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https://mathoverflow.net/questions/23004 | 12 | Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can *fail* to be abelian categories.
Poking around on the internet has lead me to believe that these categories will agree and be abelian if R is a (left?) Noetherian ring, but I'm intereste... | https://mathoverflow.net/users/184 | Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories? | [Background to the following can be found in Lam's *Lectures on Modules and Rings*, section 4.G.]
**Definition:** A finitely generated (f.g.) right module $M\_R$ is *coherent* if every f.g. submodule of $M$ is finitely presented (f.p.).
Now let $M\_R$ be a finitely presented module that is not coherent. There exis... | 5 | https://mathoverflow.net/users/778 | 23025 | 15,166 |
https://mathoverflow.net/questions/23026 | 7 | Recall the notion of **groupoid** ([Wikipedia](http://en.wikipedia.org/wiki/Groupoid), [nLab](http://ncatlab.org/nlab/show/groupoid)). An important construction of groupoids is as "action groupoids" for group actions. Namely, let $X$ be a groupoid and $G$ a group, and suppose that $G$ acts on $X$ by groupoid automorphi... | https://mathoverflow.net/users/78 | How can I understand the "groupoid" quotient of a group action as some sort of "product"? | Let $X$ and $Y$ be groupoids. An action of $Y$ on $X$ is a functor $\rho: Y \to B\operatorname{Aut}(X)$, where $B\operatorname{Aut}(X)$ is the one-object 2-groupoid such that $\operatorname{Hom}(\ast, \ast)$ is the 2-group of autoequivalences of $X$.
We define $X \rtimes Y$ as follows. Its objects are simply $\operat... | 11 | https://mathoverflow.net/users/396 | 23040 | 15,175 |
https://mathoverflow.net/questions/23029 | -3 | It is easy to show that similarity in matrices is an equivalence relation (two matrices A and B of same size being similar if there exists a matrix P such that B = PAP^(-1) )
Moreover, given a matrix, its equivalence class can be finite. E.g. The equivalence of nxn matrices containing the identity matrix I is singleton... | https://mathoverflow.net/users/5627 | Are there infinitely many equivalence classes of similar matrices? | [This is an easy question, but it doesn't feel like a homework question, so I will answer it. I have made the post community wiki to protect myself from unwanted votes, both upwards and downwards.]
For a positive integer $n$, consider the ring $M\_n(k)$ of $n \times n$ matrices with $k$-coefficients for $n \geq 1$. ... | 10 | https://mathoverflow.net/users/1149 | 23042 | 15,177 |
https://mathoverflow.net/questions/22990 | 22 | This question arose after reading the answers (and the comments to the answers) to [Why worry about the axiom of choice?](https://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice).
First things first. In *my intuitive conception of the hierarchy of sets*, the axiom of choice is obviously true. I m... | https://mathoverflow.net/users/2562 | Choice vs. countable choice | Here is one explanation of why countable choice is not problematic in constructive mathematics.
For this discussion it is useful to formulate the axiom of choice as follows:
>
> $(\forall x \in X . \exists y \in Y . R(x,y)) \implies \exists f \in Y^X . \forall x \in X . R(x,f(x))$
>
>
>
This says that a tota... | 44 | https://mathoverflow.net/users/1176 | 23043 | 15,178 |
https://mathoverflow.net/questions/23032 | 7 | The systems of the λ-cube have the axiom $\star:\square$.
I've listed a few meanings that the Curry-Howard isomorphism gives to $t : T$ below. What are the intuitive meanings of $\star$ and $\square$ in each interpretation?
$t : T : \star : \square$
**Programs:** t is a program of type T. (Possibility: T is a pro... | https://mathoverflow.net/users/5700 | What is the intuitive meaning of star and box in a pure type system? | $\star$ is a *kind*, which classifies *types*. $\square$ is a sort, and it classifies *kinds*. So this is a 4-layer deep classification. Once you get to have type-constructors, kinds get really useful. Eventually, you wish for kind-constructors too, and then you need sorts.
Turns out that you really rarely ever need ... | 9 | https://mathoverflow.net/users/3993 | 23044 | 15,179 |
https://mathoverflow.net/questions/23019 | 8 | Historically, I've checked the quality of arXiv output by reading the PDF it creates; I learned today that this actually a serious mistake. If you look at the [PDF](http://arxiv.org/pdf/1001.2020.pdf) and [DVI](http://arxiv.org/dvi/1001.2020) for my most recent article, you'll see that the PDF looks fine, and the DVI i... | https://mathoverflow.net/users/66 | Why does the arXiv produce a messed-up DVI when the PDF is fine? | Apparently the source of the problem is that TikZ behaves differently when the TeX file
is compiled with pdftex/pdflatex instead of the standard tex/latex.
Instead of simply embedding PostScript into the DVI file, as any sane system would do,
it apparently tries to draw figures using characters from some special fonts.... | 7 | https://mathoverflow.net/users/402 | 23046 | 15,181 |
https://mathoverflow.net/questions/22811 | 14 | Given natural numbers of special very composite form, like primorials or factorials, how to give some useful upper bound limit of continued fraction period length of their square roots?
I'm not a professional, any advices are welcome.
| https://mathoverflow.net/users/5539 | Upper bound of period length of continued fraction representation of very composite number square root | The continued fraction length is usually a small constant factor away from the regulator. A more precise version can also be achieved, but I don't remember a reference, so if anyone does...
Then, we know the regulator times the class number is usually a small constant factor away from the discriminant (of the order, ... | 11 | https://mathoverflow.net/users/2024 | 23054 | 15,186 |
https://mathoverflow.net/questions/23048 | 14 | Gromov-Hausdorff distance ([Wikipedia](https://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence)) between two compact manifolds measures how far away the manifolds are from being isometric.
In many cases it is possible to do coarse estimates and conclude that a sequence of manifolds converges or diverges.
H... | https://mathoverflow.net/users/3375 | What are the tricks for computing/estimating Gromov-Hausdorff distance? | I misinterpreted the question at first, sorry. Here's my new answer:
**First, an answer to the wrong question**
For two $(n-1)$ dimensional spheres of radii r and R with the metrics induced from embedding in $\mathbb{R}^n$ (note, this is the "chord metric" not the "round metric" as Zarathustra desired), the Gromov-... | 10 | https://mathoverflow.net/users/353 | 23057 | 15,188 |
https://mathoverflow.net/questions/21607 | 5 | In a recent [question](https://mathoverflow.net/questions/21314/an-example-of-a-series-that-is-not-differentially-algebraic), we learned about the existence of functions that do not satisfy any algebraic differential equation.
One nice property of such equations is that there is a good way to enumerate a basis: we c... | https://mathoverflow.net/users/3032 | beyond differentially algebraic power series | I am not quite sure whether the question is about a "natural" graded algebra which is infinitely generated, or finitely generated algebras are fine as well. Because there are nice examples of algebras of multiple zeta values, but also of multiple polylogarithms and of finite multiple harmonic sums, as well as the algeb... | 5 | https://mathoverflow.net/users/4953 | 23059 | 15,189 |
https://mathoverflow.net/questions/23061 | 9 | Recall that a space is $\delta$-hyperbolic if there is some number $\delta$ with the property that every point on an edge of a geodesic triangle lies within $\delta$ of another edge. For example a tree is $0$-hyperbolic. One of the basic facts about standard hyperbolic space is that it is $\delta$-hyperbolic for some $... | https://mathoverflow.net/users/4362 | It is well-known that hyperbolic space is delta-hyperbolic, but what is delta? | We can use the isometry group of $H^n$ to reduce to the case of an ideal triangle in the upper half plane, with vertices at -1, 1, and infinity. We want to find the distance between $i$ and the vertical geodesic with real part 1. To find the shortest geodesic, we reflect $i$ in the vertical line, and take half the dist... | 13 | https://mathoverflow.net/users/121 | 23064 | 15,192 |
https://mathoverflow.net/questions/22952 | 3 | In the K.Ueno and K.Takasaki's paper ``Toda Lattice hierarchy", advanced sdudies in pure mathematics 4(1984),pp1-95, they mentioned the Toda Lattice hierarchy of B and C type(we denote BTL and CTL hierarchies).In their paper, the bilinear relation for BTL and CTL hierarchies are almost the same as the one for Toda Latt... | https://mathoverflow.net/users/5705 | the Toda Lattice hierarchy of B and C type | I suggest
<http://arxiv.org/abs/nlin.SI/0608018>
as a first step
| 5 | https://mathoverflow.net/users/3054 | 23076 | 15,201 |
https://mathoverflow.net/questions/19477 | 6 | Is there a good reference for the structure of the Chow ring of $\mathcal{A}\_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection theory, enumerative geometry, and vector bundles on $\mathcal{A}\_g$ would be nice.
| https://mathoverflow.net/users/622 | Chow Ring of Moduli Space of Abelian Varieties | Van der Geer has written a paper computing what he calls the tautological subring of the chow ring of $\mathcal{A}\_g$. He also computes the tautological ring for a smooth toroidal compactification.
G. van der Geer, Cycles on the Moduli Space of Abelian Varieties, in "Moduli of Curves and Abelian Varieties (The Dutch... | 3 | https://mathoverflow.net/users/4910 | 23080 | 15,205 |
https://mathoverflow.net/questions/23009 | 15 | Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G\_K$. Suppose that $V \otimes W$ is crystalline. Is $V$ crystalline up to twist by a character of $G\_K$?
| https://mathoverflow.net/users/nan | If the tensor product of two representations are crystalline, are the original representations crystalline? | I'm indeed pretty sure that the answer is "yes". I'd prefer not to post the idea of the proof here because I asked one of my PhD students to write it down with all the details.
| 17 | https://mathoverflow.net/users/5743 | 23082 | 15,207 |
https://mathoverflow.net/questions/23083 | 2 | Let $k$ be a ring and $\overline{k} = k[\epsilon]/\epsilon^2$. For every $f \in k[t]$ there is a unique $f' \in k[t]$ such that $f(t+\epsilon)=f(t)+\epsilon f'(t)$ holds in $\overline{k}[t]$. It follows that $f \mapsto f'$ is a derivation. For example, the leibniz rule:
$(fg)(t+\epsilon)=f(t+\epsilon) g(t+\epsilon)=(... | https://mathoverflow.net/users/2841 | derivative in the ring k[e]/e², chain rule | The point is that you have the more general formula $f(g(t)+\epsilon h(t)) = f(g(t))+f'(g(t))h(t)\epsilon$. From that the chain rule follows:
$$(f\circ g)(t+\epsilon) = f(g(t+\epsilon)) = f(g(t)+g'(t)\epsilon) =
f(g(t))+f'(g(t))g'(t)\epsilon$$
The more general formula follows from the simpler "by substitution", i.e., ... | 6 | https://mathoverflow.net/users/4008 | 23084 | 15,208 |
https://mathoverflow.net/questions/23085 | 3 | I would like to find a reference for the following fact:
every finite dimensional complex representation of a reductive Lie algebra is semisimple.
| https://mathoverflow.net/users/4821 | On the full reducibility of representations of reductive Lie algebras | The statement is false. The standard definition of "reductive" for a finite dimensional Lie algebra $\mathfrak{g}$ over an arbitrary field of characteristic 0 is given in a number of equivalent ways by Bourbaki in Chapter 1 (1960) of their treatise on Lie groups and Lie algebras: section 6, no. 4-5. By definition, $\ma... | 20 | https://mathoverflow.net/users/4231 | 23093 | 15,212 |
https://mathoverflow.net/questions/23094 | 8 | Lets recall Platonic construction in plane geometry. It is impossible to square a circle using only ruler and callipers. But is also known that it is possible to do it with ruler which has a mark on it.
So we know by Gaois theory that we can't write expressions for the solutions to all algebraic equations using the f... | https://mathoverflow.net/users/3811 | method of finding roots of polynominal equations with arithmetic operations and roots and other functions | The answer is no: you need to keep adding more and more operations. For degree $n=5$ you can use elliptic functions (or the Jacobi $\Theta$ function, or Bring radicals - see below), for $n=6,7$ the Lauricella functions are needed (they are a 2-variable version of hypergeometric functions), and after that you need more ... | 10 | https://mathoverflow.net/users/3993 | 23099 | 15,215 |
https://mathoverflow.net/questions/23104 | 1 | It is well known that e.g. $sin(1/x)$ is of [unbounded total variation](http://en.wikipedia.org/wiki/Bounded_variation#Examples) (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded [quadratic variation](http://en.wikipedia.org/wiki/Quadratic_variation). $x\ sin(... | https://mathoverflow.net/users/1047 | Example for deterministic function with unbounded total variation and bounded quadratic variation | Function $x^a\sin(1/x)$ on $(0,1]$ has bounded variation iff $a>1$ and finite quadratic variation iff $a>1/2$. So for your example, take $1/2 < a \le 1$. It looks like more than numerical tests may be needed to decide questions like this...
I took "quadratic variation" to mean
$$
\sup \sum\_{j=1}^n |f(x\_j)-f(x\_{j-... | 4 | https://mathoverflow.net/users/454 | 23109 | 15,223 |
https://mathoverflow.net/questions/23113 | 29 | We know from elementary school that the triangle inequality holds in Euclidean geometry. Some where in High School or in Univ., we come across non-Euclidean geometries (hyperbolic and Riemannian) and Absolute geometry where in both the inequality holds.
I am curious whether the triangle inequality is made to hold in ... | https://mathoverflow.net/users/5627 | Is there any geometry where the triangle inequality fails? | There are people who seriously study quasi-normed spaces. The most natural examples are $\ell\_p$ spaces for p strictly between 0 and 1 (the "norm" given by the usual formula and the distance given by the norm of the difference). Although these spaces do not satisfy the triangle inequality, you get an inequality of the... | 32 | https://mathoverflow.net/users/1459 | 23121 | 15,232 |
https://mathoverflow.net/questions/23111 | 12 | Questions about continued fractions reminded me about a related diophantine problem. I am not quite sure that diophantine equations are still in fashion but
$$
1^k+2^k+\dots+(m-1)^k=m^k,
$$
the Erdős--Moser equation, is quite special. This seems to be the only known equation in two unknowns ($k$ and $m$ are assumed to ... | https://mathoverflow.net/users/4953 | Applications of pattern-free continued fractions | Continued fractions are used in the effective solution of Thue equations. Given such an equation, say $x^4-3x^3y+7x^2y^2-y^4 = 1$ to be specific, we can brute force find all small $x,y$ solutions, and linear forms in logarithms results can be used to eliminate solutions with $x,y$ astronomically large. In the large int... | 13 | https://mathoverflow.net/users/935 | 23138 | 15,241 |
https://mathoverflow.net/questions/23091 | 6 | I'm stuck on a technicality concerning singularities.
Basically, I have to show that the singularities of a $\mathbf{certain}$ normal projective variety over $\mathbf{C}$ are rational. (I won't bother you with the exact set-up. For, it's possible that I'm even wrong.)
My idea was to use a theorem of Viehweg and s... | https://mathoverflow.net/users/4333 | Is there an obvious way for showing singularities are quotient? | Say $X=\mathbb A^n$ and $D\_1,\dots, D\_n$, the components of $D$, are the coordinate hyperplanes $x\_i=0$, for simplicity. You can assume that WLOG, because your $(X,D)$ is isomorphic to this one in étale topology.
$\pi\_1(X\setminus D) = \mathbb Z^n$. So the cover $V\to U$ corresponds to a finite quotient of $\mat... | 8 | https://mathoverflow.net/users/1784 | 23139 | 15,242 |
https://mathoverflow.net/questions/23107 | 15 | I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,...) which are usually only stated for varieties over the complex numbers, extend to varieties over arbitrary fields of characteristic zero or to the equivariant case.
I assume that the basic finite generatio... | https://mathoverflow.net/users/519 | What is known about the MMP over non-algebraically closed fields | Both of these cases follow more or less automatically from the Minimal Model Program over an algebraically closed field $\bar k$. This is well, known, see for example the original Mori's paper *"Threefolds whose canonical bundles are not numerically effective"* or [Kollár's paper](http://www.math.princeton.edu/~kollar/... | 13 | https://mathoverflow.net/users/1784 | 23141 | 15,244 |
https://mathoverflow.net/questions/23136 | 1 | The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray originating at the point will cross before escaping. If we use projective duality, then regression depth is same as undire... | https://mathoverflow.net/users/5553 | How does one map regression depth to undirected depth of a point? | (For some context: the question here is written in terms of points in a plane, and my answer is in the same terms, but the same duality works in higher dimensions as well. The paper in question is [arxiv:cs.CG/9809037](http://arxiv.org/abs/cs.CG/9809037).)
Let P be the plane in which you are measuring regression dept... | 0 | https://mathoverflow.net/users/440 | 23146 | 15,247 |
https://mathoverflow.net/questions/23128 | 5 | Let $B$ be an infinite-dimensional Banach space, and let $M\subset\mathbb{R}^n$ be a neighborhood of the origin in $\mathbb{R}^n$.
Suppose that $I:M\to B$ is a real-analytic function with $I(0)=0$ and such that the derivative of $I$ at $0$ has maximal rank.
Is it true that there exist neighborhoods $U,V\subset B$ o... | https://mathoverflow.net/users/3651 | Local form of a real-analytic function taking values in a Banach space | I cannot give you a reference, but the answer ought to be yes.
To simplify notation, identify $\mathbb{R}^n$ with its image in $B$ under the derivative $I'(0)$. That image, being finite-dimensional, is the range $pB$ of a finite rank, bounded projection $p$. Replacing $M$ by a smaller neighbourhood if necessary, we c... | 2 | https://mathoverflow.net/users/802 | 23148 | 15,249 |
https://mathoverflow.net/questions/22662 | 4 | Suppose $V\_1$ and $V\_2$ are two $(g,K)$ modules of some reductive group $G$ with maximal compact $K$. Let $P$ be the minimal parabolic of $G$, $U$ its unipotent part, and $u$ its Lie algebra. Suppose the quotients $V\_1/uV\_1$ and $V\_2/uV\_2$ are isomorphic as modules for the Levi component of $P$, then what else do... | https://mathoverflow.net/users/1832 | what information of a representation was killed by Jacquet functor? | This is a comment, mostly on terminology that I think caused some confusion, not an answer, but I don't have enough "influence" to post this as a comment.
For real reductive groups, the Jacquet functor $V\mapsto J(V)$ (Jacquet-Casselman functor, Jacquet module, etc) is defined *differently* from the nonarchimedean ca... | 5 | https://mathoverflow.net/users/5740 | 23156 | 15,254 |
https://mathoverflow.net/questions/23154 | 6 | In [this](http://www.ams.org/journals/bull/1965-71-01/S0002-9904-1965-11265-4/S0002-9904-1965-11265-4.pdf) classic paper, Sakai proves the following Radon-Nikodym theorem:
>
> Let $M$ be a von Neumann algebra, and let $\phi$ and $\psi$ be two normal positive linear functionals on $M$. If $\psi \leq \phi$, then ther... | https://mathoverflow.net/users/2206 | A non-commutative Radon-Nikodym derivative. | Such t\_0 is unique if its support is at most p, where p is the support of ϕ.
Note that we can replace t\_0 by pt\_0p and the support of pt\_0p is at most p.
Without this additional condition t\_0 is highly non-unique, because
we can replace t\_0 by t\_0 + q,
where q is an arbitrary self-adjoint element with support ... | 6 | https://mathoverflow.net/users/402 | 23161 | 15,257 |
https://mathoverflow.net/questions/23153 | 12 | Given integers $a,b,c$ such that $\gcd(a,b,c) = 1$, it is well known that there exists only a finite set of numbers $n$ such that $n$ is not expressible as $ax+by+cz$ for non negative integers $x$,$y$,$z$.
It is also known that there exists a quadratic time algorithm for finding the maximal such $n$. However I was n... | https://mathoverflow.net/users/1737 | Frobenius number for three numbers | Simple algoritm based on continued fractions was proposed by Rödseth, O. J. On a linear Diophantine problem of Frobenius J. Reine Angew. Math., 1978, 301, 171-178
All algorithm are described in Ramrez Alfonsn, J. L. The Diophantine Frobenius problem Oxford University Press, 2005
I think that Mathematica uses Rödset... | 10 | https://mathoverflow.net/users/5712 | 23165 | 15,258 |
https://mathoverflow.net/questions/23168 | 1 | Where can I found a description of the deformation theory for modules?Is it possible to deform a free module in such way that each fibre of the deformation is still free?
| https://mathoverflow.net/users/4821 | Deformations of free modules | A free module is rigid, because $Ext^1(E, E)=0$ for any free module $E.$
For deformation theory see for example ``Functors of Artin Rings''
by Michael Schlessinger.
| 2 | https://mathoverflow.net/users/2464 | 23170 | 15,260 |
https://mathoverflow.net/questions/23171 | 10 | This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.
Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex manifold, and $L$ a totally real submanifold of $M$. A map
$u:(\Sigma, \partial\Sigma)\to(M, L)$
gives rise to a *bun... | https://mathoverflow.net/users/2819 | Maslov index of a pullback bundle | When you have a vector bundle on a manifold $X$ with boundary, trivialised over $\partial X$, there are characteristic classes valued in $H^\ast (X,\partial X)$. Here, when $L$ is orientable, the Maslov index is twice the first Chern class of $u^\ast TM$ relative to the trivialisation on the boundary induced by $L$, ev... | 10 | https://mathoverflow.net/users/2356 | 23176 | 15,264 |
https://mathoverflow.net/questions/23175 | 38 | This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
| https://mathoverflow.net/users/5756 | Mathematically mature way to think about Mayer–Vietoris | The Mayer-Vietoris sequence is an upshot of the relationship between sheaf cohomology and presheaf cohomology (a.k.a. Cech cohomology).
Let $X$ be a topological space (or any topos), $\mathcal U$ a covering of $X$. Let $\mathop{\rm Sh}X$ be the category of sheaves on $X$ and $\mathop{\rm PreSh}X$ the category of pres... | 64 | https://mathoverflow.net/users/4790 | 23180 | 15,268 |
https://mathoverflow.net/questions/23182 | 1 | $S$ is a graded ring (over non-negative integers), $f \in S\_{+}$ is a homogeneous element of positive degree, $D(f)$ the elements of Proj $S$ not containing $f$. I don't see the bijection between $D(f)$ and Spec $S\_{(f)}$. Here $S\_{(f)}$ is the zero-degree part of $S\_{f}$ obtained from $S$ by inverting f. I see the... | https://mathoverflow.net/users/5292 | Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)} | The homeomorphism $D\_+(f) \to \text{Spec } S\_{(f)}$ is given by $\mathfrak{p} \mapsto \mathfrak{p} S\_f \cap S\_{(f)}$ with inverse map $\mathfrak{q} \mapsto \oplus\_n \{x \in S\_n : x^{|f|} / f^n \in \mathfrak{q}\}$. This can be checked by simple calcuations.
| 2 | https://mathoverflow.net/users/2841 | 23183 | 15,269 |
https://mathoverflow.net/questions/23192 | 2 | The question if there is an upper bound known for Brun's constant was discussed briefly here: <http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/> but no sure answer was given.
So I thought I'd ask the question here. Can one get any upper bound for the sum of the reciprocals of the twin primes?
... | https://mathoverflow.net/users/2888 | Upper bound on Brun's constant | Crandall and Pomerance, "Prime numbers: a computational perspective" (Google books) says that Brun's constant B, the sum of the reciprocals of the twin primes, is known to be between 1.82 and 2.15.
**edited to add**: I'm aware that this isn't much of a citation. It would be nice if someone who has access to this book... | 4 | https://mathoverflow.net/users/143 | 23194 | 15,274 |
https://mathoverflow.net/questions/23202 | 65 | In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or AC) are needed. Feel free to use similar (perhaps more precise) notions of "explicit", but reasonable ones! To be hones... | https://mathoverflow.net/users/2841 | explicit big linearly independent sets | Here is a linearly independent subset of $\mathbb{R}$ with size $2^{\aleph\_0}$.
Let $q\_0, q\_1, \ldots$ be an enumeration of $\mathbb{Q}$. For every real number $r$, let
$$T\_r = \sum\_{q\_n < r} \frac{1}{n!}$$
The proof that these numbers are linearly independent is similar to the usual proof that $e$ is irration... | 124 | https://mathoverflow.net/users/2000 | 23206 | 15,280 |
https://mathoverflow.net/questions/23193 | 21 | The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ring with positive induction?
| https://mathoverflow.net/users/5761 | Axiomatic definition of integers | It's the unique commutative ordered ring whose positive elements are well-ordered.
| 28 | https://mathoverflow.net/users/5583 | 23212 | 15,284 |
https://mathoverflow.net/questions/23197 | 4 | **Question:** What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$?
Specifically, I'm thinking of the Hölder spaces $\mathcal{C}^{k,\alpha}(M, N)$ and the Sobolev spaces $\mathcal{W}^{k,p}(M, N)$.
**Some comments:**
... | https://mathoverflow.net/users/2819 | Connected components of space of maps between two manifolds | Any continuous map from *M* to *N* is homotopic to a smooth map, and if two smooth maps are homotopic, then they are also smoothly homotopic. (More generally, two homotopic functions are homotopic through a homotopy that is smooth except at the endpoints.) The proof involves convolving with Gaussians, and is standard; ... | 5 | https://mathoverflow.net/users/5010 | 23224 | 15,293 |
https://mathoverflow.net/questions/20263 | 8 | The ring of invariants $S^T$ of $k[a,b,c,d]$ under the algebraic torus action $T = k^{\*}$ with weights $(1,1,-1,-1)$
is $S = k[ac,ad,bc,bd]$ which has multigraded Hilbert series
$$
\frac{1 - abcd}{(1-ac)(1-ad)(1-bc)(1-bd)}.
$$
Is there a software package that can compute multigraded Hilbert series? Can it be computed ... | https://mathoverflow.net/users/874 | Software for computing multi-graded Hilbert series | Macaulay 2 can do multigraded Hilbert series. Let's first assume that you have a presentation of your multigraded ring. I'll mention how to calculate this below. So for your $S = k[ac,ad,bc,bd]$, we'll write it as $S = k[x,y,z,w] / (xz - yw)$.
Assuming that each of $a,b,c,d$ has its own degree direction (so the gradi... | 11 | https://mathoverflow.net/users/321 | 23235 | 15,298 |
https://mathoverflow.net/questions/23240 | 7 | A commutative noetherian ring $R$ is Gorenstein if $R$ has finite injective dimension.
Obviously, if $R$ is Gorenstein, then $R$ localized at any prime ideal $P$ is also Gorenstein. But I don't know whether the converse holds?
| https://mathoverflow.net/users/5775 | Is there a non-Gorenstein ring but locally Gorenstein? | In all of the standard references I know, Gorenstein is either only defined for local rings or a not necessarily local ring is *defined* to be Gorenstein if all of its localizations at maximal ideals are Gorenstein (which implies that its localizations at all prime ideals are Gorenstein). So I am guessing you really wa... | 15 | https://mathoverflow.net/users/1149 | 23242 | 15,303 |
https://mathoverflow.net/questions/23204 | 7 | Suppose I have distinct real numbers $a\_i \in [-1,1]$, $i \in [k]$. I want to choose real numbers $b\_j, j\in [k]$ such that the matrix $(\arccos(a\_i b\_j))\_{i,j \in [k]}$ is nonsingular.
>
> Is this always possible? Equivalently, does $\arccos$ not satisfy a functional equation of the form $\det(\arccos(a\_i x... | https://mathoverflow.net/users/3318 | Nonexistence of determinantal functional equation for $\arccos$ | A quick counter-example to the question as stated is $a\_0=0$, $a\_1=1$ $a\_2=-1$. Since $2arccos(0)-arccos(b)-arccos(-b)=0$ for all $b$, we have $2M\_1-M\_2-M\_3=0$ where $M\_1, M\_2, M\_3$ are the rows of the matrix.
So it is better to assume that the $a\_i$ are nonnegative. In this case, the answer is yes. More ge... | 12 | https://mathoverflow.net/users/4354 | 23244 | 15,305 |
https://mathoverflow.net/questions/23246 | 13 | I've been going over the extremely interesting discussions about Axiom of Choice.
It looks to me like all the "weird" consequences of AC (Banach-Tarski etc) come from using it on uncountable collections of sets.
If, instead, we only believe the Axiom of Countable Choice, do we still get unintuitive consequences in ... | https://mathoverflow.net/users/4279 | Peculiar examples with Axiom of Countable Choice ? | If you assume the existence of suitable large cardinals, then $L(\mathbb{R})$ is a model of the Axiom of Determinacy $AD$ and the Axiom of Dependent Choice $DC$. In particular, since $DC$ is stronger than the Axiom of Countable Choice $AC\_{\omega}$, it follows that $AC\_{\omega}$ is also true in $L(\mathbb{R})$. Since... | 15 | https://mathoverflow.net/users/4706 | 23249 | 15,306 |
https://mathoverflow.net/questions/23247 | 8 | Let $\mathrm{Hol}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the space of holomorphic maps of degree $d$ from a Riemann surface $\Sigma$ to complex projective space of dimension $n$. Let $\mathrm{HolEmb}^d(\Sigma, \mathbb{C} \mathbb{P}^n)$ denote the subspace of those holomorphic maps which are also embeddings, so ther... | https://mathoverflow.net/users/318 | Approximating holomorphic maps by holomorphic embeddings | I will assume that *d* is the degree of the pull-back of $\mathcal{O}(1)$ to $\Sigma$ and that it is sufficiently large with respect to the genus *g* of $\Sigma$. In this case, the dimension of the space of holomorphic maps of $\Sigma$ in $\mathbb{P}^n$ is
$$
D\_n := (n+1)d + n(1-g) ,
$$
while the dimension of the spa... | 4 | https://mathoverflow.net/users/4344 | 23251 | 15,307 |
https://mathoverflow.net/questions/23254 | 1 | Does it exist a Lie algebra $\mathfrak{g}$ that is reductive but if we consider the inclusion of Lie agebras $\mathfrak{g} \subset \mathfrak{h}$ then $\mathfrak{g}$ is not reductive in $\mathfrak{h}?$
| https://mathoverflow.net/users/4821 | Reductive Lie algebra | A reductive Lie algebra $L$ is the direct sum of a semisimple
Lie algebra $L\_1$ and an abelian Lie algebra $L\_2$. Let's consider
the case where $L\_2$ is one-dimensional.
We can embed $L$ into a larger Lie algebra $L=L\_1\oplus L\_2'$
by embedding $L\_2$ into $L\_2'$. Let $L\_2'$ be the two-dimensional
Lie subalgebra... | 4 | https://mathoverflow.net/users/4213 | 23255 | 15,308 |
https://mathoverflow.net/questions/22699 | 4 | I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) Theorem 3.1.8 of Cataldo-Migliorini's survey <http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-... | https://mathoverflow.net/users/2191 | Morphisms between pure complexes of sheaves | Dear Mikhail,
I had been hoping someone else would attempt to answer this question, as I have been wondering very similar things lately. (In fact I drove myself crazy for about a month last year trying to work out some solution to what you are asking in the second paragraph.)
I can't answer everything but here is a... | 3 | https://mathoverflow.net/users/919 | 23257 | 15,310 |
https://mathoverflow.net/questions/23229 | 23 | The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
1) f(x+1) = x\*f(x)
2) f(1) = 1
3) ln(f(x)) is convex
The Gamma function is meant to interpolate the factorial function, so I can see the importance of the first two properties. But why is log convexity important? Ho... | https://mathoverflow.net/users/5768 | Importance of Log Convexity of the Gamma Function | First, let me mention that log convexity of a function is implied by an analytic property, which appears to be more natural than log convexity itself. Namely, if $\mu$ is a Borel measure on $[0,\infty)$ such that the $r$th moment
$$f(r)=\int\_{0}^{\infty}z^r d\mu(z)$$
is finite for all $r$ in the interval $I\subset \m... | 15 | https://mathoverflow.net/users/5371 | 23262 | 15,312 |
https://mathoverflow.net/questions/23221 | 11 | Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember where I saw it, long time ago.] I would be very grateful for any (freely available, if possible) reference (Pelczynski ... | https://mathoverflow.net/users/2508 | Nonseparable Hilbert spaces as quotients of spaces of bounded functions | I don't know who first observed this (maybe Archimedes?) but it is true because $C(\{0,1 \}^\Gamma)$ is a quotient of $\ell\_1^\Gamma$ and hence $\ell\_1(2^\Gamma)$ embeds into $\ell\_\infty(\Gamma)$.
@Ady
Here is a more serious answer to your question. Take a quotient map $Q$ from $\ell\_1(2^\Gamma)$ onto $C([0,... | 8 | https://mathoverflow.net/users/2554 | 23273 | 15,319 |
https://mathoverflow.net/questions/23268 | 38 | I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects like limits, colimits, and adjoint functors, I was able to do them, although I am sure my proofs were far longer and more l... | https://mathoverflow.net/users/5094 | Geometric intuition for limits | I pick up your remarks about sheaves. Indeed, the sheaf condition is a very good example to get a geometric idea of a limit.
Assume that $X$ is a set and $X\_i$ are subsets of $X$ whose union is $X$. Then it is clear how to characterize functions on $X$: These are simply functions on the $X\_i$ which agree on the ove... | 33 | https://mathoverflow.net/users/2841 | 23276 | 15,321 |
https://mathoverflow.net/questions/23278 | 10 | It seems the notion of tensor product of abelian categories exists naturally.
Does someone know the reference of the construction?
| https://mathoverflow.net/users/5082 | Tensor product of abelian categories | Deligne's article, 'Categories Tannakiennes,' section 5 would be a good place to look. It was published in the Grothendieck Festschrift, vol. 2.
| 11 | https://mathoverflow.net/users/373 | 23284 | 15,325 |
https://mathoverflow.net/questions/22533 | 13 | Let $\mathbf{Q}\_p$ denote the field of $p$-adic numbers.
Suppose that $K/\mathbf{Q}\_p$ is a finite extension, and let $O\_K$ denote the ring of integers of $K$. Suppose that $X$ is proper over $O\_K$, with smooth generic fibre. Consider the following three statements:
A. There exists a finite extension $L/K$ such... | https://mathoverflow.net/users/nan | Potential semi-stability of etale cohomology of etale covers. | Answered to my satisfaction. Well, except for the bit about typesetting $\mathbf{Q}\_p$, that still confuses me.
| 4 | https://mathoverflow.net/users/nan | 23289 | 15,329 |
https://mathoverflow.net/questions/23228 | 9 | Consider critical edge-percolation in the induced subgraph of the square grid with vertex set {$(i,j) \in Z \times Z:\ i+j \geq 0$}, and let $p\_n$ be the probability that the cluster containing $(0,0)$ has size $> n$. How quickly does $p\_n$ fall as $n \rightarrow \infty$?
| https://mathoverflow.net/users/3621 | half-plane percolation clusters | As Leandro suggested in the comments, this should follow a power-law decay in $n$. However, Hara and Slade's rigorous work using lace expansions is only valid for dimensions $\ge 19$. Much of the rigorous work on critical exponents for two-dimensional percolation has been done only recently, via connections to Schramm-... | 4 | https://mathoverflow.net/users/238 | 23291 | 15,331 |
https://mathoverflow.net/questions/23285 | 1 | I am trying to plot the pdf of flipping heads when drawing from a bag of biased coins. Since I am interested in the % of heads flipped, not the number, I simulate 500K flips and group the results into buckets of 100, computing the % of heads in each bucket. I do this using two different methods, in method 1 I draw a co... | https://mathoverflow.net/users/5783 | simulating chances of success when drawing from a bag of biased coins | You seem to answer the question yourself in the last sentence of your first paragraph. If you draw a new coin every flip, then the experiment is equivalent to a sequence of coin tosses with a single coin (with appropriately chosen probability).
Maybe helpful to think about independence. If you draw a new coin every f... | 3 | https://mathoverflow.net/users/5784 | 23296 | 15,332 |
https://mathoverflow.net/questions/23295 | 2 | Hi all,
I have been looking at complex multiplication of elliptic curves for a course project in cryptography and the following question came up: Let $\mathcal{O}\_K$ be the maximal order in $K$ ($K$ is an imaginary quadratic field), let $h\_K (X)$ be the Hilbert class polynomial of $K$. Suppose that $\mathcal{O}$ i... | https://mathoverflow.net/users/2917 | Relation between the Hilbert Class polynomial of $\mathcal{O}_K$ and an order. | As far as I know, the best relation between the two is the following: the field generated by the hilbert class polynomial $h\_\mathcal{O} (X)$ contains the field generated by $h\_K(X)$. This is implied by Proposition 25 of [this paper](http://alpha.math.uga.edu/~pete/torspaper_FINAL.pdf).
This implies among other thing... | 3 | https://mathoverflow.net/users/3384 | 23302 | 15,334 |
https://mathoverflow.net/questions/23253 | 2 | Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory:
1. Countable models
2. Distinguished/canonical models.
I'm interested in 2 because I don't see why the standard ${\mathbb{R}}... | https://mathoverflow.net/users/nan | Reference: Countable Models of (Non-)Euclidean Geometry | Marvin Jay Greenberg got very interested in the foundations for the fourth edition(2007) of his book. This led to a survey article in the March 2010 M.A.A. Monthly. Table of contents:
<http://www.maa.org/pubs/monthly_mar10_toc.html>
It does not seem to say explicitly on the link, volume 117, number 3, March 2010, ... | 1 | https://mathoverflow.net/users/3324 | 23304 | 15,336 |
https://mathoverflow.net/questions/23306 | 5 | Suppose you have a 2-dimensional polyhedral surface with specified lengths for the edges so that the vertices all have positive curvature. I believe this has a unique isometric embedding into 3-dimensional Euclidean space as the boundary of a convex polyhedron. Could someone confirm this? If so, is there a reasonable a... | https://mathoverflow.net/users/613 | Isometric embedding of a positively curved polyhedral surface | Yes, this is the historically first version of the Alexandrov embedding theorem which was earlier discussed [here](https://mathoverflow.net/questions/22122/on-alexandrov-embedding-theorem). See the original Alexandrov's monograph (reviewed [here](http://www.math.cornell.edu/~connelly/alexandrov.pdf)), or see [my book](... | 11 | https://mathoverflow.net/users/4040 | 23309 | 15,338 |
https://mathoverflow.net/questions/23297 | 42 | I'm looking for basic examples that show the usefulness of spectral sequences even in the simplest case of spectral sequence of a filtered complex.
All I know are certain "extreme cases", where the spectral sequences collapses very early to yield the acyclicity of the given complex or some quasi-isomorphism to anoth... | https://mathoverflow.net/users/3108 | Simple examples for the use of spectral sequences | This isn't exactly what you asked, but its a very simple example that (to me) demonstrates some of the necessity of the complexities of spectral sequences. Consider the ring $R=\mathbb{C}[x,y]$, and consider the module $M=(Rx+Ry)\oplus R/x$. Then the double dual spectral sequence converges to the original module:
$$ Ex... | 9 | https://mathoverflow.net/users/750 | 23316 | 15,344 |
https://mathoverflow.net/questions/23277 | 2 | While studying some class field theory there was a lot of talk on galois extensions. Of course. When talking about non-galois number fields, usually the text will quickly take the galois closure. At this point it occurred to me that this implies many properties are shared by number fields with the same galois closure.
... | https://mathoverflow.net/users/2024 | Properties shared by number fields with the same normal closure? | There's a huge amount of literature on this problem starting with
* F. Gassmann, *Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppen*, Math. Z. 25, 661-675 (1926)
Gassmann constructed number fields with the same normal closure in which almost all primes spl... | 5 | https://mathoverflow.net/users/3503 | 23317 | 15,345 |
https://mathoverflow.net/questions/23312 | 8 | Recently, I'm tired of those theoretical parts on commuative algebra. So I hope that someone could recommend me some good textbooks on SINGULAR and Macaulay 2. And I'm wondering whether SINGULAR is better that Macaulay 2?
Thank you in advance!
| https://mathoverflow.net/users/5775 | Textbooks on SINGULAR and Macaulay 2 | Two remarks:
* Macauly 2 and Singular share the same computational engine (singular) so none if them is "better" in any real sense
* the best book+software combination I know of is COCOA plus the two volumes of "computational commutative algebra". My "issue" with the singular book is that it's too basic, and with the... | 2 | https://mathoverflow.net/users/404 | 23319 | 15,347 |
https://mathoverflow.net/questions/23181 | 3 | I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).
The sectors live in the complex plane, and for n even,
sector 0 and n/2 are bisected by the real axis, and the sectors are evenly spaced.
These branches meet at certain points, called juncti... | https://mathoverflow.net/users/1056 | Find symmetries of a tree | I don't understand how the angles of the connecting finite segments are determined, so I'll assume the angles are set so that they don't break any symmetry. First observe that the reflection wrt the real axis sends sectors 0,1,2,3,4,5 to 0,5,4,3,2,1 respectively. So in your second example, tree
(0,1,5)(1,2,5)(2,4,5)(... | 1 | https://mathoverflow.net/users/4118 | 23327 | 15,350 |
https://mathoverflow.net/questions/23325 | 5 | I know that arithmetic sentences are conserved under the addition of the axiom of choice and the continuum hypothesis to ZF (i.e. ($ZF+AC \vdash \phi$ iff $ZF \vdash \phi$) and ($ZF+CH \vdash \phi$ iff $ZF \vdash \phi$) for arithmetic sentences $\phi$) Does this result extend to analytic sentences or other hyperarithme... | https://mathoverflow.net/users/4085 | Conservation of Hyperarithmetic Sentences over AC and CH. | First, let me remark that in your question, you can combine AC and CH together, rather than having two separate conservation results as you did. In fact, you can ramp CH up to GCH and more, including such principles as $\Diamond$ or others, without any problem. That is, the conservation result is that ZFC + GCH proves ... | 7 | https://mathoverflow.net/users/1946 | 23332 | 15,353 |
https://mathoverflow.net/questions/23203 | 0 | Hi,
I've read this sentence but I can not understand what it means
[...] $\Phi'$ is the topological dual of some dense space $\Phi$ of $H\_{aux}$ [...] Notice that the choice of $\Phi$ is subject to the two conditions: [...] ,On the other hand it must be small enough so that its topological dual $\Phi$ is "sufficient... | https://mathoverflow.net/users/2597 | Topological dual and the notions of "smaller" and "larger" than... | To answer your final question: Let $\Phi \supset \Psi$. Consider $\Phi' \subset \Phi^\*$, the former is the continuous linear functionals on $\Phi$, and the latter is the set of all linear functionals on $\Phi$. Then the restriction of $\Phi'$ on $\Psi$ is obviously continuous, so $\Phi' \subset \Psi'\subset \Psi^\*$. ... | 5 | https://mathoverflow.net/users/3948 | 23344 | 15,361 |
https://mathoverflow.net/questions/23322 | 6 | Let $C$ be a compact Riemann surface, and let $U$ be a Zariski open subset in $C.$ Let $L$ be a local system (with coefficients $\mathbb C$ or $\mathbb Q\_{\ell}$) on $U.$ For each point $z\_i\in C-U,$ let $M\_i$ be the monodromy matrix of $L$ around $z\_i.$ If we identify $L$ as a representation of $\pi\_1(U),$ and le... | https://mathoverflow.net/users/370 | monodromy and global cohomology | Suppose that $C = \mathbb P^1$ and $U = \mathbb P^1\setminus \{0,\infty\}.$ Then
$\pi\_1(U)$ is cyclic, freely generated by a loop around $0$.
The local system $L$ is thus given by the vector space $L\_a$, equipped with an invertible operator,
call it $T$, corresponding to the action of the generator of $\pi\_1(U)$. ... | 10 | https://mathoverflow.net/users/2874 | 23348 | 15,363 |
https://mathoverflow.net/questions/23354 | 6 | This might turn out to be a silly question, but here goes.
Let $\mathcal{C}$ be a full additive subcategory of an abelian category $\mathcal{A}$. I'm wondering if the Grothendieck group $K(\mathcal{C})$ (defined below) depends on $\mathcal{A}$. I can't think of any examples, and I believe that it does not. Something ... | https://mathoverflow.net/users/4333 | Does the Grothendieck group depend on the embedding? | I think that the Grothendieck group DOES depend on A. Indeed, any additive category C could be embedded (by the Yoneda embedding) into the abelian category of contravariant additive functors from C to abelian groups (some call this category the category of presheaves on C). For this embedding the only exact sequence of... | 11 | https://mathoverflow.net/users/2191 | 23356 | 15,366 |
https://mathoverflow.net/questions/23349 | 17 | This ought to be a simple one to answer. Does anyone know of, or can anyone provide, an accurate English translation of the marginal remarks in Goldbach's letter to Euler
<http://upload.wikimedia.org/wikipedia/commons/1/18/Letter_Goldbaxh-Euler.jpg>
in which a statement equivalent to the Goldbach conjecture is firs... | https://mathoverflow.net/users/5575 | Translation of Goldbach's 1742 letter to Euler | I have an English translation of the letter; you can find my email address on my homepage.
Here is the relevant part:
"By the way, I take it to be not useless to note down also such propositions
that are very probable, even if a real proof is lacking; for even if
afterwards they were found to be erroneous, they coul... | 14 | https://mathoverflow.net/users/3503 | 23365 | 15,373 |
https://mathoverflow.net/questions/20007 | 9 | In Alain Roberts "Elliptic curves: notes from postgraduate lectures given in Lausanne 1971/72" page 11 (available on google books unless you already tried to read another chapter), there is a hand drawn picture of a real 2-dimensional torus and a real plane, which **topologically** represent the way a complex cubic (wi... | https://mathoverflow.net/users/404 | Visualizing a complex plane cubic together with the real plane | I found this article:
"Visualizing Elliptic Curves"
by Donu Arapura
it is available at the following URL:
<http://www.math.purdue.edu/~dvb/graph/elliptic.pdf>
In it he discusses a projection that sends sends the real
part of $x$ to $x\_1$ and the real part of $y$ to $x\_3$ thus
it would seem to preserve the entire real... | 3 | https://mathoverflow.net/users/1098 | 23367 | 15,374 |
https://mathoverflow.net/questions/23334 | 6 | The generalized Korteweg-de Vries equation is
$u\_t + u\_{xxx} + (u^p)\_x=0$
for integer $p$. (The original Korteweg-de Vries equation is the case $p=2$.) I need to understand solutions for $p=1$, but I haven't been able to find this case addressed in the literature despite extensive searching. Is there a reason wh... | https://mathoverflow.net/users/5789 | What are the interesting cases of the generalized Korteweg-de Vries equation? | For $p=1$ your equation indeed reduces to a simpler form. Put $X=x-t, T=t$. Then $u\_t=u\_T-u\_X$, $u\_x=u\_X$, and hence in the new independent variables $X,T$ your equation (with $p=1$) becomes a well-studied linear third-order equation
$$
u\_{T}+u\_{XXX}=0,
$$
which is, *inter alia*, a linearized version of the "sta... | 8 | https://mathoverflow.net/users/2149 | 23372 | 15,376 |
https://mathoverflow.net/questions/23368 | 0 | $A$ a local ring and $a\_{1}$, ..., $a\_{n}$ elements in its maximal ideal, $M$ a finitely generated $A$-module. In this case apparently the homologies from the Koszul complex are finitely generated as $A$-module. Is there a simple explanation for this? Or is it some deep result? Thanks!
| https://mathoverflow.net/users/5292 | Homology of koszul complex is finitely generated? | If your local rings are Noetherian it's obvious. The Koszul complex
consists of finitely generated free modules and the homology
modules are subquotients of it so also finitely generated.
For non-Noetherian rings it's not true, even when $n=1$. In
this case the $H\_1$ is the annihilator of $a\_1$ which may not
be fin... | 8 | https://mathoverflow.net/users/4213 | 23373 | 15,377 |
https://mathoverflow.net/questions/23361 | 12 | I am delving a bit into category theory and something has me curious about opposite categories. I have checked several books and I can't seem to find an answer.
Given a category C, the opposite category is just the abstract category with the objects of C and with the arrows of C reversed. However, the opposite catego... | https://mathoverflow.net/users/5800 | Construction Of Opposite Category as a Structure | One way of formalizing the desired categories is given by concrete categories. A category is called concrete (more precise: "concretizable") if it has a faithful functor to the category of sets $Set$. Thus the question is: Is the dual of a concrete category concrete again? The answer is yes: Since a composition of fait... | 8 | https://mathoverflow.net/users/2841 | 23379 | 15,380 |
https://mathoverflow.net/questions/23092 | 8 | Let $K$ be a quadratic imaginary field, and $E$ an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of $K$. Let $j$ be its $j$-invariant, and $c$ an integral ideal of $K$. Consider the following tower:
$$K(j,E[c])\ \ /\ \ K(j,h(E[c]))\ \ /\ \ K(j)\ \ /\ \ K,$$
where $h$ here is any ... | https://mathoverflow.net/users/5744 | Class Field Theory for Imaginary Quadratic Fields | Here is a case where it is non-Abelian. I use $K$ of class number 3. If I use the Gross curve, it is Abelian. If I twist in $Q(\sqrt{-15})$, it is Abelian for every one I tried, maybe because it is one class per genus. My comments are not from an expert.
```
> K<s>:=QuadraticField(-23); ... | 8 | https://mathoverflow.net/users/5267 | 23390 | 15,387 |
https://mathoverflow.net/questions/23378 | 7 | Franel uses the convergence of
$ \frac{\zeta(s+1)}{\zeta(s)} = \sum \frac{c(n)}{n^s}$
as an equivalent to the Riemann hypothesis.
Does anybody have a citation for this result and/or hints for computing $c(n)$?
Thanks for any insight.
Cheers, Scott
| https://mathoverflow.net/users/4111 | $\zeta(s+1)/\zeta(s)$ | Since
$$\zeta(s+1) = \sum\_{n=1}^\infty \frac{1/n}{n^s}$$
and
$$\frac{1}{\zeta(s)} = \sum\_{n=1}^\infty \frac{\mu(n)}{n^s}$$
where $\mu$ is the [Möbius function](http://en.wikipedia.org/wiki/M%C3%B6bius_function), we have
$$c(n) = \sum\_{d \mid n} \frac{d}{n}\mu(d) = \frac{1}{n}\prod\_{p \mid n} (1-p)$$
using [Dirichle... | 11 | https://mathoverflow.net/users/2000 | 23395 | 15,388 |
https://mathoverflow.net/questions/23393 | 10 | From <http://en.wikipedia.org/wiki/Analytical_hierarchy>
"If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is $\Delta^1\_2$ and is the graph of a well ordering of the Baire space. If the axiom holds then there is also a $\Delta^1\_2$ well ordering of Ca... | https://mathoverflow.net/users/nan | Set Theory and V=L | In the constructible universe $L$, there is a definable well-ordering of the entire universe. This universe is built up in transfinite stages $L\_\alpha$, and the ordering has $x\lt\_L y$ when $x$ is constructed at an earlier stage, or else they are constructed at the same stage, but $x$ is constructed at that stage by... | 16 | https://mathoverflow.net/users/1946 | 23399 | 15,389 |
https://mathoverflow.net/questions/23394 | 23 | I would like to understand at least one of the several existing approaches to algebraic geometry over $\mathbb{F}\_1$ (the field with one element). Is there an example of an "interesting" theorem that can be formulated purely in the language of ordinary schemes, but which can be proved using algebraic geometry over $\m... | https://mathoverflow.net/users/4384 | Applications of algebraic geometry over a field with one element | I'm confident that the answer to the original question is *no*. There are hardly any theorems at all in the subject, much less ones with external applications! In other words, if no further progress is ever made in any of the directions people have pursued, everything will likely be forgotten (which would not make it s... | 15 | https://mathoverflow.net/users/1114 | 23418 | 15,400 |
https://mathoverflow.net/questions/23415 | 4 | Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order?
More precisely, let $X$ be a topological space such that the map $\emptyset\to X$ factorizes as a transfinite composition of inclusions
$$
\emptyset\to\ldots\to X\_\beta\to X\_{\beta+1}\to\ldots\t... | https://mathoverflow.net/users/467 | CW structure on spaces obtained by attaching cells wildly | Fix an irrational number $\alpha$.
Let $X\_2 = [0,1]$ (built from attaching a 1-cell to two 0-cells) and, for each larger $n$, let $X\_n$ be built from $X\_{n-1}$ with a new a 1-cell by attaching the ends to $0$ and the fractional part of $n \alpha$. Take $X$ to be the union.
There are no embeddings from an open di... | 13 | https://mathoverflow.net/users/360 | 23425 | 15,403 |
https://mathoverflow.net/questions/23422 | 4 | Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}\_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of our vectors will be linearly independent (over $\mathbb{F}\_2$) too.
The question is as follows: how large $d - n$ shou... | https://mathoverflow.net/users/3448 | Random projection and finite fields | Suppose the vectors are $e\_1,\dots,e\_n$. The kernel of projection onto a random subspace of dimension $n+r$ is a random subspace of dimension $n-r$, so you want the probability that such a subspace has trivial intersection with the span of $e\_1,\dots, e\_n$. Now just count the number of choices for a basis $v\_1,\do... | 10 | https://mathoverflow.net/users/5575 | 23431 | 15,405 |
https://mathoverflow.net/questions/23398 | 5 | If $R$ is a commutative noetherian ring and $I$ is an ideal of $R$, $M$ is an $R$-module. Does $Tor\_i^R(R/I, M)$ is finitely generated for $i\ge 0$ imply $Ext^i\_R(R/I, M)$ is finitely generated for $i\ge 0$?
| https://mathoverflow.net/users/5775 | A problem on finiteness of Ext | In fact, the two are equivalent. My apologies for the length of this argument - if someone else has a shorter one, I'd be happy to hear it.
Let $(a\_1,\ldots,a\_k) = I$, and let $K\_\bullet$ be the [Koszul complex](http://en.wikipedia.org/wiki/Koszul_complex) associated to this set of generators. Note that its zero'... | 9 | https://mathoverflow.net/users/360 | 23432 | 15,406 |
https://mathoverflow.net/questions/12810 | 36 | Here is a double series I have been having trouble evaluating:
$$S=\sum\_{k=0}^{m}\sum\_{j=0}^{k+m-1}(-1)^{k}{m \choose k}\frac{[2(k+m)]!}{(k+m)!^{2}}\frac{(k-j+m)!^{2}}{(k-j+m)[2(k-j+m)]!}\frac{1}{2^{k+j+m+1}}\text{.}$$
I am confident that $S=0$ for any $m>0$. In fact, I have no doubt. I have done lots of algebraic ... | https://mathoverflow.net/users/556 | Help with a double sum, please | Olivier Gerard just told me about this wonderful website!
Regarding the question it can be done in one nano-second using the Maple
package
<http://www.math.rutgers.edu/~zeilberg/tokhniot/MultiZeilberger>
accopmaying my article with Moa Apagodu
<http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/multiZ.ht... | 44 | https://mathoverflow.net/users/5822 | 23434 | 15,408 |
https://mathoverflow.net/questions/23426 | 13 | Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ring $ \mathbb{C}[G/B]$, it is known that it is a graded polynomial ring quotient by some homogeneous ideal. For example,... | https://mathoverflow.net/users/1851 | How to Compute the coordinate ring of flag variety? | $G/B$ is most naturally a multi-projective variety, embedding in the product of projectivizations of fundamental representations: $\prod \mathbb{P}
(R\_{\omega\_i})$. So there is a multi-homogeneous coordinated ring on $G/B$. You mentioned that this ring is $\bigoplus\_{\lambda\in P\_+}$ $R\_\lambda$. This is correct, ... | 10 | https://mathoverflow.net/users/788 | 23446 | 15,415 |
https://mathoverflow.net/questions/23443 | 10 | I would like to know an example (not using the Gibbs measure Theory) of a sequence of measures $\mu\_n:\mathcal B\to[0,1]$ , where $\mathcal B$ is the $\sigma$-algebra of the borelians of a compact space $X$ such that :
1) $\mu\_n$ is ergodic, with respect to a fixed continuous function $T:X\to X$, for all $n\in\math... | https://mathoverflow.net/users/2386 | On The Convergence of Ergodic Measures | Let $X=\{0,1\}^{\mathbb{N}}$ with the infinite product topology (which is metrisable). For each $n \geq 1$, define $x\_n$ to be the sequence given by $x\_i=0$ for $1 \leq i \leq n$, $x\_i=1$ for $n+1 \leq i \leq 2n$, and $x\_{2n+i}=x\_i$ for all $i$. Let $T \colon X \to X$ be the shift transformation $T[(x\_n)]= (x\_{n... | 17 | https://mathoverflow.net/users/1840 | 23449 | 15,416 |
https://mathoverflow.net/questions/23453 | 0 | It is well-known that in average $\varphi(n)$ behaves like $\frac1{\zeta(2)}n=\frac{6}{\pi^2}n$. But it looks that in some sense it is ``asymptotically larger''. In particular, the ratio
$$
\zeta(2)(1-t)^2\sum\_{n=1}^{\infty} \varphi(n)t^{n-1}=
\zeta(2)\frac{\sum\_{n=1}^{\infty} \varphi(n)t^{n-1}}{\sum\_{n=1}^{\infty}... | https://mathoverflow.net/users/4312 | Asymptotic propeties of Euler function | Write $S(t)= \sum \varphi(n)t^n$. A standard calculation gives
$S(e^{-u})= \frac{1}{2\pi i}\int\_{(3)}\frac{\zeta(s-1)}{\zeta(s)}\Gamma(s)u^{-s} ds$,
so pushing the contour to $Re(s)=3/2$ (say) gives $S(e^{-u})=\zeta(2)^{-1} u^{-2}+O(u^{-\frac{3}{2}})$ as $u\to 1$. How many $n$ are you using in your numerical calcu... | 3 | https://mathoverflow.net/users/1464 | 23458 | 15,420 |
https://mathoverflow.net/questions/23430 | 5 | I need this for a counterexample: the multiplication in the fundamental group $\pi\_1(\Sigma X\_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X\_+$, fails to be continuous for the sort of space in the question, by a result from
>
> J. Brazas, The topological fundamental group and hoop ear... | https://mathoverflow.net/users/4177 | What is an example of a non-regular, totally path-disconnected Hausdorff space? | One of the easiest examples is the rational numbers with the subspace topology of the real line with the K-topology. Total path disconnectedness is not entirely necessary for multiplication of $\pi\_{1}(\Sigma X\_{+})$ to fail to be continuous. It just makes the path component space of $X$ equal to $X$, greatly simplif... | 6 | https://mathoverflow.net/users/5801 | 23461 | 15,423 |
https://mathoverflow.net/questions/23451 | 1 | I have constructed a hierarchical clustering of data using some proximity function. Now I would like to calculate a representative value of any given node of this clustering, such that it reflects the different contributions of the branch components.
I know that this is a very tricky and risky task, but still...
Can... | https://mathoverflow.net/users/5823 | Finding a representative of a branch in hierarchical clustering | If you do some sort of centroid linkage algorithm to construct your hierarchal clustering, you should be able to pick out the centroid values just before merging clusters as representatives at those levels. It'd require tweaking the algorithm to get the points to tag the corresponding tree, but shouldn't be too hard to... | 1 | https://mathoverflow.net/users/102 | 23467 | 15,428 |
https://mathoverflow.net/questions/23409 | 81 | My question is related to the question [Explanation for the Chern Character](https://mathoverflow.net/questions/6144/explanation-for-the-chern-character) to [this question about Todd classes](https://mathoverflow.net/questions/10630/why-do-todd-classes-appear-in-grothendieck-riemann-roch-formula), and to [this question... | https://mathoverflow.net/users/2051 | Intuitive explanation for the Atiyah-Singer index theorem | I don't think I can really give you the intuition that you seek because I don't think I quite have it yet either. But I think that understanding the relevance of Nigel Higson's comment might help, and I can try to provide some insight. (Full disclosure: most of my understanding of these matters has been heavily influen... | 42 | https://mathoverflow.net/users/4362 | 23469 | 15,430 |
https://mathoverflow.net/questions/23477 | 2 | I just used the following.
Lemma. Let $A$ be a $\mathbb{Z}\_p$-flat ring, of finite type over $\mathbb{Z}\_p$, and suppose that $A \otimes \mathbb{F}\_p$ is a domain. Then $A$ is a domain.
Proof: Suppose $ab = 0$ in $A$. Then one of $a, b$ must lie in $pA$, so we can write (without loss of generality) $a = p a\_1$.... | https://mathoverflow.net/users/1594 | Z_p flatness and irreducible components. | Your proof seems wrong to me. I might be misunderstanding some things you wrote, but surely $\mathbf{Q}{}\_p=\mathbf{Z}\_p[X]/(pX-1)$ is finite type over $\mathbf{Z}\_p$, and contains many elements which are infinitely divisible by $p$. Again if I've understood your definitions correctly, $\mathbf{Q}{}\_p\times\mathbf{... | 9 | https://mathoverflow.net/users/1384 | 23489 | 15,441 |
https://mathoverflow.net/questions/23391 | 31 | Let $S^{\lambda}$ be a Schur functor. Is there a known **positive** rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL\_n(\mathbb{C})$ irreps?
---
In response to Vladimir's request for clarification, the ideal answer would be a finite set whose cardinality is the multiplicity of... | https://mathoverflow.net/users/297 | What is known about this plethysm? | If I remember this correctly the cases $\mathrm{Sym}^k(\bigwedge^2 \mathbb{C}^n)$ and $\mathrm{Sym}^k(\mathrm{Sym}^2(\mathbb{C}^n))$ are known; and hence $\bigwedge^k(\bigwedge^2 \mathbb{C}^n)$ and $\bigwedge^k(\mathrm{Sym}^2(\mathbb{C}^n))$. I will look up the references tomorrow (if this is of interest).
**Edit** T... | 11 | https://mathoverflow.net/users/3992 | 23491 | 15,442 |
https://mathoverflow.net/questions/23476 | 2 | I was browsing in Plouffe's inverter and found that $\frac{\exp(\pi)-\ln(3)}{\ln(2)}$ is very nearly $\frac{159}{5}.$ The continued fraction is
[31, 1, 4, 12029125, ...].
Is this the same magic as $\exp(\pi \sqrt{163})$?
| https://mathoverflow.net/users/756 | Is this related to the j-function? | The trick with the modular invariant $j$ is for $\pi\sqrt D$ only (as $j(\pi\sqrt{-D})$ is rational). Your value is not the exponential of a CM-point in the upper halfplane, so nothing to do with modularity. This kind of experimental discoveries already exists in the literature; see, for example, [C. Schneider, R. Pema... | 4 | https://mathoverflow.net/users/4953 | 23496 | 15,446 |
https://mathoverflow.net/questions/23502 | 11 | *Edit: I wrote the following question and then immediately realized an answer to it, and moonface gave the same answer in the comments. Namely, $\mathbb C(t)$, the field of rational functions of $\mathbb C$, gives a nice counterexample. Note that it is of dimension $2^{\mathbb N}$.*
The following is one statement of ... | https://mathoverflow.net/users/78 | (When) does Schur's lemma break? | The idea behind Schur's Lemma is the following. The endomorphism ring of any simple $R$-module is a division ring. On the other hand, a finite dimensional division algebra over an algebraically closed field $k$ must be equal to $k$ (this is because any element generates a finite dimensional subfield over $k$, which mus... | 7 | https://mathoverflow.net/users/778 | 23503 | 15,452 |
https://mathoverflow.net/questions/23487 | 7 | A number theorist I know (who studies Galois representations) raised a question recently:
>
> Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 classes where the character takes nonzero values?
>
>
>
He has learned about a few special examples involving nona... | https://mathoverflow.net/users/4231 | Finite groups with a character having very few nonzero values? | For the case of n=2 see Gagola's paper "Characters vanishing on all but two conjugacy classes" MR0721927. Some improvements on Gagola's results can be found in Kuisch and van der Waall's papers "Homogeneous character induction [I and II]" MR1172440/MR1302857 and in my paper ["Groups with a Character of Large Degree"](h... | 8 | https://mathoverflow.net/users/22 | 23514 | 15,459 |
https://mathoverflow.net/questions/23250 | 7 | As I study category theory, I'm finding the use of universal mapping properties in defining basic concepts to be both simple and clever. Yet, the idea seems non-obvious enough that I expect quite a bit of mathematics had been done before the discovery of the technique.
> What is the chronologically earliest abstract... | https://mathoverflow.net/users/5700 | What is the earliest definition given by a universal mapping property? | I very much agree with Pete's comment that we will find incipient instances of the universal mapping property among many classical constructions in mathematics, even if the original users of those concepts would not describe the idea in those terms. Indeed, I believe that these instances will stretch back through the w... | 10 | https://mathoverflow.net/users/1946 | 23526 | 15,467 |
https://mathoverflow.net/questions/23337 | 24 | A morphism of schemes $f:X\to S$ is said to be quasi-compact if for every OPEN quasi-compact subset $K \subset S$ the subset $f^{-1}(K) \subset X$ is also quasi-compact (and open, of course!). The morphism $f:X\to S$ is said to be universally closed if for every morphism $T\to S$ the resulting base-changed morphism $X\... | https://mathoverflow.net/users/450 | Is a universally closed morphism of schemes quasi-compact ? | Yes, a universally closed morphism is quasi-compact. (I haven't yet checked whether the same approach answers question 2.)
**Proof:** Without loss of generality, we may assume that $S=\operatorname{Spec} A$ for some ring $A$, and that $f$ is surjective. Suppose that $f$ is not quasi-compact. We need to show that $f$ ... | 28 | https://mathoverflow.net/users/2757 | 23528 | 15,469 |
https://mathoverflow.net/questions/23479 | 4 | In R.O.Wells book "Differential Analysis on Complex Manifolds" p. 44 proof of Theorem 2.2 part b) the author claims that any two sections of an etale space which agree at a point agree in some neighborhood of that point (etale space is a (possibly non-Hausdorff) $Y$ with the surjection $p\colon Y \rightarrow X$ which i... | https://mathoverflow.net/users/5838 | Sections of an etale space | First note that if $p\_1:Y\_1 \to X$ and $p\_2:Y\_2 \to X$ are two etale spaces over $X$ then
any morphism of etale spaces $f:Y\_1 \to Y\_2$ (where by morphism of etale spaces I mean that $f$ is continuous and that
$p\_2\circ f = p\_1$) makes $Y\_1$ an etale space over $Y\_2$.
(Proof: Let $y \in Y\_1$, and let $V$ b... | 8 | https://mathoverflow.net/users/2874 | 23536 | 15,475 |
https://mathoverflow.net/questions/23537 | 8 | The setup for my question is as follows: $k$ is a field, $K$ a Galois extension of $k$ with group $G$, $k^\prime$ an arbitrary extension of $k$, and $K^\prime/k^\prime$ another Galois extension of fields with group $G^\prime$ (the notation, as well as the setup, is essentially verbatim from Chapter X, Section 4 of Serr... | https://mathoverflow.net/users/4351 | How canonical are localization maps in Galois cohomology? | As you note, if we choose two different embeddings $k^s \to k\_v^s$, say $\imath\_1$
and $\imath\_2$, then we get two different $G\_{k\_v}$-module structures on $M$, call
them $M\_1$ and $M\_2$, and two different restriction maps $r\_1:H^n(k,M) \to H^n(k\_v,M\_1)$
and $r\_2:H^n(k,M) \to H^n(k\_v,M\_2)$.
The point is... | 5 | https://mathoverflow.net/users/2874 | 23540 | 15,478 |
https://mathoverflow.net/questions/23534 | 7 | Suppose we have a normed vector space $V$ and its dual $V^\*$, and suppose that $X \subseteq V^\*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such that $\phi(v) = \Vert v \Vert$. Is $X$ dense in $V^\*$ (in the operator norm)? Note that this is a stronger property t... | https://mathoverflow.net/users/5852 | Is a subspace with a certain property dense in the dual of a vector space? | The answer is negative. Since the linear span of the Dirac masses is not a dense subspace of the dual of $C[0,1]$.
| 11 | https://mathoverflow.net/users/2508 | 23548 | 15,485 |
https://mathoverflow.net/questions/23547 | 42 | The motivation for this question comes from the novel *Contact* by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at the end) says that if humans compute enough digits of $\pi$, they will discover that after some point there is nothing but... | https://mathoverflow.net/users/2233 | Does pi contain 1000 consecutive zeroes (in base 10)? | Summing up what others have written, it is widely believed (but not proved) that every finite string of digits occurs in the decimal expansion of pi, and furthermore occurs, in the long run, "as often as it should," and furthermore that the analogous statement is true for expansion in base b for b = 2, 3, .... On the o... | 50 | https://mathoverflow.net/users/3684 | 23551 | 15,488 |
https://mathoverflow.net/questions/23512 | 1 | For a given integer $x>0$, I need to find all integers $a \in [0, 10^{15}]$ which have the following property: the digit sum of $a$ equals the digit sum of $x\cdot a$.
I found this link <http://mathworld.wolfram.com/CastingOutNines.html> which looks quite relevant to my task, but I can't figure out how to apply it.
... | https://mathoverflow.net/users/5846 | Finding integers whose sum of digits equals the sum of the digits of a multiple of them | If you wish to get the property $S(a)=S(xa)$ (where $S(\cdot)$ denotes the sum of decimal digits) valid for all positive integers $x$, then there are no such numbers $a$. Indeed, if such an $a$ existed, then concantination $\overline{aa}=a\cdot x$ with $x=10^n+1$, $n$ the number of digits in $a$, would give you a numbe... | 1 | https://mathoverflow.net/users/4953 | 23563 | 15,497 |
https://mathoverflow.net/questions/23564 | 30 | In the [December 2009 issue](http://www.ems-ph.org/journals/newsletter/pdf/2009-12-74.pdf) of the [newsletter of the European Mathematical Society](http://www.ems-ph.org/journals/all_issues.php?issn=1027-488X) there is a very interesting interview with Pierre Cartier. In page 33, to the question
>
> What was the on... | https://mathoverflow.net/users/394 | Is "all categorical reasoning formally contradictory"? | Note: I am not a historian. I'm just guessing as to what prompted the comments.
Here's my guess: if you do set theory naively, in the old-fashioned "anything is a set" way, then you run into Russell's paradox; the set consisting of all sets that aren't elements of themselves gives you trouble. So you then decide set ... | 36 | https://mathoverflow.net/users/1384 | 23566 | 15,499 |
https://mathoverflow.net/questions/23561 | 14 | In his answer
[here](https://mathoverflow.net/questions/2022/definition-and-meaning-of-the-conductor-of-an-elliptic-curve)
Qing Liu mentioned "the 'discriminant' of X which measures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R."
Could somebody give a referenc... | https://mathoverflow.net/users/4710 | "Nice" definition of discriminant as alluded to in an answer of Qing Liu | The "magistral paper of Takeshi Saito" mentioned by Qing Liu is the following: 'Conductor, Discriminant, and the Noether formula of Arithmetic surfaces', Duke Math Journal, vol. 57, no. 1. See here if you have a subscription to Duke: projecteuclid.org/euclid.dmj/1077306852 The definition of the discriminant is at the t... | 13 | https://mathoverflow.net/users/5830 | 23569 | 15,501 |
https://mathoverflow.net/questions/23571 | 17 | Are there non-isomorphic number fields (say of the same degree and signature) that have the same discriminant and regulator? I'm guessing the answer is no - why?
And focusing on fields of small degree (n=3 and n=4), what about a less restrictive question: can we find two such fields that have the same regulator (no d... | https://mathoverflow.net/users/5860 | Number fields with same discriminant and regulator? | Yes, see e.g. the paper "Arithmetically equivalent number fields of small degree" (Google for it) by Bosma and de Smit.
In brief: two number fields $K$ and $K'$ are said to be **arithmetically equivalent** if they have the same Dedekind zeta function. A famous group-theoretic construction of Perlis (Journal of Numbe... | 29 | https://mathoverflow.net/users/1149 | 23572 | 15,503 |
https://mathoverflow.net/questions/23145 | 6 | This might be a dumb question. If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits. However, it can happen that $c$ can be equipped with extra structure which in turn gives the morphisms out of $c$ extra structure, so that the... | https://mathoverflow.net/users/290 | When does a "representable functor" into a category other than Set preserve limits? | [Collecting my sporadic comments into one (hopefully) coherent answer.]
A more general question is as follows: For functors
$C\stackrel{F}{\to}D\stackrel{U}{\to}E$ and for an index category
$J$ such that $UF$ preserves $J$-limits, when does $F$ preserve $J$
limits?
A useful sufficient condition is that if $U$ creat... | 7 | https://mathoverflow.net/users/2734 | 23577 | 15,505 |
https://mathoverflow.net/questions/23583 | 15 | How many steps on average does a simple random walk in the plane take before it visits a vertex it's visited before?
If an exact formula does not exist (as seems likely), then I'm interested in good approximations.
I'd also like to know the standard nomenclature associated with the question, if any exists.
| https://mathoverflow.net/users/3621 | self-avoidance time of random walk | [I've corrected a stupid mistake below and added an upper bound... Please check the numerical values!]
Well, I doubt that an explicit expression exists. However, it should be possible to get good bounds. The lower bound is easy: observe that
$$
E(T) = \sum\_{k\geq 1} P(T> k-1) = 1+\sum\_{k\geq 1} c\_k 4^{-k},
$$
wher... | 12 | https://mathoverflow.net/users/5709 | 23597 | 15,518 |
https://mathoverflow.net/questions/23608 | 13 | Consider the field of fractions $K$
of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$,
where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables.
Clearly $-1$ is a sum of 4 squares in $K$.
How can one prove that $-1$ is not a sum of 2 squares in $K$?
Serre mentions without proof this... | https://mathoverflow.net/users/4149 | Is -1 a sum of 2 squares in a certain field K? | This is a special case of a theorem of A. Pfister. It is well known to quadratic forms specialists. See e.g. Theorem XI.2.6 in T.Y. Lam's *Introduction to Quadratic Forms over Fields*.
I believe the original paper is
>
> Pfister, Albrecht,
> *Zur Darstellung von $-1$ als Summe von Quadraten in einem Körper*. (... | 22 | https://mathoverflow.net/users/1149 | 23609 | 15,524 |
https://mathoverflow.net/questions/23607 | 2 | Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are allowed).
>
> I want to show that if you fix a curve class β in H2(X), then there is a bounded interval which contains the g... | https://mathoverflow.net/users/1231 | Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded? | I don't think this is true. Take $X = \mathbb P^1 \times \mathbb P^2$. Let $C$ be $\mathbb P^1 \times 0$, and let $C\_1$ be the first infinitesimal neighborhood of $C$. The curve $C\_1$ is the relative spectrum of $\mathcal O\_{\mathbb P^1} \oplus O\_{\mathbb P^1}^{\oplus 2}$, where $\mathcal O\_{\mathbb P^1}^{\oplus 2... | 8 | https://mathoverflow.net/users/4790 | 23613 | 15,528 |
https://mathoverflow.net/questions/23567 | 9 | A projective 3-manifold is a smooth manifold that admits an atlas with values in the real projective 3-space such that all transition maps are restrictions of projective transformations. A Möbius 3-manifold is defined similarly, with the projective space replaced with the standard 3-sphere and projective transformation... | https://mathoverflow.net/users/2349 | Möbius and projective 3-manifolds | The first examples of closed 3-manifolds not admitting a conformally flat (Mobius) structure [were given by Goldma](http://www.ams.org/mathscinet-getitem?mr=701512)n: 3-manifolds modeled on the Nil geometry (this link was given by Macbeth in the comments above). Sol manifolds also don't have a Mobius structure, whereas... | 13 | https://mathoverflow.net/users/1345 | 23615 | 15,529 |
https://mathoverflow.net/questions/23578 | 6 | Graphical representations of intersection of sets as logical combinations are much older than Venn.
Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I do not really find the illustrations provided in the Wiki Commons for Llull very compelling.
I expect that these kind o... | https://mathoverflow.net/users/5387 | What are the oldest illustrations of "Venn" diagrams? | You may already be familiar with Ruskey and Weston's ["A Survey of Venn Diagrams,"](http://www.combinatorics.org/Surveys/ds5/VennEJC.html) which includes a discussion of Borromean rings. Such rings are similar to the valknut and the triskelion, of which the gankyil is a type. All of these figures are quite old.
Of co... | 3 | https://mathoverflow.net/users/965 | 23616 | 15,530 |
https://mathoverflow.net/questions/23629 | 20 | This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with distinct eigenvalues are diagonalizable, i.e. conjugate to a diagonal matrix.
>
> I'd just like to see an example of a c... | https://mathoverflow.net/users/4721 | Non-diagonalizable complex symmetric matrix | $$\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}.$$
How did I find this? Non-diagonalizable means that there is some Jordan block of size greater than $1$. I decided to hunt for something with Jordan form $\left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right)$. So I want trace and determinant to be zero,... | 38 | https://mathoverflow.net/users/297 | 23631 | 15,540 |
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