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https://mathoverflow.net/questions/161834 | 0 | Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the *rank* (or dimension) of a dualizable object $V \in \mathcal{C}$ is the composite
$1 \xrightarrow{\mathrm{coev}} V \otimes \check{V} \cong \... | https://mathoverflow.net/users/2841 | Rank vanishing in tensor categories | No, I think this often fails. One way to fail would be if your category is linear over a field k of characteristic p. Then the rank of a direct sum of p copies of any fixed object will be zero. But you probably meant to exclude that kind of example. Then again there's another kind of example, the most basic of which wo... | 8 | https://mathoverflow.net/users/3931 | 161835 | 84,964 |
https://mathoverflow.net/questions/161580 | 8 | It is well known that for any lipschitz function $f:[0,1]\rightarrow [0,1]$, we can approximate it
by $\sum\_{i=1}^n f(i/n) {n\choose i} x^i (1-x)^{n-i}$, and the $L\_\infty$ error is $O(1/\sqrt{n})$. However, the best degree $n$ polynomial approximation could achieve
an additive error of $O(1/n)$. Since $\{{n\choose i... | https://mathoverflow.net/users/44539 | is there any such result about Bernstein polynomials? | It is fairly classical though I cannot guarantee that the question has ever been formally considered in this form. The problem is that if $z=x+iy$ where $x\approx \frac 12$ and $y\le 2n^{-1/2}$, then $|z^p(1-z)^{n-p}|\le Cx^p(1-x)^{n-p}$, so if $|c\_k|\le A$, then the corresponding Bernstein polynomial is bounded by $C... | 5 | https://mathoverflow.net/users/1131 | 161843 | 84,967 |
https://mathoverflow.net/questions/161748 | 2 | The **independence complex** $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.
Now, if $G$ is a **[well-covered graph](https://en.wikipedia.org/wiki/Well-covered_graph)** (where all maximal independent sets have the same cardinality $\alpha$) then $I(G)$ is ... | https://mathoverflow.net/users/22051 | Which graphs generate a matroidal independence complex? | A (finite) simplicial complex is the independent set complex of a graph $G$ if and only if its minimal nonfaces have two elements. If it is also a matroid complex, if follows that the circuits of the matroid have two elements. Thus $G$ is a disjoint union of complete graphs, not just a single complete graph.
| 7 | https://mathoverflow.net/users/2807 | 161846 | 84,968 |
https://mathoverflow.net/questions/161849 | 1 | I'm working on the amenability of some Banach algebras, and I'm wondering why $l^\infty$ is amenable ? Does any one has any idea how to start ?
Thank you in advance.
| https://mathoverflow.net/users/47369 | Amenability of $l^\infty$ | In general, commutative $C^\ast$-algebras are amenable. See [*A new proof of the amenability of $C(X)$*](http://dx.doi.org/10.1017/S0004972709001063) by Mortaza Abtahi and Yong Zhang (Bulletin of the Australian Mathematical Society, Volume 81, Issue 3, June 2010, pages 414-417).
| 3 | https://mathoverflow.net/users/47469 | 161852 | 84,970 |
https://mathoverflow.net/questions/109918 | 2 | I like Shakespeare and Greek tragedy, so let me word it as I'm doing: I desperately *need* J.H.B. Kemperman's 1956 paper *On complexes in a semigroup*, but the online archive of [*Indagationes Mathematicae*](http://www.sciencedirect.com/science/journal/00193577), where it was originally published (Vol. 18, pp. 247-254)... | https://mathoverflow.net/users/16537 | Looking for a paper of Kemperman on semigroups | For all those interested, a scan of Kemperman's paper can be found [here](http://www.math.haifa.ac.il/~seva/MathOverflow/OnComplexes.pdf).
| 3 | https://mathoverflow.net/users/9924 | 161865 | 84,973 |
https://mathoverflow.net/questions/161869 | -1 | What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)?
What is the probability that a randomly chosen number from the set of computable numbers is period(number)?
For definition of period(number),please see www.maths.ed.ac.uk/~aar/papers/kontzagi.pdf.
Further more,if... | https://mathoverflow.net/users/14024 | What is the probability that a randomly chosen number from set of c.e.number is period(number)? | [This paper](http://arxiv.org/pdf/0805.0349.pdf) by Yoshinaga shows that periods are contained within a proper subclass of the computable reals. Another proof is given in [this paper](http://wwwmath.uni-muenster.de/mjm/vol_3/mjm_vol_3_04.pdf) by Tent and Ziegler. (Disclaimer: I haven't read either paper myself.) Whethe... | 3 | https://mathoverflow.net/users/47312 | 161874 | 84,976 |
https://mathoverflow.net/questions/161820 | 39 | The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where Stands Functoriality Today? I am familiar with the following:
* Lafforgue proved the Langlands correspondence for $GL... | https://mathoverflow.net/users/48554 | Current Status on Langlands Program | As said in comments, the question has already been discussed on MO, see the links given there.
To summarize and complete, it is important to remember that in the case of number fields the correspondence between Galois representations and certain automorphic representations (the ones which are algebraic at infinity in... | 21 | https://mathoverflow.net/users/9317 | 161880 | 84,978 |
https://mathoverflow.net/questions/161884 | 13 | For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is *rational* if its homotopy groups are rational vector spaces (equivalently, the integral homology groups are rational vector spaces). The *rationalization* of a space X is a rational space Y... | https://mathoverflow.net/users/184 | The cohomology plus what characterizes the rational homotopy type? | In [this paper](http://arxiv.org/abs/0811.1655v1) by Tornike Kadeishvili, it is shown that the rational cohomology of a simply connected space carries the structure of a $C\_\infty$-algebra, and that the isomorphism type of this object determines the rational homotopy type of the space.
| 13 | https://mathoverflow.net/users/8103 | 161886 | 84,981 |
https://mathoverflow.net/questions/161847 | 6 | Let $F$ be the class of locally Lipschitz continuous functions $z=f(x,y)$, from $\mathbb R \times\mathbb R \to\mathbb R,$ such that the euclidean norm $|\ \mathrm{grad}\ f (x,y)\ |$ of its gradient vector is some function, call it $g,$ of its value, i.e. $|\ grad\ f (x,y)\ | = g(\ f(x, y)\ )\ a.e..$
Let $T$ be the subc... | https://mathoverflow.net/users/48911 | Are there nontrivial real functions of 2 real variables with gradient having constant euclidian norm on each level line? | Functions in $F$ and $T$ are, at least around a level set where the gradient is nontrivial, the same up to "reparametrizations preserving level sets." To see this, assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is in $F$, that $0$ is a nontrivial level set of $f$, and that $g > 0$ in a neighborhood of $0$. If we ... | 2 | https://mathoverflow.net/users/16659 | 161887 | 84,982 |
https://mathoverflow.net/questions/161888 | 0 | I'm trying to understand the derivation of a ML-estimator and more specifically the rewriting of the covariance matrix $\Sigma$. In this rewriting, a lemma is used to show that:
$$
\tag{1} \Omega=\sigma^2\_{c}\boldsymbol{1}\boldsymbol{1'} + \sigma^2\_{\varepsilon}I\_T=(\sigma^2\_{\varepsilon}+T\sigma^2\_{c})\boldsymbol... | https://mathoverflow.net/users/48929 | Understanding the derivation of a ML-estimator (statistics) | You can simply multiply $\Omega^{-1}$ as given by the author times $\Omega$ and see that the product is $I$. This confirms that the formula for $\Omega^{-1}$ is correct.
The key here is that
$(I-1(1'1)^{-1}1')(1(1'1)^{-1}1')=1(1'1)^{-1}1'-1(1'1)^{-1}1'=0$
so the cross terms vanish.
| 1 | https://mathoverflow.net/users/9022 | 161890 | 84,984 |
https://mathoverflow.net/questions/161894 | 8 | Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega\_X \otimes \operatorname{End} E)$. Suppose the topological Chern classes of $E$ vanish rationally. Evaluating $At(E)$ on an invariant polynomial gives a class in $\bigop... | https://mathoverflow.net/users/4622 | Atiyah classes of holomorphic vector bundles with trivial Chern classes | No. A counter-example : the vector bundle $\mathcal{O}\_{\mathbb{P}^1}(p)\oplus \mathcal{O}\_{\mathbb{P}^1}(-p)$ on $\mathbb{P}^1$ has zero Chern class, but does not admit a holomorphic connection if $p>0$ (by a theorem of Weil, a vector bundle on a curve admits a holomorphic connection iff its indecomposable summands ... | 8 | https://mathoverflow.net/users/40297 | 161895 | 84,985 |
https://mathoverflow.net/questions/161896 | 6 | A representation $\rho:G\to GL\_k(\mathbb{F})$ is called fixed-point free if for every $1\neq g\in G$ and every $0\neq v\in \mathbb{F}^k$, $\rho(g)v\neq v$. Stated differently, it is a representation where $I-\rho(g)\in GL\_k(\mathbb{F})\cup \{0\}$ for every $g\in G$. A group admiting such a representation is called a ... | https://mathoverflow.net/users/30491 | In which fixed-point free representations is the sum of every 3 elements invertible? | NEW EDIT: I MISREAD THE ORIGINAL QUESTION
If you intend to think about finite $G,$ then $|G|$ and the characteristic of $k$ must be relatively prime (for if the characteristic is $p,$ then an element of order $p$ would not act fixed point freely). Hence the representation is completely reducible. We might as well sup... | 9 | https://mathoverflow.net/users/14450 | 161901 | 84,987 |
https://mathoverflow.net/questions/161893 | 0 | Suppose $E\_1$ and $E\_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E\_1 \longrightarrow E\_2$. If $E\_1$ has no $\mathbb{Q}$-rational point of order 3, then what is Ker $\varphi$ $?$ Is it $\mu\_3$ or $\mathbb{Z}/3\mathbb{Z}$ where $\mu\_3$ denotes the 3rd roots of unit... | https://mathoverflow.net/users/33900 | Kernel of a 3-isogeny between two elliptic curves | The simplest example is $y^2=x^3+D$, which has a 3-isogeny defined over $\mathbb{Q}$ whose kernel is $\{O,(0,\sqrt{D}),(0,-\sqrt{D})\}$. As abx pointed out, as a Galois module this kernel is defined by the quadratic extension $\mathbb{Q}(\sqrt{D})$, which can be any quadratic extension. In general, if the kernel is $\{... | 7 | https://mathoverflow.net/users/11926 | 161902 | 84,988 |
https://mathoverflow.net/questions/161844 | 3 | Roughly speaking,Kolmogorov Complexity of a bits string or a description is the minimal length of programs outputing a bits string,and height of rational number is logarithm of the largest numerator or denominator of the rational.What is the relation between them? It seems that they are the same.
And more generally,w... | https://mathoverflow.net/users/14024 | What is the relation between KC and height of rational number? | After thinking about it for a bit, the relationship is as follows for a rational $q$,
$$K(\text{height}\,q) \leq^+ K(q) \leq^+ 2\,\text{height}\,q.$$
This isn't really a surprising or useful relationship, but I spell out the details nonetheless.
Upper Bound
-----------
Recall that the Kolmogorov complexity of a non... | 3 | https://mathoverflow.net/users/12978 | 161903 | 84,989 |
https://mathoverflow.net/questions/161904 | 4 | Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
>
> Is $E$ a trivial analytic vector bundle?
>
>
>
I need to the answer to this question for the following question:
[Analytic version of the Cartan lemma](https://mathoverfl... | https://mathoverflow.net/users/36688 | analytic vector bundles | Here's just a sketch, maybe someone else can fill in the details. Most of the steps will require the use of the nontrivial fact that $C^\omega(M)$ is dense in $C^\infty(M)$. Let's assume $M$ is compact.
Step 1: Construct a real-analytic embedding of vector bundles $E\to M\times\mathbb R^N$ for some finite $N$. This g... | 2 | https://mathoverflow.net/users/35353 | 161906 | 84,991 |
https://mathoverflow.net/questions/154093 | 3 | Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla\_Xs,s')+h(s,\nabla\_Xs')$$.
we know that we can write locally on open s... | https://mathoverflow.net/users/nan | Explicit form for hermitian structure $h$ with respect to $\omega$ | What you write in your question is not really correct. In fact, you need a lot more structure. If you are on a complex manifold and you have a Hermitian metric, then there exist connections satisfying your "metric invariance" condition, but in fact there are quite a few of those. If your manifold is complex, and your l... | 5 | https://mathoverflow.net/users/48933 | 161908 | 84,993 |
https://mathoverflow.net/questions/159013 | 7 | By the work of Gross and Bernard-Matessi, in dimension 3 $T$-duality should be understood as an exchange of positive and negative local model of Lagrangian torus fibrations, at least in its topological sense. Since in dimension 2, all local models of a Lagrangian torus fibration are generic, so we should expect that $X... | https://mathoverflow.net/users/43423 | Why non-compact Calabi-Yau surfaces are not self-mirror? | I agree with the above answer. In a bit more detail, let E be a smooth conic in $\mathbb{C}^2$, of the form $xy=1$. Then the mirror given by Auroux's construction states that the mirror to $\mathbb{C}^2\setminus E$ is the complement of some divisor $D$ in $\widehat{\mathbb{C}^2}$. Well, if we say that the point is $(0,... | 3 | https://mathoverflow.net/users/6986 | 161923 | 84,996 |
https://mathoverflow.net/questions/161836 | 22 | Given the binomial function $\binom{n}{k}$.
**1.** Define the following sequences,
$$\begin{aligned}
u\_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\
u\_2(k) &= \tbinom{2k}{k}\sum\_{j=0}^k (-3)^{k-3j} \tbinom{2j}{j}\tbinom{3j}{j}\tbinom{k}{3j} = 1, -6, 54, -420, 630,\dots\\
u\... | https://mathoverflow.net/users/12905 | Ramanujan's pi formulas with a twist | A general methods of construction of Ramanujan-type identities are outlined in <http://arxiv.org/abs/1211.6563> (Some conjectured formulas for 1/Pi coming from polytopes, K3-surfaces and Moonshine, by Gert Almkvist), <http://arxiv.org/abs/0712.1332> (Ramanujan-type formulae for 1/π: A second wind?, by Wadim Zudilin) an... | 10 | https://mathoverflow.net/users/32389 | 161927 | 84,998 |
https://mathoverflow.net/questions/161892 | 3 | I found two different definitions of the Moebius Ladder Graph, whose essential difference is, whether the smallest one shall be $K\_4$ or $K\_{3,3}$.
according to Wikipedia (<http://en.wikipedia.org/wiki/M%C3%B6bius_ladder>),
"*In graph theory, the Möbius ladder Mn is a cubic circulant graph with an even number n... | https://mathoverflow.net/users/31310 | Definition of the Moebius Ladder Graph | As far as I can see, the *intent* of both definitions is to give a description of the Moebius ladder as the circulant on $2n$ vertices with connection set $\{\pm 1, n\}$ which clearly includes $K\_4$.
The "verbose" definition, as a prism with a twist etc. just seems to be wrong, basically because the first half of th... | 6 | https://mathoverflow.net/users/1492 | 161943 | 85,002 |
https://mathoverflow.net/questions/156311 | 6 |
>
> All the [subfactors](http://en.wikipedia.org/wiki/Subfactor) $(N\subset M)$ are irreducible and finite index inclusions of II$\_1$ factors.
>
>
>
First recall that in [this paper](http://projecteuclid.org/euclid.pjm/1102622455), D. Bisch characterizes the Jones projections $e\_K$ of the intermediate subfacto... | https://mathoverflow.net/users/34538 | Jordan-Hölder theorem for subfactors? | The Jordan-Hölder property is true for large classes of subfactors: see class $\mathcal{C}$ and beyond, above,
but it's false in general, counter-examples are given by $(A\_n \subset S\_{n+1})$, see [this answer](https://mathoverflow.net/questions/159167/a-second-isomorphism-theorem-for-the-inclusions-of-groups/1602... | 1 | https://mathoverflow.net/users/34538 | 161951 | 85,004 |
https://mathoverflow.net/questions/161777 | 1 | My goal to write down an explicit (and simplest) contact metric structure on squashed $S\_\omega^{2n + 1}$ defined as
\begin{equation}
S\_\omega ^{2n + 1} = \left\{ {\left( {{z\_i}} \right) \in \mathbb{C}^{n + 1}\;\;|\sum\limits\_{i = 1}^{n + 1} {\omega \_i^2{{\left| {{z\_i}} \right|}^2} = 1} } \right\}
\end{equation}... | https://mathoverflow.net/users/15884 | contact metric structure on squashed spheres | After skimming through Section 2 and especially Subsection 2.3 of the paper by C. Boyer: <http://arxiv.org/abs/1003.1903> it seems to me that you have to choose a compatible almost complex structure $J$ on your contact manifold $(S^{2n+1}\_{\omega}, \mathcal{D}=\text{ker}\,\kappa)$ and set $$\Phi(X) = JX \text{ for } X... | 0 | https://mathoverflow.net/users/24221 | 161956 | 85,007 |
https://mathoverflow.net/questions/161957 | 1 | Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is,
$$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$
with the norm, $\|f\|\_{\mathcal{F}(L^{1})}=\|\hat{f}\|\_{L^{1}(\mathbb R)}.$
(By uniqueness theorem and using the fact that Fourier... | https://mathoverflow.net/users/33018 | Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$ | (1) The space is obviously complete since you defined the norm in such a way that $\mathcal{F}$ is an isometry $L^1 \cong \mathcal{F}L^1$ (maybe an isometry up to constants depending on your definition of the fourier transform). Consequently the answer to (3) is that the closed sets of $\mathcal{F}L^1$ are exactly the ... | 5 | https://mathoverflow.net/users/3041 | 161965 | 85,010 |
https://mathoverflow.net/questions/161974 | 0 | Assume $f\in L^2[0,1]$ and let $g(x)=\int\_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such integrals and their properties.
| https://mathoverflow.net/users/48438 | Analyticity of Logarithmic Integrals | No, if $f$ is the characteristic function of $[0,\frac12]$, $g$ will not be smooth at $\frac12$ by a simple explicit calculation.
We have $g=f\*h$, where $h(x)=\ln|x|$ is in $L^1$ and has singular support ${0}$. Thus $g$ is $C^\infty$ whereever $f$ is. See e.g. Hörmander, The Analysis of Partial Differential Operato... | 2 | https://mathoverflow.net/users/42207 | 161987 | 85,018 |
https://mathoverflow.net/questions/161024 | 5 | All sets are assumed to be finite subsets of the integers.
The additive energy of two sets $E(A,B)$ is defined as the number of solutions to $a+b=a'+b'$ with $a,a'\in A$ and $b,b'\in B$. The well-known Balog-Szemeredi-Gowers theorem states that there is an absolute constant $C>0$ such that (assuming $\lvert A\rvert=\... | https://mathoverflow.net/users/385 | Balog-Szemeredi-Gowers with dilates of sets | Tomasz Schoen has observed in an email that the answer to my question is no, since there is a function $f(d)\to\infty$ as $d\to\infty$ such that $\lvert A+d\cdot A\rvert\geq f(d)\lvert A\rvert$ for all $A\subset \mathbb{Z}$. In particular, Boris Bukh in 'Sums of dilates' showed that roughly $f(d)=d+1$ is permissible.
... | 2 | https://mathoverflow.net/users/385 | 161997 | 85,022 |
https://mathoverflow.net/questions/161977 | 1 | To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss discovered the constructability of the regular Heptadecagon (<http://en.wikipedia.org/wiki/Heptadecagonhttp://en.wikipedia.org... | https://mathoverflow.net/users/31310 | First Parameterized Subset of Primes that was Related to a Mathematical Result | An earlier example than the Fermat primes is the class of primes
of the form $2^n-1$, the so-called Mersenne primes. These occur in
Euclid's theorem that $2^{n-1}(2^n-1)$ is perfect when $2^n-1$ is
prime, and in the complementary theorem of Euler that these are all
the even perfect numbers.
| 4 | https://mathoverflow.net/users/1587 | 162000 | 85,023 |
https://mathoverflow.net/questions/162035 | 4 | A $2$- form $\alpha$ on a n- manifold $M$ is called anti symplectic if for every $x\in M$, $\{ v\in T\_{x} M \mid i\_{v} \alpha=0 \}$ is a $n-2$ dimensional subspace of $T\_{x}M$. So we obtain a $n-2$ dimensional subbundle of $T M$. We denote it by $\tilde{\ker} \;\alpha$
*The first question:*
>
> Under what cond... | https://mathoverflow.net/users/36688 | Anti_symplectic 2-forms | For your first question: The condition for integrability of $\mathrm{ker}\alpha$ when $\alpha$ is decomposable is that $\mathrm{d}\alpha = \lambda\wedge \alpha$ for some $1$-form $\lambda$.
For your second question: The answer is 'no' in general. If this can be done, then the tangent bundle of $M$ can be written as t... | 10 | https://mathoverflow.net/users/13972 | 162038 | 85,035 |
https://mathoverflow.net/questions/162009 | 5 | Let $X$ be a topological linear space, and let $X^\*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^\*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be a continuous linear map.
Let $\mathbb P$ be a probability measure on $(X, \mathcal B(X))$. The covariance inner pro... | https://mathoverflow.net/users/238 | Measures which exhibit the "uncorrelated implies independent" property | Associated variables satisfy the UII property – see e.g. Newman, Charles M. "Asymptotic independence and limit theorems for positively and negatively dependent random variables." Lecture Notes-Monograph Series (1984): 129. Available [here](http://books.google.pl/books?id=neFsjyQrCEwC&pg=PA129&lpg=PA129&dq=%22%20uncorre... | 3 | https://mathoverflow.net/users/34050 | 162045 | 85,037 |
https://mathoverflow.net/questions/162031 | 2 | Let $X = (X\_1, X\_2, \ldots , X\_n)$ be an $n$-dimensional random variable, where each $X\_i$ is a random variable on finite discrete set $S$. In addition, $X\_i$ are independent of each other (but not identically distributed). That is, if we denote $X \sim \mathcal{D}$ and $X\_i \sim \mathcal{D}\_i$, then $\mathcal{D... | https://mathoverflow.net/users/46938 | Empirical estimator for total variation distance between two product distributions | I have something that may or may not be useful...
Diaconis notes an interpretation of variation distance of Paul Switzer. Consider $\mu$, $\nu\in M\_p(S)$. Given a single observation of $S$, sampled from $\mu$ or $\nu$ with probability $1/2$, guess whether the observation, $o$, was sampled from $\mu$ or $\nu$. The cl... | 0 | https://mathoverflow.net/users/35482 | 162048 | 85,039 |
https://mathoverflow.net/questions/162049 | 3 | Assume two commuting $n\times n$ complex matrices $A$ and $B$ are given. Then it is in general false that if $C$ is a square root of $A$, i.e., if $C^2=A$, then $C$ commutes with $B$ (the simplest counterexample is provided by $A$ the 2-by-two identity matrix and $C=diag(1,-1)$.
Yet, when $A$ is invertible, by using ... | https://mathoverflow.net/users/8320 | Square root of a complex matrix commuting with a given one | As in Geoff Robinson's answer, let us show that there exists a square root of $A$ which is a polynomial in $A$; then surely it commutes with $B$.
Let $\mu(x)$ be the minimal annihilating polynomial for $A$, $\mu(x)=\prod\_i(x-\lambda\_i)^{\alpha\_i}$. We need to find a polynomial $P(x)$ such that $P(x)^2\equiv x\pmod... | 5 | https://mathoverflow.net/users/17581 | 162063 | 85,044 |
https://mathoverflow.net/questions/161961 | 0 | Could someone please give me a reference for the eigenvalues and eigenstates of operators related to the angular momentum of a spinless, non-relativistic 2-D quantum particle?
In particular, I'm interested to know the eigenvalues of the operator that takes the angular momentum eigenstates
$\varphi\_{q,k}(\mathbf{x}... | https://mathoverflow.net/users/33804 | Spectrum of an angular-momentum related operator | Maybe these references will be helpful:
<http://scitation.aip.org/content/aip/journal/jmp/28/2/10.1063/1.527672> (Coherent angular momentum states for the two‐dimensional oscillator, by A.J. Bracken,
D.S. McAnally and O.A. Odundun)
<http://arxiv.org/abs/1401.4985> (Lost and found: the radial quantum number of Lag... | 2 | https://mathoverflow.net/users/32389 | 162070 | 85,047 |
https://mathoverflow.net/questions/162024 | 3 | The following is quoted from the Mathematical Reviews.
MR0544896 (80j:12002) Reviewed
Bhaskaran, M.
Construction of genus field and some applications.
J. Number Theory 11 (1979), no. 4, 488–497.
12A35 (12A65)
>
> Let $k$ be a finite algebraic number field and $K$ its Hilbert class field, i.e., $K/k$ is ma... | https://mathoverflow.net/users/48985 | A question on the genus field of an algebraic number field | The setup ensures that the compositum $k(\sqrt{i})$ will be (isomorphic to) $k\otimes\_{\mathbb{Q}}\mathbb{Q}(\sqrt{i})$.
When $g$ is $2$-adically close to $x^4+1$ -- it turns out that mod $8$ (i.e., $m=3$) is good enough -- then the field $F=\mathbb{Q}\_2(\sqrt{i}) = \mathbb{Q}\_2(i,\sqrt{2},\sqrt{-2})$ contains fou... | 4 | https://mathoverflow.net/users/49003 | 162078 | 85,048 |
https://mathoverflow.net/questions/162007 | 5 | This question comes after the comments in the recent related question [Sigma-complete Lindenbaum algebras?](https://mathoverflow.net/q/161978/12976), but in its current form is sufficiently different in my opinion, and so I decided to follow Joel's suggestion. It is also something I had asked myself before but never se... | https://mathoverflow.net/users/12976 | Is every Heyting algebra the Lindenbaum algebra of an intuitionistic first order theory? | I'm not sure why you seem to disallow propositional theories. Here is a workaround... Given a Heyting algebra $H$, add a unary predicate $P\_a(x)$ for every $a \in H$ along with the axioms $$\forall x (P\_{a \land b}(x) \leftrightarrow (P\_a(x) \land P\_b(x)))$$ $$\forall x (P\_{a \lor b}(x) \leftrightarrow (P\_a(x) \l... | 5 | https://mathoverflow.net/users/2000 | 162079 | 85,049 |
https://mathoverflow.net/questions/161947 | 8 | Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and others), to my knowledge, we're still unable to prove rigorously that there exists $N>0$ such that for all integer $n>N$, $2n... | https://mathoverflow.net/users/13625 | What keeps asymptotic Goldbach's conjecture out of reach of current technology? | As far as I know there are two approaches to Goldbach type problems, the circle method and sieve methods. In the sequel I will restrict myself to the circle method, hoping that someone else writes something about sieves.
Define the exponential sum $S(\alpha)=\sum\_{n\leq N}\Lambda(n) e(n\alpha)$, where $e(x)=e^{2\pi ... | 16 | https://mathoverflow.net/users/37555 | 162085 | 85,051 |
https://mathoverflow.net/questions/162027 | 10 | I'm trying to reconstruct the proof of Godel's first theorem (Rosser's strong version) from the uncomputability of the Halting function. If we just started with the language $\mathcal{L}=\{0, S, +, \cdot\}$ and we took Robinson arithmetic $Q$, we could show $Q$ is incomplete in the following way: Let $U$ be a Turing Ma... | https://mathoverflow.net/users/47709 | What is the precise notion of "enough arithmetic" in Godel's first Incompleteness theorem? | The usual answer to this question is indeed, that you can interpret addition and multiplication which satisfy the axioms of Robinson Arithmetic.
Almost all those axioms are used if you want to show the "Rosser form" of the theorem: in particular you need to be able to show:
\begin{equation} x\leq \overline{n}\ \leftr... | 2 | https://mathoverflow.net/users/36103 | 162089 | 85,053 |
https://mathoverflow.net/questions/161966 | 2 | I want to know under what conditions does the Mittag-Leffler function ${E\_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where
${E\_{\alpha ,1}}(z) = \sum\limits\_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (\alpha k + 1)}}}$.
${E\_{1,1}}(z) = \sum\limits\_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (k + 1)}}} {\text{ ... | https://mathoverflow.net/users/48365 | Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero? | None of these functions have real zeroes, because they can be written as moment generating
functions of certain random variables $Y\_\alpha$. More precisely, for $0<\alpha <1$
$$E\_{\alpha,1}(z)=\mathbb{E} e^{zY\_\alpha}$$
where $$X\_\alpha:=(1/Y\_{\alpha})^{1/\alpha}$$ has the extreme positive stable distribution
with... | 5 | https://mathoverflow.net/users/48831 | 162090 | 85,054 |
https://mathoverflow.net/questions/162103 | 2 | Suppose $F,G$ are adjoint, and $\epsilon:F\circ G\rightarrow Id$ is the counit. Is it always true that$$
Id\_{FG}\epsilon=\epsilon Id\_{FG}
$$
as maps from $FGFG$ to $FG$?
It's true if you precompose with $F\eta G$, where $\eta:Id\rightarrow GF$ is the unit, then the result in both cases is $Id\_{FG}$, but since $\et... | https://mathoverflow.net/users/4477 | Simple technical adjunction question | No, it is not always true. For instance, suppose $G$ is the forgetful functor from monoids to sets. Then $F$ is the free monoid functor, and the two maps in question are
$$((x\_{1,1}, \ldots, x\_{1,n\_1}), \ldots, (x\_{m,1}, \ldots, x\_{m,n\_m})) \mapsto (x\_{1,1} \cdots x\_{1,n\_1}, \ldots, x\_{m\_1} \cdots x\_{m,n\_m... | 6 | https://mathoverflow.net/users/11640 | 162105 | 85,061 |
https://mathoverflow.net/questions/162061 | 3 | There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a cont... | https://mathoverflow.net/users/48222 | Existence for ODE in Banach space (accretive operators and Crandall-Liggett) | I do not think it is good to ask three questions in one, since I will only address the first now.
I would put the question back and ask you: how do you define a "solution"? Usually you have an existence proof of some object and if they are good enough, call them generalized solutions. If everything is perfect, you c... | 3 | https://mathoverflow.net/users/12898 | 162106 | 85,062 |
https://mathoverflow.net/questions/161933 | 7 | As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the property that their completion by Dedekind-cuts $\mathcal{C}(L)$ is equipotent with their powerset $\mathcal{P}(L)$?
**Edit... | https://mathoverflow.net/users/26080 | Which linearly ordered sets have the property that their completion is equipotent with their powerset? | This is a complement to Joel's answer. For uncountable orders, I don't think there will be a nice characterization in terms of embedding a canonical suborder, at least without some assumption about cardinal arithmetic, even if one only considers orders that are locally homogeneous. I'll argue that it's consistent that ... | 6 | https://mathoverflow.net/users/33768 | 162110 | 85,063 |
https://mathoverflow.net/questions/161838 | 4 | I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of view, at least). Given an (edit: equivalence invariant) algebraic structure on a groupoid $G$, say given by some 2-mon... | https://mathoverflow.net/users/4177 | Transporting algebraic structure along adjoint equivalences | Here is another answer which is perhaps more "in the spirit" of the question.
Suppose that $U: \mathcal{A} \to \mathcal{C}$ is a functor of 2-categories. Let $x$ and $y$ be objects of $\mathcal{C}$, and $f: x \to y$ a fixed morphism. Suppose that we have a lift $Y \in \mathcal{A}$ of $y \in \mathcal{C}$, that is an ... | 2 | https://mathoverflow.net/users/184 | 162113 | 85,065 |
https://mathoverflow.net/questions/162056 | 1 | In their [proof](http://arxiv.org/abs/1306.3969) of the celebrated [Kadison-Singer conjecture](http://www.siam.org/pdf/news/2123.pdf), Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on [Ramanujan graphs](http://arxiv.org/abs/1304.4132). And I have a ... | https://mathoverflow.net/users/11361 | k-th largest root in common interlacing polynomials | 1. You are correct -- we mean the the $l^{th}$ largest root where $1 \leq l \leq n$
2. The "eventually positive" condition is used to make sure all of the polynomials are oriented the same way. For a given $l$, let $r^i\_l$ be the $l^{th}$ root of $f\_i$ and assume $r^1\_l \leq \dots \leq r^k\_l$. The thing you need to... | 1 | https://mathoverflow.net/users/49021 | 162115 | 85,067 |
https://mathoverflow.net/questions/162116 | 8 | I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the [permanent](http://en.wikipedia.org/wiki/Permanent).
I would thus like to collect in this thread a list of problems specifically in real analysis and linear algebra which are proven to... | https://mathoverflow.net/users/48342 | NP-hard problems in linear algebra and real analysis | Here is one such result.
>
> Stephen A. Vavasis.
> "On the complexity of nonnegative matrix factorization."
> 2007. ([arXiv abstract link](http://arxiv.org/abs/0708.4149))
>
>
> "In this report, we define an exact version of NMF [nonnegative matrix factorization]. Then we establish several results about exact N... | 6 | https://mathoverflow.net/users/6094 | 162128 | 85,072 |
https://mathoverflow.net/questions/140134 | 4 | Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have been was Behrling's conjecture (solved). Is there is any thing else that might be of geometric significance.
For p=2, it ... | https://mathoverflow.net/users/30081 | Iwaniec's conjecture | Iwaniec conjecture is closely connected to the important Morrey's conjecture on the relationship between rank one convexity and quasiconvexity.
You will find a nice discussion in the Section 5 of the following survey by Banuelos
<http://arxiv.org/pdf/1012.4850v2.pdf>
| 4 | https://mathoverflow.net/users/48356 | 162132 | 85,075 |
https://mathoverflow.net/questions/162108 | 2 | Let's call an infinite sequence of bits $f:N\rightarrow \{0,1\}$ *absolutely random* if any computably constructed subsequence is not computable, i.e. there aren't monotonic computable function $g:N \rightarrow N$ and computable function $h:N \rightarrow \{0,1\}$ such that $\forall n ~f(g(n)) = h(n)$.
Is it known def... | https://mathoverflow.net/users/8381 | Absolutely algorithmically random infinite sequence | As Liang Yu pointed out, this notion is known as bi-immunity. Chaitin's constant is in fact bi-immune, as is any Martin-Löf random real. Indeed, bi-immunity is much weaker than Martin-Löf randomness. Again as pointed out by Liang Yu, quite weak notions of randomness and (computability theoretic) genericity suffice to e... | 6 | https://mathoverflow.net/users/47312 | 162134 | 85,076 |
https://mathoverflow.net/questions/162126 | 3 | Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?
| https://mathoverflow.net/users/34063 | Compactly supported cohomology of homotopy equivalent manifolds | Let $M$ be a punctured torus and $N$ be a twice-punctured plane. Then $M$ and $N$ are homotopy equivalent, but their one-point compactifications are not (the first being a torus and the second having the homotopy type of $S^2\vee S^1\vee S^1$). In particular, $H\_c^\*(M)$ has a nontrivial cup product but $H^\*\_c(N)$ d... | 8 | https://mathoverflow.net/users/75 | 162138 | 85,077 |
https://mathoverflow.net/questions/162137 | 5 | This is a question about two reducibility notions in computability theory. I suspect the answer is a fairly simple construction, and I'm just not seeing it.
For sets $X, Y\subseteq\omega$, we say $X$ is *Turing reducible to* $Y$ if there is a Turing machine $\Phi$ which, when given $Y$ as an oracle, yields (the chara... | https://mathoverflow.net/users/8133 | Interaction between Turing and many-one reducibility | No. A set is $m$-reducible to the Halting Problem iff it is computably enumerable, so let $Y$ be the Halting Problem and let $X$ be any $\Delta^0\_2$ set not of c.e. Turing degree.
| 12 | https://mathoverflow.net/users/47312 | 162140 | 85,078 |
https://mathoverflow.net/questions/162033 | 2 | Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form
$S(z;p)=\sum\_{k=1}^{\infty}(-1)^{k+1} (H\_k)^p z^k$
where $H\_k = \sum\_{n=1}^{k} 1/n$ are the harmonic numbers and $p \in \mathbb{N}.$
For $p=1$, $S(z;1) = \frac{\log(1+z)}{1+z}$ and at $z=1$, $... | https://mathoverflow.net/users/48990 | Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values | If we multiply $S(z;p)$ by $1+z$, we get
\begin{align\*}
(1+z)S(z;p)&=\sum\_{k\geq 1}(-1)^{k+1}\left(H\_k^p-H\_{k-1}^p\right)z^k\\
&=\sum\_{k\geq 1}(-1)^{k+1}\sum\_{n=0}^{p-1}{p\choose n}H\_{k-1}^n\frac{1}{k^{p-n}}z^k\\
&=\sum\_{n=0}^{p-1}{p\choose n}\sum\_{k\geq 1}\frac{(-1)^{k+1}H\_{k-1}^n}{k^{p-n}}z^k.
\end{align\*}... | 1 | https://mathoverflow.net/users/5263 | 162143 | 85,080 |
https://mathoverflow.net/questions/161722 | 1 | Can any one show that the following limit?
$$
\lim\_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum\_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1.
$$
If one uses the estimate based on my [previous post](https://mathoverflow.net/questions/161068/has-anyone-seen-this-series), then an upper ... | https://mathoverflow.net/users/36814 | Looking for a limit related to the series in a previous post | Assume below that $z$ is positive. Note that, using $\sqrt{z}-\sqrt{k} = (z-k)/(\sqrt{z}+\sqrt{k})$,
$$
\Big| \frac{e^z}{\sqrt{z}} - \sum\_{k=1}^{\infty} \frac{z^k}{k!\sqrt{k}} \Big|
\le \frac{1}{\sqrt{z}} + \Big|\sum\_{k=1}^{\infty} \frac{z^k}{k!} \frac{\sqrt{z}-\sqrt{k}}{\sqrt{kz}} \Big| \le \frac{1}{\sqrt{z}}+ \s... | 1 | https://mathoverflow.net/users/38624 | 162158 | 85,084 |
https://mathoverflow.net/questions/162165 | 2 | A lot has been written about the arithmetic of ordinal numbers. However, we can **also** do arithmetic with linearly ordered sets.
>
> **Question.** Is there an article or book where I can learn the basics of linearly ordered set arithmetic?
>
>
>
Here's the definitions I have in mind. Let $I$ denote a totally... | https://mathoverflow.net/users/26080 | Linearly ordered set arithmetic: reference request | Try
Rosenstein: Linear orderings
<http://books.google.com/books/about/Linear_orderings.html?id=y3YpdW-sbFsC>
| 5 | https://mathoverflow.net/users/4600 | 162166 | 85,088 |
https://mathoverflow.net/questions/162148 | 0 | Suppose $X$ is a surface, are there examples for curves on X which not 1-connected, but $h^0(C,O\_C)=1$? (Here a curve is an effective divisor, not necessarily reduced or irreducible) (1-connectedness means when it splits into two effective divisors, the two parts have intesection number greater than or equal to 1)
A... | https://mathoverflow.net/users/nan | Examples for curve not 1-connected but $h^0(C, O_C)=1$ | Here is an example with $X$ smooth : consider a regular surface $X$ with an elliptic fibration having a double fiber $2E$ (typically, an Enriques surface). Then $2E=E+E$ is not 1-connected ($E^2=0$), but one checks easily that $H^1(X,\mathcal{O}\_X(-2E))=0$, hence $h^0(\mathcal{O}\_{2E})=1$.
| 2 | https://mathoverflow.net/users/40297 | 162167 | 85,089 |
https://mathoverflow.net/questions/162087 | 4 | Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c\_1 \le \rho(A) \le 1+c\_2$ $(0<c\_1 \le c\_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such that $\|Av\| \ge \|v\|$.
I am interested in the case where $v$ is drawn uniformly at random from some discrete distr... | https://mathoverflow.net/users/46824 | Estimating the probability that $\|Av\| \ge \|v\|$ | You can have very few vectors in the discrete set satisfy the bound. For example, let $A$ be the matrix with zeros everywhere except for $1+c\_1$ in the upper left corner (let's assume $1+c\_1<\sqrt{2}$). Then there are only two vectors in $\{\pm 1\}^n$ which satisfy $\Vert Av\Vert \geq \Vert v\Vert$.
It follows that... | 1 | https://mathoverflow.net/users/327 | 162170 | 85,090 |
https://mathoverflow.net/questions/162169 | 3 | Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated
by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if
$n\neq 1$. Is this some sort of familiar surface? For example, is it normal? Does it have rational singularities? Can one construct an ex... | https://mathoverflow.net/users/3891 | An affine singular surface | Let $\mathbb{F}\_n$ be the Hirzebruch surface of index $n$. You have two open subsets on it isomorphic to $\mathbb{A}^1\times \mathbb{P}^1$, with a glueing map given by $$(t,[u:v])\mapsto (1/t,[ut^n:v])$$
The section $u=0$ has self-intersection $-n$ and the section $v=0$ has self-intersection $n$. We remove the section... | 9 | https://mathoverflow.net/users/23758 | 162174 | 85,092 |
https://mathoverflow.net/questions/162139 | 5 | In [Watson's thesis](http://arxiv.org/pdf/0810.0425v3.pdf%29) (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:
If $\phi\_n$'s are orthonormal Hecke-Maass eigenforms in some arithmetic congruence (compact/nonco... | https://mathoverflow.net/users/36735 | Generalization of Watson's triple product | The reason it works for three (but not other numbers) is the uniqueness of trilinear functionals. Let $G = \mathrm{GL}\_2(F)$, $F$ a local field and let $\pi\_i$ ($1\leq i\leq 3$) be three irreducible admissible representations. Then there is at most one $G$-invariant functional on $\pi\_1 \otimes \pi\_2 \otimes \pi\_3... | 11 | https://mathoverflow.net/users/327 | 162176 | 85,093 |
https://mathoverflow.net/questions/124722 | 3 | Let $X,Y$ be smooth irreducible algebraic varieties over $\mathbb{C}$, let $\pi: X \to Y$ be a morphism which is injective and surjective on closed points of $X,Y$. Then how to show $\pi$ is an isomorphism of $X,Y$ as varieties?
This result is not true if $X,Y$ are not smooth. One can consider the example $\mathbb{P... | https://mathoverflow.net/users/29730 | Set isomorphism of smooth varieties implies isomorphism as varieties | One way to prove this is to show that for each closed point $x\in X$ the differential $T\_\pi(x):T\_xX\rightarrow T\_{\pi(x)}Y$ is surjective, that is $\pi$ is a smooth morphism.
Let $\pi:X\rightarrow Y$ be a morphism or relative dimension $r$ of smooth varieties over an algebraically closed field. Assume that the re... | 1 | https://mathoverflow.net/users/14514 | 162208 | 85,103 |
https://mathoverflow.net/questions/161837 | 9 | My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me get started with the basics? I've seen Lang's book but that's it.
| https://mathoverflow.net/users/42140 | Infinite dimensional Riemannian geometry | Lempert, László The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces. Invent. Math. 142 (2000), no. 3, 579-603.
Lempert, László The Dolbeault complex in infinite dimensions. II. J. Amer. Math. Soc. 12 (1999), no. 3, 775-793.
Lempert, László The Dolbeault complex in infinite dimensio... | 10 | https://mathoverflow.net/users/3377 | 162217 | 85,107 |
https://mathoverflow.net/questions/162030 | 42 | This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation.
The question is in two parts. The first, as stated in the title, asks whether every Noetherian commutative ring a quotient of a Noetherian Domain? Geometrically, this que... | https://mathoverflow.net/users/48273 | Is every Noetherian Commutative Ring a quotient of a Noetherian Domain? | No for cardinality reasons.
Let $F$ a finite field and $G$ a field with cardinality strictly greater than the continuum. Then $F\times G$ is not the homomorphic image of a noetherian integral domain by lemma 2.1 in <http://spot.colorado.edu/~kearnes/Papers/residue_final.pdf>
Lemma 2.1. Let $R$ be a Noetherian integ... | 37 | https://mathoverflow.net/users/46855 | 162218 | 85,108 |
https://mathoverflow.net/questions/162220 | 1 | I know the following result is well known to the experts, but what are the good references or proofs:
Let $X,Y$ be a smooth variety with projective, birational morphism $f: X \to Y$. Let $L$ be a line bundle on $Y$, then we have the property that the higher direct image $R^if\_\*f^\* L = 0$ when $i > 0$.
This is eq... | https://mathoverflow.net/users/29730 | Nonsingular is rational singular | I think there are two issues here:
1) The "projection formula" tells you that $R^if\_\*(f^\*L)$ is isomorphic to $R^if\_\*\mathcal{O}\_X\otimes L$: see EGA chap. 0 , Prop. 12.2.3 (in EGA III.1).
2) Then the vanishing of $R^if\_\*\mathcal{O}\_X$ for $i>0$. This is one of the many consequences of Hironaka's resolutio... | 6 | https://mathoverflow.net/users/40297 | 162223 | 85,112 |
https://mathoverflow.net/questions/161578 | 1 | Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$.
In pages 10-11 of [this paper](http://www.ma.utexas.edu/mp_arc/c/09/09-101.pdf), the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the problem
$$\Delta\_{\overline g}(u) = 0\quad\text{on $M\times (0,\infty)$... | https://mathoverflow.net/users/37239 | Fractional Laplacian on compact hypersurface/manifold via harmonic extension? | The relation between the fractional Laplacian and the Dirichlet-to-Neumann map was considered by [Caffarelli and Silvestre](http://www.ams.org/mathscinet-getitem?mr=2354493) for the $\mathbb{R}^n$ case. This notion is crucial in the study of [nonlocal minimal surfaces](http://www.ams.org/mathscinet-getitem?mr=2675483).... | 2 | https://mathoverflow.net/users/3948 | 162224 | 85,113 |
https://mathoverflow.net/questions/161858 | 3 | I am not a professional mathematician so please excuse me if my question is not phrased correctly.
I am interested in the following simple sounding problem.
Consider a random $n$ by $n$ $0$-$1$ matrix $M$ where $M\_{i,j} = 1$ with probability $1/2$ and $0$ otherwise. Now choose a single random $0$-$1$ vector $v$ wh... | https://mathoverflow.net/users/45564 | Does $Mv$ converge to i.i.d in some sense? | If I understood well, you are asking this:
What is the limit distribution as $n$ tends to infinity of $u=Mv$ for $M$ a uniform random $0-1$ matrix of size $n\times n$ and $v$ a uniform random vector of size $n\times 1$ ?
If you fix a coordinate $i$ and you ask for the distribution of $u^n\_i$ is easy. $u^n\_i\dote... | 1 | https://mathoverflow.net/users/39115 | 162231 | 85,116 |
https://mathoverflow.net/questions/161948 | 8 | Denote $\operatorname{dim}(M\_k(\Gamma\_0(N)))$ by $m(k,N)$ and
$\operatorname{dim}(S\_k(\Gamma\_0(N)))$ by $s(k,N)$.
Let $N$ any positive multiple of $4$ and $j \ge 1$.
$$
a(N) := \frac1j \left(m \left(\frac{4j+1}{2},N\right)
+s\left(\frac{4j+3}{2}, N\right) \right)
$$
For $N = 4,8,12, \ldots$ this gives the first... | https://mathoverflow.net/users/48960 | A conjecture related to the Cohen-Oesterlé dimension formula of spaces of modular forms for half-integer weights | This is easy to determine by the known closed formulæ for $\dim \mathcal{S}\_k\left(\Gamma\_0(N)\right)$ and $\dim \mathcal{M}\_k\left(\Gamma\_0(N)\right)$ of Cohen and Oesterlé, namely for $k \in 1/2 + \mathbb{Z}$ with $k > 3/2$ and $N \in \mathbb{N}$,
\begin{align}
\dim \mathcal{S}\_k\left(\Gamma\_0(4N)\right) & = \f... | 6 | https://mathoverflow.net/users/3803 | 162239 | 85,118 |
https://mathoverflow.net/questions/162245 | 3 | From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and right color do not uniquely determine the top color and bottom color.
The tiles involved in practice in material science a... | https://mathoverflow.net/users/40780 | Conjecture of a subset of Wang tile which might be decidable | It is a very nice question!
But I claim, unfortunately, that the tiling problem for your special kind of tile sets
is also undecidable. Specifically, I claim that the ordinary Wang
tiling problem reduces to an instance of your kind of tiling
problem. That is, if we are given a finite list of ordinary Wang
tiles $w\_0... | 5 | https://mathoverflow.net/users/1946 | 162250 | 85,119 |
https://mathoverflow.net/questions/161860 | 2 | Gauduchon showed that every conformal hermitian structure on a compact complex $n$-fold contains an hermitian metric such that the associated 1,1-form $\omega$ satisfies $\partial {\bar{\partial}}\omega^{n-1} = 0$. Such metrics are called *Gauduchon metrics*. Popovici defines a *strongly Gauduchon metric* as a Gauducho... | https://mathoverflow.net/users/30172 | What are the known examples of compact complex $n$-folds that do not have a strongly Gauduchon metric for $n >2$? | Probably the simplest example is the following: $X$ is the Hopf manifold $({\mathbb C}^3\setminus\{0\})/<\gamma>$ where $\gamma(z\_1,z\_2,z\_3)=(2z\_1,2z\_2,2z\_3)$. It has a projection $p:X\to {\mathbb P}^2$, $p(z\_1,z\_2,z\_3)=[z\_1:z\_2:z\_3]$. It is diffeomorphic to $S^5\times S^1$, hence $H^2(X, {\mathbb R})=0$. I... | 1 | https://mathoverflow.net/users/48958 | 162253 | 85,121 |
https://mathoverflow.net/questions/87948 | 6 | Let $\mu\_t, t \geq 0,$ be a family of probability measures on the real line. One can assume whatever one wishes about them, although typically they will be continuous in some topology (usually at least the topology of weak convergence of measures), and they will be absolutely continuous with respect to Lebesgue measur... | https://mathoverflow.net/users/9610 | Markov Processes with Given Marginals | The following result was proved in
Kellerer, H.G. (1972) Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198,
99–122.
Let $p(y, t)$ be a family of marginal densities, with finite first moment, such that
for $s , t$ the density at time $t$ dominates the density at time $s$ in the convex order. Then ... | 4 | https://mathoverflow.net/users/48356 | 162257 | 85,122 |
https://mathoverflow.net/questions/162262 | 1 | Let $p\_1,...,p\_k\in\mathbb{P}^2$ be general points. Let us consider the blow-up $X\_k = Bl\_{p\_1,...,p\_k}\mathbb{P}^2$. It is clear that if $k\leq 3$ then $X\_k$ is toric and hence a Mori Dream Space. The question is:
for which $k$ is $X\_k$ a Mori Dream Space?
| https://mathoverflow.net/users/nan | When is a blow-up of $\mathbb{P}^2$ a Mori Dream Space? | If $S$ is a smooth, projective, rational surface and $-K\_S$ is big, then $S$ is a Mori Dream Space. I will prove the following: if $k\leq 8$ then $X\_k$ is a Mori Dream Space.
The anticanonical of $X\_k$ is
$$-K\_{X\_k} = 3H-E\_1-...-E\_k.$$
Therefore $-mK\_{X\_k} = 3mH-mE\_1-...-mE\_k$. Therefore the anticanonical... | 1 | https://mathoverflow.net/users/14514 | 162263 | 85,124 |
https://mathoverflow.net/questions/162249 | 2 | Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum\_{x\in X} p(x)\log p(x)$.
Given a sample of observations $x\_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x\_i=x\}}{N}$ and thus $H$.
Suppose, however, that $X=P(B)$ the [power set](http://en.wikipedia.org/wiki/Power_set) of set $B$ and t... | https://mathoverflow.net/users/36889 | How to estimate the entropy of a distribution on a power set? | I don't think the power-set structure helps you estimate entropy better.
The following extreme cases all give the same entropy for $p\_B$, but entropy of the power set observation ranges from 0 to $N$ (the full range).
* Extreme case 1, if you only observe $B$, your entropy $H(X)$ is zero, while entropy of $p\_B$ is ... | 2 | https://mathoverflow.net/users/14974 | 162265 | 85,125 |
https://mathoverflow.net/questions/162266 | -1 | Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i\_1, i\_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k\_n := c(\cdot, i\_n)$. Let $U$ be the Hilbert span of these functions; by the reproducing kernel property, $\langle k\_n, k\_{n'} \ran... | https://mathoverflow.net/users/238 | Finiteness of "novel variance" from a kernel on a compact space | You haven't specified what role is played by the topology of $I$, so my default assumption will be that the kernel is continuous, in which case the answer to your question is negative.
For a counterexample, let $I$ be the one-point compactification of $\mathbb{N}$, that is, $\{1,2,\dots,\infty\}$, and take $k\_{nm} :... | 1 | https://mathoverflow.net/users/22758 | 162268 | 85,127 |
https://mathoverflow.net/questions/161958 | 1 | (This question was originally asked <https://math.stackexchange.com/questions/725421/an-isomorphism-of-categories>, with no affirmative answer there.)
Let $C$ be an (finite) [extensive category](http://ncatlab.org/nlab/show/extensive+category) with terminal object $1$. Let $I$ be an index category. Let $j: \mathrm{Fi... | https://mathoverflow.net/users/7341 | An isomorphism of categories | I think I have finally understood your question.
$\require{AMScd}
\newcommand{\mor}[3]{#1 \colon #2 \rightarrow #3}%
\newcommand{\catl}[1]{\mathbb{#1}}%
\newcommand{\catw}[1] {\mathbf{#1}}%$
Since $\catw{FinSet}$ is a free finite coproduct cocompletion of the terminal category $1$, every functor $\mor{A}{1}{\catl{C}}... | 4 | https://mathoverflow.net/users/13480 | 162273 | 85,130 |
https://mathoverflow.net/questions/160773 | 2 | **Background**
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
1. $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
2. for every compact $K \subseteq {\mathbb R}^n$ and every $b > a \geqslant 0$, $$
\int\_a^b \|f(t,\cdot)\|\_K\,dt < \infty... | https://mathoverflow.net/users/31451 | Global Solutions of Ordinary Differential Equations | I think I got it! I turned out to be mistaken about item 2 in my Progress Notes.
**Lemma 1.** Suppose $f \colon {\mathbb R}\_{\geqslant 0} \rightarrow {\mathbb R}\_{\geqslant 0}$ is locally integrable, and let $F\colon {\mathbb R}\_{\geqslant 0} \rightarrow {\mathbb R}\_{\geqslant 0}$ be the primitive given by
$$
F(t... | 0 | https://mathoverflow.net/users/31451 | 162276 | 85,133 |
https://mathoverflow.net/questions/162270 | 8 | The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with a functor into $I$, and the essential image is the Grothendieck fibrations over $I$.
My question: if we allow $I$ to ... | https://mathoverflow.net/users/7341 | A question on the Grothendieck construction | The bicategory of elements of a Cat-valued functor is defined in e.g. Street's *Fibrations in bicategories*; it's the same as the usual one, with 2-cells as described [here](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat). Its property of classifying lax transformations is also mentioned on that page ([this page](... | 8 | https://mathoverflow.net/users/4262 | 162277 | 85,134 |
https://mathoverflow.net/questions/162285 | 1 | Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat $R$-modules. Then does $H(C\otimes\_R S)\cong H(C)\otimes\_R S$ whenever $H(C)$ is flat $R$-module. $H(C)$ is homology of chain... | https://mathoverflow.net/users/41650 | Universal coefficient theorem for local ring | It doesn't hold. Take $k$ any field (of characteristic zero if you wish), $R=k[\epsilon]/(\epsilon^2)$ its ring of dual numbers, $S=R/(\epsilon)=k$ its residue field, and $C$ the periodic complex
$$\cdots\rightarrow R\stackrel{\epsilon}\longrightarrow R\rightarrow\cdots.$$
This is a complex of projective (hence flat) $... | 4 | https://mathoverflow.net/users/12166 | 162289 | 85,135 |
https://mathoverflow.net/questions/162236 | 3 | Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||\_X)$ to vector space $Y$ and denote with $\tau\_L$ the final topology on $Y$ induced by $T$.
Is $\tau\_L$ equivalent with the topology given by the strongest norm on $Y$ in which $L$ is bounded?
| https://mathoverflow.net/users/31760 | Final topology of surjective linear map on Banach space | There exists a norm on $Y$ making $L$ bounded if and only if the kernel of $L$ is closed.
However, $\|y\|\_Y = \inf\lbrace \|x\|: L(x)=y\rbrace$ is always a *semi-norm* (that is
$\|y\|\_Y$ may be $0$ for $y\neq 0$) which induces the final topology $\tau\_L=\lbrace A\subseteq Y: L^{-1}(A)$ is open in $X\rbrace$. This ... | 2 | https://mathoverflow.net/users/21051 | 162295 | 85,137 |
https://mathoverflow.net/questions/162308 | 2 | In a search for integers $n$ whose divisors sum to a square (including $n$ itself among the divisors), it can be shown that 3 is the only odd prime whose divisor sum is square, and that no square of an odd prime has a square divisor sum. On the other hand, for odd composite $n$, square-free or not, 115 is the smallest ... | https://mathoverflow.net/users/48851 | Do more than three integers $n$ have divisor sums which are odd squares? | [Searching OEIS](https://oeis.org/search?q=3%2C+22%2C+66%2C+70%2C+81%2C+94%2C+115&sort=&language=&go=Search) for the first few terms
gives [A006532](https://oeis.org/A006532)
From the comments there:
>
> If a and b are in the sequence and relatively prime, then a\*b is also in the sequence. - Franklin T. Adams-Wa... | 7 | https://mathoverflow.net/users/12481 | 162311 | 85,141 |
https://mathoverflow.net/questions/162331 | 2 | For any graph $G$ let $\eta(G)$ be the [Hadwiger number](http://en.wikipedia.org/wiki/Hadwiger_number) of $G$.
Is there for every graph $G$ a graph $2G$ such that
-- $\chi(2G) = 2\chi(G)$, and
-- $\eta(2G) = 2\eta(G)$?
For each one of the above conditions it is easy to construct a graph $2G$ to $G$ such that th... | https://mathoverflow.net/users/8628 | Graph construction to double coloring & Hadwiger number | Define $K\_n'$ to be the graph obtained from the complete graph on $n$ vertices by subdividing each edge once. Let $G$ be a graph with $\chi(G)=c$ and $\eta(G)=h$. Define $2G$ to be the disjoint union of $K\_{2h}'$ and $K\_{2c}$. Assuming [Hadwiger's conjecture](http://en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph... | 3 | https://mathoverflow.net/users/2233 | 162339 | 85,151 |
https://mathoverflow.net/questions/144433 | 3 | Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq 1$ being a variable parameter. The axes of $E$ are not necessarily aligned with the coordinate axes.
For the case that... | https://mathoverflow.net/users/38322 | Generalising Euler's formula to ellipses and three dimensions | I have finished working on these questions and written it up: [The Euler and Grace-Daniellson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory](http://arxiv.org/abs/1404.0525).
| 3 | https://mathoverflow.net/users/38322 | 162346 | 85,153 |
https://mathoverflow.net/questions/162299 | 6 | I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with objects all formal finite direct sums $\bigoplus\_i x\_i$ of objects $x\_i$ in $\mathcal{C}$, and morphisms
$$f\colon \bigop... | https://mathoverflow.net/users/28145 | Making additive envelopes of monoidal categories monoidal | Yes, your construction will work. Additive envelope is a functor
$$\mathbf{Env} : \mathbf{Ab\text{-}Cat} \to \mathbf{Add},$$
from the category of Abelian enriched categories to the category of additive categories. This functor has a monoidal structure given by the obvious functors
$$\mathbf{Env}(A)\otimes \mathbf{Env... | 4 | https://mathoverflow.net/users/39004 | 162348 | 85,155 |
https://mathoverflow.net/questions/144296 | 3 | Assume $k\_0$ is a field with char($k\_0$) not $2$. Let us define functors from $\rm Field\_{/k\_0}\to \rm Sets$ as $\rm Pfister\_n(k):=\{\text{isomorphism classes of n-fold Pfister forms over k}\}$;
$\rm Quad\_n(k)$:= {isomorphism classes of nondegenerate $n$ dimensional quadratic form over $k$} and $\rm Quad\_{n,d... | https://mathoverflow.net/users/31747 | Set of isomorphisms of Pfister forms corresponding to first cohomology of algebraic group | For $n=3$ you have $Pfister\_3\simeq H^1(\*,G\_2)$. For greater $n$, you may use notion of cohomological invariants (like in Skip Garibaldi's lectures in Lens) to prove that any $H^1(F,G)$ have a nontrivial cohomological invariant of degree $\le 3$ (namely Tits algebras or the Rost invariant), while for $Pfister\_n$ it... | 2 | https://mathoverflow.net/users/5107 | 162350 | 85,156 |
https://mathoverflow.net/questions/162293 | 19 | Lurie, in his ICM 2010 proceedings paper [Moduli Problems for Ring Spectra](http://www.mathunion.org/ICM/ICM2010.2/Main/icm2010.2.1099.1125.pdf) (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is this:
>
> ...the initial object in the $\inf... | https://mathoverflow.net/users/4177 | On a question motivating Lurie's treatment of formal moduli problems | Yes, this has been achieved in some sense. There is a (unpublished and possibly not yet written) work of Gaitsgory and Lurie where they propose an answer to this question. Given a split reductive group scheme $G$ over $Spec(\mathbb{Z})$ they use a version of the geometric Satake correspondence to construct a stable inf... | 19 | https://mathoverflow.net/users/439 | 162357 | 85,158 |
https://mathoverflow.net/questions/162255 | 3 | Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|\_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb field $R$, determined by $d\lambda(R,\cdot)=0$ and $\lambda(R)=1$.
**Does there exist a contact form $\lambda'$ on $Y$ s... | https://mathoverflow.net/users/12310 | Extending Reeb field from contact submanifold to ambient contact manifold | This is a step of the proof of the neighbourhood theorem for contact submanifolds in Geiges's book.
The idea is find a function $f \colon Y \to \mathbb{R}$ so that $f$ is identically $1$ on $S$ and so that
\[
-df + d\lambda(R', \cdot) = 0
\]
on $TY|\_S$. If this holds, then it follows that $d( f\lambda)( R', \cdot) ... | 2 | https://mathoverflow.net/users/477 | 162362 | 85,159 |
https://mathoverflow.net/questions/162374 | 2 | There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.
* The topology inherited from projective space coincides with the Zariski topology.
* The map from the Chow ring to t... | https://mathoverflow.net/users/11541 | Two questions about the grassmannian | $\!$Hej Erik! Your first question asks why an open Schubert cell has the same closure both in the Zariski and in the classical topology. This is the same as asking why the closure of a cell is a closed subvariety. But each cell is defined by a collection of polynomial equations ($f = 0$) and inequations ($f \neq 0$), a... | 4 | https://mathoverflow.net/users/1310 | 162379 | 85,162 |
https://mathoverflow.net/questions/162368 | 2 | Let me first define stein manifolds and coadjoint orbits.
A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold:
$X$ is holomorphically convex, i.e. for every Compact space|compact subset $K \subset X$, the so-called ''holomorphic convex hull'',
$\bar K = \{z ... | https://mathoverflow.net/users/nan | Cotangent bundle of coadjoint orbit is stein manifold? | Not with the obvious complex structure. Notice that $O\_{\lambda}$ is a closed subvariety of $T^{\ast}(O\_{\lambda})$ (namely the zero section). Closed subvarieties of Stein varieties are Stein. However, positive dimensional Stein varieties are never compact, and $O\_{\lambda}$ is compact.
---
However, there is a... | 4 | https://mathoverflow.net/users/297 | 162388 | 85,164 |
https://mathoverflow.net/questions/162343 | 4 | We consider a surface (co-dimension 1) in $R^3.$ I read from the book of Stoker that for any surface there always exist patches of orthogonal curvilinear coordinates that cover the surface.
I want to know further whether we can have (better to be orthogonal) curvilinear coordinates with each coordinate curves (on wh... | https://mathoverflow.net/users/20408 | Existence of planar orthogonal curvilinear coordinates on a surface embedded in $R^3$ | The comment I wrote above is perhaps too sketchy to be useful to the OP, so here is a somewhat expanded answer:
**Initial Remark:** If one doesn't require that the planar coordinate curves be orthogonal, then every surface has such coordinates locally, and one can construct them by simply slicing the surface with two... | 6 | https://mathoverflow.net/users/13972 | 162394 | 85,170 |
https://mathoverflow.net/questions/161148 | 5 | Given two partitions M and N, with $M\_i \leq N\_i$ for all $1\leq i\leq \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum\_{\lambda: M\_i\leq \lambda\_i\leq N\_i} q^{|\lambda|}$$
I remember seeing a determinant formula before, the elements of the determinant being q-binomial coefficients. B... | https://mathoverflow.net/users/39184 | The number of partitions between two fixed partitions | Note on enumeration of partitions contained in a given shape
by Ira M. Gessel and Nicholas Loehr
link:
<http://www.sciencedirect.com/science/article/pii/S0024379509004819>
| 2 | https://mathoverflow.net/users/39184 | 162408 | 85,174 |
https://mathoverflow.net/questions/162402 | 3 | Currently, I encountered a problem of approximating the following
series:
$$
(I-X)^{-\frac{1}{2}}=I+\frac{1}{2}X+\frac{1\cdot3}{2\cdot4}X^{2}+\frac{1\cdot3\cdot5}{2\cdot4\cdot6}X^{3}+\ldots
$$
where $X$ is a diagonalizable matrix and the largest (in absolute
value) eigenvalue is less than $1-\frac{1}{\kappa},\kappa... | https://mathoverflow.net/users/49119 | Approximate the square root of (1-X) efficiently through (nested) products | Not exactly the same thing, but you can use Newton's method easily in your framework. As you can see for instance [here](https://en.wikipedia.org/wiki/Fast_inverse_square_root#Newton.27s_method), Newton's method for the inverse square root can be computed with **additions and multiplications only**, and the same algebr... | 1 | https://mathoverflow.net/users/1898 | 162414 | 85,175 |
https://mathoverflow.net/questions/162278 | 1 | I'm reading a paper and the authors applied the following sobolev type estimates
$$
||(Dv)^{2}||\_{H^{3k-2}(\Omega)}\leq C||v||\_{H^{3k-1+\alpha}(\Omega)}^{2}
$$
for $\alpha>\frac{1}{4}$,
where $v$ is a $n$ velocity vector field, $\Omega\in \mathbb{R^{n}}$ and $k$ is chosen such that $3k\geq \frac{n+3}{2}$.
I was won... | https://mathoverflow.net/users/48275 | reference needed for sobolev type estimates | Okay, the estimate actually holds. I had a "moment" yesterday when I first read gerw's argument, which is actually subtly flawed. Let me quickly illustrate:
To estimate $\| u^2\|\_{H^1}$ you need to estimate $\|u^2\|\_2$ and $\|\nabla (u^2)\|\_2$. The first term is bounded by $\|u\|\_4^2$ as claimed, but the second t... | 3 | https://mathoverflow.net/users/3948 | 162416 | 85,176 |
https://mathoverflow.net/questions/162424 | 5 | Recently I became curious about moduli spaces of linkages and so I found and began reading some papers of Kapovich and Millson. In the paper [Hodge theory and the art of paper folding](http://www.math.ucdavis.edu/%7Ekapovich/EPR/art.pdf), the Ph.D. thesis of A. Galitzer is cited. I found other citations to it in the li... | https://mathoverflow.net/users/353 | A.J. Galitzer's Ph.D. thesis: On the moduli space of closed polygonal linkages on the 2-sphere | You can find a copy [here](https://www.dropbox.com/s/1acr8l38pwx5w8v/AJ-Galitzer-PhD.pdf).
(A [note](https://matheducators.meta.stackexchange.com/a/91/262) on linking to academic papers.)
| 5 | https://mathoverflow.net/users/22971 | 162429 | 85,178 |
https://mathoverflow.net/questions/162001 | 15 | This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an $H$-space. Therefore the theorem of Cartan and Serre tells us that the rationalization $G\_{\mathbb{Q}}$ of $G$ has the homotop... | https://mathoverflow.net/users/3995 | rationalization of classifying spaces | In the paper
Arkowitz, Martin [*Categories equivalent to the category of rational H-spaces*](http://link.springer.com/article/10.1007/BF01170937), Manuscripta Math. 64 (1989), no. 4, 419–429
it is shown that the rational homotopy equivalence $G\_\mathbb{Q}\cong\prod\_n K(\pi\_n(G\_\mathbb{Q}),n)$ is an equivalence ... | 8 | https://mathoverflow.net/users/8103 | 162437 | 85,181 |
https://mathoverflow.net/questions/162435 | 5 | Consider a compact surface $\mathcal{S} \subset \mathbb{R}^3$ without boundary and assume we are given the Gaussian curvature $K(x), x\in \mathcal{S}$. It is know that Gaussian curvature does not fully determine the Riemannian metric, as for example discussed in
[Does the curvature determine the metric?](https://mat... | https://mathoverflow.net/users/43839 | Given Gaussian curvature, can one construct a metric to fulfill the Gauss-Bonnet theorem? | If $K$ is to be the Gauss curvature of some metric on a compact two-dimensional manifold $M$ with Euler characteristic $χ(M)$, then from the Gauss-Bonnet theorem one has the following conditions: (a) $K$ is positive somewhere if $χ(M)>0$, (b) $K$ changes sign or $K≡0$ if $χ(M)=0$, (c) $K$ is negative somewhere if $χ(M)... | 16 | https://mathoverflow.net/users/13268 | 162438 | 85,182 |
https://mathoverflow.net/questions/144663 | 4 | I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through these categories. So, I wonder:
1. What sorts of 'homotopy transfers' are known? At the moment I found <http://folk.uio.n... | https://mathoverflow.net/users/2191 | $T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available? | To your first question: There is sort of generalized transfer structure on $\pi\_{p,q}(E)$ as $GW$-module in $SH$, which you can find in Morel's papers. In fact, the composition
$$\pi\_{p,q}(E)(k) \rightarrow \pi\_{p,q}(E)(L) \rightarrow \pi\_{p,q}(E)(k)$$
is the multiplication with $Tr^L\_k (\langle a \rangle)$ for... | 2 | https://mathoverflow.net/users/36372 | 162448 | 85,188 |
https://mathoverflow.net/questions/162380 | 6 | An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X\_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap\_{\alpha \in y} X\_\alpha = \emptyset$. Countable incompleteness is equivalent to $(\omega,\omega)$-regularity, and for every $\kappa$, there is ... | https://mathoverflow.net/users/11145 | regularity of ultrafilters | Assuming large cardinals, the answer is no even for $\aleph\_\omega$.
Given a supercompact cardinal, Ben-David and Magidor constructed a model in which $\aleph\_{\omega+1}$ carries a uniform indecomposable ultrafilter $U$, which means that $U$ is $\theta$-indecomposable for $\aleph\_0<\theta<\aleph\_\omega$.
For re... | 6 | https://mathoverflow.net/users/18128 | 162451 | 85,190 |
https://mathoverflow.net/questions/162428 | 13 | We are interested in the following notions in the case $G=S\_N$, the symmetric group on
$\{1,\dots,N\}$.
Fix a group $G$ and a number $d$. For $(g\_1,\dots,g\_d)\in G^d$ and $x\in G$, define
$$(g\_1,\dots,g\_d)^x := (g\_1^x,\dots,g\_d^x).$$
(Throughout, $g^x:=x^{-1}gx$.)
$d$-tuples $g=(g\_1,\dots,g\_d)$ and $h=(h\_1... | https://mathoverflow.net/users/2415 | The Simultaneous Conjugacy Problem in the symmetric group $S_N$ | I'll only consider the case of $S\_N$ to start with.
Given $(g\_1,\ldots,g\_d)\in S\_N^d$, define a directed graph $G$ on vertices $\lbrace 1,\ldots,N\rbrace$ whose edges are coloured with elements of $Z\_2^d$. Edge $i{\to}j$ is coloured $(\epsilon\_1,\ldots,\epsilon\_d)$, where $\epsilon\_t=1$ iff $g\_t$ maps $i$ on... | 15 | https://mathoverflow.net/users/9025 | 162453 | 85,192 |
https://mathoverflow.net/questions/162465 | 3 | This is related to [my earlier question](https://mathoverflow.net/q/65109/5810) on isomorphism of general quotients of $\:F\hspace{.02 in}[x]\:$.
Let $F$ be a field, let $p$ and $q$ be (non-zero) monic irreducible polynomials, let $I$ and $J$ be the ideals generated by $p$ and $q$ respectively, and assume the fields ... | https://mathoverflow.net/users/nan | Can non-isomorphic field extensions be isomorphic fields? | In general, it seems to me that the answer is **no**.
In fact, take $F=\mathbb{R}, \,$ $I=(x^2+1)$ and $J=(x^2+2)$.
Then $\mathbb{R}[x]/I$ and $\mathbb{R}[x]/J$ are both isomorphic to $\mathbb{C}$.
On the other hand, for any $a, b \in \mathbb{R}$ we have $$(ax+b)^2+1 = a^2 x^2 + 2ab x +b^2+1.$$
If the right-hand... | 7 | https://mathoverflow.net/users/7460 | 162466 | 85,194 |
https://mathoverflow.net/questions/162478 | 3 | Is there any reference where I can find the character table of the symmetric group $S\_7$? A simple search in google gave me a GAP program that computes the character table, but I don't understand the table that the program produces, since it does not tell me each column in the table is associated to which conjugacy cl... | https://mathoverflow.net/users/8419 | Character table of $S_7$ | As you already discovered, GAP can easily compute this. The character table looks like this:
```
gap> G:=SymmetricGroup(7);
Sym( [ 1 .. 7 ] )
gap> ct := CharacterTable(G);;
CharacterTable( Sym( [ 1 .. 7 ] ) )
gap> Display(ct);
CT2
2 4 4 4 4 3 2 3 1 3 3 2 1 1 1 .
3 2 1 1 1 2 1 1 ... | 13 | https://mathoverflow.net/users/8338 | 162482 | 85,202 |
https://mathoverflow.net/questions/161960 | 2 | Pick an integer $b \ge 2$, and for $n \in \mathbf N$ let $s\_b(n)$ denote the sum of the base-$b$ digits of $n$. It is a nice exercise to prove that $$s\_b(n) = (b-1) \sum\_{i=1}^\infty \left\{\frac{n}{b^i}\right\}.$$
In particular, this follows at once from
$$s\_b(n) = n - (b-1) \sum\_{i=1}^\infty \left\lfloor \frac{n... | https://mathoverflow.net/users/16537 | Looking for a reference to a classical formula for the sum of the base-$b$ digits of an integer | Sorry for answering my own question, but it turns out that the second identity in the OP is explicitly stated and proved in: L. D. Yarbrough and S. Rabinowitz, *Solution to Problem E1926*, Amer. Math. Monthly **75** (Mar., 1968), No. 3, 296.
| 2 | https://mathoverflow.net/users/16537 | 162495 | 85,205 |
https://mathoverflow.net/questions/162506 | 3 | I know that Hopf fibrations (the four ones) are the only ones that have the form
$S^k \to S^m \to S^n$, but I never seen a proof. Could anyone link me a paper or text where this is proved, or prove it for me?
| https://mathoverflow.net/users/45210 | How can I prove that Hopf fibrations are the only ones with fiber, total space and base space homeomorphic to spheres? | <http://en.wikipedia.org/wiki/Adams%27_theorem>
Adams, J.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104.
<http://www.maths.ed.ac.uk/~aar/papers/adams1.pdf>
<http://dx.doi.org/10.2307%2F1970147>
| 5 | https://mathoverflow.net/users/47462 | 162508 | 85,208 |
https://mathoverflow.net/questions/162500 | 1 | Let $G$ be a finite group and let $M$ be a $G$-module that is a finite abelian $p$-group. Suppose we have extensions
$1 \rightarrow M \rightarrow E\_1 \rightarrow G \rightarrow 1$
and
$1 \rightarrow M \rightarrow E\_2 \rightarrow G \rightarrow 1$
such that the extensions have a common restriction
$1 \rightarr... | https://mathoverflow.net/users/4053 | Control of $p$-extensions by subgroups of index coprime to $p$ | I don't think this is always possible.
Let $p=3$, $M = \langle a \rangle$ with $a^3=1$, $G = D\_6 = \langle b,c \mid b^3=c^2=1, cbc=b^{-1} \rangle$, with trivial action of $G$ on $M$ and $H=\langle b \rangle$.
Let
$$E\_1 = \langle a,b,c \mid a^3=b^3=c^2=1, ab=ba, ac=ca, cbc=b^{-1} \rangle$$
and
$$E\_2 = \langle a,... | 2 | https://mathoverflow.net/users/35840 | 162515 | 85,210 |
https://mathoverflow.net/questions/162487 | 22 | Many types of equations are being used in multiple contexts, so a search for specific formulas might be one way to identify scholarly papers that are conceptually related.
Is any website or tool actually providing recommendations of related content on that basis?
The closest I can think of is Springer's LaTeX Searc... | https://mathoverflow.net/users/1759 | Which tools can identify scholarly papers that use the same types of equations? | There is [MathWebSearch](http://search.mathweb.org/); from the site:
>
> The MathWebSearch system (MWS) is a content-based search engine for mathematical formulae. It indexes MathML formulae, using a technique derived from automated theorem proving: Substitution Tree Indexing.
>
>
>
Indeed, [zbMATH](http://... | 13 | https://mathoverflow.net/users/nan | 162525 | 85,215 |
https://mathoverflow.net/questions/162520 | 3 | This is a cross-post from [MSE](https://math.stackexchange.com/questions/738094/good-book-on-analytic-continuation).
For my Bachelor's thesis, I am investigating divergent series summation methods. One of those is analytic continuation. There are quite a few books on complex analysis that include a chapter or two on ... | https://mathoverflow.net/users/93724 | Good book on analytic continuation? | English books are Hardy, Divergent series,
and P. Dienes, Taylor series:
an introduction to the theory of functions of a complex variable. Dover Publications, Inc., New York, 1957. (The title is somewhat misleading. This is a large book that indeed
contains an "introduction to complex variables" but it is also the most... | 9 | https://mathoverflow.net/users/25510 | 162528 | 85,216 |
https://mathoverflow.net/questions/161652 | 9 | I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation\*}
\mathrm{Ric}\circ\mathrm{Ric}=\lambda^2 g\circ g,
\end{equation\*}
for some $\lambda\in\mathbb{R}$, where $\circ$ is the Kulkarni-Normizu
product.
If $g... | https://mathoverflow.net/users/38600 | A question on generalized Einstein metrics on four-dimensional manifolds | I don't really know about any previous literature on the topic. But here is an example of a metric that satisfies the condition above but is not Einstein (in sloppy notation, but I hope it's clear anyway):
Let $M=\mathbb{R}\_{+}\times S^2\times S^1$ and $g=dr^2+r^2(g\_{S^2}+g\_{S^1})$ where $\text{Ric}(g\_{S^2})=2g\_... | 6 | https://mathoverflow.net/users/46176 | 162533 | 85,217 |
https://mathoverflow.net/questions/159166 | 2 | I am trying to find the leading order expression in an expansion for large $\Delta$ of ${}\_2F\_1\left(\frac{\Delta}{2},\frac{\Delta+1}{2},\Delta,z^{-2}\right)$, where $z\in\mathbb{C}$.
The only helpful relation I could find in the literature *Higher transcendental functions* by H. Bateman and A. Erdelyi, p.77. There... | https://mathoverflow.net/users/47683 | Asymptotic form of the Gauß Hypergeometric function 2F1 for three parameters approaching infinity | In fact your function is elementary and very simple, for its explicit form look at Brychkov,Marichev,Prudnikov, Integral and Series, vol.3 :
$$
F(a,a+1/2;2a;z)=\frac{1}{\sqrt{1-z}}\left(\frac{2}{1+\sqrt{1-z}}\right)^{2a-1}.
$$
Unfortunately I can not give a reference to the exact formula, this vol. 3 is the only one ... | 5 | https://mathoverflow.net/users/49208 | 162545 | 85,222 |
https://mathoverflow.net/questions/162443 | 2 |
>
> Does there exist a constant $c$ so that the lines of every Steiner
> triple system on $v$ points can be covered by $cv$ points?
>
>
>
That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then there exists a subset $P \subseteq T$, $|P|=cv$ and every line of $D$ contains a point from $P$.
| https://mathoverflow.net/users/22051 | Hitting sets (aka covers aka transversals) of Steiner triple systems | I think your question equivalently asks if there is a universal constant $c>0$ such that every Steiner triple system of order $v$ has a 'cap' (line-free set) of size at least $c v$. The complement of a cap is a hitting set, and vice-versa.
The answer to the latter question, and hence (provided I am not goofing up the... | 3 | https://mathoverflow.net/users/45255 | 162548 | 85,223 |
https://mathoverflow.net/questions/162552 | 22 | In his paper "Qualitative Distinctions Between Some Toposes of Generalized Graphs" (reproduced [here](http://conceptualmathematics.files.wordpress.com/2013/01/toposesofgeneralizedgraphs.pdf)), page 267, Lawvere says that the [idempotent-splitting completion](http://ncatlab.org/nlab/show/Karoubi+envelope) of the categor... | https://mathoverflow.net/users/2926 | Idempotents split in category of smooth manifolds? | Assume that:
$$0\in Fix(p)$$
and $U$ is a small neighborhood of $0$.
Let $$f(x)=x-p(x)$$
Identifying $\mathbb R^n$ with $T\_0\mathbb R^n$, set $\pi\_{\mathrm{ker}(dp(0))}$ to be the projection $\mathbb R^n\to\mathrm{ker}(dp(0))$.
Then $g=\pi\_{\mathrm{ker}(dp(0))}\circ f$ is smooth and regular at $0$, so $M=... | 16 | https://mathoverflow.net/users/300 | 162556 | 85,226 |
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