace 45,64,180 \rbrace$, with
$$
(t+45) (t+64) (t+180) = t^3 + 17^2 t^2 + 150^2 t + 720^2.
$$
Given any solution $(x,y,z)$ we may produce infinitely many others
(other than the trivial scaling $(c^2 x, c^2 y, c^2 z)$) by using
the theory... | 18 | https://mathoverflow.net/users/14830 | 94346 | 55,343 |
https://mathoverflow.net/questions/94322 | 9 | Let $(Q,g)$ be a (compact) Riemannian manifold with injectivity radius $\rho>0$. There is a natural metric $\tilde g$ on the tangent bundle $TQ$ which is known as the Sasaki metric and which makes $\pi:TQ\rightarrow Q$ a Riemannian submersion. Denote its injectivity radius by $\tilde\rho$. Obviously $\tilde\rho\leq\rho... | https://mathoverflow.net/users/21481 | Injectivity radius of the Sasaki metric | If the manifold is not flat then $\bar \rho=0$.
It is sufficient to show that given $\epsilon>0$ there are two tangent vectors $v,w\in T\_pQ$ such that $|v-w|=\epsilon$, but the minimizing geodesic does not lie in $T\_pQ$.
We assume that curvature at $p$ does not vanish.
Consider a loop $\gamma$ based at $p$ with l... | 6 | https://mathoverflow.net/users/10330 | 94348 | 55,344 |
https://mathoverflow.net/questions/94355 | 1 | Let $I$ be an ideal of a Noetherian ring $R$. $M$ is a finitely generated $R$-module. I question is:
Does there exist $n\_0$ such that for all $n \geq n\_0$, the short exact sequence
$$I^n/I^{n+1} \otimes\_R M \cong I^nM/I^{n+1}M$$
| https://mathoverflow.net/users/17901 | tensor of powers of an ideal | I suppose you intend to ask whether there is isomorphism as in the formula. The answer is negative, in general. For example, take $R = k[x]$, $I = (x)$, $M = R/I^2$.
[Edit] What you may have in mind is that the result holds when $M$ is flat over $R$.
| 3 | https://mathoverflow.net/users/4790 | 94367 | 55,355 |
https://mathoverflow.net/questions/94332 | 13 | In his cellebrated 1984 paper "Higher regulators and values of L-functions", Beilinson proved (among many other exciting things) that the value at the non-critical point $s=2$ of the Rankin L-function $L(f\otimes g,s)$ of the convolution of two eigenforms of weight $2$ (say of the same level $N$) is related to the imag... | https://mathoverflow.net/users/18643 | Beilinson's formula for the product of two modular curves | Dear Victor, I think an explicit formula has been worked out by S. Baba and R. Sreekantan in the following article :
MR2064735 (2005c:11073) Baba, Srinath ; Sreekantan, Ramesh. *An analogue of circular units for products of elliptic curves*. Proc. Edinb. Math. Soc. (2) 47 (2004), no. 1, 35--51.
They consider the ca... | 7 | https://mathoverflow.net/users/6506 | 94378 | 55,361 |
https://mathoverflow.net/questions/94312 | 8 | I have written a paper which involves a "cross product" in $\mathbb{R}^n$ and I would like to have a reference to point to.
Let ${\bf e\_1}, \dots, {\bf e\_n}$ be the standard basis for $\mathbb{R}^n$ and let
${\bf w\_1} = (w\_{11},\dots,w\_{1n}), \dots, {\bf w\_{n-1}}=(w\_{n-1\;1},\dots,w\_{n-1\;n}) \in \mathbb{R}... | https://mathoverflow.net/users/17263 | n-dimensional "cross product" reference request | Here's a bit of history on cross products that, if not directly useful, will hopefully provide some context. They were defined in "Beno Eckmann, *Stetige Losungen linearer Gleichungssysteme*, Comment. Math. Helv. 15(1943)" as follows: An **$r$-fold cross product** on a real vector space $V$ of dimension $n$ with inner ... | 18 | https://mathoverflow.net/users/21265 | 94385 | 55,364 |
https://mathoverflow.net/questions/94028 | 4 | We denote $\varphi:\mathbb R^2\rightarrow\mathbb R$ the addition of real numbers, and $\varphi\_\*:M\_1(\mathbb R^2)\rightarrow M\_1(\mathbb R)$ the induced push-forward map (where $M\_1(\Delta)$ stands for the set of probability measures on $\Delta$).
Note that given $\mu$, $\nu\in M\_1(\mathbb R)$, we have by defi... | https://mathoverflow.net/users/15517 | Classical convolution VS Free Convolution | The free analogue of the tensor product of measures is the free product.
Instead of the space $\mathbb{R}$, we consider the algebra $C\_0(\mathbb{R})$ of continuous functions on $\mathbb{R}$, tending to 0 at infinity. In this framework, measures are modelled by states: continuous (for supremum norm) linear functional... | 3 | https://mathoverflow.net/users/2055 | 94391 | 55,369 |
https://mathoverflow.net/questions/94387 | 1 | Let $\langle \mathbb{R}, 0, 1, +, \cdot, <\rangle$ be the standard model for $R$, and let $S$ be a countable model of $R$ (satisfying all true first-order statements in $R$). Is it true that the set $1,1+1,1+1+1,\ldots$ is bounded in $S$? My intuition says "no", but I am yet to find a counter example. I read something ... | https://mathoverflow.net/users/23006 | Non-Archimedean non-standard models for R | If $S$ is the set of real algebraic numbers then $1, 1+1, 1+1+1, \dots$ is unbounded in $S$. On the other hand, by compactness of first order logic (as Juris points out), there are models $S$ for which $1, 1+1, 1+1+1, \dots$ is bounded.
| 6 | https://mathoverflow.net/users/17836 | 94394 | 55,372 |
https://mathoverflow.net/questions/94365 | 4 | Let $M$ be a complete, non-compact, simply connected Riemannian manifold of dimension $n$ whose sectional curvatures are bounded above by $\kappa<0$. I want to prove that for any open subset $\Omega\subset M$ whose closure in $M$ is compact, the following inequality holds: $$\frac{Vol(\Omega)}{Vol(\partial \Omega)}\leq... | https://mathoverflow.net/users/18013 | Isoperimetric inequality in negative sectional curvature | I do not know if there is a way to get the isoperimetric inequality from the spectral gap, but both can be proven in almost the same way. The classical references for the linear isoperimetric inequality are S.-T. Yau, "Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold", *Ann. Sci. École ... | 7 | https://mathoverflow.net/users/4961 | 94398 | 55,375 |
https://mathoverflow.net/questions/94380 | 2 | Hello! Recently, doing my research on jet bundles, I was led to consider the following construction.
---
Let $V$ be a real vector space of dimension $n$. Consider the flag manifold $G(V,k,l)$ and the two Grassmann manifolds $G(V,k)$ and $G(V,l)$. It seems to me that there are two natural projections $\eta:G(V,k,l... | https://mathoverflow.net/users/22606 | Tautological and normal bundles over flag manifolds and jet bundles | The short answer is "yes." I'm not sure about the jet bundle stuff (not really my area), but everything else you've written is extremely well-known stuff in Lie theory. I've used them dozens of times in my own work. The quotient bundles you're calling "normal tautological bundle" i would just call "tautological bundle.... | 4 | https://mathoverflow.net/users/66 | 94399 | 55,376 |
https://mathoverflow.net/questions/94402 | 0 | I am currently working with irreducible $k[G]$-modules in MAGMA for finite fields $k$ and finite groups $G$. To construct these modules, I am using the commands **IrreducibleModules(G,k)** This results in MAGMA recognising the modules as being of type $ModGrp$. However, I wish to perform calculations such as finding th... | https://mathoverflow.net/users/19783 | Changing the Type of a Module in MAGMA | MAGMA is perfectly capable of calculating the projective cover of a member of `IrreducibleModules(G,k)`, via the `ProjectiveCover()` function. You can also just call `ProjectiveIndecomposables(G,k)`. Perhaps you are running an old version of MAGMA? Version 2.18-3 is certainly capable of this functionality.
(This woul... | 5 | https://mathoverflow.net/users/3935 | 94406 | 55,379 |
https://mathoverflow.net/questions/94323 | 10 | Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$.
While it is true that the theorem can be deduced from the Implicit Function Theorem (and I can trace those back to the ... | https://mathoverflow.net/users/22984 | Who was the first to formulate the inverse function theorem? | More probably than not it will be rather difficult to have a final word on such kind of question, since it depends on how much you insist on having all words completely respected. The question requires tracking the moment in history of math when it became more common to talk about open sets rather than neighbourhoods, ... | 14 | https://mathoverflow.net/users/6032 | 94412 | 55,383 |
https://mathoverflow.net/questions/94254 | 5 | The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order creatures. Of course, one cannot expect that all the properties (propositions of the 'language of Banach spaces') can be persev... | https://mathoverflow.net/users/22469 | Do (Banach) ultrapowers carry some sort of 'elementary equivalence'? | Say that a Banach space $X$ has property $K$ (for Kummers) provided every subspace of $X$ that is isomorphic to $X$ is complemented. The classical separable, infinite dimensional spaces that have property $K$ include $\ell\_2$, $c\_0$, and $\ell\_2 \oplus c\_0$; the first obviously, the second because $c\_0$ is separab... | 6 | https://mathoverflow.net/users/2554 | 94417 | 55,386 |
https://mathoverflow.net/questions/94420 | 3 | I am trying to find a book or a paper, which explains, how and why the **Sherali-Adams** relaxation method works. The original paper (1990) is difficult for me to understand. I need a more basic introduction, of which I suspect the existence. Thank you for the help.
| https://mathoverflow.net/users/23015 | Sherali-Adams relaxation | It isn't clear from your posting whether you're trying to understand:
Why the inequalities generated by the Sherali-Adams procedure are valid?
or
Why the procedure is complete in the sense that after enough iterations you arrive at the convex hull of the integer solutions of the original integer linear programmi... | 2 | https://mathoverflow.net/users/9022 | 94423 | 55,388 |
https://mathoverflow.net/questions/94431 | 1 | Hello,
I have a technical question. My terminology:
I - set of standard inclusions $\partial I^n \to I^n$.
I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$.
CW (CW complexes) - the usual definition (like I-cell but with "cells attached by order of dimension).
I-cof - retr... | https://mathoverflow.net/users/2095 | Technical question about cell complexes | They are exactly such. One point is that one can use classical cell complexes, stop at $\omega$ with no transfinite nonsense, in setting up the Quillen model structure: it is a compactly generated (as well as a cofibrantly generated) model category. The distinction and full details are in May and Ponto, "More concise a... | 4 | https://mathoverflow.net/users/14447 | 94443 | 55,396 |
https://mathoverflow.net/questions/94427 | 3 | Hi
I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating them could help me...
Call $\mathscr{P}$ the convex set of symmetric SDP matrices $S$ of order $N\geq 1$ such that $|... | https://mathoverflow.net/users/16934 | Mapping a subset of semi-definite matrices through arcsinus | Consider the SDP matrices
$$A = \pmatrix{1&0&-1&0\cr 0&1&0&-1\cr -1&0&1&0\cr0&-1&0&1\cr},\
B=\pmatrix{ 1&\sqrt{3}/2 & 1/2&0\cr \sqrt{3}/2&1& \sqrt{3}/2& 1/2\cr 1/2& \sqrt{3}/2&1& \sqrt{3}/2
\cr 0& 1/2& \sqrt{3}/2&1\cr}$$
where with $a=1/2$
$$ C = \Phi^{-1}(\Phi(A)/2 + \Phi(B)/2) = \pmatrix{ 1&1/2&-1/2&0\cr1/2&1&1/2&-... | 3 | https://mathoverflow.net/users/13650 | 94444 | 55,397 |
https://mathoverflow.net/questions/94426 | 6 | Given two schemes $X$ and $Y$ one can consider additive functors (eventually with some nice additional property) between the categories of $\mathcal{O}\_X$-modules and of $\mathcal{O}\_Y$ modules. Among these one has those "of a geometric nature", in particular Furier-Mukai-type functors, i.e., those of the form $\Phi\... | https://mathoverflow.net/users/8320 | Geometric natural transformations between Fourier-Mukai transforms | First a quick comment: if you work not with triangulated categories but with differential graded or $A\_\infty$ categories, then ALL (continuous) functors are given by "Fourier-Mukai" or integral transforms (a theorem of Toen - an analogue of the Schwartz kernel theorem in analysis). Moreover the (dg) category of funct... | 8 | https://mathoverflow.net/users/582 | 94450 | 55,401 |
https://mathoverflow.net/questions/94383 | 15 | When $G$ acts trivially on $M$, the first homology group is just the abelianisation of $G$ tensored with $M$, i.e. $H\_1(G;M)=(G/[G,G])\otimes\_\mathbb Z M$.
Is there any similar statement when $G$ acts not trivially on $M$? Where does the abelianisation come from?
| https://mathoverflow.net/users/11084 | First group homology with general coefficients | In the trivial case the abelianization comes from the short exact sequence
$0\to J\to \mathbb{Z}G\to \mathbb{Z}\to 0$
where $J$ is the augmentation ideal. The homology $H\_\*(G,-)$ are just derived functors and give a long exact sequence in homology, which since $H\_1(\mathbb{Z}G,\mathbb{Z})$ is always trivial, giv... | 12 | https://mathoverflow.net/users/1437 | 94457 | 55,404 |
https://mathoverflow.net/questions/92983 | 8 | **Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a *topological* triangulation with complexity $O(n)$?**
Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ triangular facets, possibly with positive genus. A *topological* triangulation of $P$ is a simplicial complex whose underl... | https://mathoverflow.net/users/6710 | Efficient topological triangulations of non-convex polyhedra | I've been thinking about the main question in the original post on and off for a few days. All of my efforts have been in the direction of finding enough examples to prove a super-linear lower bound, following Misha's suggestion to use hyperbolic volume. This hasn't worked yet - the problem appears to be tricky! In any... | 3 | https://mathoverflow.net/users/1650 | 94458 | 55,405 |
https://mathoverflow.net/questions/94424 | 9 | For a short exact sequence of groups $1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$ there is an associated fibration $K(A,1)\rightarrow K(B,1)\rightarrow K(C,1)$, which can be constructed by realizing the homomorphism $B\rightarrow C$ by a map $K(B,1)\rightarrow K(C,1)$ and the convert it into a fibration. The... | https://mathoverflow.net/users/15770 | How to Compute Transgressions in a Serre Spectral Sequence? | I can give a description in case of cohomology: Let $$1 \to H \to G \to G/H \to 1$$ be an extension of groups. Then we obtain an extension with abelian kernel
$$1 \to H\_{ab} \to G/H' \to G/H \to 1$$ Let $\varepsilon \in H^2(G/H;H\_{ab})$ be its extension class. If $M$ is a trivial $G$-module, then the differential (w... | 12 | https://mathoverflow.net/users/10194 | 94463 | 55,407 |
https://mathoverflow.net/questions/94476 | 4 | I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ansatz I know that one solution is $(x,y)=(-{\rm Erf}^{-1}(1/2),{\rm Erf}^{-1}(1/2))$, but I can not manage to prove that... | https://mathoverflow.net/users/23034 | Reducing system of equations involving Erf, Error Function | This can be solved as a problem in inequalities.
The desired implication
$$
{\rm Erf}\Bigl(\frac12(x+y)\Bigr) = \frac12\bigl({\rm Erf}(x) + {\rm Erf}(y)\bigr)
\Longleftrightarrow x + y = 0
$$
is not quite correct, because the hypothesis holds also for $x=y$;
but these are the only possibilities, which is enough to yiel... | 6 | https://mathoverflow.net/users/14830 | 94483 | 55,415 |
https://mathoverflow.net/questions/94479 | 22 | Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is it true, that the manifold B can be covered by a finite number of sets $U\_1,\dots,U\_N$ s.t. the vector bundle, restric... | https://mathoverflow.net/users/8134 | Does every vector bundle allow a finite trivialization cover? | The answer (to both questions (a) and (b)) is **YES** (assuming $B$ is a **smooth manifold**). A proof can be found on Walschap's book "[Metric Structures in Differential geometry](http://rads.stackoverflow.com/amzn/click/038720430X)", p. 77, Lemma 7.1.
For the OP's convenience, here's a sketch of the proof. Choose a... | 16 | https://mathoverflow.net/users/15743 | 94488 | 55,416 |
https://mathoverflow.net/questions/94485 | 6 | If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles are obtained depending on how the lattice 'twists' the fibers. The connection $D = d + (1/2)(pdq-qdp)$ descends to a conn... | https://mathoverflow.net/users/17913 | Connections on line bundles over the torus | The Kaehler polarisation is a choice of *complex structure*, $I$, on your torus $M\simeq S^1\times S^1$, which is *compatible* with the symplectic form. In other words, your torus becomes a *complex manifold* $X=(M,I)$, and $I$ satisfies $I^t\omega I=\omega$, $g=\omega I>0$. In particular, this tells you that $\omega$ ... | 13 | https://mathoverflow.net/users/6278 | 94496 | 55,419 |
https://mathoverflow.net/questions/94486 | 10 | The usual Fubini's theorem(see the [Wikipedia article](http://en.wikipedia.org/wiki/Fubini%27s_theorem) for example) assumes completeness or $\sigma$-finiteness on measures. However, I think I came up with a proof of the Fubini's theorem without those assumptions. Am I mistaken?
I restate the theorem to avoid confusi... | https://mathoverflow.net/users/37646 | Fubini's theorem without completeness or $\sigma$-finiteness conditions | You do not need $\sigma$-finiteness of the measure in Fubini theorem, although it is an hypothesis that can be assumed with no loss of generality, in that the support of an integrable function is, of course, $\sigma$-finite.
On the opposite, Tonelli theorem deals with non-negative measurable functions, whose support... | 14 | https://mathoverflow.net/users/6101 | 94497 | 55,420 |
https://mathoverflow.net/questions/94448 | 1 | Is it possible to find an approximate expression of $\frac{\sum\_{i=1}^{n} k\_i x\_i}{\sum\_{i=1}^{n} k\_i}$ using $\langle k \rangle$, $\langle k^2 \rangle$, $\langle x \rangle$, and $n$? Alternatively if it is possible to express the boundary (biggest and smallest possible value) of $\frac{\sum\_{i=1}^{n} k\_i x\_i}{... | https://mathoverflow.net/users/21328 | Find an approximate expression of a sum of a product using the average of each item | We must assume that $\langle k \rangle \ne 0.$ When $n=1$ we know everything.
Certainly $\langle k^2\rangle \ge\langle k\rangle ^2.$ If $\langle k^2\rangle =\langle k\rangle ^2$ then the $k\_i$ are all equal (to $\langle k \rangle$ ) and $\frac{\sum\_{i=1}^{n} k\_i x\_i}{\sum\_{i=1}^{n} k\_i}=\langle x \rangle$ exac... | 2 | https://mathoverflow.net/users/8008 | 94498 | 55,421 |
https://mathoverflow.net/questions/94221 | 4 | Let $A$ be an abelian variety over a $p$-adic field $K\_v$, i.e., $K\_v$ is a finite field extension of $\mathbb Q\_p$, for $p$ a prime number. Denote by $k\_v$ the residue field of $K\_v$ and let $\mathcal{A}\_v^{0}(k\_v)$ be the smooth part of the $k\_v$-rational points of the modulo $v$ reduced variety $A$, i.e., th... | https://mathoverflow.net/users/12668 | Why is the kernel of reduction a pro-p group? | The kernel of reduction $A\_1(K\_v)$ is isomorphic to the group $\hat A(\mathfrak{m}\_v)$ associated to the formal group $\hat A$ of $A$ defined over the valuation ring $\mathcal O\_v$ of $K\_v$ with maximal ideal $\mathfrak m\_v$. By standard properties of formal groups, the multiplication-by-$m$-endomorphism on $\hat... | 3 | https://mathoverflow.net/users/12668 | 94522 | 55,431 |
https://mathoverflow.net/questions/94462 | 3 | Let $X$ be a locally compact metric ANR (or, if preferred, a locally compact simplicial complex). If needed, assume that $X$ has finitely many [ends](http://en.wikipedia.org/wiki/End_%2528topology%2529) or is of finite dimension. My question is:
>
> What are the *necessary* conditions for the Freudenthal compactifi... | https://mathoverflow.net/users/2578 | When is the Freudenthal compactification an ANR? | Regarding simple examples: take $X$ to be the infinite mapping telescope of an inverse sequence of connected compact polyhedra $P\_i$ (so that $X$ has one end) such that for some $j$, the inverse sequence of the groups $G\_i=H\_j(P\_i)$ does not satisfy the Mittag-Leffler condition or its inverse limit does not inject ... | 2 | https://mathoverflow.net/users/10819 | 94524 | 55,432 |
https://mathoverflow.net/questions/94517 | 14 | This question has been bugging me for some time.
Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is standard fact that this is an unbounded operator which has a countable infinity of eigenvalues, all of which are negat... | https://mathoverflow.net/users/20729 | Is zero a hydrogen eigenvalue? | First of all, the Hamiltonian in question is defined on $L^2(\mathbb R^3)$, not on $L^2(\mathbb R)$. This is important because in the one-dimensional case the potential would have a non-integrable singularity which complicates things seriously. On $L^2(\mathbb R^3)$, the operator, defined as a closure from $C\_0^\infty... | 23 | https://mathoverflow.net/users/12205 | 94526 | 55,433 |
https://mathoverflow.net/questions/94459 | 3 | Hi everyone:
I have a matrix M that is positive semi-definite, i.e., all eigenvalues are non-negative. I wonder to make it invertible, what is the best strategy ?
1) add an small identity matrix: $\delta$ \* I, then compute the inverse matrix.
or
2) simple compute the pseudo-inverse $M^+$ ?
Which one is b... | https://mathoverflow.net/users/23031 | Ways to convert a Positive Semi-Definite (PSD) matrix -> Positive Definite matrix | The choice that you make can result in a huge difference in the solution. Neither method is particularly good and both can be quite unstable.
For example, suppose that
$M=[1\;0\;0\; 0;\; 0 \; 1 \; 0 \; 0; \; 0 \; 0 \; 0 \; 0;\; 0 \; 0 \; 0 \; 0]$
and
$N=[1;\; 1; \; 1; \; 1]$.
What do you want $L$ to be in ... | 4 | https://mathoverflow.net/users/9022 | 94531 | 55,436 |
https://mathoverflow.net/questions/94512 | 25 | I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced [zeta function regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization) in a compelling way. The goal is to define the determinant of a positive self-adjoint operator $A$ with "pure point spectrum" $0>\lambda\... | https://mathoverflow.net/users/2051 | Understanding zeta function regularization | Not a complete answer. First, here is an alternate derivation of the result in the finite-dimensional case which might be more enlightening. If $A$ is positive self-adjoint, we can write $A = \exp(L)$ for some self-adjoint $L$. This lets us define
$$A^s = \exp(sL)$$
for all real $s$. The trace
$$\text{tr}(A^s) = \sum... | 14 | https://mathoverflow.net/users/290 | 94540 | 55,438 |
https://mathoverflow.net/questions/94547 | 14 | Among hundreds of equivalent definitions of amenability (for discrete, countable, groups), I would like to discuss two which are most common:
A1. A group $G$ is amenable if it admits a Folner sequence.
A2. A group $G$ is amenable if it admits an invariant mean.
(See e.g. <http://en.wikipedia.org/wiki/Amenable_g... | https://mathoverflow.net/users/21684 | Amenability and ultrafilters | Of course, $ZF$ is enough to prove that $\mathbb{Z}$ has a Folner sequence. But, as you point out, $ZF$ is not enough to prove that $\mathbb{Z}$ has an invariant mean. Thus $ZF$ does not prove the equivalence of A1 and A2.
On the other hand, the Hahn-Banach Theorem is enough to prove the equivalence of A1 and A2 for ... | 14 | https://mathoverflow.net/users/4706 | 94548 | 55,442 |
https://mathoverflow.net/questions/94480 | 8 | Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic $G$.
Hilbert theorem 94 says that *if $L/K$ is everywhere unramified* (hence contained in the Hilbert class field $H... | https://mathoverflow.net/users/18238 | Generalization of Hilbert 94 and capitulation | The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in
<https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-121/issue-none/A-generalization-of-Hilberts-th... | 4 | https://mathoverflow.net/users/18238 | 94556 | 55,445 |
https://mathoverflow.net/questions/94559 | 3 | I'm finishing up a masters thesis in computer science and want to say a bit in the introduction about self-avoiding walks. My thesis looks at a random process which arose in computer science and my advisor thought it would behave like a self-avoiding walk. It turned out not to, but I still want to mention them in case ... | https://mathoverflow.net/users/11540 | Transience of self avoiding random walks on $\mathbb{Z}^d$ | There is a problem in your definition of transience: the standard definition of the self-avoiding walk is the uniform (counting) measure on the set of walks of a given length, say $n$, and one is interested in asymptotic properties of the paths, like typically the end-to-end distance, as $n\to\infty$. This is the same ... | 5 | https://mathoverflow.net/users/9430 | 94562 | 55,448 |
https://mathoverflow.net/questions/57826 | 3 | Does exists a short, simple proof of the inequality
$ \|u\|\_{L^{2}(\Omega)} \leqslant C \|Du\| \_{L^{2}(\Omega)} + \|u\| \_{L^2{(\partial{\Omega})}} $ for $u\in H^{1}=W^{1,2}(\Omega) $
(Sobolev space with one weak derivative integrable in square),
where $\Omega = \{ x\in\mathbb{R}^{n}:\ 1<|x|<2 \}$?
(we do not... | https://mathoverflow.net/users/13099 | Proof of Friedrichs inequality in a domain with simple geometry | Let $\Omega$ be an open subset of $\mathbb R^n$ with a $C^1$ boundary and $u\in H^1(\Omega)$. We compute with $D\_{x\_1}=-i\partial\_{x\_1}$,
$$
2\Re\langle D\_{x\_1}u, i x\_1u\rangle=-2\Re\int\_\Omega x\_1(\partial\_{x\_1}u)\ \overline{u} dx
=-\int\_\Omega x\_1\partial\_1(\vert u\vert^2) dx=
-\int\_\Omega \partial\_1(... | 2 | https://mathoverflow.net/users/21907 | 94565 | 55,451 |
https://mathoverflow.net/questions/94525 | 6 | Let $s(a)$ be the sum of decimal digits of a number $a$. Is it known that for any $a\ne b$ exist $n$ such that $s(na)\ne s(nb)$?
| https://mathoverflow.net/users/5712 | Equal digit sums | Yes, but if we replace condition $a\ne b$ to $a/b\ne 10^k$ for all integers $k$.
Lemma: $s(9n)=9s(n)$ iff decimal digits of $n$ are only 0's and 1's. Else $s(9n)<9s(n)$.
Call two numbers equivalent, if their ratio is a power of 10 (with integer exponent).
Consider numbers 1,11,111,... Two of them are congruent mo... | 18 | https://mathoverflow.net/users/4312 | 94567 | 55,452 |
https://mathoverflow.net/questions/94555 | 4 | Let $I^n$ be the set of vertices of the unit cube of dimension $n$ with the standard ($l\_1$) distance. Then any set of vertices in $I^n$ consisting of vertices at pairwise distance $n$ is the pair of ends of a large diagonal. There are $2^{n-1}$ of those. Thus the following property holds
* If we color $I^n$ in few... | https://mathoverflow.net/users/nan | Coloring a unit cube | I work with $I\_n$ instead $I\_{n+1}$. Consider pairs of opposite vertices. There are $2^n$ such pairs. We want to find $K=O(2^n/n)$ such pairs (call them nice) so that for any other pair $(u,u')$ there is a nice pair $(v,v')$ with distance $d(u,v)=1$. Then we may color all vertices in at most $2K$ colors so that each ... | 7 | https://mathoverflow.net/users/4312 | 94574 | 55,454 |
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