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https://mathoverflow.net/questions/137635 | 4 | I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of every k-term arithmetic progression contained in $\{ 1 , \cdots , N\}$.
This may be well-known to experts (what I'm no... | https://mathoverflow.net/users/21724 | A reference for this possibly well-known fact concerning the Kakeya conjecture? | A precise formulation of this implication may be found in Section 4 of the article
T. Wolff, Recent work connected with the Kakeya problem. Prospects in mathematics (Princeton, NJ, 1996), 129–162, Amer. Math. Soc., Providence, RI, 1999.
which may be found online at <http://www.csun.edu/~vcmth014/191dn2.ps> . The... | 8 | https://mathoverflow.net/users/766 | 137648 | 75,519 |
https://mathoverflow.net/questions/137665 | 4 | Using a probabilistic method for number theoretic purposes, I have encountered the following question (it may be very standard):
>
> Let $X\_t$ be a Gaussian process $(t>0)$ such that $X\_0=0$. What is the expected number of times for the process to cross zero in the interval $[0,c)$ as a function of $c$?
>
>
>
... | https://mathoverflow.net/users/24494 | Number of times a Gaussian process crosses zero in an interval | The magic words are "Kac-Rice formula". This is exactly the question which comes up in analyzing zeros of random polynomials.
| 4 | https://mathoverflow.net/users/11142 | 137667 | 75,527 |
https://mathoverflow.net/questions/137644 | 1 | Let $\pi$ be a group, and let $\mathcal{C}$ be the site whose underlying category is that of $\pi$-sets (with $\pi$-linear maps as morphisms). The covers are jointly surjective families of such $\pi$-linear maps (ie, maps commuting with the $\pi$-action).
Let $\langle\pi\rangle$ denote the $\pi$-set whose underlying ... | https://mathoverflow.net/users/15242 | Sheaves on the site of $\pi$-sets | I think you're right that the key is to show that all $H$-fixed elements of $F(\langle\pi\rangle)$ equalize the two maps from $F(\langle\pi\rangle)$ to $F(R)$, where I wrote $R$ to abbreviate $\langle\pi\rangle\times\_S\langle\pi\rangle$. For this purpose, let me first look more closely at $R$ and its two maps to $\lan... | 2 | https://mathoverflow.net/users/6794 | 137669 | 75,529 |
https://mathoverflow.net/questions/124020 | 5 | This question arises when studying semi-classical analysis. Since Weinstein's creed claims that "everything is Lagrangian", where a point in the phase space of classical mechanics is just a cotangent fiber, hence Lagrangian, what can we say about canonical relations, which is a Lagrangian submanifold of the twisted dir... | https://mathoverflow.net/users/27040 | What is the physical interpretation of the canonical relations? | I think I have figure this out. Consider two symplectic manifolds $M\_1$ and $M\_2$, and the canonical relation $\Gamma: M\_1\twoheadrightarrow M\_2$. Then if one consider the "point" in symplectic category, i.e. the Lagrangian submanifold $\Lambda\_i$ of $M\_i, (i=1,2)$, $\Gamma$ just maps one "point" to another "poin... | 0 | https://mathoverflow.net/users/27040 | 137670 | 75,530 |
https://mathoverflow.net/questions/133957 | 5 | Given a group $G$, we denote by $T(G)$ the subgroup generated by all (maximal) normal abelian subgroups of $G$.
Let define the series $(T\_i(G))$ by $T\_0(G)=1$ and $T\_{i+1}(G)/T\_i(G)=T(G/T\_i(G)$, and $n(G)$ to be the smallest integer such that $T\_n(G)=G$.
1) Can one produce finite p-groups $G$ with arbitrary l... | https://mathoverflow.net/users/31883 | Normal abelian subgroups in p-groups | The answer to question 1 is yes. We will demonstrate a way to add one to $n(G)$, so $n(G)$ can be any natural number.
Given a group $G$, and an abelian group $A$, take a faithful action of $G$ on $A$, and consider the semidirect product $A \rtimes G$ .
We need the following condition: If $g$ is a nontrivial element... | 4 | https://mathoverflow.net/users/18060 | 137682 | 75,538 |
https://mathoverflow.net/questions/137655 | 12 | I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.
Let $\mathcal{A} \subset M\_n(\mathbb{C})$ be an algebra which is closed under taking adjoints. (That is, $X \in \mathcal{A} \Rightarrow X^\*\in \mathcal{A}$)
Then,
$\... | https://mathoverflow.net/users/32470 | Structure theorem for finite dimensional $C^*$-algebras and their representations | $\def\CC{\mathbb C}$Suppose $A$ is a $\*$-closed subalgebra of $M\_n(\CC)$. $\CC^n$ is a left $M\_n(\CC)$-module in a canonical way, and therefore also an $A$-module. Using the fact that $A$ is $\*$-closed it is easy to see that if $S\subseteq\CC^n$ is an $A$-submodule, then $S^\perp$ is also an $A$-submodule. This imp... | 9 | https://mathoverflow.net/users/1409 | 137704 | 75,544 |
https://mathoverflow.net/questions/137701 | 6 | While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two problems as follow.
The theorem 6.6 of this chapter is to prove the $C^{2,\alpha}$ regularity of the viscosity solution... | https://mathoverflow.net/users/36582 | A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation | A linear uniformly elliptic equation of the form $F(D^2u) = 0$ can only be $\Delta u = 0$ (up to an affine transformation of the solution) since the only inputs are the second derivatives. Equations like $a^{ij}(x)u\_{ij} = 0$ are of the form $F(D^2u,x) = 0$. If $F(.,x\_0)$ is concave for each $x\_0$ (corresponding to ... | 6 | https://mathoverflow.net/users/16659 | 137707 | 75,546 |
https://mathoverflow.net/questions/137689 | 10 | The [nCatLab Grothendieck construction page](http://ncatlab.org/nlab/show/Grothendieck+construction#definition_10) gives an explicit description of the oplax colimit of any functor to Cat. Can someone give me a similarly explicit description (the objects and morphisms) of an oplax limit of any functor to Cat (or a link... | https://mathoverflow.net/users/30462 | Explicit description of the oplax limit of a functor to Cat? | **The oplax limit is the category of sections** for the functor from the Grothendieck construction to the base category.
The **strong** limit is the category of **cartesian sections** (every arrow in the base category gets mapped to a cartesian one).
Notice how this goes along very well with the interpretation as d... | 8 | https://mathoverflow.net/users/1261 | 137713 | 75,550 |
https://mathoverflow.net/questions/137678 | 86 | Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ admits a paradoxical decomposition with $n$ pieces: $\exists A\_1,\ldots,A\_k,B\_1\ldots,B\_l\subset G$, $\exists g\_1,\ldots... | https://mathoverflow.net/users/7591 | Non-amenable groups with arbitrarily large Tarski number? | It is indeed an open problem, as Misha said. But here is a solution. In *E. Golod, Some problems of Burnside type. 1968 Proc. Internat. Congr. Math. (Moscow,
1966) pp. 284-289. Izdat.* ”Mir”, Moscow, Golod announced, for every $m$ an infinite finitely generated torsion group all of whose $m$-generated subgroups are fin... | 208 | https://mathoverflow.net/users/nan | 137715 | 75,551 |
https://mathoverflow.net/questions/137714 | 4 | For a prime $p$ denote by $r(p)$ (resp. $n(p)$) the smallest prime $q$ which is a quadratic residue (resp. nonresidue) modulo $p$.
It was shown by Linnik that for any fixed $\epsilon>0$ the number of $p<x$ s.t.
$n(p)>p^\epsilon$ is bounded by $c(\epsilon)\log\log x$.
My question is what is the best known bound today ... | https://mathoverflow.net/users/9304 | Least quadratic residue and nonresidue | There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good).
*Theorem:* Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet
character $(n/p)$. If $t(p)$ is a positive function with $L(1,\chi)>t(p)/\log(p)$, then there are absolute constants $c\_1,c\_2>0$ with... | 4 | https://mathoverflow.net/users/32332 | 137723 | 75,555 |
https://mathoverflow.net/questions/137697 | 11 | There are interesting theorems about groups as union of proper subgroups. The first result in this subject is the theorem of Scorza(1926): "*a groups if union of three proper subgroups if and only it has quotient $C\_2\times C\_2$.*" In 1959, [Haber and Rosenfeld](http://www.jstor.org/stable/2310634?&Search=yes&searchT... | https://mathoverflow.net/users/6761 | Groups as Union of Proper Subgroups: References | The mentioned result of Cohn has been further extended. Let us write $σ(G) = n$ whenever $G$ is the union of $n$ proper subgroups, but is not the union of any smaller number of proper sub- groups. Thus, for instance, Scorza’s result asserts that $σ(G) = 3$ if and only if $G$ has a quotient isomorphic to $C\_2 × C\_2$. ... | 11 | https://mathoverflow.net/users/32332 | 137730 | 75,558 |
https://mathoverflow.net/questions/137719 | 6 | Consider $P\_n(x)$ polynomials defined through the recurrence relations
$$P\_n(x)=2(1-x)P\_{n-1}(x)-(1+x)^2P\_{n-2}(x),$$ with $P\_0(x)=1$ and $P\_1(x)=1-3x$.
In fact, the explicit solution of these recurrence relations is given by the formula
$$P\_n(x)=\frac{1}{2}\left[ (1+i\sqrt{x})^{2n+1}+(1-i\sqrt{x})^{2n+1}\right]... | https://mathoverflow.net/users/32389 | Collision polynomials | Writing $P\_n(x)=\frac{1}{2}\left[ (i\sqrt{x}+1)^{2n+1}-(i\sqrt{x}-1)^{2n+1}\right]$ and using that $x^u-y^u$ divides $x^v-y^v$ once $u$ divides $v$ easily shows that $P\_m$ divides $P\_n$ once $2m+1$ divides $2n+1$. With a little more effort, one should see that this sufficient condition is necessary, too.
I'm not s... | 6 | https://mathoverflow.net/users/18739 | 137731 | 75,559 |
https://mathoverflow.net/questions/137605 | 8 | Consider an order $R$ in a number field $L$. Let $C\_R$ be the set of $R$-fractional ideals modulo $L^\times$. Let $O$ be the maximal order in $L$, and $C\_O$ be the class group of $O$.
My question: Is there a formula that relates $\# C\_R$ with $\# C\_O$, that involves perhaps the conductor of $R$?
A note: Beware... | https://mathoverflow.net/users/4398 | Class numbers of orders | The ideal class semigroups mentioned in the question got studied in this setting (orders of number fields) and in other and more general ones by various authors in recent years.
A starting references is:
>
> Zanardo, P.; Zannier, U. The class semigroup of orders in number fields.
> Math. Proc. Cambridge Philos.... | 6 | https://mathoverflow.net/users/nan | 137742 | 75,562 |
https://mathoverflow.net/questions/137750 | 2 | I'm reading Kudla's Article on the Local Langlands Conjecture for $p$-adic general linear groups, and specifically I'm trying to understand how the ideas of Bernstein-Zelevinski yield show that you only need to prove the LLC for supercuspidal representations (on the automorphic side) and irreducible representatons (on ... | https://mathoverflow.net/users/30726 | L functions of Langlands Quotients of essentially-square-integrable representations | The identity $L(Q(\tau\_1,\ldots,\, \tau\_r),\,s) = \prod\_i L(\tau\_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1979), Part 2, 63-86.
| 4 | https://mathoverflow.net/users/11919 | 137759 | 75,568 |
https://mathoverflow.net/questions/137749 | 6 | I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, Jean-Pierre (1998), "Thurston's hyperbolization of Haken manifolds"
From what I understand we take hierarchy for a Haken $... | https://mathoverflow.net/users/37434 | Andreev's Theorem and Thurston's hyperbolization theorem | The sorts of hierarchies that Johannson makes use of are called "simple hierarchies", and go back to Waldhausen (used by him in the solution of the word problem and homotopy rigidity of Haken 3-manifolds). The boundary patterns associated
to these hierarchies don't contain all of the information one needs to glue up th... | 11 | https://mathoverflow.net/users/1345 | 137761 | 75,569 |
https://mathoverflow.net/questions/137748 | 3 | Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am wondering if the following is known? Let $U$ be the set of all non-$G$-paradoxical subsets of $X$. Is there a function $\n... | https://mathoverflow.net/users/26809 | Extending Tarski's Theorem on invariant measures | I would like to answer this in the negative by providing a counter example. Consider the measurable space $(\mathbb Z, 2^\mathbb Z)$ together with the action by $G := \operatorname{Aut}(\mathbb{Z}) \times \operatorname{Aut}(\mathbb{Z})$ defined by:
$(\gamma,\xi)(n):=
\left\{\begin{array}{ll}
\gamma(n)
& : n \hspace... | 2 | https://mathoverflow.net/users/31420 | 137764 | 75,571 |
https://mathoverflow.net/questions/137683 | 3 | Is there an easy classification (and proof) of the possible branched covers between Seifert fibered *integer* homology spheres which are fiber-preserving and branched over fibers (or at least what the possible relations between the orders of the singular fibers should be)?
There exist branched covers of the form $f:\... | https://mathoverflow.net/users/3405 | Classification of fiber-preserving branched covers between Seifert fibered integer homology spheres | The class of homology sphere Seifert-fibered spaces fiber over 2-orbifolds $S^2(p\_1,\ldots,p\_k)$, where $\gcd(p\_i,p\_j)=1, i\neq j$. If one has a fiber-preserving map of one homology sphere to another, then there is a corresponding non-zero degree map between the base orbifolds obtained by quotienting the fibers.
... | 1 | https://mathoverflow.net/users/1345 | 137789 | 75,583 |
https://mathoverflow.net/questions/137779 | 8 | A holomorphic vector bundle $E\to M$ over a compact Kähler manifold $M$ with Kähler form $\omega$ is called ***stable*** if for any coherent analytic subsheaf $\mathcal F$ of lower rank of $E$ there holds $\mu(\mathcal F)<\mu(E)$ where $\mu(\mathcal F)=deg\_\omega(\mathcal F)/rank(\mathcal F)$ and $deg\_\omega(\mathcal... | https://mathoverflow.net/users/5628 | are stable holomorphic bundles over compact Kähler manifolds simple? | As far as I can tell the following argument given in the case of $\mathbb{P}^n$ from Okonek-Schneider-Spindler "Vector bundles on complex projective spaces" works here as well. If someone could check this over I'd be grateful, as I'd be surprised if Uhlenbeck-Yau missed it.
First show that if $f:E\_1\to E\_2$ is a ho... | 8 | https://mathoverflow.net/users/7399 | 137793 | 75,586 |
https://mathoverflow.net/questions/137792 | 6 | My question is about terminology:
Do you know why stationary sets were named such?
Going over the [following MO question](https://mathoverflow.net/questions/37502/what-is-the-idea-behind-stationary-sets) about the intuition behind stationary sets, the only compelling argument I can think of is Fodor's lemma.
Is ... | https://mathoverflow.net/users/13694 | Why stationary sets were named such? | In Infinite Combinatorics, in: Handbook of the History of Logic, 6. Sets and Extensions in the Twentieth Century, p 226, footnote 214, Jean Larson states that the term was first used in G. Bloch: Sur les ensembles stationnaires de nombres ordinaux et les suites distinguees de fonctions regressives, Comptes Rendus Acad.... | 4 | https://mathoverflow.net/users/6647 | 137800 | 75,588 |
https://mathoverflow.net/questions/137753 | 36 | I'm supervising an undergraduate research project. Among other things, I've got the student to look at [this paper of Gene Kopp and John Wiltshire-Gordon](http://arxiv.org/abs/1102.4353). This question arose from a missing complex conjugate in something the student wrote.
Let $g$ be an element of a finite group $G$, ... | https://mathoverflow.net/users/22989 | Word evaluating to a group element and its inverse with different frequency | Yes, one can do much better than $1.7 \times 10^{244552995}$
(not surprisingly, because we're asking less than Lubotzky:
one of the two counts must be less than the other,
but not necessarily zero).
In fact a word of length $10$ suffices.
I tried $G = M\_{11}$ and $g$ an element of order $11$, and took $n=2$,
which ... | 21 | https://mathoverflow.net/users/14830 | 137802 | 75,589 |
https://mathoverflow.net/questions/137449 | 3 | Ramification of prime numbers in number fields is a topic relevant to what I'm studying (arithmetic hyperbolic 3-manifolds), and many results from algebraic number theory are used there, however most of what I need comes from the case where the number field is quadratic. Now, eventually in this life I would like to mas... | https://mathoverflow.net/users/14835 | Basic arithmetic behind ramification in quadratic number fields | See <http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/quadraticgrad.pdf>, which develops factorization into prime ideals in the ring of integers of quadratic fields from scratch. Look particularly at sections 7 and 8 (and Tables 2 and 3 on page 22).
| 3 | https://mathoverflow.net/users/3272 | 137820 | 75,595 |
https://mathoverflow.net/questions/137819 | 9 | Consider two infinite subsets $A$ and $B$ of natural numbers. I am positive that the set of prime divisors of elements of $A+B$ is infinite but I can not prove it. Maybe this is a known result. If anybody has an idea or a reference please let me know. **(by $A+B$ I mean $\lbrace a+b \; | \; a\in A, b\in B\rbrace$)**
| https://mathoverflow.net/users/nan | prime divisor of elements of $A+B$ | This was proved by Erdős and Turán when they were undergraduate students, see Theorem III [here](http://www.renyi.hu/~p_erdos/1934-03.pdf). A stronger result was proved by Győry, Stewart, and Tijdeman, see Theorem 1 [here](http://archive.numdam.org/ARCHIVE/CM/CM_1986__59_1/CM_1986__59_1_81_0/CM_1986__59_1_81_0.pdf).
| 12 | https://mathoverflow.net/users/11919 | 137821 | 75,596 |
https://mathoverflow.net/questions/137806 | -1 | I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:
For $x \in \mathbb{R}^{n}$ and $\alpha = (\alpha\_{1}, \ldots, \alpha\_{n}) \in \mathbb{N}^{n}$, we set
$$ x^{\alpha} = x\_{1}^{\alpha\_{1}}\cdots x\_{n}^{\alpha\_{n}... | https://mathoverflow.net/users/36035 | An inequality involving multi-index | The first inequality with constant 1 follows from
$
\vert x^\alpha\vert\le\Vert x\Vert\_{\infty}^{\vert \alpha \vert},\quad\text{where $\Vert x\Vert\_{\infty}$ is the sup-norm.}
$
The second equality, also with constant 1, is due to $
\Vert x\Vert\_{\infty}^k=\max\_{1\le j\le n} \vert x\_j\vert^k\le \max\_{\vert \alp... | 2 | https://mathoverflow.net/users/21907 | 137822 | 75,597 |
https://mathoverflow.net/questions/137813 | 6 | Let $k$ be an algebraically closed field of characteristic $0$, let $C\_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic closure of $K$. Let $E\_{/K}$ be an elliptic curve with $j(E) \notin k$. Let $P \in E(\overline{K}) \setminus E(K)$ be a poin... | https://mathoverflow.net/users/1149 | Mordell-Weil of an elliptic surface after adjoining a nontorsion section: as small as possible? | I do not believe you have equality in general. I sketch a counterexample below, which is a geometric version of the fact that if $E/K$ is an elliptic curves such that the quadratic twist $E^{(d)}/K$ has Mordell-Weil rank at least 2 then the Mordell-Weil of $E/K(\sqrt{d})$ is at least two bigger than the Mordell-Weil ra... | 11 | https://mathoverflow.net/users/8621 | 137823 | 75,598 |
https://mathoverflow.net/questions/137812 | 2 | Let $E$ be a Banach space, and let $(\sigma\_t)$ be a strongly continuous one-parameter group on $E$: so for $t\in\mathbb R$, we have that $\sigma\_t$ is a contraction on $E$, $\sigma\_t \sigma\_s=\sigma\_{t+s}$, $\sigma\_0$ is the identity, and for each $x\in E$, the map $\mathbb R\rightarrow E; t\mapsto\sigma\_t(x)$ ... | https://mathoverflow.net/users/406 | "Generators" of one-parameter groups of isometries | Using the semigroup property, you can extend $\sigma\_t(x)$ to a bounded entire function.
Liouville's theorem will do the rest.
| 5 | https://mathoverflow.net/users/12120 | 137828 | 75,599 |
https://mathoverflow.net/questions/137848 | 11 | Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing something, that any such sheaf extends to a sheaf on the site $\mathsf{Man}$ of all smooth manifolds. Is there a proof writte... | https://mathoverflow.net/users/25355 | Diffeology as a sheaf on the site of smooth manifolds | This would be the "comparison lemma" from SGA 4-1, Éxposé III, Thm. 4.1: if $C$ is a full subcategory of a site $D$, equipped with the induced topology, and if every object of $D$ is covered by objects of $C$, then the restriction functor $Shv(D)\to Shv(C)$ is an equivalence of categories.
More generally, for any fin... | 14 | https://mathoverflow.net/users/20233 | 137855 | 75,609 |
https://mathoverflow.net/questions/137845 | 4 | I am interested in the following question:
Given a projective plane of order $n=2^a$, is its incidence matrix must contain the incidence matrix of the Fano plane? If not, is it true that for any $n$ of the form $n=2^a$ there exists a projective plane of this order whose incidence matrix contains the incidence matrix ... | https://mathoverflow.net/users/37778 | Projective planes that contain the Fano plane | A projective plane over a field, i.e. one in which Pappos' theorem holds, does contain the Fano configuration if and only if the characteristic of the field is two. These are exactly the cases where the order is a power of two. So the latter statement is certainly true: for every such $n$ there exists the plane over $G... | 3 | https://mathoverflow.net/users/25563 | 137859 | 75,611 |
https://mathoverflow.net/questions/137850 | 7 | For each positive integer n let E(n) denote n-dimensional Euclidean space and let the term "n-dimensional convex body" mean a compact convex subset of E(n) whose interior (with respect to E(n)) is non-empty. It is easy to see that all orthogonal projections (onto a plane) of a 3-dimensional convex body are 2-dimensiona... | https://mathoverflow.net/users/4423 | A question about a question about 3-dimensional convex bodies | I think this is true. A proof would go like this:
First prove that the body must be an ellipsoid.
Without loss of generality you may assume that your body $B$ contains the origin as an interior point. Consider now the euclidean unit sphere $x^2 + y^2 + z^2 = 1$ and on each tangent plane consider the orthogonal proj... | 8 | https://mathoverflow.net/users/21123 | 137862 | 75,613 |
https://mathoverflow.net/questions/137861 | 1 | I've been trying to understand some of the idiosyncrasies associated to algebraic groups over non-algebraically closed fields $K$ of characteristic $p > 0$.
Let $G$ is a connected almost absolutely simple adjoint algebraic group over $K$ and $\widetilde{G}$ is the simply connected cover of $G$ also defined over $K$ w... | https://mathoverflow.net/users/31495 | Rank of the character group of a maximal $K$-torus for semisimple and adjoint algebraic groups | The map $\widetilde{T} \rightarrow T$ is an isogeny, and an isogeny between tori over any field $K$ induces a finite-index inclusion between the associated geometric character groups as ${\rm{Gal}}(K\_s/K)$-lattices, hence an isomorphism on associated rational vector space, so same dimension for spaces of Galois-invari... | 4 | https://mathoverflow.net/users/36938 | 137866 | 75,614 |
https://mathoverflow.net/questions/137868 | 4 | For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}\_m(\mathbb{Z}/n\mathbb{Z})$ such that the alternating group $A\_k$ is a composition factor of $G$ (i.e., writing $G$ as an iterated extension of f... | https://mathoverflow.net/users/1149 | Restricting the composition factors of subgroups of GL_m(Z/nZ) | **Latest Edit:** Below I put a sketch on how to replace the deep Larsen-Pink Theorem by easier and more familiar results.
**Edit:** There was a silly mistake in showing that we may assume that $n$ is a prime. Still, it is true, see Pete L. Clark's fix in his answer, which also enhances my rather sketchy answer. Here... | 5 | https://mathoverflow.net/users/18739 | 137875 | 75,617 |
https://mathoverflow.net/questions/137876 | 7 | The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 have significant consequences? For example, what would the statement that a number of such zeros is finite imply?
| https://mathoverflow.net/users/38448 | Consequences of a bound on possible counterexamples to Riemann hypothesis | Such bounds are generally called zero-density estimates (for the Riemann zeta function or more general $L$-functions), and they have significant consequences. Chapter 10 of Iwaniec-Kowalski's Analytic number theory is devoted to this topic.
A famous example of such a result and application is Huxley's theorem (Inven... | 12 | https://mathoverflow.net/users/11919 | 137877 | 75,618 |
https://mathoverflow.net/questions/137804 | 3 | Let $P$ be a polynomial; we ask about the existence of a non-constant analytic function $f : \Bbb{C}\longrightarrow \Bbb{C}$ such for all $z \in \Bbb{C}, f(z) = f(P(z))$. Clearly when $P$ is linear we can find such $f$. What happens when $P$ is not linear ? Any suggestion would be helpful.
| https://mathoverflow.net/users/nan | A functional equation concerning analytic functions | Here is a more conventional proof:-) Let $M(r)=\max\{|f(z)|:|z|=r\}$. Maximum principle
implies that this function is strictly increasing (unless $f$ is constant).
This gives a contradiction because $|P(z)|>|z|$ when $z$ is sufficiently large, and $P$
is of degree greater than $1$.
| 5 | https://mathoverflow.net/users/25510 | 137880 | 75,620 |
https://mathoverflow.net/questions/137816 | 5 | Let $p$ be a prime number and consider the sum $S(x)=\sum\_{n\le x}\left(\frac{n}{p}\right)\mu(n)$. For how small an $x$ in terms of $p$ is it known that $S(x)=o(x)$?
I am especially interested in unconditional results.
| https://mathoverflow.net/users/9304 | Correlation of a quadratic character with the Mobius function | In general, we can take $x>\exp\{c\_\epsilon p^\epsilon\}$ by the Prime Number Theorem for arithmetic progressions. More generally, one can use $L$-functions methods to relate $x$ to zero-free regions. In doing so it is hard to avoiding `losing logarithms'. The following elementary argument though, essentially due to G... | 11 | https://mathoverflow.net/users/4003 | 137882 | 75,622 |
https://mathoverflow.net/questions/137878 | 2 | Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(
Die Berechnung optischer und elektrostatischer Gitterpotentiale
P. P. Ewald
Annalen der Physik
Volume 369, pages 253–287, 1921
<http://onlinelibrary.wiley.... | https://mathoverflow.net/users/37526 | Ewald's generalized theta function | A step by step derivation of Ewald summation can be found in:
Williams, D. E. (1971). Accelerated convergence of crystal-lattice potential sums. Acta Crystallographica Section A, 27(5), 452–455. doi:10.1107/S0567739471000998
There's also an expanded version of the above paper:
Williams, D. E. (2006). Accelerated ... | 1 | https://mathoverflow.net/users/274 | 137888 | 75,626 |
https://mathoverflow.net/questions/137874 | 0 | Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal distributions $r(x), r(y)>0,$ for each $x,y$ respectively. Can we always find functions $f(x), g(y)$ such that
$\sum\_y p(x,y)f(x)g... | https://mathoverflow.net/users/7576 | Joint distribution with specified marginals | This is called the generalized matrix scaling problem and several other names. Both the theory and associated algorithmic problems have been studied. I suggest you start with [this paper](http://link.springer.com/article/10.1007/s10107-006-0021-4) and the papers it cites.
| 2 | https://mathoverflow.net/users/9025 | 137891 | 75,628 |
https://mathoverflow.net/questions/137774 | 18 | Math people:
I am looking for a proof of a conjecture I made. I need to give two definitions. For distinct real numbers $x\_1, x\_2, \ldots, x\_k$, define $\sigma(x\_1, x\_2, \ldots, x\_k) =1$ if $(x\_1, x\_2, \ldots, x\_k)$ is an even permutation of an increasing sequence, and $\sigma(x\_1, x\_2, \ldots, x\_k) =-1$ ... | https://mathoverflow.net/users/31107 | looking for proof or partial proof of determinant conjecture | I have a proof if all the $\mu$'s are larger than all of the $\gamma$'s. I've spent a while trying to figure out how modify this proof to work for other orderings, but I'm giving up. This argument is inspired by a [computation of the Caucy determinant by Doron Zielberger](http://tube.sfu-kras.ru/video/407?playlist=397)... | 8 | https://mathoverflow.net/users/297 | 137892 | 75,629 |
https://mathoverflow.net/questions/137887 | 10 | Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
---
*Update:* What can be said about degrees of these algebraic numbers for different $n$?
| https://mathoverflow.net/users/9550 | Do maximal polyhedra have algebraic volume? | Yes. Let $\mathbb{R}^{alg}$ be the field of real algebraic numbers. This is a [real closed field](http://en.wikipedia.org/wiki/Real_closed_field) which means that, for any statement of first order logic, using the symbols $0$, $1$, $+$, $\times$, $=$, $<$, that statement is true in $\mathbb{R}^{alg}$ if and only if it ... | 16 | https://mathoverflow.net/users/297 | 137893 | 75,630 |
https://mathoverflow.net/questions/137609 | 8 | I've read some pages on links between von neumann (VN) algebras and measurable spaces ([Spectra of $C^\*$ algebras](https://mathoverflow.net/questions/5036/spectra-of-c-algebras) and [Non-commutative geometry from von Neumann algebras?](https://mathoverflow.net/questions/3150/non-commutative-geometry-from-von-neumann-a... | https://mathoverflow.net/users/37661 | von neumann algebras and measurable spaces | I highly recommend Segal's original paper *Equivalences of Measure Spaces*
[American Journal of Mathematics
Vol. **73**, No. 2 (1951), pp. 275-313], where he introduced localizable spaces, since this was before the terminology took off.
In it he shows that an arbitrary measure space has maximal abelian (i.e. strongly c... | 8 | https://mathoverflow.net/users/10779 | 137900 | 75,633 |
https://mathoverflow.net/questions/136233 | 22 | The study of algebraic geometry usually begins with the choice of a base field $k$. In practice, this is usually one of the prime fields $\mathbb{Q}$ or $\mathbb{F}\_p$, or topological completions and algebraic extensions of these. One might call such fields **$0$-dimensional**. Then one could say that a field $K$ is *... | https://mathoverflow.net/users/33757 | To what extent can fields be classified? | I think you are lumping too many disparate kinds of fields together under the heading "zero-dimensional". As Jason says in his answer, there *are* some precise definitions of dimensions of fields (e.g. cohomological dimension but also other definitions of a field-arithmetic nature).
Another important comment is that ... | 27 | https://mathoverflow.net/users/1149 | 137905 | 75,635 |
https://mathoverflow.net/questions/137913 | 4 | Let $f: X \rightarrow Y $ be a Galois cover of with $X$ and $Y$ algebraic curves over $\mathbb{C}$. I want to compute the dimension of the subspace of $G$-invariants in $H^{0}(X,\omega^{\otimes2})$ (or by Serre duality $H^{1}(X,T\_{X})$). The point is that I always get this dimension to be zero, which I know is not tru... | https://mathoverflow.net/users/37808 | Dimension of the space of invariant quadratic differentials in Galois covers | Your short exact sequence is wrong. Unless the morphism $f$ is etale (which it is not when $Y$ equals $\mathbb{P}^1$), then the induced morphism $T\_X\to f^\*T\_Y$ is not surjective.
| 2 | https://mathoverflow.net/users/13265 | 137914 | 75,636 |
https://mathoverflow.net/questions/137757 | 8 | Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi\_2(X^{an})$ is finite.
What can we say about $X$?
I am mainly interested in the case where $\dim X = 2$, and $\Omega^1\_X$ satisfies some posit... | https://mathoverflow.net/users/37709 | Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$ | I combine user37314's answer and my comments; the claim is that any smooth projective complex algebraic surface with $\pi\_1$ abelian and $\pi\_2$ finite has a finite cover which must be an abelian surface; in particular, $X$ cannot be of general type.
By replacing $X$ by a finite cover, we may assume that $\pi\_1(X)... | 10 | https://mathoverflow.net/users/519 | 137915 | 75,637 |
https://mathoverflow.net/questions/137899 | 11 | In 'panoramic view of Riemmannian geometry' when introducing hyperkähler manifolds, Berger states, informally, that a hyperkähler manifold is a Riemmannian manifold which is Kähler for more than one different almost complex structures.
I was wondering whereas this was a theorem or just a 'catchphrase'. In other words... | https://mathoverflow.net/users/8887 | Minimum requirements for a Kähler manifold to be hyperkähler | Suppose that the metric on $M^n$ has irreducible holonomy, is simply connected (or, slightly more generally, that the restricted holonomy $H^0$ acts irreducibly), and that there exist two independent parallel complex structures. Since there is at least one parallel complex structure $I$, the holonomy group $H^0$ is a s... | 10 | https://mathoverflow.net/users/13972 | 137918 | 75,639 |
https://mathoverflow.net/questions/137911 | 5 | Let $n\geq 3$ be a natural number and Consider the following game:
Correspond an integer to each vertices of an equiangular polygon
(at least two of the numbers are unequal).
**(1) Replace the number of each vertices with the number obtained by the sum of
its neighborhood numbers minus its own number. (Edit: Do... | https://mathoverflow.net/users/nan | A game on equiangular polygons | For any $n$ there is always a (certainly non-constant) initial choice of integer labels $v=(v\_1,\dots,v\_n)$ of the vertices of the $n$-gone for which the set of numbers $A$ is unbounded.
The question is more subtle if you ask for which $n$ it is possible to find a non-constant integer choice of labels for which th... | 5 | https://mathoverflow.net/users/6101 | 137919 | 75,640 |
https://mathoverflow.net/questions/137770 | 8 | It is known that the cohomology ring of a Zoll manifold---a riemannian manifold all of whose geodesics are periodic with the same minimal period---must be the same as the cohomology ring of a compact rank one symmetric space (see Besse's book *Manifolds all of whose geodesics are closed* for references).
Is there a ... | https://mathoverflow.net/users/21123 | Easy proof of topological property of Zoll manifolds | As asked by @alvarezpaiva, I repost my remark as an answer.
After his addendum and answer showing that the fundamental group (if non-trivial) has one generator, given by a prime closed geodesic, you just have to observe that this geodesic is homotopic to itself *with reversed orientation*.
Hence the fundamental gro... | 8 | https://mathoverflow.net/users/6451 | 137925 | 75,642 |
https://mathoverflow.net/questions/137920 | 14 | Let $a,b,n$ be natural numbers such that $a!b! \mid n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must have $a+b \leq 2n$. But unfortunately I am looking for much smaller bound ! Any idea would be helpful.
| https://mathoverflow.net/users/nan | How $a+b$ can grow when $a!b! \mid n!$ | This is a small variation of Seva's argument yielding a slightly more explicit result.
Assume that $a+b>n$ with $a!b!$ dividing $n!$. Then
$$ \prod\_{n-a<m\leq b}m=\frac{b!}{(n-a)!} \ \ \text{divides}\ \ \frac{n!}{(n-a)!a!}= \binom{n}{a}. $$
Looking at the exponent of $2$ in the prime decomposition of the two sides,... | 11 | https://mathoverflow.net/users/11919 | 137928 | 75,645 |
https://mathoverflow.net/questions/137898 | 8 | Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, \mathbb{Z}).$$
Does there exist an "almost homolorphic line bundle" $L \rightarrow M $
such that its first Chern clas... | https://mathoverflow.net/users/4463 | Does there always exist a line bundle whose Chern class represents an integer symplectic form? | The answer is already 'no' in dimension $4$. The generic almost complex structure compatible with a symplectic structure in dimension $4$ does not admit any pseudoholomorphic functions (in your sense) other than the constants, so, for such data $(M,\omega,J)$, you are asking whether there is a line bundle $L$ that can ... | 10 | https://mathoverflow.net/users/13972 | 137934 | 75,648 |
https://mathoverflow.net/questions/137939 | 4 | Let $m(x) = x^n + a\_{n-1}x^{n-1} + \dots + a\_1 x+ a\_0$, $a\_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}[x] / {m(x)\mathbb{Q}[x]}$, so $K$ is an algebraic number field. Let $\mathcal{O}\_K$ be its ring of algebraic integers and $x\mathcal{O}\_K$ be the ideal in $\mathcal{O}\_... | https://mathoverflow.net/users/4760 | Prime ideals in the ring of algebraic integers | **No**. Let $p$ be a prime number with $p \equiv 3 \pmod{4}$, and let $m(x) = x^2+p^2$. Then $K = \mathbb{Q}[x]/(m) \cong \mathbb{Q}(i)$ and the element $x$ may be identified with $ip$. Since $i$ is a unit in $\mathcal{O}\_K$, we have $(ip) = (p)$ in $\mathcal{O}\_K$. Since $p \equiv 3 \pmod{4}$, $\mathcal{O}\_K/(ip) \... | 11 | https://mathoverflow.net/users/1149 | 137943 | 75,651 |
https://mathoverflow.net/questions/137725 | 7 | In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and $B(H)$ the algebra of bounded operators.
**Definition**: A von Neumann algebra is a \*-subalgebra $M \subset B(H)$ stable... | https://mathoverflow.net/users/34538 | Is the fundamental group of $II_{1}$ factors invariant under a relation? | Here is a counterexample. I don't see any easy way to augment the question to something more natural.
Let $Q$ be a $w$-rigid II$\_1$ factor with trivial fundamental group, e.g., $Q = L( \mathbb Z^2 \rtimes SL\_2(\mathbb Z) )$. Let $\mathcal S \subset \mathbb R\_+^\*$ be a non-trival subgroup. Set $M = \*\_{s \in \ma... | 11 | https://mathoverflow.net/users/6460 | 137948 | 75,652 |
https://mathoverflow.net/questions/137941 | 11 | For an $n$-element metric space $X=\{x\_1,\dots,x\_n\}$ with metric
$d$ we introduce an array containing $\frac{n(n-1)}2$ numbers
$d(x\_i,x\_j)$, $i<j$. We assume that all distances are at least
$1$. The number of *relevant scales* for the metric space $X$
is defined as the number of intervals $[2^{i-1},2^{i})$
$i=1,2,... | https://mathoverflow.net/users/37822 | The number of relevant scales for a finite metric space | First here is an $O(n\log n)$ upper bound. Make a complete graph whose vertex set is $X$ and the weights on its edges are the distances, $d(x\_i,x\_j)$. Consider a minimum weight spanning tree $T$ in this graph. Denote the weights/distances of the edges of the tree by $d\_1,\ldots, d\_{n-1}$. We claim that for every ed... | 11 | https://mathoverflow.net/users/955 | 137976 | 75,663 |
https://mathoverflow.net/questions/137975 | 1 | Consider points $a,b,c$ (not on a line) and $x\_1,...,x\_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for any convex function $f : \Bbb{R}^2 \longrightarrow \Bbb{R}$, $\frac{f(x\_1)+\cdots+f(x\_n)}{n} \leq \frac{f(a)+f(b)+f(c)... | https://mathoverflow.net/users/nan | A question concerning convex functions | The necessary and sufficient conditions are: $(1/n)\sum x\_j=(1/3)(a+b+c)$, and all $x\_j$ lie inside
the closed triangle $(a,b,c)$.
Proof of sufficiency. For every affine function, we have equality.
Let $f$ be convex. Then there exists a unique affine function $g$ which matches $f$
at $a$, $b$ and $c$. And we have $... | 6 | https://mathoverflow.net/users/25510 | 137980 | 75,664 |
https://mathoverflow.net/questions/137982 | 2 | Let $G$ be a reductive p-adic group, $\pi$ a complex smooth representation of $G$. Then it is known that if $\pi$ is irreducible, then it is admissible.
I need help to find a reference for this fact, and want to know if it is true for real reductive group. Thanks a lot.
| https://mathoverflow.net/users/1832 | reference help: irreducible implies admissible | 1) In these lecture notes:
<http://www.math.tau.ac.il/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93new-harv.lect.from-chic.pdf>
you have theorem 12, on page 37. He does it for GL\_n I think, but it should be similar for other groups...
2) For a real group, I think you should be more specific. There is... | 3 | https://mathoverflow.net/users/2095 | 137986 | 75,667 |
https://mathoverflow.net/questions/137977 | 3 | Let $F$ be an infinite field and $R$ a subring of $F$. suppose that $[F:R] < \infty$ (Index of $R$ in $F$ as a subgroup is finite). Does this force $R$ to be equal to $F$?
| https://mathoverflow.net/users/nan | Subgroups of finite index of fields | Suppose $[F : R] = n$ is finite. I first claim that the integral domain $R$ is a subfield of $F$. For if $0 \neq \theta \in R$, then we have inclusions of $R$-modules
$$R \subset R \cdot \theta^{-1} \subset R \cdot \theta^{-2} \subset \ldots$$
where in each case $R \cdot \theta^{-j-1}/R \cdot \theta^{-j} \cong R\... | 10 | https://mathoverflow.net/users/2926 | 137999 | 75,672 |
https://mathoverflow.net/questions/137432 | 7 | I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very best could be easily programmed on a Computer).
It came up to me because I am trying to construct metrics for Refinem... | https://mathoverflow.net/users/37580 | Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular | Some aspects of the standard metrization theorems are non-constructive, so there is probably no general solution for the problem you pose.
However, for your space with a basis consisting of closed regular ω-languages, there is a simpler way since this is a zero-dimensional space. A second-countable Hausdorff zero-dim... | 4 | https://mathoverflow.net/users/2000 | 138000 | 75,673 |
https://mathoverflow.net/questions/138015 | 2 | From the observation, that a bipartite graph doesn't contain odd cycles, it would seem natural to attempt to destroy all odd cycles in the most efficient way, by either removing edges or vertices of odd cycles, in order to find a maximal subset of the vertices that spans a bipartite sub graph.
**I would appreciate re... | https://mathoverflow.net/users/31310 | Reference Request for: Finding Large Bipartite Subgraphs via Destruction of Odd Cycles in Graphs | [This paper by Karabayashi and Reed](https://www.siam.org/proceedings/soda/2010/SODA10_031_kawarabayashik.pdf) seems to be closely related to what you are looking for.
| 5 | https://mathoverflow.net/users/11142 | 138018 | 75,680 |
https://mathoverflow.net/questions/138005 | 12 | I am not quite sure that my question below is appropriate for this site, probably it should be addressed to the physical commutity. But I hope that some (mathematical) physicists do attend MO. I have two questions on dimensional regularization used in the renormalization theory (they should be very basic, I am a mathem... | https://mathoverflow.net/users/16183 | Dimensional regularization in odd dimensions | There a number of papers by Alain Connes on Dimensional Regularization (Dim Reg) in the context of noncommutative field theory. Some of his papers cite
P. Breitenlohner and D. Maison, "Dimensional renormalization and the action principle,"
Comm. Math. Phys. Vol. 52, Number 1 (1977).
so I presume this might be a use... | 9 | https://mathoverflow.net/users/10475 | 138019 | 75,681 |
https://mathoverflow.net/questions/137990 | 2 | A word $w$ on the alphabet $A := \{0, 1\}$ is *factorable* if
\begin{equation}
w = u^k \mbox{ where } u \in A^\* \mbox{ and } k \geq 2.
\end{equation}
Let $L$ be the language of the set of factorable words on $A$ and $f(t)$ be its generating series, that is
\begin{equation}
f(t) := \sum\_{n \geq 0} |L \cap A^n| \; ... | https://mathoverflow.net/users/14024 | Generating function of factorable binary words | The ordinary generating function is a fairly horrible Lambert expansion (usually with a natural boundary). For more on this subject look in my old paper *[Walks on Free Groups and other Stories](http://arxiv.org/abs/1106.5947)*.
| 2 | https://mathoverflow.net/users/11142 | 138020 | 75,682 |
https://mathoverflow.net/questions/137825 | 4 | I am looking for practical error estimates for Newton-Cotes Quadrature rules.
Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective methods.
Despite an extensive (but perhaps misdrirected) search I found almost nothing dealing with error estimation when... | https://mathoverflow.net/users/24196 | Practical error-estimates for (adaptive) Newton-Cotes Quadrature | The principles behind embedded Runge-Kutta-Methods apply:
You compute the integral over a subinterval with two methods of different order. You use the difference between the two results as an estimate for the local quadrature error of the less accurate formula.
In order to meet a global tolerance, you sum up these ... | 2 | https://mathoverflow.net/users/37813 | 138024 | 75,685 |
https://mathoverflow.net/questions/138028 | 2 | Let $A = \lbrace (tr,1-t)\; | \; t \in [0,1], r \in \Bbb{Q}\rbrace$. Is it true that any continuous function from $A$ into $A$ has a fixed point?
| https://mathoverflow.net/users/37857 | A fixed point problem | Yes (I assume $A$ has the induced topology). The point $V:=(0,1)$, common endpoint of all segments $S\_r:=\{(rt,1-t): t\in[0,1]\}$, is either fixed by the continuous function $f:A\to A$, or it is mapped into some $S\_r\setminus\{V\}$. But then $f $ has a fixed point on $S\_r$.
| 3 | https://mathoverflow.net/users/6101 | 138029 | 75,687 |
https://mathoverflow.net/questions/138030 | 4 | Let $e\_1,e\_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V\_n = {\rm span}( e\_1, \dots, e\_n)$. Let $i\_n : V\_n \to V$ be the injection so that $i\_n^\dagger$ is the orthogonal projection. Finally, define $A\_n = i\_n^\dagger \c... | https://mathoverflow.net/users/16852 | Finite dimensional approximations of operators on Hilbert spaces | $A$ is in the norm closure of finite dimensional operators iff $A$ is a compact operator.
Then the spectrum and the eigenspaces of $A\_n$ converge to that of $A$.
| 5 | https://mathoverflow.net/users/26935 | 138032 | 75,688 |
https://mathoverflow.net/questions/138048 | 9 | Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)?
Bounds on Linnik's constant answer this for the first prime in each progression. Is there a known analogue for an $n$-th prime in a progression? And I found some references on an error term fo... | https://mathoverflow.net/users/38783 | How constructive is Dirichlet on primes in progressions? | Corollary 18.8 in Iwaniec-Kowalski's Analytic number theory shows the existence of an explicitly computable $L>3/2$ such that for $x>a^L$ and $(a,b)=1$, the number of primes less than $x$ and congruent to $b$ modulo $a$ is at least a constant times $\frac{x}{\varphi(a)\sqrt{a}\log x}$. So the $n$-th prime congruent to ... | 5 | https://mathoverflow.net/users/11919 | 138062 | 75,704 |
https://mathoverflow.net/questions/135968 | 35 | My interest in combinatorially motivated computational problems led me to search for simple problems that turn out to be computationally hard. In this pursuit, I came up with a problem which I hope is NP-complete. I searched the literature without finding an equivalent or close problem.
Originally motivated by this p... | https://mathoverflow.net/users/8784 | How hard is reconstructing a permutation from its differences sequence? | I tried to give a formal proof of the NP-completeness of the problem.
For the reduction details see my answer on [cstheory.stackexchange.com](https://cstheory.stackexchange.com/questions/18255/efficient-algorithm-for-existence-of-permutation-with-differences-sequence/18345#18345)
| 12 | https://mathoverflow.net/users/35419 | 138076 | 75,710 |
https://mathoverflow.net/questions/138070 | 6 | Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators.
**[Invariant subspace problem](http://en.wikipedia.org/wiki/Invariant_subspace_problem)**: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant subspace?
**Remark**: This problem is known for the Banac... | https://mathoverflow.net/users/34538 | Is there an operator algebraic reformulation of the invariant subspace problem? | To the weakest form of your question ("Is there a class of operator algebras relevant to the ISP?"), there is the "reductive algebra problem," which is not quite of the form you're asking for but is, I think, in the same spirit. An algebra of bounded operators on Hilbert space is called *reductive* if it is WOT-closed,... | 4 | https://mathoverflow.net/users/13360 | 138084 | 75,712 |
https://mathoverflow.net/questions/138061 | 14 | Let $\lbrace x\_i\rbrace\_{i=1}^\infty$ be a sequence of distinct numbers in $(0,1)$. For any $n$ after deleting $x\_1,...,x\_n$ from $[0,1]$ we get $n+1$ subintervals. Let $a\_n$ be the maximum length of these subintervals. Is there any sharp lower bound for $\limsup\_n na\_n$ ?
| https://mathoverflow.net/users/37857 | sequences of real numbers | Here is how to achieve $1/\ln 2$. It is a modification of Harald's construction. We take the sequence $1/2$, $1/4$, $3/4$, etc. and apply the transformation $x \to \ln (1+x)/\ln 2$.
So the sequence is $\ln(3/2)/\ln(2), \ln(5/4)/\ln (2),\ln(7/4)/\ln(2),\dots$
The general terms is $x\_n$ for $n=2^a+b$ with $b<2^a$ is... | 11 | https://mathoverflow.net/users/18060 | 138095 | 75,717 |
https://mathoverflow.net/questions/138086 | 3 | Let L be a Lie algebra with a one-dimensional maximal subalgebra. Is the following true?
Over a perfect field of characteristic 0 or p > 3, every such finite-dimensional Lie algebra is either 2-dimensional, or is a form of sl(2). General structure theory seems to indicate this.
| https://mathoverflow.net/users/37902 | Lie algebras with a one-dimensional maximal subalgebra | If $\mathfrak{a}$ is 1-dimensional and $\mathfrak{v}$ is an irreducible $\mathfrak{a}$-module then $\mathfrak{a}$ is maximal in $\mathfrak{a}\ltimes\mathfrak{v}$. Such $\mathfrak{v}$ of dimension $\ge 3$ indeed exists if the ground field has extensions of degree $\ge 3$, and then this Lie algebra has dimension $\ge 4$ ... | 2 | https://mathoverflow.net/users/14094 | 138103 | 75,720 |
https://mathoverflow.net/questions/138096 | 8 | I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of split groups in many sources (e.g. in Garrett's book), but there is hardly anything available in the non-split case. I kno... | https://mathoverflow.net/users/21684 | spherical buildings for non-split groups | Misha, Tits' Lecture Note "Buildings of spherical type and finite BN pairs"
gives a fairly explicit description of the buildings associated to the
classical groups (not just the split ones). I also wrote a survey called
"Buildings and classical groups" (arXiv:math/0307117)
which appeared in print some years ago.
The... | 12 | https://mathoverflow.net/users/37911 | 138105 | 75,721 |
https://mathoverflow.net/questions/138109 | 3 | I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. Of course, I could take the standard counterexample
$$
B=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}
$$
and simply change ... | https://mathoverflow.net/users/26039 | numerical range of a column-zero-sum matrix | Let $$A=\pmatrix{1&-1/2\cr-1&1/2\cr}$$ The column sums are zero and the eigenvalues are zero and $3/2$. $$A+A^t=\pmatrix{2&-3/2\cr-3/2&1\cr}$$ which has determinant less than zero, so is not positive semidefinite. [Wikipedia](http://en.wikipedia.org/wiki/Numerical_range) says the numerical range of $A$ is a closed subs... | 3 | https://mathoverflow.net/users/3684 | 138118 | 75,726 |
https://mathoverflow.net/questions/138115 | 1 | For an integer $n$ with decomposition $n=p\_1^{e\_1}...p\_k^{e\_k}$ denote
$\lambda(n)=\sum e\_i$. It follows from the prime number theorem that
$\#\{n\le x|\lambda(n)=a\pmod{2}\}\equiv x/2+O(x\exp(-c\log^{1/2}x))$ (for $a=0,1$).
What is known about $\#\{n\le x|\lambda(n)\equiv a\pmod{m}\}$ for other fixed $a,m$,
say $... | https://mathoverflow.net/users/9304 | Distribution of integers with number of prime factors lying in a given arithmetic progression | What you call $\lambda(n)$ is often denoted $\Omega(n)$ in the literature.
Hubert Delange, Sur la distribution des valeurs de certaines fonctions arithmétiques, Colloque sur la Théorie des Nombres, Bruxelles, 1955, pp. 147–161, Georges Thone, Liège; Masson and Cie, Paris, 1956, MR0085291 (19,17b) gets a very general... | 5 | https://mathoverflow.net/users/3684 | 138119 | 75,727 |
https://mathoverflow.net/questions/138114 | 5 | Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$.
I know from other results that $\mathbf C$ is cocomplete. Coproducts in $\mathbf C$ are easy to describe, but I am struggling to f... | https://mathoverflow.net/users/4814 | Coequalizers in an Eilenberg-Moore category | I'd like to add more information that is in line with Zhen's answer, but with slightly different hypotheses.
**Proposition:** If $C$ is cocomplete and a monad $T$ on $C$ preserves reflexive coequalizers, then the category of algebras $C^T$ is cocomplete.
Indeed, the forgetful functor $U: C^T \to C$ preserves and ... | 5 | https://mathoverflow.net/users/2926 | 138121 | 75,729 |
https://mathoverflow.net/questions/138098 | 5 | Is there any good reference for the Pontrjagin ring structure on
$$
H\_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H\_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)?
$$
I am familiar with Serre's theorem describing the mod 2 cohomology ring structure. I'm also aware that the action of the Dyer--Lashof operations on this ... | https://mathoverflow.net/users/8103 | Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces | By naturality and the external Cartan formula, the standard polynomial generators of $H^\*(K(\mathbf{Z}/2,k);\mathbf{Z}/2)$ given by iterated Steenrod operations on the fundamental class are primitive. Therefore the homology is a divided power algebra on the dual elements.
| 8 | https://mathoverflow.net/users/14447 | 138129 | 75,731 |
https://mathoverflow.net/questions/138031 | 1 | Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose:
\begin{align}
P1(A,B,C) &= P(A) P(B) P(C) \\
P2(A,B,C) &= P(A,B) P(C) \\
P3(A,B,C) &= P(A,B,C)
\end{align}
Is it true that $d(P1,P3) \geq d(P2,P3)$ where $d$ is the total variation distance?
In other words, is it provab... | https://mathoverflow.net/users/37799 | Distance between the product of marginal distributions and the joint distribution | I just find the following counter-example. Suppose $A,B,C$ are discrete variables. $A,B$ can each take two values while $C$ can take three values.
The joint distribution $P(A,B,C)$ is:
\begin{array}{cccc}
A & B & C & P(A,B,C) \\
1 & 1 & 1 & 0.1/3 \\
1 & 1 & 2 & 0.25/3 \\
1 & 1 & 3 & 0.25/3 \\
1 & 2 & 1 & 0.4/3 \\
1 ... | 1 | https://mathoverflow.net/users/37799 | 138135 | 75,735 |
https://mathoverflow.net/questions/138117 | 16 | The wikipedia page on Borel-Moore homology claims to give several definitions of it,
all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and which admit a sufficiently nice embedding into a smooth manifold.
My question concerns one of the definitions on tha... | https://mathoverflow.net/users/8032 | On the wikipedia entry for Borel-Moore homology | In [Chriss & Ginzburg](http://books.google.com/books?id=lwS59rR78eIC&dq=chriss+and+ginzburg&hl=en&sa=X&ei=mkH3UbavFZal4AP1m4DgAw&ved=0CDEQ6AEwAA), it's asserted that the Borel-Moore homology is given by $H\_\*(\bar X, \bar X\setminus X)$ if $\bar X$ is the one-point compactification **or** any compactification such tha... | 8 | https://mathoverflow.net/users/66 | 138138 | 75,736 |
https://mathoverflow.net/questions/138143 | 2 | Any given representation (some matrices of some algebra) will be reducible if there exists a singular, but nonzero matrix S that commutes with all elements of the representation; conversely, if no such S exists, then the representation is irreducible. This is Schur's Lemma.
**Question 1:** Does there exists some anal... | https://mathoverflow.net/users/37940 | Criterion for (non)decomposability of a representation? | I will assume our algebra to have an identity.
**Question 1.** How about: a representation is decomposable if and only if there exist two non-zero idempotent matrices $A\_1$ and $A\_2$ such that
* both commute with all elements of the representation,
* $A\_1A\_2=A\_2A\_1=0$ and
* $A\_1+A\_2=I$, the identity.
If a... | 5 | https://mathoverflow.net/users/35416 | 138145 | 75,739 |
https://mathoverflow.net/questions/138150 | 9 | Let $X$ be a projective variety over a field $k$, and $\dim X \geq 2$. By a curve $C$ on $X$, I mean a proper, reduced subscheme of $X$ of dimension $1$.
(1) If $C$ is an irreducible curve on $X$, then is $C$ numerically equivalent to some $\sum n\_i C\_i$ with $n\_i > 0$, and $C\_i$ smooth curves?
(2) If $D$ is a ... | https://mathoverflow.net/users/29730 | Curves on varieties, and a criterion for nef divisor | The answer to question $(1)$ is **no**. In fact, take an irreducible, nodal cubic curve $A \subset \mathbb{P}^2$ and take $10$ points $p\_1, \ldots, p\_{10}$ on it, different from the node. Let $X$ be the blow-up of $\mathbb{P}^2$ at the points $p\_i$ and $C$ the strict transform of $A$ in $X$. Then $C$ is an irreducib... | 13 | https://mathoverflow.net/users/7460 | 138155 | 75,741 |
https://mathoverflow.net/questions/137950 | 13 | Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^\*$. If $\dim(T)=\frac{1}{2}\dim(X)$, then $X$ is classified up to $T$-equivariant symplectomorphism by its moment polytope $\mu(X)$. If we relax t... | https://mathoverflow.net/users/25358 | Information from Moment Polytopes | Here's two natural things to ask about any compact group action on a compact manifold: (1) what are the (finitely many) conjugacy classes of stabilizer groups? Assume our group is a torus, so we can omit "conjugacy classes of". (2) For each such stabilizer, what are the (finitely many) components of its fixed point set... | 8 | https://mathoverflow.net/users/391 | 138159 | 75,742 |
https://mathoverflow.net/questions/138165 | 1 | I'm reading the paper
Lê, Hôngvân(D-MPI-NS); Wang, Guofang(PRC-ASBJ-MSY)
A characterization of minimal Legendrian submanifolds in $S^{2n+1}$. Compositio Math. 129 (2001), no. 1, 87–93.
Let $x: L^n \rightarrow S^{2n+2}$ be a minimal embedded submanifold and for fixed $a \in \mathbb R^{2n+2}$ consider the function o... | https://mathoverflow.net/users/19545 | Minimal Legendrian submanifolds and laplacian of particular functions | I think this is just for the sphere. Look at Theorem 3 in [this paper](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jmsj/1260480465) by Takahashi.
| 1 | https://mathoverflow.net/users/21123 | 138166 | 75,744 |
https://mathoverflow.net/questions/138081 | 24 | Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p$ pluses and $q$ minuses on the diagonal and with a nonzero skew symmetric $n$-form.
Now consider the corresponding spi... | https://mathoverflow.net/users/4149 | Spin group as an automorphism group | To expand on my comment to the question, we have the following algebraic construction (I think originally due to Robert Brown, 'A characterization of spin representations'):
Let $V$ be a quadratic space over a field $k$ of characteristic $\neq 2$. Attached to this is the Clifford algebra $C=C(V)$: it is equipped wit... | 25 | https://mathoverflow.net/users/7868 | 138179 | 75,753 |
https://mathoverflow.net/questions/138180 | 1 | Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:
$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class C8, and the other is $Sp(2,2^f)\wr2$ in C2. These two classes of subgroups are swapped by the graph automorphism of $G$.
... | https://mathoverflow.net/users/26700 | The action of graph automorphism of finite symplectic group on maximal subgroups | Yes it is right (for $f>1$), and it is (14.1)(2) of Aschbacher's paper
M. Aschbacher. On the maximal subgroups of the finite classical groups. Invent.
Math. 76 (1984), 469–514.
| 5 | https://mathoverflow.net/users/35840 | 138183 | 75,756 |
https://mathoverflow.net/questions/138168 | 5 | Keenan Kidwell's answer to [Place stabilizers for the absolute Galois Group](https://mathoverflow.net/questions/107052/place-stabilizers-for-the-absolute-galois-group/107057#107057) mentions that "choosing a complex conjugation" in $G\_{\mathbf{Q}}$ means choosing an embedding $\overline{\mathbf{Q}}\rightarrow\mathbf{C... | https://mathoverflow.net/users/38783 | Embeddings of $\overline{\mathbf{Q}}$ into $\mathbf{C}$ | $\def\QQ{\mathbb{Q}}\def\RR{\mathbb{R}}$A community wiki answer to record the proof sketched above. I had misread the question earlier; the comments of S.Carnahan and user36938 are correct.
Let $\sigma$ be an element of order $2$ in $Gal(\bar{\QQ}/\QQ)$ and let $R$ be the fixed field of $\sigma$. By the [Artin-Schrie... | 10 | https://mathoverflow.net/users/297 | 138197 | 75,764 |
https://mathoverflow.net/questions/138209 | 4 | I have a question about an equality involving products of central binomial coefficients. If $x\_1,...,x\_n$ and $y\_1,...,y\_n$ are positive integers, with $\sum\_i x\_i = \sum\_i y\_i$ and
$$ \binom{2x\_1}{x\_1} \cdots \binom{2x\_n}{x\_n} = \binom{2y\_1}{y\_1}\cdots \binom{2y\_n}{y\_n}\,, $$
what are the restrictions... | https://mathoverflow.net/users/37982 | Product of central binomial coefficients | Yes, there are nontrivial solutions. The first I found, with $n=5$, has
$$
\lbrace x\_i \rbrace = \lbrace 2, 5, 8, 13, 19 \rbrace,
\phantom{\infty}
\lbrace y\_i \rbrace = \lbrace 3, 4, 6, 14, 20 \rbrace,
$$
with $\sum\_i x\_i = \sum\_i y\_i = 47$ and
$$
\prod\_{i=1}^5 {2x\_i \choose x\_i} = \prod\_{i=1}^5 {2y\_i \choos... | 14 | https://mathoverflow.net/users/14830 | 138220 | 75,775 |
https://mathoverflow.net/questions/138222 | 3 | This is probably a trivial question.
While reading the paper
R. Elkik, Singularites rationnelles et deformations, *Invent. Math.* 47 Ž1978., 139147.
I came across the following short exact sequence. Consider a flat morphism $f:X\rightarrow S=Spec(R)$ of k-schemes of finite type, $X$ normal + CM, and pick a regul... | https://mathoverflow.net/users/18013 | Canonical sheaf of the fiber of a flat morphism | Sure.
Note first that we have a short exact sequence
$$
0 \to O\_X \xrightarrow{t} O\_X \to O\_{X\_t} \to 0
$$
If $D$ is the divisor corresponding to $t = 0$ on $X$, you can also view this as
$$0 \to O\_X(-D) \to O\_X \to O\_D \to 0.$$
Anyways, now apply the functor $\mathcal{H}om\_{O\_X}(\bullet, \omega\_X)$. Y... | 2 | https://mathoverflow.net/users/3521 | 138224 | 75,777 |
https://mathoverflow.net/questions/138218 | 35 |
>
> I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects.
>
>
>
I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm studying about Stanley-Reisner rings and I think I need to have a better general understanding about why I n... | https://mathoverflow.net/users/37996 | Why Cohen-Macaulay rings have become important in commutative algebra? | I think there are many reasons. Here are a few.
Practical reasons
=================
Cohen-Macaulay rings are just plain easier to work with.
### Computations in local cohomology
For example, any number of computations in local cohomology modules become much easier in the Cohen-Macaulay case (see for example Br... | 32 | https://mathoverflow.net/users/3521 | 138225 | 75,778 |
https://mathoverflow.net/questions/134272 | 4 | Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on some domain $D(x)$ around x. Moreover let $(e^k,g^k)$ converge to another distribution $(e,g)$
i). The ball-box theore... | https://mathoverflow.net/users/34518 | Ball-Box Theorem and Sequence of Distributions | I do not understand exactly what you mean by $\exp\_x$, but I guess it is something like the diffeomorphism proposed by Sergei Ivanov. I want to stress out that the ball-box theorem holds only if the coordinates are privileged w.r.t. $(e\_k,g\_k)$, so you should check that the $exp\_x$ you are using defines privileged ... | 2 | https://mathoverflow.net/users/890 | 138231 | 75,781 |
https://mathoverflow.net/questions/138232 | 9 | Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has a polar decomposition $G=KAK$, where $K$ arises from a Cartan decomposition $\mathfrak{g}=\mathfrak{k} + \mathfrak{p}$ ... | https://mathoverflow.net/users/37843 | Easy argument for "connected simple real rank zero Lie groups are compact"? | (Real rank 0) $\ \Rightarrow\ (\mathfrak a = 0) \ \Rightarrow\ (\mathfrak p=0) \ \Rightarrow\ (\mathfrak g$ is a *compact Lie algebra*).
Therefore this comes down to (e.g. [Bourbaki, Lie Groups, Chap IX, §1, no 4](http://books.google.com/books?id=oCO0xOzNLhAC&pg=PA284)):
>
> **THEOREM (H. Weyl)** *Let G be a conn... | 10 | https://mathoverflow.net/users/19276 | 138235 | 75,783 |
https://mathoverflow.net/questions/138215 | 1 | Let me define undirected graphs - slightly deviating from common usage - not to be a symmetric binary relation on an *arbitrary finite* set, but on a *finite subset of* $\,\mathbb{N}$. Anyway I write $G = (V(G), E(G))$ where $V(G)\subset \mathbb{N}$ is the underlying set of vertices of the graph $G$ and $E(G)$ is its s... | https://mathoverflow.net/users/2672 | Endofunctors of graph categories | I have to think about this a bit.
Some first remarks: Why do you need to consider propper subsets? From a category theoretic view it is easier to work with the comma-category
$$\mathrm{FinGrph}/\mathsf{blob}(\mathbb N)$$
a.k.a. finite graphs together with a graph morphism $s:G\to \mathsf{blob}(\mathbb N):=(\mathbb N,... | 2 | https://mathoverflow.net/users/1261 | 138236 | 75,784 |
https://mathoverflow.net/questions/138238 | 2 | Is it true that $$A\_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$
Here $A\_n$ is the Apery number:
$$A\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$
What is known about congruence properties modulo 3 for another set of Apery numbers $$B\_n=\sum\limits\_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}\;?$$
| https://mathoverflow.net/users/32389 | Congruence for the Apery Numbers | Yes, the first congruence was conjectured by Chowla and Cowles in $1980$ and proved by Y. Mimura in $1983$, see <http://www.sciencedirect.com/science/article/pii/0022314X83900380>. For $B\_n$ the sequence begins $(1,3,19,147,1251,11253,104959...)$, which is $(1,0,1,0,0,0,1...)$ modulo $3$.
It seems we have $B\_{2^k}\e... | 4 | https://mathoverflow.net/users/32332 | 138240 | 75,785 |
https://mathoverflow.net/questions/138241 | 5 | This concerns an assertion of Sy Friedman in [1], Lemma 2, which claims that, under certain conditions, if $\beta$ is not 0 and not $\Sigma\_1$-stable in $\alpha$, i.e. $L\_\beta\prec\_1 L\_\alpha$, then $\beta$ is in the $\Sigma\_1$ Skolem-hull of some $\beta'+1$ in $L\_\alpha$, for $\beta'<\beta$. This is used to sho... | https://mathoverflow.net/users/18628 | An ordinal not $\Sigma_1$ stable in alpha must be in the hull of smaller parameters in $L_\alpha$ | Since $\beta$ is not $\Sigma\_1$-stable in $\alpha$, there is some new $\Sigma\_1$ fact $\exists\xi\varphi(\xi,\vec a)$ true in $L\_\alpha$ but not in $L\_\beta$, for some $\vec a\in L\_\beta$. It follows that the witness $\xi$, which we may assume without loss is an ordinal, is at least as large as $\beta$. Let $\beta... | 5 | https://mathoverflow.net/users/1946 | 138248 | 75,787 |
https://mathoverflow.net/questions/134431 | 1 | Each diagonal uniform space $(X,\mathcal D)$ can be derived from the covering uniform space $(X,\Sigma\_{\mathcal D})$ and each covering uniform space $(X,\Sigma)$ can be derived from the diagonal uniform space $(X,\mathcal D\_{\Sigma})$.
The precompact reflection of the covering uniform space $(X,\Sigma)$ is denoted... | https://mathoverflow.net/users/nan | Precompact reflection in diagonal uniform spaces | You may want to look at G. Preuss, [*Foundations of topology: an approach to convenient topology*](http://books.google.ca/books?id=YY14k_ntfq0C) (Springer, 2000).
In $\S$4.3.2.19, he proves your assertion that $(X,p\Sigma)$ is in fact a precompact uniform space. Moreover, his $\S$4.3.2.15 makes it clear that in term... | 0 | https://mathoverflow.net/users/2622 | 138269 | 75,795 |
https://mathoverflow.net/questions/138213 | 15 | (I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.)
Consider the Temperley-Lieb algebra on $n$ strands, $TL\_n$, viewed as an algebra over the base ring $\mathbb{Z}[\delta]$. The closed loop evaluates to $-... | https://mathoverflow.net/users/2648 | When are Jones-Wenzl projectors defined? | **tl;dr:** *The JW projector $JW\_n$ exists if and only if the q-binomial coefficient $\binom{n}{m}\_q$ (which is actually a polynomial in $\delta$!) is non-zero in your field for all $1< m<n$.*
Your question is, in essence, one about the decomposition of the tensor product $V^{\otimes n}$ over $U(\mathfrak{sl}\_2)$ ... | 12 | https://mathoverflow.net/users/66 | 138270 | 75,796 |
https://mathoverflow.net/questions/138257 | 8 | I'm interested in a handy way, if it exists, to generate all rational solutions of $x^2 + y^2 = z (z^2 - 1)$.
Clearly there are rational solutions for $z = a/b$ (where $a, b$ are coprime integers with $0 < b < a$) iff $a, b, a - b, a + b$ are each expressible as a sum of two squares (of integers of course), or equiva... | https://mathoverflow.net/users/10454 | Rational solutions of $x^2 + y^2 = z (z^2 - 1)$ | I doubt there is a simple complete parametrization. Here are a number of comments:
**There is a two-dimensional family of solutions** Start with two solutions that do not have $x^2+y^2=0$. For concreteness sake, I'll choose $$\left( \frac{3}{8} \right)^2+ \left( \frac{6}{8} \right)^2 = \left( \frac{5}{4} \right)^3 - ... | 7 | https://mathoverflow.net/users/297 | 138286 | 75,804 |
https://mathoverflow.net/questions/138247 | 21 | Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the [Gelfond–Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem).
We know that ${\sqrt2}^{\sqrt2}$ is a transcendental number by the Gel'fond-Schneider's theorem. I've tried to prove that ${\sqrt2}^{\sqrt2}$ is an irr... | https://mathoverflow.net/users/34490 | Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem | You do not need to use Gelfond-Schneider theorem, you can just repeat one of the well known easy proofs of that theorem. For example, a much stronger theorem is proved in an Appendix to Lang's "Algebra", the proof is only 4 pages long. The proof uses only elementary linear algebra, some calculus and first notions of Ga... | 26 | https://mathoverflow.net/users/nan | 138291 | 75,806 |
https://mathoverflow.net/questions/138285 | 7 | I am trying to get a concrete handle on the isomorphism $H^4(K(\pi\_2,2),U(1)) \simeq \{$quadratic forms $\pi\_2 \to U(1) \}$. This is explained in Eilenberg and Maclane's <http://www.jstor.org/stable/1969702> and its companion but I am having a hard time getting just what this 4-cocycle should assign to a 4-simplex in... | https://mathoverflow.net/users/36591 | On quadratic forms, Pontryagin Squares, and H^4(K(pi,2),U(1)) | To a quadratic form $q: \pi\_2 \to U(1)$, we get a corresponding Pontryagin square operation $H^2(-; \pi\_2) \to H^4(-; U(1))$, and such cohomology operations are given by elements of $H^4(K(\pi\_2, 2); U(1))$. Unfortunately, I don't know if it's possible to get an explicit cochain-level description of the Pontryagin s... | 6 | https://mathoverflow.net/users/396 | 138294 | 75,808 |
https://mathoverflow.net/questions/138282 | 2 | What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be "none". The generalization by Ramanujan only seems to cover non-negative coefficients.
| https://mathoverflow.net/users/20757 | Four-Square Theorem for Negative Coefficient | With indefinite forms, it is possible for ternary forms to be universal. Indeed, all are known. References are given in *Modern Elementary Theory of Numbers* by Leonard Eugene Dickson, (1939). With any integer $M$ and any **odd** $N,$ they are equivalent to (by an invertible linear change of variables) one of four:
$... | 2 | https://mathoverflow.net/users/3324 | 138298 | 75,810 |
https://mathoverflow.net/questions/138293 | 1 | Consider $T$ to be a standard Young tableau of shape $\lambda$ (in English notation). The descent set of $T$, $Des(T)$, is defined to the set of all positive integers $i$ such that $i+1$ lies strictly south (and weakly west) of $i$ in $T$.
A flip on $T$ is defined to be a move where we exchange the positions of $i$ a... | https://mathoverflow.net/users/nan | Flips on standard Young tableaux and descent sets | Here is a counterexample copied out of a Sage session:
```
sage: T = StandardTableau([[1,2,3,7,9],[4,5,8],[6,11],[10]]); T.pp()
1 2 3 7 9
4 5 8
6 11
10
sage: T.standard_descents()
[3, 5, 7, 9]
sage: T = StandardTableau(T.bender_knuth_involution(9)); T.pp()
... | 3 | https://mathoverflow.net/users/2530 | 138319 | 75,817 |
https://mathoverflow.net/questions/138320 | 2 | Let $F$ be a field $\{\otimes,\oplus,R\_i\}$ with the one-element $R\_1$ (the null-
element $R\_0$ isn't very relevant). Let $T\_{ijk}$ be a "triangle condition"
(symmetric in the indices and integer-valued) which is used to define
${R\_i}\otimes{R\_j}=\oplus\_k T\_{ijk}\*R\_k$. Clearly $T\_{0jk}=0, T\_{1jk}=\delta^j... | https://mathoverflow.net/users/11504 | Infinite fusion categories (sort of) | For BCD what you want is called the BMW category.
For the Vogel plane, E7, exceptional, etc. everything is totally open.
| 1 | https://mathoverflow.net/users/22 | 138323 | 75,819 |
https://mathoverflow.net/questions/138321 | 25 | Let $p(n)$ denote the number of partitions of a positive integer $n$. It seems to me that we have for all $n>25$
$$
p(n)^2>p(n-1)p(n+1).
$$
In other words, the sequence $(p(n))\_{n\in \mathbb{N}}$ is log-concave, or satisfies $PF\_2$, with
$$
\det \begin{pmatrix} p(n) & p(n+1) \cr p(n-1) & p(n) \end{pmatrix}>0
$$
for $... | https://mathoverflow.net/users/32332 | Is the sequence of partition numbers log-concave? | The first two terms of the Hardy-Ramanujan formula give
$$p(n) = \frac{1}{4 \sqrt{3} n} \exp(\pi \sqrt{2n/3}) + O \left(\exp(\pi \sqrt{n/6} ) \right)$$
so
$$\log p(n) = \pi \sqrt{2/3} \sqrt{n} - \log n - \log (4 \sqrt{3}) + O(\exp(-\pi \sqrt{n/6} ) ).$$
So
$$\log p(n+2) - 2 \log p(n+1) + \log p(n) = $$
$$ \pi \sqrt{2/3... | 22 | https://mathoverflow.net/users/297 | 138326 | 75,820 |
https://mathoverflow.net/questions/138325 | 3 | Let $S\_n$ be the permutation group on $n$ elements.
Denote by $K(n)$ the largest $k$ s.t. $S\_n$ has a $k$-transitive subgroup (w.r.t. its action on the $n$-element set on which $S\_n$ acts) different from
$S\_n,A\_n$.
I heard that it has been proved that $K(n)\le 7$ for all $n$ but the proof uses
the classification... | https://mathoverflow.net/users/9304 | Maximal $k$-transitivity for a proper subgroup of $S_n$ | Actually $K(n)\le5$, with equality only for $n=12$ and $n=24$ and subgroup the Mathieu group $M\_n$. If $G\le S\_n$ is at least $2$-transitive, then an old theorem of Burnside shows that either $G$ has a regular elementary abelian normal subgroup, or $G$ has a simple non-abelian normal subgroup.
In the former case, i... | 8 | https://mathoverflow.net/users/18739 | 138328 | 75,821 |
https://mathoverflow.net/questions/137903 | 2 | Hanna Neumann in
[American Journal of Mathematics, 1948,
<http://www.jstor.org/discover/10.2307/2372201?uid=2&uid=4&sid=21102497379451> ]
introduced a notion of generalized free product of groups $G\_i$ with amalgamated subgroups $H\_{ij}< G\_i$. Did anybody consider this construction for semigroups? Thank you... | https://mathoverflow.net/users/18814 | Generalized free product of semigroups with amalgamated subsemigroups | OK, here is my comment expanded to an answer. I looked at Hanna Neumann's definition again. What she defined is *not* a special case graph of groups, but of what is now called (again, in a special case) the "fundamental group of a complex of groups". Indeed, the modern definition of the latter is category-theoretic one... | 2 | https://mathoverflow.net/users/21684 | 138331 | 75,822 |
https://mathoverflow.net/questions/138223 | 6 | Let $1<\alpha<\beta<3/2$. Set
$$
S(n)= \sum\_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.
$$
One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, how does $S(n)$ behave asymptotically, e.g., if it is asymptotically a power function? If yes, what is the exponent?
Remarks:... | https://mathoverflow.net/users/37987 | Certain asymptotics involving double infinite sum | I can show without too much work that there exist absolute constants $c, C > 0$ so that $$c \leq \frac{S(n)}{n^{1-2\beta + \frac{\beta}{\alpha}}} \leq C.$$ This at least shows what the exponent must be if there is an asymptotic formula for $S(n)$. When $\alpha = \beta$ this gives $S(n) \asymp n^{2-2\alpha}$, which unco... | 4 | https://mathoverflow.net/users/2627 | 138335 | 75,825 |
https://mathoverflow.net/questions/138329 | 2 | Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism over $\mathbf{Z}$) then one gets a holomorphic differential (on the Neron model) which is well defined up to $\pm 1$. Th... | https://mathoverflow.net/users/11765 | What is the exact meaning of the real period in the $p$-adic formulation of BSD? | Mazur-Tate-Teitelbaum have written their p-adic BSD paper for a modular form of even weight $k\geq 2$ so unless we are dealing with a elliptic curve, we might want to avoid choosing a period. They view modular symbols are taking values in $L = L\_f\otimes \mathbb{Q}$, which is a $2$-dimensional $\mathbb{Q}$-vector spac... | 5 | https://mathoverflow.net/users/5015 | 138344 | 75,830 |
https://mathoverflow.net/questions/138348 | 42 | I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states
Suppose that for each $N \in \mathbb{N}\_{>0}$ the function $\displaystyle \sum\_{n=1}^N n^{-s}$ has no zeroes for $\mathfrak{R}(s) > 1$. Then... | https://mathoverflow.net/users/10898 | Famous vacuously true statements |
| 86 | https://mathoverflow.net/users/3324 | 138349 | 75,831 |
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