parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/171415 | 11 | For the purposes of this question a *topological space* will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I would be interested to know if things are any different if we consider the category of $\Delta$-generated spaces, for exam... | https://mathoverflow.net/users/184 | Is the geometric realization of a level-wise weak equivalence a weak equivalence? | To get such a result we typically need that the degenerate subspaces include via cofibrations, and we can get a counterexample by picking a standard non-cofibration.
Let $X\_0 = \{0\}$, and let $X\_1 = \{0, 1, 1/2, 1/3, \ldots\} \subset \Bbb R$, with degeneracy $s^0: X\_0 \to X\_1$ being the natural inclusion. Build ... | 13 | https://mathoverflow.net/users/360 | 171423 | 87,619 |
https://mathoverflow.net/questions/169183 | 3 | Using a small modification to Turáns theorem we can find the minimum amount of edges a graph $G$ on $n$ vertices must have so it does not have an independent set of size $k$. Is there a similar result if we add the restriction that $G$ is a connected graph?
| https://mathoverflow.net/users/24478 | Turan's theorem for connected graphs? | Just a partial answer, expanding Tony's:
The statement is true for $n=2k$ (i.e. you need exactly $(2k-1)$ edges to avoid an independent set of size $k+1$ in a connected graph), and the extremal graphs are exactly the trees with a perfect matching.
Proof: Clearly, you need $2k-1$ edges to make it connected. And $2k... | 3 | https://mathoverflow.net/users/12487 | 171425 | 87,620 |
https://mathoverflow.net/questions/171424 | 5 | Let $H^\infty$ denote the Banach space of all bounded analytic functions on the open disc $\mathbb{D}$. It is easy to see that $H^\infty$ is a dual space. However, is there a Banach sapce $Y$ such that $H^\infty$ and $Y^{\*\*}$ are isomorphic as Banach spaces?
| https://mathoverflow.net/users/52808 | Is $H^\infty$ a second dual space? | If you only require isomorphism in the sense of an invertible, continuous linear bijection, then the answer is yes. If you require isometric linear isomorphism, the answer is no (because the unique *isometric* predual of $H^\infty$ is $L^1/H^1\_0$, and $L^1/H^1\_0$ is not isomorphic to any dual Banach space).
These r... | 6 | https://mathoverflow.net/users/763 | 171430 | 87,621 |
https://mathoverflow.net/questions/171437 | 5 | Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O\_S)=0$ and $P\_2(S)=h^0(\mathcal O\_S(2K\_S))=0$. If $S$ is smooth, Castelnuovo's rationality criterion (proved by Zariski in characteristic p) tells us that $S$ is rational.
Does this extend to singular surfaces if... | https://mathoverflow.net/users/1887 | Castelnuovo's rationality criterion on singular surfaces? | It does not hold in general: a cone over a smooth plane cubic satisfies $q=P\_2=0$ but is not rational.
On the other hand if $S$ has canonical singularities and $\tilde{S} \rightarrow S$ is any resolution, one has $P\_2(S)=P\_2(\tilde{S} )$ and $q(S)=q(\tilde{S} )$, hence $\tilde{S} $ and therefore $S$ are rational.
... | 7 | https://mathoverflow.net/users/40297 | 171440 | 87,626 |
https://mathoverflow.net/questions/168510 | 1 | Given a pair of strictly increasing functions $f,g:\mathbb{N}\to \mathbb{N}$
define:
$P\_N(f,g)\doteq \left(z\in \mathbb{C}\mapsto \prod\_{i=1}^{f(N)}\left(1+\frac{z}{v\_i(N)}\right)\in \mathbb{C}\right),$
where $v\_i:\mathbb{N}\to \mathbb{C}, |v\_i(N)|\geq g(N) \mbox{ for } i=1,2,\ldots,f(N);$
and such that $\li... | https://mathoverflow.net/users/39115 | Infinite product's question | $S$ is the set of all entire zero-free functions $F$ with $F(0)=F'(0)=1$. To approximate such an $F$, just cut off its Taylor series $F(z)=1+z+\sum\_{n\ge 2} a\_n z^n$ at high enough degree $N\_1=f(1)$. Make sure this polynomial $p$ approximates $F$ well enough on $|z|\le 1=g(1)$ (say) so that it will be zero-free ther... | 1 | https://mathoverflow.net/users/48839 | 171447 | 87,631 |
https://mathoverflow.net/questions/171405 | 2 | So, I feel like I'm missing something obvious, but I have the following situation:
Let $X\to Y$ be a finite group quotient of schemes (in fact, varieties) by the finite group $G$. Let $\tilde{Y}\to Y$ be any resolution of singularities. Then we have a natural map $X\times\_Y \tilde{Y}\to X$ which is birational, and $... | https://mathoverflow.net/users/622 | Finite Quotients and Resolutions of Singularities | I suspect that you will have trouble with trying to "simultaneously" resolve $X$ and $X/G$. We actually run into this issue in the joint paper with Anatoly Libgober arXiv:math/0206241 (although it may not be evident from the paper) and consequently had to settle for working with $\hat X\to\hat Y$ which was $G$-equivari... | 5 | https://mathoverflow.net/users/38468 | 171448 | 87,632 |
https://mathoverflow.net/questions/169305 | 6 | I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.
So I re-post it below.
Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$):
$$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\... | https://mathoverflow.net/users/33672 | Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms? | I've been hoping someone else would take a stab at Question 2. For myself, the answer relates to the work of de Bruijn (and Newman) mentioned in (3) above; see this question
[The Riemann zeros and the heat equation](https://mathoverflow.net/questions/115447/the-riemann-zeros-and-the-heat-equation)
for more on the de Br... | 6 | https://mathoverflow.net/users/6756 | 171454 | 87,634 |
https://mathoverflow.net/questions/171453 | 9 | My questions concern the following quote from “The HOD Dichotomy”, page 8.
"… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean
something else there. Also, $\{S\subseteq\lambda\mid S\in HOD \text{ and S is stationary}\}$ belongs
to $HOD$ even though there might be sets which are stati... | https://mathoverflow.net/users/5697 | Stationary sets in HOD | The broader point here is that $\text{HOD}$ has all the sets of ordinals that are definable in $V$, and in this way it is able to know some things about what is going on in $V$, even if it cannot see the full reasons for those facts. For example, $\text{HOD}$ has the function giving the cofinality in $V$ of any ordinal... | 13 | https://mathoverflow.net/users/1946 | 171455 | 87,635 |
https://mathoverflow.net/questions/171452 | 3 | Let $\mathbb{B}$ denote the groupoid of finite sets and bijections.
A functor $F : \mathbf{Set} \to \mathbf{Set}$ is *analytic* if it is the left Kan extension of some functor $G : \mathbb{B} \to \mathbf{Set}$ (*i.e.* a *species*) along the inclusion functor $\iota : \mathbb{B} \to \mathbf{Set}$; that is, if there is... | https://mathoverflow.net/users/856 | Examples of functors $\mathbf{Set} \to \mathbf{Set}$ which are not analytic | In his article *Foncteurs analytiques et espèces de structures* (Lecture Notes in Mathematics 1234), Joyal characterizes analytic functors as those which preserve filtered colimits, cofiltered limits, and weak pullbacks. So it's just a matter of finding functors which violate one of these properties.
A functor that ... | 11 | https://mathoverflow.net/users/2926 | 171456 | 87,636 |
https://mathoverflow.net/questions/171457 | 16 | The classic Hurwitz theorem for rational approximations (in simplest form; the constant can of course be improved) gives infinitely many approximations $\frac mn$ to an irrational $\alpha$ with $|\frac mn-\alpha|\lt\frac1{n^2}$. Just recently, in trying to answer a question related to rational approximation of $\pi$ I ... | https://mathoverflow.net/users/7092 | Can we get good rational approximations in all residue classes? | The answer is no. Take $\alpha=\sqrt{2}$ and note that if $|\sqrt{2}-m/n|\le 1/n^2$ then we have $0<|2n^2-m^2| \le (\sqrt{2}n+m)/n \le 3$. Now suppose we want $n\equiv 4\pmod p$ say. Then we must have that $32-m^2 \equiv b \pmod p$ for some $|b|\le 3$. But we can find a prime $p$ for which the numbers $29$ to $35$ are ... | 24 | https://mathoverflow.net/users/38624 | 171461 | 87,637 |
https://mathoverflow.net/questions/168772 | 6 | I kindly would like to ask you the following- I am refering to page
175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms
..."
My question to which I would kindly ask you to answer me is : Are
there unitary representations of SL(2, Q\_P) of type I (that is with
spherical functions) that are not in th... | https://mathoverflow.net/users/51506 | Spherical functions for sl(2,Q_p) | Be careful that among the irreducible unitary reps also the trivial representation has this property. That's why Paul Garrett says "embeds" into a prinicpal series, so you get not only unramified unitary principal series, but also the trivial representations.
Paul Garrett doesn't address the trace formula, so I will ... | 1 | https://mathoverflow.net/users/10400 | 171471 | 87,640 |
https://mathoverflow.net/questions/168883 | 7 | To my knowledge, usually there are two ways to construct supercuspidal representations over p-adic fields. The first is via theory of types (for GL(n) and classical groups), notably by Bushnell, Kutzko, Stevens etc. The other is the construction given by Yu,Jiu-Kang (for more general groups).
Both constructions are v... | https://mathoverflow.net/users/1832 | Questions on constructions of supercuspidal representations | There are certainly links between the two; a good place to start would be looking at the theory for $GL\_2$ and $SL\_2$. Henniart's appendix to Breuil-Mezard's *Multiplicités modulaires et représentations de $GL\_2(Z\_p)$ et de $Gal(\bar{Q\_p}/Q\_p)$ en $l=p$* makes explicit the construction for $GL\_2$ via strata, and... | 3 | https://mathoverflow.net/users/29273 | 171483 | 87,648 |
https://mathoverflow.net/questions/171484 | 1 | Let $S = Spec(O\_K)$ be the spectrum of the rings of integers of a number field $K$. Let $A/S \setminus T$ be an Abelian scheme over an open subscheme $S \setminus T \subseteq S$. Does the kernel of $n$-multiplication $A[n]$ become constant after a (non empty) étale base change $S' \to S \setminus \{x\_1,\ldots,x\_n\}$... | https://mathoverflow.net/users/38981 | kernel of isogeny becomes constant after base change | Do you really mean to consider an abelian scheme over the *entire* ring of integers, and not just a localization thereof? Either way, every finite flat group scheme $G$ over a domain $R$ with fraction field $F$ such that ${\rm{char}}(F)$ does not divide the order $n$ of $G$ is etale over $R[1/n]$ (as can be checked on ... | 4 | https://mathoverflow.net/users/52824 | 171489 | 87,650 |
https://mathoverflow.net/questions/171492 | 2 | Let $M$ denote a well-founded set-sized model of ZFC. The *descent value* of $M$ will be defined as the value of $n$ returned by the following process.
**Initialization.** Let $n$ equal $0$ and $X$ equal $M$.
**Step.** If $L\_{\omega\_1^X}^X$ doesn't satisfy ZFC according to $M$, halt and output $n$. Otherwise, inc... | https://mathoverflow.net/users/26080 | Do models of ZFC have arbitrarily large descent values? | No, the descent value is always at most $1$ in any model of ZFC, whether it is well-founded or not. To see this, observe that if the descent value isn't $0$, then on the next step you have $X=L\_{\omega\_1^M}^M$, and so $X$ satisfies $V=L$, and so $\omega\_1^X$ is the $\omega\_1^L$ inside $M$, and $L\_{\omega\_1^L}$ ne... | 6 | https://mathoverflow.net/users/1946 | 171494 | 87,652 |
https://mathoverflow.net/questions/171495 | 6 | Let $X$ be a complex algebraic variety, possibly singular and/or non-compact. It is well known that if $X$ is smooth then its Euler characteristic is equal to its Euler characteristic with compact support: $\chi(X)=\chi\_c(X)$.
**Questions.** (1) Is the same equality true if $X$ is singular? (I think it is still true... | https://mathoverflow.net/users/16183 | Euler characteristics with and without compact support of algebraic varieties | Let $e(-)$ denote ordinary Euler characteristic and $\chi\_c(-)$ the compactly supported version.
Complex varieties admit Whitney stratifications. In particular, each closed stratum in such a stratification is a strong deformation retract of a tubular neighborhood. It follows (Mayer-Vietoris) that if $Y\subseteq X$ i... | 6 | https://mathoverflow.net/users/23907 | 171501 | 87,656 |
https://mathoverflow.net/questions/169227 | 8 | Suppose we take, for example, the $C^\*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This is the commutative $C^\*$-algebra of almost periodic functions whose spectrum lies in $\mathbb{Z} + \theta \mathbb{Z}.$... | https://mathoverflow.net/users/22781 | C* algebras of Almost Periodic Functions | To answer my own question: One can see that the $C^\*$-algebra of almost periodic functions with spectrum in $\mathbb{Z}+ \theta \mathbb{Z}$ is isomorphic to the $C^\*$-algebra of a torus. It is generated by two commuting unitaries $e^{2\pi i x}$ and $e^{2 \pi i \theta x}.$ This same argument shows that for any finitel... | 1 | https://mathoverflow.net/users/22781 | 171502 | 87,657 |
https://mathoverflow.net/questions/171479 | 10 | Let $(X\_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is
$$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X\_i-X\_{i-1}|>c\_i) \,\leq\, \epsilon$$
for suitable constants $(c\_i)$ and $\epsilon>0$.
I read in Dubhashi-Panconesi book that for all $t>0$
$$\mathbb{P}(X\_n... | https://mathoverflow.net/users/45002 | Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability) | The general idea behind such inequalities is to follow the martingale $X$ until you lose control over the differences, then force it to be constant. This defines a new martingale $Y$ with bounded differences, which is therefore concentrated. You then add to your probability of error the probability that the differences... | 6 | https://mathoverflow.net/users/25485 | 171505 | 87,658 |
https://mathoverflow.net/questions/171458 | 15 | There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1\_1$" – indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in general (see Sacks' wonderful book [Link](https://web.archive.org/web/20200813160253/https://projecteuclid.org/euclid.pl... | https://mathoverflow.net/users/8133 | Higher recursion theory and reverse mathematics: What is to $\Pi^1_1$-$CA_0$ as $RCA_0$ is to $ACA_0$? | I am getting the feeling there is some slight miss-match of terminology, or perhaps application of terminology is a better way of putting it? It is true that the (lightface) $\Pi^1\_1$-$CA\_0$ sets of integers correspond to the meta-r.e. sets of integers, but as you say, the old meta-recursion theory lies on $\omega\_1... | 8 | https://mathoverflow.net/users/6942 | 171509 | 87,660 |
https://mathoverflow.net/questions/171512 | 16 | Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
Crossposted on [StackExchange](https://math.stackexchange.com/q/829417/60713).
| https://mathoverflow.net/users/44134 | Equivariant version of Morse theory | The answer to your question is *yes, of course*. The theory has been around at least since the late 60s! See Wasserman's [paper](http://www.maths.ed.ac.uk/~aar/papers/wasser001.pdf)
>
> A Wasserman. *Equivariant differential topology*, Topology 1969; 8(2):127-150.
>
>
>
I think the first "big application" is... | 17 | https://mathoverflow.net/users/18263 | 171515 | 87,663 |
https://mathoverflow.net/questions/171477 | 3 | Does anybody knows a reference for the following statement?
Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in a neighborhood of $0$ with a non-degenerate critical point at 0.
Then, there exists a local $S^1$-equivariant diffeomo... | https://mathoverflow.net/users/14547 | A reference for an equivariant Morse Lemma | Check Wasserman, Arthur G. Equivariant differential topology. Topology 8 1969 127--150. MR0250324 (40 #3563). In particular see Lemma 4.1 for an equivariant Morse lemma.
| 4 | https://mathoverflow.net/users/1822 | 171517 | 87,664 |
https://mathoverflow.net/questions/169216 | 15 | The paper of Bryan and Leung ["The enumerative geometry of $K3$ surfaces and modular forms"](http://arxiv.org/abs/alg-geom/9711031/) provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing a primitive homology class. If $N\_g(n)$ is the number of curves of geometric ... | https://mathoverflow.net/users/19436 | Curves on K3 and modular forms | The answer to your first question: "What is the relationship between $G\_2$ and $\Delta$?" is
$$q\frac{d}{dq} \log \Delta = -24G\_2 $$
where
$$\Delta = q\prod\_{m=1}^\infty (1-q^m)^{24}$$ and
$$G\_2 = -\frac{1}{24} +\sum\_{d=1}^\infty \sum\_{k|d}k q^d$$
(note that I've included the constant -1/24 which is in us... | 19 | https://mathoverflow.net/users/9617 | 171521 | 87,666 |
https://mathoverflow.net/questions/171506 | 3 | **Background:**
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}\_k\{H; k\},
$$
we recall (see Abe Chapter 4 for example) that the Hopf algebra structure of $H$ induces in a canonical way a group structure on $\widehat{H}$.... | https://mathoverflow.net/users/51325 | Algebraic Groups, Modules, and Comodules | A more comprehensive reference than Waterhouse is the book *Representations of Algebraic Groups* by J.C. Jantzen (2nd ed., AMS, 2003). Though he aims after a while at prime characteristic, his foundational material in Part I is much more general and often follows the treatment in the older book (in French) by Demazure-... | 1 | https://mathoverflow.net/users/4231 | 171526 | 87,667 |
https://mathoverflow.net/questions/171525 | 5 | Let $X$ be a *smooth* complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a *finite* group $\Gamma$ on $X$.
>
> Can we say that each irreducible component of the fix point set $X^{\Gamma}$ is smooth?
>
>
>
Here $X^{\Gamma}:=\{ x\in X\; | \; g(x):=\varphi\_g (x)=x\... | https://mathoverflow.net/users/4721 | Smoothness of fix point components of finite group action on smooth variety | Yes. The reason is that locally around a fixed point, the action linearizes, i.e. is analytically equivalent to a linear action of $\Gamma $ onto a vector space -- so the fixed locus is locally isomorphic to a linear space. This fact, which follows easily from an averaging process, goes back (at least) to H. Cartan, in... | 9 | https://mathoverflow.net/users/40297 | 171529 | 87,668 |
https://mathoverflow.net/questions/171519 | 6 | Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such.
Sorry if this is too silly, but so far I'm stuck.
| https://mathoverflow.net/users/25602 | Injective flat module | Here's a counterexample. Let $k$ be a field and $R=k[x,y]/(x^2,xy,y^2)$; this is a local ring with maximal ideal $m=(x,y)$. Suppose $F$ is a flat $R$-module. Then there is a short exact sequence $$0\to m\otimes F\to F\to F/m\to 0$$
But $m\cong R/m\oplus R/m$ (with $x$ and $y$ generating the summands), so this sequen... | 10 | https://mathoverflow.net/users/75 | 171530 | 87,669 |
https://mathoverflow.net/questions/171550 | 0 | What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in reasonnable time.
But I need to solve several different MIP problems, and in the end, it takes too much.
Is there someth... | https://mathoverflow.net/users/nan | Approximate solution to large mixed integer programming problem | You might try a local optimization approach. Guess values for the integer variables and solve the remaining continuous linear programming problem. Using sensitivity
analysis, see if the solution can be improved by changing one of the integer variables. Repeat as many times as you can or until no further improvements ca... | 1 | https://mathoverflow.net/users/13650 | 171561 | 87,681 |
https://mathoverflow.net/questions/169141 | 10 | This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and assume the existence of a $1$-morphism $f:x \to y$ in $\mathbf{C}$ from some object $x$ to some object $y$ and moreover ... | https://mathoverflow.net/users/18263 | Localizing 2-categories about a single morphism | Well, if you are happy with everything maximally strict and finite, then you are really looking for a description of the coinverter (a weighted colimit) in the 1-category of finite 2-categories enriched over the cartesian monoidal category of finite 2-categories. See section 2.1 of [John Bourke' thesis](http://www.math... | 2 | https://mathoverflow.net/users/4177 | 171563 | 87,682 |
https://mathoverflow.net/questions/171555 | 11 | About the order of finite simple groups there exists a very interesting result which stated as follows:
Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ is a prime, then $p>3$ and either $G\cong L\_2(p)$ or $p$ is a Fermat prime and $G\cong L\_2(p-1)$.
This result proved by... | https://mathoverflow.net/users/31045 | On the order of finite simple groups | I think you will just have to work through the orders of the finite simple groups and show that only $L\_2(p)$ is possible.
Let's try $L\_3(q)$. This has order $q^3(q^2+q+1)(q+1)(q-1)^2/(q-1,3)$. Your prime $p$ must divide one of these factors, so the largest possible value of $p$ is $q^2+q+1$, and you will find that... | 11 | https://mathoverflow.net/users/35840 | 171572 | 87,686 |
https://mathoverflow.net/questions/171568 | 1 | I'm reading Burago, Burago and Ivanov's book, and I'm on the section about Strainers. The authors say that it is obvious that the set of $(m,\varepsilon)$-strained points for any fixed natural number $m$ and $\varepsilon >0$ is open. I'm failing to see why is it so obvious.
As I understand, I have to prove that for ... | https://mathoverflow.net/users/52863 | The set of strained points in an Alexandrov space is open | Note that this is an open condition, which depends on the distances $|a\_ia\_j|$,
$|a\_ib\_j|$, $|b\_ib\_j|$, $|pa\_i|$ and $|pb\_i|$, hence the result.
| 4 | https://mathoverflow.net/users/1441 | 171585 | 87,692 |
https://mathoverflow.net/questions/171590 | 3 | I'm looking for the following result:
>
> Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. The map
> $$u \mapsto \int\_0^T \int\_{\Omega} f(u(t))$$
> is lower semicontinuous for $u \in L^2(0,T;L^2)$ where $f:\mathbb R \to \mathbb R$ is convex.
>
>
>
Does anyone know how to prove this, or a reference fo... | https://mathoverflow.net/users/52874 | Lower semicontinuity of a Bochner integral of a convex function | Use the following definition for lower semi continuity. That is, $f$ is lower semi continuous at $x\_0$ is $\liminf\_{x\rightarrow x\_0} f(x) \geq f(x\_0)$. This is equivalent to
For all $\epsilon > 0$ there exists $\delta > 0$ so that $\epsilon \geq f(x\_0) - f(x)$ for all $x \in B\_\delta(x\_0)$
Fix $u \in L^2(0,... | 1 | https://mathoverflow.net/users/49404 | 171594 | 87,695 |
https://mathoverflow.net/questions/171589 | 10 | I would like to know the structure of the fixed set of an order $p$ automorphism [Edit: induced by a matrix in $GL\_2(K)$] on the Bruhat-Tits tree for a p-adic field $K$, specifically in the case where the order divides (you can assume equals) the residue characteristic.
I have convinced myself that when the order i... | https://mathoverflow.net/users/50846 | Fixed set of order p automorphism of Bruhat-Tits tree | First a caveat: there is a difference between "an automorphism of the Bruhat-Tits tree $T$ of a $p$-adic field $K$" and "an automorphism of $T$ induced by a matrix in $GL\_2(K)$". The first set is much larger: contrarily to higher rank buildings, Bruhat-Tits trees are very "soft" and have a lot of automorphisms. I beli... | 9 | https://mathoverflow.net/users/9317 | 171597 | 87,697 |
https://mathoverflow.net/questions/171602 | 2 | Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p\_n}$, where $p\_n$ is the n-th prime and $0\le a \le b \le \frac{p\_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-decreasing? Data for the first thousand values of the sequence supports this conjecture.
Here is an example for $n=5$:... | https://mathoverflow.net/users/40145 | How does this sequence grow | The answer is yes, and the number of solutions with a prime $p$ is $\lfloor \frac{p+5}{8} \rfloor$ when $p \not\equiv 1 \pmod{8}$ and is $\lfloor \frac{p+5}{8} \rfloor + 1$ when $p \equiv 1 \pmod{8}$.
The equation $a^{2} + b^{2} + c^{2} = 0$ defines a conic in $\mathbb{P}^{2}/\mathbb{F}\_{p}$. If $p > 2$ this conic ... | 6 | https://mathoverflow.net/users/48142 | 171608 | 87,698 |
https://mathoverflow.net/questions/168126 | 13 | The numbers $2^{n(n+1)/2}$ come up in various enumerative contexts. In addition to the trivial example (bit-strings of length $n(n+1)/2$) and the old example of domino tilings of Aztec diamonds (Elkies, Kuperberg, Larsen, and Propp: see <http://www.emis.de/journals/JACO/Volume1_2/x9m7n00g384067u3.fulltext.pdf> and <htt... | https://mathoverflow.net/users/3621 | Two to the power of a triangular number: bijections | "Domino-shuffling on Novak Half-Hexagons and Aztec Half-Diamonds" by Nordenstam and Young (<http://arxiv.org/abs/1103.5054>) discusses the classes AD($n$) (domino tilings of Aztec diamonds), NILP($n$) (non-intersecting lattice paths), LT($n$) (lozenge tilings of trapezoids), HH($n$) (perfect matchings of the half hexag... | 5 | https://mathoverflow.net/users/3621 | 171625 | 87,704 |
https://mathoverflow.net/questions/166035 | 3 | My question is motivated by the following question.
[How transitive are the actions of symplectomorphism groups ?](https://mathoverflow.net/questions/61994/how-transitive-are-the-actions-of-symplectomorphism-groups)
A subset $X$ of a symplectic manifold $M^{2n}$ is called $\it displaceable$ if there is a Hamiltonia... | https://mathoverflow.net/users/11846 | Displaceability of submanifolds | There is a result of Basak Gurel which says that a nowhere coisotropic submanifold of a symplectic manifold is displaceable (by a Hamiltonian diffeomorphism) as soon as it's infinitesimally displaceable (meaning its normal bundle admits a nowhere vanishing section). A submanifold $X$ of a symplectic manifold $(M,\omega... | 3 | https://mathoverflow.net/users/39725 | 171630 | 87,706 |
https://mathoverflow.net/questions/171623 | 9 | Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact nowhere dense sets $\{f\in\mathbb{R}^\omega:\forall m(|f(m)|\leq n)\}$, and hence is not Baire.
Recall a topological grou... | https://mathoverflow.net/users/16107 | Is $\ell^\infty$ Polishable? | The answer is no.
Su Gao $\textit{Invariant Descriptive Set Theory}$, Lemma 9.3.3 has a direct proof.
---
As people usually do, denote the equivalence relation $\ell\_\infty : = \mathbb{R}^\omega \backslash \ell\_\infty$ (where of course the latter $\ell\_\infty$ refers to the group).
$\ell\_\infty$ is unive... | 7 | https://mathoverflow.net/users/43354 | 171632 | 87,707 |
https://mathoverflow.net/questions/171638 | 4 | In *[Le group de Brauer II](http://www.numdam.org/numdam-bin/item?id=SB_1964-1966__9__287_0)*, Grothendieck states
>
> Proposition 1.4.- Soit $X$ a préschéma noetherien. Supposon que les anneaux hensélisés stricts des anneaux locaux de $X$ soient factoriels, [...] Alors les groupes $H^q(X,\underline{G\_m})$ sont de... | https://mathoverflow.net/users/4177 | Unravelling some hypotheses on a variety | The henselization of a regular local ring is regular, and hence by Auslander-Buchsbaum it is a UFD. So any nonsingular variety satisfies the hypotheses.
| 4 | https://mathoverflow.net/users/396 | 171639 | 87,710 |
https://mathoverflow.net/questions/171644 | 4 | Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality:
**Q1.** There exists a set $\ B\ $ of $\ k\ $ consecutive primes such that $\ q\equiv r \mod 2\!\cdot\! a\ $ for every $\ q\in B$.
(The con... | https://mathoverflow.net/users/8385 | Prime residua races and two views on primes | The answer to question 1 is yes; see Tristan Freiberg's Ph.D. thesis [Strings of congruent primes in short intervals](https://papyrus.bib.umontreal.ca/xmlui/bitstream/handle/1866/4556/freiberg_tristan_m_2010_these.pdf;jsessionid=770A3B94C710637E4D59BA5A5D53DBD5?sequence=4). There it is mentioned that even the case $|B|... | 7 | https://mathoverflow.net/users/23008 | 171655 | 87,717 |
https://mathoverflow.net/questions/171646 | 5 | Given two real closed fields $R\_1$ and $R\_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the language of rings).
Under which conditions can we get that they are isomorphic? Isomorphic to $\mathbb{R}$?
| https://mathoverflow.net/users/38200 | Isomorphism of real closed fields | Any two back-and-forth equivalent archimedean ordered fields (or domains) are isomorphic. (This refers to the language of ordered rings, but in your case, this makes no difference, as the order is definable in the ring structure for real-closed fields).
Archimedean fields are canonically isomorphic to subfields of $\... | 9 | https://mathoverflow.net/users/12705 | 171663 | 87,720 |
https://mathoverflow.net/questions/171662 | 16 | I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to prove this when the space is sufficiently nice (metric spaces). However I do not understand the proof in full generalit... | https://mathoverflow.net/users/184 | Why does the singular simplicial space geometrically realize to the original space? | You can just write down the required homotopy.
A point in $|\text{Sing}(X)|$ is an equivalence class $[\sigma,u]$ where $u\in\Delta\_n$ and $\sigma:\Delta\_n\to X$. Define $\theta^n\_{u,t}:\Delta\_n\to\Delta\_n$ by $\theta^n\_{u,t}(x)=tx+(1-t)u$. Then define $\phi\_t[\sigma,u]=[\sigma\circ\theta^n\_{u,t},u]$. To see... | 19 | https://mathoverflow.net/users/10366 | 171664 | 87,721 |
https://mathoverflow.net/questions/171665 | 8 | *(I've asked the same question [at the MSE](https://math.stackexchange.com/questions/816155/have-you-seen-this-property-of-tolerance-relations-before), so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)*
Let $A$ be a set equippe... | https://mathoverflow.net/users/47071 | Do you have examples of such "transitive" elements? | Let $A$ be the integers bigger than $1$ and let $a$ be related to $b$ if and only if $\gcd(a,b)>1$. Then $x$ has property $(\star)$ exactly when $x$ is a prime power.
| 7 | https://mathoverflow.net/users/17836 | 171681 | 87,726 |
https://mathoverflow.net/questions/171624 | 10 | Let $A$ be a $C^\*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits\_{i=1}^n a\_i\otimes b\_i \mapsto \sum\limits\_{i=1}^n a\_i b\_i$$
is continuous with respect to the minimal/maximal tensor product of $C^\*$-algebras?
| https://mathoverflow.net/users/8699 | Continuity of the product map | I think the answer is: It's always continuous if $A$ is subhomogeneous and never continuous otherwise(min or max).
First notice that if the product map is continuous for min, then it's continuous for max because we can factor the product map as $A\otimes\_{max}A\rightarrow A\otimes\_{min}A\rightarrow A$ where the fir... | 10 | https://mathoverflow.net/users/34640 | 171683 | 87,727 |
https://mathoverflow.net/questions/171680 | 8 | When browsing the literature, I have found the following theorem of E. Tokarev:
>
> Let $X$ be a Banach lattice with weakly sequentially complete dual space. Then for any Banach space $Y$, every unconditionally converging operator $T\colon X\to Y$ is weakly compact. (In other words, Banach lattices with weakly sequ... | https://mathoverflow.net/users/15129 | Tokarev's theorem on Banach lattices which are Grothendieck spaces | There is a counterexample in
Figiel, T.; Ghoussoub, N.; Johnson, W. B.
On the structure of nonweakly compact operators on Banach lattices.
Math. Ann. 257 (1981), no. 3, 317–334.
| 7 | https://mathoverflow.net/users/2554 | 171685 | 87,729 |
https://mathoverflow.net/questions/171669 | 3 | Let $X$ be a projective variety over a field $k$ of characteristic $0$ and let $f: X \to \mathbb{P}^d$ be a $k$-morphism. Can we always find an embedding $i: X \hookrightarrow \mathbb{P}^N$ such that the morphism $g=f \circ i^{-1}: i(X) \to \mathbb{P}^d$ is given by $g(x)=(g\_0(x): \ldots : g\_d(x))$ for all $x \in i(X... | https://mathoverflow.net/users/36563 | Representing projective morphisms by homogeneous polynomials | I agree, the answer is no in general. Let's take the usual bijection: rational maps from $X$ to projective space correspond to subspaces of the global sections of line bundles on $X$.
Now you're looking for a way of embedding your variety $X$ into projective space so that when we represent the map $f$ in coordinates... | 4 | https://mathoverflow.net/users/52918 | 171687 | 87,730 |
https://mathoverflow.net/questions/171701 | 2 | I am wondering if the following ODE belongs to a well-studied class and if anything is known about its solutions:
$\partial\_t\theta = \sin(\theta)\cos(2\pi t) + \kappa$.
This equation roughly corresponds to an overdamped oscillator with applied constant torque. I am particularly interested in the values of $\kappa... | https://mathoverflow.net/users/19673 | First order ODE and possible periodic solutions | By putting $y=w\exp(\int\cos(2\pi t)/2dt)$ you kill the first derivative term in the
second order linear equation, and obtain the equation of the form $w^{\prime\prime}+Qw=0$,
where $Q$ is a trigonometric polynomial. This is called Hill's equation and your problem is
an eigenvalue problem for it. Such eigenvalue proble... | 2 | https://mathoverflow.net/users/25510 | 171710 | 87,737 |
https://mathoverflow.net/questions/171709 | 4 | This may be basic, and if it is I apologize, but I have found no references to it in literature. I would appreciate a reference at least if I am wrong. I have supplied background for those interested, but you are more than welcome to skip it - I have indicated where it begins and ends.
**BACKGROUND STARTS HERE**
I ... | https://mathoverflow.net/users/33647 | Predual of a Direct Sum of Banach Spaces | The answer is **no** as $X$ need not have a predual. Indeed, take a space $X$ which is compelemnted in its bidual but not isomorphic to a dual space (*e.g.* $X=L\_1$) and consider $X^{\*\*} = X \oplus Y$. [You can produce even more weird examples](https://math.stackexchange.com/questions/966208/are-there-spaces-smaller... | 8 | https://mathoverflow.net/users/15129 | 171713 | 87,739 |
https://mathoverflow.net/questions/171688 | 7 | At the end of [this preprint](http://arxiv.org/pdf/1306.4246.pdf), I make the following conjecture concerning the roots of the matching polynomial:
>
> If a graph $G$ is connected and contains a cycle, then the spectral radius of $G$ strictly exceeds the largest root of the matching polynomial $\mu\_G(z)$.
>
>
> ... | https://mathoverflow.net/users/34341 | Roots of matching polynomial of graph | The moments (power symmetric functions, sums of powers of the roots of) the characteristic polynomial enumerate all closed walks in the graph. Chris Godsil proved that the moments of the matching polynomial count a particular type of closed walk, called tree-like. Graphs with cycles have some closed walks that are not ... | 9 | https://mathoverflow.net/users/9025 | 171721 | 87,744 |
https://mathoverflow.net/questions/171553 | 12 | Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, where the last one is the axiom schema of induction yielding an axiom for each wff $\Phi(a)$:
* $a+0=a$
* $a\times1=a$
* $... | https://mathoverflow.net/users/9550 | Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$? | A stark demonstration of why precisely defining how you form $PA\_{\lambda +1}$ for $\lambda$ a limit ordinal: in 1939 Turing showed that if $\varphi$ is a true $\Pi^0\_1$ statement, there is a notation for $\omega+1$ according to which $PA\_{\omega+1}$ proves $\varphi$.
Less pathologically, I believe (although I can... | 8 | https://mathoverflow.net/users/8133 | 171723 | 87,745 |
https://mathoverflow.net/questions/169053 | 12 | Over a decade ago Alexander Postnikov developed his own way of looking at perfect matchings of bipartite plane graphs. As I recall, he starts with a 2-coloring of the square grid and creates a new graph whose vertices are the monochromatic patches in the grid and whose edges correspond to adjacencies of patches.
Can ... | https://mathoverflow.net/users/3621 | Postnikov's approach to perfect matchings of graphs | Lauren Williams pointed me toward <https://arxiv.org/abs/math/0609764> (the original reference for Postnikov's work) as well as <https://arxiv.org/abs/0706.2501> (an article by Postnikov, Speyer, and Williams with more of an emphasis on matchings and flows). In this setting one has Plucker relations that are essentiall... | 5 | https://mathoverflow.net/users/3621 | 171726 | 87,747 |
https://mathoverflow.net/questions/171724 | 30 | This is perhaps unanswerable,
or perhaps I am too algebraically ignorant to phrase it cogently, but:
>
> Is there some identifiable reason that polynomials over
> $\mathbb{C}$,
> $\mathbb{R}$, $\mathbb{Q}$, $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$
> are so pervasively useful in mathematics?
>
>
>
Is it becaus... | https://mathoverflow.net/users/6094 | Why are polynomials so useful in mathematics? | Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R$ are precisely the morphisms in the Lawvere theory of commutative $R$-algebras. So in some sense caring about polynomia... | 35 | https://mathoverflow.net/users/290 | 171728 | 87,748 |
https://mathoverflow.net/questions/171731 | 4 | Let $G$ be a connected complex reductive group with a maximal compact subgroup $K$.
Let $\lambda$ be a dominant weight in the interior of the positive Weyl chamber. Let $V\_\lambda$ denote the irreducible representation with highest weight $\lambda$ and fix a highest weight vector $v\_\lambda$. Then $gB \mapsto [g. v\_... | https://mathoverflow.net/users/43696 | Kostant-Kirillov form versus Fubini-Study form on Plucker embedding | The answer is yes, due to formal properties of moment maps (that depend on almost no details of your situation). Namely, suppose $K$ acts transitively on any symplectic manifold $(X,\omega\_X)$, with *equivariant* moment map $\mu:X\to\mathfrak k^\*$. Then $\mu(X)$ is a coadjoint orbit $\mathcal O$ and $\omega\_X=\mu^\*... | 2 | https://mathoverflow.net/users/19276 | 171732 | 87,751 |
https://mathoverflow.net/questions/171677 | 9 | This is a follow-up to a recent question asked by Peter Crooks [here](https://mathoverflow.net/questions/171383/). The answer by Ben Webster includes a helpful [link](https://arxiv.org/abs/math/0205048) to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper *Symplectic resolutions for nilpotent orbits*.... | https://mathoverflow.net/users/4231 | Examples of Richardson orbit closures not having a symplectic resolution? | The relevant information is in the article of Hesselink: [Polarizations in the classical groups](https://research.rug.nl/en/publications/polarizations-in-the-classical-groups) ([Wayback Machine](https://web.archive.org/web/20211226101336/https://pure.rug.nl/ws/portalfiles/portal/3442454/1978MathZHesselink.pdf)).
Fix ... | 4 | https://mathoverflow.net/users/66 | 171736 | 87,753 |
https://mathoverflow.net/questions/171738 | 3 | My question is quite simple and elementary.
Let $A(x)=\sum\_{1}^{x}a(n)$ and $\alpha(s)=\sum\_{1}^{\infty}a(n)n^{-s}$. Then, as we know,
$$ A(x)= \int\_{\gamma-i\infty}^{\gamma+i\infty}\frac{\alpha(s)}{s}x^sds$$
for some $\gamma$ such that $\alpha(\gamma)<\infty$.
Now the question comes, first, we denote $R\_i$ $(i... | https://mathoverflow.net/users/49625 | A question on the big-O value of the complex integral especially in the number theory | Here is a counterexample assuming the Riemann Hypothesis. Let $a(n):=\Lambda(n)-1$, where $\Lambda$ is the von Mangoldt function. Then $A(x)=\psi(x)-[x]$, where $\psi$ is the Chebyshev function, and $\alpha(s)=-\frac{\zeta'(s)}{\zeta(s)}-\zeta(s)$, where $\zeta$ is the Riemann zeta function. So the biggest residue of $... | 5 | https://mathoverflow.net/users/11919 | 171752 | 87,758 |
https://mathoverflow.net/questions/171686 | 6 | I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.
More specifically, let $R$ be a ring, let $M$ be a left $R$-module, and let $C: 0 \rightarrow C^0 \rightarrow C^1 \rightarrow C^2 ... | https://mathoverflow.net/users/7932 | Interpretations of differentials in hypercohomology spectral sequences as Yoneda products | Let me give an interpretation for $d\_2$ along the lines that you want. Let $\tau\_{\le p}C$ be the truncation operator which sets all terms $C^k=0$ for $k>p$, keeps $C^k$ for $k<p$ and replaces $C^p$ with $\ker d$. This operator passes to the derived category $D= D^+(R\text{-}mod)$. We have a distinguished triangle
$$... | 5 | https://mathoverflow.net/users/4144 | 171762 | 87,762 |
https://mathoverflow.net/questions/171708 | 7 | Suppose I have a hyperelliptic curve of genus $2$ over $\mathbb Q$. I want to get information about its Jacobian reduction at prime $p$ (especially, in case $p=2$). Also I'm interesting in the group of connected components of the Neron model of the Jacobian.
Is it possible to get such information using some computer ... | https://mathoverflow.net/users/10300 | Calculate reduction of Jacobian of hyperelliptic curve | The relevant information can be obtained from a regular model of the curve over ${\mathbb Z}\_p$. Such a model can be computed by repeatedly blowing up points or components of the special fiber that are singular on the arithmetic surface one obtains from the original curve. More or less detailed examples of this can be... | 4 | https://mathoverflow.net/users/21146 | 171767 | 87,764 |
https://mathoverflow.net/questions/171717 | 25 | Grothendieck's homotopy hypothesis, is, as the $n$lab states:
**Theorem:** *There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$.*
>
> What are the applications of this hypothesis? Why is it so fundamental? Can it be "generalized", perhaps by using the following defi... | https://mathoverflow.net/users/nan | Grothendieck's Homotopy Hypothesis - Applications and Generalizations | For me this result fits in a context of other results that give complete algebraic invariants for homotopy types. The broad program sometimes goes under the rubric [Whitehead's algebraic homotopy program](http://ncatlab.org/nlab/show/algebraic+homotopy).
If we define a homotopy $n$-type (for $n \geq 1$) as an object... | 32 | https://mathoverflow.net/users/2926 | 171770 | 87,765 |
https://mathoverflow.net/questions/171520 | 2 | Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$.
Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a *finite* subset of the class of singleton P-null sets in $\mathcal{F}$.
I am trying to use $S$ to construct a new probability space from $... | https://mathoverflow.net/users/nan | Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets | The condition you have is not sufficient. Although the question has some wording issues, I can answer it as I think you intended it.
Suppose $(\mathbb{R},\mathcal{B},P)$ is our probability triple, where $P([0,\frac{1}{2^n}])=4^{-\lceil\frac{n}{2}\rceil}$. In this case, there can be no limit defining $P'(\{0\})$ if th... | 0 | https://mathoverflow.net/users/52931 | 171784 | 87,770 |
https://mathoverflow.net/questions/171749 | 1 | Suppose we have a $m \times n$ matrix $M$ with $n > m$ with entries over a finite field, say $\mathbb{F}\_q$ with $q$ considered to be large compared to $m,n$. Suppose that $M$ has the property that every $m \times m$ minor of $M$ is invertible. Is there an efficient way to adjoin a column to $M$ to obtain a $m \times ... | https://mathoverflow.net/users/10898 | Extending a matrix with a certain property over a finite field | I don't know an efficient algorithm answering your question but presumably it would have to break down for large $n$ - the MDS conjecture gives the maximal $n=q+1$ for odd $q\geq m$ with minor change in the even $q$ case. See Simeon Ball's work e.g. here <http://www-ma4.upc.es/~simeon/jems-mds-conj-revised.pdf> proving... | 3 | https://mathoverflow.net/users/3143 | 171792 | 87,777 |
https://mathoverflow.net/questions/171806 | 6 | I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.
Question 1:
It is true that there exists $\epsilon>0$ such that for all $r < \epsilon$, there exists $c\_g >0$ (indep. of $x\in M$) su... | https://mathoverflow.net/users/34942 | Volume of geodesic balls | The answer to both questions is 'yes'.
To see this, just consider the exponential map $\exp:TM\to M$ and look at the pullback of the Riemannian volume form $dV$, say $\Omega = \exp^\*(dV)$ on $TM$. By the usual expansion in normal coordinates, there will be a smooth function $\phi$ on $TM$ that vanishes to order $2$ ... | 10 | https://mathoverflow.net/users/13972 | 171808 | 87,783 |
https://mathoverflow.net/questions/171807 | 11 | Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-l... | https://mathoverflow.net/users/3324 | Positive primes represented by indefinite binary quadratic form | Class field theory promises such a polynomial (more properly,
such a number field $H$, since a polynomial generating $H$ might have to err
on the first few primes, though in our case it turns out there's a
polynomial with no exceptional primes). The proof is effective,
though the recipe is often hard to carry out. So I... | 16 | https://mathoverflow.net/users/14830 | 171810 | 87,784 |
https://mathoverflow.net/questions/154528 | 2 | I am interested in the sequence
$$a(n)=\sum\_{k=0}^n {p(n-k) \choose k}$$
where $p(n)$ is a polynomial equation.
When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is quadratic?
For example when $p(n)=n^2$, it can be seen that $a(n)$ has superexponential growth by considering only ... | https://mathoverflow.net/users/45523 | Asymptotic behaviour of sequence | The easiest way to answer this question is with the steepest descent method, which is a standard techinque for calculating such asymptotic expansions. Here is the case $p(x)=x^2$. Write the sum as
$$ S = \sum\_{0\leq k\leq n} \binom{p(k)}{n-k} = \sum\_{0\leq k\leq n} s(k). $$
For the binomial coefficients one useful ... | 3 | https://mathoverflow.net/users/10423 | 171815 | 87,787 |
https://mathoverflow.net/questions/171785 | 1 | I want to modify my question mathoverflow.net/questions/50922. Let $C\_{p^e}$ be a cyclic group of order $p^e$, $p$ prime. Denote by $\text{Cext}(G,C\_p)$ the group of all central extensions of $C\_p$ by $G$, that is extensions with central subgroup $C\_p$ and the quotient $G$. Pick $G=C\_{p^e}^n$ for some natural $e,n... | https://mathoverflow.net/users/50922 | Number of non isomorphic groups in Cext(G,C_p) | The answer is very similar to that of <http://www.mathoverflow.net/questions/167446/> which was the case $e=1$. In fact it is a little more straightforward when $e>1$, because $p=2$ is no different from any other $p$. (This is because the $(xy)^{p^2} = x^{p^2}y^{p^2}$ for any $x,y$ in a group of this type, whereas $(xy... | 4 | https://mathoverflow.net/users/35840 | 171824 | 87,791 |
https://mathoverflow.net/questions/170347 | 6 | Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the [Bass conjecture](http://en.wikipedia.org/wiki/Bass_conjecture) states that $K\_0(A)$ is a finitely generated abelian group. As far as I am aware, this is wide open in general. My question is about whether something weaker... | https://mathoverflow.net/users/5101 | A weak version of Bass' conjecture | The following two statements might clarify the relation between Bass conjecture (for $K\_0$) and its weak version in the question. I only consider the case of commutative rings.
1. Contrary to the formulation of the question, the weak Bass conjecture implies the Bass conjecture for $A$ finitely generated over $\mathb... | 9 | https://mathoverflow.net/users/50846 | 171827 | 87,794 |
https://mathoverflow.net/questions/85193 | 6 | (Re)Reading [Lawvere theories versus classical universal algebra](https://mathoverflow.net/questions/69086/lawvere-theories-versus-classical-universal-algebra), I was reminded of a question I have had for quite some time:
>
> What is the best way to **present** Lawvere theories?
>
>
>
By *present*, I mean give... | https://mathoverflow.net/users/3993 | Presenting Lawvere theories? | Developing @AndrejBauer's suggestion to present them as single-sorted equational theories, you could write something like:
>
> Let $\mathsf{Grp}$ denote the Lawvere theory with the following
> presentation.
>
>
> Generators:
>
>
> 1. $c : 2 \rightarrow 1$
> 2. $e : 0 \rightarrow 1$
> 3. $i : 1 \rightarrow 1$
>... | 4 | https://mathoverflow.net/users/26080 | 171833 | 87,796 |
https://mathoverflow.net/questions/171769 | 0 | I need to prove the statement below. Since my background on Lie theory is rather weak, I post it here.
Let $\frak{g}$ be a complex semi-simple Lie algebra. Fix a Cartan subalgebra $\frak{h}$ with roots $\Delta\subset\frak{h}^\*$ and positive roots $\Delta^+\subset\Delta$, so that we have a decomposition
$$
\frak{g}=\... | https://mathoverflow.net/users/17294 | A property of compact involutions of semi-simple Lie algebras? | Actually the proof is just one line: because
$$
[\frak{h}\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}, \frak{h}\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}]\subset \mathbb{C}h\_\alpha\oplus\frak{g}\_\alpha\oplus\frak{g}\_{-\alpha}.
$$
As noted by Allen, this works for any root and any map preserving the Lie bra... | 0 | https://mathoverflow.net/users/17294 | 171841 | 87,798 |
https://mathoverflow.net/questions/171846 | 5 | In his lovely answer at [Positive primes represented by indefinite binary quadratic form](https://mathoverflow.net/questions/171807/positive-primes-represented-by-indefinite-binary-quadratic-form/171810#171810) Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if an... | https://mathoverflow.net/users/3324 | Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4 | I think the answer to the question as asked is no, but with $(205|p)=1$ it is yes, for then taking subfields of the Elkies field, look at $x^4 +x^3 + 3x^2 + 2x + 4$.
| 2 | https://mathoverflow.net/users/52997 | 171847 | 87,799 |
https://mathoverflow.net/questions/171836 | 23 | Let $G=\{g\_1,g\_2,...,g\_n\}$ be a group with $e=g\_1$ and $n$ is odd,
Set $$a\_1=g\_1$$
$$a\_2=g\_1g\_2$$
$$a\_3=g\_1g\_2g\_3$$
$$a\_n=g\_1g\_2...g\_n$$
I am looking for example that all $a\_i$ are different from each other i.e. $G=\{a\_1,a\_2,...,a\_n\}$. By the way it is clear that $a\_i\neq a\_{i+1}$.
**Note... | https://mathoverflow.net/users/47344 | Enumeration of a finite group | A group $G$ with the property is called sequenceable. For a survey, see [this paper](http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/DS10/pdf) by M. A. Ollis, which also tells that sequenceable groups are related to constructing row-complete latin squares. It is conjectured by Keedwell that $D\_6,D\_8$... | 49 | https://mathoverflow.net/users/23008 | 171849 | 87,800 |
https://mathoverflow.net/questions/171733 | 88 | I asked [a question](https://math.stackexchange.com/questions/500589/integrals-of-sqrtx-sqrt-phantom-dots-sqrtx1-in-elementary-functions) at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express ... | https://mathoverflow.net/users/9550 | Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions? | The answer is 'no'. Making the substitution
$$
x = \frac{(t-1)(t-5)(t^2+2t+5)}{16t^2},
$$
one finds
$$
{\textstyle\sqrt{x+\sqrt{x+\sqrt{x+1}}}\,\mathrm{d}x}
= \frac{(t^2-2t+5)(t^2-5)\sqrt{t^4{-}2t^2{-}40t+25}\ \mathrm{d}t}{32t^4}.
$$
Denote the right hand side of the above equation by $\beta$. Now, setting
$$
Q(t) = ... | 159 | https://mathoverflow.net/users/13972 | 171856 | 87,805 |
https://mathoverflow.net/questions/171839 | 3 | Does anyone have a reference that explains the technique of doubling of variables as introduced by Kruzkov? It seems to be a necessary tool for contraction estimates when we have weak solutions. However all the papers I have come across merely use this method without giving any explanation of it.
| https://mathoverflow.net/users/52929 | Doubling of variables method for parabolic equations | [This paper](http://www.sciencedirect.com/science/article/pii/S0022039696901552) by Felix Otto applies the method to quasilinear parabolic equations, and explains the steps in detail.
| 3 | https://mathoverflow.net/users/16530 | 171859 | 87,807 |
https://mathoverflow.net/questions/171851 | 8 | A
[matrix polynomial](http://en.wikipedia.org/wiki/Matrix_polynomial)
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on solving such equations.
For example, $X^2 =0$ has infinitely many s... | https://mathoverflow.net/users/6094 | Source for roots of matrix polynomials? | As Geoff Robinson says, a Jordan form takes you quite far. Evaluating a scalar polynomial (or an analytic function) $f(x)$ at a Jordan block $J\_{\lambda,t}$ of size $t$ and eigenvalue $\lambda$ gives the triangular Toeplitz matrix
$$
f(J\_{\lambda,t})=
\begin{bmatrix}
f(\lambda) & f'(\lambda) & f''(\lambda) & \dots & ... | 9 | https://mathoverflow.net/users/1898 | 171863 | 87,808 |
https://mathoverflow.net/questions/171527 | 4 | I'm looking for "tail-bound-like" inequalities that look like this (I state a specific setting but more general settings are interesting):
>
> Let $D$ be a distribution on a set of "nice" functions $g$: $[0,1]^d \to [0,1]$. Define
> $$\bar{g}(x) = \mathbb{E}\_{g \sim D} g(x) ,$$
> and let $\hat{g}\_n$ be a random... | https://mathoverflow.net/users/29697 | Concentration inequalities in $\ell_{\infty}$ for sums of iid random ("nice") functions? | What you have is called an *empirical process*, although it is usually written with the points and the functions reversed: let $\mathcal{F}$ be a family of functions $\Omega \to \mathbb{R}$ and let $X\_1, \dots, X\_n$ be i.i.d. elements of $\Omega$. The empirical process indexed by $\mathcal{F}$ is the collection
of ra... | 5 | https://mathoverflow.net/users/21652 | 171865 | 87,809 |
https://mathoverflow.net/questions/171828 | 1 | Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to
$$\frac{\partial}{\partial t}b(u) - \Delta u = f$$
where $b$ is continuous, increasing and locally Lipschitz and $f \in L^2(0,T;L^2)$.
How to obtain a comparison principle for this equations of this form? So I want to show ... | https://mathoverflow.net/users/52929 | Getting a comparison principle for parabolic equation when solution is not that smooth | You can use Holmgren's dual method for this kind of problems. Changing variables as $\Phi=b^{-1}$, $v=b(u)$ you can rewrite the PDE as the Generalized Porous Media Equation
$$
\partial\_t v=\Delta \Phi(v) +f,\hspace{2cm}(\text{GPME})
$$
which has been studied intensively and for which I recommend [Vazquez's book](http:... | 1 | https://mathoverflow.net/users/33741 | 171878 | 87,816 |
https://mathoverflow.net/questions/171882 | 8 | Let $q$ be a positive integer. Is it true there exists a constant $C\_q$ such that the following inequality holds for any finite set $A$ of reals:
$$\displaystyle |A+qA|\ge (q+1)|A|-C\_q\qquad (1)$$
I got this idea from the well-known inequality $|A+A|\ge 2|A|-1$, so I was thinking about the general case, but no idea a... | https://mathoverflow.net/users/50068 | Sumsets and a bound | Inequality (2) is true provided $(k,\ell)=1$ - this is a recent result of Balog and Shakan in 'On the sum of dilates of a set', <http://arxiv.org/pdf/1311.0422.pdf>.
They show that for any finite $A\subset\mathbb{Z}$ and integers $k,\ell$ such that $(k,\ell)=1$.
$$ \lvert kA+\ell A\rvert \geq (k+\ell)\lvert A\rve... | 12 | https://mathoverflow.net/users/385 | 171884 | 87,818 |
https://mathoverflow.net/questions/171880 | 5 | When $X$ is a smooth projective variety, one can use Mori's bend-and-break trick to establish the cone theorem. However, when $X$ has singularity (say klt. singularity), the cone theorem is obtained by a series of hard results: vanishing theorem -> non-vanishing theorem -> rationality theorem -> cone theorem.
I was w... | https://mathoverflow.net/users/29730 | What goes wrong to use "bend-and-break" trick for singular varieties? | When we use the bend-and-break technique in the proof of the Cone Theorem, we not only need to know that under certain conditions there are rational curves through a point of our variety $X$, but we also require an *upper bound on their degree* (with respect to a given polarization $H$).
Such a bound is only availabl... | 6 | https://mathoverflow.net/users/7460 | 171885 | 87,819 |
https://mathoverflow.net/questions/171845 | 1 | Following this question [here](https://mathoverflow.net/questions/27494/separable-sigma-algebra-equivalence-of-two-definitions) this question come to mind.
Consider a measured σ-algebra $(S,\mu)$ . Assume that μ is normalized to have total weight 1, and that S is complete (contains all subsets of null sets).
>
> ... | https://mathoverflow.net/users/42161 | Question on separability of a measure | Indeed, S1 holds if and only if the measure is a countable sum of atoms (I will understand the inclusion in the weak sense, i.e., ignoring sets of measure $0$, which makes the part directly relevant to the original question harder, but any other consistent interpretation will lead to the same proof and conclusion after... | 3 | https://mathoverflow.net/users/1131 | 171887 | 87,821 |
https://mathoverflow.net/questions/171896 | 4 | The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a meta-language for)ZFC might be quite unwieldy. Given a large cardinal axiom, set theorists can immediately classify it as local or global. Loosely speakin... | https://mathoverflow.net/users/4423 | A question about "local" versus "global" large cardinal axioms | I don't agree that it is difficult to formalize the local/global distinction, and indeed, I think that there is a largely agreed-upon technical meaning for these notions.
Specifically, a property is locally verifiable if it can be verified inside any sufficiently large rank initial segment $V\_\theta$ of the univers... | 12 | https://mathoverflow.net/users/1946 | 171902 | 87,826 |
https://mathoverflow.net/questions/171907 | 1 | Please accept my apologies in advance for my simple question.
Let $W(2)$ be a simple Lie algebra over $\mathrm{GF}(2)$. We know that it has a basis with three elements like ${w\_1,w\_2,w\_3}$. I cannot understand how to compute the Lie bracket for this Lie algebra? $[w\_1,w\_2]=?$ $[w\_3,w\_1]=?$
I have already stu... | https://mathoverflow.net/users/40491 | Understanding lie bracket of simple Lie algebra $W(2)$ | The simple Lie algebra $W(1; \underline{2})^{(1)}$ over $GF(2)$ has a basis $\{w\_1,w\_2,w\_3 \}$ such that $[w\_i,w\_j]=w\_k$, where $\{i,j,k\}=\{1,2,3\}$.
| 3 | https://mathoverflow.net/users/14653 | 171908 | 87,828 |
https://mathoverflow.net/questions/171545 | 3 | Suppose that $\kappa$ is a cardinal, $X$ is a set with $|X|>\kappa$, and $\mathcal{U}\subseteq P(P\_{\kappa}(X))$ is a normal ultrafilter. We say that a collection $C\subseteq P\_{\kappa}(X)$ is a conditional closure system if whenever $D\subseteq C$ and $D\neq\emptyset$, then $\bigcap D\in C$. Let's say that $\mathcal... | https://mathoverflow.net/users/22277 | Are normal ultrafilters generated by conditional closure systems? | I claim that if $\lambda,\kappa$ are cardinals with $\lambda>\kappa$ and $\mathcal{U}\subseteq P(P\_{\kappa}(\lambda))$ is a normal ultrafilter, then $\mathcal{U}$ is not generated by conditional closure systems. In fact, I shall now prove that $\mathcal{U}$ is not generated by sets closed under taking finite intersect... | 2 | https://mathoverflow.net/users/22277 | 171910 | 87,830 |
https://mathoverflow.net/questions/171912 | 13 | The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv\_T$-invariant are demanded to be determined.
In "Turing determinacy and the continuum hypothesis" (published in 1989), Ramez Sami writes:
>
> "The main question so far unse... | https://mathoverflow.net/users/8133 | Does Turing determinacy imply full determinacy? | This is open. In $L(\mathbb R)$ the answer is yes. Hugh has several proofs of this, and it remains one of the few unpublished results in the area. The latest version of the statement (that I know of) is the claim in your parenthetical remark at the end. This gives determinacy in $L(\mathbb R)$ using, for example, a ref... | 12 | https://mathoverflow.net/users/6085 | 171913 | 87,831 |
https://mathoverflow.net/questions/171840 | 3 | As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry [here](http://ncatlab.org/nlab/show/monoid)). What I would like to know is if the notion of generating subset of a monoid can be generalised to this context - precisely, by gene... | https://mathoverflow.net/users/41562 | How to define a generating subset for algebra in a category? | Here is a proposal that avoids requiring a notion of free monoid. Let $M$ be a monoid in some monoidal category $C$ and let $s : S \to M$ be a morphism (there is really no reason to restrict our attention to subobjects / monomorphisms).
**Definition #1:** $S$ *weakly generates* $M$ if, for any parallel pair of morph... | 5 | https://mathoverflow.net/users/290 | 171915 | 87,833 |
https://mathoverflow.net/questions/169136 | 0 | If I have a [compound Poisson process](http://en.wikipedia.org/wiki/Compound_Poisson_process)
$$Y(t) = \sum\_{i=1}{N(t)}D\_{i}$$
where $ \{\,N(t) : t \geq 0\,\}$ is a Poisson process with rate $\lambda$, and $ \{\,D\_i : i \geq 1\,\}$ are i.i.d random variables with distribution function $G$, which are also independe... | https://mathoverflow.net/users/51669 | Compound Poisson process and central limit theorem | I think the following shows what you're after. Similar problems/exercises can be found in many texts on probability theory (I guess it's more common in textbooks to consider the limit as time grows to infinity).
By the stationary and independent increments of the Poisson process, together with the independence of the... | 1 | https://mathoverflow.net/users/15752 | 171928 | 87,836 |
https://mathoverflow.net/questions/171922 | -3 | $B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum\_{i=1}^{\infty}a\_i x^i,a\_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if $f(x)$ has a natural boundary, is $$\lim\_{n\rightarrow \infty}K(a\_1a\_2\cdots a\_i \cdots a\_n)\rightarrow \infty$$ $K(a... | https://mathoverflow.net/users/14024 | Randomness about coefficients of series | This question is not well formed.
First, I assume by $K(a\_1 a\_2 \ldots)$ you mean $K(\langle a\_1, a\_2, \ldots \rangle)$. As the size of the sequence increases (regardless of the choice of $a\_i$), the complexity must increase to infinity.
On the other hand, there is nothing computationally complex about the co... | 3 | https://mathoverflow.net/users/12978 | 171930 | 87,838 |
https://mathoverflow.net/questions/171919 | 5 | Let $x\_1,\cdots , x\_n$ be a sequence of real number such that $x\_i\geq 1$ for all $1\leq i\leq n$, $S=\{\alpha\_1x\_1+\cdots +\alpha\_nx\_n | \alpha\_i\in\{0,+1,-1\}\}$ and $I=[a,b)$ be a Interval with length $2$. So I was wondering if there was any subsequent upper bound on $|I \cap S|$. Is there a general bound wh... | https://mathoverflow.net/users/50068 | A bound on a set | For a given $A\subset\{1,\dots,n\}$, let $S\_A$ denote the multiset of $\alpha\_1 x\_1+\cdots+\alpha\_n x\_n$ with $\alpha\_i=\pm 1$ for $i\in A$ and $\alpha\_i=0$ for $i\not\in A$. Note that, as multisets,
$$ S=\bigcup\_{A\subset\{1,\dots,n\}} S\_A,$$
so that
$$ I\cap S=\bigcup\_{A\subset\{1,\dots,n\}} (I\cap S\_A).$$... | 7 | https://mathoverflow.net/users/11919 | 171931 | 87,839 |
https://mathoverflow.net/questions/171628 | 13 | Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A **braided deformation** of $\mathcal{C}$ over a local artin ring $R$ with residue field $k$ is an $R$-linear braided monoidal category $\mathcal{C}'$, whose hom-sets are free $R$-modules, together with a... | https://mathoverflow.net/users/344 | Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory | I'm not very familiar with $\infty$-categories, but it's probably worth mentionning that the Drinfeld-Cartier result you quote is essentially equivalent to the formality of the $E\_2$ operad. Every infinitesimal deformation of the kind you mention turns the trivial deformation of your category into an algebra over the ... | 4 | https://mathoverflow.net/users/13552 | 171937 | 87,842 |
https://mathoverflow.net/questions/171917 | 2 | I was looking at the following page that attempts to prove that any Galois extension of a subfield $F$ of $\mathbb R$ contained in $F(\sqrt[n]{a})$ for some real $a$ with a real $n$th root must have degree at most $2$. <http://planetmath.org/node/40163>
I'm not sure about a step in their proof, where they conclude fr... | https://mathoverflow.net/users/53031 | "Galois subfields of real radical extensions are at most quadratic" | Let $N$ be a normal closure of $K$, so $N$ is $K$ adjoin an $n$th root of unity. Then you know the Galois group of $N$ over $K$, and you know the Galois group of $N$ over $F$, and the group of $N$ over $L$ has to be a normal subgroup of the group of $N$ over $F$, containing the group of $N$ over $K$. A little group the... | 1 | https://mathoverflow.net/users/3684 | 171942 | 87,846 |
https://mathoverflow.net/questions/171916 | 10 | Suppose $K\subset \mathbb{C}^n$ is a compact subset and $f:\mathbb{C}^n\setminus K\to \mathbb{C}$ is a holomorphic function. Then, provided $n>1$, $f$ extends to a holomorphic function defined on the whole $\mathbb{C}^n$. This is the Hartogs' extension theorem, and a proof can be found e.g. somewhere in the very beginn... | https://mathoverflow.net/users/2349 | Extending holomorphic functions | Yes, there are several generalizations of Hartogs Extension Theorem that hold on Stein spaces.
For a good survey you can look at the paper by Øvrelid and Vassiliadou [Hartogs Extension Theorems on Stein Spaces](http://link.springer.com/article/10.1007/s12220-010-9134-3), *Journal of Geometric Analysis* **20** (2010),... | 12 | https://mathoverflow.net/users/7460 | 171945 | 87,848 |
https://mathoverflow.net/questions/171944 | -2 | are all NP problems made up of P problems? that is, can NP problems be thought of as an accumulation of P problems? or can NP problems be divided up into a series of P problems?
| https://mathoverflow.net/users/49871 | are all NP problems made up of P problems? | The answer is yes. Suppose that $A$ is any NP problem, so there is a polynomial time algorithm $p$ such that $a\in A$ just in case there is some $b$ (of size at most $q(|a|)$, where $q$ is a fixed polynomial) such that $p$ accepts the pair $(a,b)$. Let $A\_b$ be the set of $a$ such that $b$ has size at most $q(|a|)$ an... | 5 | https://mathoverflow.net/users/1946 | 171947 | 87,849 |
https://mathoverflow.net/questions/171957 | 12 | I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives a ring homomorphism $$\alpha\colon\Omega\_\*^{spin}\rightarrow KO^{-\*}(pt),$$ from the spin-bordism ring to real K-th... | https://mathoverflow.net/users/32022 | Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant | If $M$ is a homotopy sphere of dimension $4k>0$, then the signature is clearly zero. By the Hirzebruch signature theorem, you get $0=\langle L\_k (TM); [M] \rangle = b\_k \langle p\_k (TM); [M] \rangle$ for a certain number $b\_k \neq 0$. Therefore, the Pontrjagin classes of $TM$ are all trivial, and hence the $\hat{A}... | 16 | https://mathoverflow.net/users/9928 | 171958 | 87,852 |
https://mathoverflow.net/questions/171920 | 29 | I have been wondering lately what makes simplicial sets 'tick'.
**Edited**
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of simplicial sets is to build spaces out of these building blocks by gluing, and allow maps to be defined simplex by... | https://mathoverflow.net/users/nan | What's special about the Simplex category? | Intuitively, I see the product-preservation or indeed finite limit preservation of geometric realization $\hat{R}: [\Delta^{op}, \mathbf{Set}] \to \mathbf{kSpace}$ as lifting (through the forgetful functor $U: \mathbf{kSpace} \to \mathbf{Set}$) a more basic left exact left adjoint $[\Delta^{op}, \mathbf{Set}] \to \math... | 21 | https://mathoverflow.net/users/2926 | 171960 | 87,853 |
https://mathoverflow.net/questions/171592 | 1 | Let $P$ be a polynomial with real coefficients, and $\deg P=d$. There is Markov-Berenstein inequality: $P′(x)\leq\frac{d\|P\|}{\sqrt{1-x^2}}$,where $\|P\|=\max\_{|x|\le1} |P(x)|$ and $|x|\leq1$. Are there any improvements when $P$ is increasing in the interval $[-1,1]$. I am particularly interested in bounding $|P'(x)|... | https://mathoverflow.net/users/38136 | Markov-Bernstein like inequalities for monotone polynomials | The following paper seems to fully answer the question:
<http://arxiv.org/pdf/1205.0846.pdf>.
| 3 | https://mathoverflow.net/users/38136 | 171962 | 87,854 |
https://mathoverflow.net/questions/145680 | 6 | Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by
$$
g\_{mn} = \left(
\begin{array}{cc}
g\_{\mu\nu} & g\_{\mu 5} \\
g\_{5\nu} & g\_{55} \\
\end{array}
\right)
$$
where $g\_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $g\_{\mu 5}$ is the ordinary four dimensional Maxwel... | https://mathoverflow.net/users/30967 | Why does closed string theory have only one dilaton field instead of $22$? | You would actually *not* expect 22 dilatons. Let me try to explain.
As you have pointed out, a putative field theory limit of the closed bosonic string would consist of a metric, a 2-form (which is the potential for a 3-form) and a dilaton.
Let us assume that such a theory exists and let us dimensionally reduce to ... | 4 | https://mathoverflow.net/users/394 | 171976 | 87,859 |
https://mathoverflow.net/questions/171977 | 9 | Krein–Rutman theorem is a generalization of Perron–Frobenius theorem, I know that things could be more subtle in infinite dimension, yet there's an important result in Perron–Frobenius that's missing in Krein-Rutman and I don't quite understand.
In Perron–Frobenius theorem, we know that for a irreducible non-negativ... | https://mathoverflow.net/users/51690 | Comparing Krein-Rutman theorem and Perron–Frobenius theorem | Beware of Wikipedia! It is true that the infinite dimensional setting makes things slightly more delicate, but actually not so much.
Assuming that the positive cone $C\subset X$ under consideration is solid (i-e has non empty interior) and that your operator $T:X\to X$ is compact and strongly positive (i-e maps the p... | 11 | https://mathoverflow.net/users/33741 | 171981 | 87,862 |
https://mathoverflow.net/questions/171971 | 3 | This is related to a question asked on mathstackexchange <https://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo>. This question is inspired by Remark 7.4 from the paper $\textit{Structure of Null Sets in the Plane and Applications}$.
The remark i... | https://mathoverflow.net/users/49404 | For Every Measure Zero Set $E$ There Exists a Positive Measure with Lower Lebesgue Density 0 and Upper Lebesgue Density 1 | Yes, we can do this by a small modification of the original argument, see the [linked question.](https://math.stackexchange.com/questions/831184/for-every-null-set-e-there-is-a-measurable-set-f-with-different-upper-and-lo) I'll describe the whole argument again here, but if you already read those answers, then the shor... | 4 | https://mathoverflow.net/users/48839 | 171985 | 87,863 |
https://mathoverflow.net/questions/171983 | 4 | There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with different integral coefficients,different power of nomials but they have the same natural boundary.
As we know, some functions ... | https://mathoverflow.net/users/14024 | How to classify the complex function with same natural boundary in complex plane? | There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides [here](http://www.mth.kcl.ac.uk/~pushn/ebd65/simon.pdf) ). They give, among other things, the definition of "strong natural boundary". This concept relates to "right limits" of the sequence of Taylor coefficients and give a crit... | 5 | https://mathoverflow.net/users/24309 | 172000 | 87,870 |
https://mathoverflow.net/questions/171996 | 4 | Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$?
Here $d(H)$ denotes the minimal cardinality of a generating set of a group $H$. It is enough to consider the case of $G$ having a trivial Frattini... | https://mathoverflow.net/users/38889 | Generators of Sylow subgroups | I don't think so. Let $p$ and $q$ be primes with $q|p-1$. Then there are $q$ inequivalent $1$-dimensional modules for $C\_q$ over ${\mathbb F}\_p$, If we take the semidirect product of the direct sum of these modules by $C\_q$, then we get a supersolvable group $G$ of order $p^qq$ with $d(G)=2$, with an elementary abel... | 7 | https://mathoverflow.net/users/35840 | 172002 | 87,871 |
https://mathoverflow.net/questions/171622 | 16 | Let $M$ be a closed connected manifold and fix a basepoint $q \in M$ and a Riemannian metric on $M$. Let $F(M)$ denote the orthonormal frame bundle of $M$. This is a principal $O(n)$-bundle over $M$ ($n = \dim M$). The homotopy sequence of this bundle reads
$$\dots \to \pi\_2(M,q) \to \pi\_1(O(n),I) \to \pi\_1(F(M),F\_... | https://mathoverflow.net/users/39725 | Cohomology class of the group extension from a principal bundle | Let $M$ be an orientable manifold, with chosen base point $q$ and chosen Riemannian metric. The extension of the question arises from the principal bundle $SO(n)\to F(M)\to M$.
There is another bundle giving rise to the same extension, namely $\mathbb{RP}^\infty\to \tilde{M}\to M$, and there are three equivalent way... | 4 | https://mathoverflow.net/users/50846 | 172003 | 87,872 |
https://mathoverflow.net/questions/171995 | 15 | Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with dimension $n$ over $F$?
This questions is motivated by the classification of low dimensional algebras. It seems that at least... | https://mathoverflow.net/users/27976 | Are there only finitely many associative algebras of fixed dimension? | Even for $4$-dimensional algebras with identity it's not true.
For $a\in F$ let $B(a)=F\langle x,y|x^2=y^2=0,xy=ayx\rangle$. Then $B(a)\not\cong B(b)$ unless $a=b$ or $a=b^{-1}$. This is quite easy to see by considering which elements of $B(a)$ square to zero:
If $z=\lambda\_11+\lambda\_xx+\lambda\_yy+\lambda\_{yx}... | 30 | https://mathoverflow.net/users/22989 | 172011 | 87,873 |
https://mathoverflow.net/questions/172009 | 17 | It is well known that the set $\{(n,m) \in \Bbb N^2 : \gcd(n,m) = 1\}$ of coprime integers has a natural density of $\zeta(2)^{-1}$ in $\Bbb N^2$.
It seems reasonable to think that the density of the $\{(n,m) \in \Bbb N^2 : \gcd(n,m(m+1))=1\}$ is still positive. I am no specialist of this kind of questions so I fail ... | https://mathoverflow.net/users/35098 | Probability that $n$ is coprime to both $m$ and $m+1$ | The density exists and equals
$$ C:=\sum\_d\frac{\mu(d)\tau(d)}{d^2}=\prod\_p\left(1-\frac{2}{p^2}\right)\approx 0.322634\ . $$
Note that the right hand side is the product of local densities over the primes.
Indeed, the number of pairs $(n,m)\in\mathbb{N}^2$ with $1\leq n,m\leq x$ and $\gcd(n,m(m+1))=1$ equals
$$ \s... | 21 | https://mathoverflow.net/users/11919 | 172015 | 87,874 |
https://mathoverflow.net/questions/171999 | 5 | Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram.
Then for any two simple roots $\alpha$ and $\beta$, whether a linear combination $n\alpha+m\beta$ is a root can be judged easily from the Dynkin diagra... | https://mathoverflow.net/users/17294 | Which linear combinations of simple roots are roots | My favorite answer to #2 and #3 is Kostant's "Find the highest root game", which is written up in detail in section 5.4 of [Balázs Elek's notes on reflection groups](http://pi.math.cornell.edu/%7Eallenk/elek_reflection_groups_2016.pdf). It is not hard to show that all plays of the game (from all starting positions, i.e... | 7 | https://mathoverflow.net/users/391 | 172033 | 87,882 |
https://mathoverflow.net/questions/172028 | 2 | Let $(N,J\_N)$ and $(M, J\_M)$ be two compact almost complex manifolds with non integrable almost complex structures (i.e., the Nijenhuis tensor is non zero
for both $J\_N$ and $J\_M$).
Does it imply that there can not exist any non constant pseudo-holomorphic
map $~f:N\rightarrow M$? Pseudo-holomorphic means
$$ ... | https://mathoverflow.net/users/4463 | Is the Nijenhuis tensor an obstruction to the existence of non constant pseudo-holomorphic maps? | The brief answer to your question is 'no': For example, take $N=M$ and $J\_N=J\_M$. Then the identity map of $N$ is a nonconstant pseudo-holomorphic map.
What *is* true is that the nonvanishing of the Nijnhuis tensors of the two manifolds puts nontrivial conditions (beyond merely being complex linear) on the induced ... | 10 | https://mathoverflow.net/users/13972 | 172035 | 87,884 |
https://mathoverflow.net/questions/171794 | 5 | Im sure this is a beginners question.
Let $k$ be a field and $I(k)$ the fundamental ideal in the Witt-ring W(k).
The Arason-Pfister-Hauptsatz states:
"If $\varphi$ is any anisotropic class in $I^n(k)$, then $rank(\varphi) \geq 2^n$."
It is well known that for the kernel of the discriminant $ker(e\_1) = I^2$ hol... | https://mathoverflow.net/users/51251 | Rank four quadratic Form with non trivial discriminant in I(k) | The answer depends on your field $k$. For example if $k$ is the $p$-adic field $\mathbb{Q}\_p$, $p\neq 2$, it is known that the only anisotropic form of dimension $4$ over $k$ is isomorphic to the norm form of the unique quaternion algebra over $k$ which is of course in $I^2$ since it is a Pfister form.
By contrast yo... | 3 | https://mathoverflow.net/users/30062 | 172050 | 87,891 |
https://mathoverflow.net/questions/171867 | 3 | Suppose we have two real-valued functions $f(x)$ and $g(x)$, both equal to their Newton series expansion:
$$f(x) = \sum\_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$
$$g(x) = \sum\_{k=0}^\infty \binom{x}k \Delta^k g\left (0\right)$$
Is their composition $F(x)=f(g(x))$ also equal to its Newton series expans... | https://mathoverflow.net/users/10059 | If two functions are equal to their Newton series, is their composition also equal to its Newton series? | Here goes, as promised.
Let $f$ be entire of order less than $1$, so $|f(z)|\le Ce^{|z|^p}$, $p<1$. Write the Newton polynomial
$$
P(x)=\sum\_{k=0}^n\Delta^kf(0) {x \choose k}
$$
Note that $g(k)=f(k)-P(k)=0$ for $k=0,1,\dots,n$. On the other hand, we can crudely estimate $|g|$ in a disk of radius $R>2n$ by $Ce^{R^p... | 10 | https://mathoverflow.net/users/1131 | 172066 | 87,896 |
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