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https://mathoverflow.net/questions/186645 | 12 | For given $n$, consider a polynomial $\sum\_{k=0}^na\_kz^k$ with all coefficients $a\_k\in\{\pm1\}$. I am interested in the following:
>
> **How big can the modulus of a non-real root of such a polynomial be?**
>
>
>
Wlog we can assume $a\_n=1,a\_{n-1}=-1$. Then a systematic search for small $n$, looking eac... | https://mathoverflow.net/users/29783 | $\pm1$-polynomials with a maximal non-real root | A closely related problem was considered by [Odlyzko and Poonen](http://retro.seals.ch/digbib/view;jsessionid=508577B1E580B9C6A779A3ECA50D6EF6?rid=ensmat-001:1993:39::181) who looked at the class of polynomials with $0$ or $1$ coefficients. On page 330, they discuss the question of the smallest (in size) non-real root ... | 10 | https://mathoverflow.net/users/38624 | 186669 | 92,682 |
https://mathoverflow.net/questions/186672 | 8 | * If $A$ is any $n \times m$ matrix and $B$ is any $m \times n$ matrix then one familiar form of the Sylvester's identity is $\det(I + AB) = \det(I + BA)$.
Now somehow curiously this above identity is often enough used along side another statement which says that for any vector $v$ and any invertible square matrix $... | https://mathoverflow.net/users/38852 | About Sylvester's determinant | For the first point, note that by Sylvester's identity
$$\det(I\_n + v^TA^{-1}v) = \det(I\_n + (v^T)(A^{-1}v)) = \det(I\_n+(A^{-1}v)(v^T)) = \det(I\_n + A^{-1}vv^T),$$
so
$$\det(A)\det(I\_n+v^TA^{-1}v) = \det(A)\det(I\_n + A^{-1}vv^T) = \det(A + vv^T).$$
Given $u \in \mathbb{R}^m$ and $v \in \mathbb{R}^n$, the... | 11 | https://mathoverflow.net/users/21564 | 186674 | 92,683 |
https://mathoverflow.net/questions/186668 | 2 | Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) to $\mathbb{R}$ and let $F\_T$ denote the pullback of $T$, i.e. the map $F\_T: \mathcal{F}(N,\mathbb{R}) \to \mathcal{F}(M,... | https://mathoverflow.net/users/61523 | Inverse Problem for Pullback | Concerning (i):
If $F\colon\mathcal{F}(N,\mathbb R)\to \mathcal{F}(M,\mathbb R)$ is of the form $F\_T$, $F$ is a homomorphism of algebras, this is an obvious necessary condition. It turns out, it is also a sufficient one.
There is a contravariant functor $\mathcal A$ from the category of smooth manifolds and smooth... | 4 | https://mathoverflow.net/users/40950 | 186675 | 92,684 |
https://mathoverflow.net/questions/186679 | 3 | Two notions that occur often in representation theory seem to be that of a "characteristic variety" and that of an "associated variety". The former term seems exclusive to D-module theory while the latter is common in the study of primitive ideals. While I am still a beginner in terms of learning these ideas, it would ... | https://mathoverflow.net/users/61567 | Characteristic Varieties and Associated Varieties | A possible way to describe the relation is as follows.
Let $G$ be a complex semi-simple group with the Lie algebra $\mathfrak{g}$, $B\subset G$ be a Borel subgroup, $X=G/B$ be the flag variety. Let $\mathcal{D}\_X$ be the sheaf of rings of differential operators on $X$.
We have the $G$-equivariant moment map $\pi\... | 5 | https://mathoverflow.net/users/16183 | 186691 | 92,687 |
https://mathoverflow.net/questions/186706 | 5 | Me and my advisor were looking for a specific proof of the disorder in $2d$ harmonic crystals. We could not find a paper or a textbook with it, so I thought trying my luck here.
Basically, it is a proof of the instability in an harmonic lattice crystal that uses the idea of random walk and the discrete lagrangian, a... | https://mathoverflow.net/users/42864 | Harmonic Crystal using Random Walk | The object you look at is called the Gaussian Free Field (on your graph, with zero boundary
conditions) in dimension $2$. There is a lot known about it. For some pointers see the Wiki page
<http://en.wikipedia.org/wiki/Gaussian_free_field>, my lecture notes
<http://www.wisdom.weizmann.ac.il/~zeitouni/notesGauss.pdf> an... | 10 | https://mathoverflow.net/users/35520 | 186724 | 92,700 |
https://mathoverflow.net/questions/186737 | 6 | Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set.
**Question: Is there a conformal bijection between $\mathbb{D}/G$ and $\mathbb{C} \setminus K$ for some Cantor set $K \subset ... | https://mathoverflow.net/users/7631 | Uniformization of a plane minus cantor set | Koebe uniformisation theorem says that any planar Riemann surface is biholomorphic to a domain in the Riemann sphere .See George Springer's book on Riemann surfaces.At the risk of self promotion you can also look at my book with T Napier titled An Introduction to Riemann Surfaces .
| 6 | https://mathoverflow.net/users/4696 | 186743 | 92,707 |
https://mathoverflow.net/questions/186754 | 2 | The [Borel--Bott--Weil Theorem](http://en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem) gives the dimensions of the cohomology groups of the equivariant line bundles over flag manifolds. Does there exist an analogous result for general equivariant vector bundles over the flag varieties, or even just for c... | https://mathoverflow.net/users/42100 | Analogue of Borel--Bott--Weil for General Equivariant Vector Bundles | The [article](http://www.math.tamu.edu/~jml/kostant61.pdf) Lie Algebra Cohomology and the Generalized Borel-Weil theorem by Kostant contains generalization of the BBW theorem to equivariant vector bundles over $G/P$ associated to a $G$-representation, where $P$ is a parabolic subgroup. Actually, it contains a bit more,... | 9 | https://mathoverflow.net/users/6818 | 186757 | 92,711 |
https://mathoverflow.net/questions/186758 | 4 | What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the sense that there exist sets $F\_{k} \subset \mathbb{R}^{n}$, $k \in \mathbb{N}$, (resp. $G\_{k} \subset \mathbb{R}^{n}$) wi... | https://mathoverflow.net/users/33804 | Approximation of sets by sets with regular border | In the case of a compact set $E\subset\mathbb{R}^n$, we can arrange that the covers $F\_k$ are each a finite union of open balls, with the closure of the next contained in the previous $\bar F\_{k+1}\subset F\_k$, and $\bigcap\_k F\_k=E$. To get these, simply cover $E$ with suitable tiny balls centered at each point of... | 3 | https://mathoverflow.net/users/1946 | 186762 | 92,712 |
https://mathoverflow.net/questions/186764 | 10 | Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is proper, that is, the preimage of a compact set is compact?
It is also posted on <https://math.stackexchange.com/q/100454... | https://mathoverflow.net/users/16323 | The properness of a submersion | This is true in greater generality. If $M$ and $N$ are locally compact, and the fibers of $f$ are compact connected and $f$ is a quotient map then f is proper. This falls under the rubric of monotone light factorization. Look at the book of G.T. Whyburn and E. Duda titled Dynamic Topology. Monotone Light factorization ... | 9 | https://mathoverflow.net/users/4696 | 186767 | 92,715 |
https://mathoverflow.net/questions/186768 | 16 | Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all $x \in \mathbb{Q}$ ?
| https://mathoverflow.net/users/38889 | Images of polynomials | We don't need Faltings theorem here. See Theorem 1 in:
H. Davenport, D. J. Lewis, A. Schinzel, Polynomials of certain special types. Acta Arith. 9 (1964), 107-116.
| 14 | https://mathoverflow.net/users/26218 | 186773 | 92,718 |
https://mathoverflow.net/questions/186744 | 10 | cLet $a,b,c,d\in \mathbb{Z}$ and suppose we have the equation $ac+bd=1$. One way of thinking about this equation is it expresses the fact $\gcd(c,d)=1$. It is well-known that all other similar equations expressing that fact are of the form $(a+td)c+(b-tc)d=1$ for some $t\in\mathbb{Z}$.
Letting $a'=a+td,b'=b-tc$, one ... | https://mathoverflow.net/users/3199 | A back and forth Euclidean algorithm over the integers--does it have bounded length? | We get an isomorphic problem by switching $c$ with $d$, and replacing $b$ with $-b$. Then we are considering matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ in $SL\_2(\mathbb{Z})$. Passage from $(a,b)$ to $(a',b')$ amounts to multiplication by a matrix $T^t = \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix}$ for so... | 9 | https://mathoverflow.net/users/121 | 186781 | 92,722 |
https://mathoverflow.net/questions/186787 | 2 | Let $U=\cup\_{n=1}^{\infty} U(n)$ be endowed with the weak topology, let $BU$ be its classifying space. Then for any compact CW complex $X$, $[X,BU]$ classifies all the vector bundles on $X$ up to stable isomorphisms, which is in one-one correspondence with $\tilde{K}(X)$.
On the other hand, let $\mathrm{Fred}\_0(H)... | https://mathoverflow.net/users/37354 | What is the relationship between $BU$ and $\textrm{Fred}_0(H)$? | See a paper by G. Segal, "K-homology theory and algebraic K-theory" (I'm sure it's not the original source, though). There is a homotopy equivalence between $BU\times \Bbb Z$ and the space $Fred$ of Fredholm operators. We can consider $BU \times \Bbb Z$ as a classifying space of the category of virtual vector bundles. ... | 8 | https://mathoverflow.net/users/10605 | 186790 | 92,727 |
https://mathoverflow.net/questions/186786 | 2 | In the [Wikipedia article on the Enriques-Kodaira classification](http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#Hodge_numbers_and_Kodaira_dimension), before the classification itself, the following sentence appears:
>
> For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} − ... | https://mathoverflow.net/users/21564 | For compact complex surfaces $h^{1,0}$ is either $h^{0,1}$ or $h^{0,1} - 1$. Do we need to use the Enriques-Kodaira classification? | This is contained in Theorem 2.6 in chapter IV in the book of Barth, Peters, Van de Ven (in the new edition with Hulek, it is Theorem 2.7, chapter IV).
It does not rely on classification, and it was first proved by Kodaira.
| 4 | https://mathoverflow.net/users/13168 | 186796 | 92,728 |
https://mathoverflow.net/questions/186792 | 9 | I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta\_n$ of h-cobordism classes of homotopy $n$-spheres (with group operation coming from the connected sum). It occurred to me that there ought to be a whole $(\infty,0)$-category of homotopy $... | https://mathoverflow.net/users/47658 | K-theory of the h-cobordism category | I don't know if this is the sort of thing you have in mind, but using [surgery theory](http://en.wikipedia.org/wiki/Surgery_theory) you can in fact write down a single connective spectrum / infinite loop space whose $n^{th}$ homotopy group is $\Theta\_n$. This is the space $PL/O$, one definition of which is that it is ... | 10 | https://mathoverflow.net/users/290 | 186798 | 92,730 |
https://mathoverflow.net/questions/186219 | 2 | Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and
$f(a)=\inf\{b : (a,b)\in R\}$
$g(b)=\inf\{a : (a,b)\in R\}$
then what can we call $f$ and $g$? Perhaps there is some standard terminology.
| https://mathoverflow.net/users/4600 | Uniformizing a relation on ordered sets | The maps $f,g$ constitute a Galois connection, but you have to take $\leq := \supseteq$ in both cases.
| 0 | https://mathoverflow.net/users/8628 | 186802 | 92,733 |
https://mathoverflow.net/questions/186812 | 5 | I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the answer to the following question is well known, but I couldn't work it out this morning.
Let $X\subset\{0,1\}^{\mathbb... | https://mathoverflow.net/users/24586 | Approximating Subshifts From Below | If $X$ is minimal and not a periodic orbit then it cannot contain a periodic orbit and hence in particular cannot contain a Markov shift. A classical construction by Grillenberger shows that one can construct uniquely ergodic (hence in particular minimal) subshifts with arbitrary entropy. (This result also follows from... | 10 | https://mathoverflow.net/users/1840 | 186814 | 92,738 |
https://mathoverflow.net/questions/186807 | 13 | Suppose that $E$ an elliptic curve defined over $\mathbb{Q}$ and $p$ an odd prime. Let $G=\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$. I am wondering whether the cohomology group $H^1(G, E[p])$ can be nontrivial. If $G=GL\_2(\mathbb{F}\_p)$ (which is the case for all but finitely many primes $p$ if $E$ does not have compl... | https://mathoverflow.net/users/61621 | For an elliptic curve $E/\mathbb{Q}$ can the cohomology group $H^1(\text{Gal}(\mathbb{Q}(E[p])/\mathbb{Q}), E[p])$ be nontrivial? | Fix elements $\zeta$ and $\alpha$ with $\zeta$ a primitive third root of unity and $\alpha^3 = -4$. These generate a field $K = \Bbb Q(\zeta,\alpha)$ which is the splitting field of $x^3 + 4$, with Galois group $G$ the symmetric group on three letters.
Consider the elliptic curve $y^2 = x^3 + 1$. Unless I have miscal... | 9 | https://mathoverflow.net/users/360 | 186840 | 92,743 |
https://mathoverflow.net/questions/186823 | 14 | $\newcommand\Tr{\text{Tr}}$My question is whether there can be a nonstandard model of PA having a unique inductive truth predicate.
**Background.** If $\mathcal{N}=\langle N,+,\cdot,0,1,<\rangle$ is a model of the
first-order PA axioms, then a *truth predicate* on $\mathcal{N}$, also commonly called a *satisfaction ... | https://mathoverflow.net/users/1946 | Is there a nonstandard model of arithmetic having precisely one inductive truth predicate? |
>
> The answer to the question is in the positive.
>
>
>
Let $(\cal{N}^\*,\textrm{Tr\*})$ be a *rather classless* elementary extension of $(\cal{N},\textrm{Tr})$, where $\cal{N}$ is a model of PA, and $\textrm{Tr}$ is a full truth predicate on $\cal{N}$, e.g., let $\cal{N}$ be the standard model of PA, and $\tex... | 13 | https://mathoverflow.net/users/9269 | 186844 | 92,746 |
https://mathoverflow.net/questions/186367 | 6 | Let $Z\_1$ and $Z\_2$ be two closed subschemes of a smooth, complex, algebraic variety $X$. We will be interested in the Grothendieck ring of coherent sheaves of $X$, i.e. isomorphism classes of such sheaves modulo exact sequences and a product given by the tensor product of $\mathcal{O}\_X$-modules.
Denote by $[\mat... | https://mathoverflow.net/users/9947 | Why does a flat degeneration induce equality in the K-Theory? | As Allen says, we are talking about flat subfamilies of $X$. I.e. $Y$ is a closed subscheme of $X \times \mathbb{A}^1$, flat over $\mathbb{A}^1$, with $Z\_0 \times \{ 0 \}$ and $Z\_1 \times \{ 1 \}$ the fibers over $0$ and $1$.
Replacing $Y$ by its closure in $X \times \mathbb{P}^1$, we may assume that we instead hav... | 2 | https://mathoverflow.net/users/297 | 186845 | 92,747 |
https://mathoverflow.net/questions/186829 | 3 | I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form:
$X\_{i}= [1, x\_{i}, y\_{i}, x\_{i}^2, x\_{i}y\_{i}, y\_{i}^2, x\_{i}^3, x\_{i}^2y\_{i},...,x\_{i}y\_{i}^{d-1}, y\_{i}^{d}]$
so that the matrix $X$ c... | https://mathoverflow.net/users/61349 | When does a Vandermonde-like matrix have full rank | This is not an answer but rather a very week sufficient condition. (The true condition is that the points in question should not lie on a degree $d$ curve, but that would be merely a restatement.) So, it suffices to assume that the set of points $\{(x\_i,y\_i)\}$ contains a product $X\times Y=\{x\_0,\ldots,x\_d\}\times... | 2 | https://mathoverflow.net/users/44953 | 186846 | 92,748 |
https://mathoverflow.net/questions/186857 | 1 | I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input from someone.
Fix $k \in \mathbb{N}$ and some $\epsilon > 0$. I would like a function $f$ defined for all $\mathbb{Z}... | https://mathoverflow.net/users/48408 | Existence of arithmetic function satisfying a certain property | Suppose that $f$ satisfies your conditions: $|f(n\_1)\dotsb f(n\_k)|\le
A|n\_1+\dotsb+n\_k|^{-\epsilon}$ if $n\_1+\dotsb+n\_k\ne 0$, and
$|f(n\_1)\dotsb f(n\_k)|\ge B$ if $n\_1+\dotsb+n\_k=0$, with positive $A$ and
$B$ depending only on $k$ and $\epsilon$. The second condition implies
that $\max\{|f(n)|,|f(-n)|\}\ge \s... | 3 | https://mathoverflow.net/users/9924 | 186867 | 92,756 |
https://mathoverflow.net/questions/186851 | 58 | To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of Lee's series) but I don't have any background in categorical logic or model theory.
I've recently come across some int... | https://mathoverflow.net/users/56938 | Synthetic vs. classical differential geometry | One point of [synthetic differential geometry](http://ncatlab.org/nlab/show/synthetic+differential+geometry) is that, indeed, it is "synthetic" in the spirit of traditional [synthetic geometry](http://ncatlab.org/nlab/show/synthetic%20geometry) but refined now from incidence geometry to differential geometry. Hence the... | 44 | https://mathoverflow.net/users/381 | 186875 | 92,760 |
https://mathoverflow.net/questions/186876 | 5 | Let $G$ be a finite group and $\phi\colon G\to \mathrm{GL}\_d(\mathbb C)$ be an irreducible representation, with character $\chi$. Recall that
* $\phi$ is *complex type* if $\chi$ is not real-valued,
* $\phi$ is *real type* if $\phi$ is the complexification of a representation $G\to\mathrm{GL}\_d(\mathbb R)$,
* $\phi... | https://mathoverflow.net/users/61662 | Can groups of twice-odd order have quaternionic representations? | I'm not sure how acceptable it is to give answers referring to theorems in textbooks. If you'd like me to reproduce the relevant argument, then please let me know.
Chapter III.11 of Simon, *Representations of finite and compact groups*, shows the following (in the notation there): if $H \le G$ with $H$ finite and $[G... | 5 | https://mathoverflow.net/users/2383 | 186878 | 92,762 |
https://mathoverflow.net/questions/186881 | 0 | I know the combinatorial interpretation of first, and second order Stirling numbers (#of k cycles of n items, and #of partitions n items into k subsets). Is there an interpretation for the generalized Stirling numbers?
| https://mathoverflow.net/users/61664 | Combinatorial Interpretation of Generalized Stirling numbers | [Combinatorial Interpretation of Generalized Stirling Numbers](https://cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.pdf) (2009)
>
> A combinatorial interpretation of the earlier studied generalized
> Stirling numbers, emerging in a normal ordering problem and its
> inversion, is given. It involves unordered fores... | 1 | https://mathoverflow.net/users/11260 | 186887 | 92,766 |
https://mathoverflow.net/questions/186886 | 16 | **Edit**: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.
Consider the local rings
$$R = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz\rangle$$
and
$$S = \mathbb{C}[[x,y,z]]/\langle xy+xz+yz+xyz\rangle.$$
Is $R$ isomorphic to $S$?
**Some cont... | https://mathoverflow.net/users/10273 | Two rings...are they isomorphic? | Yes, they are isomorphic.
More generally, if $k$ is any algebraically closed field of characteristic not $2$, and $n$ is given, then all $k$-algebras of the form $k[[x\_1,\ldots,x\_n]]/(f\_2+f\_3+\cdots)$, where each $f\_i$ is homogeneous of degree $i$, and $f\_2$ is a nondegenerate quadratic form, are isomorphic. (I... | 18 | https://mathoverflow.net/users/2757 | 186888 | 92,767 |
https://mathoverflow.net/questions/186899 | 7 | The first few unstable homotopy groups of the unitary groups $U(n)$ were calculated by Borel-Hirzebruch, Toda, and Kervaire, and they are all torsion. There is a paper by Matsunaga (details below) in which the p-primary parts of the next few homotopy groups are calculated. This paper claims to be giving a complete desc... | https://mathoverflow.net/users/4042 | Are all unstable homotopy groups of $U(n)$ torsion? | Being an $H$-space, $U(n)$ has the rational homotopy type of a product of odd-dimensional spheres. As we know its cohomology, these are $S^1 \times S^3 \times S^5 \times \cdots \times S^{2n-1}$. In particular, its rational homotopy groups vanish above degree $2n-1$.
| 16 | https://mathoverflow.net/users/318 | 186901 | 92,770 |
https://mathoverflow.net/questions/186771 | 5 | [Szemeredi's regularity lemma](http://en.wikipedia.org/wiki/Szemer%C3%A9di_regularity_lemma) is a well-known result about partitioning large graphs into pieces such that most pairs of pieces are "regular". The precise statement takes a bit of detail so I'll just refer to the the Wikipedia link.
My question is about t... | https://mathoverflow.net/users/61605 | irregular pairs in half graphs - Szemeredi regularity | Your guess at the quantification for (1) is correct. I don't know anywhere where this is explicitly worked out. It is a fairly tedious analysis, more or less you need to argue that the partition sets essentially respect the bipartition and that if a part among the $v\_i$ isn't adjacent to any irregular pairs then its l... | 4 | https://mathoverflow.net/users/59289 | 186910 | 92,774 |
https://mathoverflow.net/questions/184260 | 2 | Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific Series in Optimization and Neural Computation, 6)" and I wish to read further trying to resolve the very large scale linear ... | https://mathoverflow.net/users/40780 | books on very large scale linear optimization | Have a look at Marti, Reinelt: *The Linear Ordering Problem*. This books gives an up-to-date overview.
| 1 | https://mathoverflow.net/users/60435 | 186920 | 92,778 |
https://mathoverflow.net/questions/186870 | 5 | What is an example of a $C^{\*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$.
>
> To what extent such algebras are classified?
>
>
>
| https://mathoverflow.net/users/36688 | A nilpotency question on $C^{*}$ algebras | I'm not sure if this is useful: If your $C^\*$-algebra also has a non-zero nilpotent element, then it will have nilpotent elements of all orders, it is not (algebraically) of bounded index, so does not satisfy a polynomial identity. $C^\*$-algebras which *do* satisfy a polynomial identity are the subject of the followi... | 3 | https://mathoverflow.net/users/5734 | 186921 | 92,779 |
https://mathoverflow.net/questions/186922 | 5 | Can someone recommend a book about the history of mathematics being used for weather prediction, preferable one which covers recent developments?
| https://mathoverflow.net/users/60435 | Book about the history of mathematics for weather prediction | [Invisible in the Storm: The Role of Mathematics in Understanding Weather](http://press.princeton.edu/titles/9957.html)
one [review (AMS)](http://www.ams.org/notices/201308/rnoti-p1051.pdf) a second [review (EMS)](http://www.euro-math-soc.eu/review/invisible-storm-role-mathematics-understanding-weather),
>
> "I... | 7 | https://mathoverflow.net/users/11260 | 186929 | 92,780 |
https://mathoverflow.net/questions/186928 | 2 | According to [1]
>
> Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
> $S(\mathcal{X})$ denote the class of all sequences
> $f=(f\_j)=(f\_0,f\_1,...)$ of Bochner-integrable random vectors in
> $\mathcal{X}$, with $f\_0=0$ defined on a probability space
> $(\Omega,F,P)$.
>
>
> For $f\in S(\ma... | https://mathoverflow.net/users/42531 | How does Azuma's Inequality result from Pinelis Inequality? | For a fixed integer $n$, we consider $d\_j=X\_j$ if $j\leqslant n$ and $d\_j=0$ otherwise. If $|X\_j|\leqslant r\_j$ almost surely, then we have
$\sum\_{j=1}^\infty\mathrm{ess\,sup}\lVert X\_j\rVert^2\leqslant \sum\_{j=1}^nc\_j^2=:c$, hence we use the result with $d'\_j:=d\_j/c$ and $r':=r/c$.
| 2 | https://mathoverflow.net/users/17118 | 186940 | 92,782 |
https://mathoverflow.net/questions/173505 | 2 | Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is $R$-linear in its first argument and $h(v,w)=h(w,v)^\theta$. Denote
$$H(R) = \{v\in R^n\mid h(v,v)=0\}\;,$$
and look at a map... | https://mathoverflow.net/users/41178 | Projecting solutions of Hermitian forms over local rings | I will answer the question in a slightly more specific context, where
* We restrict ourselves to the projective plane $\mathbb{P}\_2(R)$ over $R$, i.e. triples $(x,y,z)\in R^3$ such that $xR+yR+zR=R$, up to multiples in $R^\ast$.
* We use the given Hermitian form $h\big((x\_1,y\_1,z\_1),(x\_2,y\_2,z\_2)\big) = x\_1z\... | 1 | https://mathoverflow.net/users/41178 | 186944 | 92,783 |
https://mathoverflow.net/questions/186913 | 2 | For a (symmetric) random walks on countable groups generated by $\mu$, there is a "brute-force computation" argument of Avez (1974) that shows that if the entropy $h\_\mu$ is trivial then there are no non-constant $\mu$-harmonic functions on the group. [Definitions are below.]
Given it is fairly easy to show that zer... | https://mathoverflow.net/users/18974 | Speed and absence of non-constant bounded harmonic functions | Here is an alternative simple proof (without any entropy) of non-Liouville $\Rightarrow$ positive speed using some of the theory of the Martin boundary/positive harmonic functions. It's quite high-level so not really what you're looking for, but is too long for a comment.
Let $G$ be a transient graph with associated... | 3 | https://mathoverflow.net/users/41827 | 186959 | 92,786 |
https://mathoverflow.net/questions/186957 | 0 | An integer-valued polynomial is a polynomial $p(x)$ such that $\forall x \in \mathbb{Z}, p(x) \in \mathbb{Z}$.
Theorem: For any $n$-degree polynomial $p$, if $p(x) \in \mathbb{Z}$ for all $x \in \{0, 1, ..., n\}$, then $p$ is an integer-valued polynomial.
The proof is pretty simple for $n = 0, 1$
I have a proof f... | https://mathoverflow.net/users/61697 | Integer-valuedness of a polynomial determined by output of first n integers? | This follows from the calculus of finite differences (see, for example, the eponymous book).
If you start with a n'th degree polynomial p(x) evaluated at n+1 consecutive integers, and then take the 1st differences, then differences of those (i.e., 2nd differences), and so forth, up to the n'th differences, you will ... | 4 | https://mathoverflow.net/users/44797 | 186960 | 92,787 |
https://mathoverflow.net/questions/186951 | 6 | The operad $\mathcal{D}\_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB\_n,1)$ where $PB\_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}\_2$ have a categorical analogue called braided monoidal category. More precisely, there is an operad in groupoids $\mathcal{P}aB$ whose ... | https://mathoverflow.net/users/10707 | Framed version of braided monoidal category | First of all let me correct you: it is not true that the groups $PB\_n$ assemble into an operad in groups.
The point is that $PB\_n$ is the fundamental group of $D\_2(n)$, which requires the choice of a base point. But it is impossible to choose basepoints on the spaces $D\_2(n)$ simultaneously in a way that is comp... | 12 | https://mathoverflow.net/users/1310 | 186962 | 92,789 |
https://mathoverflow.net/questions/186908 | 9 | The definition of a **gerbe on a smooth manifold** that I know is that - after fixing an open cover $U\_i$, a gerbe consists of the data of line bundles $L\_{ij}$ on two-fold-intersections $U\_{ij}$, isomorphisms $\alpha\_{ijk}: L\_{ij} \otimes L\_{jk} \longrightarrow L\_{ik}$ on three-fold intersections that satisfy a... | https://mathoverflow.net/users/16702 | Gerbes and Stacks | There is a canonical equivalence of $2$-categories
$$St\left(Man/M\right) \simeq St\left(Man\right)/M$$
between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a map to $M$ (regarding $M$ as a representable sheaf). Given a map $\pi:\mathscr{Y} \to M$ for $\mathscr{Y}$ some stack on ... | 3 | https://mathoverflow.net/users/4528 | 186966 | 92,791 |
https://mathoverflow.net/questions/186936 | 10 | I would like to know to which extent the theory developed for smooth projective varieties in the following articles
>
> A. Białynicki-Birula, Some theorems on actions of algebraic groups.
> *Ann. of Math. (2) , 98:480–497, 1973.*
>
>
> A. Białynicki-Birula, Some properties of the decompositions of
> algebraic v... | https://mathoverflow.net/users/4721 | Białynicki-Birula theory for non-complete varieties | Everything holds for a smooth quasiprojective $X$ as long as there exists a $\mathbb{C}^\*$ action so that
$$
\lim\_{t \to 0} t \cdot x
$$
exists for every $x \in X$. This is guaranteed when $X$ is projective but holds more generally. The theory really depends on a local analysis of torus actions at fixed points s... | 9 | https://mathoverflow.net/users/12402 | 186976 | 92,796 |
https://mathoverflow.net/questions/186979 | 3 | In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only if it is $\Sigma^{1}\_2$
ii) A set $\mathtt A$ of reals is Ordinal Turing Machine-computable if and only if it is $\... | https://mathoverflow.net/users/20597 | At what level of the analytic hierarchy do Cohen reals lie? | You had asked about the perfect set $P$, but before treating that, let me first explain what is the complexity of the set of all $L$-generic Cohen reals:
**Theorem.** The set of reals that are $L$-generic for Cohen
forcing is defined by a $\Pi^1\_2$ definition. If non-empty, it is
not defined by any $\Sigma^1\_2$ def... | 3 | https://mathoverflow.net/users/1946 | 186985 | 92,798 |
https://mathoverflow.net/questions/186977 | 2 | Let $h:\mathbb{N}\rightarrow\mathbb{C}$ be a bounded multiplicative function with $h(p)=0$. The motivation for this question is just a general enquiry and, since I suppose it has already been considered, it seems worthwhile asking it here.
>
> What is the best possible bound on the modulus of the sum
> $$S(x)=\su... | https://mathoverflow.net/users/10980 | Bound for sums of bounded multiplicative functions that are zero at primes | To expand on Lucia's comment, we note that we can assume without loss of generality that each coefficient of $h$ is bounded by $1$, in which case $S(x)$ is largest when $h(p^k) = 1$ for all $k \geq 2$ (and of course $h(p) = 0$). So
\[S(x) = \#\left\{n \leq x : p | n \implies p^2 | n\right\}.\]
This is the number of [po... | 4 | https://mathoverflow.net/users/3803 | 186989 | 92,800 |
https://mathoverflow.net/questions/186856 | 4 | I am curious whether there are any Liouville theorems for the following pde:
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$
where $a(x)$ is a smooth bounded vector field with $ |x| |a(x)| \rightarrow 0$ as $ |x| \rightarrow \infty$. As far as the solutions $ \phi$ I am assuming that $ \p... | https://mathoverflow.net/users/29444 | A Liouville theorem involving an advection term | If $\Delta u + b(x) \cdot \nabla u = 0$ and $u$ is bounded, then $u$ is constant provided $|b| = O(1/(1+|x|))$. This follows from scaling, the Harnack inequality and the maximum principle.
Indeed, add a constant so that $\inf\_{\mathbb{R}^n} u = 0$. Let $v(x) = u(Rx)$ for $R$ large. Then $v$ solves the equation
$$\De... | 7 | https://mathoverflow.net/users/16659 | 186991 | 92,802 |
https://mathoverflow.net/questions/185739 | 11 | I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them there are idealizations and valuation domains. But the first non-noetherian ring we are thinking about is $K[X\_1,\dots,X\_n,\dots]$, $K$ a field. The finitely generated ideals of this ring have primary decomposition,... | https://mathoverflow.net/users/23950 | Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$? | Another example using idealization construction (i.e. $(a,b)(c,d)=(ac,ad+bc)$):
1. start with $R=k[T]\_{(T)}+k(T)$, with $k=$prime field contained in $K$;
2. primary ideals of $R$ are $(0)+(0)$, $(0)+k(T)$, and $(T^n)+k(T)$ $(n>0)$;
3. the ideal $I=(0)+k[T]\_{(T)}$ is irreducible and not primary;
4. $R$ is countable ... | 2 | https://mathoverflow.net/users/59248 | 186999 | 92,806 |
https://mathoverflow.net/questions/186943 | 6 | There is a classical theorem of Jackson stating that the $N$-th partial sum $S\_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - S\_N f(x)| \leq c \frac{K \log N}{N}
$$
uniformly for all $x \in [0,1]$, where $c$ is an absolute constant and $K$ is t... | https://mathoverflow.net/users/46852 | Jackson's theorem for partial sum of Fourier series | The estimate in the wikipedia article you linked does what you want because the modulus of continuity satisfies $\omega(2\pi/N)\le 2\pi L/N$ ($L$ = Lipschitz constant), but here's a derivation from scratch:
For convenience, let's focus on $x=0$, and assume that $f(0)=0$. We want to bound
$$
(S\_Nf)(0) = \int\_{-\pi}^... | 6 | https://mathoverflow.net/users/48839 | 187007 | 92,811 |
https://mathoverflow.net/questions/186907 | 3 | Let $A$ be a $C^\*$-algebra. An *extension of $A$ by the compact operators $K$* is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$.
Two embeddings $\epsilon\_1$ and $\epsilon\_2$ are *weakly equivalent* if $\;u \epsilon\_1 (\cdot)u^\* = \epsilon\_2(\cdot)$ for some unitary $u \in B(H)/K$ and *strongly... | https://mathoverflow.net/users/33290 | Strong and weak equivalence of C$^∗$-extensions by compacts | Let $C(H) = B(H)/K(H)$ be the Calkin algebra.
>
> Apparently weak equivalence classes are strictly larger in general.
>
>
>
Yes, the easiest example will be $A = C(H)$ with $\epsilon\_1: C(H) \to C(H)$ the identity map and $\epsilon\_2: x \mapsto s^\*xs$ where $s \in C(H)$ is the image of a unilateral shift $S... | 4 | https://mathoverflow.net/users/23141 | 187011 | 92,813 |
https://mathoverflow.net/questions/187017 | 0 | I'm not sure whether the level of this question is suitable for Mathoverflow.
Let $M$ be a smooth manifold, $E$ and $F$ are finite dimensional (smooth) vector bundles on $M$. Let $\phi: E\rightarrow F$ be a bundle map.
We know that $\ker\phi$ is not necessarily a vector bundle if $\phi$ is not surjective. But is it... | https://mathoverflow.net/users/24965 | Is the kernel of a map between finite dimensional vector bundles still of finite type? | No. Namely, the rank of $\Phi:E\to F$ cannot drop locally (some minor, in local frames, has $\det\ne 0$). But it can increase locally. So the rank of $ker(\Phi)$ can drop locally. But the rank of $\Psi:G\to E$ cannot drop locally.
| 2 | https://mathoverflow.net/users/26935 | 187019 | 92,816 |
https://mathoverflow.net/questions/120817 | 9 | A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler compact complex manifold admits a hermitian metric such that the wedge product of two ${\bar{\partial}}$-harmonic forms is ${\bar{\partial}}$-har... | https://mathoverflow.net/users/30172 | ${\bar{\partial}}$-geometrically formal ? | In a paper by S. Torelli and A. Tomassini, "On Dolbeault formality and small deformations" (to appear in Internat. J. Math.), the authors study (geometrically) Dolbeault formality. In particular, they investigate the behaviour of (geometrically) Dolbeault formality under small deformations of the complex structure.
I... | 3 | https://mathoverflow.net/users/29341 | 187027 | 92,820 |
https://mathoverflow.net/questions/187023 | 3 | I am looking for a good reference for the uniformization theorem for Riemann surfaces, which states that each simply connected Riemannian surface is conformally equivalent to the complex plane $\mathbb{C}$, the Riemann sphere $\hat{\mathbb{C}}$ or the unit disk $\mathbb{D}$.
I know one proof from Ahlfors book, where ... | https://mathoverflow.net/users/26608 | A good reference for uniformization theorem for compact and non-compact Riemann surface | Good reference in English is Hubbard, Teichmuller theory, Matrix editions, Ithaca, NY, 2006, MR2245223. There is a very good reference in French,
H. P. de Saint Gervais, Uniformisation des surfaces de Riemann, ENS Editions, 2010.
It is a whole book dedicated to the Uniformisation theory and its history.
A short easil... | 7 | https://mathoverflow.net/users/25510 | 187054 | 92,827 |
https://mathoverflow.net/questions/187052 | 1 | Let $E\to X$ be a complex vector bundle over a compact Kahler surface $X$. Assume $c\_{i}(E)\in H^{i,i}(X)$ for all i. Does the bundle $E$ admit a holomorphic structure?
| https://mathoverflow.net/users/58022 | On holomorphic vector bundles over compact Kahler surfaces | For line bundles, this is the Lefschetz (1,1)-theorem which says that the map $H^1(X,\mathcal{O}^\times)\to H^2(X,\mathbb{Z})$ from the exponential sequence surjects onto the $H^{1,1}(X)$-part.
For higher rank bundles, the answer is positive whenever $X$ is additionally assumed to be projective. In this case, a theor... | 3 | https://mathoverflow.net/users/50846 | 187058 | 92,829 |
https://mathoverflow.net/questions/181828 | 11 | **Background:**
The little $k$-cubes operad is the $(\infty,1)$-operad defined by embedding disjoint unions of $k$-dimensional open cubes rectilinearly into one another, that is using maps $(0,1)^k\rightarrow (0,1)^k$ of the form $(x\_i)\mapsto (a\_i i\_k+b\_i)$ for $x\_i\in (0,1)$ and $a\_i\geq 0$ for $i=1,...,k$. 2... | https://mathoverflow.net/users/27870 | $k$-Disk algebras versus $E_k$ algebras | There is an unfortunate clash of terminologies here. Traditionally, the little discs operad comes in two variants:
* the "usual" $\mathtt{D}\_n$: the space of arity $r$ operations consists of embeddings of that do not allow rotations (with some other conditions). In other words, such embeddings $D^n \hookrightarrow D... | 11 | https://mathoverflow.net/users/36146 | 187059 | 92,830 |
https://mathoverflow.net/questions/187063 | 4 | Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional to the square of the mutual distance. Assume that the sphere itself poses no friction, so that points are free to move. ... | https://mathoverflow.net/users/54625 | Stable equilibria of points on the 2-sphere | This is the famous [Thomson problem](https://en.wikipedia.org/wiki/Thomson_problem). You can find a list of optimal configurations and many references on the Wikipedia page. Your intuitions for $n=7, 8, 9, 20$ are wrong, and $n=5$ is not that obvious. By the way, your description of the interaction between electrons is... | 5 | https://mathoverflow.net/users/20595 | 187066 | 92,833 |
https://mathoverflow.net/questions/187062 | 1 | in this question <https://mathoverflow.net/a/55528/61732> it is stated that a normal variety is CM outside a set of codim at least 3. That would imply that normal surfaces are CM. [edited:] I wanted to ask if under the assumption that the variety is CM, the canonical divisor being Cartier is equivalent to the local rin... | https://mathoverflow.net/users/61732 | Normal surface is Cohen-Macaulay - reference | Serre's theorem tells you that normal implies $S\_2$, that is, Cohen-Macaulay in codimension 2.
I am not sure I understand your second question, but of course Gorenstein implies Cohen-Macaulay. Finally I am sure that Eisenbud's "Commutative Algebra: with a View Toward Algebraic Geometry" contains all of these well-kno... | 1 | https://mathoverflow.net/users/40297 | 187067 | 92,834 |
https://mathoverflow.net/questions/187047 | 3 | Let $X$ be a smooth projective variety over a scheme $S$ being the spectrum of a discrete valuation ring of mixed characteristic $(0,p)$. Let $X\_n$ be the respective thickenings of the reduced special fiber $X\_1$. Then there is the following algebraization isomorphism:
$$Pic(X)\xrightarrow{\sim} \varprojlim Pic(X\_n)... | https://mathoverflow.net/users/61632 | Algebraization isomorphism, formal existence, mod p | As you imply, you can identify $\varprojlim\_n \mathrm{Pic}(X\_n)$ with $\mathrm{Pic}(\hat{X})$, the Picard group of the completion of $X$, but think that you can deduce that the natural functor $\mathrm{Pic}(X)\rightarrow \mathrm{Pic}(\hat{X})$ is an equivalence from the theorem you state. What does follow is that the... | 1 | https://mathoverflow.net/users/13647 | 187074 | 92,836 |
https://mathoverflow.net/questions/186964 | -4 | We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and let $\chi\_\omega$ denotes the indicator function on a subdomain $\omega$ of $\Omega$.
Do we have
$$\chi\_\omega\Delta... | https://mathoverflow.net/users/24060 | Does the Laplacian commutes with the indicator function | This fails even in the case of the one dimensional case on an interval since the rhs picks up $\delta$ functions at the endpoints due to the jump singularities there and these are abseent in the lhs.
| 2 | https://mathoverflow.net/users/61738 | 187077 | 92,837 |
https://mathoverflow.net/questions/187056 | 2 | I'm looking at the following game:
2 symmetric players $\{A,B\}$, each choose a number $x\_a,x\_b\in [0,1]$.
The utility of player $A$ is:
$$U\_A = \
\begin{cases} p\cdot x\_a &\mbox{if } x\_a> x\_b \\
\frac{p\cdot x\_a+ (1-p)\cdot (1-x\_a)}{2} &\mbox{if } x\_a= x\_b\\
(1-p)(1-x\_a)\ \ \ \ \ \ \ \ \ \ & \mbox{else... | https://mathoverflow.net/users/47499 | How to find equilibrium of the following game? | Since you say it's symmetric, I'm assuming the utility of player $B$ is
$$U\_B = \
\begin{cases} p \cdot x\_b &\mbox{if } x\_b> x\_a \\
\frac{p\cdot x\_b+ (1-p)\cdot (1-x\_b)}{2} &\mbox{if } x\_a= x\_b\\
(1-p)(1-x\_b)\ \ \ \ \ \ \ \ \ \ & \mbox{else}\end{cases}
$$
For $1/2 \le p \le 2/3$, consider the mixed strate... | 1 | https://mathoverflow.net/users/13650 | 187079 | 92,838 |
https://mathoverflow.net/questions/187061 | 1 | I'm reading an article I wrote my doctoral supervisor. In this article he states that if $G$ be a hypercentral group and suppose that $G$ is generated by (a finite number of) Prufer subgroups. Then $G$ is abelian.
I asked him about this statement. He told me it was true. He said that in the good books of reference I ... | https://mathoverflow.net/users/61637 | A condition for Hypercentral Groups to be Abelian | This should follow along the lines of Robinson's book "Finiteness conditions and generalized soluble groups", Part 2, Section 9.2. I will try to sketch an argument using Robinson's terminology. Since $G$ is hypercentral, it is locally nilpotent. By your assumption it is therefore periodic. Since it is generated by Prue... | 3 | https://mathoverflow.net/users/6339 | 187090 | 92,839 |
https://mathoverflow.net/questions/187087 | 7 | The numbers game is a (one-player) game played on a finite graph with an initial assignment of numbers to its vertices, studied by Alon, Bj\"orner, Brenti, Donnelly, Eriksson, Krasikov, Mozes, Peres, Proctor, Wildberger, and probably others as well (see <http://arxiv.org/abs/math/0610702> for a detailed bibliography). ... | https://mathoverflow.net/users/3621 | Origin of the numbers game | The problem is due to [Elias Wegert](http://www.mathe.tu-freiberg.de/ana/mitarbeiter/elias-wegert), see [Relaxation procedures on graphs](http://www2.math.ou.edu/~jalbert/putnam/DAM6960.pdf).
[This blog post](http://mattbakerblog.wordpress.com/2014/02/25/the-pentagon-problem/) discusses the problem.
| 4 | https://mathoverflow.net/users/12674 | 187092 | 92,841 |
https://mathoverflow.net/questions/187082 | 11 | Who is the most ancient mathematician of which we have a photograph?
(or, in the same vein, what is the oldest photograph of a mathematician)
A quick search on MacTutor History of Mathematics gives Binet (b.1786) as a pretender with Cauchy (b.1789) coming close...
| https://mathoverflow.net/users/61737 | Oldest photographed mathematician | Most ancient: Wikipedia has a [daguerreotype of Gauss](https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss) (1777–1855) on his deathbed. Or possibly [Farkas Bolyai](https://archive.org/stream/jstor-2968693#page/n2) (1775–1856) in what look like similar circumstances.
Less ancient, but allegedly photographed earlier: G... | 27 | https://mathoverflow.net/users/19276 | 187093 | 92,842 |
https://mathoverflow.net/questions/187102 | 0 | Green's function of a differential operator contains a lot of information of that operator. In particular, if we have a differential operator on a compact manifold with discrete spectrum, then Green's function satisfies
$$G(x,x',\lambda)=\sum\_n\frac{\psi\_n^\*(x)\psi\_n(x)}{\lambda-\lambda\_n}$$
where $\lambda\_n... | https://mathoverflow.net/users/18261 | Green's function and eigenvalues with multiplicity | I assume that your differential operator is linear unbounded with compact resolvent.
Eigenvalues of higher multiplicity have eigenspaces: any basis of the eigenspace form the eigenfunctions for this eigenvalue. They are not unique! But the expression in the Greens function is independent of the choice of an orthonormal... | 2 | https://mathoverflow.net/users/26935 | 187103 | 92,847 |
https://mathoverflow.net/questions/187099 | 3 | Let we have a complex of abelian topological or lie groups $$\ldots \to G\_{n}\to G\_{n+1}\to \ldots$$ such that the image of $G\_{n}$ is a closed subgroup of $G\_{n+1}$. Then we have a complex of fundamental groups $$\ldots \to \pi\_{1}(G\_{n})\to \pi\_{1}(G\_{n+1})\to \ldots$$
>
> Are there some standard theorems... | https://mathoverflow.net/users/36688 | A Comparison between $\pi_{1}$ of cohomology and cohomology of $\pi_{1}$ | If you consider an exact sequence of groups, it's a fibration of topological spaces, so you get a homotopy long exact sequence. So that makes me think $\pi\_n$ behaves like a derived functor from topological abelian groups to abelian groups. In particular this is the derived functor of $\pi\_0$.
So there is the spect... | 2 | https://mathoverflow.net/users/18060 | 187104 | 92,848 |
https://mathoverflow.net/questions/187105 | 5 | Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
| https://mathoverflow.net/users/38889 | Disjoint images of polynomials | I guess you won't be satisfied with the answer $f=0$ and $g=1$. :)
But the answer is yes even if you assume that $f$ and $g$ are nonconstant. For example, consider $f(x)=2x^3$ and $g(x)=(x^3-2)^3$. If $f(a)=g(b)$ for some $a,b \in \mathbb{Q}(\zeta)$, then one finds that $2$ is a cube in $\mathbb{Q}(\zeta)$, which is ... | 18 | https://mathoverflow.net/users/2757 | 187107 | 92,849 |
https://mathoverflow.net/questions/187083 | 17 | How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we assume the boundary is described by the equation $f(x,y,z)=0$, then we have a system of $5+5+3=13$ equations in $15$ variab... | https://mathoverflow.net/users/35959 | Does the boundary of a convex body contain a regular planar pentagon? | A MathSciNet search turns up the following paper: V. V. Makeev, Polygons inscribed in a closed curve and in a three-dimensional convex body, *Journal of Mathematical Sciences* **161** (2009), 419–423 (translated from Russian). The final result in the paper is:
>
> **Corollary**. Each convex body $K\subset \mathbb{R... | 11 | https://mathoverflow.net/users/3106 | 187119 | 92,850 |
https://mathoverflow.net/questions/187121 | 3 | Suppose we have two automorphisms on a graph $G$ such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both pairs simultaneously? More specifically, let $A$ and $B$ be two automorphisms on $G$ and let $x$, $y$, $z$, $w$ be four dis... | https://mathoverflow.net/users/61759 | Graph automorphism that swaps two pairs of nodes | Label the vertices of $C\_5$ cyclically: $x,y,z,w,v$. There is a (unique) automorphism that swaps $x$ and $y$, and another (unique) automorphism that swaps $z$ and $w$, but there is no automorphism that swaps both pairs.
| 8 | https://mathoverflow.net/users/43266 | 187124 | 92,851 |
https://mathoverflow.net/questions/187118 | 2 | In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
& \min \text{ # nonzero components of } x \cr
& \text{s.t.} \cr
& {\left\| {Ax - b} \right\|\_\infty } \le \varepsi... | https://mathoverflow.net/users/40780 | Standard names and methods for this type of fitting minimization | I am not sure if there is a standard name for these type of problems. I would call problems of this type **sparse approximation problems** because you want to solve a linear equation approximately (assuming that $\epsilon$ is small) and sparsely. In statistics these problems (where one really aims to minimize the numbe... | 1 | https://mathoverflow.net/users/9652 | 187127 | 92,852 |
https://mathoverflow.net/questions/187088 | 0 | Consider a complex-oriented multiplicative generalized cohomology theory $h^{\*}(X)$. It is complex-oriented, if by the definition the following two conditions hold:
1) There exists an element $t\in h^{2}(\mathbb{C}P^{\infty})$, restricting (the abuse of notation) to the canonical generator $t$ of $h^{2}(\mathbb{C}P... | https://mathoverflow.net/users/61739 | Definition of Milnor exact sequence and complex-oriented generalized cohomology of $\mathbb{C}P^{\infty}$ | Milnor's paper is called *On axiomatic homology theory* and is published in Pacific J. Math.
Volume 12, Number 1 (1962), 337-341. It is also reprinted in Frank Adams' *Algebraic Topology: A Student's Guide*. It is wonderfully lucid (as with all of Milnor's writings) and should tell you all you need to know.
(The answ... | 2 | https://mathoverflow.net/users/8103 | 187130 | 92,854 |
https://mathoverflow.net/questions/187134 | 12 | I'm studying **Bernard Aupetit: A Primer on Spectral Theory**
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Thank you.
| https://mathoverflow.net/users/52860 | What is the best reference for Spectral theory? | Quite expansive and well detailed are the two books :
* Dunford & Schwartz, *Linear Operators*, Wiley Classics Library, 1971
* Rudin, *Functional Analysis*, McGraw-Hill, 1991
Less specialized but treating very well some parts of the topic is the Brezis, *Functional Analysis* too, and there are many exercises.
Hop... | 8 | https://mathoverflow.net/users/43737 | 187138 | 92,856 |
https://mathoverflow.net/questions/186896 | 9 | Let $T$ be a complete first-order theory. Recall that a formula $\phi(\overline{x},\overline{y})$ has the *finite cover property* (fcp) if for all $n$, there exist $\overline{a}\_1,\dots,\overline{a}\_n$ such that $$T\models \lnot\exists \overline{x} \bigwedge\_{i = 1}^n \phi(\overline{x},\overline{a}\_i),$$
but for al... | https://mathoverflow.net/users/2126 | Is nfcp equivalent to stable + eliminates $\exists^\infty$? | In fact, this theorem is from Classification Theory: see Theorem 4.4 there (the "f.c.p. theorem").
Assertion (8) exactly says that $T^{eq}$ does not eliminate $\exists^{\infty}$ and the theorem says that for stable $T$, this is equivalent to having the f.c.p.
| 7 | https://mathoverflow.net/users/19534 | 187144 | 92,857 |
https://mathoverflow.net/questions/187150 | -1 | Is it possible to convert irrational p-adic numbers to a standard number? Rationals and negative rationals are relatively straightforward, but is there a way to know that for instance $\ldots 2100121201\_{p=3} = 10.0111010220\ldots\_{3}$?
| https://mathoverflow.net/users/61778 | Converting p-adic to decimal | If by "standard" you mean "real", it is not possible : $p$-adic numbers and real ones are really different. By the way, what do you really mean by the "decimal" $10.0111010220...\_{3}$ ?
$\mathbf{R}$ i s a completion of $\mathbf{Q}$ for a precise metric, the usual one. $\mathbf{Q}\_p$ is the completion for an other m... | 1 | https://mathoverflow.net/users/43737 | 187152 | 92,858 |
https://mathoverflow.net/questions/187070 | 6 | Consider an $n$-simplex with vertices given by
$(0,0,\dots,r\_i,r\_{i+1},\dots,r\_n)$ where $r\_1,\dots,r\_n$ are given natural numbers,
and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, unimodular triangulation?
| https://mathoverflow.net/users/1056 | Regular unimodular triangulation for a certain simplex | The answer is yes. Consider the hyperplanes $x\_i=x\_j+m$ and $x\_i=m$ for all $i$ $j$
and integers $m$. These hyperplanes triangulate the entire space into unimodular simplices, and the hyperplanes are compatible with the hyperplanes that determine the polytope in the question (meaning each face is determined by an in... | 0 | https://mathoverflow.net/users/1056 | 187154 | 92,860 |
https://mathoverflow.net/questions/187157 | 11 | **Question:** Is there a group $G$ and a CW-complex $X$ such that
1) $X$ is homotopy equivalent to the circle $S^{1}$.
2) $G$ acts on $X$
3) the space of fixed points $X^{G}$ is weakly equivalent to $S^{2}$ ?
| https://mathoverflow.net/users/61328 | from a circle to higher spheres | A very general answer is given by Tony Elmendorf's paper *Systems of Fixed Point Sets*, <http://www.ams.org/journals/tran/1983-277-01/S0002-9947-1983-0690052-0/>. Very loosely speaking, it says that, if you can write down a reasonable system of fixed sets, it can be realized up to homotopy equivalence.
Take, for exam... | 13 | https://mathoverflow.net/users/58888 | 187170 | 92,866 |
https://mathoverflow.net/questions/187173 | 0 | Is $\lim\_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta\_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta\_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number of $\mathbb{Q}^{\text{ab}}$-rational points of $y^3 - x^3 - x - 1$ is finite.
If the answer is negative, then a slight... | https://mathoverflow.net/users/38889 | The number of solutions of a Diophantine equation | To answer your concrete question, first, your curve (or rather, its projective closure) is isomorphic to the elliptic curve 8649b1, $y^2 + y = x^3 - 8$. Already over $\mathbb Q$, it has rank two and therefore infinitely many rational points.
More generally, for any elliptic curve $E$ over $\mathbb Q$, you should be a... | 8 | https://mathoverflow.net/users/21146 | 187175 | 92,868 |
https://mathoverflow.net/questions/187178 | -3 | I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these constants could be called Rcof, +cof, .cof,
<cof, 0cof, 1cof, etc.
For sure, in this ZFC axiomatic system, the Real numbers... | https://mathoverflow.net/users/51212 | An axiomatic system with a set of constants that form a complete ordered field | Assuming that ZFC is consistent, then your system will not settle the question of whether 1cof is an element of 2cof or vice versa. One can see this by observing that from any model of ZFC we can make various models of your system, by interpreting your new constant symbols 1cof, 2cof etc. in any of the various ways tha... | 0 | https://mathoverflow.net/users/1946 | 187181 | 92,869 |
https://mathoverflow.net/questions/187117 | 4 | I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A\_\infty$) ring spectra. One thought I had was to think of cocommutative Hopf-algebras in this setting as functors $H:Alg(\mathbb{S})\to E\_\infty Spc$, in other words, functors from associative algebra... | https://mathoverflow.net/users/11546 | Hopf-algebras in associative ring spectra | Okay, I'll take a crack at it, although this is basically just a translation of Tyler's comments from the [chat room](http://chat.stackexchange.com/rooms/9417/homotopy-theory). Recall that given a topological group (or group like $A\_\infty$-space) $A$, we can form the bar complex associated to $A$, $BA$ such that $\Om... | 6 | https://mathoverflow.net/users/11546 | 187189 | 92,871 |
https://mathoverflow.net/questions/186451 | 9 | That's a question from MSE ([here](https://math.stackexchange.com/questions/1004179/space-of-borel-measurable-maps)) that did not receive any answer for some days. I migrate it to MO.
Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f: X \to Y$. Is $M(X,Y)$ also standard ... | https://mathoverflow.net/users/58682 | Space of Borel measurable maps | With the $\sigma$-algebra specified to be the trace of the product $\sigma$-algebra, as in yadaddy's comment: No.
For let $X=[0,1]$ and $Y=\{0,1\}$, let $\mathcal A$ be the product $\sigma$-algebra on $Y^X$, and let $\mathcal B$ denote the trace of $\mathcal A$ on $M(X,Y)$. Then $\mathcal B$ is not countably generate... | 4 | https://mathoverflow.net/users/26591 | 187191 | 92,873 |
https://mathoverflow.net/questions/187180 | 5 | For example, how to prove
$\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$
where $\sigma > 0$ is a parameter, is positive definite? I have tried to construct the target kernel base on some negative definite kernel, such as the square distance $||x-y||^2$, but doesn't se... | https://mathoverflow.net/users/61795 | How to prove that a kernel is positive definite? | Here is a simple trick that relies only on elementary ideas.
First, we use the observation that for $t>0$
\begin{equation\*}
\exp(-t\|x-y\|^2) = \exp(-t\|x\|^2)\exp(2t\langle x, y\rangle)\exp(-t\|y\|^2),
\end{equation\*}
which is clearly positive definite (use $\langle x,y\rangle^k$ is pd for $k\ge0$).
Now, just u... | 10 | https://mathoverflow.net/users/8430 | 187192 | 92,874 |
https://mathoverflow.net/questions/187199 | 5 | Consider a Riemannian manifold $M$ with sectional curvatures $K\ge 0$ and let $\Pi$ be a 2-plane in the tangent space of $M$ at a point $p$. In a small enough neighborhood $U$ of 0 the exponential map will be a diffeomorphism and $S=\text{exp}(U\cap \Pi)$ will be an open surface in $M$. The Gaussian curvature of this s... | https://mathoverflow.net/users/14454 | Gaussian Curvature of Exponentiated 2-Planes | The answer is "no".
Say, there is a non-negatively curved Riemannian metric on $\mathbb R^3$ with a point $p\in M$ and a sectional direction $\Pi$ such that the surface $S$ has negaive curvature at the points arbitrary close to $p$.
First notice that $S$ is a ruled surface, in particular it has nonpositive Gauss cur... | 2 | https://mathoverflow.net/users/1441 | 187210 | 92,875 |
https://mathoverflow.net/questions/187214 | 1 | Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that maps the area to a m-sphere? Bounded means the area $A<\infty$ with the boundary defined by the polynomials. To partition m... | https://mathoverflow.net/users/14024 | Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that | See for instance:
S. Łojasiewicz, *Triangulation of semi-analytic sets*, Ann. Scuola Norm. Sup. di Pisa, ser. 3, 18.4 (1964), pp. 449–474,
or start a search with the key-words "**Triangulation**" and "**semi-algebraic**", for more recent results, or more suitable to your needs.
| 1 | https://mathoverflow.net/users/6101 | 187219 | 92,876 |
https://mathoverflow.net/questions/187209 | 3 | Does there exist an elliptic curve over a number field $K$ such that $WC(E/K)\cong H^1(G\_K, E)$ is trivial?
| https://mathoverflow.net/users/56286 | Trivial Weil-Châtelet group | Weil-Chatelet groups are huge. A theorem of Shafarevich states that if $n \ge 2$ and if $E$ is an elliptic curve (or an abelian variety) over a number field $k$ then $H^1(G\_k,E)$ has infinitely many elements of order $n$. See Section 5 of Pete Clark's lecture on WC groups:
[http://alpha.math.uga.edu/~pete/wcnotes.pdf]... | 10 | https://mathoverflow.net/users/4140 | 187224 | 92,880 |
https://mathoverflow.net/questions/187145 | 5 | Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in $\Omega$ are bounded and uniformly continuous in $\Omega$. My question is about the natural differentiation operator $D:C^... | https://mathoverflow.net/users/61771 | When is a `1-form' with continuous coefficients exact? | $\def\ssp{\kern.4mm}
$Here is a sketch of proof of sufficiency of $d\ssp f=0$ , i.e. of $\partial\_i f\_j=\partial\_j f\_i$ (in the distributional sense) assuming that the case where $f$ is $C^1$ is known, for which I refer e.g. to the Poincaré lemma in §V.5 on pages 124−125 in Serge Lang's *Differential Manifolds*, Sp... | 4 | https://mathoverflow.net/users/12643 | 187235 | 92,885 |
https://mathoverflow.net/questions/187233 | 4 | The primes do a nice job of intersecting an arithmetic progression $\{a+dn\}\_{n=0}^\infty$ when $a$ and $d$ are coprime (see [Dirichlet's theorem](http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions)).
I would like a set of integers $S$ such that
* the asymptotic density of $S$ is zero, a... | https://mathoverflow.net/users/29873 | Thin sets that are well-distributed over arithmetic progressions? | Sets of the form $[n^\alpha]$, $\alpha\in(1, \infty)\setminus\mathbb{N}$ are well distributed in arithmetic progressions. More generally we have that for every $\alpha\in(1, \infty)\setminus\mathbb{N}$ and every $\beta\in(0,1)$ the set $\beta n^\alpha\bmod 1$ is equidistributed in $[0,1]$. By Weyl's criterion for equid... | 8 | https://mathoverflow.net/users/37555 | 187238 | 92,887 |
https://mathoverflow.net/questions/187243 | 4 | Let $\alpha, \beta \in \mathbb{R}$. Let $\{x\}$ denote the fractional part of $x$ and let $\|x\| = \min(\{x\}, 1-\{x\})$.
If we assume that $\alpha$ is irrational, then there exists an increasing sequence of integers $(n\_k)\_{k \in \mathbb{N}}$ such that $\|n\_k \alpha - \beta\| \to 0$. Is it possible that the rate... | https://mathoverflow.net/users/61820 | Rate of convergence of an irrational rotation | For the algebraic case a lot is known. There is a deep result of Gelfond (see A. O. Gelfond, Transcendental and Algebraic Numbers, Dover, New York, 1960) that if $\lambda$ is an algebraic number of absolute value 1, then for every $\epsilon>0$ there is a constant $C>0$ such that $$|\lambda^n-1|>Ce^{-n\epsilon}$$ for al... | 7 | https://mathoverflow.net/users/8112 | 187256 | 92,895 |
https://mathoverflow.net/questions/187254 | 8 | **Question: Let $G,H$ be profinite groups of cardinality $|\mathbb{R}|$, with the same finite quotients (here I only consider quotients by normal, *open* subgroups). Then are $G$ and $H$ isomorphic?**
Background: First, note that one needs some sort of cardinality assumption, as otherwise one could take $G=\mathbb{F}... | https://mathoverflow.net/users/4181 | Are profinite groups of cardinality $|\mathbb{R}|$ determined by their finite quotients? | Without the extra finiteness conditions, but also not relying on any set-theoretic assumptions, take $G = \widehat{\mathbb{Z}}^{\mathbb{N}}$ and $H = G \times A$ for a nontrivial finite abelian group $A$. Both $G$ and $H$ have every possible finite abelian quotient, but only $H$ has torsion.
| 15 | https://mathoverflow.net/users/290 | 187257 | 92,896 |
https://mathoverflow.net/questions/187249 | 2 | [Wikipedia](http://en.wikipedia.org/wiki/Slender_group) states that every homomorphism from the Baer–Specker group ${\bf Z}^{\bf N}$ into a slender group factors through ${\bf Z}^n$ for some natural number $n$. Where can I find a proof?
If this is not trivial, then I would also need a reference to insert in my paper... | https://mathoverflow.net/users/49372 | Every homomorphism from the Baer–Specker group into a slender group factors through ${\bf Z}^n$, why? | The key word is **every** in the definition.
Assume that the image of $f\colon\Bbb Z^{\Bbb N}\to G$ is infinitely generated (as otherwise we are done). Denote by $p\_i$ the projections of $\Bbb Z^{\Bbb N}=\prod\Bbb Z$ to its factors and let $\inf a:=\min\{i\,|\,p\_i(a)\ne0\}$ for $a\in\Bbb Z^{\Bbb N}$. Then, playing... | 5 | https://mathoverflow.net/users/44953 | 187258 | 92,897 |
https://mathoverflow.net/questions/187250 | 7 | [Wikipedia](http://en.wikipedia.org/wiki/Slender_group) states that every free abelian group is slender. Where can I find a proof?
If this is not trivial, then I will also need a reference to use in my paper.
| https://mathoverflow.net/users/49372 | Every free abelian group is slender, why? | This follows fairly straightforwardly from the fact that every map from $A=\mathbb{Z}^\mathbb{N}$ to $\mathbb{Z}$ factors through a finite subproduct (let me call this "Specker's theorem"). Suppose $F$ is a free abelian group and $f:A\to F$ is a homomorphism. Since a subgroup of a free abelian group is free, we may ass... | 8 | https://mathoverflow.net/users/75 | 187259 | 92,898 |
https://mathoverflow.net/questions/187172 | 13 | Fundamental sequence for a countable limit ordinal $\alpha$ is an increasing sequence $\{\alpha[i]\}$ of ordinals of length $\omega$ such that $\lim\_{i\rightarrow\omega}\alpha[i]=\alpha$. There are many (continuum many, in fact) possible choices for fundamental sequence for any ordinal, some are quite natural, like $\... | https://mathoverflow.net/users/30186 | Peano arithmetic vs. fast-growing hierarchy with pathological fundamental sequences | The answer is no. Choose a fundamental sequence for $\epsilon\_0$ itself in the usual way, which I think is $\epsilon\_0[n]=\omega^{\omega^{{\vdots}^\omega}}$, and then modify the earlier fundamental sequences for $\alpha=\epsilon\_0[n]$ by making it start with $0,1,2,\ldots,n$, before resuming with the usual values. I... | 13 | https://mathoverflow.net/users/1946 | 187261 | 92,899 |
https://mathoverflow.net/questions/186938 | 22 | Consider the local rings
$$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$
and
$$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$
Is $R$ isomorphic to $S$?
**Some context**: I am trying to understand formal neighborhoods of points on certain varieties. I expect one answer, and I'm g... | https://mathoverflow.net/users/10273 | Two (other) rings...are they isomorphic? | Consider the map $\varphi:S\to R$ given by
$$\varphi(x) = \frac{4x}{4-x},$$
and similarly for $y$, $z$, and $w$. One needs to check that this is well-defined;
indeed, $\varphi$ maps $xyz+xyw+xzw+yzw+xyzw$ to
$$\frac{4^4(xyz+xyw+xzw+yzw)}{(4-x)(4-y)(4-z)(4-w)}.$$
Since $\varphi$ obviously induces an isomorphism on the ... | 11 | https://mathoverflow.net/users/10273 | 187264 | 92,901 |
https://mathoverflow.net/questions/187269 | 6 | Let $G$ be a nontrivial finite group. Given $n\in\mathbb{Z}\_{\geq 1}$,
let $G^n$ be the cartesian product of $n$ copies of $G$.
Further let $S\subseteq G^n$ be a generating set of $G^n$.
**Question:** Do we always have $|S|\geq n$?
| https://mathoverflow.net/users/11765 | Bounding from below the cardinality of a set of generators of the $n$-fold cartesian product of a finite group | The answer is no. -- For example, ${{\rm A}\_5}^3$ is $2$-generated.
-- We have e.g.
$$
\langle (2,5)(3,4)(6,7,8,9,10)(11,12,15), (1,3,2,4,5)(6,7,9,10,8)(11,13,12,14,15) \rangle \ \cong \
{{\rm A}\_5}^3.
$$
Finding such generator pair is easy -- just pick two random elements until you find some
which generate.
Act... | 9 | https://mathoverflow.net/users/28104 | 187270 | 92,902 |
https://mathoverflow.net/questions/187057 | 5 | A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid groups are residually nilpotent.
Consider the pure homotopy braid group which is a factor group of the pure braid gro... | https://mathoverflow.net/users/15770 | Are homotopy braid groups residually nilpotent? | The pure homotopy braid group is torsion-free nilpotent. This is Proposition 1.12 in N. Habegger and X.-S. Lin, The classification of links up to link homotopy, Journal of the American Mathematical Society 3 (1990), 389-419, where $\mathcal{A}(K)$ is the pure homotopy braid group. However, the authors do not use the te... | 5 | https://mathoverflow.net/users/46142 | 187273 | 92,904 |
https://mathoverflow.net/questions/187278 | 14 | If we start with the **category of finite complexes** and continuous maps, and then identify two morphisms iff they are homotopic, we get the **homotopy category of finite complexes**, and it is trivial to observe that a continuous map $f:X\to Y$ is an isomorphism in this category iff it is a homotopy equivalence. Does... | https://mathoverflow.net/users/35353 | Is there a category whose isomorphisms are precisely the simple homotopy equivalences? | Unfortunately not. Let's say $D$ is a category and $F$ is a functor from finite complexes to $D$ that takes simple homotopy equivalences to isomorphisms.
We note that for any finite complex $X$, the inclusion $i\_0: X \to [0,1] \times X$ is a simple homotopy equivalence. We can factor it as a composition of elementar... | 13 | https://mathoverflow.net/users/360 | 187283 | 92,908 |
https://mathoverflow.net/questions/186828 | 7 | This is a subsequence of primes $p$ for which $p^2-1$ has at most 6 prime divisors counted with multiplicity.
This sequence described in the question is the sequence [A079153 in OEIS](https://oeis.org/A079153).
I could not find references on the first sequence, but I found a mention of the second sequence in anoth... | https://mathoverflow.net/users/7060 | Are there infinitely many primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity? | At the present time, the best (general) result is due to James Maynard, who has proven that any admissible 3-tuple takes at most 7 prime factors infinitely often (see [this paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9030782&fileId=S0305004113000339)). In particular, (using a certain... | 11 | https://mathoverflow.net/users/3199 | 187284 | 92,909 |
https://mathoverflow.net/questions/187149 | 11 | In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
*Let $A$ be an abelian variety [over an alg. closed field $k$, say - my assumption]. Let $X$ be a non-singular subvariety of $A$.
Assume that every curve in $X$ generates $A$. ... | https://mathoverflow.net/users/17308 | On a proposition in Hartshorne's paper "Ample vector bundles on curves" | The assertion (X) is false in any characteristic $p > 0$ for any $Y$ with genus at least 2. (It is true and easy for $Y$ of genus 1, and true and easy and uninteresting for $Y$ of genus 0.)
To see this, we may and do choose a subgroup scheme $G \subset J := {\rm{Jac}}(Y)$ of the nonzero infinitesimal Frobenius kerne... | 7 | https://mathoverflow.net/users/52824 | 187291 | 92,911 |
https://mathoverflow.net/questions/185891 | 4 | I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, probably at the cost of carrying out a Monte Carlo simulation. Any pointers?
| https://mathoverflow.net/users/1580 | Stochastic methods for solving very high-dimensional PDE | It seems to me this question "has not received enough attention" because of the conflation of two issues: dimensional reduction of a high-dimensional PDE and stochastic (Monte Carlo) integration of the PDE.
The socalled ["curse of dimensionality"](http://en.wikipedia.org/wiki/Curse_of_dimensionality) refers to the f... | 5 | https://mathoverflow.net/users/11260 | 187304 | 92,914 |
https://mathoverflow.net/questions/187303 | 4 | In complex analysis, we're constantly faced with problems about the analyticity of a function, on which many theorems are developed. I of course know a bunch of formulas and theorems, but could not have any intuitions as I do in real-numbered world. So, what, intuitively, does the property of analyticity mean? Could yo... | https://mathoverflow.net/users/61853 | What does analyticity imply in complex analysis? | EDIT: 11.16.2014, 3:40pm ET. Let me incorporate Liviu Nicolaescu's answer. (I am not sure whether this is legitimate:-)
Analyticity is a property encountered not only in complex analysis: a function of a real variable (or of a p-adic variable) can be analytic. 'Analytic' really means that in a neighborhood of every p... | 17 | https://mathoverflow.net/users/25510 | 187305 | 92,915 |
https://mathoverflow.net/questions/186837 | 6 | According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably equivalent to V.
My question is whether one can transfer the result also to uniform oriented matroids of rank 4 and hig... | https://mathoverflow.net/users/51596 | The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher | Yes, the Universality Theorem works for any rank $r\geq3$. One way to prove this is to consider a rank $3$ oriented matroid $M$ with the desired realization space, then take the dual $M^\*$, do a lexicographic extension in general position and then take the dual again. Repeat this $r-3$ times.
Lexicographic extension... | 8 | https://mathoverflow.net/users/21482 | 187306 | 92,916 |
https://mathoverflow.net/questions/164269 | 3 | Having access to those references, accumulating many results in one domain is always a bless, like Feller's book in probability, Dembo-Zeitouni's large deviation, Grimmett's percolation and recent Optimal Transport of Villani.
There are variants of asymptotic results in probability theory: law of large numbers, cent... | https://mathoverflow.net/users/39212 | References on law of large numbers, CLT and iterated logarithm laws | There is a very recent book, October 2014 if I am not mistaken, by Oleg Klesov, titled "Limit Theorems for Multi-Indexed Sums of Random Variables". It has a fascinating content with good survey of many different limit problems.
Here is the table of content.
* Some Remarks on the Theory of Limit Theorems for Multi-I... | 3 | https://mathoverflow.net/users/39212 | 187318 | 92,922 |
https://mathoverflow.net/questions/187317 | 0 | There are three questions:
1. Please let me know your proof of the following theorem:
If $CP^3$ can be immersed in $R^8$ with an Euler class $W\_{2}(\nu)$ for the normal bundle of $CP^3$ respect to $R^8$ then
$$\int\_{CP^3} \!W\_{{2}} \left( \nu \right) c^2$$
is divisible by 3.
Where $c$ is the cohomological gene... | https://mathoverflow.net/users/61860 | An integrality theorem for immersions of complex projective spaces in the euclidean space | I may have miscalculated, but writing $e(\nu)$ for the Euler class of the normal bundle to such an immersion I find that
$$\int\_{\mathbb{CP}^3} e(\nu) c^2 = 2,$$
which appears to contradict 1).
To see this, write $\nu$ for the oriented 2-plane bundle occurring as the normal bundle of the immersion, so that $T\mathbb... | 2 | https://mathoverflow.net/users/318 | 187325 | 92,926 |
https://mathoverflow.net/questions/89990 | 1 | In Terry Tao's [notes](http://terrytao.wordpress.com/2008/03/28/285g-lecture-1-ricci-flow/#more-293) on the Poincare Conjecture, he makes a jump I can't understand.
>
> From differentiating the identity $g^{\alpha \beta}g\_{\beta \gamma} = \delta^\alpha\_\gamma$ we obtain the variation formula $\frac{d}{dt}g^{\alph... | https://mathoverflow.net/users/17532 | Variation formula of a metric | As mentioned in the comments, once you're comfortable with indices, this is just an exercise in calculus. Differentiating both sides of the identity $g^{\alpha\gamma}g\_{\gamma\zeta} = \delta^{\alpha}\_{\zeta}$ with respect to the parameter $t$, we see that
\begin{align\*}
\frac{d}{dt}\left(g^{\alpha\gamma}g\_{\gamma... | 7 | https://mathoverflow.net/users/21564 | 187339 | 92,930 |
https://mathoverflow.net/questions/187297 | -1 | I am interested in the type of program, which is given as input to a Universal Turing Machine (UTM) with language $L$, and for which it holds that every possible finite string $s$ of symbols in $L$ appears in the output. I am writing a paper in which I call such a program a Universal Turing Program (UTP). There are inf... | https://mathoverflow.net/users/61387 | Can an algorithm decide whether a program computes all strings? | The answer seems to be no, this is not decidable.
You seem to have a concept in mind of what it means for a program
$p$ to be UTP, and it involves the idea that pieces of any given
computation history of any other program (on some fixed input)
appear on the tape during the computation of $p$ on the trivial
input.
A... | 4 | https://mathoverflow.net/users/1946 | 187354 | 92,937 |
https://mathoverflow.net/questions/187110 | 19 | So my question refers to families of elliptic curves over the $\mathbb{A}^1\_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal).
Some sources give an equation for such a family, namely $$E\_1 := y^2 + xy = x^3 - \frac{36}{j-1728}x - \frac{1}{j... | https://mathoverflow.net/users/15242 | Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728 | Question 1: Yes, there is a nontrivial section, namely
$s: (x,y) = (-1/36, y\_0)$ where $y\_0$ is either solution of $y^2-y/36=-1/36^3$
(i.e. $y = 1/72 \pm 1/18^{3/2}$).
[Note that the numerator $36$ in the equation for $E\_1$
is a typo should be $36x$.]
Using the theory of elliptic surfaces we can show that in fact... | 20 | https://mathoverflow.net/users/14830 | 187364 | 92,939 |
https://mathoverflow.net/questions/84823 | 5 | As is well-known, with the finite set of the first $n$ primes $p\_1,\dots ,p\_n$ one can find all primes exactly (i.e. without false positives) with the Eratosthenes Sieve in the interval $[p\_n+1;p\_{n+1}^2]$. Of course, that requires computing $p\_{n+1}$ first to know where $p\_{n+1}^2$ is, but then one can sieve wit... | https://mathoverflow.net/users/469 | Recursivity of the primes | Just to close off this old question, noting $(q\_n)$ the sequence of first primes produced (2, 3, 11, 127, 16139, 260 467 367, 67 843 249 271 912 789, ...) I'll make the guess that $\ln(\ln(q\_n))$ is asymptotic to $n\ln(2)$.
Supporting data:
```
n q_n ln(l(q_n)) ln(ln(q_n))-ln(ln(q_{n-1}))
1 ... | 1 | https://mathoverflow.net/users/469 | 187369 | 92,940 |
https://mathoverflow.net/questions/187363 | 2 | Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$, working in the space $H\_0^1(\Omega)$ with the inner product
$$(u,v)\_{H\_0^1} = \int\_\Omega \nabla u \cdot \nabla v$$
for $u\in H\_0^1$ and $\mu(\{|\nabla u|>1\}) >0$, is there a way to define a function $v$ such that
$$(u,v)\_{H\_0^1} = \int\_{\{|\nabla u|>... | https://mathoverflow.net/users/49551 | Sobolev Space, "characteristic function" for the weak derivative | There is generally no $v\in H^1\_0$ with $\nabla v=\chi\_{\{|\nabla u|>1\}}\nabla u$.
Consider for example $u:(-1/2,2)\to\mathbb R$, $u(x)=\min\{1-x/2,1+2x\}$. Now $u\in H^1\_0((-1/2,2))$ but there is no $v\in H^1\_0$ with $v'=2\chi\_{(-1/2,0)}$ by the fundamental theorem of calculus.
If $u\in C^1\_0$ and $u$ is cons... | 4 | https://mathoverflow.net/users/55893 | 187374 | 92,942 |
https://mathoverflow.net/questions/187174 | 2 | Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$.
As one could find in G.M. Troianello "Elliptic Differential Equations and Obstacles Problems" (lem. 1.5) there exists a linear and continuos extension operator of $C^{k+\alpha}(\partial\Omega)$ to... | https://mathoverflow.net/users/58541 | Extensions in parabolic Hölder spaces | It is possible to extend the function first to $ Q=[0,T]\times \Omega$ and then to $\mathbb{R}^{N+1}$. The second step is covered by a general statement for anisotropic Besov spaces, theorem 18.5 in [*O. V.Besov, V. I.Il'in, S. M.Nikolskii, Integral representations of functions and imbedding theorems*].
To extend a ... | 1 | https://mathoverflow.net/users/14551 | 187376 | 92,943 |
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