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https://mathoverflow.net/questions/232783 | 4 | Given a finite set $\Lambda=\{\lambda\_1,\dots,\lambda\_d\}\in \{1,2,\dots\}^d$
of $d$ strictly positive integers, we consider
the real number
$$\mu(\Lambda):=\sup\_{l\in\{2,3,\dots\}}\frac{1}{l}\sum\_{i=1}^d (\lambda\_i\ \ \text{mod}\ \ l),$$
where $(\lambda\_i\ \ \text{mod}\ \ l)\in\{0,\dots,l-1\}$.
Does $\mu(\Lam... | https://mathoverflow.net/users/4556 | Integers with small residues modulo all integers | No $\kappa>1$ implies any bound on $d$. Without loss of generality, let $2\geq\kappa>1$. Assume that we have already found a $\Lambda$ of cardinality $d-1$ satisfying $\mu(\Lambda)\leq\kappa$. We shall construct a $\Lambda'$ of cardinality $d$ satisfying $\mu(\Lambda')\leq\kappa$.
Let $\lambda\_1,\dots,\lambda\_{d-1}... | 4 | https://mathoverflow.net/users/11919 | 232793 | 108,108 |
https://mathoverflow.net/questions/232779 | 8 | Recently, I got interested in the study of the combinatorial aspects of continued fractions. Precisely, I read of the following lemma of Flajolet (see [here](http://www.stat.purdue.edu/~mdw/ChapterIntroductions/ContinuedFractionsUpdateViennot.pdf)):
**Lemma.** It holds
$$\sum\_{\omega} \nu(\omega) \, z^{|\omega|} = \... | https://mathoverflow.net/users/nan | Combinatorial aspects of continued fractions | One book that I know which covers this topic is (available online):
[Analytic Combinatorics](http://algo.inria.fr/flajolet/Publications/book.pdf), P. Flajolet and R. Sedgewick. In particular, (as the OP now also notes in a comment), the relevant section is V.4.
The [following slides](http://algo.inria.fr/flajolet/Pub... | 5 | https://mathoverflow.net/users/8430 | 232796 | 108,109 |
https://mathoverflow.net/questions/232797 | 9 | I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post.
Most of the details below are fairly elementa... | https://mathoverflow.net/users/6085 | Two elementary inequalities for real-valued polynomials | One generalization of (1), which is not quite an inequality, is the so-called Hawaiian conjecture. It states that the number of real zeroes of $(f'/f)'$ does not exceed the number of **nonreal** zeroes of $f$. This paper claims a proof: Mikhail Tyaglov, *On the number of real critical points of logarithmic derivatives ... | 4 | https://mathoverflow.net/users/44953 | 232801 | 108,112 |
https://mathoverflow.net/questions/104059 | 78 | The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner that will be made precise below.
Setup
-----
Consider a nested family of simplicial complexes $K(p)$ indexed by prime ... | https://mathoverflow.net/users/18263 | The topology of Arithmetic Progressions of primes | No.
In fact, for $p = 435052917615787$, this will absolutely be false, as the second homology group will not vanish.
Note that a 2-dimensional "hole" in your complex is a 3-simplex all of whose faces are 2-simplices in $K(p)$, i.e. 4 primes such that each 3 of them lie in some arithmetic progression.
Of course, y... | 31 | https://mathoverflow.net/users/74819 | 232820 | 108,119 |
https://mathoverflow.net/questions/232821 | 49 | This is a naive question that could justifiably be quickly closed.
Nevertheless:
***Q***. Why is
[Péter Frankl's](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture)
conjecture so difficult?
>
> If any two sets in some family of sets have a union that also belongs to the family, must some element belong t... | https://mathoverflow.net/users/6094 | Why is the Frankl conjecture hard? | (**Migrated** by request from the comments.)
Bruhn and Schaud's (2013) [**The journey of the union-closed sets conjecture**](https://arxiv.org/abs/1309.3297) provides a rather readable write-up. Particularly relevant is the section **Obstacles to a proof**; for example, you may check just after *Conjecture 15* in whi... | 33 | https://mathoverflow.net/users/22971 | 232823 | 108,120 |
https://mathoverflow.net/questions/231559 | 0 | In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex game or not? If not, could you give a counterexample?
Thank you.
| https://mathoverflow.net/users/87870 | Is the linear production game a convex game? | I don't think it is. Consider the following LP
$$\max x$$
$$s.t.$$
$$x \leq \sum\_{s\in S} b\_s^1$$
$$x \leq \sum\_{s\in S} b\_s^2$$
Now consider $S=\{1,2,3\}$ with $b\_1=(1,0)$, $b\_2=(0,1)$, $b\_3=(0,1)$.
Then $V(\{1\}) = 0$, $V(\{1,2\}) = 1$, $V(\{1,2,3\}) = 1$, $V(\{1,3\}) = 1$,
then $V(\{1,2,3\})-V(\{1,2\}) = ... | 1 | https://mathoverflow.net/users/39187 | 232825 | 108,121 |
https://mathoverflow.net/questions/232454 | 4 | Assume $n$ points $P\_i \in \mathbb{R}^2, i \in {1,2,...,n}$. For each point there is a $k$ nearest neighbour $(k<n)$, or equivalently for each point $P\_i$ there is one circle with center the point $Pi$ and radius $r\_i$ such that the circle contains exactly $k+1$ points (considering also the center $Pi$). The questio... | https://mathoverflow.net/users/88352 | k nearest points | What you're asking is: what is the ply of the system of $k$-nearest-neighbor balls? Here the ply is the maximum number of balls that have a common intersection. It's not quite the same as the degree of the $k$-nearest-neighbor graph (Fischler's answer) because the ply can be maximized at a point that's not one of the g... | 1 | https://mathoverflow.net/users/440 | 232828 | 108,123 |
https://mathoverflow.net/questions/232816 | 1 | I am following up on the answer of Denis Serre to this same question here [Short time existence on nonlinear parabolic PDE](https://mathoverflow.net/questions/99994/short-time-existence-on-nonlinear-parabolic-pde)
I have tried to generalise the proof of the Picard-Lindelof theorem, as suggested by Denis, to prove sho... | https://mathoverflow.net/users/66025 | Proving short time existence for semi-linear parabolic PDE | You would recommend the beautiful book of M. TaylorPartial Differential Equations, Vol 3 : Nonlinear Equations, Applied Math Sciences series, Vol 117, Springer. In particular the chapter 15 : Nonlinear Parabolic Equations.
| 1 | https://mathoverflow.net/users/48525 | 232833 | 108,126 |
https://mathoverflow.net/questions/232842 | 2 | Let $\phi:A \to B$ be a flat ring homomorphism, $M$ be a $B$-module which is flat when considered as an $A$-module. Is the tensor product $M \otimes\_B M \otimes\_B ... \otimes\_B M$ flat over $A$? If not true in general, is there any known cases of $\phi$ (other than etale morphisms) when it holds true? Furthermore, i... | https://mathoverflow.net/users/58203 | Is flatness preserved under exterior power | Let $A=\mathbb{C}[x^2]$, $B=\mathbb{C}[x^2,x^3]$, and $M=\mathbb{C}[x]$.
Then $B$ and $M$ are free as $A$-modules, but $M\otimes\_BM$ (and also $\wedge^2M$) has a one-dimensional $A$-submodule spanned by $x\otimes 1 - 1\otimes x$, and so isn't flat.
| 6 | https://mathoverflow.net/users/22989 | 232847 | 108,127 |
https://mathoverflow.net/questions/208615 | 9 | By "nice curve", I mean a smooth, projective, geometrically integral curve over $\newcommand{\Q}{\mathbb{Q}}\newcommand{\Jac}{\operatorname{Jac}}\Q$ with at least one $\Q$-rational point. The Mordell–Weil rank $r(C)$ of a nice curve $C$ is the rank (as an abelian group) of $\Jac(C)(\mathbb{Q})$, the group of $\Q$-ratio... | https://mathoverflow.net/users/31308 | Distribution of Mordell–Weil ranks of higher genus curves | General Katz-Sarnak heuristics suggest that the analogue of the minimalist conjecture should still be true. Let me sketch the reason why from two perspectives - the function field model, where we can establish a version of the conjecture, and the Sarnak-Shin-Templier conjectures on families of automorphic forms.
Firs... | 5 | https://mathoverflow.net/users/18060 | 232852 | 108,129 |
https://mathoverflow.net/questions/232817 | 1 | **Background:**
The strong law of large numbers (SLLN) is a powerful result in
probability, and there has been extensive literature on when the SLLN holds.
However, constructing nontrivial examples for which the SLLN fails to hold seems to be (very) hard.
K.L. Chung's famous paper "The strong law of
large numbers... | https://mathoverflow.net/users/14390 | Dependent Bernoulli sequence for which the strong law fails to hold | Let $(S\_n, n\geq 0)$ be the one-dimensional simple random walk started at the origin. Set $X\_n=\mathbf{1}\{S\_n\geq 0\}$. Then $n^{-1}(X\_1+\cdots+ X\_n)$ is the proportion of time that the SRW is non-negative; it doesn't converge a.s. anywhere (although converges in distribution to a non-trivial r.v., cf. the arcsin... | 4 | https://mathoverflow.net/users/81488 | 232858 | 108,133 |
https://mathoverflow.net/questions/232741 | 7 | **Setup**: Let $\mathcal{O}$ be an operad in the category of sets, and let $\mathcal{O}\text{-Alg}$ denote the category of algebras on it (i.e., operad functors $\mathcal{O}\to\mathbf{Set}$. This category is cocomplete; let $\kappa$ denote an initial object. Call an algebra $X$ *initially monic* if the unique map $!\_X... | https://mathoverflow.net/users/2811 | Monomorphisms in operad algebras | Even for single-sorted operads, the coproduct of initially monic algebras need not be initially monic.
First a general construction. Let $A$ be a commutative monoid. Then the comma category $A \downarrow \mathrm{CMon}$ (aka the undercategory or co-slice under $A$) is the category of algebras of an operad $\mathcal{O... | 4 | https://mathoverflow.net/users/2926 | 232864 | 108,134 |
https://mathoverflow.net/questions/232860 | 4 | Let $\mu$ be some ergodic measure of our compact Riemannian manifold $M$, which is preserved by $f\in Diff^{1+\beta}(M)$. Is it possible that all the Lyapunov exponents of $\mu$ will be positive? Intuitively this seems wrong, but I couldn't find any general proof without assuming that $h\_\mu(f)>0$ (which I don't want ... | https://mathoverflow.net/users/70853 | Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents? | As Will shows, the case in which $\mu$ is absolutely continuous with respect to Lebesgue measure and has density bounded away from zero and infinity is constrained in that the Lyapunov exponents of $\mu$ must sum to zero. If $\mu$ is an arbitrary ergodic measure then the Lyapunov exponents can all be positive, for exam... | 5 | https://mathoverflow.net/users/1840 | 232866 | 108,136 |
https://mathoverflow.net/questions/232853 | 1 | Let $n$ be a positive integer and $W\_n$ be the linear subspace of the real vector space $\mathbb{R}[X\_1,\cdots,X\_n]$ generated by the following set
$$S\_n=\{X\_1^{i\_1}\cdots X\_n^{i\_n}:i\_1+\cdots+i\_n=n\ \text{and}\ i\_k\in \{0,1,2\},k=1,\cdots,n\}.$$
Let
$$T\_n=\{(X\_{i\_1}-X\_{i\_2})(X\_{i\_2}-X\_{i\_3})\cdots... | https://mathoverflow.net/users/58096 | A linear subspace of $\mathbb{R}[X_1,\cdots,X_n]$ and its generated set | $\let\eps\varepsilon$Substitute $X\_k\leftarrow\eps\_k:=\exp(2\pi ki/n)$. Notice that $\arg(\eps\_k-\eps\_\ell)\equiv (\pi+\arg \eps\_k+\arg\eps\_\ell)/2\pmod{\pi}$. Summing up, we get that the argument of every element of $T\_n$ is congruent to a constant modulo $\pi$; so all their real lnear combinations also share t... | 0 | https://mathoverflow.net/users/17581 | 232868 | 108,137 |
https://mathoverflow.net/questions/232676 | 8 | Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\mathfrak{g}$-modules belongs to precisely one of the reduced enveloping algebras $U\_\chi(\mathfrak{g})$ with $\chi \in \math... | https://mathoverflow.net/users/83211 | Can we count the number of simple modules for a reduced enveloping algebra? | The answers to your several closely related questions are not yet known, though many parts of the story have emerged. In particular, there is no "formula" for the number of simple $U\_\chi(\mathfrak{g})$-modules in a typical (meaning regular) block, and such a formula probably doesn't exist. It's quite difficult in gen... | 6 | https://mathoverflow.net/users/4231 | 232874 | 108,140 |
https://mathoverflow.net/questions/232840 | 0 | I have the following function that I would like to optimize over the value A
$$f(A)=\sum\_k \frac{\mathbf{y}\_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{x}\_k\mathbf{x}\_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{y}\_k}{\mathbf{x}\_k^H\left[\begin{array}{cc} 1&0\\ 0& A^2 \end... | https://mathoverflow.net/users/88565 | a sum of ratios of quadratic forms | The property you mention does not always happen.
$f(A)$ always reduces to the form
$$
f(A) = \sum\_k \frac{\alpha\_k A^2+\beta\_k A + \gamma\_k}{\delta\_k A^2 + \epsilon\_k}
$$
with $\{ \alpha\_k,\beta\_k,\gamma\_k,\delta\_k,\epsilon\_k\}$ real-valued degree-4 expressions involving the components of $\mathbf{x}\_k$... | 1 | https://mathoverflow.net/users/82067 | 232883 | 108,144 |
https://mathoverflow.net/questions/232479 | 1 | Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma\_1(t),\dots,\gamma\_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve with finite arc-length.
Define a possibly transfinite partition $P=\{a=x\_0<x\_1<\dots<x\_\Omega\le b\}$ of an initial ... | https://mathoverflow.net/users/88148 | Ordinal of injectivity for a smooth regular curve with a finite arc-length | If I understand the question correctly, the answer is that there is no better bound than $\Omega=\omega\_1$, the first uncountable ordinal. I'll sketch the solution for $d=1$, I believe the others are similar. For any countable ordinal $\alpha$, there is an order-preserving embedding of $\alpha$ into $[a, b]$ with clos... | 1 | https://mathoverflow.net/users/8133 | 232893 | 108,149 |
https://mathoverflow.net/questions/232713 | 3 | *Edit: all sets / theories considered below are supposed to be recursively enumerable, although I'd also be interested in any possible generalizations to non-enumerable theories.*
In the comments on [this question](https://mathoverflow.net/questions/232209/relationship-between-first-and-second-incompleteness-theorems... | https://mathoverflow.net/users/4336 | Theories of arithmetic from recursively inseparable sets | Here's an obstacle to such a construction:
>
> Suppose $T$ is any theory in the language of arithmetic extending $PA$. Then the set $Pr(T)$ of sentences proved by $T$ computes a complete consistent extension of $PA$.
>
>
>
Proof: We can use a "greedy algorithm" to build such a complete consistent extension. Le... | 1 | https://mathoverflow.net/users/8133 | 232897 | 108,150 |
https://mathoverflow.net/questions/228077 | 7 | Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma\_i\in A$. Let $P$ be the "pure braid group", the kernel of the natural homomorphism from $A$ to $W$. Obvious elements in $P$ include the squares $\sigma\_i^2$ of the standard Coxeter gen... | https://mathoverflow.net/users/18269 | Generators of pure braid groups of arbitrary Coxeter groups | This note
<http://arxiv.org/pdf/1511.08731v3.pdf>
(unfortunately written in French), Corollaire 3.7, answers your question.
Writing $\bf{W}$ for the canonical positive lift of $W$ in the Artin-Tits group $B\_W$, the pure braid group is generated by the elements of the form $\bf{w}\bf{s}^2 \bf{w}^{-1}$ where $\bf{... | 4 | https://mathoverflow.net/users/26751 | 232904 | 108,154 |
https://mathoverflow.net/questions/232751 | 12 |
>
> If $h(T)$ denotes the number of (directed) [Hamiltonian paths](https://en.wikipedia.org/wiki/Hamiltonian_path) in the [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) $T,$ what is the range of $h(T)$ as $T$ ranges over all (finite) tournaments $T$?
>
>
>
By a classical theorem of [Rédei]... | https://mathoverflow.net/users/43266 | The number of Hamiltonian paths in a tournament | This sounds bizarre, but my computations suggest that 21 might be a possibility..
I checked the tournaments on up to 9 vertices (10 vertices is under way) and got the following results:
* by 6 vertices, every odd number (except 7 and 21) less than 33 had occurred at least once
* by 7 vertices, every odd number (exc... | 6 | https://mathoverflow.net/users/1492 | 232910 | 108,157 |
https://mathoverflow.net/questions/232911 | 1 | Let $(\Omega,\Sigma,\mu)$ be a countably generated probability space. Must $(\Omega,\Sigma,\mu)$ be isomorphic modulo null sets to a [standard probability space](https://en.wikipedia.org/wiki/Standard_probability_space)?
I assume not, so here is a more specific question. Let $\Omega$ be an ultraproduct of finite sets... | https://mathoverflow.net/users/20598 | Countably generated $\sigma$-algebra | For the first question, the answer is yes. There is an isomorphism between measure algebras. Let $B$ be the Boolean algebra of all measurable sets modulo the collection of null sets. Then define a metric $\rho$ on $B$ by letting $\rho(x,y)=\mu((x\wedge y')\vee(y\wedge x'))$.
$\mathbf{Theorem}$:(Caratheodory, see Royd... | 3 | https://mathoverflow.net/users/22277 | 232913 | 108,158 |
https://mathoverflow.net/questions/232849 | 8 | I am looking for a good name for the following problem:
>
> Given elements $g\_1,\dotsc,g\_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g\_1^G\dotsb g\_n^G$ contains the identity element $1$.
>
>
>
In some situations it might be more natural to pose this problem ... | https://mathoverflow.net/users/20995 | Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem | There is a name for this problem: (the solvability of) a genus 0 quadratic equation over G. Check out Sections 2.1 and 3.3 in <http://arxiv.org/abs/0802.3839>.
This terminology is also consistent with your topological interpretation. Also, whether the set $$
g\_1^G\cdots g\_n^G
$$ contains a commutator is equivalent ... | 3 | https://mathoverflow.net/users/38698 | 232927 | 108,163 |
https://mathoverflow.net/questions/232928 | 0 | If $X$ is a non-empty set, we say that $M\subseteq X$ is a *majority* if $|M| > |X\setminus M|$.
Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N(v)=\{x\in V: \{x,v\} \in E\}$.
Let $n$ be a positive integer. We say that a map $c:V(G) \to \{1,\ldots,n\}$ is a *majority coloring* if the fol... | https://mathoverflow.net/users/8628 | Majority colorings | I’m basing my answer on [this MSE answer](https://math.stackexchange.com/questions/24083/graph-coloring-problem-possibly-related-to-partitions) to a similar question.
Theorem: Let $G$ be a simple graph. If $G$ has no edges, $\chi\_\mbox{maj}(G)=1$; otherwise, $\chi\_\mbox{maj}(G)=2$.
Proof: The no-edge case is tri... | 3 | https://mathoverflow.net/users/8201 | 232943 | 108,167 |
https://mathoverflow.net/questions/232941 | 8 | Fix an algebraic closure $\overline{\mathbb Q}$ of the field $\mathbb Q$ of rational numbers. For a prime $p$ let $K\_p$ the field of all algebraic elements in ${\mathbb Q}\_p$.
1. Question: Is $K\_p$ normal over $\mathbb Q$?
If so, it defines a unique subfield of $\overline{\mathbb Q}$ which we might want to write... | https://mathoverflow.net/users/nan | Is the intersection of all p-adic fields equal to Q? | 1. No: the field $\mathbb{Q}\_5$ has a cube root of $2$, but does not contain a square root of $-3$. (It's easy to form such examples for all primes.)
2. An alternate version of the question is as follows: Given an algebraic number $\alpha$ such that there is an inclusion $\mathbb{Q}(\alpha) \rightarrow \mathbb{Q}\_p$ ... | 13 | https://mathoverflow.net/users/88608 | 232946 | 108,169 |
https://mathoverflow.net/questions/232950 | 6 | Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists?
Start with the observation that any given diffeomorphism $\rho:M\rightarrow M$ of a Riemannian manifold $M$ allows for pulling the... | https://mathoverflow.net/users/10909 | Local differential geometry and invariant theory | This question was answered decisively by Weyl and Cartan. The essential point is that there is a canonical way to reduce this nonlinear action to the linear action of $\mathrm{O}(n)$ on a finite dimensional vector space: Use normal coordinates.
The point is that a metric $g$ can be written in $p$-centered coordinates... | 13 | https://mathoverflow.net/users/13972 | 232954 | 108,173 |
https://mathoverflow.net/questions/232930 | 19 | It is a well-known result that all profinite groups arise as the Galois group of *some* field extension.
>
> What profinite groups are the absolute Galois group
> $\mathrm{Gal}(\overline{K}|K)$ of some extension $K$ over
> $\mathbb{Q}$?
>
>
>
The answer is simple enough in the finite case:
* (Artin-Schreie... | https://mathoverflow.net/users/43108 | Profinite groups as absolute Galois groups | This is a very good question which is a big open problem. There are a number of theorems, some of them easy and some very difficult, and also a number of conjectures, restricting the class of groups which may turn out to be absolute Galois groups. But it seems that nobody has any idea about how a precise description of... | 23 | https://mathoverflow.net/users/2106 | 232964 | 108,175 |
https://mathoverflow.net/questions/232956 | 6 | Where could I find a detailed exposition in *English* of Godel's proof (not Henkin's) of Completeness Theorem for first order logic? The [wikipedia article](https://en.wikipedia.org/wiki/Original_proof_of_G%C3%B6del%27s_completeness_theorem) omits certain details that I am not clear about, and Godel's original disserta... | https://mathoverflow.net/users/87975 | Godel's proof of Completeness | In addition to an English translation of the original version contained in the Collected Works referred to by Andrés, there is an English translation of a rewritten version of the paper contained in *From Frege to Gödel* edited by Jean van Heijenoort. The latter is also contained in the Collected Works.
| 11 | https://mathoverflow.net/users/18939 | 232972 | 108,179 |
https://mathoverflow.net/questions/232963 | 3 | Let $n$ be a natural number, $u\_+,v\_+,u\_-,v\_-$ be real or complex column vectors of length $n$, and $M\_1,M\_2,\ldots,M\_k$ be a finite collection of $n\times n$ real or complex matrices.
Consider the problem of determining whether there exists a tuple $(\lambda\_1,\lambda\_2,\ldots,\lambda\_\ell)$ of any finite ... | https://mathoverflow.net/users/70454 | Complexity class of matrix generalization of knapsack problem | It is undecidable. I'll use this result: Given an finite set of integer matrices, it is undecidable whether there is a product with 0 in the upper-right corner. See for example [this article](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.45&rep=rep1&type=pdf).
So take such a finite set $S$ of $n\times n... | 2 | https://mathoverflow.net/users/9025 | 232974 | 108,180 |
https://mathoverflow.net/questions/232757 | 6 | Trying to solve a problem, I fell on the following statement :
If $k$ and $r$ are natural numbers such that $r \leq k$, if a union closed family of sets ("union closed" means that the union of two sets from the family is always a member of the family) has at least ${k \choose r} + 1$ members with cardinality $r$, the... | https://mathoverflow.net/users/82840 | Literature about a property of union closed families? | This is similar to your proof but without induction.
We prove that there are at least 3 such sets. For $r=k$ this is clear, so assume that $k>r$. Consider our $\binom{k}{r}+1$ $r$-sets. Call an element $v\in V$ appropriate if $v$ belongs to at most $\binom{k-1}{r-1}$ our sets. Then there exist at least $\binom{k}{r}+... | 3 | https://mathoverflow.net/users/4312 | 232988 | 108,185 |
https://mathoverflow.net/questions/232562 | 5 | Ax-Grothendieck Theorem states that if $\mathbf K$ is an algebraically closed field, then any injective polynomial map $P:\mathbf K^n\longrightarrow \mathbf K^n$ is bijective.
> **Question 1.** What does the inverse map of $P$ look like ? What kind of map is that ?
$P^{-1}$ need not be polynomial, as the example $... | https://mathoverflow.net/users/18583 | Inverse of a polynomial map | In light of abx's comment, I came up with the following argument. I'm sure some version of this must be in the literature somewhere.
Recall that a dominant morphism $Y \to X$ of varieties is *separable* if $K(X) \to K(Y)$ is a separable field extension. In characteristic $0$, this is automatic.
>
> **Theorem.** L... | 4 | https://mathoverflow.net/users/82179 | 232992 | 108,186 |
https://mathoverflow.net/questions/232687 | 2 | Standard Morse lemma states that if the singularity of a smooth function $f$ is non-degenerate, one can choose coordinates such that function has a "quadratic" form $f = \sum \limits\_i x\_i^2 - \sum \limits\_j x\_j^2$. To prove this one uses an integral representation $f(x) - f(0) = \int \limits\_0^1 \frac{d}{dt}f(tx)... | https://mathoverflow.net/users/84714 | Modification of Morse lemma with two functions | In general, this is not possible. One needs a neighbourhood $U$ of the origin such that on every level set $f^{-1}(c)\cap U$, the minimum of $g|\_U$ is exactly $(c/\alpha)^2$. If this condition is satisfied, then regard $g$ as a fibrewise Morse function for the family $f\colon U\to V\subset\mathbb R$. Because the Morse... | 3 | https://mathoverflow.net/users/70808 | 232998 | 108,189 |
https://mathoverflow.net/questions/232976 | 2 | I'm working on a stochastic differential equations research problem and I have come across this second order ODE, my gut tells me it has an analytic or close to analytic solution, but I just can't find it. Any help would be appreciated.
$$y''(x)-(A+B\,\sin 2x)\, y'(x)-\lambda y(x)=0\quad$$
for $x\in(0,\pi)$ with boun... | https://mathoverflow.net/users/88618 | Second order linear ODE question with boundary conditions | A standard change of the variable killing the term $y'$ reduces this to Hill's equation (see Whittaker-Watson, chapter "Mathieu functions", in vol. II).
It is not clear what you mean by "analytic or semi-analytic", and what you mean
by "solution": the problem that you stated seems to be an eigenvalue problem not a prob... | 2 | https://mathoverflow.net/users/25510 | 233011 | 108,192 |
https://mathoverflow.net/questions/233018 | 5 | My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The *generalized Cantor space* is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is *$\kappa$-compa... | https://mathoverflow.net/users/2415 | When is the generalized Cantor space $\kappa$-compact? | A cardinal $\kappa$ is weakly compact if and only if $2^{\kappa}$ is $\kappa$-compact and $\kappa$ is strongly compact if and only if $2^{I}$ is $\kappa$-compact for all sets $I$ where $2^{I}$ is given the topology with basis of open sets of the form $[\sigma]$ where $\sigma:J\rightarrow 2$ and $|J|<\kappa$.
To prove... | 10 | https://mathoverflow.net/users/22277 | 233026 | 108,194 |
https://mathoverflow.net/questions/233027 | 1 | Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ on $G/K$, given by left-multiplication.
We now identify the cotangentbundle $T^\*N = G \times\_K \mathfrak{k}^\circ$... | https://mathoverflow.net/users/75382 | Existence of left-invariant metric on the cotangentbundle of homogeneous spaces? | No. The simplest example is $G = \mathrm{SL}(2,\mathbb{R})$ acting on $\mathbb{RP}^1 = G/P$, where $P$ is the (noncompact) subgroup of upper triangular matrices. It's easy to show that $G$ cannot not preserve any Riemannian metric on $T^\*(G/P)$: If it did preserve a metric $g$, then, since $G$ preserves the zero secti... | 2 | https://mathoverflow.net/users/13972 | 233030 | 108,196 |
https://mathoverflow.net/questions/233023 | 2 | Consider the co-presheaf $\mathcal{F}$ of continous real-valued functions with relatively-compact support on a topological space $X$. Consider a point $x\in X$.
1) When $\mathcal{F}$ is considered a co-presheaf with values in the category of sets, what is the co-stalk $\mathcal{F}\_x$?
2) For X paracompact a partit... | https://mathoverflow.net/users/66824 | Co-stalk of co-presheaves and cosheaves | Projective limits in vector spaces and in sets are the same so the stalk does not depend on whether you consider this as a co-presheaf of sets or vector spaces.
in both case it is just the directed projective limit of the $\mathcal{F}(U)$ for $U$ among neighbourhood of $x$, the corestriction maps $\mathcal{F}(U) \rig... | 9 | https://mathoverflow.net/users/22131 | 233031 | 108,197 |
https://mathoverflow.net/questions/233016 | 1 | Let $f: X \to S$ be a proper morphism (not necessarily smooth!, but perhaps flat) and $S$ be an excellent, regular, affine, integral variety over a finite field (I do not know which of these properties are really necessary).
Does $\{s \in S : X\_s := f^{-1}(s) \text{ is (geometrically) integral}\}$ contain an open su... | https://mathoverflow.net/users/nan | locus of integral fibres open | If $f$ is proper and flat, the "geometrically integral locus" is open by EGA IV, (12.2.1)(x). This works for any morphism of schemes which is proper, flat and finitely presented.
The geometrically connected locus is not open in general: think of a ramified double cover.
| 7 | https://mathoverflow.net/users/7666 | 233036 | 108,198 |
https://mathoverflow.net/questions/233015 | 6 | Let $a\_{n,k}$ be the coefficient of $$X\_1^{\frac{k(n-1)}{2}}X\_2^{\frac{k(n-1)}{2}}\cdots X\_n^{\frac{k(n-1)}{2}}$$ in the expansion of the real polynomial $$\left(\prod\limits\_{1\leq i<j\leq n}(X\_j-X\_i)\right)^k,$$where $n,k$ are positive integers such that $n>1$ and $k(n-1)\equiv0 \pmod 2$.
Since $$\prod\limit... | https://mathoverflow.net/users/58096 | The coefficient of a specific monomial in the expansion of the following polynomial | Let's call your polynomial $\Delta$. Then $\Delta$ is anti-symmetric in the $x\_i$ (if you switch $x\_i$ to $x\_j$ then $\Delta$ turns into $-\Delta$). If $k$ is odd, then $\Delta^k$ also has the property, so $a\_{n,k}=0$ whenever $k$ is odd.
I will now show that $(-1)^{\binom{n}{2} m} a\_{n,2m}>0$. Write $k=2m$. Let... | 7 | https://mathoverflow.net/users/297 | 233037 | 108,199 |
https://mathoverflow.net/questions/233035 | 0 | Suppose I have two discrete long-time series from two dynamical systems. I assume these two systems have compact attractor. How do I measure the closeness or distinctness of the two attractors from the two time series? It is possible that these two attractors are disjoint. In that case I hope the measure still can tell... | https://mathoverflow.net/users/22391 | How to quantify the closeness/distinctness of the attractors? | The Hausdorff distance between two sets $A,B$ is defined by
$$
d\_H(A,B)=\inf\_{\epsilon>0}\big((\forall a\in A\exists b\in B : d(b,a)<\epsilon) \& (\forall b\in B\exists a\in A : d(a,b)<\epsilon)\big).
$$
It is often used to measure distances between compact subsets of a metric space.
| 0 | https://mathoverflow.net/users/50457 | 233039 | 108,200 |
https://mathoverflow.net/questions/232951 | 2 | I need to use the following theorem:
Let $\mathfrak{g}$ be a semisimple real Lie algebra, $\Sigma$ a set of **restricted** roots for $\mathfrak{g}$. Let $\rho$ be any finite-dimensional representation of $\mathfrak{g}$. Then any **restricted** weight $\lambda$ of $\rho$ satisfies
$$
\forall \alpha \in \Sigma,\quad 2\... | https://mathoverflow.net/users/39348 | Characterization of restricted weights of representations of real semisimple Lie groups | As nfdc23 indicates in comments, the basic source is $\S5$ of the 1965 IHES paper on reductive groups by Borel and Tits [*here*](http://www.numdam.org/numdam-bin/item?id=PMIHES_1965__27__55_0), in particular 5.7-5.8. Note however the essential list of corrections in $\S5$ of their 1972 IHES paper [*here*](http://www.nu... | 2 | https://mathoverflow.net/users/4231 | 233041 | 108,201 |
https://mathoverflow.net/questions/233044 | 2 | In fact, I'm intrested in such a question:
---
If $\sigma$ and $\pi$ are two permutations of $\{1,2,\ldots,n\}$
what is necessary and sufficient condition to be the following statement true. For all $a\_1\geqslant a\_2\geqslant \ldots \geqslant a\_n$ and $b\_1\geqslant b\_2\geqslant \ldots \geqslant b\_n$
$$
a\_1... | https://mathoverflow.net/users/66586 | On generalization of rearrangement inequality | Necessary and sufficient condition is the following: for all $k\in \{1,\dots,n\}$, the sequence $\sigma(1),\dots,\sigma(k)$ strongly minorates $\pi(1),\dots,\pi(k)$, that means: if $u\_1\leqslant \dots \leqslant u\_k$ is increasing permutation of $\sigma(1),\dots,\sigma(k)$ and $v\_1\leqslant \dots \leqslant v\_k$ is i... | 4 | https://mathoverflow.net/users/4312 | 233050 | 108,203 |
https://mathoverflow.net/questions/233038 | 1 | Assume that $X\_n$ is a sequence of a zero-mean and unit variance random variables (and maybe having density w.r.t. to Lebesgue). Can we conclude that $ P(X\_n \in [0,R\_n]) $ is bounded away from zero eventually, say $$\liminf\_{n \to \infty} P(X\_n \in [0,R\_n]) > 0$$ assuming that $R\_n \to \infty$. Intuitively, $X\... | https://mathoverflow.net/users/36687 | Lower bounding the probability that a zero-mean sequence of random variables stays positive | Here's a proof if a third moment condition is satisfied.
Suppose that $\mathbb EX=0$, $\mathbb EX^2=1$ and $\mathbb E|X|^3\le K$.
Then let $Z=|X|$. We use Cauchy-Schwarz: $Z^2=Z^{1/2}Z^{3/2}$, so that $(\mathbb EZ^2)^2\le \mathbb EZ\cdot \mathbb EZ^3$. This gives $\mathbb EZ\ge \frac 1K$. Hence $\mathbb EX\mathbf 1... | 3 | https://mathoverflow.net/users/11054 | 233052 | 108,204 |
https://mathoverflow.net/questions/233040 | 3 | Let $R$ be a unital ring. We define the Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $a=xy,\;b=yx$ for some $x,y\in R$. (This is inspired by the usual Murray Von Neumann equivalent relation in K theory, which is defined on the set of idempotents of a ring). The relation $M$ is a reflexive and... | https://mathoverflow.net/users/36688 | An isomorphic invariant in ring theory | If you have a finite ring $R$ that is [von Neumann regular](https://en.wikipedia.org/wiki/Von_Neumann_regular_ring), then $[0]=N(R)$ or more generally if you have a von Neumann regular ring $R$ such that each element has a power that belongs to a subsemigroup which is a group. This follows from a result on semigroups i... | 2 | https://mathoverflow.net/users/15934 | 233053 | 108,205 |
https://mathoverflow.net/questions/233051 | 2 | We know every cubic surface in $\mathbb{P}^3$ is obtained by blowing up $\mathbb{P}^2$ at 6 points in general position. Hence they are all birational to $\mathbb{P}^2$.
My question is: Do we have more refined ways to classify these cubics? I heard we can do something with configurations?
Thanks!
| https://mathoverflow.net/users/48616 | Classification of cubic surfaces in $\mathbb{P}^3$ | Well, you can classify smooth cubic surfaces up to isomorphism instead. This gives rise a 4-dimensional moduli space.
This can be seen by noting that $\mathrm{PGL}\_3$ acts transitively on collections of $4$ points in $\mathbb{P}^2$ in general position, so one may assume that the first $4$ blown-up points are $[0:0:1... | 3 | https://mathoverflow.net/users/5101 | 233059 | 108,206 |
https://mathoverflow.net/questions/232999 | 13 | Consider the first-order language $\mathcal{L}\_{\text{OA}}:=(+,\cdot,0,1)$; in this language, we can formulate statements of ordinal arithmetic. Clearly, the theory $T\_{\text{OA}}$ of $(\text{On},+,\cdot,0,1)$ is not recursive, as $\omega$ is definable in $\mathcal{L}\_{\text{OA}}$ (as being the only ordinal $\gamma$... | https://mathoverflow.net/users/49044 | Is ordinal arithmetic more complicated than classical arithmetic? | Using what are now called Ehrenfeucht–Fraïssé Games and extensions thereof, Ehrenfeucht showed that
$$(\mathrm{Ord},{<}) \sim (\omega^\omega,{<})$$
$$(\mathrm{Ord},{<},{+}) \sim (\omega^{\omega^\omega},{<},{+})$$
$$(\mathrm{Ord},{<},{+},{\cdot}) \sim (\omega^{\omega^{\omega^\omega}},{<},{+},{\cdot})$$
where $\sim$ ... | 12 | https://mathoverflow.net/users/2000 | 233062 | 108,208 |
https://mathoverflow.net/questions/233058 | 22 | In Thurston's book *The Geometry and Topology of Three-Manifolds* it is proven that the underlying space of a two-dimensional orbifold is always a topological surface.
Are there any easy examples of higher dimensional orbifolds whose underlying spaces are not topological manifolds?
| https://mathoverflow.net/users/88649 | What is an example of an orbifold which is not a topological manifold? | It is quite easy to give an example in real dimension $4$.
In fact, it was shown by D. Mumford in the paper
[**The topology of normal singularities of an algebraic surface and a criterion for simplicity**](http://www.ams.org/mathscinet-getitem?mr=153682), *Inst. Hautes Etudes Sci. Publ. Math.* (1961), no. 9, 5 - ... | 16 | https://mathoverflow.net/users/7460 | 233063 | 108,209 |
https://mathoverflow.net/questions/233073 | 5 | Let $F$ be a $p$-adic field.
In "Tamely ramified supercuspidal representations of $Gl\_n$" (Am. J. Math **73** (1977)), Howe constructs a supercuspidal representation $\pi\_{\psi}$ of $GL\_n(F)$ from the following data:
* A tamely ramified extension $F'/F$ of degree $n$, and
* An 'admissible' character $\psi: F'^\t... | https://mathoverflow.net/users/30726 | Is Howe's construction of tame supercuspidal representations independent of additive character? | The only real use of $\chi$ is to identify Moy–Prasad quotients with their character lattices; but notice that this is done *twice*, first to produce from $\theta$ an element $y$ (on p. 442), then to produce from this element $y$ a character $\theta$ of a 'wider' but 'deeper' group (in Lemma 12 on p. 450). These two id... | 7 | https://mathoverflow.net/users/2383 | 233074 | 108,213 |
https://mathoverflow.net/questions/233077 | 5 | Let $T$ be pruned subtree of $\omega^{<\omega}$. For my cases of interest, we may assume that $T$ is infinitely branching at every node, and consists of increasing sequences.
Let $A=\{x\in\omega^{\omega}:\exists y\in[T](x \text{ is a subsequence of } y)\}$, where $[T]$ denotes the (closed) set of infinite branches th... | https://mathoverflow.net/users/16107 | Is the set of subsequences of branches through a tree Borel? | With your assumption that the tree consists of increasing
sequences only, then the answer is yes, this is Borel. The reason
is that we can identify whether or not $x$ is a subsequence of a
branch through the tree simply by checking the arithmetic-in-$T$
condition that every finite initial segment of $x$ is a
subsequenc... | 6 | https://mathoverflow.net/users/1946 | 233079 | 108,214 |
https://mathoverflow.net/questions/233084 | 6 | Let $\Sigma\_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma\_k$-action. Then we can form a $k$-dimensional vector bundle
$$
\xi:\mathbb{R}^k\longrightarrow M\times\_{\Sigma\_k}\mathbb{R}^k\longrightarrow M/\Sigma\_k.
$$
We notice that the transition functions of $\xi$ are given in t... | https://mathoverflow.net/users/41075 | non-orientability of vector bundles induced from a symmetric group action | Yes if $M$ is connected, no otherwise.
If $M$ is connected, the covering gives a surjective homomorphism $\pi\_1(M/\Sigma\_k)\to \Sigma\_k$. The vector bundle comes from a representation of $\pi\_1$ defined by composing this with the permutation representation of $\Sigma\_k$. Taking determinants, the determinant line... | 2 | https://mathoverflow.net/users/18060 | 233086 | 108,216 |
https://mathoverflow.net/questions/233071 | 0 | From [Wikipedia](https://en.wikipedia.org/wiki/Hilbert_symbol#Properties_2): given $a\in K^\times$,
`(a,b)=1 for all b [in K*] if and only if a is in K*ⁿ`
So suppose that $(\frac{a\ ,\ K^\times\!}{p})\neq 1$ [assume $n$ above generates the prime ideal $p\unlhd{\cal O}\_K$]. Given $h\in\Bbb N$, are there any hypothese... | https://mathoverflow.net/users/57771 | Norm Residue Symbol refinement? | For $h=0$ there must be a witness to $(a,b/n)\not=1$ because if all integers $b$ satisfied $(a,b/n)=1$ then by multiplicativity all $b\in K^\times$ would.
| 1 | https://mathoverflow.net/users/85322 | 233092 | 108,217 |
https://mathoverflow.net/questions/233078 | 0 | Let $X, Y$ be two different Banach spaces, and let $T: X \to Y$ be a compact linear operator. Suppose the identity $I : X \to Y$ **is well-defined**. (For example, we could have $X = L^2([0,1])$ and $Y = L^1([0,1])$, both equipped with the Lebesgue measure).
Do most/all elementary results in spectral theory hold in ... | https://mathoverflow.net/users/88657 | Spectrum of compact operator between different Banach spaces | By "$I$ is well-defined", I presume you mean you have a continuous injection $\iota$ of $X$ into $Y$. If $X$ and $Y$ are not isomorphic it will not be a bijection, because there is no continuous linear bijection from $X$ to $Y$, and the same applies to $T - \lambda \iota$. Thus $T - \lambda \iota$ will never be inverti... | 2 | https://mathoverflow.net/users/13650 | 233095 | 108,219 |
https://mathoverflow.net/questions/233057 | 18 | The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate parabolic bundles (i.e., vector bundles with flags at finitely many points)? Said differently, how do parabolic bundles a... | https://mathoverflow.net/users/41301 | What are parabolic bundles good for? | Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set
up of a Riemann surface with cusps. They were trying to generalize the
Narasimhan-Seshadri correspondence on a compact Riemann surface (between
polystable bundles of degree $0$ and unitary representations of the
fundamental group). In the no... | 18 | https://mathoverflow.net/users/11682 | 233096 | 108,220 |
https://mathoverflow.net/questions/233009 | 5 | Let us call a partition *odd* if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$.
Let $H(n)$ denote the set of all *hook* partitions of $n$. I have made the following surprising observation (to me, at least): For every $\mu\in H(m)$,
$$... | https://mathoverflow.net/users/78061 | Sum of skew characters over hooks and "odd" partitions | I know you were asking for a reference, and there may be better approaches, but just to offer one proof of your statement based on the Murnaghan–Nakayama formula. Assume $m>0$ then any skew tableaux $\lambda/\mu$ has shape $(a,1^b)$ with $a+b=n$. This means $a$ boxes in the first row and $b$ boxes in the first column b... | 5 | https://mathoverflow.net/users/22846 | 233097 | 108,221 |
https://mathoverflow.net/questions/233067 | 19 | Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D\_1$, $D\_2$, ..., $D\_N$. For a set of indices $I$, let $D\_I = \bigcap\_{i \in I} D\_i$ and let $D^{\circ}\_I = D\_I \setminus \bigcup\_{J \supsetneq I} D\_J$. I would like to compute $H^{\ast}(X... | https://mathoverflow.net/users/297 | A spectral sequence for computing cohomology of a space from that of its strata | Let $X = T\_n \supset T\_{n-1} \supset \cdots \supset T\_{-1} = \varnothing$ be a topological space filtered by closed subspaces, where for simplicity I assumed the filtration bounded. Then there is a spectral sequence
$$ E\_1^{pq} = H^{p+q}\_c(T\_p \setminus T\_{p-1}) \implies H^{p+q}\_c(X).$$
(Some mild point-set ass... | 17 | https://mathoverflow.net/users/1310 | 233103 | 108,224 |
https://mathoverflow.net/questions/72321 | 27 | An infinite dimensional Banach space $X$ is **decomposable** provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on $X$ whose rank and corank are both infinite. The first separable indecomposable Banach space was constructed by Gowe... | https://mathoverflow.net/users/2554 | Decomposable Banach Spaces | According [to the recent preprint](http://arxiv.org/abs/1603.01753) by Koszmider, Shelah and Świętek under the generalised continuum hypothesis there is no such bound. In particular, one cannot prove the existence of such a bound working merely within the ZFC.
| 12 | https://mathoverflow.net/users/15129 | 233110 | 108,227 |
https://mathoverflow.net/questions/232886 | 4 | **Does the following class of graphs have a name?**
I'm interested in directed graphs with the following property: for every cycle (of the underlying undirected graph) half of the edges go in one direction and half in the other direction.
More formally, let $\vec{G}$ be a directed graph and $G$ the corresponding un... | https://mathoverflow.net/users/37211 | Name for directed graphs with "balanced cycles" | Yes, these are known as **graded graphs**.
(This terminology is used in Bela Bollobas's *Modern Graph Theory*, inter alia.)
| 3 | https://mathoverflow.net/users/39521 | 233111 | 108,228 |
https://mathoverflow.net/questions/232689 | 8 | Given a smooth compact $n$-dimensional manifold $M^{n}$, let $\operatorname{Diff}(M)$ denote the group of smooth diffeomorphisms $M \rightarrow M$ equipped with the Whitney $C^{\infty}$-topology. Let $h\_{\phi} \colon [0, 1] \times M \rightarrow M$ denote the homotopy associated to a path $\phi \colon [0, 1] \rightarro... | https://mathoverflow.net/users/88487 | Can any path in the diffeomorphism group of a smooth compact manifold be approximated by a smooth path? | Here is an example of a very hands-on (and standard) construction of the path that you're looking for. The goal will be take a sequence of diffeomorphisms in the path that are located sufficiently close to each other in the $C^\infty$ topology, and then to join these diffeomorphisms by smooth paths.
Choose an auxilia... | 4 | https://mathoverflow.net/users/48067 | 233112 | 108,229 |
https://mathoverflow.net/questions/150347 | 6 | A countable discrete group $\Gamma$ is said to be weakly amenable with Cowling-Haagerup constant 1 if there exists a sequence of finitely supported functions $(\phi\_n)$ on $\Gamma$ such that $\phi\_n\rightarrow 1$ pointwise and $\sup\_n ||\phi\_n||\_{cb}\leq 1$, where $||\phi||\_{cb}$ denotes the (completely bounded) ... | https://mathoverflow.net/users/9401 | Weakly amenability and exactness for discrete groups | See Theorem B/Theorem 3.4. in "Exactness of locally compact groups" <http://arxiv.org/abs/1603.01829>
| 2 | https://mathoverflow.net/users/9401 | 233113 | 108,230 |
https://mathoverflow.net/questions/233106 | 1 | Do we have that$$\|Du\|\_{L^{2p}} \le C\|u\|\_{L^\infty}^{1\over2} \|D^2u\|\_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C\_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$.
| https://mathoverflow.net/users/nan | $L^p$-bounding inequality | Let $u$ be a smooth compactly supported real-valued function defined on $\mathbb R^n$.We have
$$
\int \vert\partial\_j u\vert^{2p} dx=\langle\partial\_j u,\text{sign}({\partial\_j}u )\vert\partial\_j u\vert^{2p-1}\rangle=
-\langle u,\partial\_j\bigl(\text{sign}({\partial\_j}u )\vert{\partial\_j}u\vert^{2p-1}\bigr) \ran... | 2 | https://mathoverflow.net/users/21907 | 233116 | 108,232 |
https://mathoverflow.net/questions/233019 | -1 | Given are $2n$ random vectors $x\_i,y\_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is almost surely linearly independent. What can be said about the vector $a\_i=[a\_{i,1},\ldots,a\_{i,n}]^\top$ which consists... | https://mathoverflow.net/users/81838 | Are the coefficients of a linear combination of random vectors as random? | So $A = Y X^{-1}$, where $X$ and $Y$ are the $n \times n$ matrices with these vectors as columns, and $A$ is the matrix with entries $a\_{ij}$.
Almost surely, $A$ has full rank because $Y$ and $X^{-1}$ do. Thus the rows of $A$ are almost surely linearly independent, as are the columns.
| 2 | https://mathoverflow.net/users/13650 | 233125 | 108,235 |
https://mathoverflow.net/questions/233120 | 2 | I once read some books about Nevanlinna theory, most of them will discuss the Nevanlinna main theorem small function theorem under some conditions. While, I know little motivation of small function theorem in Nevanlinna theory. And also I want to know elegant application of this theorem. I also have the same question a... | https://mathoverflow.net/users/11966 | The motivation and application of Nevanlinna second main theorem for small functions | One motivation is mentioned in Yamanoi's paper where the second main theorem for small functions is proved: a theorem of Picard says that if we have meromorphic solutions $x(z),y(z)$ of $F(x,y)=0$, where $F$ is a polynomial then the genus of $F(x,y)=0$ is at most one. Then one wants a generalization to equations of
the... | 4 | https://mathoverflow.net/users/25510 | 233147 | 108,240 |
https://mathoverflow.net/questions/232881 | 3 | **Conjecture:** Let $\pi: X\to X'$ be a proper flat surjective morphism of complex spaces.
If $X$ is Kahler, is $X'$ Kahler?
This conjecture when $X$ and $X'$ are smooth solved by Jean Varouchas from Nancy and by some assumption of geometric flatness of $\pi$ solved also by him on singular setting.
But Is there an... | https://mathoverflow.net/users/nan | A conjecture from Jean Varouchas on Kahler varieties | This problem was solved by Barlet and Varouchas in
[this paper.](http://www.numdam.org/item?id=BSMF_1989__117_3_327_0)
The base $X'$ is assumed to be reduced (surely this is OK for you), and the fibers pure dimensional (also certainly acceptable).
| 5 | https://mathoverflow.net/users/13168 | 233151 | 108,241 |
https://mathoverflow.net/questions/233122 | 8 | Let $G$ and $H$ be graphs on the vertex set $\{1, \ldots, n\}$ and let $(e\_i)$ be the standard basis of $\mathbb{R}^n$. For each edge $\{i,j\}$ define *edge vectors* $e\_i - e\_j$ and $e\_j - e\_i$ in $\mathbb{R}^n$.
>
> Question 1: If there is a linear isomorphism of $\mathbb{R}^n$ with itself that takes the edge... | https://mathoverflow.net/users/23141 | Non-isomorphic graphs with isomorphic edge vectors | Attempt to show that there is no example for Question 2.
Let $G$ be a graph with central vertex $v\_0$, $H$ be a graph with central vertex $u\_0$ and cycle structures (cyclic matroids) of $G$ and $H$ are isomorphic. I claim that $H$ and $G$ themselves are isomorphic as graphs. Let $T$ be a spanning tree in $G$ forme... | 2 | https://mathoverflow.net/users/4312 | 233155 | 108,244 |
https://mathoverflow.net/questions/232695 | 6 | I am reading Miles Reid's notes on weighted projective spaces, and I'm a little confused about a particular paragraph (notes [here](http://homepages.warwick.ac.uk/~masda/surf/more/grad.pdf), page 8):
>
> A famous case is the $E\_8$ singularity $X: (x^2+y^3+z^5=0)$, which is
> naturally weighted homogeneous with we... | https://mathoverflow.net/users/86235 | Resolution of the $E_8$ singularity with a weighted blowup | Edit: Your computation is correct. The weighted blowup $Y \to X$ as defined in Example 3.7 of Reid's notes (i.e. the graph of the quotient morphism $X \to \mathbb{P^1}$) is singular at all the points of the exceptional line. However, the statement about $Y$ having 3 cyclic singularities becomes true when you replace $Y... | 4 | https://mathoverflow.net/users/1508 | 233156 | 108,245 |
https://mathoverflow.net/questions/233124 | 5 | Let $f=\sum\_{n=1}^\infty a\_nq^n$ be a newform of level $N$ and weight $k\ge 2$. Suppose that $f$ is a CM modular form in the sense of §3 of Ribet's paper [Galois representations attached to eigenforms with nebentypus](http://link.springer.com/chapter/10.1007%2FBFb0063943): i.e. there exists a quadratic character $\va... | https://mathoverflow.net/users/54339 | Field cut out by a CM modular form is imaginary | As Joel says, $a\_p=0$ if $p$ is inert in $K$, so this means that the Galois representation attached to $f$ must be induced from a 1-dimensional representation of the absolute Galois group of $K$. If $K$ were real quadratic then the grossencharacter corresponding to this 1-dimensional representation would have to be eq... | 2 | https://mathoverflow.net/users/85322 | 233162 | 108,247 |
https://mathoverflow.net/questions/233167 | 1 | I'm looking for a text I could cite that *explicitly* states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma \in \mathfrak{S}\_n$ and $\lambda \vdash n$,
$\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),... | https://mathoverflow.net/users/5621 | Reference request: $\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$ for characters of the symmetric group | This is example 2 in page 116 of MacDonald's book, "Symmetric Functions and Hall Polynomials"
| 6 | https://mathoverflow.net/users/78061 | 233168 | 108,249 |
https://mathoverflow.net/questions/233172 | 1 | I know the fact that (undirected) exchange graphs of quivers with the same underlying undirected graph but have different orientations are isomorphic (i.e. quivers that are just finitely many arrow-flipping away from each other such as An of different orientations)..I want to ask several questions:
1.Why are exchange... | https://mathoverflow.net/users/73892 | Why are exchange graphs of quivers with the same underlying graph but have different orientations isomorphic? | You don't say how you are thinking about exchange graphs, but I am going to interpret this as a question about cluster algebras. (If it is a question about cluster categories, then the final answers are the same, but there is more preliminary background to explain why the answers are the same as for cluster algebras.)
... | 2 | https://mathoverflow.net/users/297 | 233176 | 108,253 |
https://mathoverflow.net/questions/233188 | 22 | I have very little experience with Galois representations, mostly as they relate to class field theory, elliptic curves, and modular forms, but they seem to have quite a reputation in number theory as one of the most important objects of study, in particular because Tannakian philosophy states that they allow us to rec... | https://mathoverflow.net/users/58443 | Concrete Applications of knowing $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ | Of course, Galois theory intervenes as a basic tool in the study of some diophantine equation. But deeper aspects, in the form of Galois representations, were crucial for the proof of at least 4 fantastic theorems in the last century.
* The Mordell-Weil theorem concerning rational points of elliptic curves over the f... | 24 | https://mathoverflow.net/users/10696 | 233189 | 108,257 |
https://mathoverflow.net/questions/230626 | 7 | Let $\mathbb{N}$ denote the set of positive integers, and let's say that $A\subseteq \mathbb{N}$ is *numerically dense* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$
Is there a topology $\tau$ on $\mathbb{N}$ such that the collection of numerically dense subsets of $\mathbb{N}$ equals the coll... | https://mathoverflow.net/users/8628 | Numerical and topological density | Suppose we have such a topology on $\mathbb{N}$. Let $\{A\_\alpha : \alpha < \mathfrak{c}\}$ be an uncountable almost disjoint family of subsets of $\mathbb{N}$. For each $\alpha$, let
$$B\_\alpha = \bigcup\{[2^n,2^{n+1}) : n \in A\_\alpha\}.$$
Notice that $\{B\_\alpha : \alpha < \mathfrak{c}\}$ is still an almost disj... | 4 | https://mathoverflow.net/users/70618 | 233197 | 108,259 |
https://mathoverflow.net/questions/233114 | 8 | Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the Lyapunov exponents of $\mu$ will be positive?
This is a version of a [previous question](https://mathoverflow.net/q/232860/1... | https://mathoverflow.net/users/18060 | Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set? | The answer is no. In fact, the following result holds: if $M$ is a compact manifold, $f\in\mathrm{Diff}^{1+\alpha}(M)$ and $\mu$ is an ergodic $f$-invariant probability measure such that all its Lyapunov exponents are positive, then $\mu$ is a periodic measure, i.e. there exists a periodic source $\{p,f(p),\ldots,f^{n-... | 6 | https://mathoverflow.net/users/889 | 233206 | 108,261 |
https://mathoverflow.net/questions/233203 | 1 | Suppose we have an algebra $A$ (unital, associative), with an ideal $I \leq A$ and a finitely generated module $M$ over $A$.
It is possible to obtain both $\mathrm{End}\_A(M)$ and $\mathrm{End}\_A(M/IM)$ as subquotients of $M\_n(A)$ ($n\times n$ matrices with values in $A$), as given for example [here](https://mathov... | https://mathoverflow.net/users/88722 | Under what assumptions can endomorphisms of $M/IM$ be realized as a subquotient of endomorphisms of $M$? | Let $A=\mathbb{C}[x,y]$, $I=(x,y)$, and $M$ the $3$-dimensional $A$-module $(x,y)/(x^2,y^2)$ (with basis $\{x,y,xy\}$).
Then $M/IM$ is isomorphic to a direct sum of two copies of $\mathbb{C}=A/I$, so $\operatorname{End}\_A(M)\cong\mathbb{C}$, but $\operatorname{End}\_A(M/IM)\cong M\_2(\mathbb{C})$.
| 2 | https://mathoverflow.net/users/22989 | 233208 | 108,262 |
https://mathoverflow.net/questions/233105 | 10 | In this post, when I talk about bounded arithmetic theories,
I mean the theories of arithmetic according to "Logical Foundations of Proof Complexity", which capture the complexity classes between $AC^0$
and $PH$, and the theories capturing $PSPACE$ and $EXPTIME$ (see Sam
Buss's PhD thesis ([Bounded Arithmetic](http:... | https://mathoverflow.net/users/39685 | Bounded Arithmetic vs Complexity Theory | If $T\_1$ and $T\_2$ are theories corresponding to complexity classes $C\_1$ and $C\_2$ (resp.), then separation of $C\_1$ from $C\_2$ from $C\_2$ implies separation of $T\_1$ from $T\_2$, but not necessarily vice versa. (This is already mentioned in T. Chow’s answer.) In fact, with details somewhat dependent on the pa... | 7 | https://mathoverflow.net/users/12705 | 233211 | 108,263 |
https://mathoverflow.net/questions/233207 | 22 | I am teaching an introductory group theory course, and it has come to the inevitable proof that $A\_n$ is simple for $n\geq 5$. Now, there seem to be a number of proofs that I can find – one the "standard" one with $3$-cycles, and the others using primitivity or conjugacy class size estimation. Does anyone have a list ... | https://mathoverflow.net/users/11142 | Simplicity of alternating group $A_n$ | Just turning comments into an answer.
1. **Iwasawa's Criterion**. This will do $A\_5$ (using the natural action), $A\_n$ with $n> 6$ (by considering the action on the set of all 3-subsets of $\{1,\dots, n\}$) and a whole bunch of the classical groups (using transvections acting on the associated polar space). Some no... | 19 | https://mathoverflow.net/users/801 | 233215 | 108,265 |
https://mathoverflow.net/questions/227766 | 6 | *This is a question that arose a while ago in work with Damir Dzhafarov on some pieces of reverse mathematics. As far as I know, it has no deep significance; however, it feels like the sort of thing we ought to know. The solution is probably very simple, but I don't see it.*
Fix a computable total function $f$ of two... | https://mathoverflow.net/users/8133 | Finding limit-nondecreasing sets for certain functions | You're right. This is the kind of things we ought to know.
Let $\mathsf{LNS}$ be the statement that every such function f has an f-good set.
Results
-------
I claim that $\mathsf{LNS}$ admits
* cone avoidance (if A is non-computable, then any computable instance of $\mathsf{LNS}$ has a solution X such that A is ... | 4 | https://mathoverflow.net/users/8833 | 233218 | 108,267 |
https://mathoverflow.net/questions/232899 | 9 | Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum $\xi\oplus\xi$, is trivial or not? I want to try different methods as many as possible.
My attempt: To prove $\xi\oplus\xi... | https://mathoverflow.net/users/82774 | non-triviality of the underlying real vector bundle of the complexification of a real vector bundle | It is a bit unclear what the question is asking exactly, so let me try to give some examples what can happen to a vector bundle $E$ when taking $E\oplus E$ (for simplicity, let me call this Whitney doubling).
Certainly, as already mentioned in the question, the non-vanishing of the Stiefel-Whitney classes is a suffi... | 5 | https://mathoverflow.net/users/50846 | 233248 | 108,272 |
https://mathoverflow.net/questions/233244 | 1 | Trying to obtain some exchangeability-related results, I ended up with the following questions, which I couldn't answer (at least, not in the negative); this is also related to this [MO thread](https://mathoverflow.net/questions/152907/convex-hulls-of-families-of-probability-measures) (edit: actually, not really)
Giv... | https://mathoverflow.net/users/32898 | Representation of probability measure over product spaces | It's true; we can use the following rather trivial construction.
Notation: for measurable $A \subset S^2$, let $F\_A : \mathcal{P}(S)^2 \to [0,1]$ be the "evaluation map" $F\_A(\alpha, \beta) = (\alpha \otimes \beta)(A)$.
Consider the map $T : S^2 \to \mathcal{P}(S)^2$ defined by $T(x,y) = (\delta\_x, \delta\_y)$. ... | 1 | https://mathoverflow.net/users/4832 | 233253 | 108,274 |
https://mathoverflow.net/questions/233225 | 5 | Fix a quiver $Q$ without loop. Denote the set of vectices of $Q$ by $I$.
Let $\Lambda\_V$ be the Lusztig nilpotent scheme with associated vector space $V$ over $I$. Briefly speaking, when $Q$ is a $ADE$ quiver, $\Lambda\_V$ is just the scheme of $\Lambda$-module of dimension $|V|$ where $\Lambda$ is the preprojective a... | https://mathoverflow.net/users/41979 | analog of Lusztig nilpotent scheme | One thing that fits the role you're looking for is the Nakajima quiver varieties. These are discussed [later in the seminar you linked to](http://webpages.math.luc.edu/~ptingley/oldseminars/QuantumGroupsSpring2011/lecture10.pdf). There's a natural map of $U(\mathfrak g)$ to functions on these spaces (defined by [Nakaji... | 2 | https://mathoverflow.net/users/66 | 233259 | 108,277 |
https://mathoverflow.net/questions/233258 | 2 | There are common definitions of [series-parallel (SP) graphs and digraphs](https://en.wikipedia.org/wiki/Series-parallel_graph): the basic idea is as follows. A SP graph (or digraph) has two distinguished vertices $s$ ("source") and $t$ ("target"). The graph with a single edge is SP by definition. Identifying the targe... | https://mathoverflow.net/users/1847 | Generalizing series-parallel digraphs with feedback | A slightly more general class is discussed in
Dan Dougherty, Claudio Gutiérrez.
Normal Forms and Reduction for Theories of Binary Relations.
LNCS 1833, pp 95-109 (2000)
(<http://link.springer.com/chapter/10.1007/10721975_7>)
In Definition 4 they introduce four operations:
1. parallel composition
2. series composi... | 2 | https://mathoverflow.net/users/12674 | 233262 | 108,278 |
https://mathoverflow.net/questions/233245 | 3 | Let $H\_1,H\_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix
$$H\_\mu:=\mu H\_1+(1-\mu)H\_2.$$
I'm looking for a description of $\text{Eig}(H\_\mu,\mu)$.
Clearly, $\text{img}(H\_1)\cap\ker(H\_2)\subseteq\text{Eig}(H\_\mu,\mu)$, but under which con... | https://mathoverflow.net/users/85651 | Eigenspace of convex combination of two idempotent matrices | In fact, we always have equality.
Suppose $v \in \text{Eig}(H\_\mu, \mu)$. Write $v = v\_1 + v\_2$ where
$v\_1 = H\_1 v$, $v\_2 = (I-H\_1) v$ are orthogonal.
We have
$$ \mu v\_2 = H\_\mu v - \mu v\_1 = \mu H\_1 v + (1-\mu) H\_2 v - \mu v\_1 = (1-\mu) H\_2 v$$
i.e. $$H\_2 v = \frac{\mu}{1-\mu} v\_2$$
Now since $H\_2$... | 4 | https://mathoverflow.net/users/13650 | 233264 | 108,279 |
https://mathoverflow.net/questions/233221 | 15 | There seems to be a huge discrepancy in what people refer to when they speak of "game theory". I tend to think of it as including, among other things:
* *Combinatorial* game theory dealing with certain games of perfect information, i.e., things like the Sprague-Grundy theory of (terminating, perfect information) impa... | https://mathoverflow.net/users/17064 | What does "game theory" cover and how should it be called? | If we include the larger research community -- economics, computer science, social sciences, business schools, operations research, etc -- I think there really is a partition between combinatorial game theory and what I would propose to call "equilibrium" game theory. Most in econ and related fields who study/use game ... | 11 | https://mathoverflow.net/users/29697 | 233279 | 108,281 |
https://mathoverflow.net/questions/233273 | 1 | Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow M\times\_{G} \mathbb{R}^n\longrightarrow M/G.
$$
And by the covering space theory, we have an epimorphism
$$
r(M,G): \pi\_1(M... | https://mathoverflow.net/users/41075 | vector bundles induced by an action of a finite subgroup of $O(n)$ | By the universal property of the universal covering $U$ of $M\_1/G$ you have epimorphisms from $U$ to both $M\_1$ and $M\_2$.
The groupp of deck transformations $D$ of the covering $U\to M\_1$ is the kernel of $r(M\_1,G)$ and likewise for $M\_2$, so the two kernels are equal, hence $M\_1\cong U/D\cong M\_2$ and the vec... | 2 | https://mathoverflow.net/users/nan | 233281 | 108,282 |
https://mathoverflow.net/questions/233282 | 4 | The question is completely contained in the title :)
I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential closeness. However, I don't think that this argument can be extended beyond sequences.
Thank you.
| https://mathoverflow.net/users/53155 | Is the topological dual of a Banach space weakly* closed in its algebraic dual? | By the bipolar theorem, this is the case precisely when the topological and algebraic duals coincide. Whether this is true or not for infinite dimensional Banach spaces depends on the set theory you are using.
Edit in response to the comments. Using results of Solovay and Schwartz, the belgian mathematician Garnir sh... | 5 | https://mathoverflow.net/users/88761 | 233285 | 108,283 |
https://mathoverflow.net/questions/233275 | 5 | We know that we can have a noncompact surface with a so called Monkey saddle (<https://en.wikipedia.org/wiki/Monkey_saddle>), on which there is a isolated flat umbilic (where the Gaussian curvature vanishes), and the Gaussian curvature is strictly negative everywhere else.
So my question is, is there an example where... | https://mathoverflow.net/users/85168 | Isolate flat umbilic on compact Riemannian surface with nonpositive curvature | Compact surfaces with non-negative Euler characteristic cannot carry such metrics because of Gauss-Bonnet. However, this is the only obstruction.
First, such metrics exist on any compact orientable surface of genus greater than 1. For example, let $M$ be a hyperelliptic Riemann surface of genus $g\ge 2$, and let $\om... | 6 | https://mathoverflow.net/users/13972 | 233294 | 108,285 |
https://mathoverflow.net/questions/233290 | 4 | Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}\_k(S^q) \simeq \pi\_{q - 1}(\text{O}(k))/\mathbb{Z}\_2,$$and the isomorphism classes of the complex vector bundles are given by$$\text{Vect}\_k(S^q, ... | https://mathoverflow.net/users/83593 | Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ | This is general for a suspension of a space. A suspension is the union of two cones $SX=CX\_+\cup CX\_-$. A cone is contractible, hence a vector bundle trivializes over each cone. The only information about the vector bundle is contained in how the two trivializations are glued along $C\_-X \cap C\_+X\cong X$. This can... | 5 | https://mathoverflow.net/users/12156 | 233295 | 108,286 |
https://mathoverflow.net/questions/233091 | 20 | Let $d\in\mathbb{N}$
be squarefree.
Let $\mathcal{O}\_d$
be the ring of integers of $\mathbb{Q}(\sqrt{-d})$.
Let $\Gamma\_d=\mathrm{PSL}\_2(\mathcal{O}\_d)$.
Let $\mathcal{H}^3$
be the upper half-space model for hyperbolic 3-space.
Let $X\_d=\mathcal{H}^3/\Gamma\_d$
be the orbifold obtained by the usual action of $\Gam... | https://mathoverflow.net/users/14835 | What was a "cusp" to Hurwitz in 1892? | It is very difficult to find a paper of Hurwitz dealing in any way with the geometry or topology of $\mathbb{H}^3/PSL\_2(\mathcal{O}\_{\mathbb{Q}(\sqrt{-D})})$. (And I am not sure I like what this implies about our ways of keeping knowledge alive, attribution and such.) The closest two things I could find were the foll... | 21 | https://mathoverflow.net/users/50846 | 233296 | 108,287 |
https://mathoverflow.net/questions/233283 | 4 | To state the question and fix conventions I will introduce some notation from e.g. (Lin-Trudinger, Bull. Aust. Math. Soc. 1994, ``On some inequalities for elementary symmetric functions")
Given $\lambda\_1, \dots, \lambda\_n$ let
\begin{align\*}
\sigma\_k(\lambda) = \sum\_{1 \leq i\_1 < i\_2 < \dots < i\_k \leq n} ... | https://mathoverflow.net/users/40460 | Unusual inequality concerning elementary symmetric functions | We use the following trick, which is standard in such questions. For numbers $x\_1,\dots,x\_n$ consider the polynomial $F(t)=\prod\_{i=1}^n (t-x\_i)$, let $F'(t)=n\prod\_{i=1}^{n-1}(t-y\_i)$ be its derivative ($y$'s are real provided that $x$'s are --- by Rolle theorem). Note that normalized elementary symmetric functi... | 6 | https://mathoverflow.net/users/4312 | 233299 | 108,288 |
https://mathoverflow.net/questions/233237 | 0 | Some weeks ago I asked the same question at [math.stackexchange][1] but I have not gotten any feedback. The flavour of the question (but see the details later) is about whether to understand congruences in abelian monoids we can always consider the enriched structures obtained by considering polynomial rings.
Let us ... | https://mathoverflow.net/users/36185 | Congruences of abelian monoids which can be extended to (ideal) congruences of polynomials | If $K$ is a commutative ring with unit (I guess even this is not needed) an $M$ and $N$ are monoids, then given a homomorphism $\varphi\colon M\to N$, there is an induced homomorphism of monoid algebras $\Phi\colon KM\to KN$ that on the basis agrees with $\varphi$. If $m,m'\in M$ with $m-m'\in \ker \Phi$ if and only if... | 1 | https://mathoverflow.net/users/15934 | 233315 | 108,291 |
https://mathoverflow.net/questions/233322 | 3 | Let $p$ be a prime and $\mathbb{Q}\_p$ denotes the $p$-adic numbers. Is it true that the degree of the nontrivial $\mathbb{Q}\_p$-irreducible representations of a cyclic group of order $p^n$ is divisible by $p-1$ ?
Proofs or references are appreciated.
| https://mathoverflow.net/users/33900 | Degree of irreducible representations of a finite cyclic group over $\mathbb{Q}_p$ | Yes, it is true. Clearly, it suffices to consider the case that the representation is faithul. Let $G$ be the cyclic group, and $h \in G $ be an element of order $p$. By Clifford's Theorem and Maschke's Theorem ( and since $G$ is Abelian), the representation restricts to $\langle h \rangle$ as a direct sum of equivalen... | 11 | https://mathoverflow.net/users/14450 | 233328 | 108,293 |
https://mathoverflow.net/questions/233331 | 9 | A (±1)-matrix is a matrix whose entries are 1 and −1.
An $n \times n$ (±1)-matrix is called an Hadamard matrix if the rows are
orthogonal.
Equivalently,
An $n \times n$ (±1)-matrix $H$ is Hadamard ⇔ $H H^t = nI\_n$,
where $I\_n$ denotes the $n \times n$ identity matrix.
In this paper : <http://www.sciencedirect.co... | https://mathoverflow.net/users/33047 | spectrum of Hadamard matrices | For a symmetric Hadamard matrix $H$ of order $m$ we have the minimal polynomial $x^2-m$, i.e. eigenvalues $\pm\sqrt{m}$. Indeed, by Cayley-Hamilton theorem $HH^\top=H^2=mI$, as $HH^\top=mI$ for any Hadamard matrix.
In general, eigenvalues are not $\pm\sqrt{m}$. E.g. if I take a skew-symmetric Hadamard matrix of order... | 7 | https://mathoverflow.net/users/11100 | 233333 | 108,295 |
https://mathoverflow.net/questions/233300 | 2 | Let K be an algebraic closed field, $gl\_n$ be the general linear Lie algebra over K, and $sl\_n$ be the special linear Lie algebra.
Let $\chi\in gl\_n^\*$. Let $U\_\chi(gl\_n)$ be the corresponding reduced enveloping algebra for $gl\_n$. If M is an indecomposable module for $U\_\chi(gl\_n)$, then is M still indecompos... | https://mathoverflow.net/users/88769 | indecomposable modules restricted from $gl_n$ to $sl_n$ | This is actually true (in somewhat more generality), as remarked by Jantzen in a recent updating of his unpublished 2011 notes on restrictions of modular representations of $\mathfrak{gl}\_n$ to $\mathfrak{sl}\_n$. Here he works with representations attached to reduced enveloping algebras relative to various linear fun... | 3 | https://mathoverflow.net/users/4231 | 233335 | 108,296 |
https://mathoverflow.net/questions/233324 | 4 | Given a connected topological space $X$, its space of path $PX$ is again a topological space. On the other hand, for a simplicial set $K\_{\bullet}$, its path space is given by
$$
PK\_{n}:=\operatorname{Map}\_{sSet}(\Delta[1], K)\_{n}.
$$
Here my question: given a simplicial topological space $X\_{\bullet}$, what is i... | https://mathoverflow.net/users/41970 | Path space of a simplicial topological space? | It is sensible to restrict to simplicial {\em based} spaces $X\_\*$ and then apply the functor $P$ levelwise. This is used, for example, to compare $|\Omega X\_\*|$ with $\Omega |X\_\*|$ in The Geometry of iterated loop spaces <http://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf>
| 2 | https://mathoverflow.net/users/14447 | 233339 | 108,298 |
https://mathoverflow.net/questions/232777 | 4 | Let $F$ be an ordered field.
What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?
| https://mathoverflow.net/users/83742 | Which ordinals can be embedded into an ordered field? | This parameter of a field does not equal any its common cardinal characteristic that I could think of, though it is related in several ways.
Let me first introduce some notation. Assume $F$ is an ordered field. As noted in the comments, if an ordinal embeds in $F$, it embeds in every interval $(a,b)$ of $F$, so we ca... | 8 | https://mathoverflow.net/users/12705 | 233340 | 108,299 |
https://mathoverflow.net/questions/233332 | 7 | Let $\lambda$ be a partition with $\leq n$ rows and let $L\_{\lambda}$ be the corresponding irreducible representation of ${\rm GL}\_n(\mathbb{C})$. Let $e\_m(X\_1,\dots,X\_n)$ be the $m$th [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial). Is there a known formula for the... | https://mathoverflow.net/users/4002 | Explicit formulas for certain elements in $Z(U(\mathfrak{gl_n}))$ | I believe the elements of $\mathcal{Z}(\mathfrak{g})$ you are looking for are (suitably renormalised) the [Capelli elements](https://en.wikipedia.org/wiki/Capelli%27s_identity); these come from the (renormalised) Capelli determinant
\[C(u) = \det\left(E\_{jk} + \left(u - \frac{n - 2j + 1}{2}\right) \delta\_{jk}\right),... | 5 | https://mathoverflow.net/users/3803 | 233343 | 108,301 |
https://mathoverflow.net/questions/233303 | 7 | In this [paper](http://www.sciencedirect.com/science/article/pii/0166218X89900474?np=y), Friedland shows (in Lemma 3.4) that if $\phi$ is an isomorphism of coherent algebras, then there exists a unitary $U$ such that
$$ \phi(M) = UMU^\dagger$$
for all $M$. I am wondering if the same is true of any *trace-preserving* is... | https://mathoverflow.net/users/18606 | Can every trace preserving isomorphism of unital self-adjoint matrix algebras be realized as conjugation by a unitary? | It is true that every unital self-adjoint algebra $A$ is semisimple. For $A$ contains no non-zero nilpotent right ideal (given any such ideal $I$ and any $M \in I$, we have trace($MM^{\ast}) =0$ since $MM^{\ast} \in I$ is nilpotent, and this forces $M = 0$). Thus $A$ is semisimple, and is a direct sum of full matrix al... | 4 | https://mathoverflow.net/users/14450 | 233351 | 108,303 |
https://mathoverflow.net/questions/233359 | 3 | It is true that the axiom of choice is equivalent to the statement that every linear space has a Hamel basis. There are some linear spaces which definitely don't need axiom of choice to possess (rather canonical) basis: for example $c\_{00}$, the space of all sequences with compact supports. Is it possible to give an e... | https://mathoverflow.net/users/24078 | Linear space with (Hamel) basis and the axiom of choice | No. It is not possible.
Suppose that $V$ is a specified vector space, then it is consistent that the axiom of choice fails very far above $V$ in the hierarchy of sets (the von Neumann hierarchy). In particular it would mean that $V$ has a basis, but still the axiom of choice fails, as it fails very far above $V$.
G... | 6 | https://mathoverflow.net/users/7206 | 233362 | 108,306 |
https://mathoverflow.net/questions/233304 | 2 | Suppose everything below is defined over $k=\overline{\mathbb{F}}\_q$.
Let $H$ be a connected algebraic group acting on a separated variety $Y$. Denote the morphism $H\times Y\rightarrow H\times Y; (h,y)\mapsto (h,hy)$ by $f$, and denote by $\pi$ the left projection of $H\times Y$ to $H$. By applying proper base chan... | https://mathoverflow.net/users/56217 | A homotopy argument in etale topology | It follows from functoriality. For three varieties $X, Y, Z$ with maps $f:X \to Z$ and $g: Y\to Z$, for every isomorphism $h: X \to Y$ forming a commutative triangle with $f$ and $g$, we get an induced isomorphism between $R^i f\_! Y$ and $R^i g\_! Y$.
Apply this to your morphism $f$ and you get the desired action.
... | 2 | https://mathoverflow.net/users/18060 | 233376 | 108,308 |
https://mathoverflow.net/questions/233365 | 8 | I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is, its intersections with planes through the origin) are affine equivalent planar bodies. Then $K$ is an ellipsoid.
Equ... | https://mathoverflow.net/users/4354 | Convex body with affine-equivalent cross-sections | For given integers $n>2$ and $k\in \{2,3,\dots,n-1\}$ the question of Banach asks whether any $n$-dimensional real Banach space with isometric $k$-dimensional sections is a Hilbert space. For $k=2$ this is proved by
[H. Auerbach, S. Mazur, S. Ulam](http://link.springer.com/article/10.1007%2FBF01733278) (here I am not ... | 6 | https://mathoverflow.net/users/4312 | 233392 | 108,313 |
https://mathoverflow.net/questions/233394 | 2 | in this paper <http://arxiv.org/pdf/1412.1626.pdf> it says that Lemma 3.1/(3.1) follows from Theorem 1.3 in <http://arxiv.org/pdf/math/0402192.pdf> without extra details. Can somebody please explain that?
I can see, that these two estimates are every similar as $q=2$, $r=\infty$ , $n=3$ are admissable in Theorem 1.3.... | https://mathoverflow.net/users/88808 | Estimates for Klein-Gordon-Equation follow directly from Wave equation Estimates | I don't think they mean that you can literally just plug in Theorem 1.3 of Sterbenz-Rodnianski to get Lemma 3.1. I think they mean that the proof is basically the same, with suitable adjustments.
A rough sketch:
The key to Theorem 1.3 of Sterbenz-Rodnianski is the derivation of the $L^\infty$ estimate (14). If you... | 1 | https://mathoverflow.net/users/3948 | 233410 | 108,315 |
https://mathoverflow.net/questions/233409 | 0 | Let $A\_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A\_m Z^m + \cdots + A\_1 Z + A\_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if $\det P(Z ) = 0$ and
$s\_1 \ge s\_2 \ge \cdots \ge s\_n$ are singular values of $P(Z)$.
Let $s\_n$ have multipli... | https://mathoverflow.net/users/78479 | $P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$? | The smallest singular value $s\_n$, which has been stated to have multiplicity one, is a simple eigenvalue of the matrix $M(z) = P(z)P(z)^\*$, or equivalently, a simple root to the characteristic polynomial $\chi(\lambda;z) = \det (\lambda I - M(z))$.
As defined, each coefficient in the matrix $M(z)$ is polynomial wi... | 4 | https://mathoverflow.net/users/60984 | 233414 | 108,316 |
https://mathoverflow.net/questions/233418 | 4 | A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. Who first came up with this definition? I had thought it was M. Hochster [Ann. Math. 96 (1972), 318-337], but a number o... | https://mathoverflow.net/users/8604 | Who first considered constructibility of simplicial complexes? | If you want the first use of the term "constructible" in this context, then your reference to Mel Hochster's work is right-on. But if you want the actual notion, then things get slightly hazy. I think the oldest (implicit) use is Newman's two papers from 1926 called something like The foundations of combinatorial analy... | 7 | https://mathoverflow.net/users/18263 | 233419 | 108,317 |
https://mathoverflow.net/questions/45177 | 10 | Perhaps there are none with integral coefficients; so let us admit rational coefficients. The map $(x, y) \mapsto x + \frac{1}{2}(x + y)(x + y + 1)$ is well known, and swapping $x$ and $y$ in the formula yields another, so we have two for starters.
| https://mathoverflow.net/users/7458 | What polynomials biject from $\mathbb{N}^{2}$ to $\mathbb{N}$? | Describing such bijections is an open problem. Maximal result (there is no other bijections among polynomials of degree not higher than 4) are contained in
>
> John S. Lew, Arnold L. Rosenberg,
> *Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials*, J. Number Theory **10** ... | 6 | https://mathoverflow.net/users/88700 | 233431 | 108,321 |
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