parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/153064 | 12 | A semisimple braided category with duals is called modular when a certain matrix $S$ is invertible. The components $S\_{AB}$ are indexed by (isomorphism classes of) simple objects of the category and one computes $S\_{AB}$ by colouring the Hopf link with (representants of) $A$ and $B$ and evaluates the resulting diagra... | https://mathoverflow.net/users/13767 | Is "being a modular category" a universal or categorical/algebraic property? | The Drinfeld center $Z(C)$ of a braided category "contains" $C$ and $\bar C$ (the category with opposite braiding) and therefore also $C\boxtimes \bar C$ and you can show that the following is equivalent
1) $C$ is modular
2) $Z(C)$ is equivalent with $C\boxtimes \bar C$.
In other words, for a braided category $C$... | 16 | https://mathoverflow.net/users/10718 | 153068 | 81,489 |
https://mathoverflow.net/questions/153067 | 4 | I'm interested in the following diophantine eqaution: $(5^n-1)/4=y^2$.
It turns out that this is a special case of the Nagell-Ljunggren equation, where $x=5$ and $q=2$
It has been shown that for x=5 this has no solutions but I'm looking for an elementary solution of this special case.
| https://mathoverflow.net/users/44801 | Special Case of famous Equation | This kind of problem usually requires a little algebraic number theory. Joe Silverman sketches one possible approach in the comments. Here is another. Let's rewrite as
$$
(2y)^2-5^n=-1.
$$
If $n$ is even then the left-hand side is a difference of two squares, which quickly gives a contradiction. So write $n=2m+1$. Then... | 6 | https://mathoverflow.net/users/4140 | 153076 | 81,495 |
https://mathoverflow.net/questions/152953 | 10 | Is there a Riemannian metric on $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ which is not conformally equivalent to a product metric?
More generally, assume that $M$ and $N$ are two manifolds. What obstructions are there for a metric $g$ on $M \times N$ to be conformally equivalent to a product metric for metrics
$... | https://mathoverflow.net/users/36688 | Obstructions for a metric to be conformally equivalent to a product metric | I'll give a partial answer to the OP's second question, which I take to be asking for the obstructions for a metric in dimensions greater than $2$ to be conformal to a product metric.
This is, first of all, a local question, since, even locally, most metrics in dimension $3$ or more are not conformally equivalent to... | 17 | https://mathoverflow.net/users/13972 | 153082 | 81,497 |
https://mathoverflow.net/questions/153078 | 8 | The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form of the Generalized Riemann Hypothesis (that is, the analogue of RH for Dirichlet L-functions). So, would the full EH co... | https://mathoverflow.net/users/13625 | Would Elliott-Halberstam conjecture follow from GRH? | The Elliott-Halberstam conjecture is not known to follow from GRH.
Even the weak version of EH (which is with $Q=x^{1/2+\epsilon}$ for any fixed $\epsilon>0$)
does not follow from GRH. On the other hand, it is known that the Elliott-Halberstam conjecture almost implies the twin primes conjecture, i.e., it implies that... | 11 | https://mathoverflow.net/users/32332 | 153089 | 81,500 |
https://mathoverflow.net/questions/146358 | 3 | Let $A$ be a unital associative algebra over a commutative noetherian ring $R$. Assume that $A$ is homologically smooth, which means that $A\in D\_{perf}(A\otimes\_R^L A^{op})$, which also means that $A$ is a compact object in $D(A\otimes\_R^L A^{op})$ (for simplicity, we assume that $A$ is cofibrant; in this case, the... | https://mathoverflow.net/users/41650 | Homological smoothness implies projectivity? | When you write $HH\_\*(A)$ I will assume that you mean the $R$-linear Hochschild homology of $A$. (If you'd meant the absolute version, there would be no hope -- e.g., consider $A = R = \mathbb{Z}[x]/x^2$.)
Then, under your hypothesis, it is easy to show that the Hochschild chain complex $HH^R\_\bullet(A)$ is *perfe... | 4 | https://mathoverflow.net/users/1921 | 153101 | 81,505 |
https://mathoverflow.net/questions/153092 | 26 | The *undecidability of the halting problem* states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ and interesting computer languages (say C++, Scheme) for which we have a constructive halting algorithms for program... | https://mathoverflow.net/users/40919 | Can We Decide Whether Small Computer Programs Halt? | As you noticed in your question, for any particular value of $n$, there is a constructive algorithm that solves the halting problem for instances of size at most $n$. Since for a particular value of $n$, there are only finitely many instances, one may simply hard-code the finitely many answers into the program itself. ... | 24 | https://mathoverflow.net/users/1946 | 153106 | 81,507 |
https://mathoverflow.net/questions/143806 | 7 | I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?
In my precise situation, I work with $X \subset \mathbb{A}^n$ Gorenstein of codimension 3 and I have a resolution:
$$0 \rightarrow \mathcal{O... | https://mathoverflow.net/users/37214 | Computing cotangent complex | Your X is not a local complete intersection in A^n. Then the cotangent complex is going to have cohomology in infinitely many degrees. (Follows from a conjecture by Quillen proved by Avramov IIRC.) Your best bet would be to use Quillen's spectral sequence relating Tor\_\*(O/I, O/I) to the cohomology of the cotangent co... | 5 | https://mathoverflow.net/users/44817 | 153121 | 81,515 |
https://mathoverflow.net/questions/153032 | 2 | Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D\_P$ is a principal ideal domain?
| https://mathoverflow.net/users/44785 | Domains $D$ for which for any prime $P$, $D_P$ is a PID | It's called an "almost Dedekind domain" in the literature on non-Noetherian commutative algebra. Every almost Dedekind domain is a Prüfer domain, or equivalently, locally a valuation domain. However, there exist Prüfer domains that are not almost Dedekind, e.g. any valuation domain that's not a PID. Both classes of dom... | 7 | https://mathoverflow.net/users/17218 | 153128 | 81,519 |
https://mathoverflow.net/questions/153111 | 1 | One particularity of the Generalized Riemann Hypothesis seems to deserve some clarification. In particular, what is included in the commonly accepted version of the conjecture?
GRH states that
$$\displaystyle \pi(x; a, d) = \frac{1}{\phi(d)} \int\_2^x \frac{dt}{\log t} + O\left(x^{1/2 + \epsilon}\right),$$
where... | https://mathoverflow.net/users/10898 | The implicit constant in GRH | You seem to confuse GRH with its consequences (or equivalent formulations). GRH is a statement about the zeros of Dirichlet $L$-functions, and there is no implicit constant in the statement.
At any rate, your displayed formula with any implicit constant depending on $a$ and $d$ implies GRH for Dirichlet $L$-functions... | 10 | https://mathoverflow.net/users/11919 | 153129 | 81,520 |
https://mathoverflow.net/questions/153077 | 3 | I usually find it difficult to check irreducibility of polynomials in $K[[X,Y]]$ ($K$ algebraically closed). Does anyone know about generic methods that can be used ? And especially of ones that can be applied to polynomials like $XY-(X+Y)(X^2+Y^2)$ (I found it reducible by using Hensel's lemma after a suitable change ... | https://mathoverflow.net/users/3333 | Generic methods to check irreducibility of polynomials in $K[[X,Y]]$ | I have a paper giving an irreducibility test (and factoring "algorithm", in some sense of the word) for formal power series over a PID, which should eventually appear in Trans. of the AMS. In particular it applies to $K[[X]][[Y]]$.
See Theorem 1 of <http://arxiv.org/abs/1107.4860>
In essence: Write any polynomial $... | 2 | https://mathoverflow.net/users/17218 | 153130 | 81,521 |
https://mathoverflow.net/questions/153107 | 1 | Let $M$ be a (fine) moduli space which parametrizes certain varieties (The moduli space in my mind is the moduli space of abelian surfaces with certain polarization -- but I don't know what is the moduli problem of this moduli space and whether it is fine or not). Then is there a variety $V$ over $M$ such that the fibr... | https://mathoverflow.net/users/29730 | The variety associated to moduli space | You should have a look at Chapter 8 of Lange and Birkenake *Complex Abelian Varieties*. I will refer to this book in what follows.
Fix a type $D= \textrm{diag}(d\_1, \ldots, d\_g)$. Then the moduli space of polarized complex abelian varieties of dimension $g$ and type $D$ with symplectic basis is precisely the Siegel... | 2 | https://mathoverflow.net/users/7460 | 153133 | 81,524 |
https://mathoverflow.net/questions/153146 | 2 | Given two positive integers $a,b$, and an odd prime $p$, I want to know whether the number of solutions to the following equation is finite:
$X^2=a+bp^{Y}$
where $X,Y$ are variables and are integers.
I checked with google, and in the case $b=1$ this seems to follow from a result of A Baker on logrithmetic forms (... | https://mathoverflow.net/users/39388 | A problem on the finiteness of solutions to a Diophantine equations | Yes, there are only finitely many solutions. Even more is true. For fixed nonzero $a$ and $b$, the equation
$$ x^2 = a + bz^y $$
has only finitely many solutions in integers $(x,y,z)$ with $y\ge3$. See the article on the Ramanujan-Nagell equation (and its generalizations) <https://en.wikipedia.org/wiki/Ramanujan-Nagell... | 4 | https://mathoverflow.net/users/11926 | 153147 | 81,528 |
https://mathoverflow.net/questions/153127 | 8 | Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$.
Let $G$ be an algebraic group over $k$. The affine Grassmannian $Gr\_G$ is the functor that associates to a $k$-algebra $A$ the set of isomorphism classes of pairs $(E, \va... | https://mathoverflow.net/users/11877 | Relations between affine Grassmannian and Grassmannian | I am not an expert, but the affine Grassmannian is intimately related to the representation theory of the Langlands dual group $G^{\vee}$. Assume that $G$ is complex semisimple and simply-connected (for simplicity, so to speak). Recall that the dominant weights of $G^{\vee}$ index the finite-dimensional irreducible com... | 3 | https://mathoverflow.net/users/25358 | 153148 | 81,529 |
https://mathoverflow.net/questions/152983 | 11 | Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the non-principal types).
Now let $\mathfrak{M}$ be a countable transitive model of set theory, and assume that $T$ is also (countable... | https://mathoverflow.net/users/38200 | Is "approximate categoricity" absolute? | This is really a comment, but I need a bit more space.
If $\phi$ is a sentence of ${\cal L}\_{\omega\_1,\omega}$ that is $\aleph\_0$-categorial, there is a complete first order theory $T$ in an expanded vocabulary such that the models of $\phi$ are exactly the reducts atomic models of $\phi$. The expansion is done in... | 16 | https://mathoverflow.net/users/5849 | 153168 | 81,536 |
https://mathoverflow.net/questions/153039 | 3 | Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the maximal component: $P\left(\mathrm{argmax}\_{i}X\_{i}=k\right)$. Is there some closed-form expression, or a recursive formul... | https://mathoverflow.net/users/44790 | Maximal component of a multivariate Gaussian distribution | Without loss of generality, the problem is equivalent to computing the probability
$P(X\_1 \geq \max(X\_2, \ldots, X\_n))$. We can transform the coordinates as $X\_2-X\_1, X\_3-X\_1, \ldots, X\_n-X\_1$ which is a fully general multivariate normal distribution of degree $n-1$. The problem is thus equivalent to finding t... | 2 | https://mathoverflow.net/users/8737 | 153172 | 81,539 |
https://mathoverflow.net/questions/153150 | 7 | Let $F$ be a field and $A$ be an $F$-central simple algebra of degree $n$. Let $0< k< n$ and let $SB\_k(A)$ denote the generalized Severi-Brauer variety: if $E/F$ is a field extension, $SB\_k(A)(E)$ consists of the right ideals of dimension $kn$ of $A\_E=A\otimes\_F E$.
If $A$ is split, i.e. $A\simeq M\_n(F)$, then $... | https://mathoverflow.net/users/6249 | When are generalized Severi-Brauer varieties Grassmannians? | Let's first phrase this in terms of non-abelian cohomology. You have a map $$H^1(F, PGL\_n)\to H^1(F, \operatorname{Aut}(Gr(k, n)))$$ given by sending a central simple algebra to the associated generalized Severi-Brauer. You'd like to know if this map has trivial kernel (it's just a map of pointed sets, so by this I ju... | 11 | https://mathoverflow.net/users/6950 | 153177 | 81,541 |
https://mathoverflow.net/questions/153178 | 8 | Assume that we are working in a set theory T, formalized in the language of ZFC, whose axioms---in addition to those of ZFC---include also the negation of "V=OD".
For any set X, let CARD(X) denote the cardinal number of X, let ORDEF(X) denote the set of ordinal definable elements of X and let P(X) denote the the set... | https://mathoverflow.net/users/4423 | A question about the Ordinal Definable elements of Power Sets | $
\newcommand\N{\mathbb{N}}
\newcommand\R{\mathbb{R}}
\newcommand\ZFC{\text{ZFC}}
\newcommand\HOD{\text{HOD}}
\newcommand\OD{\text{OD}}
$For your initial remark, yes, it is relatively consistent with ZFC that there
are only countably many ordinal-definable subsets of $\N$. To see
this, suppose that $V$ is any model of ... | 11 | https://mathoverflow.net/users/1946 | 153184 | 81,544 |
https://mathoverflow.net/questions/152834 | 12 | Consider a functional equation of the following form:
$$\sum\_{k=0}^n a\_k\,\underbrace{f(f(\cdots f}\_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a\_i\in \mathbb{R},\;\text{and}\;f^0=\text{id}\big)$$
A well-known strategy for producing continuous solutions to this is to take the real roots $\lambda\_1... | https://mathoverflow.net/users/44678 | Relating the roots of polynomials to the solution sets of certain functional equations | Let $P(z)=a\_nz^{n}+\ldots+a\_0$ denote the characteristic polynomial associated to this functional equation. We prove that if
$P$ does not have real roots, then the functional equation does not have a continuous solution. We may assume that $a\_0\neq 0$, else $0$ would be a
root of $P$.
Suppose there is a conti... | 14 | https://mathoverflow.net/users/38624 | 153205 | 81,555 |
https://mathoverflow.net/questions/153183 | 7 | Let $X$ be a complex variety. It is well-known that $X$ is smooth if and only if the sheaf of Kähler differentials $\Omega\_X^1$ is locally free (Hartshorne p. 177).
Question: What happens for forms of higher degree? I.e. define $\Omega\_X^p := \bigwedge^p \Omega\_X^1$ (no reflexive hull or something).
For what value... | https://mathoverflow.net/users/44860 | Smoothness and Kähler differentials | I think that if you do not take reflexive hulls, then all $p\leq \dim X$ should work. Since you said "variety" I assume you mean "reduced". In that case, $\mathscr F$ being locally free is equivalent to $\dim \mathscr F\_x\otimes \kappa(x)$ being constant. Since tensor operations commute, it seems to me that a coherent... | 5 | https://mathoverflow.net/users/10076 | 153224 | 81,564 |
https://mathoverflow.net/questions/149621 | 25 | It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn ([On discrete harmonic functions.](http://zbmath.org/?q=an:0033.06303) - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
>
> If a bounded function $f : \mathbb Z^2 \rightarrow \mathbb{R}$ satisfies the fol... | https://mathoverflow.net/users/5712 | The origin of Discrete `Liouville's theorem' | 
[Mathematica (Cluj), **6**, 146-151 (1932)](https://ilorentz.org/beenakker/MO/Capoulade.pdf)
>
> In a recent article from this journal, Mr. Bouligand has indicated the
> possibility of a modified proof of a theorem of Mr. Picard: *"a
> harmonic function that ... | 15 | https://mathoverflow.net/users/11260 | 153228 | 81,565 |
https://mathoverflow.net/questions/153235 | 4 | Let $\kappa$ be an uncountable cardinal of a c.t.m $M$ of $ZFC$.
Is there a generic extension of $M$ like $M[G]$ such that all uncountable cardinals of ground model collapse except $\kappa$?
| https://mathoverflow.net/users/nan | How can I collapse all cardinals of ground model except one of them? | Obviously we cannot do this and preserve ZFC in the extension, since $M[G]$ will have only one uncountable cardinal. But if you give up power set in the extension, then for many cardinals $\kappa$ you can do this with class forcing. For example, $\text{Coll}(\omega\_1,\lt\text{Ord})$ is the forcing having as conditions... | 5 | https://mathoverflow.net/users/1946 | 153236 | 81,567 |
https://mathoverflow.net/questions/98787 | 1 | $$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus U.$ Elements are of the form
$ (v; x,y). $ The norm on $L$ is given by
$$ (v; m,n)^2 = v^2 + 2 xy.$$ We infer the inner product
$$ (v\_1; x\... | https://mathoverflow.net/users/3324 | Lorentz quotient and orientation | This particular question (``consistently orienting the reductions") and all those arising from your above link are interesting.
However, the answer to this question seems `easy' -- and i'm wondering if i'm not missing something.
First, let us fix an orientation on the total lorentz space $V':=V \oplus U$ and let ... | 1 | https://mathoverflow.net/users/20516 | 153238 | 81,569 |
https://mathoverflow.net/questions/153219 | 0 | Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda\_{i}$, and the probability that player $i$ defeats player $j$ is $\frac{\lambda\_{i}}{\lambda\_{i}+\lambda\_{j}}$ (ordinary Bradley-Terry comparison). Once a tournament has concluded th... | https://mathoverflow.net/users/44886 | Expected rank of players in a Bradley-Terry round-robin tournament | The rank for player $i$ is $R\_i=1 + \sum\_{j \ne i} I(i,j)$ where $I(i,j)$ is an indicator random variable for whether player $j$ has a higher score than player $i$, so $E[R\_i] = 1 + \sum\_{j \ne i} E[I(i,j)]$. The expected value $E[I(i,j)]$ can be computed by considering the $2^{2n-3}$ possible results of players $i... | 2 | https://mathoverflow.net/users/2954 | 153242 | 81,571 |
https://mathoverflow.net/questions/153239 | -3 | In the [Wikipedia article for Formula](https://en.wikipedia.org/wiki/Formula) (which has no references), it is claimed that:
"The informal use of the term formula in science refers to the general construct of a relationship between given quantities....
In mathematics, a formula is an entity constructed using the sym... | https://mathoverflow.net/users/27933 | Is there a precise definition of "mathematical formula"? | Every book of mathematical logic should be a good reference where to find the notion of formula.
Usually when one refers to formulas it means formulas of a first order language.
A first order language is specified by a family of symbols of three types: constants symbols, function symbols and predicate or relation sy... | 1 | https://mathoverflow.net/users/14969 | 153243 | 81,572 |
https://mathoverflow.net/questions/153230 | 2 | I start my question with a definition and some motivation.
Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
1. $P$ is Lagrangian, i.e. Maximal isotroic, dim$P\_m=n$, $\forall m\in M$, and $\omega(P,P)=0$
2. P involutive... | https://mathoverflow.net/users/nan | The space of holomorphic sections are finite dimensional? | $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\Hom}{Hom}$ Suppose that $V$ is a finite dimensional real space equipped with an almost complex structure $J$. Let $P\subset V^{\bC}=V\otimes\_{\bR}\bC$ be a **complex** vector subpace.
Then $P^\*=\Hom\_{\bC}(P,\bC)$ is a quotient of ... | 1 | https://mathoverflow.net/users/20302 | 153246 | 81,575 |
https://mathoverflow.net/questions/112306 | 19 | Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define
$$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle \langle \tilde{v}, \tilde{v} \rangle}$$
where $\langle \ , \ \rangle$ is the standard Hermitian inner product and $\tilde{v}$... | https://mathoverflow.net/users/297 | Are the only local minima of $\angle(v, Av)$ the eigenvectors? | Thank you for this interesting question! (Long time I was expecting the opposite answer.)
**True motivation.** Let $V$ be a finite-dimensional $\mathbb C$-linear space equipped with a positive-definite hermitian form $\langle-,-\rangle$. Pick a $1$-dimensional $\mathbb C$-linear subspace $p\subset V$. The orthogonal ... | 7 | https://mathoverflow.net/users/40352 | 153271 | 81,582 |
https://mathoverflow.net/questions/115516 | 12 | Suppose $G$ is a finite group and $A$ an abelian subgroup. Suppose for some natural number $n\geq 2$, elements of $\gamma\_n(G)$ have the form $[a, x]$ where $a\in A$ and $x\in G$. Then $G$ is solvable.
| https://mathoverflow.net/users/44949 | A conjecture on solvablity of finite groups | I claim that the conjecture is false, and there is a counterexample with $G=S\_5$, $n=2$, and $A$ cyclic of order 6. In this case $\gamma\_2(G)=[G,G]=G'=A\_5$ has order $5!/2=60$. Take $A$ to be the abelian group $A=\langle(1,2,3),(4,5)\rangle$, but any conjugate of $A$ will also work. A simple computation shows that
t... | 14 | https://mathoverflow.net/users/23827 | 153273 | 81,583 |
https://mathoverflow.net/questions/153269 | 0 | **Theorem** $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.
>
> In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The Jordan-Brouwer Separation Theorem" intuitively. Then what happens when we replace topological sphere of topology to... | https://mathoverflow.net/users/36119 | The Jordan-Brouwer Separation Theorem | The magic words are: [Alexander duality.](http://en.wikipedia.org/wiki/Alexander_duality)
| 3 | https://mathoverflow.net/users/11142 | 153275 | 81,584 |
https://mathoverflow.net/questions/153272 | 15 | Let $T\_0$ be $\mathsf{ZFC}$ and, for $n\in\omega$, set $T\_{n +1}=T\_{n}+\mathrm{Con}(T\_{n})$.
>
> **Question 1:** Is there a natural number $n$ such that $T\_{n}$ is equiconsistent with $\mathsf{ZFC}+$ some large cardinal axiom?
>
>
> **Question 2:** Is the consistency strength of the $T\_{n}$ bounded by some ... | https://mathoverflow.net/users/nan | How strong is the iterated consistency of ZFC? | The procedure you suggest really cannot get too far. Here is an abstract result explaining what I mean (see [**Aspects of incompleteness**](http://projecteuclid.org/euclid.lnl/1235416274) by Per Lindström): Given an r.e. sequence of r.e. theories that interpret arithmetic and whose union is consistent, there is a $\Pi^... | 24 | https://mathoverflow.net/users/6085 | 153276 | 81,585 |
https://mathoverflow.net/questions/153278 | 3 | As many mathematicians know, each person has an Erdős number (see: <http://en.wikipedia.org/wiki/Erd%C5%91s_number>). That is, Erdős himself has Erdős number zero, each person who published anything with Erdős (mathematical or not) has an Erdős number one, and each person who published something with someone with Erdős... | https://mathoverflow.net/users/10898 | Systems similar to Erdős numbers? | This is not really an answer and rather an extended comment. But there is a lot of serious research on network systems that formalize or are related to collaboration networks of authors. A good starting point may be [this webpage on research on collaboration in research by the Erdős Number Project.](http://www.oakland.... | 5 | https://mathoverflow.net/users/27829 | 153280 | 81,587 |
https://mathoverflow.net/questions/153277 | 45 | There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on the moduli space of elliptic curves. I gather that the homotopy theorists regard it as a major achievement.
My question ... | https://mathoverflow.net/users/44912 | Why should I care about topological modular forms? | One of the closer connections to geometric topology is likely from invariants of manifolds. The motivating reason for the development of topological modular forms was the [Witten genus](http://en.wikipedia.org/wiki/Genus_of_a_multiplicative_sequence#Witten_genus). The original version of the Witten genus associates pow... | 32 | https://mathoverflow.net/users/360 | 153313 | 81,601 |
https://mathoverflow.net/questions/151579 | 0 | I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible.
Let $\varphi:R\to S$ be a surjective ring homomorphisms (rings are commutative with identity). Let $X$ be an $S$-module, and regard it as an ... | https://mathoverflow.net/users/4800 | Behavior of duality under pull-back | To to do this, one just has to replace the $\text{Hom}$ by an $\text{Ext}$. Let me explain. You already assumed that both $R$ and $S$ are Gorenstein, so that makes things easier.
Since $R$ and $S$ are Gorenstein, they are locally equidimensional, and so let us assume that $R$ is pure dimension $d$ and $S$ is pure di... | 3 | https://mathoverflow.net/users/3521 | 153318 | 81,604 |
https://mathoverflow.net/questions/152907 | 6 | Let $X$ be a standard Borel space, so that the space of Borel probability measures on $X$ is also a standard Borel space. We denote it by $\mathcal P(X)$.
In [this paper](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Game-theory-optimal-stopping-probability-and-s... | https://mathoverflow.net/users/11768 | Convex hulls of families of probability measures | Many thanks to Gerald Edgar and D. Kelleher for their answers. I am (and was) a little familiar with the Choquet's theory and tried to find the answer to OP and similar questions there (mostly in "Lecture notes on Choquet's theorem"), but my search was not very successful. By no means it implies that the answer can't b... | 2 | https://mathoverflow.net/users/11768 | 153319 | 81,605 |
https://mathoverflow.net/questions/153316 | 3 | In studying automourphic representation, I want to know whether my understanding is on the right way.
Let $\pi$ be a irreducible cuspidal representation of $GL\_2(A\_F)$.
Then $\pi\_v$, the local component of $\pi$, is the unique subquotient of the induced representation of $\rho(\chi|\cdot|^s, \chi|\cdot|^{-s})$ ... | https://mathoverflow.net/users/29422 | Local component of global irreducible representation of GL_2(A_F) | Not quite. The local component $\pi\_v$ is one of four types.
1. If $\pi\_v$ belongs to the *tempered principal series*, then it is of the form $\rho(\chi\_1,\chi\_2)$, where $\chi\_1$ and $\chi\_2$ are arbitrary unitary characters (not necessarily equal).
2. If $\pi\_v$ belongs to the *complementary series*, then i... | 5 | https://mathoverflow.net/users/11919 | 153322 | 81,606 |
https://mathoverflow.net/questions/153325 | 2 | For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with finite volume). However, not every link complement can be endowed with such a structure. Does anybody know if there is ... | https://mathoverflow.net/users/37354 | Is there a criterion for a link complement to have a hyperbolic structure with finite volume | There is a topological criterion due to Thurston. Using the JSJ machine (and work of many others) this criterion can also be phrased algebraically. I'll essay these below. Please note that the situation is much simpler for knots. To answer your question most directly, here is the desired reference to Wikipedia.
<htt... | 7 | https://mathoverflow.net/users/1650 | 153327 | 81,607 |
https://mathoverflow.net/questions/153324 | 8 | Let $G$ be a connected, simply-connected complex semisimple group with affine Grassmannian $\mathcal{G}r$. Fix a maximal torus and Borel $T\subseteq B\subseteq G$. I am reading "Loop Grassmannian Cohomology, the Principal Nilpotent and Kostant Theorem" by Ginzburg. He says that there exists a one-parameter subgroup $\n... | https://mathoverflow.net/users/25358 | The Bialynicki-Birula Stratification of the Affine Grassmannian | If I understand what's written Ginzburg correctly, this claim is incorrect, but very easily fixed.
**Why is this incorrect:** Because there's no cocharacter into $T$ which has finite dimensional BB cells. If there were, then there would be a character whose action on tangent space at the identity coset $[e]$ had a f... | 10 | https://mathoverflow.net/users/66 | 153330 | 81,609 |
https://mathoverflow.net/questions/152984 | 8 | Let $A$ be a Banach algebra with the following property:
*For every two nets $ x\_{\alpha}$ and $y\_{\alpha}$ in $A$, $x\_{\alpha}y\_{\alpha}$ converges if and only if $y\_{\alpha}x\_{\alpha}$ converges.*
Is $A$ necessarily commutative?
1)Note that we do not assume that the above two nets converge to the same val... | https://mathoverflow.net/users/36688 | characterization of commutative Banach algebras | No. Some algebras satisfy the identity $xy=-yx$ but are not commutative. Such an algebra clearly satisfies your condition. Obviously, it never has an approximate identity.
To construct one, let $V,W$ be two Banach spaces and let $f: V \times V \to W$ be a continuous symplectic bilinear form. Then define a multiplicat... | 10 | https://mathoverflow.net/users/18060 | 153331 | 81,610 |
https://mathoverflow.net/questions/153323 | 6 | I think that there is a metric on the huge space of all $C^{\*}$ algebras. What is the explicit
definition of this metric?may you introduce me a reference?
Moreover is the restriction of this metric to commutative $C^{\*}$ algebras gives us a discrete metric? by discrete I mean "every commutative $C^{\*}$ algebra h... | https://mathoverflow.net/users/36688 | Metrics on the space of $C^{*}$ algebras | Probably the most standard metric is [Banach-Mazur distance](http://en.wikipedia.org/wiki/Banach-Mazur_compactum), and there is indeed [a theorem due to Amir](http://link.springer.com/article/10.1007%2FBF03008398#page-1) which says that if the Banach-Mazur distance between $C(K)$ and $C(L)$ is less than $2$ then $K$ an... | 14 | https://mathoverflow.net/users/23141 | 153340 | 81,614 |
https://mathoverflow.net/questions/153338 | 0 | I am studying first order deformations and a natural question arises.
**Situation:** Let $X\_1$ be a scheme. $\pi: X\_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an algebraically closed field. $X\_0$ is the central fiber, i.e., $X\_1 \times\_{{\rm Spec}~ k} {\rm Spec~} k[t]/(t^2)$. T... | https://mathoverflow.net/users/40852 | The closure of an effective Cartier divisor in a special situation | Sorry, I do not understand what you say in 5. What if A = k[x, y] and X\_0 = Spec(A) and U\_0 = D(x) and the effective Cartier divisor is given by the zero locus of y + t/x ? Then (y + t/x)(y - t/x) = y^2 and the scheme theoretic closure of Z\_1 is given by an ideal containing both y^2 and xy + t which cannot be princi... | 0 | https://mathoverflow.net/users/44817 | 153341 | 81,615 |
https://mathoverflow.net/questions/153351 | 1 | Let $f:X \to Y$ be a flat, family of smooth projective varieties. Assume futher that $Y$ is smooth. Suppose there exists a scheme $Y'$ such that the associated reduced scheme $Y'\_{\mathrm{red}} \cong Y$. Does there exist a flat family of schemes over $Y'$ such that the pullback of this family to $Y$ is isomorphic to $... | https://mathoverflow.net/users/43198 | Making a family of schemes non-reduced | As formulated, there are many counterexamples, although I suspect the OP has a slightly different question in mind. For instance, let $Y$ be $\mathbb{P}^2$, let $X$ be the blowing up of the diagonal in $Y\times Y$ with either of its natural morphisms to $Y$, and let $Y'$ be the nonreduced scheme from Exericse II.5.9 of... | 3 | https://mathoverflow.net/users/13265 | 153356 | 81,620 |
https://mathoverflow.net/questions/153346 | 3 | I have a question about the finite analog of the puzzle proposed [here](https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice) involving mathematicians guessing the contents of boxes.
Specifically, suppose there are $k$ unopened boxes each containing a single symbol from an al... | https://mathoverflow.net/users/44653 | Simple reason that a mathematician cannot do better than random when guessing contents of a box? | It follows from the fact that, for any strategy $\sigma$, then the average over configurations of a correct guess is precisely $1/r$:
$$\frac{1}{r^k}\sum\_CP\_{\sigma}(C)=\frac{1}{r}.$$
This is true for deterministic strategies because if you partition configurations into sets of $r$ that differ only in the content of ... | 3 | https://mathoverflow.net/users/22989 | 153366 | 81,624 |
https://mathoverflow.net/questions/153350 | 0 | Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going into detail with how it comes up in the paper, I am curious as to how this connection comes about, more or less how to con... | https://mathoverflow.net/users/nan | Birkhoff decomposition vanishing of the Chern numbers | The proof is straightforward if you accept some standard results on cohomology over $\mathbb{P}^1$. Let me try. We want to prove that any vector bundle $E$ on $\mathbb{P}^1$ is a direct sum of line bundles $\mathcal{O}(k)$.
The proof is by induction on the rank of $E$.
Replacing $E$ by $E(k)$ for some integer $k$ we ma... | 2 | https://mathoverflow.net/users/40297 | 153368 | 81,625 |
https://mathoverflow.net/questions/132600 | 8 | I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs.
Let me review the most common definition and basic properties first.
Let $ G $ be a (finite simple) graph. For any natural number $ n $, we define the multi-chromatic number $ \chi\_n(G) $ as the least... | https://mathoverflow.net/users/5340 | Fractional chromatic number, find reference to a particular alternate definition for | This result, or something that sounds very similar, is in the paper "On the fractional chromatic number and the lexicographic product of graphs", Sandi Klavẑar (1998). Within that paper, a few references are cited for similar results as well.
| 3 | https://mathoverflow.net/users/18969 | 153385 | 81,633 |
https://mathoverflow.net/questions/153384 | 3 | A function f is *diagonaly non-recursive* (DNR) if for every Turing index $e$, $f(e) \neq \Phi\_e(e)$.
A set is *strongly hyperhyperimmune* if there is no r.e. set of disjoint r.e. set intersecting it.
In their paper "A cohesive set which is not high", the authors claim that the jumps of the strongly hyperhyperimmu... | https://mathoverflow.net/users/8833 | Jump of strongly hyperhyperimmune degrees and DNR relative to 0' | Carl Jockusch and Frank Stephan. A cohesive set which is not high. Mathematical Logic Quarterly, 39:515-530, 1993. A corrective note (keeping the main results intact) appeared in the same journal, 43:569, 1997.
| 3 | https://mathoverflow.net/users/4600 | 153386 | 81,634 |
https://mathoverflow.net/questions/153392 | 3 | We know that there is a map from $h:\pi\_{i}^{st}(pt)\rightarrow KO\_{i}(pt)$ and we know all the $KO\_{i}(pt)$ by Bott periodicy: they are $Z, Z\_{2},Z\_{2},0,Z,0,0,0$. We also know $\pi\_{i}^{st}(pt)$ for i=0~7.
My question is how to determine the image of $h$?
In particular (which is the case I care most), how to... | https://mathoverflow.net/users/44651 | How to compute the Hurewicz image of a stable map into real K theory | For $i = 1$ you can argue using the fact that $S\_n \to O(n)$ induces an isomorphism on abelianizations. $\pi\_i^{st}$ are the homotopy groups of the group completion of $\coprod\_n BS\_n$, and at least for $i$ positive $KO\_i$ are the homotopy groups of the group completion of $\coprod\_n BO\_n$.
For $i = 2$, you c... | 6 | https://mathoverflow.net/users/44966 | 153395 | 81,638 |
https://mathoverflow.net/questions/70722 | 12 | <http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm>
.$\quad$1. Can the independence assumption be weakened, similar to [this](http://en.wikipedia.org/wiki/Central_limit_theorem#CLT_under_weak_dependence)?
.$\quad$2. Can the identically distributed assumption be dropped/weakened, in the l... | https://mathoverflow.net/users/nan | Can the Law of the Iterated Logarithm be strengthened? | For your third question, see a [paper of Erdös](http://www.ams.org/mathscinet-getitem?mr=6630), where he proves an even more precise result (at least, in the special case $S\_n=\sum\_{i=1}^nY\_i$ where $Y\_i$ (independent) are $\pm 1$ with probability $\frac 12$), namely that for $\delta>0$, the following holds with pr... | 6 | https://mathoverflow.net/users/35353 | 153396 | 81,639 |
https://mathoverflow.net/questions/153394 | 4 | For some positive integer $n$, recall that the Quot scheme $Quot(\mathcal{O}\_{\mathbb{P}^n})$ parametrizes ideal sheaves of subschemes in $\mathbb{P}^n$. As far as I understand (from a previous post) that the subset of $Quot(\mathcal{O}\_{\mathbb{P}^n}) \times Quot(\mathcal{O}\_{\mathbb{P}^n})$ consisting of pairs $(\... | https://mathoverflow.net/users/32151 | Generalization of Hilbert/Quot schemes | For convenience, denote $\text{Quot}\_{\mathcal{O}\_{\mathbb{P}^n}/\mathbb{P}^n/\mathbb{Z}}$ by $H$; of course this is the same as the Hilbert scheme of $\mathbb{P}^n$, but it is indeed a Quot scheme as well. On the scheme $H\times \mathbb{P}^n\_{\mathbb{Z}}$, there is a universal surjective homomorphism of quasi-coher... | 5 | https://mathoverflow.net/users/13265 | 153397 | 81,640 |
https://mathoverflow.net/questions/153402 | 5 | Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H\_1(\Sigma)$ can be *realized by a simple closed curve* if there exists an oriented simple closed curve $\gamma$ on $\Sigma$ such that $[\gamma] = h$.
If $\Sigma$ has $0$ or $1$ boun... | https://mathoverflow.net/users/44973 | Realizing homology classes on surfaces with boundary by simple closed curves | This is *related to* a nontrivial question, address in [this](http://arxiv.org/abs/0801.3944) paper of Chas and Krongold (there are other related papers of Moira Chas with Fabiana Krongold and Dennis Sullivan, which a google search will bring up).
The original question, however, *is* trivial, since if we take some cu... | 2 | https://mathoverflow.net/users/11142 | 153407 | 81,643 |
https://mathoverflow.net/questions/153399 | 3 | In this book on page 82 I found an estimate $$\sum\_{j=1}^{n-2}\frac{\sin(k+1)\theta\_{j}}{\sin\theta\_{j}}=O\_{\epsilon}\left( p^{k\epsilon}\right)$$ as $k$ goes to infinity, for all $\epsilon>0$. Here $p$ is a prime number and $\theta\_{1},...,\theta\_{n-2}$ are complex numbers. After this estimate, the authors says:... | https://mathoverflow.net/users/44967 | On an asymptotic in Sarnak's book: "Some applications of modular forms" | A much more detailed writeup of this is is Davidoff/Sarnak/Vallette, see page 127 and on, especially p. 130.
| 6 | https://mathoverflow.net/users/11142 | 153409 | 81,644 |
https://mathoverflow.net/questions/153408 | 14 | The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations.
In other words, denoting by $ \mathfrak{lie}\_2 $ the free Lie algebra on two variables, an associator is a grouplike element in the completed hopf al... | https://mathoverflow.net/users/36098 | Associators, Grothendieck-Teichmüller group and monoidal categories | 0) This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded,... | 8 | https://mathoverflow.net/users/13552 | 153411 | 81,645 |
https://mathoverflow.net/questions/153002 | 2 | Specifically, I am looking for a proof that for squarefree $d\in\mathbb{N}\setminus\{ 1,3\}$, there exists some Kleinian $\Gamma$ and some $\alpha\in$Aut$(PSL(2,\mathbb{Z}))$, such that the Bianchi group $\Gamma\_d\cong \Gamma\*\_{\alpha}$. I have a reference that this proof appears in the article:
Charles Frohman an... | https://mathoverflow.net/users/14835 | Frohman & Fine's proof about Bianchi groups as HNN extensions (or anyone else's) | The second reference listed in the question,
Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, *Proceedings of the American Mathematical Society*, Vol. 102, No. 2, pp. 221-229,
does in fact include a proof of the statement in question. The article is available on [Jstor](http://... | 0 | https://mathoverflow.net/users/14835 | 153423 | 81,654 |
https://mathoverflow.net/questions/21279 | 4 | I will begin by stating my question, and then write down some related thoughts.
Let $\mathfrak{g}$ be a finite dimensional nilpotent Lie algebra over $\mathbb{C}$. Choose an ideal $\mathfrak{h}$ in $\mathfrak{g}$ of codimension 2. The quotient $\mathfrak{g}/\mathfrak{h}$ is then abelian. If $L$ is any 1-dimensional s... | https://mathoverflow.net/users/4384 | Isomorphism classes of nilpotent Lie algebras | I have found a counterexample while studying Lie algebra degenerations. Consider the following filiform nilpotent Lie algebra $\mathfrak{g}$ of dimension $13$, given by the brackets with respect to a basis
$(e\_1,\ldots ,e\_{13})$:
\begin{align\*}
[e\_1,e\_i] & = e\_{i+1},\quad 2\le i\le 12 \\[0.2cm]
[e\_2,e\_3] & = e\... | 6 | https://mathoverflow.net/users/32332 | 153457 | 81,665 |
https://mathoverflow.net/questions/153094 | 4 | Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r:=G((t))/G[[t]]$$ be its affine Grassmannian. I have read that $\mathcal{G}r$ possesses a natural very ample line bundle/invertible sheaf (see "A Polytope Calculus for Semisimple Groups" by J. Anderson, for instance). This allows one t... | https://mathoverflow.net/users/25358 | Reference for the Natural Ample Line Bundle on the Affine Grassmannian | You may look at Proposition 13.2.19 in S. Kumar's book "Kac-Moody Groups, their Flag Varieties and Representation Theory".
| 2 | https://mathoverflow.net/users/45000 | 153463 | 81,669 |
https://mathoverflow.net/questions/153436 | 13 | Given a system of 2nd-degree polynomials, $P=\{p\_1,\dots,p\_m\}$ where $p\_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find a common zero of all of these polynomials? In other words, given $P$, can you find $x\_1,\dots,x\_n\in\mathbb{R}$ such that $p\_i(x\_1,\dots,x\_n)=0$ for all $i$? I only need to ... | https://mathoverflow.net/users/11723 | Can you efficiently solve a system of quadratic multivariate polynomials? | There is an exact algorithm that needs $n^{O(m)}$ operations (cf. <http://arxiv.org/abs/cs/0403008>). One cannot expect anything better than that, unless P=NP. Indeed, it is easy to formulate several NP-complete problems as testing solvability of linear equations in 0-1 variables $x\_i$, and the latter can be enforced ... | 19 | https://mathoverflow.net/users/11100 | 153473 | 81,672 |
https://mathoverflow.net/questions/153431 | 5 | Let
$$S\_a(N)=\sum\_{n\le N}\frac{\varphi(an)}{n^2}.$$
The usual machinery gives an asymptotic formula
$$S\_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi\_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\varepsilon}),$$
where $C(a)$ some complicated function and
$$\varphi\_+(a)=a\prod\_{p\mid a}\left(1+\frac1p\right).$$
Is it... | https://mathoverflow.net/users/5712 | On a sum involving Euler totient function | I have found a proof of more general formula in the book *Postnikov, A. G. Introduction to analytic number theory American Mathematical Society, 1988,* (section 4.2). This proof is simple but it has a small mistake inside. (For arithmetic progression starting from $0$ this mistake vanishes.)
Please give more referenc... | 2 | https://mathoverflow.net/users/5712 | 153477 | 81,673 |
https://mathoverflow.net/questions/153439 | 3 | Suppose you have a general $n$-th dimensional random Gaussian vector with probability distribution function $\mathcal{N}\left(\mathbf{x}|\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$.
What is the computational complexity (both space or time) of calculating the cumulative distribution function
$\int\_{-\infty}^{y\_1}d... | https://mathoverflow.net/users/44790 | Computation complexity of calculating the cdf of an n-th dimensional gaussian random vector | To some extent this depends on the model of computation. Some remarks:
1. The problem must always involve an accuracy parameter $\epsilon$, since in general the answer will not be rational
2. Even in the case of a standard 1-dimensional normal, the question is not so easy. I believe this can be done to accuracy $\eps... | 5 | https://mathoverflow.net/users/658 | 153480 | 81,674 |
https://mathoverflow.net/questions/153476 | 0 | Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square root with $\operatorname{Im}\sqrt{z^2 - 1}>0$. Then $g$ can be extended across $(-\infty , -1]\cup [1,\infty )$ to the ... | https://mathoverflow.net/users/19433 | Use of Jensen's inequality on a Riemann surface | In general, Jensen's formula holds with the integral taken over both sheets (and zeros counted on both sheets). See, for example,
MR1069755
Lang, S., Cherry, W.
Topics in Nevanlinna theory.
Lecture Notes in Mathematics, 1433.
I don't know what exactly are you trying to do but if $z\to\infty$ in your problem, then
i... | 2 | https://mathoverflow.net/users/25510 | 153482 | 81,676 |
https://mathoverflow.net/questions/153461 | 0 | The modified Szpiro conjecture is described in
[Wikipedia](https://en.wikipedia.org/wiki/Szpiro_conjecture)
and [here](http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01/node2.html) and [here](http://www.encyclopediaofmath.org/index.php/Szpiro%27s_conjecture).
>
> The modified Szpiro co... | https://mathoverflow.net/users/12481 | Does the modified Szpiro conjecture require minimal model? | The conductor **does not** depend on the model. The $c\_4$ and $c\_6$ do. In fact as $E$ varies among the different integral models for the same elliptic curve, the left-hand side of the inequality differs by an arbitrary $12$-th power of an integer. The conjecture certainly allows for $a\_1 \ne 0$ and $a\_3 \ne 0$. If... | 5 | https://mathoverflow.net/users/4140 | 153485 | 81,678 |
https://mathoverflow.net/questions/153484 | 22 | Let $M$ be a compact manifold and $\varphi : M \longrightarrow M$ be a diffeomorphism which is isotopic to the identity. Does there exist a vector field $ X $ on $M$ such that $\varphi$ is the flow at time $1$ of $X$? If that is not always the case, where does the obstruction for such a $\varphi$ to be a flow lives in ... | https://mathoverflow.net/users/25511 | Flows of vector fields and diffeomorphisms isotopic to the identity | The answer is no. In fact, there are diffeomorphisms arbitrarily close to the identity which are not contained in flows (which are also often called $1$-parameter subgroups; here one is thinking of the set of vector fields on $M$ as a sort of "Lie algebra" for the diffeomorphism group).
Here's an example which I lear... | 26 | https://mathoverflow.net/users/317 | 153488 | 81,680 |
https://mathoverflow.net/questions/153475 | 1 | I'll use another way to give the question besides the title.
Define
$$ A\_k(n) = \frac{1}{2 \pi \mathrm{i}} \int\_{|q|=1} \left( \frac{1}{2 \pi \mathrm{i}}\int\_{|z|=1}\prod\_{j=-n}^{n}(1+qz^j) \frac{dz}{z} \right) \frac{1}{q^k} \frac{dq}{q}.$$
By some numeral calculation buy a simple program, I note that if we fre... | https://mathoverflow.net/users/22954 | A problem on counting k-subsets of {-n,-n+1,...,n-1,n} satisfying that sum of elements equal to 0 | Here is a sketch of a proof. First I'll switch $q$ and $z$, since this is more consistent with standard “$q$-series” notation.
Let $C\_k(n)$ be the coefficient of $z^k$ in $\prod\_{j=-n}^n (1+q^jz)$, so that $A\_k(n)$ is the constant term in $q$ in $C\_k(n)$.
By the $q$-binomial theorem
$$\prod\_{j=0}^{m-1} (1+q^iz)... | 7 | https://mathoverflow.net/users/10744 | 153502 | 81,684 |
https://mathoverflow.net/questions/153509 | 5 | Is there an established German translation of "locale"? The term appears mostly untranslated as "Locale"; a single time I've seen "Lokal". Where I'm located, we say "Örtlichkeit" or "Ort".
Quoting the nLab: A [locale](http://ncatlab.org/nlab/show/locale) is, intuitively, like a topological space that may or may not h... | https://mathoverflow.net/users/31233 | German translation of "locale" (from pointless topology) | [Wikipedia](https://de.wikipedia.org/wiki/Heyting-Algebra) uses "Locale", capitalized as a German noun but otherwise unchanged. It seems to resonate with how the word was originally introduced by [John Isbell](http://ojs.statsbiblioteket.dk/index.php/math/article/viewFile/11409/9426):
>
> "an inspired choice, which... | 7 | https://mathoverflow.net/users/11260 | 153515 | 81,688 |
https://mathoverflow.net/questions/153511 | 0 | This is just a reference request for a result which is very general, useful and should be well-known, but I've failed to find a good reference to cite.
The problem is to define the "most natural" stationary distribution of a finite Markov Chain with specified initial state. What I mean here by "stationary distributio... | https://mathoverflow.net/users/21059 | Stationary distribution in general Markov Chains |
>
> I call a Strongly Connected Component (SCC) "ergodic"
> if we cannot get out of it, and "transient" otherwise.
>
>
> It suffices to compute the probability ρ(C)
> to end up in each ergodic component C
>
>
>
Although your case is not technically an absorbing Markov chain (because not every state will eventu... | 3 | https://mathoverflow.net/users/39754 | 153519 | 81,690 |
https://mathoverflow.net/questions/153403 | 5 | I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least picks out something relevant.
Dream 2. Picks out the top spaces homotopic to finite CW-complexes from the category of ... | https://mathoverflow.net/users/30909 | Small objects in categories | The subcategory of spaces equivalent to finite CW-complexes is the smallest one containing a point and closed under finite homotopy colimits. (In fact, this is the universal homotopy theory generated under finite hocolims by a single object).
Someone more patient than I could might fill in the details to make the fo... | 5 | https://mathoverflow.net/users/6936 | 153520 | 81,691 |
https://mathoverflow.net/questions/153506 | 3 | Let $f:X \to Y$ be a proper surjective morphism of projective surfaces such that there exists a curve $C \subset X$ for which $f|\_{X\backslash C}$ is an isomorphism and $f(C)$ is a set of points. Suppose $X$ is a closed subscheme of $\mathbb{P}^n$ for some integer $n$ and $C$ contracts to a rational singularity i.e., ... | https://mathoverflow.net/users/43198 | Contractibility of curves and embedding into projective space | Let $X\subset \mathbb P^3$ be an arbitrary smooth projective surface of degree $d>2$ and assume that $X$ contains a line $\ell\simeq \mathbb P^1$ If $d=4$, assume in addition that $X$ is general among such surfaces. This implies that its Picard number is $2$. A simple adjunction computation shows that $\ell^2=2-d<0$ ($... | 5 | https://mathoverflow.net/users/10076 | 153525 | 81,693 |
https://mathoverflow.net/questions/153505 | 3 | In [a paper](http://link.springer.com/article/10.1023/A:1004157323726#page-1) the author lists, without justification, generators for a Lie algebra. *I would be grateful if someone could justify these choices and perhaps suggest how I might have found them for myself.*
Consider the three-dimensional affine space with... | https://mathoverflow.net/users/44642 | How was this Lie algebra found? | I would try the following: observe that $E$ can also be described as the subgroup of $\text{Aff}\_7$ under which the pullback of the polynomial $p:=z-x^2-y^2$ vanishes up to oder 3 along the zero set $z-x^2-y^2=0$. This translates into: $p$ is pulled back to an element in the ideal generated by $p$ up to terms of oder ... | 4 | https://mathoverflow.net/users/745 | 153527 | 81,695 |
https://mathoverflow.net/questions/153532 | 8 | Here is a problem that has been bugging me for a while.
Let $\| \|$ be a norm over $\mathbb{R}^n$, let $C$ be a convex subset of $\mathbb{R}^n$ with non-empty interior, and let $f: (C,\|\|) \rightarrow (\mathbb{R}^n,\|\|)$ be a distance-preserving map.
Is it true that there exists an isometry $g$ of $(\mathbb{R}^n,... | https://mathoverflow.net/users/45005 | Restricted isometry | So the problem is that your map $f$ is not surjective.
However, $f$ is locally surjective in the following sence:
If $\Omega$ is the interior of $C$, $x\in \Omega$ and $\varepsilon>0$ is such that $B\_\varepsilon(x)\subset \Omega$ then the restriction $f|\_{B\_\varepsilon(x)}$ is a bijection from $B\_\varepsilon(x)... | 3 | https://mathoverflow.net/users/1441 | 153533 | 81,696 |
https://mathoverflow.net/questions/153521 | 7 | Single-particle billiards systems in a domain with corners, or multi-particle billiards in a domain with smooth boundary, can exhibit singularities in finite time. (The former phenomenon is well known; for an example of the latter, see e.g. section 1 of Charles Radin's article "Dynamics of Limit Models", available at <... | https://mathoverflow.net/users/3621 | Well-definedness of single-particle smooth billiards flow | I think the paper you want is [B. Halpern, "Strange Billiard Tables." *Transactions of the AMS* Vol 232, 1977.](http://www.ams.org/journals/tran/1977-232-00/S0002-9947-1977-0451308-7/S0002-9947-1977-0451308-7.pdf)
Thanks to Carl for pointing out that Halpern considers tables with the additional condition of nonvanish... | 6 | https://mathoverflow.net/users/2954 | 153540 | 81,701 |
https://mathoverflow.net/questions/153508 | 0 | I'm trying to understand a few things about automorphic L-functions. In page 5 of <http://arxiv.org/pdf/1401.0390.pdf>, the author mentions the isobaric sum decomposition $\pi=n\_{1}\pi\_{1}\boxplus\cdots\boxplus n\_{r}\pi\_{r}$. Let $d(\pi)$ denote the dimension of the representation space of $\pi$. Does the following... | https://mathoverflow.net/users/13625 | degree of an isobaric sum | No, it has nothing to do with the dimension of the representation space. (Automorphic representations of $GL\_n$ are always infinite-dimensional for $n > 1$.) The isobaric sum operation **changes the group**: each $\pi\_i$ is an automorphic representation of $GL(d\_i)$ for some $d\_i$, and the isobaric sum is an automo... | 5 | https://mathoverflow.net/users/2481 | 153554 | 81,706 |
https://mathoverflow.net/questions/153400 | 12 | I want to see if it is possible to force the existence of a function
$F:\aleph\_2 \times \aleph\_2\rightarrow \aleph\_1$ such that:
a) $F(a,b)=F(b,a)$, for all $a,b\in \aleph\_2$ and
b) for all distinct $a,b$, the set $\{x|F(a,x)=F(b,x)\}$ is finite.
What is known:
1) Under CH, there is no such function.
2... | https://mathoverflow.net/users/13694 | Can we force the existence of this function? | First notice that we can separate the condition depending the relative position of $a,b$, and $x$ in (b): let $F\_0$ be such that for any $a,b$ there are finitely many $x<\min(a,b)$ with
$F\_0(x,a)=F\_0(x,b)$, let $F\_1$ be likewise for $a<x<b$ and let $F\_2$ be for $a,b<x$. Then $F(a,b)=\langle F\_0(a,b),F\_1(a,b),F\... | 10 | https://mathoverflow.net/users/6647 | 153574 | 81,711 |
https://mathoverflow.net/questions/153568 | 4 | Loosely speaking, are elliptic Kummer extensions big? More concretely:
Let $E$ be an elliptic curve over $\mathbb{Q}$, let $p$ be a prime, and
let $F$ be a subfield of $\overline{\mathbb{Q}}$ containing the
coordinates of all the $p$-power torsion of $E$. Given $c > 0$, does
there
exist $N > 0$ (depending only on $E$... | https://mathoverflow.net/users/37644 | Are elliptic Kummer extensions big? | $Gal(F(P)/F)$ is indeed a subgroup of $E[p^{n}]$. We see that it is not contained in $E[p^{n-1}]$, because otherwise $p^{n-1}P\in E(F)$. Thus, since $E[p^{n-1}]$ consists of all the elements of order less than $p^n$, order of $Gal(F(P)/F)$ is at least $p^N$. So we may take any $N \leq \log\_p c$
| 5 | https://mathoverflow.net/users/18060 | 153576 | 81,713 |
https://mathoverflow.net/questions/153572 | 2 | Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian.
Some books consider the Laplacian on the initial domain $\lbrace f\in C^{\infty}(\Omega) \vert f\_{\vert\partial\Omega}=0 \rbrace$... | https://mathoverflow.net/users/21870 | What is the right initial domain for the Dirichlet-Laplacian on a bounded domain? | If you take $C\_0^\infty$ as the initial domain, then there are many self-adjoint extensions and the Dirichlet Laplacian is only one of them. The Neumann Laplacian is another one.
| 3 | https://mathoverflow.net/users/12120 | 153579 | 81,714 |
https://mathoverflow.net/questions/153581 | 15 | Let $v$ be a vector field. Does there exists a volume form $\Omega$
such that its Lie derivative is proportional to itself with a constant coefficient:
$$\mathcal{L}\_v \Omega= C \cdot \Omega? \ \ \ \ \ (\ast)$$
A simplification of the question: assume that the divergence $\sum\_i \frac{\partial v^i}{\partial x\_i}$... | https://mathoverflow.net/users/14515 | Does for every vector field there always exist a volume form for which the vector field is a homothety? | If there is such a volume form $\Omega$, by Moser's theorem we can pick local coordinates in which $\Omega=dx^1 \wedge \dots \wedge dx^n$. If in some coordinates $v$ vanishes to order $k$, for some $k>1$, but not at order $k$, then the same is true in any coordinates. For example, we can suppose that $v=f^2w$ with $f=0... | 16 | https://mathoverflow.net/users/13268 | 153587 | 81,718 |
https://mathoverflow.net/questions/153586 | 5 | This is an extension of [this question](https://math.stackexchange.com/questions/627180/submission-of-papers-to-arxiv-or-similar) and [this question](https://math.stackexchange.com/questions/607895/asymptotic-formula-for-almost-primes) on MathStackExchange.
I have developed a formula for almost primes which is far mo... | https://mathoverflow.net/users/45057 | Submission of papers to ArXiv or similar | I don't know anything about almost primes, but I think you're mistaken not to send your paper to people in the field. If you're afraid of someone stealing your idea, I wouldn't worry --- having already submitted it to a journal establishes your priority (besides the fact that having your idea stolen seems very unlikely... | 12 | https://mathoverflow.net/users/23141 | 153588 | 81,719 |
https://mathoverflow.net/questions/153571 | 1 | My question is about connectedness of the Riemann zeta function and the theory of Bergman spaces. Because I'm working in this area of Bergman spaces, and I had Analytic number theory as a subject on my graduate studies, I would like to know more about applications of the theory of Bergman spaces to Analytic number theo... | https://mathoverflow.net/users/44967 | Riemann zeta function in the framework of Bergman spaces | I am not aware of such an "Riemann" conjecture in the theory of Bergman spaces. However,
a result which lies perhaps "in the intersection" of both areas is *Voronin's Universality Theorem*, about the universality of zeta-functions, i.e., the property of zeta-functions (and Dirichlet $L$-functions) to approximate arbitr... | 2 | https://mathoverflow.net/users/32332 | 153590 | 81,720 |
https://mathoverflow.net/questions/72651 | 8 | If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$ is a fusion category and $M$ is a semisimple module category over $C$, then the dual of $C$ over $M$ is fusion if $\dim... | https://mathoverflow.net/users/22 | If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion? | We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in [*Dualizable Tensor Categories*](http://arxiv.org/abs/1312.7188) (joint with Christopher Douglas and Chris Schommer-Pries). Note that (for C semisimple) separability is equivalent to semisimplicity of Z(C) (see Corollary 3.5.9.), so T... | 7 | https://mathoverflow.net/users/22 | 153598 | 81,723 |
https://mathoverflow.net/questions/153618 | 3 | Let $X$ be a normed space (not necessarily complete) and $\{w\_n\}\_{n\in\mathbb N}\subset X$ be
a linearly independent set. Let $\{x^\*\_n\}\_{n\in\mathbb N}$ be a set of linear functionals on $X$ with the property that $x\_i^\*(w\_j)=\delta\_{ij}$. These linear functional are not necessarily bounded.
My question is... | https://mathoverflow.net/users/43681 | Continuous dual basis in a normed space | Sure. We can ensure that ${\rm span}(u\_1, \ldots, u\_k) = {\rm span}(w\_1, \ldots, w\_k)$ and $\|u\_k^\*\| = 1$ for each $k$. The construction goes by induction on $k$. For $k = 1$ let $u\_1 = \frac{1}{\|w\_1\|}w\_1$ and use Hahn-Banach to find $u\_1^\* \in X^\*$ such that $\|u\_1^\*\| = u\_1^\*(u\_1) = 1$. Inductivel... | 5 | https://mathoverflow.net/users/23141 | 153626 | 81,738 |
https://mathoverflow.net/questions/153624 | 4 | Any one knows a reference for computing K\_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?
Thanx!
| https://mathoverflow.net/users/42736 | K-Theory of Algebra of Zeroth Order Pseudo differential operators | I do not have a reference, except that Higson-Roe's book ''Analytic K-homology'', p. 47ff should give enough information.
Let $P(M)$ be the algebra of order zero scalar pseudodifferential operators on the closed manifold $M$. There is a short exact sequence
$$
K(L^2(M)) \to P(M) \to C(SM)
$$
where $C(SM)$ are the... | 4 | https://mathoverflow.net/users/9928 | 153651 | 81,744 |
https://mathoverflow.net/questions/153650 | 5 | Consider the first Cohen model, i.e. let $M$ be a countable transitive model of ZFC + $V=L$, let $\mathbb P$ be the poset consisting of finite partial functions from $\omega\times\omega$ to $2$, let $M[G]$ the generic extension adding a sequence $\{a\_n\}\_{n\in\omega}$ of generic subsets of $\omega$, and let $M\subset... | https://mathoverflow.net/users/41274 | A question about the first Cohen model | While not an answer to your question directly, here is some information which you might consider useful. (See edit for that part.)
In Jech **The Axiom of Choice** he points out that the proof of Halpern-Levy is much more difficult than the proof of Halpern that $\sf BPI$ holds in the Mostowski permutation model. He p... | 6 | https://mathoverflow.net/users/7206 | 153655 | 81,747 |
https://mathoverflow.net/questions/153648 | 6 | 1) What are the examples of elliptic curves over $\mathbb{Q}$ with good reduction and $\mu$-invariant $\geq 2$ at $p = 3$ and how to find them $?$
2) Let $\Lambda = \mathbb{Z}\_{p}[[T]] $ and $ K=\mathbb{Q}\_{\infty} $ be the cyclotomic $ \mathbb{Z}\_{p} $-extension of $ \mathbb{Q} $. Then the Pontrjagin dual $ X\_{... | https://mathoverflow.net/users/30999 | $\mu$-invariant and Pontryagin dual of Selmer group of elliptic curves 1 | 1) 91b3 is an example with $\mu=2$ at $p=3$. Recall that the $\mu$-invariant (with respect to a prime $p$) only changes when there is an isogeny of degree $p$. More precisely it changes just be the quotient of the real Néron periods. Hence we are looking here for curves with cyclic isogenies of degree 9 defined over $\... | 8 | https://mathoverflow.net/users/5015 | 153667 | 81,751 |
https://mathoverflow.net/questions/153635 | 6 | The motivation to ask this question is some proposition of flasque sheaves.
Let's recall the definition of flasque sheaf:A sheaf $F$ on a topological space $X$ is flasque if for every inclusion $V\rightarrow U$ of open sets,the restriction map $F(U)\rightarrow F(V)$ is surjective.
It is known that the proposition of ... | https://mathoverflow.net/users/41650 | How to characterize flasque sheaves in more functorial way? | **Yes — by the Yoneda lemma, flasque sheaves can indeed be seen as $E$-injectives, where $E$ consists of the inclusion maps $\newcommand{\O}{\mathcal{O}}\newcommand{\restr}{\mathord{\upharpoonright}}\O\_X\restr\_U \to \O\_X\restr\_V$, for all pairs of opens $U \subset V$.**
Here $\O\_X \restr\_U$ is the sheaf of modu... | 11 | https://mathoverflow.net/users/2273 | 153670 | 81,752 |
https://mathoverflow.net/questions/153669 | 11 | Since the days of Aristotle and Descartes, it has been known that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the [Mpemba effect](https://en.wikipedia.org/wiki/Mpemba_effect), named after a student who rediscovered the effect in the sixties. Several theori... | https://mathoverflow.net/users/36090 | On mathematical studies of the Mpemba effect | Try this reference:
O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox, by Xi Zhang Yongli Huang, Zengsheng Ma and Chang Q Sun <https://arxiv.org/abs/1310.6514>
P.S. I see you have already found this reference. Some useful information about Mpemba effect can be found here <https://math.ucr.edu/home/baez/physi... | 6 | https://mathoverflow.net/users/32389 | 153671 | 81,753 |
https://mathoverflow.net/questions/153676 | 3 | Consider an elementary chain of models of some first-order theory $T$:
$$ (M\_\alpha)\_{\alpha < \kappa}, M\_\alpha \prec M\_\beta \; {\rm for} \; \alpha < \beta .$$
Let also $(N\_\alpha)$ be another elementary chain of models of the same theory.
Assume, that for every $\alpha < \kappa$ ($\kappa$ a cardinal) we have a... | https://mathoverflow.net/users/38200 | Isomorphism of a chain of structures | No. Consider the following two linear orderings: $\mathcal{M}=\mathbb{Z}\cdot\mathbb{N}$, and $\mathcal{N}=\mathbb{Z}\cdot\mathbb{N}^\*$. (Recall that $\mathcal{L}\cdot\mathcal{L}'$ is the linear order gotten by replacing each point in $\mathcal{L}'$ with a copy of $\mathcal{L}$, and that $\mathcal{L}^\*$ is just $\mat... | 4 | https://mathoverflow.net/users/8133 | 153679 | 81,756 |
https://mathoverflow.net/questions/153659 | 15 | I have seen different definitions of a rank of a module $M$ over a commutative ring $R$.
1. [Here in nLab](http://ncatlab.org/nlab/show/rank), for quite general modules, the rank is defined locally at $p\in \mathrm{Spec}(R)$ as the dimension over the residue field $\kappa(p)$ of the $\kappa(p)$ vector space $M\_p/pM\... | https://mathoverflow.net/users/3333 | Different definitions of the rank of a module | For a finitely generated module $M$ over a commutative ring $R$, the first definition gives a rank *function* $r\_M: \operatorname{Spec} R \rightarrow \mathbb{N}$, whereas the third definition gives $r\_M((0))$.
When $M$ is projective, $r\_M$ is locally constant, so when $\operatorname{Spec} R$ is connected -- so whe... | 12 | https://mathoverflow.net/users/1149 | 153681 | 81,758 |
https://mathoverflow.net/questions/153683 | 6 | The Rado graph contains every finite graph as induced subgraph, and its also holds for countable graphs. So it is an universal graph of size $\aleph\_0$, which contains all graphs of size $\aleph\_0$ as induced subgraph.
**Question:** Is it also true for higher cardinals? Does there exist a graph of size $\kappa$, w... | https://mathoverflow.net/users/40458 | Universal graphs on higher cardinals | Following up on Asaf's comment, indeed, if $\kappa^{\lt\kappa}=\kappa$, then we may build a $\kappa$-universal graph of size $\kappa$. Proceed in $\kappa$ many stages. At each stage, we have a graph of size $\gamma$, less than $\kappa$. There are precisely $2^\gamma\leq\kappa$ many ways to add a single new point to thi... | 6 | https://mathoverflow.net/users/1946 | 153684 | 81,759 |
https://mathoverflow.net/questions/153685 | 12 | Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free.
In them he describes a number of two-sorted systems (with sorts for integers and finite sets or sequences of integers) weaker than th... | https://mathoverflow.net/users/26143 | Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$ | If I understand the notation correctly, Theorem 8.28 in [1] shows that FSTZ is a notational variant of $V^0$ (and the same holds for all the theories Friedman shows to be definitionally equivalent to FSTZ in later sections, but see below for a caveat).
Specifically, FSTZ is biinterpretable with $V^0$ as follows. (I w... | 8 | https://mathoverflow.net/users/12705 | 153688 | 81,761 |
https://mathoverflow.net/questions/153687 | 2 | Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some knowledge as to the basic heuristics. I know this is a really trivial question, but again I am a not very experienced yet... | https://mathoverflow.net/users/nan | Quantization by cohomology | (The following is likely not the answer you are looking for, but it is an answer to what you actually ask.)
The case can be made that *all* of quantization is via cohomology, namely that quantization is fundamentally pull-push in some generalized cohomology theory, hence is index theory.
For [geometric quantization... | 10 | https://mathoverflow.net/users/381 | 153694 | 81,763 |
https://mathoverflow.net/questions/153695 | 8 | Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum\_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does there exist any better bound conditionally or unconditionally?
I am expecting a result like $A(x)=O(\frac1{\sqrt x})$ i... | https://mathoverflow.net/users/36735 | Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$ | As Alexey has pointed out the problem can be reduced, via [summation by parts](http://en.wikipedia.org/wiki/Summation_by_parts), to understanding the asymptotic of Mertens' sum
$$M(x) := \sum\_{n\leq x} \mu(n).$$
Conditional on the Riemann hypothesis, the best bound to date on Mertens' sum, [due to Soundararajan](http... | 12 | https://mathoverflow.net/users/630 | 153704 | 81,766 |
https://mathoverflow.net/questions/153700 | 2 | 4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim round sphere. Here uniformly is in the obvious sense of the volume form associated with the round metric. What is the probability the two curves cross?
| https://mathoverflow.net/users/41654 | Probability of random geodesics on the round sphere intersecting | The probability is 1/8. With each (generic) pair of geodesic segments you can associate another 15 pairs which you get by replacing each of the 4 end points by its antipodal point. You can easily check that of these 16 pairs, exactly 2 intersect.
| 8 | https://mathoverflow.net/users/2991 | 153705 | 81,767 |
https://mathoverflow.net/questions/153710 | 3 | I would like to know more about the tools in Harmonic analysis, but the ones that give a really good results in other theories. One of them are decompositions, like Whitney, Calderon-Zygmund etc...The reason that I would like to know more about this is that these decomposition tools give a better results in other field... | https://mathoverflow.net/users/44967 | Strong decomposition tools from Harmonic Analysis in other fields | There are at least two other decomposition that (I would argue) have been the foundation for fundamental advances in harmonic analysis.
(1) The wave packet decomposition. This decomposition underlies the proof of [Carleson's theorem](http://www.ams.org/mathscinet-getitem?mr=199631) (this is more explicit in [Fefferma... | 6 | https://mathoverflow.net/users/630 | 153713 | 81,774 |
https://mathoverflow.net/questions/153673 | 16 | One of the standard conjectures in algebraic geometry is that an operator $\Lambda$ on the cohomology algebra of a projective variety is algebraic. To my lying eyes it looks like there are two definitions of the operator $\Lambda$ out there and that the two don't agree.
On the algebraic side, let $X$ be a projective ... | https://mathoverflow.net/users/4054 | Why don't the algebraic and geometric adjoints of the Lefschetz operator agree? | In addition to Kleiman's "Algebraic cycles and the Weil conjectures" mentioned by abx, I can highly recommend reading the first article in the Motives proceedings (Jannsen/Kleiman/Serre) by Kleiman: "The standard conjectures".
It introduces several $\Lambda$ and $\star$ operators, and shows that when one is algebraic... | 6 | https://mathoverflow.net/users/21815 | 153714 | 81,775 |
https://mathoverflow.net/questions/153721 | 18 | Call an integer sequence $\mathbf{x}=\left( x\_1,x\_2,\cdots \right)$ *feasible* if it is $f(r)=\left(\lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots \right)$. for some real $r \gt 1$.
>
> Q: if the members of $f(r)$ all have the same parity, must $r$ be an integer?... | https://mathoverflow.net/users/8008 | Floors of powers of reals, how much do the first few determine the next? | This is related to [OEIS 014217 Floor(phi^n), where phi = (1+sqrt(5))/2 is the golden ratio](https://oeis.org/A014217).
From the comments:
$$a(n) =\lfloor \phi^n \rfloor = L(n)-(1+(-1)^n)/2$$
Where $L(n)$ are Lucas numbers.
Since $L(6n)$ are well known to be even, take $r=\phi^6$ where $\phi$ is the golden rati... | 14 | https://mathoverflow.net/users/12481 | 153723 | 81,777 |
https://mathoverflow.net/questions/153745 | 35 | What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
<http://www.math.cmu.edu/~eschimme/AIM/LongDescription.html>
But I couldn't find any more details with respect to a new core model.
Also... | https://mathoverflow.net/users/17176 | Latest stand of core model theory? | ${}$Hi Ioanna,
**I.**
The answer probably depends on how we define core model. At the level of "there are no Woodin cardinals in any inner model", we can finally show that core models exist, provably in $\mathsf{ZFC}$. The result was known before, of course, but we needed extra assumptions (such as: There is a meas... | 29 | https://mathoverflow.net/users/6085 | 153756 | 81,789 |
https://mathoverflow.net/questions/153665 | 4 | For each $k\ge 1$ there is a sequence $x\_{1,k},\ldots,x\_{k,k}$ of positive integers such that for all nonnegative integers $ a\_i $,
$$
\sum\_{i=1}^k a\_i x\_{i,k}
=2\sum\_{i=1}^k x\_{i,k}
\quad\Longrightarrow\quad a\_1=\cdots= a\_k=2
$$
(For $k=3$ the minimal example seems to be $(9, 12, 17)$.)
This is easy t... | https://mathoverflow.net/users/4600 | Number-theoretic dot-product property? | Many results of this type can be proven by a basic existence of rational points argument. For any finite set of nontrivial linear equations, we can find a set of integers that does not satisfy any of them. To replace the infinite set of equations in the question with a finite set, we need some kind of bound on the $a\_... | 4 | https://mathoverflow.net/users/18060 | 153760 | 81,793 |
https://mathoverflow.net/questions/153763 | 7 | The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h\_1, \cdots, h\_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a complete residue system modulo $p$, then there exist infinitely many positive integers $n$ such that the tuple
$$\dis... | https://mathoverflow.net/users/10898 | Higher dimensional generalization of the Hardy-Littlewood conjecture? | This is sometimes called Dickson's conjecture (questions of this form were raised in his 1904 paper, available [here](http://oeis.org/w/images/2/22/A_new_extension_of_Dirichlet%27s_theorem_on_prime_numbers.pdf)). Tao and Green's paper [Linear Equations in the Primes](http://arxiv.org/abs/math/0606088) give unconditiona... | 8 | https://mathoverflow.net/users/630 | 153765 | 81,795 |
https://mathoverflow.net/questions/153764 | 2 | The space $SL\_3(\mathbb{R})/SO\_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space.
Fix any copy of $SO\_2(\mathbb{R})$ sitting inside $SO\_3(\mathbb{R})$, then the coset space $SL\_3(\mathbb{R})/SO\_2(\mathbb{R})$ can be though of as a bundle over $SL\_3(\mathbb{R})/SO\... | https://mathoverflow.net/users/45156 | Is this sphere bundle over SL3/SO3 trivial? | As you remark, the quotient $SL\_n(R)/SO\_n(R)$ is contractible, and so any smooth bundle over it is smoothly trivial. So, in particular your bundle is diffeomorphic to a product.
The contractibility of $SL\_n(R)/SO\_n(R)$ stems from the Gram-Schmidt orthogonalization process; presumably you can use this to exhibit a... | 10 | https://mathoverflow.net/users/3460 | 153767 | 81,796 |
https://mathoverflow.net/questions/153743 | 4 | Suppose $A$ is an algebra of signature $\mathcal{L}$ and $V=Var(A)$ is the variety generated by $A$. I want to know is it possible to classify relatively free elements of $V$? As a special case, for a group $G$, under what conditions $G$ is free in $Var(G)$?
| https://mathoverflow.net/users/44949 | Relatively free algebras in a variety generated by a single algebra | Suppose that $A$ is a finite universal algebra with minimal cardinality of a generating set $d$. Then $A$ is relatively free in some variety iff it is relatively free on $d$ generators in the variety it generates, in which case it is free on any generating set of $d$ elements. Moreover, this occurs iff each map from a ... | 4 | https://mathoverflow.net/users/15934 | 153773 | 81,800 |
https://mathoverflow.net/questions/151704 | 4 | I am interested in the existence of a set of vectors $\{ v\_{ij} \}\_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^\*\_{ij} v\_{ij'} \right\vert = 1/N$ for $j \ne j'$ and
$v^\*\_{ij} v\_{i'j} = \delta\_{ii'}$.
For real vectors, the condition $\left\vert v^\*\_... | https://mathoverflow.net/users/18969 | Set of orthogonal simplexes or partial mutually unbiased bases | Vern Paulsen has provided me an answer, and I reproduce it here with his permission. For $j \in \{1,\dots,N+1\}$ define the matrix with columns $U\_j = (v\_{1j};\dots;v\_{Nj})$. The condition $v^\*\_{ij} v\_{i'j} = \delta\_{ii'}$ is equivalent to $U\_j$ being a unitary matrix. The other condition, $\lvert v^\*\_{ij} v\... | 2 | https://mathoverflow.net/users/18969 | 153775 | 81,802 |
https://mathoverflow.net/questions/153748 | 3 | Let $\tau\_0$ be the element of dual Steenrod algebra $A\_p^{\*}$ at a prime $p$ which is dual to Bockstein $\beta \in A\_p$. It is well known $\tau\_0^2 =0$. Is it true/known that the elemnet $\xi\_1$ belong to $p$-fold Massey product of $ \tau\_0$, i.e.
$$ \langle \tau\_0, \ldots, \tau\_0 \rangle \ni \xi\_1 ?$$
If ... | https://mathoverflow.net/users/19186 | Massey product in Dual Steenrod Algebra | So far as definitions of Massey products, the dual Steenrod algebra is the homology $H\_\*(H\Bbb Z/p)$ of an $E\_\infty$ ring spectrum, and so it has both Massey products and power operations.
Kraines, in Theorem 14 of "Massey higher products" from 1966, shows that the restricted power $\langle u \rangle^p \subset \l... | 8 | https://mathoverflow.net/users/360 | 153801 | 81,811 |
https://mathoverflow.net/questions/153808 | 9 | Let $\overline{M}\_{g,n}$ be the moduli space of $n$-pointed genus $g$ Deligne-Mumford stable curves. This is a normal projective scheme. Then
$$codim\_{\overline{M}\_{g,n}}Sing(\overline{M}\_{g,n})\geq 2.$$
For instance for $g = 1, n = 2$ wa have that $\overline{M}\_{1,2}$ is a rational surface with four singular poi... | https://mathoverflow.net/users/14514 | Singularities of moduli spaces of curves | No. In fact more is true: the locus of all $n$-pointed curves of genus $g-1$ with a single elliptic tail $E$, such that $\mathrm{Aut}(E)=\mathbf Z/6$, has codimension two in $\overline M\_{g,n}$ and consists of *noncanonical* singularities. This was famously determined by Harris and Mumford in their paper on the Kodair... | 14 | https://mathoverflow.net/users/1310 | 153809 | 81,814 |
https://mathoverflow.net/questions/153530 | 5 | Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold. Using the metric to identify the tangent and cotangent bundles defines a natural symplectic
structure on the tangent bundle, $(TM, \omega)$, and a natural volume form $\Omega=\omega^{n}$ on $TM$.
For every $r>0$, let $D\_{r}(M)$ be the open disc bundle on... | https://mathoverflow.net/users/36688 | What is the geometric interpretation of this quantity? | Both limits (1) and (2) are equal to $C(1)/V(1)$ because of the homogeneity of the volume and the symplectic capacity. Namely, the symplectic form is homogeneous of degree $1$ with respect to dilations on the tangent bundle and so it makes no difference what the radius of your disc bundle is.
What is then geometric m... | 7 | https://mathoverflow.net/users/21123 | 153820 | 81,817 |
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