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https://mathoverflow.net/questions/63381 | 10 | I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R \to S$. Because I am interested in computing some derived tensor products $S \otimes\_R$, I have worked out a "Koszul" reso... | https://mathoverflow.net/users/78 | Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it? |
>
> Question 1: Does there necessarily
> exist a resolution of S that computes
> the derived $S\otimes\_R$ and that is
> Gerstenhaber in a compatible way?
>
>
>
Yes. As pointed out in the comments, the category of dg Gerstenhaber algebra admits a model structure in which the weak equivalences are the quasi-is... | 9 | https://mathoverflow.net/users/7031 | 105937 | 61,239 |
https://mathoverflow.net/questions/105934 | 3 | Imagine that we're trying to define the expression
$$\int\_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a measurable subset of $X$. That most typical approach to making this integral rigorous is the method of Lebesgue, whereb... | https://mathoverflow.net/users/26080 | Defining the integral of a function using the product measure | Linearity of this integral is very mysterious. Moreover, the definition of the product measure *using* integration, i.e. $\mu \otimes \lambda (M) =\int \int I\_M(x,y) d\mu(x) d\lambda(y)$,
is very easy (up to a technical problem concerning measurability) and can be understood without
knowing Caratheodory's construction... | 5 | https://mathoverflow.net/users/21051 | 105940 | 61,240 |
https://mathoverflow.net/questions/105931 | 9 | Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric boundaries of hyperbolic 4-manifolds" by Long and Reid in Geom. Topol. 4 (2000), 171–178 that there are 3-manifolds are ... | https://mathoverflow.net/users/1190 | Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold? | Yes. Every (compact) flat $n$-manifold is diffeomorphic to a cusp cross section of a hyperbolic $(n+1)$-manifold. This is a theorem of McReynolds, Controlling manifold covers of orbifolds, Math. Res. Lett. 16 (2009), 651-662.
In your case, $n=3$, this is a theorem of Nimershiem. B. E. Nimershiem, All flat three-manifo... | 11 | https://mathoverflow.net/users/1335 | 105948 | 61,243 |
https://mathoverflow.net/questions/105814 | 9 | [Metastuff: I asked this question in a slightly different way on mathSE last week, and it didn't go anywhere, which is why I am asking here. I added the DST tag because it's basically a problem about Borel equivalence relations stripped of all the Borelness constraints. I do need help, so helpful redirection is appreci... | https://mathoverflow.net/users/20475 | Constructing an injective reduction of equivalence relations | This is a very nice problem, which I like very much.
The answer is that yes, indeed, there is such an injective reduction of $E$ to $F$. And one can give a recursive construction. (This answer now incorporates several simplifications to my original construction.)
Specifically, let $\delta\_\kappa^E$ be the number o... | 4 | https://mathoverflow.net/users/1946 | 105949 | 61,244 |
https://mathoverflow.net/questions/105932 | 12 | We all know plenty of examples of multilinear forms in finitely many variables (e.g. determinants). However, I am missing an interesting example of a form in infinitely many variables, linear in each. The important point here is the word "interesting" ; the example should occur naturally in some nice situation in algeb... | https://mathoverflow.net/users/17988 | Example of a form linear in infinitely many variables ? | Consider the Hilbert space $H=L^2(S^1, \mathbf{C}^n)$ and the subspaces $H\_+, H\_-$ of positive and negative Fourier modes. One can construct the Hilbert space of infinite wedge products $V=\bigwedge(H\_+)\hat{\otimes}\bigwedge(H\_-)^\*$.
Just as $H$ carries an action of $LSU(n)=Maps(S^1, SU(n))$ by pointwise multip... | 5 | https://mathoverflow.net/users/18512 | 105954 | 61,247 |
https://mathoverflow.net/questions/105938 | 4 | Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is:
$$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)})}.$$
A space $\langle X,\mathscr{O}\rangle$ is *normal* iff for any disjoint closed sets $A$ and $B$ there are $O\_... | https://mathoverflow.net/users/22019 | Is normality of a Hausdorff space consequence of some property of open domains? | No.
For example let $X=\{(x,y): x,y \in \mathbb{Q}, y \geq 0 \}$ with the [irrational slope topology](http://brubeck.jdabbs.com/spaces/irrational-slope-topology/). The clousure of any two non-empty open subsets of $X$ have non-empty intersection. This implies that $X$ satisfies ($\dagger$) trivially but it also impl... | 6 | https://mathoverflow.net/users/17836 | 105956 | 61,248 |
https://mathoverflow.net/questions/105955 | 3 | The first time I tried to understand [Euler's formula](http://en.wikipedia.org/wiki/Euler%2527s_formula) was about 2 years ago. I didn't need it, I just randomly ran across it, when trying to understand a Fourier transformation. The problem was, that I didn't know what exponentiation by complex number means.
Now I h... | https://mathoverflow.net/users/26083 | Is Euler's formula a theorem or a definition? | The question seems to presuppose that there is only one way to develop this part of complex analysis. There are several, among which some may take Euler's formula as a definition of complex exponentiation. As far as I know, most define complex exponentiation in a different way --- by the Taylor series of the exponentia... | 6 | https://mathoverflow.net/users/6794 | 105957 | 61,249 |
https://mathoverflow.net/questions/105933 | 2 | I'm trying to understand a part of the PhD thesis of Kenji Lefèvre-Hasegawa (e.g. available [here](http://arxiv.org/abs/math/0310337)). My question is about the proof of Lemma 1.3.2.3b stating:
>
> Remarquons que nous avons un isomorphisme de complexes $\bigoplus\_{i\geq 1}Gr\_i(\Omega BA)\to \Omega \overline{T^c}V... | https://mathoverflow.net/users/15887 | Associated graded of a filtration of a tensor product | OK let me try a naive answer, and then maybe you will elaborate a bit on what it is that worries you?
The key idea is very simple - the bar differential has a part coming from the differential on $A$, and the remaining part, encoding the product of $A$. The first part preserves the number of tensor factors, the rema... | 3 | https://mathoverflow.net/users/1306 | 105960 | 61,251 |
https://mathoverflow.net/questions/105952 | 10 | Dear community,
assume that $f\colon X\to Y$ is a homology equivalence, i.e. induces isomorphisms on all integer homology groups. Is there a map $g\colon Y\to X$ which is also a homology equivalence?
Actually, I am only interested in the case if $f\colon X\to X^+$ is obtained by Quillen´s plus construction.
Thank... | https://mathoverflow.net/users/26086 | Reverse map of a homology equivalence. | As I was writing this answer, Oscar beat me to the punch. I will keep it posted anyway.
Let $X^3$ be the Poincare homology sphere. Let $\tilde X \to X$ be the universal cover (note: $\tilde X$ is $S^3$). As in my comment, if there were a map $S^3 \to X$ inducing a homology equivalence, then that map
must necessarily ... | 13 | https://mathoverflow.net/users/8032 | 105962 | 61,252 |
https://mathoverflow.net/questions/105951 | 17 | **Question** Does anyone know of a good reference which I can cite for the *finite* dimensional version of Mountain Pass Lemma?
**Motivation** I am writing a paper and found myself using the following result:
>
> Let $f$ be a proper smooth real-valued function on $\mathbf{R}^3$ such that $f(0) = 0$, $f|\_{B\_1(0)... | https://mathoverflow.net/users/3948 | Finite dimensional "Mountain Pass Lemma" | My book *An Invitation to Morse Theory*, 2nd Edition, Springer Verlag 2011 describes the finite dimensional Mountain Pass Lemma in Example 2.53. There I work on a compact manifold, but the compactness of the manifold can be substituted by a properness assumption on the function. In the same section I explain a more gen... | 17 | https://mathoverflow.net/users/20302 | 105963 | 61,253 |
https://mathoverflow.net/questions/105945 | 7 | Let $E/\mathbb C$ be an elliptic curve. It is known that if $E$ has CM, then $j(E)$ is an algebraic integer. My first question is: what about the converse? Is there a way to identify the subset of algebraic integers whose elements correspond to [isomorphism classes of] elliptic curves with CM?
The second question is: f... | https://mathoverflow.net/users/26082 | elliptic curves with and without CM | 1)If an elliptic curve has integral $j$-invariant it absolutely DOES NOT NEED to have CM. The class of curves with integral $j$-invariant (let's call that the class of IM Elliptic curves for Integral Modulus) is MUCH MUCH larger than the class of CM Elliptic curves. In fact, one can use Heilbronn's Theorem that class n... | 5 | https://mathoverflow.net/users/3384 | 105981 | 61,261 |
https://mathoverflow.net/questions/105968 | 2 | If $U$ is a smooth algebraic variety, then one can give a simple description of the lowest weight part of its cohomology: if $X$ is a smooth compactification and $j \colon U \to X$ the inclusion, then
$$ W\_k H^k(U,\mathbf Q) = \mathrm{Im} \left( j^\ast \colon H^k(X,\mathbf Q) \to H^k(U,\mathbf Q)\right). $$
Is the... | https://mathoverflow.net/users/1310 | Second lowest weight piece of the cohomology of an algebraic variety | Here is a geometric description of $W\_{k+1}H^k(U)/W\_k H^k(U)$. Set $d=\dim U$. Suppose $X$ is a compactification of $U$ such that the complement $X\setminus U$ is the union of normal crossing divisors, or, more generally, union of smooth varieties $D\_i$ all of whose intersections are smooth. Take submanifolds $Y\_i\... | 5 | https://mathoverflow.net/users/2349 | 105982 | 61,262 |
https://mathoverflow.net/questions/105971 | 40 | (And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of Iwaniec and Kowalski, says the following. Let $r(m)$ be the number of representations of $m$ as two squares, and suppose tha... | https://mathoverflow.net/users/1050 | How should an analytic number theorist look at Bessel functions? | Radial Fourier transforms provide a good, consistent perspective on most of the theory. The Fourier transform $\widehat{f}(t)$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}$ is given by the integral of $f(x) e^{2\pi i \langle x,t \rangle} \, dx$ over $x \in \mathbb{R}^n$. If $f$ is a radial function (i.e., $f(x)$... | 32 | https://mathoverflow.net/users/4720 | 105990 | 61,268 |
https://mathoverflow.net/questions/105987 | 0 | If I start with a commutative rig, and apply the [Grothendieck Group](http://en.wikipedia.org/wiki/Grothendieck_group) construction to it, twice, once to the additive structure and once to the multiplicative structure, is the result well-known? Does the order of application of the construction matter?
In particular, ... | https://mathoverflow.net/users/3993 | commutative rigs and the Grothendieck Group | The Grothendieck group of a semiring is a ring. So you are just asking what will happen if you take the multiplicative Grothendieck group of a ring.
Well, it depends on whether you include $0$ in the multiplicative structure! If you include $0$, then $a/b=0a/0b=0/0$ so you get the trivial group. Similarly, inverting ... | 4 | https://mathoverflow.net/users/18060 | 105994 | 61,272 |
https://mathoverflow.net/questions/105756 | 22 | Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
>
> Is $SL(2,\mathbb C)/SL(2,\mathbb Z)$ a quasi-projective variety?
>
>
>
The natural generalization of this question s... | https://mathoverflow.net/users/35353 | Is SL(2,C)/SL(2,Z) a quasi-projective variety? | No, the quotient is not quasi-projective. In the paper [Invariant meromorphic functions on complex semisimple Lie groups](http://digreg.mathguide.de/cgi-bin/ssgfi/anzeige.pl?db=reg&ti=I%2a&nr=092504&ew=SSGFI) by D. N. Ahiezer you can find the following result.
>
> **Theorem.** Let $G$ be a connected semisimple line... | 26 | https://mathoverflow.net/users/605 | 105997 | 61,273 |
https://mathoverflow.net/questions/106000 | 1 | Hi, I am really struggling with this question.
The question is :
Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta u=f$ , where $ u $ is given by $u(x)= \frac{1}{4\pi}\int\_{R^3}\frac{1}{|x-y|}f(y)\,dy$.
1. Show that $L^2$ norm of... | https://mathoverflow.net/users/25634 | The integrability of fundamental solution of laplace equation follows from integrability of f ? | Your question text suggests that yo know how to do (1). (Interchange the order of integration).
Now (2) is easy: outside the support of $f$ we have $\Delta u=0$, that is $u$ is harmonic outside
the support of $u$. Harmonic functions are surely $C^\infty$.
For (3), expand the kernel $1/|x-y|$ in a series of spherical ha... | 1 | https://mathoverflow.net/users/25510 | 106004 | 61,276 |
https://mathoverflow.net/questions/106013 | 3 | Motivation
There are several rules about what makes a rack legal for a game of eight-ball: the top ball has to be a solid, the eight-ball is in the middle, the two bottom vertices have to be one solid and one stripe, etc. But a rule I learned when I first learned to play pool was that there should be no three balls w... | https://mathoverflow.net/users/25028 | Hypergraph coloring problem motivated by legal billards racks | I calculated the first few terms recursively.
$1,
2,
6,
24,
130,
960,
9702,
134512,
2562516,
67152240,
2422643366,
120395521752, \\\
8245524190254,
778511553019200,
101361018574446630$
They weren't in the OEIS, but taking off the initial $1$ and dividing the other terms by $2$ produced [A007017](http://oeis.org/A0... | 3 | https://mathoverflow.net/users/2954 | 106016 | 61,284 |
https://mathoverflow.net/questions/106017 | 8 | Is it consistent intuitionistically (in the sense of topos theory) for there to be a surjection from the natural numbers to the (Dedekind, let us say) real numbers? [I've managed to convince myself this happens in the effective topos, but not so convincingly as I'd like; I suspect others will be able to answer more con... | https://mathoverflow.net/users/3902 | Intuitionistic consistency of surjection from naturals to reals | **Theorem** (Intuitionistic logic with Dependent Choice):
For every sequence $a : \mathbb{N} \to \mathbb{R}$ there is $x \in \mathbb{R}$ such that $|x - a\_n| > 0$ for all $n \in \mathbb{N}$.
*Proof.* Define a nested sequence of closed intervals $I\_0 \supseteq I\_1 \supseteq I\_2 \supseteq \cdots$ such that the len... | 11 | https://mathoverflow.net/users/1176 | 106024 | 61,287 |
https://mathoverflow.net/questions/105961 | 0 | I heard a conjecture "3-dim positively curved Alexandrov space is of the form S^3/J.(I cannot make sure my statement is accurate).
What is the classification of n-dim positively curved Alexandrov space? And if a n-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification?
Maybe... | https://mathoverflow.net/users/24637 | positively curved Alexandrov space | Again, I guess you want a topological classification.
Such classification would include classification of all smooth positively curved manifolds
which is too much to ask.
For the (quasi)geodesic subset, one should be able to say something if it has big dimension, say codimension 1;
otherwise there is no chance.
| 5 | https://mathoverflow.net/users/1441 | 106031 | 61,290 |
https://mathoverflow.net/questions/106014 | 2 | Let $G$ be a real semisimple Lie group (say $SL(2,\mathbb{R})$) and $H$ be its Cartan subgroup (say torus or diagonal subgroup of $SL(2,\mathbb{R})$).
My questions is: it is always true that we have a natural symplectic structure on the quotient space $G/H$?
If it is not true, could we consider this weaker version:... | https://mathoverflow.net/users/24965 | Does $G/H$ (quotient of a real semisimple Lie group by a Cartan subgroup) have a natural symplectic structure? | To add to Robert Bryant's answer, the coadjoint orbits of a Lie group always have a natural symplectic structure, the (Lie-)Kirillov-Kostant(-Souriau) form. These are the symplectic leaves of the natrual Lie-Poisson structure on $\mathfrak{g}^\ast$. For $x \in \mathfrak{g}^\ast$, we can identify the coadjoint orbit $\m... | 11 | https://mathoverflow.net/users/14628 | 106034 | 61,292 |
https://mathoverflow.net/questions/106022 | 2 | Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be the subspace of maps $f:X\to Y$ such that $f(A)\subseteq B$. Suppose the inclusions $A\hookrightarrow X$ and $B\hookrighta... | https://mathoverflow.net/users/25828 | When is the inclusion of a relative mapping space into a mapping space a cofibration? | The inclusion $\operatorname{Map}(X,A;Y,B)\hookrightarrow \operatorname{Map}(X;Y)$ will be a cofibration whenever $(X,A)$ and $(Y,B)$ are cofibrations, under some mild point-set hypotheses. My argument requires $Y$ to be Hausdorff and $A$ and $X$ to be locally compact Hausdorff (but I'm not sure if these restrictions a... | 6 | https://mathoverflow.net/users/8103 | 106038 | 61,294 |
https://mathoverflow.net/questions/106033 | 3 | One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (*i.e.* entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is dropped in both cases, of course.
My question concerns the seemingly natural generalization of this to the case that ... | https://mathoverflow.net/users/22051 | Hadamard product and inertia | Here's my second attempt at an answer.
As before, I claim that in general, anything can happen. That is, the inertia of $A\circ B$ can be almost anything (i.e., can be very different from the inertia of either $A$ or $B$).
Here is a line of thought.
1. Let $A$ be psd, and let $B=ee^T$ (the all ones matrix). Then,... | 2 | https://mathoverflow.net/users/8430 | 106041 | 61,295 |
https://mathoverflow.net/questions/106040 | 4 | Let $\lambda$ be a partition and define $T\_\lambda^n(k)$ to be the number of semi-standard young tableaux of shape $(k\lambda\_1,k\lambda\_2,\dots,k \lambda\_n).$
Now, one can prove that $T\_\lambda^n(k)$ is a polynomial in $k.$
**Is this well-known? Is there a known formula for $T\_\lambda^n(k)$? Would it be intere... | https://mathoverflow.net/users/1056 | Sequence of semi-standard Young tableaux, counting | If my memory serves me correctly, the number of semi-standard Young tableaux of shape $\lambda$ and with entries in $[n]$ (I suppose you mean that) can be interpreted as the number of lattice points in a certain polytope determined by $n$ and $\lambda$, and scaling up the parts of $\lambda$ is tantamount to refining th... | 8 | https://mathoverflow.net/users/19077 | 106043 | 61,296 |
https://mathoverflow.net/questions/105883 | 47 | Background
----------
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful [book](http://rads.stackoverflow.com/amzn/click/0387001735): *fixed point theorems fall into two broad categories*. The first class is usually functional analytic and imposes strong cond... | https://mathoverflow.net/users/18263 | Can we actually find any fixed points with Brouwer's theorem? | The paper "Exponential lower bounds for finding Brouwer fixed points"
*Addendum by original poster:* It was non-trivial to find a copy of this great paper of Hirsch, Papadimitriou and Vavasis. It does answer my general question quite clearly: finding Brouwer fixed points is exponentially hard in the worst case no mat... | 18 | https://mathoverflow.net/users/16671 | 106044 | 61,297 |
https://mathoverflow.net/questions/106047 | 0 | Consider $\tau$ and $\tau'$ in the upper half plane such that $j(g \tau) = j(g \tau')$ for all $g \in GL\_2^{+}(\mathbb{Q})$, where $j$ is the modular $j$-function and $GL\_2^{+}(\mathbb{Q})$ acts as Mobius transformations. Then does $\tau = \tau'$?
| https://mathoverflow.net/users/16858 | A question regarding the j-function | Unless you are proposing some action of $GL\_2(\mathbb Q)^{+}$ on the upper-half plane other than the obvious one, every value of $\tau$ is fixed by every scalar. In $PGL\_2(\mathbb Q)^+$, the fixed points are CM points.
But regardless, this is true for every value of $\tau$. Multiply both $\tau$ and $\tau'$ by $n$ u... | 2 | https://mathoverflow.net/users/18060 | 106048 | 61,298 |
https://mathoverflow.net/questions/105983 | 6 | Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
>
> For large enough $p$, the $j$-invariant $j(E)$ must be in $O\_K$.
>
>
>
I would also like to know exactly how large $p$ must be. (This lower bound may very w... | https://mathoverflow.net/users/5744 | Must the $j$-invariant of an elliptic curve with an isogeny be integral? | If you are given K, and you want to find a p such that J\_0(p) has a rank-0 quotient over K, you might try proceeding as follows. Let $\chi$ be the quadratic twist whose kernel is the Galois group of K. Then what you really want is to show that there exists a newform f in S\_2(p) with the property that both the corresp... | 7 | https://mathoverflow.net/users/431 | 106052 | 61,299 |
https://mathoverflow.net/questions/106049 | 4 | One can make a model of the hyperbolic plane inside the Euclidean plane, either using the conformal model or projective model.
How does one make a model of the Euclidean plane inside the hyperbolic plane?
| https://mathoverflow.net/users/25762 | Euclidean inside Hyperbolic | I believe there is no good model of $\mathbb{E}^2$ in $\mathbb{H}^2.$ However, there is an excellent model in $\mathbb{H}^3:$ any horosphere will work.
**Also** This is not particularly interesting, but if you use the hyperboloid model of $\mathbb{H}^2,$ you can project it (from, e.g., the point $(2, 0, 0)$ onto the ... | 5 | https://mathoverflow.net/users/11142 | 106055 | 61,301 |
https://mathoverflow.net/questions/106046 | 0 | If you define 6j symbols completely formally via trivalent graphs
(take <http://math.ucr.edu/home/baez/qg-fall2000/qg10.2.html> for a start,
but be careful - looks like Racah coefficients to me...well, he already
mentions "fudge factors" :-) I see no -1^whatever. Now in any standard
work (e.g. M. Rotenberg et al.) you ... | https://mathoverflow.net/users/11504 | Phase choice for 6j symbols | There are two real forms of $SL(2,C)$, the compact real form $SU(2)$ and the split real form $SL(2,R)$. Mathematicians prefer the split real form and physicists the compact real form. The two cases have different signs. There is also a choice of sign conventions involved even once you have decided which real form you w... | 2 | https://mathoverflow.net/users/3992 | 106056 | 61,302 |
https://mathoverflow.net/questions/106042 | 0 | Hi everyone! I have a question about how to find the closed form of a function defined by
$$\phi(\theta)=\inf\_{x\geq 2}f(x;\theta)\equiv\inf\_{x\geq 2}\frac{(x+2)^2}{\frac{1}{\theta}\left(\frac{x-1}{2}-\frac{1}{x}\right)+\frac{x^2-1}{16}},\ \ \theta>0$$
Since finding the minimum of $f(x;\theta)$ leads to solving a... | https://mathoverflow.net/users/25005 | question about the closed form of a function | If $\theta>2$ then $\phi(\theta)=16$. For $\theta<2$, let
$$t(x)=\frac{4(x^3-4x^2-6x-4)}{x^2(1+2x)},$$
$$g(x)=f(x,t(x)).$$
The function t is invertible from $[x\_m,\infty)$ to $[0,2)$, where $x\_m$ is the root of
t(x)=0, approximately 5.28. We have $\phi(\theta)=g(t^{-1}(\theta))$.
| 1 | https://mathoverflow.net/users/12120 | 106058 | 61,303 |
https://mathoverflow.net/questions/106059 | 0 | Hi,
Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$.
Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$?
Thanks!
| https://mathoverflow.net/users/26112 | Is a jointly convex function of x and y convex as a function of x when y=z(x)? | Here is a counterexample for $m=n=1$. Let $f(x,y)=x^2+y^2$ which is convex, and let $z(x)$ be continuous such that $z(0)=0$, $z(1)=2$, $z(2)=1$. Then
$$ 2f(1,z(1))=10>0+5=f(0,z(0))+f(2,z(2)), $$
hence $f(x,z(x))$ is not convex.
| 2 | https://mathoverflow.net/users/11919 | 106060 | 61,304 |
https://mathoverflow.net/questions/106057 | 8 | This is a cross-post of the following math.stackexchange question: <https://math.stackexchange.com/questions/188760/cup-product-and-hypercohomology>
I always found the cup product slightly mysterious. Recently I discovered the following interesting theorem (in Voisin's book *Hodge theory and complex algebraic geometr... | https://mathoverflow.net/users/5181 | Cup products and hypercohomology | Any two natural approaches will likely yield the same product *up to sign*, but this is not much of an answer. If you want a precise reference comparing the Cech-style cup product with the product using resolutions, take a look at Godement's Théorie des Faisceaux.
---
After rereading the post-edited question, I ... | 2 | https://mathoverflow.net/users/4144 | 106062 | 61,305 |
https://mathoverflow.net/questions/106068 | 5 | Hi All,
I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:
I am trying to understand the structure (e.g., decomposition) of the unitary representation $L^2(N/M)$ where $N$ is a nilpotent Lie group acting by left translation on this Hilbert space (coming fro... | https://mathoverflow.net/users/26116 | Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group | The representation you're looking at is $\mathrm{Ind}\_M^N1$ and as such, its decomposition into irreducibles is very well understood using Kirillov's orbit method. (Essentially, the irreducibles that enter correspond to the coadjoint orbits in the image of the moment map
$T^\*(N/M)\to\mathfrak n^\*$.)
I'd say the ba... | 8 | https://mathoverflow.net/users/19276 | 106073 | 61,309 |
https://mathoverflow.net/questions/106021 | 10 | Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K\_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by
$$\operatorname{Fil}^p\_{\mathrm{cod}} K\_m(X) :=\bigcup\nolimits\_Y \text{Ker}(K\_m(X)\to K\_m(X-Y))$$
where the limi... | https://mathoverflow.net/users/2481 | When do the $\gamma$-filtration and codimension filtration of K-theory agree? | At the time when I constructed those maps, the motivic spectral sequence was not known to exist. I gave a conjectural construction for that spectral sequence in an American Journal of Math paper called "Relative Cycles and Algebraic K-Theory", but I still don't know whether the gaps in that construction can be (or have... | 8 | https://mathoverflow.net/users/10503 | 106074 | 61,310 |
https://mathoverflow.net/questions/106061 | 1 | I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should especially contain $\mathbb{R}$ and $\mathbb{C}$.
If e.g. $f \circ g = h$ holds for formal power series $f, g, h \in R[[X]... | https://mathoverflow.net/users/25840 | Lifting identities of formal power series | Take $g(x)=x-x^2\in\mathbb{Z}[[x]]$. There is an $f(x)\in x\mathbb{Z}[[x]]$ which is an inverse to $g(x)$ under composition (this is true because $g(x)$ has constant term 0 and linear term a unit; we could also write down $f(x)$ explicitly). We have $g(1)=0$, and $f(0)=0$ (both expressions converge in any topological r... | 5 | https://mathoverflow.net/users/5263 | 106075 | 61,311 |
https://mathoverflow.net/questions/86987 | 2 | If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula
\[ \int \int\_D h(z)\, dx dy = \pi r^2 h(0) \]
The disk is the simplest example of a [quadrature domain](http://www.math.kth.se/~gbjorn/schottky.pdf) since the integral of a holomorphic function over the domain is determined by t... | https://mathoverflow.net/users/1358 | quadrature domains from circles? | A bounded simply connected domain with analytic boundary
is a quadrature domain if and only if the
inverse of the Riemann mapping
function (mapping the disc onto the domain) is rational.
See, for example,
P. J. Davis, The Schwarz function and its applications, The Mathematical
Association of America, Bualo, N. Y., 1... | 6 | https://mathoverflow.net/users/25510 | 106079 | 61,314 |
https://mathoverflow.net/questions/106080 | 9 | This is the first in what may be a series of questions on the theme "a Banach algebraist/Bear Of Little Brain needs help with algebraic geometry".
$\newcommand{\Cplx}{{\mathbb C}}\newcommand{\fg}{{\mathfrak g}}$
Let $\fg$ be a complex semisimple Lie algebra. I have two objects on my mind:
-- the algebraic group ove... | https://mathoverflow.net/users/763 | How to get Haar measure on a compact Lie group, given the complexification? | I'll write $K$ for the compact group. A correction to start with: You write $\int\_K f g$, but I assume you mean $\int\_K f \bar{g}$.
**Complex conjugation** We need to identify the map $\sigma: R \to R$ which has the property that $\sigma(f)|\_K = \overline{\sigma(f)}$. Note that this map will NOT be complex conjug... | 6 | https://mathoverflow.net/users/297 | 106085 | 61,315 |
https://mathoverflow.net/questions/106070 | 5 | The [Grothendieck Group](http://en.wikipedia.org/wiki/Grothendieck_group) construction seems very similar to the Int construction over [traced monoidal categories](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf) (pdf).
[It may help to either rewrite the group construction multiplicatively, or to use additive not... | https://mathoverflow.net/users/3993 | Int construction (traced monoidal categories) and Grothendieck Group | I believe that the Grothendieck group of a commutative monoid can be viewed as a special case of the Int construction. Namely, you can think of a commutative monoid $M$ as a discrete symmetric monoidal category (i.e., a symmetric monoidal category with no non-identity morphisms). This category admits a unique trace, an... | 4 | https://mathoverflow.net/users/396 | 106087 | 61,316 |
https://mathoverflow.net/questions/106088 | 1 | Consider a random variable $F$ with a distribution parameterized by $\theta$ and another random variable $G$ with a distribution parameterized by a variate of $F$, denoted $f$. Note that $F$ is dependent on $\theta$, but $G$ is dependent on $\theta$ only through $f$.
Consider the likelihood for $\theta$ given observa... | https://mathoverflow.net/users/26120 | Is there a notion of likelihood that incorporates information content? | What you are missing is normalization: your likelihood functions $L(\theta|f,g)$ and $L(\theta|f)$ shouldn't be comparable, because adding up the likelihood for all the possible $\theta$s gives different totals. If you instead define
$\displaystyle L(\phi|x) = \frac{Pr[X=x|\phi]}{\sum\_{\phi'} P[X=x|\phi']}$,
the... | 4 | https://mathoverflow.net/users/1474 | 106089 | 61,317 |
https://mathoverflow.net/questions/106111 | 0 | Basically, its a 2-sequence longest common sub-sequence (LCS) problem.
What's so special?
1: each alphabet only occurs 2 times, one in each sequence, which means with no same alphabet in one sequence.
2: we can execute 2 (or k) "LCS"es at a time, and we want the sum length of the 2 "LCS"es is the maximum.
Take... | https://mathoverflow.net/users/25958 | A simplified/harder 2-sequence longest common sub-sequence (LCS) problem | If one sequence is $(a\_1,\ldots, a\_n)$ with all $a\_i$ distinct, then the other one is a permutation of it of the form $(a\_{w(1)},\ldots, a\_{w(n)})$. A common subsequence corresponds therefore to an increasing subsequence of the permutation $(w(1),\ldots,w(n))$.
Now the maximal total length of $k$ disjoint increa... | 0 | https://mathoverflow.net/users/730 | 106116 | 61,325 |
https://mathoverflow.net/questions/106112 | 13 | Has a notion of "category of Riemannian manifolds" been defined and used in the literature?
For which reasons is it or would it (not) be a useful notion?
I think the objects should be all (perhaps complete) Riemannian manifolds, and two objects should certainly be isomorphic if they are isometric as Riemannian mani... | https://mathoverflow.net/users/4721 | Convenient definition of "category of Riemannian manifolds"? | I'm sure the answer to your question is "it depends on the application". Here are three categories that come to (my idiosyncratic) mind.
Perhaps the most general category in the direction you're looking is a version of Lawvere's category of metric spaces. Recall that $\mathbb R\_{\geq 0}$ is a category, on account of... | 17 | https://mathoverflow.net/users/78 | 106119 | 61,326 |
https://mathoverflow.net/questions/106104 | 19 | Dear MO\_World,
I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning cyclotomic polynomials that I'm unable to tackle.
The question is this:
>
> Let $p$ be a prime and let $p|n$. When is ... | https://mathoverflow.net/users/11054 | Cyclotomic polynomials evaluated at roots of unity | Set $\zeta=e^{2\pi i/p}$. By what Anthony wrote, we may assume $n=pm$ where $p$ does not divide $m$. Note that if $\Phi\_n(\zeta)=\pm\zeta^j$, then $\Phi\_n(\zeta^k)=\pm\zeta^{jk}$ for each $k$ prime to $p$, because the absolute Galois group of $\mathbb Q$ acts transitively on the $\zeta^k$s (by irreducibility of $\Phi... | 15 | https://mathoverflow.net/users/18739 | 106123 | 61,328 |
https://mathoverflow.net/questions/106120 | 5 | What is the maximum number of spheres that can be placed in 3D such that all inter-touch?
One can of course place four unit spheres tetrahedrally and then add a smaller sphere in the
middle, so this number must be at least 5.
[By the way, I was trying to extend the "five points in 2D cannot be inter-connected witho... | https://mathoverflow.net/users/20757 | Inter-Kissing Number for Spheres of Different Sizes | In $\mathbb R^n$, the answer is $n+2$.
You can apply an inversion which sends two of the spheres in to two parallel hyperplanes.
The rest of the spheres will have the same radii and their centers lie in a hyperplane.
Hence everything follows.
| 12 | https://mathoverflow.net/users/1441 | 106124 | 61,329 |
https://mathoverflow.net/questions/101915 | 7 | Let $\pi:X\rightarrow Y$ be a smooth proper (assume projective if needed) morphism of schemes with $Y$ locally noetherian, and let $Z\subset X$ be an irreducible integral closed subscheme containing no fiber of $\pi$.
* Is the locus $Pic\_\pi(Z)=\{y\in Y:Z\_y \text{ is Cartier in }X\_y\}$ closed in $Y$?
* If not, wha... | https://mathoverflow.net/users/1939 | Irreducible "family" of relative effective divisors of a smooth morphism | In fact both sets are constructible in $Y$.
Suppose for simplicity that $X$ is connected. Then the dimension of the (non-empty) fibers of $X\to Y$ is constant (EGA IV.12.1.1(i), and flatness is enough), denote it by $d$. Let $Y'$ be the (integral) image of $Z$ in $Y$. We can replace $X\to Y$ by $X\times\_Y Y'\to Y'$... | 4 | https://mathoverflow.net/users/3485 | 106130 | 61,332 |
https://mathoverflow.net/questions/59463 | 16 | In ''[The Probabilistic Method](http://books.google.com/books?id=V8YgNioxF6AC&printsec=frontcover&dq=probabilistic+method&hl=en&src=bmrr&ei=55eLTZ6YNM-WhQeE94yyDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDAQ6AEwAA#v=onepage&q&f=false)'' by Alon and Spencer, the following **unbalancing lights** problem is discussed.... | https://mathoverflow.net/users/7767 | Unbalancing lights in higher dimensions | Belatedly answering my own question: Pellegrino and Seoane-Sepulveda have [shown](http://arxiv.org/pdf/1010.0461.pdf) the lower bound that $m(A) \ge n^{(d+1)/2}/\text{poly}(d)$. As far as I know, it is still open whether the $\text{poly}(d)$ term can be replaced with a universal constant.
| 7 | https://mathoverflow.net/users/7767 | 106141 | 61,338 |
https://mathoverflow.net/questions/106143 | 6 | A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex structures with respect to which the metric is Kähler.
A K3 surface is a hyperkähler manifold of real dimension $4$. It is cla... | https://mathoverflow.net/users/25713 | Complex structures on a K3 surface as a hyperkähler manifold | These $2$-spheres are called 'twistor lines'. They indeed cover the moduli space (in the non-polarized case) : more precisely, any two points of the moduli space may be linked by a chain of twistor lines.
A reference where this is nicely explained (and used !) is Huybrecht's Bourbaki talk about Verbitsky's Torelli th... | 10 | https://mathoverflow.net/users/2868 | 106144 | 61,339 |
https://mathoverflow.net/questions/106096 | 14 | This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ... My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me force you to work in my favorite model, I ask only that you say with some precision what model you are using, and that your ... | https://mathoverflow.net/users/78 | What are the higher morphisms between enriched higher categories? | For $F,G:C\to D$, an $S$-enriched transformation $F\to G$ will consist of, for each $x\in C$, a morphism $1\to D(F x, G x)$ in $S$, together with for each $x,y\in C$, an equivalence between the composites
$$ C(x,y) \to D(F x, F y) \to D(F y, G y) \otimes D(F x, F y) \to D(F x, G y)$$
and
$$ C(x,y) \to D(G x, G y) \to D... | 9 | https://mathoverflow.net/users/49 | 106145 | 61,340 |
https://mathoverflow.net/questions/106150 | 5 |
>
> Let $G$ be an algebraic group over an
> algebraically closed field. If $H$ and
> $K$ are closed subgroups and one of
> them is connected, then their
> commutator $[H,K]$ is also connected.
>
>
>
Is there an easy way to see this fact?
The proof that I see in Springer's book on Linear Algebraic Groups is... | https://mathoverflow.net/users/2720 | Commutator of algebraic subgroups is connected | The case for $G$ a topological group doesn't look hard; it just comes down to two facts:
* A union of connected sets which have a point in common is also connected. For example, suppose $K$ is a connected subgroup and $H$ is any set. For each fixed $h \in H$, the set of commutators $[h, K]$ is the image of $K$ under... | 12 | https://mathoverflow.net/users/2926 | 106152 | 61,344 |
https://mathoverflow.net/questions/106139 | 3 | Let $A$ be a commutative ring with $1$, and $B\subseteq A$ be a subring.
>
> Is there a simple condition on $B$ and $A$ guaranteeing that $B\to A\rightrightarrows A\otimes\_B A$ is an equalizer?
>
> In other words, when does $a\otimes\_B 1 = 1\otimes\_B a$ imply that $a\in B$?
>
>
>
This always holds, for... | https://mathoverflow.net/users/1474 | When is this diagram of tensor powers an equalizer? | I think what you want is exactly that the dominion (in the sense of Isbell) of $B$ in $A$ be equal to $B$.
Recall that if you are in a category of algebras (in the sense of Universal Algebra), and $B\subseteq A$ is a subalgebra of $A$, then the **dominion of $B$ in $A$** (relative to the category of context) is the ... | 8 | https://mathoverflow.net/users/3959 | 106154 | 61,346 |
https://mathoverflow.net/questions/106148 | 13 | Let $X$ be a Kähler manifold of dimension $n$, and $Z \subset X$ an analytic subset of codimension $k$. I have read in a paper the following result, a proof of which I cannot find:
$$c\_k(\mathscr{I}\_Z) = (-1)^k(k-1)![Z]$$
The form of the expression suggests using GRR, but I cannot figure out how.
>
> Does an... | https://mathoverflow.net/users/25566 | Chern classes of ideal sheaf of an analytic subset | In Fulton's book *Intersection Theory*, Theorem 15.3 and Example 15.3.1 (p. 297 of my edition) it is proven, by using GRR, that $$c\_k(\mathscr{O}\_Z)=(-1)^{k-1}(k-1)![Z].$$
By using the short exact sequence
$$0 \to \mathscr{I}\_Z \to \mathscr{O}\_X \to \mathscr{O}\_Z \to 0$$
we obtain $1=c(\mathscr{O}\_X)=c(\mathscr{O... | 14 | https://mathoverflow.net/users/7460 | 106166 | 61,351 |
https://mathoverflow.net/questions/106162 | 7 | Context
-------
According to the FAQ, questions of the form "the sorts of questions you come across when you're writing or reading articles or graduate level books" are acceptable. This falls into the "reading graduate level books."
Problem Statement
-----------------
Let $N$ be the natural numbers.
$B \subsete... | https://mathoverflow.net/users/26147 | Elementary Proof of Basis of Order k | If $B$ is a basis of order $k$ such that every integer $n$ can be written as a sum of $k$ elements from $B$ in $\asymp n^{o(1)}$ ways, then a simple counting argument yields $|B \cap [1 , X]| \asymp X^{\frac{1}{k}+o(1)}$. Thus a stronger estimate $|B \cap [1 , X]| \asymp (X \log X)^{\frac{1}{k}}$ in your problem is cer... | 6 | https://mathoverflow.net/users/21724 | 106170 | 61,354 |
https://mathoverflow.net/questions/106169 | 2 | Peter Peterson's book "Riemannnian Geometry" p351 says:
1. Closed manifold has no nontrivial totally convex subset. Using the energy functional if $A\subset M$ is totally convex, then $A\subset M$ is $k$-connected for any $k$.
2. It is however not possible for a closed n-manifold to have $n$-connected nontrivial sub... | https://mathoverflow.net/users/24637 | Closed manifold has no nontrivial totally convex subset? | The inclusion from a closed totally convex subset to the ambient manifold is a homotopy equivalence. Details can be found in
["Totally convex sets in complete Riemannian manifolds"](http://www.intlpress.com/JDG/archive/1981/16-2-333.pdf) by Bangert, JDG, 1981.
For the second assertion,
Cheeger-Gromoll prove in thei... | 9 | https://mathoverflow.net/users/1573 | 106172 | 61,355 |
https://mathoverflow.net/questions/106173 | 1 | How does one prove that for a smooth projective variety $X$ over an algebraically closed field $k$, an algebraic cycle that is homologically equivalent to zero is also numerically equivalent to zero? In all the papers I have seen, this is just stated without proof. Is there a simple proof or a good reference?
| https://mathoverflow.net/users/22823 | Homological equivalence for algebraic cycles | This just says cup product is well defined. Let $Z \subset X$ be an algebraic cycle; let $n=\dim X$ and $k=\dim Z$. $Z$ is homologically equivalent to zero if the image $[Z]$ of $Z$ in $H^{2n-2k}(X)$ is zero, and $Z$ is numerically zero if $[Z] \cup [Y]=0$ for all algebraic cycles $Y$ of dimension $n-k$.
It's possibl... | 4 | https://mathoverflow.net/users/297 | 106176 | 61,356 |
https://mathoverflow.net/questions/106174 | 1 | Hello all,
I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain".
In Stephen Boyd and Lieven Vandenberghe book ("Convex Optimization") they present the following example:
"For the function $f(X) = \log... | https://mathoverflow.net/users/25265 | Question regard checking convexity by "restriction to any line that intersects the function domain" | Think of $V$ as a tangent vector at $Z.$ you can see that, since the set of psd matrices is an open subset of the set of all symmetric matrices, there is absolutely no restriction on $V.$ On the other hand, the result (concavity of log det ) only holds in the psd cone, so $Z$ better be an element of the cone.
| 1 | https://mathoverflow.net/users/11142 | 106177 | 61,357 |
https://mathoverflow.net/questions/106171 | 2 | That is, how to calculate it given the size of N(that is, logN) and assuming that logN is much greater than M. Its an approximation. There is no exact formula.
I do know that according to the prime number theorem the probability for N being prime is 1/ln(N) for large positive integers.
Thanks.
| https://mathoverflow.net/users/21733 | The probability that a random number N has at least M factors | The count of numbers up to $x$ having at most $k$ *prime* divisors (counted with mutiplicity) is
asymtotically equal to (see for example [Almost prime](http://en.wikipedia.org/wiki/Almost_prime) on Wikipedia)
$$ \frac{x}{\log x} \frac{(\log \log x)^{k-1}}{(k-1)!} .$$
Thus, since the number of divisors is definitely ... | 3 | https://mathoverflow.net/users/nan | 106184 | 61,363 |
https://mathoverflow.net/questions/106193 | 2 | The following seems to be the "official" notion of torsion-freeness in the context of semigroups:
>
>
> >
> > **TF1.** A (multiplicatively written) semigroup $\mathfrak A$ is torsion-free if there do not exist $a,b \in \mathfrak A$ and $n \in \mathbb N^+$ such that $a \ne b$ and $a^n = b^n$.
> >
> >
> >
>
> ... | https://mathoverflow.net/users/16537 | On the notion of torsion-freeness in semigroup theory | I believe your first definition is primarily used by people in commutative semigroup theory. Most people in non-commutative semigroup theory that I know prefer the second. An important example is the free profinite monoid. Every periodic element is an idempotent so it is torsion-free in the second sense. This is a non-... | 4 | https://mathoverflow.net/users/15934 | 106195 | 61,367 |
https://mathoverflow.net/questions/106191 | 35 | This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$, which has the same eigenvalues) is similar to a symmetric matrix, so has real eigenvalues. Take the vectors of eigen... | https://mathoverflow.net/users/6494 | Eigenvalues of the product of two symmetric matrices | Here are the results that you are probably looking for.
The first one is for positive definite matrices only (the theorem cited below fixes a typo in the original, in that the correct version uses $\prec\_w$ instead of $\prec$).
>
> **Theorem** (Prob.III.6.14; Matrix Analysis, Bhatia 1997). Let $A$ and $B$ be Her... | 37 | https://mathoverflow.net/users/8430 | 106199 | 61,368 |
https://mathoverflow.net/questions/103815 | 7 | I want to know why we add an intensional equality in type theory to definitional equality ?
What is the aim with this intensional equality ?
thanks
| https://mathoverflow.net/users/nan | intensional equality in type theory | The key thing to notice is that the definitional equality is a *judgement* and not a proposition (so in particular, definitional equalities can not be part of your assumptions, nor can they be proved, they can only be checked by the type-checker). The intensional equality, on the other hand, *is* a proposition, so it c... | 7 | https://mathoverflow.net/users/11035 | 106203 | 61,371 |
https://mathoverflow.net/questions/103760 | 4 | I've been looking at curves of the form $y^2=x^3+k$ (where k is 6th power free and not divisible by 3^3) and I've noticed that there seems to be distinct grouping in residues classes modulo 504.
One effect that I noticed, in these residue classes, was that the populations of positive and negative k values seem to inv... | https://mathoverflow.net/users/25232 | Parity dependent population inversion in Mordell elliptic curves | The parity of the analytic rank of any elliptic curve $E\_k: y^2 = x^3 + k$
over ${\bf Q}$ was determined in the paper
>
> Liverance, Eric:
> A formula for the root number of a family of elliptic curves.
> *J. Number Theory* **51** #2, 288--305 (1995).
>
>
>
(This reference was posted a few weeks ago by Larr... | 8 | https://mathoverflow.net/users/14830 | 106214 | 61,376 |
https://mathoverflow.net/questions/106217 | 9 | Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!)
Let $D>9$ be a rational number which is not a square, and consider the quadratic irrational
$$\xi = \frac{-3 + \sqrt{D}}{2},$$
(I'... | https://mathoverflow.net/users/7399 | Last term of repeating continued fraction expansion | It's known that a quadratic irrational has a purely periodic continued fraction expansion if and only if it is greater than $1$ and its conjugate is between $-1$ and $0$. Your observation amounts to proving that $r+2 + (-3+\sqrt{D})/2$ has this property (note that adding $r + 2$ makes it start with $2r+1$). This amount... | 13 | https://mathoverflow.net/users/4720 | 106219 | 61,377 |
https://mathoverflow.net/questions/106228 | 8 | Let $(a\_n)\_{n=0}^\infty$ be an arbitrary sequence in a real Banach space $X$. Does there exist a smooth function $f: \mathbb {R} \rightarrow X$ such that $f^{(n)}(0)=a\_n$ for $n=0,1,2,\ldots $?
| https://mathoverflow.net/users/25921 | Borel's theorem for Banach's space valued functions | According to *The Convenient Setting of Global Analysis* (Kriegl and Michor), this is due to Wells (1973). The statement given is:
>
> **15.4. Borel's Theorem.** [Wells, 1973]. Suppose a Banach space $E$ has $C^\infty\_b$-bump functions. Then every formal power series with coefficients in $L\_{\text{sym}}^n(E;F)$ f... | 9 | https://mathoverflow.net/users/45 | 106229 | 61,381 |
https://mathoverflow.net/questions/106231 | 7 | Hello all,
I come across the following problem.
Is it true that for a **positive definite** matrix $X^{n\times n}$, the following holds
$\text{trace}(X^{-1})\geq\text{trace}([\text{diag}(X)]^{-1})$,
where $\text{trace}(\;\cdot\;)$ and $\text{diag}(\;\cdot\;)$ are the matrix trace and diagonal operators, respect... | https://mathoverflow.net/users/25265 | Trace matrix inequality | It is quite well known that the diagonal of an Hermitian matrix is majorized by sequence of its eiqenvalues (this is known as Schur's theorem). Since all of these numbers are positive, we can use [Karamata's inequality](http://en.wikipedia.org/wiki/Karamata%27s_inequality) for function $x \mapsto \frac{1}{x}$ to conclu... | 19 | https://mathoverflow.net/users/24953 | 106233 | 61,383 |
https://mathoverflow.net/questions/106237 | 7 | The signature of compact oriented $4k$-manifolds has the following additivity property, first observed by S. P. Novikov: If two manifolds are glued by an orientation-preserving diffeomorphism of their boundaries, then the signature of their union is the sum of their signatures.
C.T.C. Wall showed that this additivity p... | https://mathoverflow.net/users/1190 | Additivity of Signature | Yes, the signature is additive in this case. Wall gave a precise description of the failure of additivity in gluing two $4k$-manifolds along a subset of their boundaries. (I think the name of the paper is "On the non-additivity of the signature", or something like that, but I don't have it in front of me.) Let $A$ and ... | 8 | https://mathoverflow.net/users/284 | 106241 | 61,386 |
https://mathoverflow.net/questions/106212 | 8 | I am a physicist and would like to understand the section 1 of
[this math paper](http://www.math.colostate.edu/~renzo/mg/paper28.pdf), which explains how the SYZ conjecture implies topological mirror symmetry. I have some technical problem and would appreciate your help. My question is purely mathematical and nothing ... | https://mathoverflow.net/users/26161 | Elementary computation of direct image sheaves. | *Edited following algori's comment.*
In the context of the paper I think one can make these deductions on rather formal grounds - deep understanding of higher direct images is not needed.
Gross assumes that the base $B$ is a simply connected closed 3-manifold (i.e., it's $S^3$).
Let $i\colon B\_0\to B$ be the incl... | 5 | https://mathoverflow.net/users/2356 | 106244 | 61,388 |
https://mathoverflow.net/questions/106238 | 10 | Let $K$ be a $p$-adic field, $X$ a smooth proper algebraic variety over $K$, and $0 \le i \le 2 \dim X$. For a prime $\ell \ne p$ one can consider the $\ell$-adic cohomology $H^i(\overline{X}, \mathbb{Q}\_\ell)$, and massage this in the usual way (via Grothendieck's abstract monodromy theorem) to get a Weil–Deligne rep... | https://mathoverflow.net/users/2481 | Is there a "universal" cohomology theory for varieties over p-adic fields? | The following is an adaptation of an argument of Serre, explaining why there shouldn't be a universal cohomology theory for $\mathbb{F}\_p$ varieties taking values in $\mathbb{Q}$ vector spaces. Be warned that this is not my field, so I may be missing something basic.
Let $p$ be a prime which is $3 \bmod 4$, let $X$ ... | 17 | https://mathoverflow.net/users/297 | 106246 | 61,389 |
https://mathoverflow.net/questions/106243 | 4 | Let $ G,H $ are simple finite graphs and $A = G \times H$. Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.
Let $G$ has smaller chromatic number. Experiments suggest that given a coloring $f$ of $G$ one can color
$A = G \times H$. Color the vertices $(a,b)$... | https://mathoverflow.net/users/12481 | Coloring tensor products of graphs | The only way that two vertices $(u,v)$ and $(u',v')$ end up getting the same color is if $f(u)=f(u')$. But then there is no edge between $u$ and $u'$ in $G$ so there is no edge between $(u,v)$ and $(u',v')$ in $G\times H$. So your conjecture is true for all graphs.
In fact, as explained in the page you linked to abov... | 8 | https://mathoverflow.net/users/2384 | 106247 | 61,390 |
https://mathoverflow.net/questions/106234 | 5 | Let $ (X,\mathcal{F},\mu) $ and $ (G,\mathcal{G},\nu) $ be two measure spaces with $ \mu $ and $ \nu $ being $ \sigma $-finite. Per definition, the linear span of
$$
\{
\mathbf{1}\_{C}
~|~
C \in \mathcal{F} \otimes \mathcal{G} ~ \text{and} ~ (\mu \otimes \nu)(C) < \infty
\}
$$
is dense in $ {L^{p}}(X \times Y,\mathcal{... | https://mathoverflow.net/users/19874 | Simple functions on a product measure space | A useful density lemma is the following.
>
> Let $(X,\mathcal{A}, \mu)$ be a
> measure space and let $\Gamma\subset\mathcal{P}(X)$ a ring of sets
> of finite measure that generates the
> $\sigma$-algebra $\mathcal{A}$. Then,
> the linear span of the characteristic
> functions of sets in $\Gamma$ is dense
> i... | 6 | https://mathoverflow.net/users/6101 | 106252 | 61,393 |
https://mathoverflow.net/questions/105881 | 3 | Hey there,
several weeks ago, there was a discussion on the Brauer hom (see [Is the Brauer correspondence injective ?](https://mathoverflow.net/questions/99106/is-the-brauer-correspondence-injective/99117#99117) ). I like to investigate this hom when being applied to simple modules:
Let $k$ be an alg. closed field o... | https://mathoverflow.net/users/23104 | Brauer homomorphism and simple modules | I can point to two places where Brauer quotients of simple modules have been considered. The earliest appears in section 15 of the paper
P.J. Webb and J. Thévenaz, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), 1865-1961,
especially Example 15.8 and results 15.4 - 15.7 which lead up to it. Those ... | 8 | https://mathoverflow.net/users/26171 | 106254 | 61,395 |
https://mathoverflow.net/questions/106260 | 0 | Given a group G acting both on the left and the right of a set X, say the actions are compatible if $g \cdot (x \cdot g') = (g\cdot x)\cdot g'$ for all $g,g'\in G$ and $x\in X$.
Here are some examples of compatible $G$-actions:
1) For any left action (resp. right action), that action and the trivial right action (... | https://mathoverflow.net/users/3345 | Sets with compatible right and left G-actions | Suppose we have a group $G$ acting on the right and on the left as above on a set $X$. These actions give rise to a left action of $G\times G$ on $X$ by setting $(g\_1,g\_2)\cdot x=g\_1\cdot x \cdot g\_2^{-1}$. Conversely, if we have a left action of $G\times G$ on a set $X$, we can construct compatible left and right ... | 10 | https://mathoverflow.net/users/2349 | 106261 | 61,396 |
https://mathoverflow.net/questions/106259 | 6 | This note appears in "Proceedings of Symposia in pure mathematics" vol.33 1979 part 1 pp. 26-69.
The question will be about materials on page 31-32.
Let $G$ be a reductive algebraic group (not necessarily connected) defined over a local field $K$.
We fix a maximal $K$-split torus $S$ of $G$ and take N(resp. Z) to b... | https://mathoverflow.net/users/11056 | a question on TITS' note "Reductive groups over local fields" | Are you sure that $G$ isn't required to be connected? I think this is needed in order to construct the "valued root datum" structure which underlies Bruhat-Tits structure theory. Anyway, the key point is that there is the concept of "valuation" on the root datum, which is really a collection of "valuations" on the poss... | 8 | https://mathoverflow.net/users/26145 | 106264 | 61,398 |
https://mathoverflow.net/questions/106268 | 7 | Hey all,
I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E\_2$ page of the classical mod-2 version, especially for $t-s=3$ (mod 4). Since the order of this image is known and it is known t... | https://mathoverflow.net/users/24021 | Image of J in the classical Adams Spectral Sequence | The image of $J$ is pretty easy to see in the Adams $E\_2$ term: it consists of the elements along the vanishing line, plus, in dimensions 8k-1, of the towers that end near the vanishing line.
This identification is due to Mahowald, see *The order of the image of the J-homomorphism* for the announcement ( <http://pr... | 7 | https://mathoverflow.net/users/8824 | 106270 | 61,399 |
https://mathoverflow.net/questions/106272 | 2 | Let $C$ be a (quasi-projective) variety over an algebraically closed field $k$ and let $k(C)$ be its field of rational functions. Then for any birational map $\sigma: C \dashrightarrow C$ there is an automorphism of $k(C)$ that induces an automorphism of the Galois cohomology group $H^1(k(C),A)$ for any $Gal(k(C)^{alg}... | https://mathoverflow.net/users/2234 | action of automorphisms on the Galois cohomology of the function field of a variety | since $\left(k(C)^\times\right)^n)$ certainly includes $k$, $\sigma(h)/h$ lies in it if and only if its divisor class is an $n$-fold multiple of a principal divisor.
One can easily compute the divisor class of $\sigma(h)$ by pulling back (or pushing forward?) the divisor class of $h$ along the automorphism $\sigma$.
... | 3 | https://mathoverflow.net/users/18060 | 106274 | 61,400 |
https://mathoverflow.net/questions/106277 | 1 | I'm using the LM algorithm to do gradient descent in a model fitting context. I'm minimizing:
$$
c(x) = \sum ( f\_i(x) - y\_i )^2
$$
I'm noticing that after a few steps when I'm close to the minima, I often wind up with a zero somewhere on the diagonal of the Hessian -- which I approximate in the usual Gauss-Newton... | https://mathoverflow.net/users/19899 | Levenberg-Marquadt near the minima for non-zero-residual problems | It's not true that you've reached the true minimum if the Hessian has a zero on the diagonal. It just means that the problem doesn't depend on the corresponding variable.
One popular LM implementation uses a QR decomposition (probably of $J$) with pivoting. If a diagonal element of $R$ becomes very small, it will be ... | 2 | https://mathoverflow.net/users/20781 | 106281 | 61,403 |
https://mathoverflow.net/questions/106279 | 2 | Here's what I hope is a final question outside of my area that I need to understand a problem about stable vector bundles on $\mathbb{P}^2$. Thank you everybody for your help so far!
Suppose I have a rational number $0<\alpha<1$ with a palindromic continued fraction expansion, i.e.
$$\alpha = [0; a\_1,\ldots,a\_k]... | https://mathoverflow.net/users/7399 | Palindromic continued fraction | It's straightforward to prove by induction on $k$ that $[a\_k; a\_{k-1},\ldots,a\_1] = q\_k/q\_{k-1}$. (Let the left-hand side be $r\_k$, observe that $r\_{k+1} = a\_{k+1} + 1/r\_k$, and then note that the sequence $q\_k/q\_{k-1}$ satisfies the same recurrence.) Then $[0; a\_k,\ldots,a\_1] = q\_{k-1}/q\_k$; deduce for ... | 5 | https://mathoverflow.net/users/379 | 106284 | 61,405 |
https://mathoverflow.net/questions/106278 | 4 | I'm trying to work with the following sum:
$$f:=\sum\_{d\leq x}\mu(d)\tau(d) \Big[ \frac{x}{d}\Big] $$
Where $\mu$ is the Mobius function, $\tau(n)$ is the number of positive divisors of $n$ and $h(x)=[x]$ is the floor function.
We know that $$\sum\_{d\leq x} \mu(d)\Big[ \frac{x}{d}\Big]=1,$$ $$ \sum\_{d\leq x} \Big[... | https://mathoverflow.net/users/10920 | Problems with the divisor function in a summation | First, unraveling the floor function your sum is the same as
$$\sum\_{d\leq x}\left(1\*\mu\tau\right)(d)$$ where $\*$ represents mobius convolution. Let $f(n)$ denote the above multiplicative function. Then $f(1)=1$, and $f(p^k)=-1$ for $k\geq 2$.
This means that $f=\mu$, Mobius function, on all but the prime powers,... | 5 | https://mathoverflow.net/users/12176 | 106287 | 61,407 |
https://mathoverflow.net/questions/106285 | 12 | I have sometimes wondered about the following:
(1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to work in an $RO(G)$-graded context?
(2) Who was the first to realize that the correct set up for equivariant stable homot... | https://mathoverflow.net/users/8032 | Equivariant homotopy theory: some history questions | Well if you insist John :) (1) The first explicit formulation I know of is in the nice paper:
Klaus Wirthmuller. Equivariant homology and duality. Manuscripta Math. 11 (1974), 373–390.
He writes: "The ideas developed here partly originate from suggestions made by T. tom Dieck, who introduced me to the subject." They we... | 20 | https://mathoverflow.net/users/14447 | 106290 | 61,409 |
https://mathoverflow.net/questions/106230 | 8 | I am looking for an (incomplete) list of partial flag varieties for
which all Schubert cells admit small resolutions of singularities.
This is interesting, for many reasons. My motivation is, that a description of a small resolution will give the corresponding IC sheaves very explicitly and hence explicit formulas fo... | https://mathoverflow.net/users/2837 | Schubert varieties which admit small resolutions of singularities | For the Hermitian symmetric $G/P$, Nicolas Perrin explicitly classified all the Schubert varieties admitting a small resolution. (More explicitly, he classifies all the minimal models and quotes a theorem that says a small resolution is a smooth minimal model.) If I remember correctly, except for some very small rank e... | 8 | https://mathoverflow.net/users/3077 | 106293 | 61,411 |
https://mathoverflow.net/questions/106282 | 7 | The ABP maximum principle states (roughly) that, if $a^{ij} \partial \_i \partial \_j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient regularity of the coefficients), $\sup \_{\Omega} u \leq \sup \_{\partial \Omega} u + C (\int \_{\Omega} \vert f \vert^n )^{1... | https://mathoverflow.net/users/3709 | Alexandrov-Bakelmann-Pucci maximum principle | In my view, the key point the ABP estimate is that it ties pointwise information (the PDE) to information in measure (the contact set). This is crucial to regularity theory for non-divergence equations; it enables the proof of a Harnack inequality (due to Krylov and Safonov), and $C^{1,\alpha}$ regularity for viscosity... | 16 | https://mathoverflow.net/users/16659 | 106294 | 61,412 |
https://mathoverflow.net/questions/106298 | 12 | Let $(C, \otimes)$ be a symmetric monoidal category (maybe braided is also okay). Then the category $\text{Mon}(C)$ of monoid objects in $C$ is also a symmetric monoidal category with the same monoidal product. The category $\text{Mon}(\text{Mon}(C))$ is, by the Eckmann-Hilton argument, just the category of commutative... | https://mathoverflow.net/users/290 | Iterating monoid categories | Your example suggests the characterization of those $C$ which are already stabilized: they are exactly the monoidal categories whose monoidal structure is cocartesian. To see this, it suffices to observe that the monoidal structure on $\mathrm{CMon}(C)=\mathrm{Mon}(\mathrm{Mon}(C))$ is always cocartesian, and that coca... | 12 | https://mathoverflow.net/users/49 | 106300 | 61,413 |
https://mathoverflow.net/questions/106236 | 3 | A right crossed module is a homomorphism of groups
$\partial\colon F\to G$
together with a right action of $G$ on $F$, written
$(g,f)\mapsto f^g$,
satisfying certain conditions.
The question is, whether a crossed module $F\to G$ is in some sense abelian.
Following
[Norrie](http://archive.numdam.org/article/BSMF_199... | https://mathoverflow.net/users/4149 | Non-quasi-abelian braided crossed modules | Recall that free simplicial groups model all connected homotopy types. A simplicial group $G$ is free if each $G\_n$ is free and degeneracies are defined by maps between the bases.
A simplicial group is $n$-reduced if $G\_i=0$ for $i\leq n$. They model $(n+1)$-connected homotopy types.
A simplicial group $G$ is $n$... | 2 | https://mathoverflow.net/users/12166 | 106306 | 61,417 |
https://mathoverflow.net/questions/106310 | 3 | **Question:** What is decomposition of the representation k[Flag(F\_q)] as bimodule over GL\_n(F\_q) , Hecke(q) ?
(Let k=Complex numbers. Further question: is there any change for char k = p ? )
**Remark:** Hecke(q) is deformation of k[S\_n] - which is semisimple, so there are no non-trivial deformations for generi... | https://mathoverflow.net/users/10446 | Decomposition of k[Flag(F_q)] as bimodule over GL_n(F_q) , Hecke(q) ? | If you take the space $X\_\lambda$ of flags of shape $\lambda$ (here $\lambda$ is a partition of $n$, and a flag of shape $\lambda$ is one where the $i$th subspace has dimension $\lambda\_1+\dotsc+\lambda\_i$), then there exists a family of irreducible representations of $V\_\lambda$ of $GL\_n(\mathbf F\_q)$ such that
... | 3 | https://mathoverflow.net/users/9672 | 106315 | 61,420 |
https://mathoverflow.net/questions/106313 | 6 | While reading Peter Shor's paper *Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer*, I came across the following quote:
>
> "This scheme will thus work as long as $n$ is odd and not a prime power; finding factors of prime powers can be done efficiently with classical ... | https://mathoverflow.net/users/13410 | Algorithm for detecting prime powers | If N is a prime power it is of the form $p^i$ where $i \leq \log\_2(N)$. One can thus compute each of the first $\log\_2(N)$ roots of $N$, and test if the resulting number is (first an integer) and then if it is a prime using the [AKS algorithm](https://en.wikipedia.org/wiki/AKS_primality_test) (although perhaps Shor m... | 11 | https://mathoverflow.net/users/630 | 106316 | 61,421 |
https://mathoverflow.net/questions/106297 | 5 | Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. Given a Borel subgroup $B\subset G$ and a maximal torus $T \subset G$, one has an associated system of simple roots $\Delta(... | https://mathoverflow.net/users/4494 | Rational automorphisms of semisimple algebraic groups | I think ${\rm Aut}(\Delta(B,T))$ doesn't always split: there are some very serious obstructions for that.
According to a recent result of Skip Garibaldi ad Philippe Gille on algebraic groups with few subgroups, published in PLMS, there exist groups $G$ of type ${\rm D}\_4$ whose all proper connected subgroups defined... | 5 | https://mathoverflow.net/users/24386 | 106320 | 61,424 |
https://mathoverflow.net/questions/106323 | 2 | Let $f: \mathbb{R} \rightarrow X$, where $X$ is a real Banach space, $g: \mathbb{R}\rightarrow \mathbb{R}$.
Is it then true the Faa di Bruno's formula on $(f\circ g)^{(n})(x)$ ?
| https://mathoverflow.net/users/26191 | Faa di Bruno's formula for vector valued functions | For $g:B\_1\rightarrow B\_2$, $f:B\_2\rightarrow B\_3$, $B\_j$ Banach spaces , $g,f$ smooth,
we have for $n\ge 1$
$$
\frac{(f\circ g)^{(n)}}{n!}=\sum\_{n\_1+\dots+n\_r=n\atop
r\ge 1, n\_j\ge 1} \frac{f^{(r)}\circ g}{r!}\frac{g^{(n\_1)}}{n\_1!}\dots
\frac{g^{(n\_r)}}{n\_r!},\tag{FB}
$$
where the symmetric $n$-multilinea... | 4 | https://mathoverflow.net/users/21907 | 106339 | 61,431 |
https://mathoverflow.net/questions/106336 | 8 | Suppose I have an operad $P(n)$ in topological spaces in the sense of the book by Markl, Shnider and Stasheff, i.e. there is *no* space $P(0)$. In agreement with some of the answers to [this](https://mathoverflow.net/questions/29636/operad-terminology-operads-with-and-without-o0) question, I will call such an operad *... | https://mathoverflow.net/users/3995 | Delooping and unreduced operads | The answer to your first question is no: for example, take $X$ to be any connected pointed space, and regard $X$ as a nonunital commutative monoid by saying that the product of
any two points of $X$ is equal to the base point. This endows $X$ with the structure of
an algebra over any operad $\mathcal{O}$ with $\mathcal... | 15 | https://mathoverflow.net/users/7721 | 106345 | 61,435 |
https://mathoverflow.net/questions/106330 | 2 | Hi everyone,
let $A$ be a PID, let $\mathfrak{m}$ be a maximal ideal of $A[X]$.
I would like to find a direct simple proof of that fact that $\mathfrak{m}\cap A\neq 0$.
For the moment, I only know to prove it using the following fact : if $I$ is a prime ideal of $A[X]$ such that $I\cap A=0$ ($A$ PID), then one sh... | https://mathoverflow.net/users/26194 | Maximal ideals of $A[X],$ $A$ PID | As Inta's example shows, the problem as stated is not true. Here is a true restatement: Suppose that $A$ is a UFD with infinitely many primes. Then any maximal ideal $\mathfrak{m}$ of $A[x]$ has nontrivial intersection with $A$.
The morally right condition on $A$ is "for every nonzero $f \in A$, there is a nonzero p... | 6 | https://mathoverflow.net/users/297 | 106353 | 61,438 |
https://mathoverflow.net/questions/106349 | 5 | Suppose $f:X\rightarrow Y$ is a birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blowup sequence on $Y$ which dominates $f$ such that the preimage of D and all other subsequent preimages involved are snc divisors? Thanks for any comments or references.
| https://mathoverflow.net/users/3525 | Can a birational morphism between smooth varieties be dominated by smooth blowup sequences? | Here's an easy way to do it for morphisms in characteristic zero.
Suppose that $f : X \to Y$ is projective and thus the blowup of some ideal $I$ on $Y$, see Harthsorne Chapter II, Section 7 (if $f$ is not projective, make it projective by compactifying and using Chow's lemma).
The nice thing about setting things u... | 9 | https://mathoverflow.net/users/3521 | 106355 | 61,440 |
https://mathoverflow.net/questions/106113 | 5 | I'm in search of a set theory that satisfies the following requirements.
1. There is a universal set $V$ such that $\forall x(x \in V)$. So for example, $V \in V$.
2. Sets whose elements are 'large' exist. e.g. I want $\lbrace V,\emptyset\rbrace$ to be a well-defined set with cardinality $2$.
3. [Edit] Sets form a bo... | https://mathoverflow.net/users/26080 | In search of a set theory with specific properties | Here is a CW answer incorporating the answers in the comments. Both NF (New Foundations) and NFU (New Foundations with urelements, obtained from NF by weakening the axiom of extensionality) satisfy conditions (1) through (3).
| 5 | https://mathoverflow.net/users/1682 | 106365 | 61,445 |
https://mathoverflow.net/questions/106366 | 7 | Suppose $G$ is a semisimple $\mathbb{R}$-algebraic group with finite center, and suppose $G$ acts irreducibly on a vector space $V$. Suppose $U \subset V$ and $W \subset V$ are subspaces.
$\mathbf{Question 1}.$
Is it always possible to find $g \in G$ such that $g U$ and $W$ are in general position, i.e.
$$\dim(g U... | https://mathoverflow.net/users/16143 | Separating subspaces in an irreducible representation | Let $G=SL\_n(\mathbb R) \times SL\_m(\mathbb R)$. Let $V$ be the irreducible representation given by taking the defining representation $V\_n$ of the first tensor the defining representation $V\_m$ of the second. Let $U$ be a copy of of $V\_n$ inside $V\_n \otimes V\_m$ and let $W$ be a copy of $V\_m$. The intersection... | 8 | https://mathoverflow.net/users/18060 | 106369 | 61,447 |
https://mathoverflow.net/questions/106367 | 0 | Hello all,
I am dealing with some SDP optimization, and I come across the following problem.
The optimization problem is given by
\begin{aligned}
&\operatorname\*{min}\_{t\_1,\ldots,t\_m,X}\ \sum t\_i \\\
&\;\text{s.t.}\ \
\begin{bmatrix}
A(X)+B & e\_i \\\
e\_i^T & t\_i
\end{bmatrix} \succeq 0,\ \ \ i=1,\ldot... | https://mathoverflow.net/users/25265 | Is these two optimization problems share the same solution? | I don't see any reason why the two problems should have the same optimal solution.
Consider both SDPs with $A(X)=X$, $B=[25\ \ {-10};\ {-10}\ \ 20]$, and $C=1$, and $m=1$. Let the matrices be $2\times 2$.
Then, the first SDP has the optimal solution
\begin{equation\*}
t = .1502,\qquad X = \begin{bmatrix} 0.4257 & 0... | 2 | https://mathoverflow.net/users/8430 | 106375 | 61,450 |
https://mathoverflow.net/questions/106360 | 7 | I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d \times \mathbb{S}^1$. Is there a classification of these manifolds?
| https://mathoverflow.net/users/26207 | Examples of manifolds with effective circle actions? | As already pointed out, whenever you have a compact group $G$ acting on a manifold $M$, this action can be made isometric by constructing a metric on $M$ via a standard averaging process. In particular, if you have a (effective) circle action on $M$, you can find a metric $g$ on $M$ such that this action is isometric. ... | 8 | https://mathoverflow.net/users/15743 | 106376 | 61,451 |
https://mathoverflow.net/questions/106356 | 1 | Dear All,
I want to calculate the distance between two sets in which the maximum distance between the sets are minimized. Formally problem defined as,
$\displaystyle \min\_{a \in A} \max\_{b \in B}$ d(a,b)
This problem looks similar to Hausdorff distance, but the order of $min$ and $max$ are swapped. While a lot ... | https://mathoverflow.net/users/26203 | Distance calculation in metric space | This sort of thing is considered under the name "minimax facility location problem". See <http://en.wikipedia.org/wiki/Facilities_location>.
| 4 | https://mathoverflow.net/users/13650 | 106377 | 61,452 |
https://mathoverflow.net/questions/106332 | 4 | Is there any bounds for the norm of sub-stochastic matrix? (But it's not doubly stochastic matrix, I mean only the row sum is less than 1, while the column sum may not.
| https://mathoverflow.net/users/25747 | norm of (sub)stochastic matrix | A useful and easy to compute bound is given by the reasonably well-known relation (see e.g., [this Wikipedia section](http://en.wikipedia.org/wiki/Matrix_norm#Examples_of_norm_equivalence))
\begin{equation\*}
\|A\| \le \sqrt{\|A\|\_\infty \|A\|\_1}
\end{equation\*}
between the spectral norm, and the induced $1$ and $\i... | 4 | https://mathoverflow.net/users/8430 | 106379 | 61,454 |
https://mathoverflow.net/questions/106363 | 4 | The Kazhdan Lusztig map gives a correspondence between conjugacy classes of Weyl groups and nilpotent orbits. So, let $w$ be a conjugacy class and
$N$ be the nilpotent orbit it gets mapped to under the KL map. My question is
Is there a simple relationship between $n(w)$ and $dim(N)$ ? (alternatively, is this data o... | https://mathoverflow.net/users/26208 | Kazhdan Lusztig Map and conjugacy classes of Weyl groups. | The question itself is not precisely enough stated (or documented) to permit a real answer, I think. It's an old problem to specify some "natural" correspondence between conjugacy classes in a Weyl group and nilpotent orbits in a related semisimple complex Lie algebra (or perhaps unipotent classes in a corresponding al... | 8 | https://mathoverflow.net/users/4231 | 106380 | 61,455 |
https://mathoverflow.net/questions/106382 | 5 | A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do my best to answer questions raised in the comments. Also, please feel free to retag if you want.
"Let $\Sigma$ and $\L... | https://mathoverflow.net/users/11540 | Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid | If you can generate sufficiently many points $S$ to "delimit" the ellipsoid
defined by $\Sigma$, then the results of
this paper yield an approximation algorithm:
* Kumar & Yildirim, "Computing Minimum Volume Enclosing Axis-Aligned Ellipsoids,"
*Journal of Optimization Theory and Applications*, 136 (2), pp. 211 - 228 ... | 3 | https://mathoverflow.net/users/6094 | 106391 | 61,458 |
https://mathoverflow.net/questions/106335 | 9 | When talking about a single random variable, knowing only its distribution, the construction of a probability space is quite easy. Namely, let $(X,\mathscr A)$ be a measurable space and let $\mathsf Q$ be some probability measure over this space which we refer to as a distribution of some random variable. The usual def... | https://mathoverflow.net/users/11768 | Symmetries of probability distributions | This is a fascinating topic. One impressive systematic study of symmetries is in the book by Olav Kallenberg (2005)
* [Probabilistic Symmetries and Invariance Principles](https://rads.stackoverflow.com/amzn/click/com/0387251154)
In there, though, the measurable space has to have some structure to get the most out o... | 9 | https://mathoverflow.net/users/25326 | 106393 | 61,459 |
https://mathoverflow.net/questions/106400 | 2 | Is every countable Dedekind domain the ring of integers of some number field? I tried googling different keywords, but did not find anything. Does anyone know of research in this area?
| https://mathoverflow.net/users/17173 | Is every countable Dedekind domain the ring of integers of some number field? | Nope. $\mathbb F\_p[x]$, $\mathbb Q[x]$, and all other affine algebraic curves over countable fields, are countable Dedekind domains. None are the ring of integers of a number field.
| 11 | https://mathoverflow.net/users/18060 | 106401 | 61,462 |
https://mathoverflow.net/questions/106387 | 4 | Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius theorem.
Of course one can define a $C^1$ foliation of a $C^0$ sub-bundle $V$ as well. However the conditions of the Fr... | https://mathoverflow.net/users/17965 | Frobenius theorem with lesser regularity | Not an answer, but a pointer to the difficulties:
If you restrict yourself to just one dimensional distributions, you immediately see a problem.
Let $V$ be a continuous (non-vanishing) vector field on $\mathbb{R}^n$, $n > 1$, it clearly generates a $C^0$ sub-bundle of the tangent bundle. By Peano's existence theor... | 7 | https://mathoverflow.net/users/3948 | 106412 | 61,468 |
https://mathoverflow.net/questions/106409 | 7 | Here, $E\_1$ denotes the set of arithmetic formulas starting with a bounded existential quantifier, followed by a quantifier-free formula. Is there an $E\_1$-formula $\phi$ such that $\phi(n)$ holds
iff $n$ is prime? If yes, it is likely to be rather complicated to obtain, as this apparently implies that PRIMES is in P... | https://mathoverflow.net/users/15814 | Is there an $E_1$-definition of primality? | I don’t see how an $E\_1$ formula would give a polynomial-time algorithm. Primality testing has been known to be in NP long before AKS (due to Pratt), and this gives a definition of primes by a $\Sigma^b\_1$ formula (a bounded existential quantifier followed by a formula with logarithmically bounded quantifiers); if we... | 10 | https://mathoverflow.net/users/12705 | 106419 | 61,470 |
https://mathoverflow.net/questions/106416 | 7 | As is well known, the Hilbert scheme of two points on a given smooth projective variety X are blow up along diagonal of product of X and then quotient the Z2 action. It is smooth.
My question is whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth. If so, why and how to describe the ... | https://mathoverflow.net/users/26222 | Whether Hilbert schemes of 3 points on arbitrary smooth projective varieties are smooth | Yes, the Hilbert scheme of 3 points on a smooth variety is smooth. I don't know of a global description for the resulting Hilbert scheme, but here's the local reason this is true.
1. Every length 3 scheme is abstractly isomorphic to a subscheme of the plane.
2. For a zero dimensional subscheme $\text{Spec} A$ of a sm... | 8 | https://mathoverflow.net/users/4 | 106420 | 61,471 |
https://mathoverflow.net/questions/106429 | 5 | We take an ordered ring to be a structure of type $(+ - \times < 0\,\, 1)$ satisfying the usual axioms. If $A$ is an ordered ring then we say that an element $a$ of $A$ is infinitesimal if for all integers $n$ it holds that $-1<na<1$.
Let $T$ be the set of sentences that hold in every ordered ring that has no infinit... | https://mathoverflow.net/users/5229 | Is the first order theory of ordered rings without infinitesimals effectively enumerable? | The answer to the first question is no. First, let $\chi=\forall x>0\\,\exists y\\,(xy=1)$ and $T'=T+\chi$, so that $T'$ is the first-order theory of archimedean ordered fields. Let $\phi(x)$ be a formula defining $\mathbb Z$ in $\mathbb Q$, and let $\psi$ be the sentence “$\phi(x)$ defines a discretely ordered ring”. ... | 7 | https://mathoverflow.net/users/12705 | 106434 | 61,475 |
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