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https://mathoverflow.net/questions/133568 | 1 | Let $G$ a semisimple group and $B$ a Borel subgroup.
We denote by $Bun\_{G}$ the stack of G-bundles.
Is it true that a certain open subset $Bun\_{B,r}$ maps smoothly to $Bun\_{G}$?
My question comes from Lemma 14 .2.1 from
<http://arxiv.org/pdf/math/0611323.pdf>
but I'm not sure to understand well.
| https://mathoverflow.net/users/27398 | Smooth map to the stack of G-bundles | The short answer is the following: for a $G$-torsor $E\_G$ over $C$ and for the associated projective scheme $E\_{G,B} := E\_G/B$, a lift $E\_B$ of $E\_G$ to a $B$-torsor over $C$ is the same thing as a section $\sigma:C\to E\_{G,B}$ of the projection $\pi:E\_{G,B}\to C$. Via infinitesimal deformation theory of the Hil... | 5 | https://mathoverflow.net/users/13265 | 134149 | 74,087 |
https://mathoverflow.net/questions/134145 | 16 | What do people mean by the "cyclic cover trick"? I have found this expression a couple of times with no complete explanation, both talking about curves and surfaces...
| https://mathoverflow.net/users/4096 | what is the cyclic cover trick? | The "cyclic cover trick" can refer to more than one thing. One example is as follows. Let $L$ be an invertible sheaf on a smooth projective scheme such that some power $L^{\otimes d}$ has a global section $s$ whose zero scheme $D$ is a smooth Cartier divisor (all of these smoothness conditions are not strictly necessar... | 26 | https://mathoverflow.net/users/13265 | 134150 | 74,088 |
https://mathoverflow.net/questions/134151 | 3 | Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction.
Now consider a quasi-category(i.e., weak Kan complex) $\mathcal{C}$, in which every $k$-arrows are invertible if $k>1$. Given a fami... | https://mathoverflow.net/users/7341 | Localisation in a quasi-category | Some aspects and plenty of further pointers (mostly to Jacob Lurie's book, of course) are collected here:
<http://ncatlab.org/nlab/show/localization+of+an+(infinity,1)-category>
<http://ncatlab.org/nlab/show/reflective+sub-(infinity,1)-category>
| 3 | https://mathoverflow.net/users/381 | 134153 | 74,090 |
https://mathoverflow.net/questions/134117 | 3 | Assume we have a real-analytic function $f(x, y)>0$ in some neighborhood of 0. When does there exist a complex-analytic function $w(z)$ such that $|w(z)|=f(x,y)$ for $z=x+iy$.
One necessary condition is that $\Delta\ln f=0$. Is there anything else?
| https://mathoverflow.net/users/34984 | Which real analytic functions of two variables locally are magnitudes of complex-analytic functions | To formally complete the question: the answer is by Anthony Quas in the comment below.
| 1 | https://mathoverflow.net/users/34984 | 134154 | 74,091 |
https://mathoverflow.net/questions/134156 | 3 | Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This $\alpha$, by its powers, gives rise to a co-ordinatisation for the affine space $K=\mathbf{Q}^n$. Now what is known about s... | https://mathoverflow.net/users/22878 | Lower Degree Elements in an Algebraic Number Field | Write an arbitrary element of $K$ as $x = x\_0 + x\_1 \alpha + \dots + x\_{n-1} \alpha^{n-1}$ with $x\_0, \dots, x\_{n-1} \in \mathbb{Q}$. Then you can write $1, x, \dots, x^m$ as
$$
x^d = y\_{d,0} + y\_{d,1} \alpha + \dots + y\_{d,n-1} \alpha^{n-1},
$$
where the $y\_{d,k}$ are polynomial expressions in $x\_0, \dots, ... | 6 | https://mathoverflow.net/users/2698 | 134161 | 74,093 |
https://mathoverflow.net/questions/134056 | 4 | Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient conditions, and also in references!
| https://mathoverflow.net/users/2191 | When an exact embedding of abelian categories induces a full embedding of their derived categories? | It suffices to require that through any epimorphism in $A'$ from an object of $A'$ onto an object of $A$ some epimorphism in $A$ (onto the same object) would factorize; or the dual condition for monomorphisms. The natural generality is that of exact categories (or perhaps even wider, but at least so).
The standard re... | 6 | https://mathoverflow.net/users/2106 | 134172 | 74,097 |
https://mathoverflow.net/questions/134171 | 4 | Let $PB\_n$ be the pure braid group on $n$ strands. The group $PB\_n$ has every conceivable finiteness property. Also, it has a large abelianization. My question is whether the commutator subgroup $[PB\_n,PB\_n]$ is finitely-generated or not.
| https://mathoverflow.net/users/35094 | Finite generation of the commutator subgroup of the pure braid group | No, the group $PB\_n$, $n\ge 3$ has an [epimorphism](http://arxiv.org/pdf/1106.4602v1.pdf) onto the free group $F\_2$. Since the commutator subgroup $[F\_2,F\_2]$ is not finitely generated, the commutator subgroup $[PB\_n,PB\_n]$ is not finitely generated either.
| 12 | https://mathoverflow.net/users/nan | 134176 | 74,098 |
https://mathoverflow.net/questions/134174 | 5 | Suppose I have a collection of "elements" together with operations that satisfy the axioms for a commutative ring with identity --- except that these elements form not a set, but a proper class.
Must such a thing contain a maximal ideal (where an "ideal" is allowed to be proper-class-size)? Obviously, the usual Zorn'... | https://mathoverflow.net/users/10503 | Proper-class sized "ring" with no maximal ideals | The [axiom of global choice](http://en.wikipedia.org/wiki/Axiom_of_global_choice) is sufficient. More generally, any proper class ring whose elements can be enumerated along the ordinals will have a maximal ideal. The argument is the usual one...
Suppose $x\_\alpha$, $\alpha \in \mathrm{Ord}$, enumerates the elements... | 6 | https://mathoverflow.net/users/2000 | 134179 | 74,100 |
https://mathoverflow.net/questions/134177 | 0 | How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z\_p^\*$ contains an element $a$ of order $n$? No ideas at all...
| https://mathoverflow.net/users/35115 | For any n and some prime p there is an elemnet in Zp* of order n | $Z\_p^\*$ is of order $p-1$ so what you are really asking is for a prime in the arithmetic progression $n+1, 2n+1, 3n+1, \ldots$. This is true by Dirichlet's theorem, see <http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions>
| 1 | https://mathoverflow.net/users/28128 | 134180 | 74,101 |
https://mathoverflow.net/questions/134160 | 3 | Assume we have a smooth Riemannian metric $g$ on a small one-sided neighborhood $U$ of $0$ on the plane, say $U\_\epsilon=\lbrace(x, y): x^2+y^2<\epsilon, y\geq 0\rbrace$.
When does there exist a diffeomorphism $f$ of ( may be smaller) one-sided neighborhood of zero $U\_\delta$ such that the resulting Riemannian metr... | https://mathoverflow.net/users/34984 | Local boundary symmetrisation of Riemannian metrics by coordinate changes | In other words, you are asking the following:
>
> Assume we cut a surface by a curve $\gamma$ and take doubling of one piece.
> When the doubling has smooth metric?
>
>
>
Once you reformulate it this way, the answer is evident.
First note that $\gamma$ has to be geodesic otherwise the doubling has infinite c... | 5 | https://mathoverflow.net/users/1441 | 134189 | 74,106 |
https://mathoverflow.net/questions/134159 | 5 | My question concerns the name for determinants of Hankel-matrices $H = (s\_{i+j})\_{i,j = 0}^n$.
In the classical textbook of Shohat and Tamarkin (1943) "The Problem of Moments", these determinants are defined without a name (on page viii).
In several math articles, I found the name Hankel determinant. In the field... | https://mathoverflow.net/users/35109 | How did Hankel determinants get the name Hankel-Hadamard? | Some background is given in the appendix of this 1988 [paper](http://pra.aps.org/pdf/PRA/v37/i12/p4557_1) by Handy and Bessis, who apparently introduced this terminology: *Hankel-Hadamard matrix, Hankel-Hadamard determinant, Hankel-Hadamard positivity, Hankel-Hadamard inequality*. It refers to a class of matrices of th... | 2 | https://mathoverflow.net/users/11260 | 134193 | 74,107 |
https://mathoverflow.net/questions/62764 | 18 | The short version
-----------------
Here is an extremely natural hyperplane arrangement in $\mathbb{R}^n$, which I will call $R\_n$ for *resonance arrangement*.
Let $x\_i$ be the standard coordinates on $\mathbb{R}^n$. For each nonempty $I\subseteq [n]=\{1,\dots,n\}$, define the hyperplane $H\_I$ to be the hyperpla... | https://mathoverflow.net/users/1102 | A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable? | Here's what I know about this arrangement.
Regarding the number of chambers in this arrangment, Zuev obtained the lower bound $2^{(1-o(1))n^2}$. The proof uses Zaslavsky's theorem and a difficult estimate due to Odlyzko.
<http://www.doiserbia.nb.rs/img/doi/0350-1302/2007/0350-13020796129K.pdf> describes an improve... | 8 | https://mathoverflow.net/users/11541 | 134194 | 74,108 |
https://mathoverflow.net/questions/134197 | 3 | Given $n\in\mathbb{N}$, consider the $\ell\_2$ unit sphere $\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ equipped with its "geodesic" metric $\rho\_n$ defined as:
$\rho\_n(x,y)=\arccos \Big(\langle x,y\rangle\Big)$,
where $\langle\cdot,\cdot\rangle$ is the usual Euclidean inner product in $\mathbb{R}^{n+1}$. I have seen ... | https://mathoverflow.net/users/nan | Hausdorff measure on the sphere is well defined? | This [MSE question](https://math.stackexchange.com/questions/107099/volume-form-and-hausdorff-measure) contains a slick proof of the fact that
>
> If $i:M^n\hookrightarrow \mathbb{R}^{n+1}$ is an embedded submanifold, then writing $g=i^\*\delta$ as the induced metric on $M$, the volume form $vol\_g$ and restriction... | 6 | https://mathoverflow.net/users/1540 | 134200 | 74,110 |
https://mathoverflow.net/questions/134201 | 3 | My earlier related question
[Lower Degree Elements in an Algebraic Number Field](https://mathoverflow.net/questions/134156/lower-degree-elements-in-an-algebraic-number-field)
has been given a clean answer for the first part. My present question is below:
Take a number field $K=\mathbf{Q}(\alpha)$ of degree $n$ admit... | https://mathoverflow.net/users/22878 | Polynomials giving Lower Degree Elements in an Algebraic Number Field | Let $\beta=2^{1/4}$, and let $\alpha=\beta^3+\beta^2+1$. Then ${\bf Q}(\alpha)={\bf Q}(\beta)$ has degree 4 over the rationals. Then $\alpha^2=2\beta^3+4\sqrt2+4\beta+3$, and there is no quadratic polynomial $f$ with rational coefficients such that $f(\alpha)$ has degree 2 over the rationals.
For another example, le... | 5 | https://mathoverflow.net/users/3684 | 134206 | 74,111 |
https://mathoverflow.net/questions/134140 | 3 | I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset $P$ as naturally turns up in set theory, then every constant presheaf is separated. My naive extrapolation is that this i... | https://mathoverflow.net/users/4177 | For which sites are all constant presheaves separated? | OK, I've found the answer in the Elephant. Given the map of sites $p\colon S\to \ast$, we get a functor of toposes $Set = Pre(\ast) \to Pre(S)$ by precomposition with $p$, which lands in separated presheaves precisely when $p$ preserves covers. Since $\ast$ has only the cover $id\colon \ast \to \ast$, then we see that ... | 3 | https://mathoverflow.net/users/4177 | 134209 | 74,113 |
https://mathoverflow.net/questions/134203 | 6 | Let $H$ be a separable infinite dimensional Hilbert space. Denote by $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathcal{K}(H)$ the ideal of compact operators. When endowed with the strong (or weak) operator topology, is $\mathcal{K}(H)$ Borel in $\mathcal{B}(H)$?
Remarks:
* It is well known that... | https://mathoverflow.net/users/16107 | Is the ideal of compact operators strongly Borel? | Yes.
This can be deduced from the argument given for Corollary 3.2 in G. A. Edgar, [Measurability in a Banach space](http://dx.doi.org/10.1512/iumj.1977.26.26053), Indiana Univ. Math. J. 26 (1977), 663-677, [MR542944](http://www.ams.org/mathscinet-getitem?mr=542944).
The proof (attributed to Talagrand) is easy, so ... | 7 | https://mathoverflow.net/users/29555 | 134210 | 74,114 |
https://mathoverflow.net/questions/134211 | 3 | This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We know that $\mathfrak{t}\subset \mathfrak{g}$. The Lie group $G$ acts on $\mathfrak{g}$ by by the adjoint action.
For any... | https://mathoverflow.net/users/24965 | The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular | Yes, it's still true that $G\cdot X\cap\mathfrak t=W\cdot X$.
The inclusion $\supset$ is clear. Conversely, suppose $g\cdot X\in\mathfrak t$. Since $\mathfrak t$ consists exactly of all $T$-fixed points in $\mathfrak g$, it follows that $t\cdot g\cdot X=g\cdot X$ for all $t\in T$. Hence $g^{-1}Tg$ is contained in the... | 3 | https://mathoverflow.net/users/19276 | 134219 | 74,116 |
https://mathoverflow.net/questions/134227 | 1 | For a natural number $n$ we denote by $\pi(n)$, for the set of prime divisors of $n$.
Let $q=p^\alpha$ and $q'=r^\beta$ where $r$ and $p$ are odd prime numbers and $\alpha,\beta$ are natural numbers.
Let $p\mid (q'-1)$ and $r\mid (q-1)$. I need to prove that this is impossible that $\pi(q-1)\cup \pi(p)=\pi(q'-1)\cu... | https://mathoverflow.net/users/31045 | On the set of divisors of $q-1$ and $q'-1$ | $q=27=3^3$, $q'=13=13^1$. $3\mid13-1$, $13\mid27-1$. $$\pi(26)\cup3=\pi(12)\cup13=\{2,3,13\}$$
| 3 | https://mathoverflow.net/users/3684 | 134229 | 74,120 |
https://mathoverflow.net/questions/134205 | 8 | Let $C\_n=\{0,1\}^n$ be the hypercube and denote by $\operatorname{G}(N,p)$ the Erdos-Renyi random graph (edges appear independently with probability $p$). Assume that $N=2^n$. Could one pin down $p=p(n)$ such that we can embed the hypercube into $\operatorname{G}(N,p)$ and vice versa? Is there anything known about sim... | https://mathoverflow.net/users/24494 | Embedding a hypercube into the Erdos-Renyi random graph | $C\_n$ has $N n/2$ edges, while $G(N,p)$ has approximately
$p N^2/2$ edges. So you're almost surely not going to embed $G(N,p)$ into $C\_n$.
For any of the $N!$ possible maps of $C\_n$ onto the vertices of $G$, the probability that the edges of $C\_n$ all appear is $p^{N n/2}$. The expected number of embeddings is t... | 10 | https://mathoverflow.net/users/13650 | 134235 | 74,122 |
https://mathoverflow.net/questions/134148 | 8 | In Aubin's book (nonlinear problems in Riemannian Geometry), starting from p. 106, it is shown that a Green's function of a compact manifold without boundary satisfies
$$G(P,Q) \leq k \rho(P,Q)^{2-n},$$
where $\rho(P,Q)$ denotes the geodesic distance between $P$ and $Q$.
a) I do not understand their proof. In fa... | https://mathoverflow.net/users/34942 | Aubin's book - construction of Green's function on compact manifold | Your have two nice texts of Frederic Robert (a descendant of Aubin) about the construction and estimate of Green function
one in french: <http://www.iecl.univ-lorraine.fr/~Frederic.Robert/ConstrucGreen.pdf>
an other in english where you will find an answer to (b) when assuming Neumann boundary condition: <http://ww... | 9 | https://mathoverflow.net/users/9253 | 134236 | 74,123 |
https://mathoverflow.net/questions/134241 | 2 | Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site.
Let $S$ be some orientable surface obtained by removing finitely many points from a closed surface.
The Riemann moduli space for $S$ is just the set of isomorphism classes of co... | https://mathoverflow.net/users/15242 | Elementary question about Isotopy (in the definition of a Teichmuller space) | 1. D.B.A. Epstein proved in his paper "Curves on 2-manifolds and isotopies", Acta Math. 1966, that homotopy implies isotopy for homeomorphisms of surfaces of finite type.
2. There are conformal automorphisms of Riemann surfaces which are not isotopic to the identity and there is the trivial automorphism, which is the i... | 4 | https://mathoverflow.net/users/21684 | 134247 | 74,125 |
https://mathoverflow.net/questions/134250 | 2 | Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on condition $\tau$ is invertible) number $n$. It easily follows from Jordan theorem, that *every* eigenvalue of $\tau^n$ has... | https://mathoverflow.net/users/13480 | Eigenvalues of powers of linear mappings | Perhaps you can use:
$$\det(\lambda - \tau^n) = \det((-1)^n\prod\_{\omega\_i:\text{ nth roots of } \lambda} \omega\_i - \tau)= (-1)^n\prod\_{\omega\_i:\text{ nth roots of } \lambda} \det(\omega\_i - \tau) $$
by the multiplicativity of the determinant. This righthand side is only zero if for an $i$
$$\det(\omega\_i - \t... | 3 | https://mathoverflow.net/users/33927 | 134252 | 74,127 |
https://mathoverflow.net/questions/134213 | 2 | Let $p\_1, p\_2, \ldots$, be the power sum symmetric functions. Let $p\_n^\* = n \frac{\partial}{\partial p\_n}$. Then $$ p\_n^\* p\_m - p\_m p\_n^\* = \delta\_{m, n} 1. $$
Where could I find this result in some books or papers? Thank you very much.
| https://mathoverflow.net/users/11877 | References request: representations of Heisenberg algebra. | In Nakajima's annals paper on the homlogy of Hilbert Schemes and the representation theorx of the Heisenberg algebra.
This is also in his book on Hilbert schemes of points in chapter 8.
| 2 | https://mathoverflow.net/users/10400 | 134258 | 74,131 |
https://mathoverflow.net/questions/134259 | 13 | Due to Andreas Blass's answer to my question "Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?", i have now investigated real closed fields (RCF), because i had never studied them before. My background is logic-based cognitive robotics, not model theory or foundations of mathe... | https://mathoverflow.net/users/35046 | Is the first-order theory (with =) of real numbers with addition and multiplication complete and decidable? | Yes, all those assertions are expressible in the language of real closed fields, and all such assertions are decidable by Tarski's theorem. You can refer to addition, subtraction, multiplication, division, etc., plus the order; you can refer to any specific rational number (such as .223 etc.) because you have $1$ in th... | 23 | https://mathoverflow.net/users/1946 | 134260 | 74,132 |
https://mathoverflow.net/questions/134261 | 2 | Hello,
let's consider a compact and connected Riemannian manifold with the Schrödinger Operator $L=-\Delta +V:dom(H)\subset L^2(M)\rightarrow L^2(M)$ whereas $dom(L):=\lbrace f\in C^{\infty}(M,\mathbb{R}) \vert f\_{\vert \partial M}=0 \rbrace$ and $V\in L^2(M)$ bounded.
The spectrum of the Friedrichs extension of $... | https://mathoverflow.net/users/21870 | Eigenfunctions of Schrödinger Operators on the boundary | In addition to Michael Renardy's apt comment about the physical sense of the situation, one can also see the boundary vanishing from the characterization of the Friedrichs extension. Namely, by construction, the domain of the Friedrichs extension is inside the +1 Levi-Sobolev space closure of the original domain. With ... | 1 | https://mathoverflow.net/users/15629 | 134266 | 74,136 |
https://mathoverflow.net/questions/134265 | 7 | Hello,
I am looking for a reference to something like that: if $f\colon X\to Y$ is a finite (i.e., proper with finite fibers) morphism of reduced and irreducible normal (or at least smooth) complex spaces such that $f$ is 1-1 over $U\subset Y$, where $U$ is open and dense, then $f$ is an isomorphism.
Could somebody... | https://mathoverflow.net/users/29992 | Zariski's main theorem in the complex analytic category | One reference is Proposition 14.7 in Remmert's paper *Local Theory of complex analytic spaces*, Several complex variable VII, Encyclopaedia of Math. Sci. vol **74**. For the reader's convenience I will restate the result here.
Recall that a finite, surjective, holomorphic map $\eta \colon X \to Y$ between complex spa... | 6 | https://mathoverflow.net/users/7460 | 134274 | 74,141 |
https://mathoverflow.net/questions/133383 | 6 | This is to be confronted with Joseph Gubeladze' paper : "Toric varieties with huge Grothendieck group" !
| https://mathoverflow.net/users/34304 | Is the algebraic Grothendieck group of a weighted projective space finitely generated ? | Adam Massey showed that K°(P(1,...,1,q)) = K°(P(1,...,1)).On the other hand P(1,...,1,q) is the cone with wertex (0, ...,0,1) which projects the Veronese variety Vq. Who knows any other particular nice geometrical exemples (small dimensions) of weighted projective spaces whith finitely generated algebraic GROTHENDIECK ... | 1 | https://mathoverflow.net/users/34304 | 134281 | 74,145 |
https://mathoverflow.net/questions/134142 | 17 | If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X\_t$ and $\Fl^Y\_t$ their local flows, let $[\Fl^X\_t,\Fl^Y\_t]:= \Fl^Y\_{-t}\Fl^X\_{-t}\Fl^Y\_t\Fl^X\_t$ denote the group commutator of the flows. Then
$$
\partial\_t|\_0 [\Fl^X\_{t},\Fl^Y\_{t}] =0, \qquad
\frac12 \partial\_t^2|\_0 [\Fl^X\_{t},\Fl^Y\_{t... | https://mathoverflow.net/users/26935 | Where did Sophus Lie write the group commutator for two one parameter groups | Looking around on the internet, I found an English translation of Lie's 1891 paper *Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen. I.* (I.e., The foundations of the theory of infinite continuous transformation groups - I) at the location
<http://neo-classical-physics.info/upl... | 16 | https://mathoverflow.net/users/13972 | 134282 | 74,146 |
https://mathoverflow.net/questions/134190 | 1 | Hello all,
I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand.
Specifically, I am attempting to estimate the amount of variance in my Monte Carlo estimates for the mean of my function $g$. The first and simplest approach I've seen mentioned... | https://mathoverflow.net/users/20652 | Understanding the rationale behind "batch means" estimation | The variance of the mean of $n$ random variables is
$\mbox{Var}(\bar{x})=\mbox{Var}(\sum\_{i=1}^{n} x\_{i}/n)$
$\mbox{Var}(\bar{x})=\sum\_{i=1}^{n} (1/n)^{2} \mbox{Var}(x\_{i})$
$\mbox{Var}(\bar{x})=n(1/n)^{2} \mbox{Var}(x\_{i})$
$\mbox{Var}(\bar{x})=(1/n) \mbox{Var}(x\_{i})$
Thus
$\mbox{Var}(x\_{i})=n\mbo... | 2 | https://mathoverflow.net/users/9022 | 134288 | 74,149 |
https://mathoverflow.net/questions/134291 | 0 | Does anyone have a reference for the following result? I am pretty sure that it is true, and should not be hard to prove, but i would surprise me if it is not already proven in many places:
>
> Let $G$ be a locally compact Abelian group and $U$ an open precompact set in $G$. Then for all $f \in C\_C(G)$ we can find... | https://mathoverflow.net/users/11552 | Can any compactly supported continuous function be written as a linear combination of functions with small support | Regarding partitions of unity: Every locally compact group is paracompact, thus it is normal, thus there exist partitions of unity. But you do not need this argument: For general locally compact groups and any compact subset $A$ and any finite precompact open cover of $A$ there exists a continuous function with the val... | 0 | https://mathoverflow.net/users/33842 | 134296 | 74,151 |
https://mathoverflow.net/questions/134295 | 3 | This question want to be a follow up of the [following question](https://mathoverflow.net/questions/66641/relation-between-monads-operads-and-algebraic-theories).
In that thread I was interested in understanding relation between various presentation of algebraic theories. In particular in [Eduardo Pareja Tobes](https:/... | https://mathoverflow.net/users/14969 | Further relation between monads and theories | As noted by Zhen Lin, monads (on $\mathbf{Set}$) should be thought of as morally equivalent to *algebraic* theories of fairly general type. By an algebraic theory I mean a single-sorted theory specified by operations and universally quantified equations between terms. The operations are allowed to be infinitary. Certai... | 7 | https://mathoverflow.net/users/2926 | 134307 | 74,153 |
https://mathoverflow.net/questions/130525 | 8 |
>
> QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces?
>
>
>
Recall that a space is *perfect* if every closed set is a $G\_\delta$ set (that is, a countable intersection of open sets... | https://mathoverflow.net/users/11647 | On the cardinality of perfect spaces with the countable chain condition | The reason why my Hausdorff non-regular counterexample above is perfect is that it is a countable union of closed discrete sets. Now, Uspenskij constructed $\sigma$-closed discrete ccc regular spaces of arbitrarily large cardinality. So that answers my question in the negative.
See this paper: <http://dml.cz/dmlcz/1... | 5 | https://mathoverflow.net/users/11647 | 134320 | 74,159 |
https://mathoverflow.net/questions/134321 | 12 | Let PD stand for projective determinacy, and consider the two claims:
(1) For each n=1,2,..., Con(ZFC+PD) implies Con(ZFC + there are n Woodin cardinals)
(2) Con(ZFC+PD) implies Con(ZFC + there are infinitely many Woodin cardinals).
Clearly (2) implies (1).
As far as I know, (1) is true, and I have seen casual ... | https://mathoverflow.net/users/33399 | Consistency strength of projective determinacy (PD) | (2) is false. (1) is true. In fact, $\mathsf{ZFC}+\mathsf{PD}$ (with $\mathsf{PD}$ stated as an axiom schema) implies that for every $n$ there is an inner model of $\mathsf{ZFC}$ with $n$ Woodin cardinals. Also, $\mathsf{ZFC}+\mathsf{PD}$ is equiconsistent with the theory that result from adding to $\mathsf{ZFC}$ the a... | 19 | https://mathoverflow.net/users/6085 | 134333 | 74,166 |
https://mathoverflow.net/questions/134303 | -1 | Can a Busemman space be CAT(1)?
| https://mathoverflow.net/users/35166 | CAT(K) and Busemann | Answering a sensible question appeared in comments: If a geodesic space is Busemann and CAT(1), then it must be CAT(0).
Indeed, CAT(1) implies that the space has well-defined metric angles between geodesic segments. And if a space is Busemann and has well-defined angles, then it is CAT(0).
Indeed, let $X$ be the sp... | 9 | https://mathoverflow.net/users/4354 | 134340 | 74,168 |
https://mathoverflow.net/questions/134339 | 3 | My question is about a fairly artificial preorder on functions from $\omega$ to $\omega$, which for simplicity I'll call "reals."
For $r, s\in {}^\omega\omega$, write $r\le\_E^\*s$ if for each real $f$ computable from $r$, there is some real $g$ computable from $s$ such that $g$ escapes $f$: $$ \forall n\exists m>n (... | https://mathoverflow.net/users/8133 | A computability-theoretic preorder on reals | This order has only one element.
Indeed, let $h$ be any function. Let $G$ be Cohen generic relative to $h$ (something like 3-generic relative to $h$ will do). Then
$$
h \le^\*\_E G \le^\*\_E 0
$$
| 6 | https://mathoverflow.net/users/4600 | 134341 | 74,169 |
https://mathoverflow.net/questions/134287 | 3 | I attempted to get some information on this from MSE, but did not even receive a comment, so i'm trying my luck here:
Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K\_d/\mathbb Q$ of degree d ramified only at p. It is contained in the cyclot... | https://mathoverflow.net/users/35157 | Gaussian Periods | The constant term of the minimal polynomial for $\omega$ is (up to sign) the product of the conjugates of $\omega$ over the rationals. I studied it in my papers, A combinatorial problem in finite fields, I, Pacific J. Math. 82 (1979), no. 1, 179–187, MR0549842 (80i:05010) and A combinatorial problem in finite fields, I... | 6 | https://mathoverflow.net/users/3684 | 134342 | 74,170 |
https://mathoverflow.net/questions/134304 | 4 | Hello there,
This is probably very easy but I can't find an argument.
Call a function $f: R^n \to R$ monotone increasing if $x\_i \le y\_i$ for each $1 \le i \le n$ implies $f(x) \le f(y)$.
I'd like to show that such a function is measurable; I'd be very surprised if this is not the case.
If $n=1$ it's ok, for ... | https://mathoverflow.net/users/33563 | Monotone functions are measurable | We use induction on $n$. For $n=1$ this is trivial. Otherwise, suppose $f : \mathbb{R}^{n+1} \to \mathbb{R}$ is monotone increasing, fix $a \in \mathbb{R}$, and define $g : \mathbb{R}^n \to \mathbb{R}$ by $g(x) = \inf\{t \in \mathbb{R} : f(x,t) \ge a\}$. Then $g$ is monotone decreasing. By the induction hypothesis, $g$... | 7 | https://mathoverflow.net/users/4832 | 134344 | 74,172 |
https://mathoverflow.net/questions/134354 | 3 | Can a positive curvature smooth metric on the disk be $C^0$ approximated by smooth flat metrics?
Similar question under extra assumption: the same conformal class.
If there are positive examples in both cases, then are there some relatively weak extra-assumptions that the answer is no? Also I am curious how the ans... | https://mathoverflow.net/users/34984 | Continuous approximation of smooth metrics. | No, because too many geometric quantities are continuous in the $C^0$ topology: e.g. length of curves, volume.
Take the standard metric on the radius $1$ sphere, remove a small open disk near the south pole, and push forward stereographically the metric to a metric on the disk. Then the length of the boundary in this... | 2 | https://mathoverflow.net/users/4961 | 134360 | 74,179 |
https://mathoverflow.net/questions/134369 | 4 | Let $X$ be a complex manifold and $L$ a line bundle on it. Define $Y:=\mathbb{P}(L\oplus \mathcal{O}\_X)$ be the projective bundle over $X$. Here is a statement I don't understand:
>
> The summands $L$ and $\mathcal{O}\_X$ respectively determine divisors on $Y$, each of which is isomorphic to $X$.
>
>
>
How ca... | https://mathoverflow.net/users/50973 | A question on a projective bundle $\mathbb{P}(L\oplus \mathcal{O}_X)$ | This is a rather standard fact.
Let $\mathscr{E}$ be a rank $2$ locally free sheaf on $X$ and let $\pi \colon \mathbb{P}(\mathscr{E}) \to X$ be the corresponding $\mathbb{P}^1$-bundle. Then there is a natural correspondence between sections $\sigma \colon X \to \mathbb{P}(\mathscr{E})$ of $\pi$ and suriections $\mat... | 5 | https://mathoverflow.net/users/7460 | 134370 | 74,181 |
https://mathoverflow.net/questions/134366 | 5 | Consider the sequence $\lbrace \frac{\phi(i)}{i}\rbrace\_{i=1}^\infty$ where $\phi$ is the Euler's function. The Sequence is clearly dense in $[0,1]$. What can be said about the limsup of its average sequence ?? I mean the sequence $\frac 1n\sum\_{i=1}^n \frac{\phi(i)}{i}$. Its value or an upper bound would be helpful.... | https://mathoverflow.net/users/nan | A Sequence of Real numbers | It is well known that $\sum\_{k\le x} \phi(k)=\frac{3}{\pi^2}x^2+O(x\log(x))$. In a similar way we obtain
$$
\sum\_{k\le x} \frac{\phi(k)}{k}=\frac{6}{\pi^2}x+O(\log(x)),
$$
by using the Moebius function, i.e., $\sum\_{k\le x} \frac{\phi(k)}{k}=\sum\_{k\le x}\sum\_{d\mid k}\frac{\mu(d)}{d}=\sum\_{d\le x}\frac{\mu(d)}{d... | 7 | https://mathoverflow.net/users/32332 | 134374 | 74,184 |
https://mathoverflow.net/questions/133977 | 4 | After a quick internet search I found no method for incremental entropy computation.
### Question 1
Let $\{x\_i\}\_{i=1}^n$ and $\{x\_i\}\_{i=1+n}^{n+m}$ be two samples and let $S\_i^j:=\sum\_{k=i}^j x\_k$. Define entropy $$H\_i^j:=-\sum\_{k=i}^j\frac{x\_k}{S\_i^j}\log\_2\frac{x\_k}{S\_i^j}.$$ Is there an algorith... | https://mathoverflow.net/users/34944 | Incremental entropy computation | Using Andre's identitiy, I derived more general formulas (and algorithms). I made the note with derivations [available on arXiv](http://arxiv.org/abs/1403.6348). Comments are more than welcome.
For convenience I also give proofs of (some of) the results here.
**Lemma.** Let $\{x\_i\}\_{i=1}^n$ be a sample of posit... | 2 | https://mathoverflow.net/users/34944 | 134376 | 74,186 |
https://mathoverflow.net/questions/134119 | 2 | Recall that a complex manifold is Stein if it is holomorphically convex and separable. If we assume holomorphically convex alone, then there is Cartan-Remmert reduction to say how far it is from being Stein. If we assume holomorphic separation alone, I learnt from Th 9.9(b) in p41 of Demailly's lecture notes [Here](htt... | https://mathoverflow.net/users/12904 | Holomorphic separation and the existence of strictly plurisubharmonic functions | I realized that my comment was misleading, so I prefer to put a complete answer.
In Demailly's proof at some point you end up with a countable open cover $\{V\_j\}$ of your complex manifold $X$ and a family of non-negative smooth psh functions $\{v\_j\}$ such that $v\_j$ is strictly psh on $V\_j$. Then you want to s... | 1 | https://mathoverflow.net/users/9871 | 134377 | 74,187 |
https://mathoverflow.net/questions/134382 | 6 | All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed *projective* tensor product
$X \tilde{\otimes}\_ \pi Y$ has a representation as a series $\sum\limits\_{n=1}^\infty x\_n \otimes y\_n$ which converges with respect to the $\pi$-norm (in an a... | https://mathoverflow.net/users/21051 | series representation in injective tensor products | Given $T: Y\to X$ of finite rank, let $(x\_i,x\_i^\*)$ be an Auerbach basis (meaning they are biorthogonal and both the vectors and the biorthogonal functionals all have norm one) for the range of $T$. For $1\le i \le n$ let $T\_i y = n^{-1} x\_i^\*(Ty) x\_i$. For $j = kn +i$, $1 \le k \le n$ and $1\le i \le n$, let $T... | 5 | https://mathoverflow.net/users/2554 | 134391 | 74,194 |
https://mathoverflow.net/questions/131554 | 7 | Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of **almost-orthogonal sets** asks, for each $\epsilon> 0$, for the construction (or at the very least, bounds on the cardinality) of a **maximal** s... | https://mathoverflow.net/users/31084 | Almost orthogonal vectors in subsets of euclidean space | Nearly two months after I asked the above question, I have come across a paper where Noga Alon has given an explicit result in [Problems and results in Extremal Combinatorics, Part I](http://www.tau.ac.il/~nogaa/PDFS/extremal1.pdf) in section 9, *On ranks of perturbations of identity matrices*. He says
**Lemma 9.1.**... | 4 | https://mathoverflow.net/users/31084 | 134408 | 74,199 |
https://mathoverflow.net/questions/134418 | 1 | Does anyone have a reference for the fact that the uniformity of pointwise convergence on real functions of $[0,1]$ (that is, the uniformity generated by the sets $\lbrace (f,g) : |f(x) - g(x)| < \epsilon \rbrace$ for every $x\in[0,1]$, $\epsilon>0$) does not have a countable base?
It does not appear that either Bour... | https://mathoverflow.net/users/34823 | Reference: uniformity of pointwise convergence has no countable base | If it is not in bourbaki, I don't have a reference in mind, but here is a proof :
Let $(B\_n)$ for $n \in \mathbb{N}$ be any countable family of entourage.
Then because each $B\_n$ is an entourage one has that (for each $n$) there exists a finite set of point $A\_n$ of $[0,1]$ and an $\epsilon\_n$ such that if for ... | 1 | https://mathoverflow.net/users/22131 | 134421 | 74,202 |
https://mathoverflow.net/questions/134401 | 3 | My current research requires some knowledge on the eigenvectors of elements (of infinite order) of Coxeter groups view as reflections in their geometric representation.
After some reading, my impression is that many has been done for the spectrum of "Coxeter elements" or "Coxeter transformations", which are (if I un... | https://mathoverflow.net/users/20595 | Eigenvalues for elements of (infinite) Coxeter groups | The eigenvalues of elements of infinite order are certainly not trivial to study,
and as far as I know little has been determined about them.
Keep in mind that arbitrary infinite Coxeter groups are quite varied and hard to study systematically beyond the most basic theory.
Maybe I can clarify at least what you've... | 5 | https://mathoverflow.net/users/4231 | 134424 | 74,203 |
https://mathoverflow.net/questions/134420 | 4 | I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My questions have to do with constructing a general hyperbolic punctured torus. I would like to think of it as the quotient... | https://mathoverflow.net/users/4325 | Hyperbolic structures on once punctured tori | Question 1: I would call them "rectangular", because they are conformally equivalent to gluing opposite sides of a Euclidean rectangle by Euclidean translation and then removing the point obtained by gluing the four corners.
Question 2: I believe this is true. The way to prove it would be to cut the quadrilateral int... | 7 | https://mathoverflow.net/users/20787 | 134428 | 74,204 |
https://mathoverflow.net/questions/134433 | 5 | Let $E\to C$ be a rank $2$, degree $2g-2$, holomorphic vector bundle over a curve of genus $g$.
By Riemann-Roch theorem,
$$H^0(E)-H^1(E)= \deg(E)+2.(1-g)=0. $$
Question: For which $g$, there is such $E$ with $H^0(E)=0$ (and thus $H^1(E)=0$ as well)?
| https://mathoverflow.net/users/5259 | Looking for a special rank 2 vector bundle | Every curve admits a degree g-1 line bundle with $h^0=h^1=0$ -- in fact a generic degree g-1 line bundle has this property, since the space of degree g-1 divisors is g-1 dimensional, but the space of line bundles is g dimensional. Taking the direct sum of two such line bundles will give you a bundle of the type you are... | 5 | https://mathoverflow.net/users/14848 | 134437 | 74,206 |
https://mathoverflow.net/questions/134409 | 8 | For a finite set $X$ let $X^\* $ be the set of all non-empty proper subsets of $X$. Let $f : X^\* \longrightarrow X^\* $ be an increasing function such that for some $A \in X^\* $, $|f(A)| \not = |A|$. It is true that $f$ must have a fixed point ?
(By increasing I mean when $A\subseteq B$ then $f(A)\subseteq f(B)$ )... | https://mathoverflow.net/users/nan | Increasing functions on the set of all non-empty proper subsets of a finite set | Take $A$ maximal among sets such that $|f(A)|>|A|$. For each one-element set $\{x\}$, $f(\{x\})\subseteq f(A)$, because otherwise $|f(A \cup \{x\})| > |A \cup \{x\}|$. Draw a directed graph with one arrow leaving each element $x\in X$ pointing to an element $y\in X$ with $y \in f(\{x\})$. Let $T$ be the set of points i... | 6 | https://mathoverflow.net/users/18060 | 134441 | 74,209 |
https://mathoverflow.net/questions/109817 | 21 | A standard Carleman-type estimate is of the form
$$
\sum\_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C\_{0}^{\infty},
$$
where $\phi$ is some weight function. This formula turns to be very useful in the study of uniqueness of Cauchy problems,... | https://mathoverflow.net/users/23078 | What's the idea behind Carleman estimates? | The weight function $\phi$ indeed plays an (I would even say *the*) essential role in Carleman estimates, and constructing one such that the estimate holds is one of the major challenges in proving and applying Carleman estimates. I will give a very informal overview, and refer to the references below for the technical... | 22 | https://mathoverflow.net/users/30516 | 134473 | 74,215 |
https://mathoverflow.net/questions/134475 | 4 | This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, see e.g. <http://www.cut-the-knot.org/proofs/3ColorsBichromaticLines.shtml#solution>
However, there are more interesting... | https://mathoverflow.net/users/955 | What are interesting 3-colorings of the plane without rainbow lines? | There's a definitive answer given by Hales and Straus in <http://msp.org/pjm/1982/99-1/pjm-v99-n1-p03-s.pdf> (but note their comments re priority in the introduction). The brief summary is that the colorings you want for Desarguesian planes correspond to valuations of the underlying field.
| 3 | https://mathoverflow.net/users/1266 | 134489 | 74,218 |
https://mathoverflow.net/questions/134384 | 2 | I would like to understand some trivialities about log-structures. Given a log-scheme $(X,M\_X)$ the log-structure $M\_X$ is defined via push-out. Are there stupid examples in which this push-out is actually a direct sum $\mathbb{G}\_m\oplus P$ for some monoid $P$?
Furthermore assume we have a ring $R$, an element $... | https://mathoverflow.net/users/35184 | trivialities on log-structures | For the first question, given a sharp monoid $P$ (i.e., the group of units of $P$ is the
zero group), there is a log structure on any scheme $X$ given by $M\_X={\mathcal O}\_X^\*\oplus P$ with the structure map given by $(f,p)\mapsto f$ if $p=0$ and $(f,p)\mapsto 0$ if $p\not =0$. This comes from the prelog structure $... | 1 | https://mathoverflow.net/users/23917 | 134491 | 74,219 |
https://mathoverflow.net/questions/134426 | 2 | Let $X$ be a Riemann Surface and $K$ a compact subset of $X$. Every holomorphic function in $K$ be uniformly approximable on $K$ by holomorphic functions on $X$ if $X-K$ have no connected component with compact closure in $X$.
Is the conversely also true?
| https://mathoverflow.net/users/35199 | Approximation Runge's Theorem | Yes .Suppose U is a relatively compact component of the complement of K and p a point in U.The frontier of U lies in K .By Weierstrass Let g be a holomorphic function that vanishes at
p and nowhere else on X and f the reciprocal of g.By assumption we can find a sequence of holomorphic functions on X converging uniforml... | 1 | https://mathoverflow.net/users/4696 | 134500 | 74,220 |
https://mathoverflow.net/questions/134499 | 1 | is there any example of function which is computable on some set and uncomputable on other set? That is for example function f(n) which is computable on some (finite, or for example for even numbers) set A of N and, uncomputable on N\A ?
By nontrivial example I mean function which is not defined as computable functio... | https://mathoverflow.net/users/3811 | Nontrivial, partially uncomputable function | How about the characteristic function of the Halting Problem? That is, $f(x)=1$ iff $\Phi\_x(x)\downarrow$ and $f(x)=0$ otherwise.
This function is clearly incomputable, but is computable on large infinite sets since by the padding lemma we can produce large (computable) sets $X$ of indices for equivalent programs wh... | 5 | https://mathoverflow.net/users/8133 | 134501 | 74,221 |
https://mathoverflow.net/questions/134508 | 10 | The "extension" (or "analytic") form of the theorem of Hahn-Banach has a natural and yet elegant proof. In just any textbook I have ever seen, it is proved first; the "separation" (or "geometric") version of Hahn-Banach's theorem is proved as a kind of corollary of the former.
Question: Are the two theorems actually ... | https://mathoverflow.net/users/26039 | Direct proof of the separation theorem of Hahn-Banach | Yes, the two theorems are equivalent in the sense that one can easily be deduced from the other and both have direct proofs from scratch.
A standard textbook starting with a direct proof of the geometric version is Schaefer's Topological Vector Spaces, [Chapter II, Section 3](http://books.google.com/books?id=9kXY742p... | 14 | https://mathoverflow.net/users/29555 | 134519 | 74,228 |
https://mathoverflow.net/questions/134523 | 1 | Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic system in infinite ways.Could any one give an example or proof?
**EDIT**:We define a proof or a reduced proof as one wh... | https://mathoverflow.net/users/14024 | Is there an recursively axiomatized system with infinitely many proofs for some propositions or a proposition | How about the theorem "There exists at least one prime number?" There are infinitely many distinct proofs of this result, none of which includes another as a subproof. Certainly this example is trivial in some sense, but I think it is not obvious how to pin down why this shouldn't be counted.
Perhaps a better example... | 7 | https://mathoverflow.net/users/8133 | 134527 | 74,233 |
https://mathoverflow.net/questions/134525 | 18 | By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!
| https://mathoverflow.net/users/6779 | Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n? | Let $n > 1$ be odd.
The curve:
$$X: \ x^n + 2 y^n + 4 z^n = 0$$
does not have any points over $K/\mathbf{Q}\_2$ unless the ramification index $e(K/\mathbf{Q}\_2)$ is divisible by $n$. **Proof:** At least two of the terms $x^n$, $2 y^n$, $4 z^n$ must have the same $2$-adic valuation. On the other hand, the ramificat... | 29 | https://mathoverflow.net/users/nan | 134528 | 74,234 |
https://mathoverflow.net/questions/134518 | 2 | Let $F\_n$ be the free group on $n\geq 2$ generators and let $H < F\_n$ be a finite index normal subgroup. Let $P\subset F$ be the subset consisting of primitive elements (an element of $F\_n$ is called primitive if it is a member of a free basis). Furthermore, we define $\tilde P\subset H$ to be the subset consisting ... | https://mathoverflow.net/users/35229 | Generating certain subgroups of free groups via primitive elements | If $x\_1 \in H$ then, since $H \unlhd F\_n$, all conjugates of $x\_1$ in $F\_n$ are in $H$ and hence also in $\overline{H}$. So the normal closure $N$, say, of $x\_1$ in $F\_n$ lies in $\overline{H}$.
Now any element of $H$ can be written as $nh$ with $n \in N$ and $h \in \langle x\_2,\ldots,x\_n \rangle$. But $x\_1 ... | 4 | https://mathoverflow.net/users/35840 | 134538 | 74,237 |
https://mathoverflow.net/questions/134504 | 6 | For the context of this question, a progression-free set is a subset of integers that does not contain length-three arithmetic progressions.
For large $N$, it is known that $[N] = \{1, \ldots, N\}$ contains progression-free sets of size $O (N \log^{1 / 4} N / 2^{2 \sqrt{2 \log\_2 N}} )$ (Elkin, 2011). However, when $... | https://mathoverflow.net/users/18705 | Non-asymptotically densest progression-free sets | More a comment than and answer, but a bit too long.
First, two things to keep in mind:
On the one hand, people conjectured for a while that the thing actually *is* $N^{\log\_3 2}$ or at least $N^c$ for $c<1$ so there should not be any "simple" (general) constructions that beats this.
On the other hand, all con... | 4 | https://mathoverflow.net/users/nan | 134540 | 74,238 |
https://mathoverflow.net/questions/134532 | 1 | I came across this statement in [the autobiography by Saunders Mac Lane](http://rads.stackoverflow.com/amzn/click/1568811500).
>
> It was the interaction between solenoids and group extension that got our collaboration started, and this first work of collaboration revealed much else to be done, some stimulated by a... | https://mathoverflow.net/users/34151 | Where is a proof of "2 is more than 1 plus 1" said by Saunders Mac Lane? | The paper Mac Lane is referring to must be "Group Extensions and Homology" from May, 1942. (This fits the description about "interactions between solenoids and group extensions".) The main result in that paper is a form of the universal coefficient theorem for cohomology. I don't see how this can be interpreted as sayi... | 12 | https://mathoverflow.net/users/10503 | 134542 | 74,239 |
https://mathoverflow.net/questions/134543 | 2 | Let $f:X\rightarrow Y$ a faithfully flat morphism between $k$-schemes. We assume that the fibers are locally of finite type, do we have that $f$ is locally of finite type?
| https://mathoverflow.net/users/27398 | flat and finite type morphisms | No. Let $Y$ be $\text{Spec} \mathbb{Z}$. Let $X$ be $\text{Spec} (\mathbb{Z} \times \mathbb{Q})$.
$\textbf{Edit}.$ As pointed out, the OP wants an example over a field. As the commenters explain, the same idea works over a field $k$ with $Y = \text{Spec} k[t]$ and with $X=\text{Spec}(k[t] \times k(t))$.
| 5 | https://mathoverflow.net/users/13265 | 134544 | 74,240 |
https://mathoverflow.net/questions/134541 | 1 | There is a ring $R$ and its subring $K$ with unit. We have a matrix $A$ of order $n$ over $R$. Someone said, that if $A^m$ for $m=1,...,n$ can be represented as a sum of matrices over $R$ which a integral over $K$, than $A$ is integral over $K$. Why is that?
| https://mathoverflow.net/users/35115 | When powers of matrices are represented as a sum of integral matrices | A matrix is integral iff its eigenvalues are integral. Thus A is integral if and only if A^m is integral.
| 0 | https://mathoverflow.net/users/35245 | 134551 | 74,243 |
https://mathoverflow.net/questions/134555 | 1 | I have a continuous function $\mu(x,y)=G(h\_1(x+y), h\_2(|x-y|))$ such that $\mu(x,x)=0$ and $\mu(x,y)+\mu(y,z)=\mu(x,z)$ for $x\le y\le z$. I want to show that the only possible case is $\mu(x,y)=c\cdot (x+y)|x-y|$ for some constant $c$.
If I assume that I have a polynomial, no problem, since I have elementary symme... | https://mathoverflow.net/users/17460 | Symmetric function | If $g(s,t) = a(s+t) - a(s-t)$ for an arbitrary continuous function $a$, $u(x,y) = g(x+y, y-x) = a(2y) - a(2x)$ satisfies $u(x,x) = 0$ and $u(x,y) + u(y,z) = u(x,z)$ for all $x,y,z$.
Then $\mu(x,y) = g(x+y,|x-y|)$ agrees with $u(x,y)$ when $x \le y$, and so
$\mu(x,x) = u(x,x) = 0$ and $\mu(x,y) + \mu(y,z) = u(x,y) + ... | 3 | https://mathoverflow.net/users/13650 | 134556 | 74,247 |
https://mathoverflow.net/questions/133346 | 8 | Let $\{ G\_n \}\_{n \ge 1}$ be a sequence of graphs such that the number of vertices of $G\_n$ tends to $\infty$ as $n \to \infty$. We say that $\{ G\_n \}\_{n \ge 1}$ is an [expander family](http://en.wikipedia.org/wiki/Expander_graph) if
$\lambda\_2( G\_n)$ is bounded away from $0$ as $n \to \infty$. Here $\lambda\_2... | https://mathoverflow.net/users/4558 | Do there exist "expanding" $1$-skeletons of simple $4$-polytopes? | As far as I know there are no examples of graphs of simple 4-polytopes (or simple $d$-polytopes, for fixed $d$) that are expanders, and not even such examples for the dual graphs for triangulations of spheres of fixed dimensions. Some information on expansion properties of graphs of polytopes can be found in my chapter... | 8 | https://mathoverflow.net/users/1532 | 134560 | 74,249 |
https://mathoverflow.net/questions/134232 | 2 | This is question follows on from [this one](https://mathoverflow.net/questions/133347/decomposition-of-hermitian-form-used-in-the-definition-of-griffiths-nakano-positi).
In the linked question, the hermitian form $\theta\_E$ on $T^{1,0}X\otimes E$ is defined globally as $$\theta\_E(v\otimes\sigma,v\otimes\sigma):=h(i... | https://mathoverflow.net/users/21564 | Local expression involved in the definition of positivity of vector bundles | My answer will consist mainly of a collection of trivial facts but which nonetheless often generate some confusion.
I begin by fixing some notation. Let $V$ be a complex vector space of complex dimension $n$, and $V^{\mathbb R}$ its underlying real vector space, together with the complex structure $J$ given by the mu... | 2 | https://mathoverflow.net/users/9871 | 134569 | 74,253 |
https://mathoverflow.net/questions/134405 | 14 | I recently came across this problem:
Let $T=\mathbb R /2 \pi \mathbb Z$ the circle, with its proabability Haar measure $\mu$. Any integrable function $f : T \rightarrow \mathbb C$
has Fourier coefficients $c\_n(f)= \int\_T f(t) e^{-int} \frac{dt}{2\pi}$, and we define
$$||f||\_{A(T)} = \sum\_{n \in \mathbb Z} |c\_n(f... | https://mathoverflow.net/users/9317 | A question in Fourier analysis | Yemon Choi (henceforth "YC") answered the question to within a constant factor
with a function $f\_\delta$ showing that $\alpha(E\_\delta) \leq 2/\delta$
where $E\_\delta$ is an interval of measure $1-\delta$ (YC's notation).
That's optimal asymptotically $-$ and even exactly if $2/\delta \in {\bf Z}$
$-$ for functions... | 14 | https://mathoverflow.net/users/14830 | 134570 | 74,254 |
https://mathoverflow.net/questions/134581 | 28 | The Symmetric groups $S\_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in S\_n$).
**Question:** What are other families of (finite) groups where all complex irreducible characters are ration... | https://mathoverflow.net/users/35261 | Groups in which all characters are rational. | Here's one characterization that I learned from Serre
(see Definition 7.1.1 in his
[*Topics
in Galois Theory*](http://www.math.mcgill.ca/~darmon/pub/Articles/Serre/c.pdf) (p.65)): an element $g$ of a finite group $G$
satisfies $\chi(g) \in {\bf Q}$ for all characters $\chi$ **iff**
$g$ is conjugate in $G$ to $g^m$ for ... | 34 | https://mathoverflow.net/users/14830 | 134585 | 74,258 |
https://mathoverflow.net/questions/134558 | 16 | I am very far removed from being an expert on derived categories. Every few months, however, I read a different introductory text with the hope that eventually I will have some basic grasp on this concept.
This has the added benefit that it puts various aspects of the technicalities of homological algebra, algebraic ... | https://mathoverflow.net/users/5756 | How can one interpret homology and Stokes' Theorem via derived categories? | There is an interpretation of homology using derived categories, and you can find a brief treatment around remark 3.3.10 in Kashiwara-Schapira "Sheaves on Manifolds". If $M$ is an $n$-manifold and $a: M \to \ast$ is the tautological map to a point, then the homology of $M$ with coefficients in an abelian group $F$ is t... | 9 | https://mathoverflow.net/users/121 | 134601 | 74,265 |
https://mathoverflow.net/questions/134595 | 4 | Let $V = \mathbb{C}^n$, $A = \Lambda^{\bullet}(\mathbb{C}^n)$ is a graded algebra (with $A\_0 =
\mathbb{C}, A\_1 = V$, etc).
Consider $A\_0$ as a left $A$-module, how do we compute the graded ring $\text{Ext}^{\bullet}\_A(A\_0, A\_0)$? (Doing the $n=3$ example should be enough; then it would be easy to generalize.)
... | https://mathoverflow.net/users/2623 | Computing Ext in Exterior algebra (related to Koszul duality) | Consider the Koszul complex
$$
\dots \to S^3V\otimes A(-3) \to S^2V\otimes A(-2) \to V\otimes A(-1) \to A \to A\_0 \to 0,
$$
where $(-i)$ is the shift of grading. This is a free resolution of $A\_0$. Using this to compute $Ext$ you obtain $Ext^\bullet(A\_0,A\_0) = S^\bullet(V^\*)$.
| 7 | https://mathoverflow.net/users/4428 | 134615 | 74,270 |
https://mathoverflow.net/questions/134618 | -4 | Let $(M,J)$ be a Kaehler manifold. How can one describe the opposite complex structure? What is the precise definition of the opposite complex structure? Can one describe the opposite complex structure in terms of $J$?
| https://mathoverflow.net/users/35273 | Opposite complex structure on Kaehler manifold | The opposite complex structure is $-J$.
| 8 | https://mathoverflow.net/users/13268 | 134619 | 74,271 |
https://mathoverflow.net/questions/134459 | 5 | Let $G$ ba a compact Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{g}^{\*}$ be the dual of $\mathfrak{g}$. We known the Weil algebra is
$$W(\mathfrak{g})=\wedge(\mathfrak{g}^{\*})\otimes S(\mathfrak{g}^{\*}).$$
Choose a basis $e\_{1},\cdots,e\_{n}$ for $\mathfrak{g}$ and let
$e\_{1}^{\*},\cdots,e\_{n}^{\*}... | https://mathoverflow.net/users/3896 | A question about Weil algebra | If $\tilde e\_i = \sum\_j A\_{ij} e\_j$ then for the dual basis we have
$\tilde e\_k^\ast = \sum\_\ell ((A^t)^{-1})\_{kl} e\_l^\ast$, so the the base change matrix cancels out of the formula.
| 3 | https://mathoverflow.net/users/26935 | 134627 | 74,277 |
https://mathoverflow.net/questions/129975 | 4 | Sorry for asking such a basic question, but this is not my area of expertise.
In my work I'm using the coarea formula: for $\Omega \subseteq \mathbb{R}^n$ open and bounded, and $u : \Omega \to \mathbb{R}$ Lipschitz,
$\int\_{\Omega} |\nabla u| = \int\_{-\infty}^\infty \mathcal{H}\_{n-1}(u^{-1}(t))dt$.
I can calcul... | https://mathoverflow.net/users/658 | Technical question on perimeter of level sets | The answer to your first question is yes. You can take as surface area of a set the (total) variation of its characteristic function. This is the "standard definition" of [perimeter](http://en.wikipedia.org/wiki/Caccioppoli_set#Caccioppoli_definition) for general measurable sets in $\mathbb{R}^n$. See, for example, [th... | 3 | https://mathoverflow.net/users/22302 | 134628 | 74,278 |
https://mathoverflow.net/questions/134636 | 7 | Given a torsion-free hyperbolic group $G$, does there exist a number $n(G)$ such that for any $x,y,z\in G$, $x^n y^n z^n =1$ implies that $x$, $y$, and $z$ commute pairwise?
Some musings/questions...
When $G$ is free, the result is true for $n=2$.
Clearly, this does not generalize to hyperbolic groups (e.g. non-or... | https://mathoverflow.net/users/31040 | Lyndon-Schützenberger for torsion-free hyperbolic groups | The second statement is true if $x,y,z$ are not torsion as proved by Arzhantseva and, independently, by Kapovich and Weidmann (see Arzhantseva, Goulnara N.
A dichotomy for finitely generated subgroups of word hyperbolic groups. Topological and asymptotic aspects of group theory, 1–10,
Contemp. Math., 394, Amer. Math. ... | 8 | https://mathoverflow.net/users/nan | 134639 | 74,283 |
https://mathoverflow.net/questions/134641 | 0 | Let $K\subset L$ be a field extension and let $K\subset F\_1,F\_2,...,F\_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}\_{(F\_i)}:=\prod\_{i=1}^n F\_i$ instead of $\mathbb{A}\_K^n$ (or $\mathbb{A}\_L^n$ for that matter) and consider zero sets of polynomials $f\in K[X\_1,X\_2,..... | https://mathoverflow.net/users/1849 | Algebraic varieties in "mixed" affine spaces | The product of Weil restrictions $P = \prod\_{i=1}^n {\rm{R}}\_{F\_i/K}(\mathbf{A}^1\_{F\_i})$ is naturally a closed subscheme of the direct product $\prod\_{i=1}^n {\rm{R}}\_{L/K}(\mathbf{A}^1\_L) = {\rm{R}}\_{L/K}(\mathbf{A}^n\_L)$. For any closed subscheme $Z \subset \mathbf{A}^n\_K$, we also get a closed subscheme ... | 5 | https://mathoverflow.net/users/35361 | 134657 | 74,286 |
https://mathoverflow.net/questions/134650 | 10 | As in the title, say $\lambda$ is some irrep of the symmetric group $S\_n$, and $Br\_n$ the braid group on $n$ strands,
>
> What is $H^\*(Br\_n, \lambda)$?
>
>
>
| https://mathoverflow.net/users/4707 | H*(braid group, irrep of symmetric group) = ? | I claim that the dimension of $H^\*(Br\_n;\lambda)$ is twice the dimension of the subspace of $\lambda$ fixed by the subgroup $S\_2\leq S\_n$.
Indeed, there is a spectral sequence
$$ H^p(S\_n;H^q(PBr\_n;\lambda)) \Longrightarrow H^{p+q}(Br\_n;\lambda). $$
As a $PBr\_n$-module, $\lambda$ is just a sum of finitely many... | 18 | https://mathoverflow.net/users/10366 | 134667 | 74,290 |
https://mathoverflow.net/questions/134668 | 4 | This is my first post here, so bear with me ;)
In wikipedia and other references, [**Schwartz space**](http://en.wikipedia.org/wiki/Schwartz_space) is defined as the set of infinitely differentiable functions on $\mathbb{R}^n$. On the other hand, A locally convex space X is named a **LS-space** if there is a Fréchet-S... | https://mathoverflow.net/users/35442 | Schwartz space defined on locally convex spaces | *The* Schwartz space (of infinitely differentiable functions) is just an example of *a* Schwartz space. The term “Schwartz space” is used for two concepts. The German Wikipedia has two articles: [Schwartz-Raum](http://de.wikipedia.org/wiki/Schwartz-Raum) and [Schwartz-Raum (allgemein)](http://de.wikipedia.org/wiki/Schw... | 6 | https://mathoverflow.net/users/33842 | 134670 | 74,292 |
https://mathoverflow.net/questions/133530 | 5 | I am trying to study (finite) spherical buildings from a very combinatorial point of view : Every rank 3 spherical building is a finite simplicial complex of dimension 3, so one can define its density as the ratio #triangles/#vertices, expressed as a function of the number $n$ of vertices. Maybe it is the wrong name fo... | https://mathoverflow.net/users/6325 | Density/Thickness of rank 3 spherical buildings | While I have never seen this notion of "density" of a finite building before, let's see what we can do...
A thick finite $C\_3$ building corresponds to a thick finite rank 3 polar space. These are well-known, they are $W(5,q)$, $Q(6,q)$, $Q^-(7,q)$, $H(5,q^2)$ and $H(6,q^2)$, corresponding to the groups $Sp(6,q)$, $O... | 5 | https://mathoverflow.net/users/8338 | 134681 | 74,296 |
https://mathoverflow.net/questions/134390 | 5 | Hi
Let $R = K[X\_1,\ldots, X\_n]$ where $K$ is a computable field.
Suppose we are given two modules with presentations
$$ R^n \rightarrow R^m \rightarrow M \rightarrow 0 $$
and
$$ R^l \rightarrow R^p \rightarrow N \rightarrow 0 $$
Then is it possible to verify whether $M$ is isomorphic to $N$ (using a computer alge... | https://mathoverflow.net/users/31532 | Checking whether modules are isomorphic, via a computer algebra software | You can almost do. There is a theorem (due essentially to Bongartz, but in this form perhaps can be found in Yongwei Yao's thesis) that if $\rm{length}(M\otimes L)) = \rm{length}(N\otimes L))$ for all finite length module $L$, then $M\cong N$ (this is a local result, but under reasonable assumptions, for example if bot... | 3 | https://mathoverflow.net/users/2083 | 134702 | 74,304 |
https://mathoverflow.net/questions/134707 | 1 | Suppose $G$ is a locally finite group such that $G=\bigcup\_{i=1}^\infty S\_i$, where $S\_i$ is a finite group and $S\_i \triangleleft S\_{i+1}$ for all $i \in \mathbb{N}$. Let $P$ be a Sylow (maximal with respect to inclusion) $p$-subgroup of the group $G$. For each $i \in \mathbb{N}$ put $P\_i=S\_i \cap P$. **Then $P... | https://mathoverflow.net/users/35603 | Sylow p-subgroups of locally finite groups | Yes. Fix $i$. For $n\ge i$, let $L\_n$ be a $p$-Sylow of $S\_n$ containing $P\_n$. Note that $S\_i$ is subnormal in $S\_n$. By the lemma below, $L\_n$, the intersection $M\_n=L\_n\cap S\_i$ is a $p$-Sylow of $S\_i$. Now there exists an infinite set $N$ of $n$ such that the Sylow $M=M\_n$ does not depend on $n\in N$. Si... | 1 | https://mathoverflow.net/users/14094 | 134713 | 74,307 |
https://mathoverflow.net/questions/28849 | 27 | There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:
1. A closed subgroup of $GL(n,\mathbb C)$ closed under conjugate transpose.
2. The set of real points $G(\mathbb R)$ of a real algebraic group such that $G(\mathbb C)$ is [reductive](http://en.wikiped... | https://mathoverflow.net/users/1626 | Definitions of real reductive groups | This is an old question, probably abandoned because its answer would require a short article (complete with references). While I can't supply such an article, I can point to some of Borel's writings which involve a definition of "real reductive group". In an old comment, Brian Conrad already referred to the added surve... | 14 | https://mathoverflow.net/users/4231 | 134717 | 74,308 |
https://mathoverflow.net/questions/134706 | 1 | Let $A$ be a square Matrix and $||\cdot ||\_p$ the induced Matrix norm for $1 \leq p \leq \infty$. Is it true that
$$||A||\_p\leq \max(||A||\_1,||A||\_{\infty})?$$
For $p=2$ the answer is yes because $||A||\_2^2\leq ||A||\_1||A||\_{\infty}$.
The motivation is that I have a family of matrices where I can bound the inf... | https://mathoverflow.net/users/35593 | Matrix-Norm equivalence with p-Norm | Yes, by the [Riesz-Thorin theorem](https://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
| 4 | https://mathoverflow.net/users/1044 | 134722 | 74,311 |
https://mathoverflow.net/questions/134726 | 4 | In the Wikipedia article on Hilbert manifolds, it is claimed that every Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space. However, no explicit reference is given. Does anybody know a reference.
I must honestly say that I find that result somewhat hard to believe...
| https://mathoverflow.net/users/16702 | Hilbert manifolds and embedding | [MR0263120](https://mathscinet.ams.org/mathscinet-getitem?mr=263120) (41 #7725)
Eells, J.; Elworthy, K. D.
Open embeddings of certain Banach manifolds.
Ann. of Math. (2) 91 1970 465–485.
| 8 | https://mathoverflow.net/users/26935 | 134728 | 74,313 |
https://mathoverflow.net/questions/134603 | 29 | Recall that a Serre fibration between topological spaces is a map which has the homotopy lifting property (HLP) for all CW complexes (equivalently for all disks $D^k$). The Serre fibrations are the fibrations in a model category structure on topological spaces in which the weak equivalences are the weak homotopy equiva... | https://mathoverflow.net/users/184 | Is the counit of geometric realization a Serre fibration? | $\newcommand{\real}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Sing}[1]{\operatorname{Sing}(#1)}$$\newcommand{\counit}{\epsilon}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\proj}{\mathrm{proj}}$$\newcommand{\NN}{\mathbb{N}}$$\newcommand{\RR}{\mathbb{R}}$**Yes, the map $\real{\Sing{X}} \to X$ is a Serre fibratio... | 21 | https://mathoverflow.net/users/21095 | 134731 | 74,314 |
https://mathoverflow.net/questions/134712 | 5 | In the paper **Stability in Yang-Mills Theories** (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured:
*"The inclusion $\mathcal{M}\_n\hookrightarrow\mathcal{B}\_n$ induces an isomorphism of the pointed homotopy groups $\pi\_k(\cdot)$ for $k\le 2|n|$."*
... | https://mathoverflow.net/users/12310 | Conjecture on homotopy groups of moduli space of self-dual connections | On afterthought I now asked Taubes himself, who pointed me to the paper **The Topology of Instanton Moduli Spaces** (Boyer-Hurtubise-Mann-Milgram). This conjecture is a form of the *Atiyah-Jones Conjecture*. The paper is essentially what is known about this problem, and in particular the above conjecture is still open.... | 6 | https://mathoverflow.net/users/12310 | 134736 | 74,317 |
https://mathoverflow.net/questions/131067 | 3 | After Jun-Ichi Igusa' talk at ICM 1962, H.J. Tramer computed the ring structure of the integral cohomology of such a space ( not yet called WPS ).
In 1971 M.F. Atiyah called it WPS and established a Riemann-Roch theorem for it .
This means that, more than ten years before Tetsuro Kawasaki, H.J. Tramer determined
H\*... | https://mathoverflow.net/users/34304 | Who first computed the integral cohomology ring of a weighted projective space (WPS) ? | Lastly I got a copy of H.J. Tramer's Ph.D.Thesis on the cohomology ring of "pseudo-projective sapaces" (1965), i.e. weighted projective spaces WPS (as they are called since the seventies).The question was posed to him by J. Igusa who,in his study of Siegel modular forms, had to do with WPS(2,3,5,6).The computation of t... | 2 | https://mathoverflow.net/users/34304 | 134748 | 74,318 |
https://mathoverflow.net/questions/134686 | 2 | We know that if $i:R\_n\rightarrow C\_n$ is an isometry then for any $n$-dimensional operator space E, there is a factorization $i=uv$ with $v:R\_n\rightarrow E$, $u:E\rightarrow C\_n$ such that $\|u\|\_{cb}\|v\|\_{cb}=n^{1/2}$. The question is, if E is a homogeneous Hilbertian operator space can we find $u$ and $v$ su... | https://mathoverflow.net/users/35535 | Isometries between Hilbertian homogeneous finite dimensional operator spaces | Yes. See the proof of Proposition 10.1 in Pisier's book.
The idea is simple: if you start from any factorization $uv$, by the polar decomposition you can assume that both maps are diagonal in an orthonormal basis. Averaging these diagonal maps over all permutations of the basis does not increase the cb norm, and yields... | 2 | https://mathoverflow.net/users/10265 | 134755 | 74,321 |
https://mathoverflow.net/questions/134550 | 0 | Hi,
I wanted to ask, under which conditions can one rewrite the optimization objective
$\min\_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$
as
$\min\_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$
I have particular interest in the case where $f(x) = \|x\|\_1$ and $g(x) = \|y - Ax\|\_2$ (i.e. for the Lasso!), but would like to know ... | https://mathoverflow.net/users/17243 | Rewrite optimization objective | Here's a solution for the specific case where $f(x)=\| x \|\_{1}$ and $g(x)=\| y-Ax \|\_{2}$. The same appraoch applies to $g(x)=\| A^{T}(y-Ax) \|\_{\infty}$.
There are two cases.
1. If $g(0) \leq s$, then $x^{\*}=0$ is feasible for the first problem and because $f(x) > 0$ for all $x \neq 0$, $x^{\*}=0$ is the uni... | 1 | https://mathoverflow.net/users/9022 | 134766 | 74,324 |
https://mathoverflow.net/questions/134447 | 3 | Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K\_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow
$$\int\_{\mathbb{R}^{n}} K\_{t}(x, \cdot)\,dy = 1?$$
Or would this at least follow for the kernel $\tilde{K}\_{t}$ after transference of $H$ to an appropriate $L^{2}... | https://mathoverflow.net/users/12968 | Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator? | If $V$ is a *positive* polynomial then $\int\_{\mathbb{R}^n} K\_t(x,y)dy$ is *strictly less than $1$.* This follows from the Feynman-Kac formula,
$$ \int\_{\mathbb{R}^n} K\_t(x,y)dy = \mathbb{E}\_x \left ( \exp\left ( -\int\_0^t V(b(s))ds\right ) \right )$$
where the expectation on the right hand side is over Brownian... | 3 | https://mathoverflow.net/users/6781 | 134772 | 74,328 |
https://mathoverflow.net/questions/134299 | 7 | Do any of the standard methods of acceleration convergence of series, when applied to
the series $1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + ...$, give convergence to 0 with error $o(1/n)$?
I tried applying Euler's method to the series, and found that the estimates fall like $1/n$; in fact, the $n$th estimate seems to be to $2/... | https://mathoverflow.net/users/3621 | accelerating convergence of a class of sequences | The Aitken delta squared method gives $O(1/n^2)$.
| 1 | https://mathoverflow.net/users/12120 | 134775 | 74,329 |
https://mathoverflow.net/questions/134758 | 1 | I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^\*$ be its dual space of continuous linear functionals. (if $X$ is not locally convex then $X^\*$ is trivial, but that's fine for these purposes). Let $K... | https://mathoverflow.net/users/238 | Agreement of two topologies on a linear space | What do you think about this rather trivial example: Let $X=c\_0$ be the Banach space of
(real) null sequences, $X^\*=\ell\_1$ its dual (where $y=(y\_n)\_{n} \in \ell\_1$ is considered as the functional $(x\_n)\_n \mapsto \sum\_n x\_n y\_n$) and $K:X^\* \to X$
the inclusion. Then $A\_K=\ell\_2$ is dense in $c\_0$ so t... | 4 | https://mathoverflow.net/users/21051 | 134780 | 74,331 |
https://mathoverflow.net/questions/134031 | 6 | Let $X$ be a smooth scheme over $\mathbb{F}\_{p}$ for a prime number $p$. As far as I understand,
there is a surjective morphism from
$\Omega^\bullet\_{W\mathcal{O}\_X} \to W \Omega\_{X}^\bullet$ which induces an isomorphism
$\Omega^\bullet\_{W\mathcal{O}\_X}/(T+Fil^n \Omega^\bullet\_{W\mathcal{O}\_X}) \to W\_{n}\Ome... | https://mathoverflow.net/users/32151 | Vanishing cohomology of de-Rham Witt complex | If $A$ is a finitely generated ring then so is each $W\_n(A)$, and in fact one can bound the number of generators needed by a function of only $p$, $n$, and the number $d$ of generators of $A$. Write $N\_p(d,n)$ for such a function. (See below.) Then $\Omega^i\_{W\_n(A)}$ vanishes for $i>N\_p(d,n)$. In particular, this... | 5 | https://mathoverflow.net/users/1114 | 134785 | 74,333 |
https://mathoverflow.net/questions/134756 | 8 | **NOTATION**: $P(x)$ stands for the product of all primes which do not exceed real $x$; e.g. $P(10)=210$.
**QUESTION**: Given any real $x\ge 3$, compute or estimate the smallest natural number $n:=n(x)\ge 3$ such the remainder of the division of $n$ by any odd prime $p\le x$ is $1$ or $2... | https://mathoverflow.net/users/8385 | Remainders $\quad 1\quad 2\quad $ only | The values $n(p)$ for primes $13 \leq p < 100$, found by computation: $n(13) = n(17) = 716$,
$n(19) = 62987$,
$n(23) = 367082$, $n(29) = 728366$, $n(31) = 64822396$, $n(37) = 1306238012$,
$n(41) = 11182598506$, $n(43) = 715041747422$, $n(47) = 51913478860882$,
$n(53) = 454746157008782$, $n(59) = 9314160363311806$, $n(6... | 6 | https://mathoverflow.net/users/28104 | 134786 | 74,334 |
https://mathoverflow.net/questions/134789 | 2 | i hope my question is not too trivial.
Let's suppose we have a vector space $V$ with a unimodular quadratic form $q$ of signature $(m,n)$.
My question is: which is the maximum dimension of an istropic subspace of $V$?
I would say it is $min\{m,n\}$. This is why. Suppose $min\{m,n\}=m$, if $min\{m,n\}=n$ the proof ... | https://mathoverflow.net/users/35761 | Maximum dimension of an isotropic subspace in a quadratic space | This is the Witt index of the quadratic form. I'm assuming you're working over $\bf{R}$, in which case it is indeed $\min\{m,n\}$ (use Witt cancellation). It can be smaller over other fields (like $\bf{Q}$); for instance, you could have an indefinite quaternary form which is anisotropic. For more details see Chapter 1 ... | 3 | https://mathoverflow.net/users/2698 | 134798 | 74,340 |
https://mathoverflow.net/questions/134795 | 12 | Suppose $X\subset \mathrm{GL}\_n(p)$ is a set of invertible matrices such that for every $A,B\in X$ then also $A-B\in \mathrm{GL}\_n(p)\cup \{0\}$. (If anyone knows a name for such sets I would be grateful).
I was trying to figure out what is the size of the largest such $X$. Since $\mathbb{F}\_{p^n}$ can be embedded... | https://mathoverflow.net/users/30491 | Largest subset of $GL_n(p)$ in which pairwise subtraction is also in $GL_n(p)$ | The upper bound can be obtained as follows: Fix a nonzero vector $v$ from $\mathbb F\_p^n$. Then the vectors $Bv$ for $B\in X$ are pairwise distinct and nonzero, for if $Bv=B'v$, then $B-B'$ is not invertible. Thus the map $X\to\mathbb F\_p^n\setminus\{0\}$, $B\mapsto Bv$ is injective, hence $\lvert X\rvert\leq p^n-1$.... | 13 | https://mathoverflow.net/users/18739 | 134800 | 74,341 |
https://mathoverflow.net/questions/134773 | 2 | I posed the question [here](https://math.stackexchange.com/questions/428982/homocyclic-primary-module-over-pid), but get no answers yet.
Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess... | https://mathoverflow.net/users/26700 | Homocyclic primary module over PID | Choose generators $n\_1,\dotsc,n\_r$ for $N$ with $p^{s\_i}n\_i=0$. It is not hard to see that the annihilator of $p^{s\_i}$ on $M$ is $p^{s-s\_i}M$, so we can choose $m\_i$ with $p^{s-s\_i}m\_i=n\_i$. If we can prove that the elements $m\_i$ form a basis for $M$ over the ring $R/p^s$, then everything else is clear.
... | 3 | https://mathoverflow.net/users/10366 | 134801 | 74,342 |
https://mathoverflow.net/questions/134643 | 11 | The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal transformations, as if I had chosen Euclidean signature, but of course this story also works with $\mathrm{SO}(n-1,1)$, .... In... | https://mathoverflow.net/users/78 | Is there a version of supersymmetry for homogeneous spaces? | Homogeneous superspaces generalizing the "Super-Poincaré\Lorentz =
Superspace" exist. These superspaces are given as coset spaces of super-Lie groups $G/H$. Please see for example, the following [article](http://arxiv.org/abs/hep-th/0409257) by:
Schunck and Wainwright containing an explicit construction of the
supersp... | 3 | https://mathoverflow.net/users/1059 | 134803 | 74,343 |
https://mathoverflow.net/questions/134799 | 7 | Let $A\in M\_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, satisfying
$AA^{\dagger}A=A, A^{\dagger}AA^{\dagger}=A^{\dagger}$ and $AA^{\dagger}$ Hermitian). This was again gener... | https://mathoverflow.net/users/32332 | Existence of a generalized matrix inverse over an arbitrary field? | The condition $AA'A=A$ means that the semigroup of matrices is regular. This is really true: see Clifford - Preston, The Algebraic Theory of Semigroups, sec.2.2, ex. 6(g).
| 7 | https://mathoverflow.net/users/18814 | 134807 | 74,345 |
https://mathoverflow.net/questions/134735 | 6 | Let $R$ be a regular local ring, not necessarily of dimension one. Let $X \to Spec R$ be a nodal curve, i.e. all the geometric fibers are reduced and has at most nodal singularities. Assume $X$ admits a stable model $Y$ over $Spec R$. Is it necessairily true that there is morphism $X \to Y$ over $Spec R$, which is "con... | https://mathoverflow.net/users/35655 | stable reduction and blow down | As user61789 points out, there are two natural questions related to this post. First, under the given hypotheses on the (proper) morphism $\pi\_X:X\to \text{Spec}(R)$, and assuming that the generic fiber is stable (in the sense of Deligne, Mumford, Mayer, Grothendieck, Knudsen, ...), does there exist a stable model $\p... | 4 | https://mathoverflow.net/users/13265 | 134811 | 74,347 |
https://mathoverflow.net/questions/134763 | 2 | Consider a grid of points $T=\{t\_0,t\_1,\ldots,t\_m\}$ with $0\le t\_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation\*}
f(t)=\sum\_{k=-n}^n c\_k e^{2\pi k it}
\end{equation\*}
(with n as small as possible) such that
1) $f(t\_0)=1$, and $f(t\_k)=0$ for $k\ge1$... | https://mathoverflow.net/users/34919 | Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay | Some initial thoughts to get us started:
Consider
$$
g(z) = \bigg ( \frac{\sin(2\pi(z - t\_0))}{2\pi(z - t\_0)} \cdot \prod\_{k \neq 0} \frac{\sin(2\pi(z - t\_k))}{\sin(2\pi(t\_0 - t\_k))} \bigg )^2
$$
Note that the Fourier transform of $g$ is compactly supported (by Paley-Wiener).
Now define
$$
h(x) := \sum\_{n \in ... | 3 | https://mathoverflow.net/users/35810 | 134812 | 74,348 |
https://mathoverflow.net/questions/134783 | 3 | I am aware of the following question: [Definitions of Reductive and Semisimple Groups](https://mathoverflow.net/questions/109629/definitions-of-reductive-and-semisimple-groups)
So let me phrase a precise question:
>
> Is there a standard technique by which one can translate the unitary/smooth admissible represent... | https://mathoverflow.net/users/10400 | How to translate the representation theory of semisimple to reductive groups? | The derived group $G'$ of $G$ always works for your second question (i.e., $G'(F)Z(F)$ is closed and cocompact in $G(F)$). Indeed, by using local class field theory and Kneser-Bruhat-Tits we know that ${\rm{H}}^1(F,H)$ is finite for any connected reductive $F$-group $H$, so ${\rm{H}}^1(F,G')$ is finite. If $X \rightarr... | 5 | https://mathoverflow.net/users/35361 | 134815 | 74,349 |
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