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https://mathoverflow.net/questions/25642 | 7 | It's a big, famous, hard problem in operator algebras to determine if the von Neumann algebras $L(F\_2)$ and $L(F\_3)$ are isomorphic, or not. Here $F\_n$ is the free group on n generators and $L(F\_n)$ is the weak-operator-topology closure of the group algebra $\mathbb C[F\_n]$ acting naturally on the Hilbert space $\... | https://mathoverflow.net/users/406 | Telling group algebras apart | Well, yes. Imagine that you have an algebra $A$ over $\mathbb{C}$ and you want to find out whether it is $\mathbb{C}[F\_2]$ or $\mathbb{C}[F\_3]$. Pick any one-dimensional $A$-module $M$ and compute $\operatorname{Ext}^1\_A(M,M)$. If $A=\mathbb{C}[F\_2]$, you'll get a $2$-dimensional vector space over $\mathbb{C}$, whi... | 10 | https://mathoverflow.net/users/2106 | 25653 | 16,811 |
https://mathoverflow.net/questions/25592 | 49 | While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into the notion of *dual Coxeter number* but am uncertain about the extent of its influence in Lie theory. The term was prob... | https://mathoverflow.net/users/4231 | What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Kac number")? | The dual Coxeter number comes up naturally as a normalization factor for invariant bilinear forms on the Lie algebra: according to Kac's book which you quote, $2h^{\vee}$ is the ratio between the Killing form and the "minimal" bilnear form (the trace form for $sl\_n$), which has the property that the square of the leng... | 26 | https://mathoverflow.net/users/582 | 25680 | 16,827 |
https://mathoverflow.net/questions/25655 | 15 | Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The biggest $k$ such that $\text{Ext}^k(M,.)$ doesn't vanish
Why do we expect this intuitively? Why should lengths of injectiv... | https://mathoverflow.net/users/2837 | Why do modules with small support have high Exts? | To understand what "nice" is in your sense has been a very interesting question in commutative algebra.
In the following discussion I will assume, unless otherwise notice, that $(R,m,k)$ is Noetherian local, and $M$ is finitely generated. Let $(1)$ be the codimension of support of
$M$ and $(2)$ be the biggest non-va... | 14 | https://mathoverflow.net/users/2083 | 25681 | 16,828 |
https://mathoverflow.net/questions/25674 | 3 | I would like to find those integers $x,y$ that satisfies $y^2=x^3+1$. Is there some elementary way to find those?
| https://mathoverflow.net/users/6266 | Integer points of an elliptic curve | I'm not sure the following counts as "elementary" but it's certainly not too difficult.
First show your curve has rank 0. To do this, I asked SAGE, but it's not so hard to do by hand. If you rewrite your curve $C$ as $y^2 = x^3-3x^2+3x$ by renaming $x+1$ as $x$,
then you can write down the 2-isogeneous curve $\overli... | 8 | https://mathoverflow.net/users/412 | 25690 | 16,834 |
https://mathoverflow.net/questions/25664 | 6 | What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions?
I guess I am sometimes bothered by the thought that any random choice over an uncountable set by definition would seem to almost ... | https://mathoverflow.net/users/1320 | Axiom of Computable Choice versus Axiom of Choice | The Axiom of Choice is a principle that applies to arbitrary families of arbitrary sets, and this is a realm where the concept of recusive functions or Turing Turing computability simply does not apply. For example, mathematicians may use AC to select elements of subsets of a (possibly uncountable dimension) vector spa... | 14 | https://mathoverflow.net/users/1946 | 25694 | 16,837 |
https://mathoverflow.net/questions/25534 | 20 | Consider the following optimization problem:
**Problem:** find a monic polynomial $p(x)$ of degree $n$ which minimizes $\max\_{x \in [-1,1]} |p(x)|$.
The solution is given by Chebyshev polynomials:
**Theorem:** Let $T\_n(x) = cos (n \cdot cos^{-1} x)$. Then $(1/2^{n-1}) T\_n$ is a monic polynomial of degree $n$ ... | https://mathoverflow.net/users/1407 | Is there an intuitive explanation for an extremal property of Chebyshev polynomials? | Well, let's try to avoid the hat.
Consider the dual (and obviously equivalent) problem: find the polynomial $p(x):[-1,1]\rightarrow [-1,1]$ of degree $n$ with the greatest possible leading coefficient. We have some information on values of $p$, and need something about its coefficient. Let's try Lagrange's interpolat... | 9 | https://mathoverflow.net/users/4312 | 25697 | 16,839 |
https://mathoverflow.net/questions/25637 | 2 | Does anybody know a translation of [Föllmer: Calcul d'Ito sans probabilités](http://archive.numdam.org/ARCHIVE/SPS/SPS_1981__15_/SPS_1981__15__143_0/SPS_1981__15__143_0.pdf) in English or German?
It seems to be a very interesting text - Abstract: "It is shown that if a deterministic continuous curve has a 'quadratic ... | https://mathoverflow.net/users/1047 | Föllmer: "Calcul d'Ito sans probabilités" in English or German? | Föllmer's approach was mainly adopted by specialists in Mathematical Finance.
Have a look at [*Introduction to Stochastic Calculus for Finance*](http://books.google.co.uk/books?id=KKLa1j-diXwC&printsec=frontcover&dq=Introduction+to+Stochastic+Calculus+for+Finance%3A+A+New+Didactic+Approach&source=bl&ots=-L3eKT8OjQ&s... | 6 | https://mathoverflow.net/users/5371 | 25706 | 16,847 |
https://mathoverflow.net/questions/25707 | 8 | If $p$ is a prime of the form $4n+3$, the class number $h$ of $Q[\sqrt{-p}]$ can be expressed using the number $V$ of quadratic residues and $N$ nonresidues in the interval $[1,\frac{p-1}{2}]$:
* If $p=8n+7$ then $h=V-N$
* If $p=8n+3$ then $h=\frac{1}{3}(V-N)$
This result seems so simple and elegant, but its proof... | https://mathoverflow.net/users/5860 | Intuition for a formula that expresses the class number of an imaginary quadratic field by counting quadratic residues | I'll have to be brief; I can think of two reasons "why":
1. Cauchy and Jacobi proved that for a prime ideal ${\mathfrak p}$ in a complex quadratic number field with prime discriminant, the h-th power of ${\mathfrak p}$ (with $h$ as in your question) is principal. Their technique was what we nowadays know as the Stick... | 4 | https://mathoverflow.net/users/3503 | 25709 | 16,849 |
https://mathoverflow.net/questions/25208 | 6 | Let $\omega^\omega$ be Baire space. If $A,B\subseteq\omega^\omega$ we say that $A$ is Wadge reducible to $B$ (written $A\leq\_w B$) if there is a continuous function $f:\omega^\omega\rightarrow\omega^\omega$ with $x\in A$ if and only if $f(x)\in B$. (In other words $A$ is a continuous preimage of $B$). By identifying s... | https://mathoverflow.net/users/2436 | A subset of Baire space Wadge incomparable to a Borel set? | The same type of diagonalization should allow you to do this.
Suppose $B$ is a Borel set that is not $F\_\sigma$.
Let $(f\_\alpha:\alpha<2^{\aleph\_0})$ list all continuous functions.
At any stage $\alpha<\aleph\_0$ we will have $A\_\alpha$ and $C\_\alpha$ disjoint
subsets of $\omega^\omega$ of size less than $2^... | 6 | https://mathoverflow.net/users/5849 | 25712 | 16,852 |
https://mathoverflow.net/questions/25714 | 6 | I am looking for a reference to the following result.
>
> Let $f:\mathbb R^m\to\mathbb R$ be a convex function.
> Then $f$ is differentiable at all points of outside of a countable union of $(m-1)$-rectifiable sets.
>
>
>
**Comments:**
* $n$-rectifiable set is an image of Lipschitz map from bounded domain ... | https://mathoverflow.net/users/1441 | The set of non-smooth points of a convex function is (m - 1)-rectifiable | The paper [On the differentiation of convex functions in finite and infinite dimensional spaces](http://dml.cz/dmlcz/101616) by Zajíček primarily deals with the general question in Banach space, but it looks like it has a summary of the situation (as of 1979) that tells you everything that you might want to know. It ha... | 6 | https://mathoverflow.net/users/1450 | 25716 | 16,854 |
https://mathoverflow.net/questions/25620 | 10 | Given any two points on a hyperbolic paraboloid ($xy = z$ or $z = (x^2 - y^2)/2$) how does one find the geodesic between them?
I know that since the hyperbolic paraboloid is doubly ruled, some of the geodesics are lines. However, I have very little idea of how to find the geodesics between arbitrary points.
If an ... | https://mathoverflow.net/users/6270 | Geodesics on a hyperbolic paraboloid | It is standard differential geometry to find the differential equation for the geodesics on this surface. (But I could easily have made a mistake in the calculation anyway.) Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. You have a curve
$\vec{p}(t) = (x(t),y(... | 8 | https://mathoverflow.net/users/1450 | 25721 | 16,858 |
https://mathoverflow.net/questions/25726 | 4 | I have an array of points with their coordinates X and Y. Each point represents a bus stop.
I need to sort the points in a sequence by giving them sequence numbers, so that the path from the first to the last is the shortest.
For a convention, let's call the array "points", so we access and write by calling
```
po... | https://mathoverflow.net/users/6275 | Algorithm for the shortest path through all the points of a 2D cloud | If you only care about the length of the path between the first and last bus stops, then it looks like you are trying to solve the shortest Hamiltonian path problem (HPP). This is related to the more widely studied traveling salesman problem, see [TSP](http://en.wikipedia.org/wiki/Travelling_salesman_problem). Since yo... | 7 | https://mathoverflow.net/users/2233 | 25730 | 16,865 |
https://mathoverflow.net/questions/25687 | 9 | Is it true that if $\operatorname{Ext}^{1}\_{A}(P,A/I)=0$ forall $I$ then $P$ is projective?
Similar statements are true for flat and injective modules, but I'm beginning to suspect that projective modules cannot be characterized solely by ideals.
| https://mathoverflow.net/users/5292 | Is it true that if $\operatorname{Ext}^{1}_{A}(P,A/I)=0 $ for all $ I$ then $P$ is projective? | Answer is yes if $A$ is Noetherian and $P$ is finitely generated. Indeed, your
condition
implies that $Ext^1(P, N)=0$ for any finitely generated $A$-module $N$, which implies that
$P$ is projective.
| 10 | https://mathoverflow.net/users/6277 | 25733 | 16,868 |
https://mathoverflow.net/questions/25740 | 0 | My question is local and coordinate-full: I have an open neighborhood $0 \in U \subseteq \mathbb R^n$, and I'm allowed to make it smaller around $0$. On this neighborhood, I have a constant-rank-$k$ smooth integrable distribution. "Smoothness" means that I have $k\leq n$ many vector fields $v\_1,\dots,v\_k$ — I will wo... | https://mathoverflow.net/users/78 | Does every smooth integrable constant-rank distribution have a basis in which the structure constants are traceless? | Am I missing something? By the Frobenius theorem, there exist co-ordinates $y^1, \dots, y^n$ such that $\partial/\partial y^1, \dots, \partial/\partial y^k$ span the same distribution. So these $k$ co-ordinate vector fields form a basis where the structure constants vanish identically.
| 3 | https://mathoverflow.net/users/613 | 25741 | 16,871 |
https://mathoverflow.net/questions/25734 | 7 | By the fundamental theorem of algebra, the algebraic closure $\mathbb{K}$ of $\mathbb{Q}$ decomposes as $\mathbb{K} = F \oplus i F$ where $F = \mathbb{R} \cap \mathbb{K}$ (the intersection is in $\mathbb{C}$). I want to know if there is a purely algebraic way to characterize $F$, i.e. without invoking any analysis, top... | https://mathoverflow.net/users/4362 | Is there a purely algebraic criterion which characterizes the real algebraic numbers? | Paul: you ask if there is a way to algebraically characterize the field of real algebraic numbers. As a specific field in $\mathbf C$, no there's not a good algebraic characterization, but as an abstract field *yes there is a characterization*. This field is one particular example (and the only concrete one at that) of... | 15 | https://mathoverflow.net/users/3272 | 25742 | 16,872 |
https://mathoverflow.net/questions/10282 | 27 | Other than the standard baby Rudin, Spivak, and Stein-Shakarchi, are there other alternative and comprehensive analysis texts at the undergraduate level? For example something that has general results that would serve as a very good reference book for specialist analysts in any field, whether functional, complex and me... | https://mathoverflow.net/users/nan | Alternative undergraduate analysis texts | Nobody has mentioned Folland's "Real Analysis with Applications"?? This was the textbook for my undergraduate real analysis course (measure theory, Banach spaces, Hilbert spaces), and I still go back to it all the time. I am not yet all that experienced (I just finished my third year of graduate school), but overall I ... | 14 | https://mathoverflow.net/users/4362 | 25746 | 16,876 |
https://mathoverflow.net/questions/25747 | 13 | We have no topologists on our faculty, and from time to time I get to teach our topology course. I know that there are examples of inequivalent knots with the same Homfly polynomial, and I know that there are non-trivial knots with trivial Alexander polynomial, but I don't know whether the question has been settled as ... | https://mathoverflow.net/users/3684 | Is there a non-trivial knot with trivial Homfly polynomial? | All I know is that in their 2003 paper Eliahou, Kauffman and Thistlethwaite
write that they did not find any links with trivial HOMFLY-PT. Although they do find links
with both trivial Jones and Alexander.
<http://www.math.uic.edu/~kauffman/ekt.pdf>
My guess would be there exist links with trivial HOMFLY-PT but no ... | 6 | https://mathoverflow.net/users/692 | 25760 | 16,882 |
https://mathoverflow.net/questions/25758 | 15 | Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater than one, the obstruction to factoriality is the nonvanishing of the (Weil) divisor class group $\operatorname{Cl}(R)$,... | https://mathoverflow.net/users/1149 | Seeking Noetherian normal domain with vanishing Picard group but not a UFD | If the group $C\_2$ acts on $\mathbb{C}[x,y]$ by sending $x$ to $-x$ and $y$ to $-y$, then the fixed subalgebra is the domain $\mathbb{C}[x^2,y^2,xy]$. This is Noetherian and normal. A theorem of Nakajima gives the isomorphism type of the divisor class group (in this case, $C\_2$.) A theorem of Kang asserts for such ex... | 19 | https://mathoverflow.net/users/1446 | 25763 | 16,885 |
https://mathoverflow.net/questions/25782 | 0 | I am reading something where this is used extensively, but it is not defined anywhere and no references are given, and I can't find any.
| https://mathoverflow.net/users/2300 | In a k-linear category, what is the tensor product between a hom space and an object? | If $\mathcal{C}$ is a $k$-linear category, $X \in \mathcal{C}$, and $V$ is any $k$-vector space (in particular, it could be a hom of two objects in $\mathcal{C}$), then $V \otimes X$ (sometimes written $V \odot Y$ to avoid confusion with a monoidal structure) is the object representing the functor $\mathcal{C} \to \ope... | 6 | https://mathoverflow.net/users/396 | 25783 | 16,897 |
https://mathoverflow.net/questions/25035 | 4 | I'm responsible for a charity donation site. We're about the change the site design, and we want to know the best way of detecting if the distribution of donations changes after the design. The problem is the data is quite clumpy, particularly around \$5, \$10, \$20, \$25 and \$50 values, with \$15 being relatively rar... | https://mathoverflow.net/users/6141 | Detecting a shift in distribution where the distribution is clumpy | The [Kolmogorv-Smirnov](http://en.wikipedia.org/wiki/Kolmogorov%25E2%2580%2593Smirnov_test) test suggested by Steve Huntsman seems to have done the job.
| 1 | https://mathoverflow.net/users/6141 | 25786 | 16,900 |
https://mathoverflow.net/questions/25774 | 6 | Where can we find Deligne's paper " Theorie de Hodge I"?
| https://mathoverflow.net/users/3945 | Where can we find Deligne's paper " Theorie de Hodge I"? | [Voici](http://math.harvard.edu/~tdp/Deligne-Theorie.de.Hodge-1-single-page.pdf).
| 19 | https://mathoverflow.net/users/307 | 25787 | 16,901 |
https://mathoverflow.net/questions/25794 | 66 | Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing of $L(1,\chi)$ plays an essential role in the proof of Dirichlet´s theorem on primes in arithmetic progressions. In hi... | https://mathoverflow.net/users/1849 | Shortest/Most elegant proof for $L(1,\chi)\neq 0$ | I like the proof by Paul Monsky:
'Simplifying the Proof of Dirichlet's Theorem'
American Mathematical Monthly, Vol. 100 (1993), pp. 861-862.
Naturally this does maintain the distinction between real and complex
as whatever you do, the complex case always seems to be easier
as one would have two vanishing L-functions ... | 34 | https://mathoverflow.net/users/4213 | 25797 | 16,907 |
https://mathoverflow.net/questions/25756 | 3 | In [[Föllmer 81]](http://archive.numdam.org/ARCHIVE/SPS/SPS_1981__15_/SPS_1981__15__143_0/SPS_1981__15__143_0.pdf) (English translation to be found [here](http://books.google.co.uk/books?id=KKLa1j-diXwC&printsec=frontcover&dq=Introduction+to+Stochastic+Calculus+for+Finance%3A+A+New+Didactic+Approach&source=bl&ots=-L3eK... | https://mathoverflow.net/users/1047 | Examples of deterministic processes of quadratic variation which are of unbounded variation | Take $f:[0,1]\to\mathbb{R}$ such that $f(0)=0$ and it interpolates linearly between $f(1/n)=\frac{(-1)^n}{n}$ for any natural $n$.
| 2 | https://mathoverflow.net/users/2968 | 25806 | 16,914 |
https://mathoverflow.net/questions/25800 | 6 | I'd be grateful for a reference for the following result, which I believe to be true, and
should be well-known.
Let the continuous functions $f\_0,f\_1,\cdots,f\_n: [0,1]\rightarrow [0,\infty)$ be given
and consider the problem of maximizing the integral
$$\int\_0^1 f\_0(x)d\mu(x)$$
over all *positive* Borel measure... | https://mathoverflow.net/users/5365 | A simple infinite dimensional optimization problem | This is a particular case of the Generalized Moment Problem.
The result you are looking for can be found in the [first chapter](http://www.worldscibooks.com/etextbook/p665/p665_chap01.pdf) of *Moments, Positive Polynomials and Their Applications* by Jean-Bernard Lasserre (Theorem 1.3).
The proof follows from a gener... | 6 | https://mathoverflow.net/users/5371 | 25808 | 16,915 |
https://mathoverflow.net/questions/24693 | 24 | I heard this puzzle from Bob Koca. Suppose we play misere tic-tac-toe (a.k.a. noughts and crosses) where both players are X. Who wins?
That particular puzzle is easy to solve, but more generally, has $n \times n$ impartial tic tac toe, in both normal and misere forms, been studied before?
---
EDIT: Thane Plambe... | https://mathoverflow.net/users/3106 | Neutral tic tac toe | It's possible to give a complete theory of 3x3 misere "X-only" tic-tac-toe disjunctive sums by introducing the 18-element commutative monoid $Q$ given by the presentation
$Q = \langle\ a,b,c,d\ | \ a^2=1,\ b^3=b,\ b^2c=c,\ c^3=ac^2,\ b^2d=d,\ cd=ad,\ d^2=c^2 \rangle\ $.
Such a "disjunctive" game of 3x3 neutral tic... | 10 | https://mathoverflow.net/users/6295 | 25811 | 16,917 |
https://mathoverflow.net/questions/25825 | 9 | Recall that the (first) Weyl algebra over $\mathbb{C}$ is the algebra generated by $x,y$ with the relation $yx-xy=1$. It can be realized as the algebra of polynomial differential operators in 1 variable, i.e. $\mathbb{C}[x]$ is a faithful representation, where $x$ acts by multiplication by $x$ and $y$ acts by $\frac{\p... | https://mathoverflow.net/users/135 | Why is this algebra called the q-Weyl algebra? | Suppose that $x$ and $y$ obey $xy-yx=h$, where $h$ is a central element. Set $X$ and $Y$ to be $e^x$ and $e^y$. For now, don't worry too much about what this exponentiation means. Then $XY=q YX$, where $q=e^h$.
If we interpret $X$ and $Y$ as operators on functions then $(Xf)(x)=e^x f(x)$ and $(Yf)(x) = f(x+h)$. You ... | 18 | https://mathoverflow.net/users/297 | 25828 | 16,930 |
https://mathoverflow.net/questions/25829 | 2 | I would like to know if there is a equation for the maximum number of shortest paths that pass through *r* where *r* is a node contained in any path from node *s* (a fixed node, i mean, *s* is the only source of paths) to any node *t* in an unweighted undirected acyclic graph. I've searched and found [this work](http:/... | https://mathoverflow.net/users/6299 | Maximum number of shortest-paths | Judging from the link you provide, you have three distinct vertices s,t,d and want to compute the number of shortest walks P(s,d,t) from s to d that contain t. The reason I use "walks" instead of "paths" is because of graphs like:
```
s----d----t
```
where we must reuse edges.
If you really mean acyclic graph, t... | 1 | https://mathoverflow.net/users/2264 | 25838 | 16,937 |
https://mathoverflow.net/questions/25836 | 4 | By the Pati-Salam group I refer to SU(2) x SU(2) x SU(4). It can be obtained as the group of isometries of the 8 dimensional manifold $S^3 \times S^5$, but I wonder if this is the only 8 dimensional manifold having this group of isometries.
This particular manifold is interesting because a quotient by any U(1) will ... | https://mathoverflow.net/users/4037 | Manifolds whose isometry group is Pati-Salam? | $S^3 \times S^5$ has isometry group $SO\_4(\mathbb{R}) \times SO\_6(\mathbb{R})$, which has $SU(2) \times SU(2) \times SU(4)$ as a four-fold cover. Since it appears that you aren't worrying too much about central terms, we can replace $S^3$ with $\mathbb{R}P^3$, $S^5$ with $\mathbb{R}P^5$, or take a quotient by a diago... | 5 | https://mathoverflow.net/users/121 | 25839 | 16,938 |
https://mathoverflow.net/questions/25803 | 1 | Hi,
Can anyone familiar with the book 'A Probabilistic Theory of Pattern
Recognition' or the theory described help me out?
See quote from chapter 12, 'Vapnik-Chervonenkis Theory', of 'A
Probabilistic Theory of Pattern Recognition' below. I'm not following
the geometric setup outlined there.
Some specific question... | https://mathoverflow.net/users/6293 | A question about Chapter 12 (Vapnik-Chervonenkis Theory) of 'A Probabilistic Theory of Pattern Recognition' | Yes, a hyperrectangle is a generalisation of rectangle to higher dimensions. Here the data is given by points in $\mathbb{R}^d$, so the hyperrectangles are all of that dimension. As with all such algorithms you need to find a way to get a handle on the set of classifiers, so rather than the infinite class of all hyperr... | 2 | https://mathoverflow.net/users/447 | 25856 | 16,949 |
https://mathoverflow.net/questions/25846 | 9 | Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start the random walk with one particle at the origin. For each $n \geq 1$ and $x \in \mathbb{Z}^2$ let $p\_n(x)$ be the probabi... | https://mathoverflow.net/users/4345 | Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance | As written the statement is false for $n=3$: note that $p\_3(2,2) = 0$ but $p\_3(3,0) > 0$, while $|(2,2)| < |(3,0)|$. Similar counterexamples exist for all $n\geq 5$. So for larger $n$ you would at least need some extra condition about $L^1$ norms to guarantee that you can't have $|x|<|y|$ with $p\_n(x)=0$ and $p\_n(y... | 3 | https://mathoverflow.net/users/5963 | 25859 | 16,951 |
https://mathoverflow.net/questions/25862 | 15 | This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in the past written a paper about (amongst other things) compact endomorphisms of $p$-adic Banach spaces (and indeed of Bana... | https://mathoverflow.net/users/1384 | Naive questions about "matrices" representing endomorphisms of Hilbert spaces. | Chapter V of Halmos' "A Hilbert space problem book" is called "Infinite matrices".
It contains lots of nice results and problems, and also the statement that "there are no elegant and usable necessary and sufficient conditions [for a matrix to be the matrix of an operator]".
| 19 | https://mathoverflow.net/users/5743 | 25864 | 16,954 |
https://mathoverflow.net/questions/25870 | 1 | I know coefficients of some function in basis $p\_j,j=1...K$ where
$p\_{j}(x)=\sum\_{s\in Z}a\_{s,j}\exp\left(-2\pi i(j+sK)x\right)$
With respect to inner product $(f,g)=\int\_{0}^{1}f(x)\overline{g(x)}dx$ this basis is orthogonal.
What kind of basis is it? Can I somehow use Fourier analysis framework for this basi... | https://mathoverflow.net/users/3589 | some strange orthogonal basis and an integral equation with it | If $a\_{s,j}$ is in $\ell\_2$ for any fixed $j$, $p\_j$ forms an orthogonal basis for some finite dimensional sub-space of $L^2(\mathbb{T})$. If $f(x) = \sum\_{j=0}^K b\_j p\_j(x)$, then its fourier coefficients are just given by $\hat{f}(\xi) = a\_{s,j} b\_j$ where $j = \xi \mod K$ and $s = \lfloor \xi / K \rfloor$, s... | 1 | https://mathoverflow.net/users/3948 | 25872 | 16,958 |
https://mathoverflow.net/questions/25866 | 15 | A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where
$E,M$ are smooth manifolds and $p$ is smoothly locally trivial. Also suppose that the fiber $F$ of the bundle is smoothly contractible.... | https://mathoverflow.net/users/6206 | smooth sections of smooth fiber bundles | The answer is: Yes (at least for finite dimensional manifolds).
In fact you only need that the fiber is contractible not smoothly contractible. Take any continuous section $s\_0 \colon B \to E$. cover $B$ by open sets $U\_i$ such that the bundle is trivializable over each $U\_i$, also make sure that the closure of ea... | 9 | https://mathoverflow.net/users/4500 | 25885 | 16,967 |
https://mathoverflow.net/questions/25863 | 7 | [Torsors](http://math.ucr.edu/home/baez/torsors.html) are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.
In general I'm interesting in the notion of 'subtraction/division' induced by having a torsor. My application is... | https://mathoverflow.net/users/2620 | Torsors for monoids | One common definition of torsor under a group $G$ is a (**Edit:** nonempty) set $X$ together with an action $act: G \times X \to X$, such that the map $(act,id): G \times X \to X \times X$ is a bijection of sets. The definition is still meaningful if you replace "group" with "monoid".
**Edit:** As Torsten has pointed... | 2 | https://mathoverflow.net/users/121 | 25886 | 16,968 |
https://mathoverflow.net/questions/25873 | 16 | Let $A$ be a commutative algebra, say over $\mathbb{C}$.
Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^\*$ action on spec(A): $A\_i$ can be thought of as those functions on which $t$ acts by multiplication with $t^i$. Similary a graded $A$ module is just a $\mathbb{C}^\*$ equivariant s... | https://mathoverflow.net/users/2837 | Geometric interpretation of filtered rings and modules | To a filtered algebra $(A,F)$ one can assign its Rees algebra $R=\bigoplus\_i F\_iA$. It is a graded algebra containing the algebra of polynomials in one variable $\mathbb{C}[t]$ naturally embedded as the subalgebra generated by the element $t\in R\_1$ corresponding to the element $1\in F\_1A$. So the algebra $R$ defin... | 26 | https://mathoverflow.net/users/2106 | 25888 | 16,970 |
https://mathoverflow.net/questions/25821 | 0 | In [a previous question](https://mathoverflow.net/questions/25740/does-every-smooth-integrable-constant-rank-distribution-have-a-basis-in-which-the/25741), I asked an utterly trivial question, which Deane Yang correctly pointed out was utterly trivial. I will now ask a similar question, which is the one I meant to ask ... | https://mathoverflow.net/users/78 | Does a smooth, constant-rank, integrable distribution have a basis in which the traces of the structure constants are the divergences of the corresponding basis elements? | The answer to the question in the title is yes. More precisely I claim one may always find a divergence free basis where also all structure constants vanish. The following is a hopefully valid proof by induction on the rank k of the distribution: if k=1 then by an answer to your previous question one may always find a ... | 2 | https://mathoverflow.net/users/745 | 25896 | 16,977 |
https://mathoverflow.net/questions/25875 | 2 | Given $k$ an algebraically closed field, I know that that a maximal ideal $\mathfrak{m}$ of $A = k[X\_1,\cdots,X\_n]$ is just a $\langle X\_1-a\_1,\cdots,X\_n-a\_n \rangle $ (Nullstellensatz).
Knowing that, it seems intuitive that $\mathfrak{m}$ can not be generated by less than $n$ elements. Is that true? In that case... | https://mathoverflow.net/users/6187 | Generators of a maximal ideal of $k[X_1,\cdots,X_n]$ | Since you mentioned Krull's height theorem (= the generalized principal ideal theorem) and having difficulty applying it, I thought you or someone else might appreciate seeing how this works: it is quite straightforward.
The generalized principal ideal theorem is as follows: let $R$ be a Noetherian ring and $I$ a pro... | 7 | https://mathoverflow.net/users/1149 | 25898 | 16,978 |
https://mathoverflow.net/questions/20704 | 8 | In many cases, the recurrence equations that people are solving involves index of only non-negative values. Here I have a recurrence equation which arises from transport of light in an infinite 1D chain:
$a\_m=\sum \_{j=1}^{\infty } \left(T\_ja\_{m+j}+T\_ja\_{m-j}\right) + \delta \_{m,0}$
where $\delta\_{m,0}$ is t... | https://mathoverflow.net/users/5217 | Solving recurrence equation with indexes from negative infinity to positive infinity | I see your problem more like a linear equation on an infinite dimensional space of doubly infinite sequences than as a recurrence equation, since there are no initial values from which start to build up the solution. In the following I will assume that $\sum\_{j=1}^\infty|t\_j|<\infty$ and that $\mathbf{a}=(a\_m)$ is b... | 4 | https://mathoverflow.net/users/1168 | 25900 | 16,980 |
https://mathoverflow.net/questions/25897 | 0 | **Problem**
Consider the following data set:
```
YEAR;AMOUNT;MEASUREMENTS
1985;9.53013698630137;365
1986;11.086301369863;365
1987;13.0712328767123;365
1988;11.9248633879781;366
1989;10.2191780821918;365
1990;7.41933085501859;269
1991;12.1751396648045;358
1992;9.7037037037037;108
1993;13.1452261306533;199
1994;8.70... | https://mathoverflow.net/users/5908 | When to throw away data | I cannot comment due to rep requirements but Alekk's comment reg weight- If you use hierarchical bayesian (also known as multi-level models) ideas to process the data then the observations coming from years 1990 and 1992 to 1994 will receive lower weight when estimating the parameters of interest.
In general, throwin... | 4 | https://mathoverflow.net/users/4660 | 25908 | 16,985 |
https://mathoverflow.net/questions/25901 | 9 | This might be standard, but I have not seen it before:
Let $K$ be an algebraically closed field (of characteristic 0 if necessary). Let $G$ be the orthogonal group ${\bf O}(m)$ or the symplectic group ${\bf Sp}(2n)$, and let $\mathfrak{g}$ be its Lie algebra. Is there a classification of the adjoint orbits of $G$ act... | https://mathoverflow.net/users/321 | Classification of adjoint orbits for orthogonal and symplectic Lie algebras? | "Classification" can mean more than one thing, but it's useful to be aware of the extensive development of *adjoint quotients* by Kostant, Steinberg, Springer, Slodowy, and others. This makes sense for all semisimple (or reductive) groups and their Lie algebras over any algebraically closed field, perhaps avoiding a fe... | 6 | https://mathoverflow.net/users/4231 | 25914 | 16,990 |
https://mathoverflow.net/questions/25894 | 9 | Apologies for not knowing exactly what I'm looking for, but I'd appreciate any general pointers to get me started.
I'm interested in efficient representations (graphical and otherwise) of finite graphs with repeated structure --- something like coding theory for graph structures. For example, an $N$ by $N$ grid graph... | https://mathoverflow.net/users/2785 | Representing repeated structure in graphs | I am not sure what exactly is meant by a *repeated structure* but at least some covering graphs should qualify. Covering graphs may be quite large, however they admit concise description via *voltage graphs*. The edges of a small base graph or voltage graph are equipped with group elements, called *voltages*. For insta... | 3 | https://mathoverflow.net/users/4400 | 25930 | 16,998 |
https://mathoverflow.net/questions/25922 | 14 | Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the Hirzebruch-Riemann-Roch theorem, this boils down to showing that $$ \chi(X,\mathcal{O}\_X) = \chi(Y,\mathcal{O}\_Y).$$
... | https://mathoverflow.net/users/4333 | Is the Euler characteristic a birational invariant | If you are willing to stick to characteristic zero, then you can assume that there is actually a morphism $f\colon X\longrightarrow Y$ realizing the birational equivalence (reason: look at the graph $\Gamma\subset X\times Y$ realizing the birational equivalence and take its closure, use resolution of singularities to r... | 14 | https://mathoverflow.net/users/1055 | 25934 | 17,000 |
https://mathoverflow.net/questions/25878 | 12 | This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is enough to capture everything on its space. In fact, by the Gelfand-Neumark theorem, it is enough to consider the commuta... | https://mathoverflow.net/users/6031 | Relevance of the complex structure of a function algebra for capturing the topology on a space. | Here is a slightly different, perhaps simpler take on showing that $C(X,\mathbb{R})$ determines $X$ if $X$ is compact Hausdorff. For each closed subset $K$ of $X$, define $\mathcal{I}\_K$ to be the set of elements of $C(X,\mathbb{R})$ that vanish on $K$. The map $K\mapsto\mathcal{I}\_K$ is a bijection from the set of c... | 7 | https://mathoverflow.net/users/1119 | 25942 | 17,007 |
https://mathoverflow.net/questions/25944 | 8 | Suppose $F\_1$ and $F\_2$ are free groups, and suppose $\alpha:F\_1 \to F\_2$ is a surjective homomorphism. Then, because $F\_2$ is free, the homomorphism splits, and we get a subgroup $H$ of $F\_1$ isomorphic to $F\_2$ and a retraction of $F\_1$ onto $H$, i.e., a surjective map to $H$ that restricts to the identity on... | https://mathoverflow.net/users/3040 | Does every retraction of free groups arise from projection to a subset of a freely generating set? | No. This is explicitly stated in the paragraph above Theorem 1 of:
[Turner, Edward C, Test words for automorphisms of free groups.
Bull. London Math. Soc. 28 (1996), no. 3, 255--263.](http://blms.oxfordjournals.org/cgi/reprint/28/3/255.pdf)
The author refers to Proposition 1.
**EDIT:**
Let's give an explicit ex... | 13 | https://mathoverflow.net/users/1463 | 25945 | 17,008 |
https://mathoverflow.net/questions/25943 | 16 | The following questions occurred to me.
This is not research mathematics, just idle curiosity.
Apologies if it is inappropriate.
1. Suppose you have a fixed volume *V* of maleable material,
perhaps clay.
The goal is to form it into a shape *S* (convex or nonconvex) that would roll down an inclined
plane as fast as p... | https://mathoverflow.net/users/6094 | Fastest Rolling Shape? | There is one thing that's way more important than the precise definition of how the race starts and ends: the moment of intertia of $S$, which determines how much energy is wasted on the rotation. Let us assume that $S$ is cylindrically symmetrical with mass $m$, radius $R$, moment of intertia $I$, and let $k=I/mR^2$. ... | 11 | https://mathoverflow.net/users/5740 | 25950 | 17,012 |
https://mathoverflow.net/questions/25956 | 10 | Let $\chi$ be a real nonprincipal Dirichlet's character modulo $m$.
In [my
answer](https://mathoverflow.net/questions/25794/shortest-most-elegant-proof-for-l1-chi-neq-0/25822#25822) to the question on $L(1,\chi)$, I explain a trick for showing that $L(1,\chi)>0$ on the simplest examples
of the real characters modulo ... | https://mathoverflow.net/users/4953 | Positivity of $L(1,\chi)$ for real Dirichlet's character | These are called Fekete polynomials, and you can find out a great deal about them [here](http://www.dms.umontreal.ca/~andrew/PDF/fekete.pdf). Unfortunately they tend to have lots of real zeros in $(0,1)$ when $m$ is large.
| 5 | https://mathoverflow.net/users/1464 | 25958 | 17,017 |
https://mathoverflow.net/questions/25911 | 7 | For some context see [Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance](https://mathoverflow.net/questions/25846/random-walks-in-z2-z2-intrinsic-characterization-of-euclidean-distance)
As per Noah's answer and JBL's comment this was false as stated. However, I think the following reformula... | https://mathoverflow.net/users/4345 | Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II | As I mentioned in my comment - what you are suggesting implies that the probability of being at (3,4) is the same as being at (5,0) for all large $n$. That seems unlikely, and would guess that $C\_n=5$ for $n$ large.
---
The answer to your modified question is yes! $\tilde C\_n$ tends to infinity as $n$ goes to i... | 8 | https://mathoverflow.net/users/1004 | 25959 | 17,018 |
https://mathoverflow.net/questions/25826 | 2 | The Nakai-Moishezon criterion states that a line bundle $L$ over a surface $X$ is ample iff $L \cdot L > 0$ and $L \cdot C > 0$ for every curve $C$.
We can use this criterion to check that if $X$ is the product of two elliptic curves, then lots of divisors of $X$ are not ample. The fibers of the projection maps of $X... | https://mathoverflow.net/users/3314 | Is there an Abelian surface such that every effective divisor is ample? (Together with a boil down version to a question in Complex Lie group theory) | This answer exists simply to record that BCnrd and Bjorn Poonen gave excellent answers in the comments above. If someone votes up my answer, this will be removed from this list of unanswered questions. (And, as I have made this answer CW, I will not gain any reputation.)
| 5 | https://mathoverflow.net/users/297 | 25962 | 17,021 |
https://mathoverflow.net/questions/25919 | 0 | Suppose $f$ is a continuous function of infinitely many real variables, and that 0 is an "identity element" for $f$ in the sense that
$$ f(0,\alpha,\beta,\gamma,\dots) = f(\alpha,\beta,\gamma,\dots). $$
Has anyone thought about the following limit (in particular, is there anything in the literature on it)?
$$ \li... | https://mathoverflow.net/users/6316 | A derivative of sorts? | I think Willie Wong's answer here suffices. This is just a case of looking at something from a suddenly different point of view and missing the obvious since it's not the way I'd been looking at it before.
I had an occasion to think about the difference
$$
f(\alpha,\beta,\gamma,\delta,\dots) - f(\alpha+\beta,\gamma,\... | 0 | https://mathoverflow.net/users/6316 | 25969 | 17,025 |
https://mathoverflow.net/questions/25977 | 16 | the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I found the explanation on wikipedia : "In mathematics, more specifically general topology and metric topology, a compact spa... | https://mathoverflow.net/users/6324 | How to understand the concept of compact space | Some heuristic remarks are helpful only to a subset of readers. (Maybe that's true of all heuristics, as a meta-heuristic - if everyone accepts a rough explanation, it's something rather more than that.)
Non-compactness is about being able to "move off to infinity" in some way in a space. On the real line you can do... | 20 | https://mathoverflow.net/users/6153 | 25981 | 17,032 |
https://mathoverflow.net/questions/25978 | 0 | I heard that there is a result which is proved that RL\subseteq L^{4/3}, but I don't which paper have proved it.
Can someone tell me this paper?
| https://mathoverflow.net/users/6326 | A result about LSpace and RLSpace | I think that the currently best known bound is L^{3/2} in
Michael E. Saks, Shiyu Zhou: RSPACE(S) \subseteq DSPACE(S3/2). FOCS 1995 344-353
There was a paper showing Symmetric Log space in L^{4/3}
R. Armoni, A. Ta-Shma, A. Wigderson, S. Zhou.
A (log n )^{4/3} space algorithm for (s,t) connectivity in undirected gra... | 6 | https://mathoverflow.net/users/6327 | 25985 | 17,035 |
https://mathoverflow.net/questions/25948 | 9 | In commutative Iwasawa theory, the main conjecture states that the p-adic L-function generates the characteristic ideal of an algebraic object. Non-commutative Iwasawa theory seems to mimik this - except that the existence of the object on the analytic side (to my knowledge) is still conjectural. My question is: why is... | https://mathoverflow.net/users/5730 | non-commutative iwasawa theory | First a short answer. I don't think one can say that the commutative analytic side is known, as you do. It is fully known only in the cyclotomic $\mathbb{Z}\_{p}$ situation, assuming the ETNC and in the crystalline case (see below). The answer to your "why is it so hard" question is: because $p$-adic $L$-functions are ... | 5 | https://mathoverflow.net/users/2284 | 25989 | 17,039 |
https://mathoverflow.net/questions/25993 | 23 | Consider a compact subset $K$ of $R^n$ which is the closure of its interior. Does its boundary $\partial K$ have zero Lebesgue measure ?
I guess it's wrong, because the topological assumption is invariant w.r.t homeomorphism, in contrast to being of zero Lebesgue measure. But I don't see any simple counterexample.
| https://mathoverflow.net/users/6129 | Sets with positive Lebesgue measure boundary | Construct a Cantor set of positive measure in much the same way as you make the `standard' Cantor set but make sure the lengths of the deleted intervals add up to 1/2, say.
Let $U$ be the union of the intervals that are deleted at the even-numbered steps and let $V$ be the union of the intervals deleted at the odd-numb... | 30 | https://mathoverflow.net/users/5903 | 26000 | 17,044 |
https://mathoverflow.net/questions/26001 | 53 | I asked myself, which spaces have the property that $X^2$ is homeomorphic to $X$. I started to look at some examples like $\mathbb{N}^2 \cong \mathbb{N}$, $\mathbb{R}^2\ncong \mathbb{R}, C^2\cong C$ (for the cantor set $C$). And then I got stuck, when I considered the rationals. So the question is:
Is $\mathbb{Q}^2$ ... | https://mathoverflow.net/users/3969 | Are the rationals homeomorphic to any power of the rationals? | Yes, Sierpinski proved that every countable metric
space without isolated points is homeomorphic to the rationals:
<http://at.yorku.ca/p/a/c/a/25.htm> .
An amusing consequence of Sierpinski's theorem is that
$\mathbb{Q}$ is homeomorphic to $\mathbb{Q}$. Of course here one
$\mathbb{Q}$ has the order topology, and the ... | 91 | https://mathoverflow.net/users/4213 | 26009 | 17,050 |
https://mathoverflow.net/questions/25971 | -4 | What is this video trying to tell us?
<http://www.youtube.com/watch?v=JX3VmDgiFnY>
The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic projection is wrong (since for example some fractional linear transformations have only one fixed point, which is imposs... | https://mathoverflow.net/users/2260 | Meaning of the Mobius transformations video | Any Möbius transformations is a composite of a rotation of *S*2 (3 degrees of freedom), along with a translation and dilation of ℝ2 (3 degrees of freedom), adding up to the six dimensions of the Lie group PSL(2,ℂ) = group of Möbius transformations.
In the video, the translations are depicted by letting the sphere mov... | 4 | https://mathoverflow.net/users/5690 | 26014 | 17,053 |
https://mathoverflow.net/questions/26013 | 4 | Let $R$ be a local artinian Gorenstein ring and $M$ a finitely generated $R$-module, then
$\mathrm{Ext}\_R^1(M,M) = 0$ if only if $M$ is projective?
| https://mathoverflow.net/users/5775 | Selforthogonal modules over Artinian Gorenstein rings | This would be a very strong version of the Auslander-Reiten Conjecture ([see here](https://www.leuschke.org/research/AR-Conjecture/), for example) in the Gorenstein case. The Conjecture is still open, though many partial results are known.
By the way, [this paper](https://doi.org/10.1016/j.jalgebra.2008.04.027) (Scie... | 7 | https://mathoverflow.net/users/460 | 26017 | 17,055 |
https://mathoverflow.net/questions/18902 | 5 | Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, and preferably sparse) matrices arise naturally?
I am especially interested in problems that can be mapped onto a setup... | https://mathoverflow.net/users/1847 | Are there interesting problems involving arbitrarily long time series of small matrices? | A very significant application in the context of communications engineering is the modelling of multiple-input-multiple-output (MIMO) communications channels.
These channels are typically modeled by complex $n \times m$ matrices where $n$ is the number of receive antennas and $m$ is the number transmit antennas. The... | 2 | https://mathoverflow.net/users/5378 | 26022 | 17,056 |
https://mathoverflow.net/questions/25991 | 3 | Suppose we have a full exceptional collection (F1,...,Fn) of coherent sheaves on a smooth projective variety X. The number n of sheaves in this collection is equal to the rank of the Grothendieck group K0(X).
Is there any relation between n and the rank of the Picard group Pic(X) or the dimension of X?
| https://mathoverflow.net/users/6330 | Number of sheaves in a full exceptional collection | As a far as I know, there is no obvious relation between the number of objects in a full exceptional collection and the dimension of $X$, unless you impose extra conditions on the full exceptional collection, as in Bondal and Polishchuk's 'Homological properties of associative algebras', where they prove that when the ... | 12 | https://mathoverflow.net/users/4659 | 26030 | 17,058 |
https://mathoverflow.net/questions/26020 | 2 | Hi,
Please recommend a good book on wave equations and fourier series / transforms at 3rd year undergraduate level.
Our course text is a bit dense and can be hard to follow - see the course text at <http://www.ouw.co.uk/bin/ouwsdll.dll?COURSEMS324_Mathematics_-_Pure_and_Applied#> - Block 1 - Waves.
As mentioned ... | https://mathoverflow.net/users/6332 | Good books on wave equations and fourier analysis | Hum, unfortunately I am not familiar with the Open University course, so I am just making a guess based on the course description you linked to.
Insofar as Fourier Analysis is concerned, a decent text is Stein and Shakarchi's Fourier Analysis: an introduction. ( <http://press.princeton.edu/titles/7562.html> ) You wi... | 4 | https://mathoverflow.net/users/3948 | 26036 | 17,060 |
https://mathoverflow.net/questions/26025 | 4 | Hi people!
This my first question, here. I don't sure if it has a trivial answer, or not.
Let G a group, N normal subgroup in G. In which cases there is a subgroup in G isomorphic to G/N?
TIA
| https://mathoverflow.net/users/6334 | Given a normal subgroup N⊆G, when does G contain a subgroup isomorphic to G/N? | Assuming you're looking at the case where the isomorphism is induced by the quotient $G \to G/N$ (as per George McNinch's comment), then this should be if and only if the sequence
$$ 0 \to N \to G \to G/N \to 0$$
splits. i.e. there is a section $\sigma : G/N \to G$. This is then seen to be equivalent to $G$ being isomo... | 8 | https://mathoverflow.net/users/1703 | 26038 | 17,061 |
https://mathoverflow.net/questions/26032 | 4 | This is related to an [older question](https://mathoverflow.net/questions/17597/prime-numbers-with-given-difference) about [prime k-tuples and constellations](http://en.wikipedia.org/wiki/Prime_k-tuple), but takes a slightly different direction.
Given an integer k, we want to find n such that the interval {n+1, ..., ... | https://mathoverflow.net/users/5701 | Intervals with large numbers of primes | For fixed $k$ this is definitely hopeless, since it would imply that for some $b$ there are infinitely many primes $p$ such that $p + b$ is prime, and this is a well-known open problem that seems out of reach of the latest techniques for finding small gaps between primes (see this survey article of Soundararajan for ex... | 6 | https://mathoverflow.net/users/5575 | 26042 | 17,064 |
https://mathoverflow.net/questions/26043 | 10 | Consider a compact differentiable manifold $M$. We say that $f:M\to M$ and $g: M \to M$ are topologically conjugated if there exists $h:M\to M$ a homeomorphism such that $f\circ h= h \circ g$. The conjugacy class of a homeomorphism $f$ is the set of all $g$ such that $g$ is topologically conjugated to $f$.
If a homeo... | https://mathoverflow.net/users/5753 | Topological conjugacy between homeomorphisms and diffeomorphisms | Here is an example. Consider a map $f:\mathbb R^2\to\mathbb R^2$ given by $(r,\varphi)\mapsto (r,\varphi+\sin(1/r))$ in polar coordinates $(r,\varphi)$, $0<r\le 1/\pi$. For $r>1/\pi$, let $f$ be identity, then close up the plane to make a compact manifold.
This map has zero topological entropy but has no conjugate $C... | 13 | https://mathoverflow.net/users/4354 | 26057 | 17,074 |
https://mathoverflow.net/questions/26018 | 33 | I just caught sight on arXiv a paper by Holst and Stern titled [Geometric Variational Crimes](http://arxiv.org/abs/1005.4455). Apparently a Variational Crime is an approach to solve problems using a finite element method (e.g. Galerkin) where certain assumptions about approximate solutions are violated. As you can see,... | https://mathoverflow.net/users/3948 | What are "variational crimes" and who coined the term? | Thanks for your interest in the paper! (It's also nice to see something on Math Overflow that I know something about.) Your summary of variational crimes is actually pretty close to the mark: it refers to certain "abuses" of the Galerkin method, where some of the assumptions are violated, and thus the standard error es... | 44 | https://mathoverflow.net/users/673 | 26066 | 17,080 |
https://mathoverflow.net/questions/26031 | 16 | There are two simple and classic enumerations that still I'm puzzled about. Let's start with a simple counting problem from a well-known dynamical system.
**fact 1** Consider the "tent map" f:[0,1]→[0,1] with parameter 2, that is
>
> f(x):=2min(x,1-x).
>
>
>
Clearly, it has 2 fixed points, and more generall... | https://mathoverflow.net/users/6101 | Periodic orbits and polynomials | How I see it is every fixed point comes from an equation $T^nx=x$. Denoting $f(x)=2x$ and $g(x)=2(1-x)$ we see that this corresponds to solving equations
$$h\_1\circ h\_2\circ\cdots \circ h\_n (x)=x$$ where $h\_i\in \{f,g\}$. This is geometrically the intersection of two lines and thus gives a unique solution, and the... | 11 | https://mathoverflow.net/users/2384 | 26068 | 17,081 |
https://mathoverflow.net/questions/26059 | 14 | A Noetherian group (also sometimes called slender groups) is a group for which every subgroup is finitely generated. (Equivalently, it satisfies the ascending chain condition on subgroups).
A finitely presented group is a group with a presentation that has finitely many generators and finitely many relations.
Flipp... | https://mathoverflow.net/users/3040 | Example of Noetherian group (every subgroup is finitely generated) that is not finitely presented |
>
> [Tarski monsters](http://en.wikipedia.org/wiki/Tarski_monster_group%20) provide examples of 2-generator noetherian groups that is not finitely presented.
>
>
>
Edit (YCor): Tarski monsters, as defined in the link (infinite groups of prime exponent $p$ in which every nontrivial proper is cyclic) exist for lar... | 11 | https://mathoverflow.net/users/6339 | 26073 | 17,084 |
https://mathoverflow.net/questions/26062 | 2 | Suppose that $K/\mathbb{Q}\_l$ is a finite extension, with ring of integers $\mathcal{O}\_K$. Suppose $\mathcal{G}/K$ is a (linear) algebraic group (connected+reductive), and $\Gamma\subset \mathcal{G}(K)$ is a hyperspecial maximal compact subgroup. This just means (I believe) that we can find $\tilde{\mathcal{G}}/\mat... | https://mathoverflow.net/users/3513 | Automorphism of algebraic group preserving a hyperspecial maximal compact | As noted, there seem to be some "reductive"s missing from the question. Here's
what is known: let $R$ be a Henselian discrete valuation ring with field of
fractions $K$, and let $R^{\prime}$ be the integral closure of $R$ in the
maximal unramified extension $K^{\prime}$ of $K$; a smooth affine scheme $X$
over $R$ defin... | 3 | https://mathoverflow.net/users/930 | 26078 | 17,088 |
https://mathoverflow.net/questions/26084 | 8 | Given any two points x and y on a circle O, one can form four different lenses (regions between two circles, one of which is O) that have corners at x and y and make angles of 2π/3 at their corners. For three points x, y, and z, with O the circumcircle of triangle xyz, two of these lenses are outside O and two of them ... | https://mathoverflow.net/users/440 | Möbius-invariant triangle center? | Unless I'm mistaken you seem to describe the [isogonal conjugate](http://en.wikipedia.org/wiki/Isogonal_conjugate) of the Fermat point X(13). This is the [first isodynamic point](http://en.wikipedia.org/wiki/Isodynamic_point), or X(15) in ETC.
| 7 | https://mathoverflow.net/users/2384 | 26085 | 17,091 |
https://mathoverflow.net/questions/25757 | 6 | Are there any bounds on residues of $1/\zeta$ in roots of $\zeta$ in critical strip, which may use RH, but do not use the conjecture on simplicity of roots or something similar? I did not find such resuts in Titchmarsh, but I could miss something. Thanks!
| https://mathoverflow.net/users/4312 | Residues of $1/\zeta$ | This is actually a very difficult problem, and currently most results are highly conjectural. It essentially comes down to finding useful bounds on discrete moments of the Riemann zeta function of the form
$$J\_k(T) = \sum\_{0 < \Im(\rho) < T}{|\zeta'(\rho)|^{2k}},$$
as one can then choose the correct value of $k$ and ... | 7 | https://mathoverflow.net/users/3803 | 26092 | 17,096 |
https://mathoverflow.net/questions/26098 | 4 | I want to pick a random direction in n-dimensional space. How can I do this?
The reason I want to do this is to pick a neighbor for [hill climbing optimization](http://en.wikipedia.org/wiki/Hill_climbing).
| https://mathoverflow.net/users/6343 | How to pick a random direction in n-dimensional space | You can proceed as explained at <http://mathworld.wolfram.com/HyperspherePointPicking.html>
| 11 | https://mathoverflow.net/users/1409 | 26100 | 17,101 |
https://mathoverflow.net/questions/26086 | 12 | Analytic/meromorphic continuation is a difficult problem in general. For "motivic L-functions", the idea of proving their analytic continuations by first proving their modularity goes back, I guess, to Riemann.
Here I just want to ask a purely complex-analytic question. Let's restrict ourselves to the case of one va... | https://mathoverflow.net/users/370 | Analytic continuation of holomorphic functions | Well, in case of power series some criterions do exist. Roughly speaking, one can take the element
$$
\sum\limits\_{n=1}^{\infty}c\_nz^n,\quad z < 1,\label{1}\tag{$\ast$}$$
and consider an analytic function $\phi$, such that $\phi(n)=c\_n$ for every $n$. The element \eqref{1} can be analytically extended onto some angu... | 15 | https://mathoverflow.net/users/5371 | 26103 | 17,103 |
https://mathoverflow.net/questions/26091 | 3 | Is it true that if $\mbox{Ext}^{1}(P,M)=0$ for every *finitely generated* module $M$ then $P$ is projective? Or that if $\mbox{Ext}^{1}(M,Q)=0$ for every *finitely generated* module $M$ then $Q$ is injective?
| https://mathoverflow.net/users/5292 | Does Ext commute with direct limit? | For the first question you already have had an answer in [Is it true that if $\operatorname{Ext}^{1}\_{A}(P,A/I)=0 $ for all $ I$ then $P$ is projective?](https://mathoverflow.net/questions/25687/projective-module/25698) if $\mathrm{Ext}^1\_{\mathbb Z}(P,M)=0$, then it depends on the axioms of set-theory whether the co... | 8 | https://mathoverflow.net/users/4008 | 26105 | 17,104 |
https://mathoverflow.net/questions/26075 | 14 | There's a well known theorem due to Beck that characterizes when an adjunction is monadic, that is, if $F$ is left adjoint to $G$, $G:D \to C$, $GF:=T$ is always a monad on $C$, and the adjunction is called monadic, essentially, when $D$ is the Eilenberg–Moore category $C^T$ of $T$-algebras and $G$ is the forgetful fun... | https://mathoverflow.net/users/4528 | Characterization of Kleisli adjunctions | There is a unique functor $\mathbf{Kl}(GF) \rightarrow \mathbf{D}$ commuting with the adjunctions from $\mathbf{C}$, since the Kleisli category is initial among adjunctions inducing the given monad; and this functor is always full and faithful, since $\mathbf{Kl}(GF)(A,B) \cong \mathbf{C}(A,GFB) \cong \mathbf{D}(FA,FB)... | 18 | https://mathoverflow.net/users/2273 | 26106 | 17,105 |
https://mathoverflow.net/questions/26104 | 1 | A space $X$ is called *locally contractible* it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the path fibration $PX \to X$ space of based paths with evaluation at the endpoint as projection) admits local sections iff $... | https://mathoverflow.net/users/4177 | An example of a space which is locally relatively contractible but not contractible? | The same counterexample as for semilocally 1-connected works: namely, you can take the cone on the Hawaiian earring space. The space itself is contractible, but no sufficiently small neighborhoods of the "bad" point at the base of the cone are 1-connected (hence not contractible).
| 7 | https://mathoverflow.net/users/360 | 26107 | 17,106 |
https://mathoverflow.net/questions/26112 | 17 | *What is a good example of a fact about the moduli space of some object telling us something useful about a specific one of the objects?*
I am currently learning about moduli spaces (in the context of the moduli space of elliptic curves). While moduli spaces do seem to be fascinating objects in themselves, I am after... | https://mathoverflow.net/users/6345 | Examples of the moduli space of X giving facts about a certain X | The easiest example I can think of is the natural incidence correspondence between $\mathbb{P}^3$ and the parameter space of cubic surfaces. This can be used to show that every cubic surface contains a line; from this [it follows easily](https://mathoverflow.net/questions/20112/interesting-results-in-algebraic-geometry... | 25 | https://mathoverflow.net/users/828 | 26118 | 17,112 |
https://mathoverflow.net/questions/26119 | 15 | Let $(X,d)$ be a metric space. Banach's fixed point theorem states that if $X$ is complete, then every contraction map $f:X\to X$ has a unique fixed point. A contraction map is a continuous map for which there is an real number $0\leq r < 1$ such that $d(f(x),f(y))\leq rd(x,y)$ holds for all $x,y\in X$.
>
> Suppose... | https://mathoverflow.net/users/5952 | Converse to Banach's fixed point theorem? | The answer is no, for example look at the graph of $\sin(1/x)$ on $(0,1]$. But for more information and related questions check out "[On a converse to Banach's Fixed Point Theorem](http://www.ams.org/journals/proc/2009-137-09/S0002-9939-09-09904-3/)" by Márton Elekes.
| 11 | https://mathoverflow.net/users/2384 | 26122 | 17,115 |
https://mathoverflow.net/questions/26072 | 6 | Hello!
I'd like to understand the relation between the following two theorems:
* The "global" duality for projective schemes, as explained in [Hartshorne]: If $X$ is an equidimensional projective Cohen-Macaulay scheme of dimension $n$ over an algebraically closed field with dualizing sheaf $\omega\_X$, then for al... | https://mathoverflow.net/users/3108 | Comparison of Local and Global Duality | In the case of varieties over a perfect field this question is explained with great detail in the book by J. Lipman:
Dualizing sheaves, differentials and residues on algebraic varieties. (French summary)
Astérisque No. 117 (1984)
known by some people as "*Lipman's blue book*". If you want to get rid of the base fi... | 5 | https://mathoverflow.net/users/6348 | 26130 | 17,122 |
https://mathoverflow.net/questions/26083 | 18 | For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in understanding arithmetic questions?
| https://mathoverflow.net/users/4692 | Why are topological ideas so important in arithmetic? | Why are topological ideas so important in arithmetic? In some sense KConrad is of course spot on, but let me offer a completely different kind of answer.
Why are complex functions of one variable so important in arithmetic? (Zeta function, L-functions, Riemann hypothesis, Birch--Swinnerton-Dyer, modular forms, theta ... | 29 | https://mathoverflow.net/users/1384 | 26135 | 17,126 |
https://mathoverflow.net/questions/26081 | 14 | The metric of a Riemannian manifold determines the shortest
distance between any two points.
1. I assume the reverse holds? That is, if you are given the
shortest distance *d(x,y)* between every pair of points
of a manifold *M*, the metric for *M* is determined?
I am mainly interested in compact, connected, closed 2-... | https://mathoverflow.net/users/6094 | Shortest-path Distances Determining the Metric? | Concerning the second question. A single closed geodesic is not enough. For example, let $S$ be the equator of the standard sphere. All distances between points of $S$ are realized by paths in $S$, so you don't have any information about the metric outside $S$, except that it is sufficiently large (so that the paths ou... | 9 | https://mathoverflow.net/users/4354 | 26136 | 17,127 |
https://mathoverflow.net/questions/26134 | 9 | Hello,
in their book [Cohomology of Finite Groups](http://books.google.com/books?id=sKmshEctdw0C&dq=cohomology+of+finite+groups&printsec=frontcover&source=bn&hl=de&ei=vV3-S_3OFcSrsAb79c39CQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDIQ6AEwAw#v=onepage&q&f=false) Adem and Milgram investigate the cohomology of the ... | https://mathoverflow.net/users/3102 | Cohomology of orthogonal and symplectic groups | Direct finite group computations of cohomology of the finite groups of Lie type tend to be very sparse. The case $p=2$ has special interest for topologists and does provide some explicit results. More generally, some indirect results of interest have been found in recent decades by systematically comparing cohomology o... | 8 | https://mathoverflow.net/users/4231 | 26138 | 17,128 |
https://mathoverflow.net/questions/26137 | 17 | One of the recent questions, in fact
[the answer](https://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/25631#25631)
to it, reminded me about the binomial sequence
$$
a\_n=\sum\_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,
\qquad n=0,1,2,\dots,
$$
of the Apéry numbers. The numbers $a\_n$ com... | https://mathoverflow.net/users/4953 | Binomial supercongruences: is there any reason for them? | I learned the second congruence as a version of [Wolsteholme's theorem](http://en.wikipedia.org/wiki/Wolstenholme%2527s_theorem), and I would be a bit surprised if Kazandzidis was the first person to observe the equivalence between this form and any other form of Wolsteholme's result. As for the "reason" that this resu... | 19 | https://mathoverflow.net/users/1450 | 26147 | 17,133 |
https://mathoverflow.net/questions/26094 | 5 | Let Σp={1,...,p}ℤ be the full shift on p symbols, and let X ⊂ Σp be a subshift -- that is, a closed σ-invariant subset, where $\sigma\colon \Sigma\_p\to \Sigma\_p$ is the left shift. Then σ is expansive, and hence there exists a measure of maximal entropy (an mme) for (X,σ).
It is well known that if X is a subshift o... | https://mathoverflow.net/users/5701 | A topologically mixing subshift with multiple measures of maximal entropy | Yes. See Haydn's paper [here](http://docs.google.com/viewer?a=v&q=cache%3ANF8M7-tT3w8J%3Aciteseerx.ist.psu.edu/viewdoc/download%253Fdoi%253D10.1.1.139.9001%2526rep%253Drep1%2526type%253Dpdf+Multiple+measures+of+maximal+entropy+and+equilibrium+states+for+one-dimensional+subshifts&hl=en&gl=us&pid=bl&srcid=ADGEESjeew53q33... | 5 | https://mathoverflow.net/users/1847 | 26151 | 17,135 |
https://mathoverflow.net/questions/26117 | 17 | Prime matrices as defined in the following paper [Prime matrices P. F. RIVETT AND N. I. P. MACKINNON](https://www.jstor.org/stable/3616179) carry over many properties of factorization as in natural numbers to matrices over the field of naturals.
I quote the following:
>
> A matrix in a set M of matrices is prime ... | https://mathoverflow.net/users/5627 | Prime/undecomposable matrices | For a different point of view, you might like to take a look at Section 12.5 and Appendix A in the [free on-line version](http://iml.univ-mrs.fr/editions/preprint00/book/prebookdac.html) of the following book (in which you'll find some interesting open questions related to "prime matrices" of the type you described):
... | 6 | https://mathoverflow.net/users/3029 | 26157 | 17,137 |
https://mathoverflow.net/questions/26156 | 3 | Hello!
I'm in search of some (possibly statistical) measure for matrices. I want to classify a square matrix as having the largest numbers running along the main diagonal or along the anitdiagonal. It isn't necessarily on the diagonals, the numbers can be some rows to the left or right, so a simple trace wouldn't wor... | https://mathoverflow.net/users/6351 | "Main" diagonal of a matrix | The following very coarse invariant is perhaps useful:
The argument of the sum $\sum\_{s,t}a\_{s,t}e^{i(s-t)\pi/n}$ (where $n$ is the size of the matrix and where the sum is over all entries) should be related to a typical jump. It is close
to $0$ if the matrix is "concentrated" near the diagonal and close to $\pi$ i... | 3 | https://mathoverflow.net/users/4556 | 26163 | 17,142 |
https://mathoverflow.net/questions/26175 | 19 | Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. Namely, Tannakian categories are defined of unipotent lisse $\ell$-adic sheaves, unipotent vector bundles with a flat c... | https://mathoverflow.net/users/386 | Unipotency in realisations of the motivic fundamental group | Essentially because the Tannakian theory gives in the unipotent case (and only in
that case) a reasonably sized answer with an easy motivic interpretation.
For the size you should be aware that already in the topological situation the
group scheme associated by Tannaka theory to the fundamental group of $\mathbb
P^1$... | 16 | https://mathoverflow.net/users/4008 | 26185 | 17,156 |
https://mathoverflow.net/questions/26184 | 5 | Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?
Some explanations:
1. All the coefficients are real numbers.
2. The number of variables is the same as the number of equations, and all the equations are independent.
3. Some of the equat... | https://mathoverflow.net/users/6347 | Decide how many non-negative solutions a set of multivariate quadratic equations have | Not efficiently, at least not unless the problem has some additional structure which can be exploited. The set of mixed Nash equilibria of a two-player game can be written as the nonnegative solutions of such a polynomial system. In general it is #P-hard to count the equilibria of such a game ([Conitzer and Sandholm](h... | 4 | https://mathoverflow.net/users/5963 | 26196 | 17,164 |
https://mathoverflow.net/questions/24508 | 9 | I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.
* The basic example of such sequences are monotone sequences of real numbers.
* A more interesting examples comes from metric fixed point theory:
Let $B$ be a Banach space and $f\colon B \to B$ be a continuous mapping t... | https://mathoverflow.net/users/3365 | Sequence that converge if they have an accumulation point | The following version of the mean ergodic theorem is taken from the book of Krengel, "ergodic theorems".
Let T be a bounded linear operator in a Banach space X. The Birkhoff averages are denoted by $A\_n = {1\over n} \ \Sigma\_{k=0}^{n-1} \ T^k$.
Assume that the sequence of operator norms $||A\_n||$ is bounded indepe... | 2 | https://mathoverflow.net/users/6129 | 26200 | 17,168 |
https://mathoverflow.net/questions/26203 | 13 | The Arnold conjecture on a closed symplectic manifold $(M,\omega)$ says in the weakest version that for a non-degenerate Hamiltonian there are at least $k$ 1-periodic orbits where $k$ is the sum of the Betti numbers of $M$. It is easy to show that one can assume that $\omega$ is integral, so I do so in the following.
... | https://mathoverflow.net/users/4500 | Floer homology and status of the Arnold conjecture |
---
*V. I. Arnol'd, June 12, 1937 - June 3, 2010.*
The very sad news of his death is reported today [here](http://www.france-info.com/ressources-afp-2010-06-03-l-eminent-mathematicien-russe-vladimir-arnold-mort-subitement-en-450299-69-69.html).
---
After Floer, the main difficulty in solving the weak Arnol'... | 22 | https://mathoverflow.net/users/2356 | 26208 | 17,173 |
https://mathoverflow.net/questions/26213 | 1 | What properties of a canonical bundle are preserved under birational isomorphism between
smooth projective varieties. In particular, is triviality of the canonical bundle preserved?
| https://mathoverflow.net/users/6254 | Birational properties of canonical bundle | Generally speaking, to your specific question, the answer is no.
See for example Hartshorne, Chapter V, Section 3. On the other hand, numerous other properties are preserved (some of these are also mentioned in Hartshorne, for example, the geometric genus). There are more subtle things as well, see any book on the *... | 2 | https://mathoverflow.net/users/3521 | 26215 | 17,176 |
https://mathoverflow.net/questions/25817 | 16 | It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry of Origami.
Mathematics Magazine 68 (1995), no. 5, 357–371.
If one checks the paper by Geretschlager above, one sees... | https://mathoverflow.net/users/6270 | Origami Constructions: Intersecting two Circles | Have you found the papers by Roger Alperin? He has some very nice articles, especially in regards to relating various construction systems -- in addition to origami and compass-straightedge constructions, there are a variety of others, like the Vieten constructions, Pythagorean constructions, and even some variants on ... | 8 | https://mathoverflow.net/users/35575 | 26219 | 17,180 |
https://mathoverflow.net/questions/26209 | 2 | A question inspired by [Is the Euler characteristic a birational invariant](https://mathoverflow.net/questions/25922/is-the-euler-characteristic-a-birational-invariant):
As remarked in Mike Roth's answer to the above linked question, if $X$ and $Y$ are smooth projective varieties in characteristic zero that are birat... | https://mathoverflow.net/users/6254 | Birational correspondences and codimension where not an isomorphism | No it is not possible (see for instance Thm II:2.4 of Shafarevich: Basic Algebraic Geometry) which says that the exceptional locus is always of codimension $1$ provided the target is smooth. The crucial property is that the target variety be $\mathbb Q$-factorial. The example of Dmitri shows that this condition is nece... | 4 | https://mathoverflow.net/users/4008 | 26227 | 17,185 |
https://mathoverflow.net/questions/24859 | 3 | Please consider a two-dimensional surface populated with a set of Cartesian coordinates $(x\_i, y\_i)$ for $N$ circles with individual radii $r\_i$, where $r\_{\min} < r\_i < r\_{\max}$.
Here, the number of circles, $N$, may be large - ranging from hundreds to tens of thousands. The circles may sparsely populate the ... | https://mathoverflow.net/users/3248 | Finding the union of N random circles arbitrarily (or conspiratorially) placed on a two-dimensional surface | CGAL has a general Voronoi diagram module that's quite customizable. While I've never used it to build power diagrams, it should not be hard to add in the right kind of distance function to generate the diagrams you need:
<https://doc.cgal.org/Manual/3.5/doc_html/cgal_manual/Voronoi_diagram_2/Chapter_main.html>
| 3 | https://mathoverflow.net/users/972 | 26232 | 17,188 |
https://mathoverflow.net/questions/26220 | 22 | I hope this question is not so elementary that it'll get me banned...
In mathematics we see a lot of impredicativity. Example of definitions involving impredicativity include: subgroup/ideal generated by a set, closure/interior of a set (in topology), topology generated by a family of sets, connected/path connected c... | https://mathoverflow.net/users/6361 | Impredicativity | Yes, it is worth the effort. A predicative version of an impredicative construction is typically more explicit and informative than the impredicative one. For example, consider the construction of a subgroup $\langle S \rangle$ of a group $G$ generated by the set $S$:
* **impredicative**: $\langle S \rangle$ is the i... | 30 | https://mathoverflow.net/users/1176 | 26233 | 17,189 |
https://mathoverflow.net/questions/26169 | 6 | Let $G$ be a (connected) reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.
Under which circumstances can one say that this functor is a) full and b) faithful?
Edited for clarity:
Here we think of algebraic re... | https://mathoverflow.net/users/1594 | Points of reductive groups | A bit of Tannakian formalism clarifies the situation. Recall that for every abstract group $\Gamma$ there is a notion of "algebraic hull" $\Gamma^{alg}$ constructed as follows: Consider pairs $(\varphi,H)$ where $H$ is an algebraic group over $k$ and $\varphi:\Gamma\to H(k)$ a homomorphism of groups with Zariski-dense ... | 1 | https://mathoverflow.net/users/5952 | 26235 | 17,191 |
https://mathoverflow.net/questions/26243 | 15 | Given a non-integral real $\alpha$, is there an entire (see <http://en.wikipedia.org/wiki/Entire_function>) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with $x$ real non-negative)?
Clearly, such a function if it exists is not unique since $h(x)+e^{-x}$ and similar functions... | https://mathoverflow.net/users/4556 | Asymptotic approximation of $x^\alpha$ by entire functions | Start with an entire function $f$ such that $f(x)=1/x + O(1/x^2)$ for $x>0$, $x\rightarrow\infty$. For example $f(z)= (1-e^{-z})/z$.
Let F be some primitive for $f$: $F(z)=\int\_1^z f(s)ds$.
We have $F(x)= ln(x)+C+O(1/x)$, with C some constant ($ \ C=\int\_1^\infty \ (f(x)-{1\over x})\ dx$ ).
Then consider $h(x)... | 15 | https://mathoverflow.net/users/6129 | 26245 | 17,197 |
https://mathoverflow.net/questions/26248 | 4 | Various sources claim that a maximum norm $||A||\_{max}=\max\_{i,j}|a\_{ij}|$ is not submultiplicative, i.e. $||AB||\_{max}\not\leq||A||\_{max}||B||\_{max}$.
Where can I find what norm a,b satisfy $||AB||\_{max}\leq||A||\_{a}||B||\_{b}$?
| https://mathoverflow.net/users/6358 | Submultiplicative matrix norm: Max Norm | The inequality $\|AB\|\_{\max} \leq \|A\|\_{a}\|B\|\_{b}$ for all $A$, $B$ can be achieved or destroyed just by rescaling the norms $\|\cdot\|\_a$ and $\|\cdot\|\_b$. Let's suppose that we're considering $d \times d$ matrices. If we just make sure that the two norms $\|\cdot\|\_a$ and $\|\cdot\|\_b$ are scaled so that ... | 7 | https://mathoverflow.net/users/1840 | 26252 | 17,202 |
https://mathoverflow.net/questions/26269 | 9 | Apologies in advance if this is too elementary.
The following is well known when $A$ is an algebraically closed field:
>
> Let $X$ be an integral closed subscheme of $P^n\_A$. Then $\Gamma(X, \mathcal{O}\_X) = A$.
>
>
>
My question is: For what other rings does the above statement hold?
There are two proo... | https://mathoverflow.net/users/344 | When is it true that the ring of global regular functions on a projective variety is just the base ring? | The statement is true if and only if $A$ is an algebraically closed field.
Assume the statement is true.
Let $\mathfrak{m}$ be a maximal ideal of $A$. Then $A \to A / \mathfrak{m}$ is certainly finite. Thus, as I explain in my comment above, $\mathbb{P}^n\_{A / \mathfrak{m}}$ is an integral subscheme of some $\math... | 5 | https://mathoverflow.net/users/5094 | 26279 | 17,217 |
https://mathoverflow.net/questions/26268 | 8 | Do the fusion categories $Rep(S\_4)$ and $Rep(A\_5)$ admit non-symmetric braidings? All the other rep. cats. of finite subgroups of $SU(2)$ do (in the McKay correspondence). My guess is no.
| https://mathoverflow.net/users/6355 | Non-symmetric Braiding on finite group Representation Categories | Eric,
The answer is no for $Rep(A\_5)$ and yes for $Rep(S\_4)$,
thanks to Victor Ostrik's observation.
For a braided category $C$ let $C'$ denote its Mueger center,
i.e., the subcategory of objects $Y$ in $C$ such
that the square of braiding of $Y$ with any $X$ in $C$ is identity.
So $C$ is symmetric if $C=C'$ and $C... | 8 | https://mathoverflow.net/users/3011 | 26280 | 17,218 |
https://mathoverflow.net/questions/26266 | 1 | What is your favorite examples of spectral sequences arising naturally in arithmetic geometry?
Please explain it in some detail
| https://mathoverflow.net/users/4245 | spectral sequences in number theory | I posit the following example, in response to your ambiguous question:
The coniveau spectral sequence seems to play an important role in 'arithmetic geometry'. One instance is in class field theory for schemes:
From W. Raskind's nice survery article "Abelian class field theory of arithmetic schemes" [AMS, 1992, pgs... | 9 | https://mathoverflow.net/users/4235 | 26282 | 17,220 |
https://mathoverflow.net/questions/26272 | 15 | It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$. I'm curious as to a weaker result: that for any $\varepsilon>0$, $\pi(n)\gg n^{1-\varepsilon}$.
Chebyshev famously proved that if $\lim \pi(n)\log n$ exists it must be equa... | https://mathoverflow.net/users/6043 | Who first proved that there are at least n^(1-ε) primes up to n? | In the 1850's, Chebyshev gave the following explicit bound, for sufficiently large n.
$$0.92129 \, \frac{n}{\log n} < \pi(n) < 1.0556 \, \frac{n}{\log n}.$$
This is mentioned in "An introduction to the theory of the Riemann zeta function" by S. J. Patterson. There is an exercise in the first chapter that gives the... | 17 | https://mathoverflow.net/users/6129 | 26286 | 17,224 |
https://mathoverflow.net/questions/26287 | 3 | I've read in many places, including the [n-Lab page](http://ncatlab.org/nlab/show/Lie+algebroid), that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) derivation that makes $\Gamma(\wedge^\bullet A^\*)$ into a cochain complex (... | https://mathoverflow.net/users/78 | What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid? | I think that the relation is that if $\psi\colon A \to TM$ is the action of the algebroid, then we have that $df(a)=\psi(a)(f)$ for $f$ in degree zero of the CE complex and $a\in\Gamma(A)$ and $d\omega(a,b)=\omega([a,b])+\psi(a)(\omega(b))-\psi(b)(\omega(a))$, where $\omega\in\Gamma(A^\ast)$ and $a,b\in\Gamma(A)$. One ... | 2 | https://mathoverflow.net/users/4008 | 26293 | 17,230 |
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