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https://mathoverflow.net/questions/88048 | 8 | If $K = \mathbb{Q}(\alpha)$ is a number field, where $\alpha$ is algebraic, and $\mathcal{O}\_K$ the ring of integers in $K$, then the set of fractional ideals over $\mathcal{O}\_K$ forms a group and if we mod out by the set of principal ideals, the resulting group is finite and we call its size the class number of $K$... | https://mathoverflow.net/users/10898 | On the class number | This is from Buell, *Binary Quadratic Forms*. From page 84, the class number for a negative discriminant $\Delta$ is about $$\frac{\sqrt{|\Delta|}}{\pi},$$ which comes from an $L$-function calculation on page 83.
Let's see, on page 101, he points out that for negative field discriminants, class group and narrow clas... | 5 | https://mathoverflow.net/users/3324 | 88052 | 52,166 |
https://mathoverflow.net/questions/87941 | 4 | Let $f: X \to Y$ and $g: Y \to Z$ be morphisms of schemes\* such that f is flat and finite, g is proper and $R^{> 1}g\_\*E=0$ for all sheaves and Z is affine.
Let E be a vector bundle on Y such that $R^1 g\_\* E=0$.
Can we say anything about $H^1 (X,f^\* E)$?
By the projection formula this is the same as $R^1 g\_\* (... | https://mathoverflow.net/users/16857 | If $f: X \to Y$ is a finite flat morphism of schemes, $g: Y \to Z$ is a proper morphism of relative dimension one, $Z$ is affine and $E$ is a vector bundle on $Y$ with $R^1g_*E=0$ then $H^1(X,f^*E)=0$? | Actually, for any such projective $g$ and any $E$ there exists an $f$ such that this fails. In fact you may even assume that $f$ is a double cover. (I suspect that it also fails without the projective assumption, but this seems convincing enough).
Let $g:Y\to Z$ be a projective morphism, $Y$ smooth, $Z$ quasi-projec... | 4 | https://mathoverflow.net/users/10076 | 88058 | 52,168 |
https://mathoverflow.net/questions/88033 | 11 | I'm looking for a comprehensive reference to existing proofs of Rokhlin's theorem that a 4-dimensional closed spin PL manifold has signature divisible by 16.
I'm specifically interested in direct proofs (if any such exist) which do not rely on the fact that $\pi\_i(PL/O)=0$ for small $i$.
The most commonly cited refe... | https://mathoverflow.net/users/18050 | Existing proofs of Rokhlin's theorem for PL manifolds | Another approach to the theorem that could probably be rewritten to work in the PL category is the approach of Kirby and Melvin in Appendix C of the following paper:
MR1117149 (92e:57011)
Kirby, Robion(1-CA); Melvin, Paul(1-BRYN)
The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C).
Invent. Math. ... | 11 | https://mathoverflow.net/users/317 | 88060 | 52,169 |
https://mathoverflow.net/questions/88062 | 3 | Suppose you roll an $n$ sided die (valued $1,\dots,n$) until the sum is at least $s\in\mathbb{N}$. Which of the integers $s, s+1, \dots, s+n-1$ are you most likely to end up with?
| https://mathoverflow.net/users/20831 | Rolling a die until the sum is at least some number | Denote by $p(k)$ the probability that the cummulative sum is equal to $k$ at some point. If we denote by $f(i)$ the probability that the first sum $\geq s$ is equal to $s+i$ then observe that
$$f(i)=\sum\_{k=s+i-n}^{s-1} \frac{p(k)}{n},$$
in particular this implies that $f(0)\geq f(1)\geq \cdots \geq f(n-1)$.
| 11 | https://mathoverflow.net/users/2384 | 88065 | 52,172 |
https://mathoverflow.net/questions/88064 | 2 | That is, suppose that a subset S of the octonions $\mathbb{O}$ is a group under octonionic multiplication. Does it follow that S is contained in the Quaternions $\mathbb{H}$?
| https://mathoverflow.net/users/21280 | Are associative "subgroups" of octonions necessarily quaternionic? | Are you assuming that the subset is multiplicatively closed? If not, exactly what do you mean by "associative"? Do you mean associations of length 3, or of all lengths?
If $S$ is associative, then the $\mathbb R$-linearity of multiplication shows that the vector subspace generated by $S$ is associative (simply break ... | 2 | https://mathoverflow.net/users/18060 | 88068 | 52,173 |
https://mathoverflow.net/questions/88039 | 2 | $j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great arithmetical significance, for they generate the maximal abelian extensions of imaginary quadratic fields.
I would be intereste... | https://mathoverflow.net/users/4800 | Generalization of singular moduli | Some standard references on abelian varieties having complex multiplication:
* MR0125113 Shimura, Goro; Taniyama, Yutaka. *Complex multiplication of abelian varieties and its applications to number theory*. Publications of the Mathematical Society of Japan, 6 The Mathematical Society of Japan, Tokyo 1961 xi+159 pp.
*... | 5 | https://mathoverflow.net/users/11926 | 88070 | 52,174 |
https://mathoverflow.net/questions/88056 | 110 | I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a nutshell, is there a natural condition to impose on a structure sheaf $\mathcal{C}^k\_M$ of a topological space $M$ that can... | https://mathoverflow.net/users/2051 | Is there a sheaf theoretical characterization of a differentiable manifold? | Definition: ''A smooth manifold is a locally ringed space $(M;C^{\infty})$ which satisfies the conditions:
1. Each $x \in M$ admits a neighborhood $U$, such that $(U,C^{\infty})$ is isomorphic to $(\mathbb{R}^n,C^{\infty})$ as a locally ringed space.
2. The global sections of $C^{\infty}(M)$ separate points.
3. The s... | 62 | https://mathoverflow.net/users/9928 | 88085 | 52,184 |
https://mathoverflow.net/questions/87996 | 2 | Let $x,y,z$ be dimensions that appear in the Clebsch-Gordan series
$x\*x=1+t+u+y+z$.
(E8 family if $t=x$ (say), but there is at least another family. E.g. B4(R4) belongs to the latter.)
With the right pick of dimension ($t,u,y,z$ are not equivalent!) I got the following diophantine equation:
$$\eqalign{-27(-1+x)... | https://mathoverflow.net/users/11504 | A diophantine equation for the E8 knot polynomial family | You will find some information on this in my papers:
MR1960703 (2004g:17003a) Westbury, Bruce W.
$R$-matrices and the magic square.
J. Phys. A 36 (2003), no. 7, 1947--1959.
MR2029689 (2005g:18014) Westbury, Bruce .
Invariant tensors and diagrams.
Proceedings of the Tenth Oporto Meeting on Geometry, Topology and
P... | 1 | https://mathoverflow.net/users/3992 | 88086 | 52,185 |
https://mathoverflow.net/questions/88090 | 9 | I am trying to understand the topology (in terms of homology groups) of the free loop space $\Lambda M$ of nice spaces (Complete Riemannian connected finite dimensional manifolds $M$). I see the free loop space (of H^1 loops) as a Hilbert manifold, cf. Klingenbergs book. If the manifold $M$ has a non-trivial fundamenta... | https://mathoverflow.net/users/12156 | The connected components of the free loop space | The different components are, indeed, not all homotopy equivalent, and you are quite right in noting that the argument that works for $\Omega M$ (via concatenation of loops) does not hold here.
This is best illustrated when $M = BG = K(G,1)$ is an Eilenberg-MacLane space with precisely one (non-abelian) homotopy grou... | 19 | https://mathoverflow.net/users/4649 | 88092 | 52,188 |
https://mathoverflow.net/questions/87473 | 3 | Given a family of hypersurfaces $H\_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as
$$
R[u] (t,p) = \int\limits\_{H\_{t,p}} u(x) a(x) \omega
$$
where $\omega$ is such differential form as $d\_{x}g \wedge \omega = dx... | https://mathoverflow.net/users/17896 | On the generalized Radon transform and currents | Here is the simple explanation (credits should go to the late great V.I. Arnold who gave the explanation below in one of his books). The form $\omega$ is sometimemes referred to as the Gelfand-Leray residue.
Place yourself in the case when $t$ is a regular value of $g$. Fix a point p on the fiber $g^{-1}(t)$ so that ... | 4 | https://mathoverflow.net/users/20302 | 88095 | 52,190 |
https://mathoverflow.net/questions/88088 | 1 | Let me give an example:
1. this is a definition in object language:
R(x,y) is a symmetric formula ↔ （∀x∀y(R(x,y)→R(y,x))）
2.this is a definition in metatheory:
R(x,y) is a symmetric formula if and only if R(x,y)→R(y,x) is a theorem.
In other words:
R(x,y) is a symmetric formula if and only if ⊢ R(x,y)→R(y,x).
In ot... | https://mathoverflow.net/users/21284 | Difference about defined symbols in metatheory or in object language | To begin with the second question, everyday reasoning in mathematics is in the object language. (Though if you happen to be a logician, this object language may well get used as a meta-language for another object language.)
As for the first question, I don’t quite understand the goal. First, the definition of a symme... | 2 | https://mathoverflow.net/users/12705 | 88096 | 52,191 |
https://mathoverflow.net/questions/88017 | 11 | What is known about the Sylow 2-subgroups of $\rm{PGL}\_n(\mathbb{F}\_q)$, where $q$ is any prime power? For example, according to Theorem 7.9 in Isaacs's Character Theory, these Sylow 2-subgroups cannot be generalised quaternion. Is there a classification of all these Sylows available?
Concretely, let me make the fo... | https://mathoverflow.net/users/35416 | Sylow subgroups of projective general linear groups | The irreducibles of the Sylow 2-subgroups of ${\rm GL}(n,q)$, $q$ odd, have indeed trivial Schur indices: Let $S$ be such a Sylow 2-subgroup. First, observe that the natural module $(\mathbb{F}\_q)^n$ splits into a sum of simple modules $U\_1\oplus \dotsb \oplus U\_l$, and the dimension of each simple module is a power... | 12 | https://mathoverflow.net/users/10266 | 88098 | 52,192 |
https://mathoverflow.net/questions/88073 | 12 | Who proved the modern form of the [fundamental theorem of Galois theory?](http://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory). Was it in the original Galois' manuscript?
| https://mathoverflow.net/users/nan | The fundamental theorem of Galois theory | Galois's Proposition I (as translated by Edwards) is:
Let the equation be given whose $m$ roots are $a,b,c,\ldots$. There will always be a group of permuations of the letters $a,b,c,\ldots$ which will have the following property: 1) that each function invariant under the substitutions of this group will be known rati... | 9 | https://mathoverflow.net/users/10503 | 88099 | 52,193 |
https://mathoverflow.net/questions/88102 | 6 | I know that there is a 1965 volume containing the Euler/Goldbach correspondence, but I'm interested in looking at the original manuscripts. I'm not finding anything at University of Basel or Berlin-Brandenburg Academy of Sciences. Any help would be appreciated!
| https://mathoverflow.net/users/18044 | Where is the Euler/Goldbach correspondence? | Leonhardi Euleri Opera omnia: Descriptio commercii epistolici, Volume 1 gives detailed information where the *original* letters are, or at least were when the book was written) (I take it from your question you know of some partial printed/edited versions available at the link mentioned in a comment). Most of them are ... | 6 | https://mathoverflow.net/users/nan | 88105 | 52,196 |
https://mathoverflow.net/questions/88112 | 10 | It is unknown whether a hyperbolic group is residually finite. Is it known under the additional hypothesis of locally indicability? Namely: Is a locally indicable hyperbolic group, residually finite?
| https://mathoverflow.net/users/16804 | Locally Indicable Hyperbolic Groups | The answer is "it is unknown". There are many potential counterexamples to the conjecture that all hyperbolic groups are residually finite. For example, let $\phi, \psi$ be two injective (but not surjective) emdomorphisms of the free group $F\_2$. Consider the corresponding multiple ascending HNN extension of $F\_2$, $... | 12 | https://mathoverflow.net/users/nan | 88115 | 52,200 |
https://mathoverflow.net/questions/80021 | 5 | If $\mathcal{A}$ is a complete locally convex (Hausdorff) associative unital algebra (over $\mathbb{C}$) one is interested in defining "transcendental" functions of a given algebra element $a \in \mathcal{A}$ like e.g. exponentials $\exp(a)$ by means of the power series expansion. This works fine for complete locally *... | https://mathoverflow.net/users/12482 | Entire calculus and clmc algebras | At least for commutative and completely metrizable locally convex algebras there is a theorem of Mityagin, Rolewicz, and Zelasko (Studia Math. 21 (1962), 291-306) which says that the algebra has to be locally m-comvex if all entire functions operate on it (meaning that $f(a)=\sum\limits\_{n=0}^\infty\frac{ f^{(n)}(0)}{... | 4 | https://mathoverflow.net/users/21293 | 88116 | 52,201 |
https://mathoverflow.net/questions/87979 | 3 | Given the heat equation:
$$\partial\_{t}{\varPhi(x,t)}=k^2\partial\_{xx}{\varPhi(x,t)}$$
with the boundary conditions:
$$\Phi(x,0)=\Phi\_0$$
and a Neumann boundary condition of the kind:
$${\partial\_{x}}{\Phi(0,t)=\nu(t)+C}$$
where $\nu(t)$ is a stochastic variable with gaussian distribution ${\sigma=\sigm... | https://mathoverflow.net/users/21258 | Stochastic Heat Equation | In this case $\Phi(x,t)$ is itself a stochastic process and this equation should be rewritten in a proper way. There is some literature as [this](https://www.math.lsu.edu/cosa/2-2-04%255B140%255D.pdf) and a more general theory of stochastic pde due to John Walsh (a tutorial can be found [here](http://www.math.utah.edu/... | 5 | https://mathoverflow.net/users/19520 | 88131 | 52,208 |
https://mathoverflow.net/questions/87901 | 4 | I faced to a bit weird control problem, that is minimize cost functional
\begin{equation}
J(u) = \int\_0^Tg(t,x(t),u(t),\dot u(t))dt
\end{equation}
subject to
\begin{equation}
\dot x(t) = f(t,x(t),u(t),\dot u(t)), \quad x(0) = x\_0
\end{equation}
where $u(t)$ is control, a piecewise continuous function. And it diffe... | https://mathoverflow.net/users/19988 | Optimal control problem with control derivative. | Can be reduced to impulse optimal control problem. In the following system
\begin{align}
&\dot x(t) = f(t,x(t),u(t), v(t)) \newline
&\dot u(t) = \nu(t)
\end{align}
where $\nu(t) = v(t) + \sum\limits\_i c\_i \delta(t-\tau\_i)$ is an impulse control, and
$u(t)$ should be considered as another variable.
For furth... | 1 | https://mathoverflow.net/users/19988 | 88136 | 52,209 |
https://mathoverflow.net/questions/88132 | 6 | How smooth are the singular values of a matrix $F$ in terms of entries of $F$? I am hoping for Lipschitz continuity, but was not able to find it.
| https://mathoverflow.net/users/21298 | Lipschitz continuity of singular values | The singular values of $F$ are the (square roots of ) eigenvalues of $F F^t,$ and the regularity of the latter have been studied half-to-death. See either T. Kato (perturbation theory of linear operators, ch. 1) or Golub-van Loan (Matrix Computations -- they almost certainly talk about singular values directly, without... | 8 | https://mathoverflow.net/users/11142 | 88139 | 52,211 |
https://mathoverflow.net/questions/88122 | 3 | Let $X,Y$ be complex projective varieties with $X$ irreducible, and let $f:X\dashrightarrow Y$ be a rational map.
If $U\subseteq X$ is the largest open set where $f$ can be defined, is it true that
$\mathrm{codim}\_{X}(X\setminus U)\geq 2$.
I know this is true if $X$ is smooth.
EDIT: In view of the inkspot's answer, ... | https://mathoverflow.net/users/15606 | On the points of indeterminacy of a rational map | If $D'=D\cap$X\sing(X) is not empty then it is an irreducible divisor on X\sing(X) (which is smooth). So you can apply the very same theorem to X\sing(X) and D' to get that the thing you describe in the edit can't happen and f can be defined almost everywhere on D'.
| 1 | https://mathoverflow.net/users/12208 | 88140 | 52,212 |
https://mathoverflow.net/questions/88141 | 5 | Let $X$ be an irreducible variety and let $U \subset X$ be a proper open set. Question : can there be a morphism $f : X \rightarrow U$ such that the composition $U \hookrightarrow X \stackrel{f}{\rightarrow} U$ is the identity?
My guess is that the answer is "no", but I can't seem to prove it.
One observation is t... | https://mathoverflow.net/users/21300 | Varieties cannot be isomorphic to proper open subsets | <http://en.wikipedia.org/wiki/Ax-Grothendieck_theorem>
| 19 | https://mathoverflow.net/users/51 | 88143 | 52,213 |
https://mathoverflow.net/questions/88072 | 5 | Hi,everyone.
I want to know that how to compute the henselization of some simple rings, for example: $k[x]\_{(x)}$ and $R[X]\_{(X)}$ where $k$ is a field and $R$ is a excellent DVR.
thank you very much!
| https://mathoverflow.net/users/5274 | how to compute the henselization of some simple rings? | In both cases, the henselization $A^h$ of your ring $A$ is its algebraic closure in its completion $\hat{A}$.This follows for instance from Artin approximation (Algebraic approximation of structures over complete local rings, Publications Mathématiques de l'IHÉS, 36, 1969, p. 23-58): any system of polynomial equations ... | 10 | https://mathoverflow.net/users/4069 | 88147 | 52,215 |
https://mathoverflow.net/questions/88138 | 15 | Note: in the following, all scheme/algebra morphisms should be assumed essentially of finite type.
>
> **Geometric version:** Let $X$ be a scheme flat over $S$ (both noetherian), and let $\mathscr{F}$ be a coherent sheaf on $X$, also flat over $S$. The scheme-theoretic support $\mathfrak{X}$ for $\mathscr{F}$ is a ... | https://mathoverflow.net/users/5094 | Is the support of a flat sheaf flat? | There are many counterexamples to this. Suppose that $S$ is a smooth surface over $\mathbb C$. Let $T \to S$ be a finite morphism from another smooth surface $T$, and consider a factorization $T \to V \to S$, where $V$ is obtained by gluing two points on a fiber. Then $T \to S$ is flat, $V \to S$ is not. Embed $V$ into... | 10 | https://mathoverflow.net/users/4790 | 88149 | 52,217 |
https://mathoverflow.net/questions/88134 | 6 | Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid this by proving Con(T + CH) for all (suitable) finite fragments T of ZFC using the Reflection principle to prove the exis... | https://mathoverflow.net/users/8092 | Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC | The reflection principle is a theorem scheme; each of its instances is provable in ZFC.
The following proof works entirely in ZFC:
1. Assume Con(ZFC) together with not-Con(ZFC+non-CH) and aim for a contradiction.
2. By assumption plus completeness theorem, there is a model (M,E) of ZFC. M may be nonstandard in the... | 19 | https://mathoverflow.net/users/14915 | 88150 | 52,218 |
https://mathoverflow.net/questions/88142 | 3 | Say $X$ is a smooth algebraic variety, $U$ is a Zariski open set in $X$, $L$ is a local system on $U$, and $IC(L)$ the intersection cohomology sheaf on $X$ which restricts to $L$ on $U$. Then is:
$$\dim L\_u \ge \sum\_i \dim \mathrm{H}^i(\mathrm{IC}(L)\_x) $$
for $u \in U$ and all $x \in X$?
If not, is it true ... | https://mathoverflow.net/users/4707 | Bounding the size of stalks of IC sheaves | The answer to the first question is certainly "no": you can easily find example of X whose IC complex (for the trivial local system of rank 1) has (total) stalk of dimension $\geq 1$ (for instance this is true for most nilpotent orbit closures in a simple Lie algebra).
Edit: Sorry I missed the assumption that X is sm... | 3 | https://mathoverflow.net/users/3891 | 88152 | 52,220 |
https://mathoverflow.net/questions/88155 | 3 | This question has a few parts:
1) Is the Bousfield class of $\langle E\rangle$ the class of $E$-acyclics, i.e. $\langle E\rangle=\left\{ X:E\wedge X=0\right\}$ or is it the class of spectra which are Bousfield equivalent to $E$? It seems that both of these definitions are used. In either case, which is then the corre... | https://mathoverflow.net/users/11546 | Bousfield Classes | 1) The best convention is probably this. Identify $\langle E\rangle$ with the class of $X$ for which $E\wedge X\neq 0$. The the "Bousfield class of $E$" is the class $\langle E\rangle$, and we say that "$E$ and $F$ have the same Bousfield class" if $\langle E\rangle = \langle F\rangle$.
With this convention, the par... | 7 | https://mathoverflow.net/users/437 | 88163 | 52,225 |
https://mathoverflow.net/questions/88156 | 4 | See the definition of the Alexander-Whitney transformation:
<http://ncatlab.org/nlab/show/Alexander-Whitney+map>
and the Eilenberg-Zilber transformation:
<http://ncatlab.org/nlab/show/Eilenberg-Zilber+map>
Is there a natural or obvious way to extend them to higher tensor powers i.e to, lets say
$$
\Delta\_{A... | https://mathoverflow.net/users/21302 | Extend Alexander-Whitney and Eilenberg-Zilber map to n-fold tensor products | Yes, there is a generalization to a finite number of simplicial complexes. A reference is *Corollary 2.2* in the paper
>
> Eilenberg, MacLane: On the groups $H(\Pi,n)$, II: Methods of Computation, Ann. of. Math. 60(1954), No. 1, 49 - 139.
>
>
>
Using the definitions from nLab, the maps are given as follows:
... | 6 | https://mathoverflow.net/users/10194 | 88167 | 52,228 |
https://mathoverflow.net/questions/88161 | 6 | In answer to [A coverage question](https://mathoverflow.net/questions/41187/a-coverage-question) Cam mentions an article by [SOUND](http://arxiv.org/abs/0707.0237). I have been running a computer program for [THIS](https://mathoverflow.net/questions/88048/on-the-class-number) and would like to know if there are a reaso... | https://mathoverflow.net/users/3324 | Erdos Kac for imaginary class number | There are many papers about strong probabilistic models of $L(1,\chi\_d)$, in particular by Granville and Soundararajan. These are quite precise (basically, because the Euler product at 1 is "almost" absolutely convergent, one can model its value by a random Euler product, and even prove that this model is close to the... | 11 | https://mathoverflow.net/users/20038 | 88168 | 52,229 |
https://mathoverflow.net/questions/87709 | 6 | A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K\_g$ given by a Riemannian metric $g$ on $Q.$
If $V{<}e$ then $g\_J=(e-V)g$ is the Jacobi metric on $Q,$ and Maupertuis' principle states that, up to reparametrization, the tr... | https://mathoverflow.net/users/12617 | How the Jacobi metrics may be useful in mechanics with or without constraints? | Dear Giuseppe,
I think that the idea is that much more is known about the dynamics of geodesic flows in Riemannian and Finsler manifolds than about the dynamics of more general Hamiltonian systems. Here are two examples where the Jacobi metric is or has been useful:
**1.** A generalized spherical pendulum: *If $V$ ... | 7 | https://mathoverflow.net/users/21123 | 88190 | 52,241 |
https://mathoverflow.net/questions/88184 | 78 | This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more explanations.
As a layman, I have started searching for expositories/more informal, rather intuitive, also original account o... | https://mathoverflow.net/users/13351 | What is the significance of non-commutative geometry in mathematics? | $\DeclareMathOperator\coker{coker}$I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit skeptical about its relevance to the rest of mathematics. To this day I sometimes find it hard to get excited about pur... | 60 | https://mathoverflow.net/users/4362 | 88201 | 52,248 |
https://mathoverflow.net/questions/88202 | 3 | Let $A$ be an $n \times n$ matrix. Are there formulas that convert linear combinations of traces of powers of $A$ to determinant of $A$ and vice versa from linear combinations of determinants of powers of $A$ to trace of $A$? (I am mainly looking for the latter - conversion of determinants to trace of $A$).
| https://mathoverflow.net/users/10035 | Trace Determinant | You can find the determinant of a matrix $A$ of size $n$ in terms of the traces of $A^m$, for $m = 1, \ldots, n$. The trace of $A^m$ is the sum of the $m$th powers of the eigenvalues of $A$, and you can express the elementary symmetric polynomials (so in particular the product of the eigenvalues, which is the determina... | 16 | https://mathoverflow.net/users/21146 | 88204 | 52,249 |
https://mathoverflow.net/questions/88145 | 11 | A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{Z}$ (implicitly as a group scheme) especially when $n$ is a prime power: [here](https://mathoverflow.net/questions/8725... | https://mathoverflow.net/users/4231 | Uniform setting for computing orders of algebraic groups over finite quotients of the integers? | I think the assumptions needed on a group scheme $G$ over $\mathbf{Z}$ are that $G$
should be smooth, affine, and of finite type over $\mathbf{Z}$. Actually,
I'm going to take the point of view that $p$ is fixed and so I'll suppose that $G$ is smooth, affine, and of finite type over the local ring $\mathbf{Z}\_{(p)}$.
... | 4 | https://mathoverflow.net/users/4653 | 88205 | 52,250 |
https://mathoverflow.net/questions/88235 | 3 | $A$ is an [adjacent matrix](http://en.wikipedia.org/wiki/Adjacency_matrix) of a network. $la$ is the largest eigenvalue of $A$ and $Va$ is its corresponding eigenvector.
I am interested in the following martix: $bA+c-dI$ ($b$, $c$, and $d$ are all real constants, $bA+c$ is obtained by multiply each element of $A$ by ... | https://mathoverflow.net/users/21328 | Eigenvalues of sum of an adjacent matrix and a constant | An easy upper bound is $la+nc$, where $n$ is the dimension. This is because the matrix norm of a sum is no more than the sum of the matrix norms.
An easy lower bound is $k+nc$, where $k$ is the average vertex degree. This is because, for all $v$, $v \cdot Mv/||v||^2$ is no more than the highest eigenvalue of $M$. The... | 2 | https://mathoverflow.net/users/18060 | 88240 | 52,272 |
https://mathoverflow.net/questions/88233 | 11 | The following problem was raised in [a Mathlinks thread](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=56&t=374534):
**If $a,b,c\in\mathbb Z$ such that $a\ne0$ and $b^2-4ac\ne 0$, for how many consecutive integers $x$ can $ax^2+bx+c$ ba a perfect square ?**
The polynomial $-15x^2+64$ is obviously a squ... | https://mathoverflow.net/users/29783 | how many consecutive integers $x$ can make $ax^2+bx+c$ square ? | To resonate with Noam Elkies' comments, it is conjectured that $8$ squares is the maximum for arbitrary $a$, and $4$ squares is the maximum for $a=1$. For $5$ *symmetric* squares the smallest known leading coefficients are $a=15$ and $a=-20$, while for $5$ *increasing* squares they are $a=60$ and $a=-56$. It is known t... | 14 | https://mathoverflow.net/users/11919 | 88245 | 52,274 |
https://mathoverflow.net/questions/88247 | 8 | Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:
*Let B be an augmented algebra over a semi-simple algebra T.*
But I am stuck already with "augmented algebra"... -- can not find a definition on the web.
| https://mathoverflow.net/users/13441 | What is "augmented algebra"? | An *augmented ring* is simply a triple $(A,M,\epsilon)$ with $A$ ring, $M$ a left $A$-module and $\epsilon:A\to M$ a surjection of $A$ modules. One says then that $A$ is augmented over $M$. You can find this defined in Cartan-Eilenberg, for example.
Often, $M$ is itself a ring and the map $\epsilon$ is also a ring mo... | 15 | https://mathoverflow.net/users/1409 | 88248 | 52,276 |
https://mathoverflow.net/questions/88230 | 11 | I'm looking for a notion of an Abelian category $\mathcal{A}$ "generated" by a given category $\mathcal{C}$
More precisely I need something along the following lines. Denote $\mathcal{Ab}\_2$ the 2-category of Abelian categories and $\mathcal{Cat}$ the 2-category of categories. We have the forgetful 2-functor $\mathc... | https://mathoverflow.net/users/11146 | Is there an "Abelian envelope" 2-functor? | I'd be inclined to go in a slightly different direction in constructing the free abelian category generated by a category. I would do it in stages:
* First construct the free $Ab$-enriched category $F\_1(C)$ generated by a category $C$. This would have the same objects as $C$, but the hom $F\_1(C)(a, b)$ between two... | 13 | https://mathoverflow.net/users/2926 | 88251 | 52,279 |
https://mathoverflow.net/questions/88252 | 26 | Hi. I'll confess from the start to not being a logician. In fact this question came up not from research but during a discussion with a friend about whether the classical proof that $\sqrt{2}$ is irrational can be made acceptable to an intuitionist. (It can be.)
The question is: *Are there any "natural" statements wh... | https://mathoverflow.net/users/5621 | What can be proven in Peano arithmetic but not Heyting arithmetic? | The first example that occurs to me is (a formalization in the language of arithmetic, via coding, of) "For every Turing machine M and every input x, the computation of M on input x either terminates or doesn't terminate." With classical logic, this is trivially provable, as an instance of the law of the excluded middl... | 48 | https://mathoverflow.net/users/6794 | 88253 | 52,280 |
https://mathoverflow.net/questions/17532 | 66 | Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between two objects $x$ and $y$ is the free $R$-module on $\operatorname{Hom}\_C(x,y)$. Thus in $RC$ we allow arbitrary $R$-lin... | https://mathoverflow.net/users/4183 | Does linearization of categories reflect isomorphism? | Hi Tilman. I believe I proved that (in your language) linearization reflects isomorphism. The following is a sketch. I will send you a more detailed version. The general case may be reduced to the case of prime fields $F\_p$ and certain categories $C$ with fixed objects $x$ and $y$ and morphisms $f\_1,\dots,f\_m\colon ... | 29 | https://mathoverflow.net/users/21336 | 88258 | 52,282 |
https://mathoverflow.net/questions/88220 | 13 | Prove that there are infinitely many positive integers $a$, $b$, $c$ that are consecutive terms of an arithmetic progression and also satisfy the condition that $ab+1$, $bc+1$, $ca+1$ are all perfect squares.
I believe this can be done using Pell's equation. What is interesting however is that the following result fo... | https://mathoverflow.net/users/16321 | Special arithmetic progressions involving perfect squares | Starting from the equations in my previous answer, we get, by multiplying them in pairs,
$$(x-y)x(x+y)(x+2y) + (x-y)x + (x+y)(x+2y) + 1 = (z\_1 z\_6)^2\,,$$
$$(x-y)x(x+y)(x+2y) + (x-y)(x+y) + x(x+2y) + 1 = (z\_2 z\_5)^2\,,$$
$$(x-y)x(x+y)(x+2y) + (x-y)(x+2y) + x(x+y) + 1 = (z\_3 z\_4)^2\,.$$
Write $u = z\_1 z\_6$, $v =... | 18 | https://mathoverflow.net/users/21146 | 88263 | 52,284 |
https://mathoverflow.net/questions/88257 | 3 | Maybe I'm looking at the wrong places, but I can't find a definition of
the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the "Moore construction in the
**alternating face map complex** "
for abelian simplicial groups, for example given in
<http://n... | https://mathoverflow.net/users/21302 | Coboundary map on the cochain complex of abelian cosimplicial groups? | If the coface maps are $d^i: C^{n-1} \to C^n$ $(i=0,...,n)$ then the coboundary map is
$$\delta^n = \sum\_{i=0}^n(-1)^i d^i: C^{n-1} \to C^n.$$
It might be helpful to keep in mind that the "co" refers to a dual concept, i.e. arrows are reversed. For example a face map $d\_i: C\_n \to C\_{n-1}$ corresponds to a cofa... | 3 | https://mathoverflow.net/users/10194 | 88266 | 52,287 |
https://mathoverflow.net/questions/69552 | 2 | I have found it necessary to define the following property:
Consider a finite set $X$ and a group $H$ of permutations of $X$. Suppose for every normal subgroup $K$ of $H$ that $K$ acts faithfully on every $K$-orbit.
The property I have is that $G$ acts on $X$, not necessarily faithfully, inducing a permutation grou... | https://mathoverflow.net/users/4053 | Name for a certain kind of 'weakly primitive' permutation group | So this question is not left unanswered, I should add that I decided to call them 'subprimitive'. The context is a construction of some examples of hereditarily just infinite profinite groups, which has now appeared in my paper 'Inverse system characterizations of the (hereditarily) just infinite property in profinite ... | 2 | https://mathoverflow.net/users/4053 | 88270 | 52,288 |
https://mathoverflow.net/questions/88267 | 1 | Covering the Rationals -- A Paradox?
------------------------------------
The following seems to yield a paradox. Can anyone provide the proper resolution?
**Preamble**
It is easy to show that between any two reals there is a rational. If $\xi1$ and $\xi2$ are real numbers with $\xi2 > \xi1$, there is an integer ... | https://mathoverflow.net/users/21340 | Covering the Rationals -- A Paradox? | There are, in fact, uncountably many gaps between the intervals of your construction. Indeed, the complement $F$ of your open set is (homeomorphic to) a Cantor set. You are correct that it contains no interval. Investigate a write-up of the "Cantor set" itself in a real analysis (or topology) textbook to understand how... | 1 | https://mathoverflow.net/users/454 | 88276 | 52,291 |
https://mathoverflow.net/questions/88227 | 3 | Let's consider the following Cauchy problem:
$$u\_t=\alpha(x,t,u)u\_{xx}+\beta(x,t,u)(u\_x)^2+\gamma(x,t,u)u\_x+\phi(x,t,u)$$
$$u(x,0)=u\_0(x).$$
Assume that functions $\alpha,\beta,\gamma,\phi$ are regular (Hölder continous or even smooth but may have some sort of singularity in $t=0$ like $1/t$ term for example... | https://mathoverflow.net/users/19379 | Asymptotic expansions of solutions of nonlinear PDE's | The most direct way to get a small time expansion is using a gradient expansion. This can be worked out in the following way. Rescale your time variable as $t\rightarrow\lambda t$ being $\lambda$ a parameter taken to be arbitrary large and introduced to fix expansion order. Then, takes $\phi$ proportional to $\lambda$.... | 2 | https://mathoverflow.net/users/19520 | 88280 | 52,293 |
https://mathoverflow.net/questions/88284 | 22 | Suppose you can make infinitely many copies of yourself. Each of them starts his/her life in a Hilberts Hotel, where each room is labeled by an element in the [free group](http://en.wikipedia.org/wiki/Free_group) with two generators, and structured as the Cayley graph of the group. (All the room have the same size, so ... | https://mathoverflow.net/users/2097 | How to get rich in a Hilberts Hotel? | No, you cannot get rich with identical copies on the unlabeled tree. This is a special case of the Mass Transport Principle - take a look at the book of [Lyons and Peres](http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html), chapter 8.
| 18 | https://mathoverflow.net/users/1061 | 88291 | 52,297 |
https://mathoverflow.net/questions/88282 | 8 | Let $\operatorname{ GPF}(n)$ be the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a\_{n+1}=\operatorname{ GPF}(xa\_n+y)$, eventually enter a cycle for all positive integers $x,a\_0,y>0$?
Is there any set of positive integers $... | https://mathoverflow.net/users/21346 | The sequence $a_{n+1}=$ the greatest prime factor of $(xa_n+y)$ | There is a paper on this problem, Mihai Caragiu, Recurrences based on the greatest prime factor function, JP J. Algebra Number Theory Appl. 19 (2010), no. 2, 155–163, MR2796479 (2012a:11010). The summary begins,
We introduce and discuss a generalized ultimate periodicity conjecture for prime sequences $\lbrace q\_n\... | 4 | https://mathoverflow.net/users/3684 | 88304 | 52,303 |
https://mathoverflow.net/questions/88289 | 3 | I don't know whether this is known or not, but I was thinking of the following problem.
Let $n$ and $m$ be natural numbers. Are there $n$ polynomial $f\_1,...,f\_n\in \mathbb{C}[x]$, such that all of their intersection numbers are at least $m$?
It is also possible to state an even harder question in a more natural ... | https://mathoverflow.net/users/5756 | Are there n polynomials for which all intersection multiplicities are at least m? | Ok, so here's the long version of what I already said in the comments:
Start by picking $g\_1,\ldots,g\_n\in \mathbb C[x]$ such that $g\_i\neq g\_j$ whenever $i\neq j$ which satisfy the prescribed condition at $0$ (that's not hard). Since $g\_i-g\_j\neq 0$ for each $i\neq j$, there will be only finitely many $0\neq a... | 3 | https://mathoverflow.net/users/17498 | 88306 | 52,305 |
https://mathoverflow.net/questions/88272 | 5 | We know that if $X$ is a smooth connected variety over a field $k$, then any line bundle on $X\times\_k\mathbb{A}^1$ is from a line bundle on $X$. This is simply because they have the same Picard group. This has a generalization, in fact, $X\times\_k\mathbb{A}^1\to X$ induces an isomorphism of $CH\_r(X)\to CH\_{r+1}(X\... | https://mathoverflow.net/users/18380 | A^1-invariant for Vector Bundles? | One way to think about $\mathbb{A}^n$-families of bundles on $X$ is in terms of $Ext^1$ groups of bundles on $X$. That is, suppose that $Ext^1(E,F) \simeq k^n$. Then, as Chris Brav mentions in the comments, we can think of this as an $\mathbb{A}^n$-point of the moduli of vector bundles on $X$, which over each point of ... | 9 | https://mathoverflow.net/users/916 | 88307 | 52,306 |
https://mathoverflow.net/questions/88310 | 9 | Given any monoidal category $(\mathcal{C}, \otimes, I)$ we have an associated multicategory
$M\_{\mathcal{C}}$ with underlying category $\mathcal{C}$, and $k$-morphisms $M\_{\mathcal{C}}(A\_1,\ldots,A\_k; B) = \mathcal{C}(A\_1\otimes\cdots\otimes A\_k, B)$.
Given any (lax) monoidal functor $F\colon \mathcal{C}\rightar... | https://mathoverflow.net/users/1378 | Which functors between multicategories that come from monoidal categories are monoidal? | I guess that you can take the image under $F$ of the identity in $M\_C(;I)$ (the arity 0 is allowed).
| 7 | https://mathoverflow.net/users/10217 | 88312 | 52,309 |
https://mathoverflow.net/questions/88269 | 2 | The problem is that we want to approximate domain-bounded functions (that is, functions restricted to a domain such as [0,10]) as (probably infinite) series of other functions. We know that, for integrable functions in the interval, we can definitely do that with sin waves (that's the fourier transform), but we wondere... | https://mathoverflow.net/users/21338 | Can we approximate an arbitrary function as a (probably infinite) sum of bell shapes? | You can't approximate arbitrary square-integrable functions in the same way as a Fourier series or wavelet series, since your functions cannot make a basis for the function space.
But what you described is a quadratic spline (almost but not quite a usual [basis B-spline](http://en.wikipedia.org/wiki/B-spline)). So yo... | 4 | https://mathoverflow.net/users/13785 | 88319 | 52,312 |
https://mathoverflow.net/questions/88317 | 6 | I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101.
The setup is as follows. Let $G$ be a finite group, considered as an algebraic group over a field $k$. The coordinate ring of $G$ is then just the set of func... | https://mathoverflow.net/users/21356 | Coproduct on coordinate ring of finite algebraic group | I think the author accidentally described the dual of the Hopf algebra you're thinking of.
Finite group rings are usually endowed with multiplication $(g,h)\mapsto gh$ and comultiplication $g \mapsto g\otimes g$ (see [here](http://en.wikipedia.org/wiki/Group_Hopf_algebra)).
The coordinate ring $k[G]$ is obtained by... | 6 | https://mathoverflow.net/users/17498 | 88320 | 52,313 |
https://mathoverflow.net/questions/88268 | 5 | Let $T$ be a measure-preserving invertible transformation of a Lebesgue space, and let $P$ be the partition of the Lebesgue space into the orbits of $T$.
1) Is it true that $P$ is nonmeasurable (in the sense of Rokhlin) when $T$ is ergodic ? Why ?
2) If true, is it a necessary and sufficient condition for ergodicity ? ... | https://mathoverflow.net/users/21339 | partition into the orbits of a dynamical system | Although it appears you've already settled matters with the information in Jon's answer, I'll offer a quick summary and elaboration.
Let $(X,\mathcal{B},\mu)$ be a Lebesgue space (set + $\sigma$-algebra + probability measure) and $P$ the partition into orbits $\mathcal{O}(x) = \{T^n x \mid n\in \mathbb{Z}\}$ for an i... | 7 | https://mathoverflow.net/users/5701 | 88329 | 52,318 |
https://mathoverflow.net/questions/88323 | 2 | EDIT: Gerhard Paseman has given some wonderful answers to this question below. Thank you. This is an attempt to revisit this to hopefully make the question more rigorous with some notation and try to provide motivation from the context of the Bateman-Horn conjecture, and ask whether this is fruitful or not. (It seems n... | https://mathoverflow.net/users/18494 | Analogues of Jacobsthal's function | I will register a lengthy opinion here, which represents my feelings
on the subject. I have no proofs to offer at this time.
I have not taken the perspective of viewing the set of numbers coprime to
a squarefree integer $m$ as the Cartesian product of smaller sets. This
perspective may be useful, but it is hard for m... | 6 | https://mathoverflow.net/users/3402 | 88332 | 52,319 |
https://mathoverflow.net/questions/88335 | 6 | Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. Consider the strict transform $\widetilde X$ of $X$. Then, it is easy to verify that $Y=\widetilde X\cap E$ is a quadric hyp... | https://mathoverflow.net/users/9871 | Blowing-up an ordinary double point, then contracting the exceptional locus to a curve | I guess the answer in **no** when $\deg(X) \geq 3$.
In fact, in this case Griffiths showed that the two rulings $L\_1$ and $L\_2$ in $E$ are *numerically equivalent* in $\widetilde{X}$ (although not algebraically equivalent in general), so the corresponding $K\_{\widetilde{X}}$-negative rays $R\_1$ and $R\_2$ in the... | 8 | https://mathoverflow.net/users/7460 | 88338 | 52,321 |
https://mathoverflow.net/questions/88349 | 3 | It's known that for a random vector $(X\_1,\dots,X\_n)\in \mathbb{R}^n$ with a log-concave distribution, any subvector has a long-concave distribution. I'm wondering if there are any results about its converse, in particular, when $(X\_1,\dots,X\_n)$ is isotropic, $X\_i$'s are identically distributed (but not necessari... | https://mathoverflow.net/users/21367 | marginal log-concave distributions and joint log-concave distributions | No. Let $X$ be, say, a standard normal random variable, $Z$ an independent random variable with $P[Z=1] = P[Z=-1] = 1/2$, and $Y=XZ$. Then $X$ and $Y$ are uncorrelated standard normal (in particular log-concave) random variables, but the distribution of $(X,Y)$ is not log-concave.
This is of course also a counterexam... | 3 | https://mathoverflow.net/users/1044 | 88350 | 52,326 |
https://mathoverflow.net/questions/88360 | 5 | Let $R$ be a reduced algebra of finite type over a field $k$ of characteristic 0. Let $S$ be a reduced finite $R$-algebra. Is $S \otimes\_R S$ reduced?
(In positive characteristic one can get non-reduced tensor products of reduced algebras even over a field.)
I have failed to find a counterexample so I thought that... | https://mathoverflow.net/users/2234 | is tensor square of a reduced ring reduced? | No. Let $R = k[x,y]/(y^2-x^3)$. Let $S = k[t]$, with the map $R \to S$ given by $(x,y) \mapsto (t^2, t^3)$. So $S \otimes\_R S = k[t,u]/(t^2-u^2, t^3-u^3)$. The element $t-u$ is not zero in the tensor product, because all the relations are in degree $>1$. But $(t-u)^3 = 3 (t-u)(t^2-u^2) - 2 (t^3-u^3)$ is in the ideal.
... | 13 | https://mathoverflow.net/users/297 | 88363 | 52,331 |
https://mathoverflow.net/questions/88362 | 6 | Hi everybody,
I have a lack of references concerning projective limits and injective limits. Up to my faults in Bourbaki there are only proj and inj limits indexed by a partially ordered set (not a category), and this set is almost systematically directed (i.e. for all $i,j$ there exists $k$ such that $k\geq i$, and ... | https://mathoverflow.net/users/21374 | permutation of projective limits with inductive limits | The canonical morphism $\alpha : \mathrm{colim}\_{i \in I} ~ \mathrm{lim}\_{j \in J} M\_{ij} \to \mathrm{lim}\_{j \in J} ~ \mathrm{colim}\_{i \in I} M\_{i,j}$ is injective in the following situation:
1) $I$ is a directed set.
2) For every $j \in J$ and every $i \to i'$ in $I$ the morphism $M\_{i,j} \to M\_{i',j}$ i... | 7 | https://mathoverflow.net/users/2841 | 88369 | 52,333 |
https://mathoverflow.net/questions/88361 | 10 | Let $X$ denote the largest subset of odd integers with the property that
every exponent in the prime factorization of any $x \in X$ belongs to $X$.
The conjecture states that the density of $X$ among the integers is
$1-\frac{1}{\sqrt{3}}$.
Is this conjecture correct ?
| https://mathoverflow.net/users/15997 | A Conjecture on the Density of a subset of integers | This is an update of my original response which was incorrect.
The conjecture is false. I will show that any $X$ in the conjecture has density at most
$$ 0.4226496 < 1-\frac{1}{\sqrt{3}}= 0.4226497...$$
Let $Y$ be the set of odd numbers whose prime exponents are odd and different from $9$. Clearly, any $X$ in the... | 13 | https://mathoverflow.net/users/11919 | 88371 | 52,335 |
https://mathoverflow.net/questions/88368 | 80 | This question was inspired by this [blog post](http://quomodocumque.wordpress.com/2012/02/12/is-there-a-noncommutative-siegels-lemma) of Jordan Ellenberg.
Define a "computable group" to be an at most countable group $G$ whose elements can be represented by finite binary strings, with the properties that
* There is... | https://mathoverflow.net/users/766 | Can a group be a universal Turing machine? | **Update.** Here is a more direct construction. (See edit history for previous version.)
There is such a universal computable group as you request. Let $F$
be the free group on infinitely many generators $\langle
a\_p\rangle\_p$, indexed by the Turing machine programs $p$. Let $G$
be the quotient of this group by all... | 55 | https://mathoverflow.net/users/1946 | 88376 | 52,339 |
https://mathoverflow.net/questions/88351 | 7 | I've been reading Voight's paper on Shimura curves and it prompted the following question; see <http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf> for which notes I'm talking about
Let $F$ be a totally real number field, $B$ a quaternion algebra which splits at exactly one real place, and $... | https://mathoverflow.net/users/4333 | Zograf's bound on the index of a modular curve for Shimura curves | Here are the answers to some of your questions. If $P$ is a (non-zero) prime ideal of the integers of $F$ then $B$ will either be split or ramified at $P$, depending on whether $B\otimes\_F F\_P$ is isomorphic to $M\_2(F\_P)$ or not. Now $B$ will only be ramified at a finite set of primes, and if $P$ is a prime which s... | 6 | https://mathoverflow.net/users/1384 | 88381 | 52,341 |
https://mathoverflow.net/questions/88372 | 7 | The question is in the title.
I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A embeds into the category $R$-mod. The derived category is not abelian, of course, but I have a particular subcategory that i... | https://mathoverflow.net/users/1231 | Is the bounded derived category of coherent sheaves of a variety a small category? | The answer to your question really depends on what you mean by the word "the". An unhelpful answer is that the coherent sheaves over any variety form a proper class (hence "no"). A more useful answer is (as mentioned in the comments) that there exists a small category that is equivalent to any category that can be reas... | 5 | https://mathoverflow.net/users/121 | 88394 | 52,346 |
https://mathoverflow.net/questions/88356 | 5 | Let $X$ a variety. Say $X=\operatorname{Spec} A$.
Consider two ideals of $A$, say $I$ and $J$, with equal radical ; and consider the blow-ups of X with centre $I$ and $J$, say $Y\_I$ and $Y\_J$. **How can I decide if the blow-up map $Y\_I \to X$ factors through $Y\_J$ ?**
For example, if $X = \operatorname{Spec}k[x... | https://mathoverflow.net/users/19205 | Comparing blow-ups with comparable centres | There are *two* conditions I can think of.
1. Perhaps $I = J \cdot J'$ for some other ideal $J'$. Indeed, blowing up a product of ideals is the same as blowing up one ideal (say $J$) and then blowing up the total transform of the other (say $J'$).
2. The other is requiring that $I = \overline{J}$, here this is integr... | 4 | https://mathoverflow.net/users/3521 | 88400 | 52,349 |
https://mathoverflow.net/questions/88380 | 6 | Consider Higher order predicate logic over dependent type theory (DPL) as defined in Chapter 11 of B. Jacobs's book "Categorical Logic and Type Theory" (though I think this question applies to first-order predicate logic too).
In the category $\mathbb{P}$ of propositions-in-dependent-type-contexts, an object is a wel... | https://mathoverflow.net/users/4085 | Why no morphisms from the contradictory proposition to the inconsistent context? | My understanding of the situation is that you have to correctly understand what the morphisms do. A geometric picture might be helpful.
Let us interpret a context $\Gamma = x\_1 : A\_1, \ldots, x\_n : A\_n$ as a cartesian product $|\Gamma|$ of topological spaces. A proposition in a context $\Gamma \vdash \phi$ is int... | 6 | https://mathoverflow.net/users/1176 | 88407 | 52,351 |
https://mathoverflow.net/questions/88408 | 1 | Let $R$ be an Henselian discrete valuation ring with field of fractions $K$. Let $M$ be a torsion-free $R$-module of finite rank (i.e. $dim\_K(M\otimes\_RK)<+\infty$). Let $D$ be the maximal divisible $R$-submodule of $M$, then $M$ is said to be reduced if $D=0$. If I am not wrong if $M$ is reduced, of finite rank and ... | https://mathoverflow.net/users/18942 | Is a reduced, torsion-free module of finite rank over an Henselian ring free? | Theorem 19 in Kaplansky's Infinite Abelian Groups gives an example of a torsion-free, reduced, indecomposable rank 2 module for any incomplete discrete valuation ring.
In short, the construction is as follows: choose $\lambda\in\hat R\setminus R$. This induces a homomorphism $\tilde\lambda\colon K\to\hat R/R$. The sh... | 5 | https://mathoverflow.net/users/2035 | 88411 | 52,354 |
https://mathoverflow.net/questions/88399 | 20 | Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S\_n$ with $A\_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's decomposition. I'm interested in $|A|$. More specifically I aim to prove that all of its roots as of a polynomial in $x$ are ... | https://mathoverflow.net/users/19864 | An $n!\times n!$ determinant | This determinant came up, and was evaluated, in the [comments of the Secret Blogging Seminar](http://sbseminar.wordpress.com/2009/02/25/a-warmup-gl_t-for-t-not-an-integer/#comment-4870). The motivation there was that it vanishes if and only if $V^{\otimes n}$ has neglible endomorphisms in Deligne's category of "$GL\_x$... | 16 | https://mathoverflow.net/users/297 | 88421 | 52,360 |
https://mathoverflow.net/questions/88420 | 18 | I've looked on the web and haven't found a simple example.
| https://mathoverflow.net/users/16371 | Example of a weak Hausdorff space that is not Hausdorff? | The one-point compactification of $\mathbb{Q}$ has the property that every compact subset is closed. So it is certainly a weak Hausdorff space. But it isn't Hausdorff, as $\mathbb{Q}$ isn't locally compact.
**Addendum**
Another example is the [cocountable topology](http://en.wikipedia.org/wiki/Cocountable_topology)... | 24 | https://mathoverflow.net/users/12362 | 88422 | 52,361 |
https://mathoverflow.net/questions/88433 | 4 | Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following classic result.
>
> If $p\_t(x,y)$ is the kernel of the semigroup $e^{t\Delta}$, then there exist $C,c>0$, such that
> $$p\_{t}(x,y)\leq... | https://mathoverflow.net/users/16326 | Reference for estimation gaussian of the heat kernel | A probabilistic proof is given in the book: "Stochastic Analysis on Manifolds" by E. Hsu.
See Theorem 5.3.4, which also gives the lower bound.
| 2 | https://mathoverflow.net/users/21061 | 88436 | 52,365 |
https://mathoverflow.net/questions/88413 | 11 | Let $V$ and $I$ be two closed subsets of a Banach space $A$.
The set $V$ is a convex cone, and $I$ is a linear subspace of $A$. I also know that $V\cap I=\{0\}$.
I would like to know whether $I+V$ is closed. I've seen that there is a criterion of Dieudonné which I can't use here because I know that neither $V$ nor ... | https://mathoverflow.net/users/20341 | A criterion for the sum of two closed sets to be closed ? | When $V$ is also a subspace, the standard equivalence to the sum being closed is that the unit spheres of $V$ and $I$ are a positive distance apart. I bet this is true when $V$ is just a convex cone but don't have time right now to think about it (the given condition is clearly sufficient for closedness of the sum; nec... | 8 | https://mathoverflow.net/users/2554 | 88438 | 52,366 |
https://mathoverflow.net/questions/88449 | 1 | The first of my questions may be entirely elementary, but the second (closely related) question may be of appropriate interest for this site.
Suppose that we are given $w\_1, w\_2, \cdots, w\_n$ of positive integers which are co-prime. Let $H > 1$ be a positive parameter. I am interested in evaluation the sum
$$\di... | https://mathoverflow.net/users/10898 | Values of various weighted sums | **new answer** I think that the sum will be $$\frac{n}{w\_1w\_2\dots w\_n}\binom{H+n}{n+1}+o(H^{n}).$$ I do not think that it matters if the positive values $w\_i$ are integers nor (if they are) if they are relatively prime. If my calculations are correct, then in the special case that $w\_1=w\_2=\dots=w\_n=1$ and $H$ ... | 3 | https://mathoverflow.net/users/8008 | 88455 | 52,374 |
https://mathoverflow.net/questions/88045 | 4 | **Definition:** Let $B$ be a boolean algebra. Say $X \subseteq B$ is *quasi-dense* in $B$ if for all $b \in B$, there is $x \in X \setminus$ { $0,1$ } such that either $x \leq b$ or $b \leq x$.
**Question:** Suppose $A \subseteq B \subseteq C$ are atomless boolean algebras, $A$ is quasi-dense in $B$, and $B$ is **den... | https://mathoverflow.net/users/11145 | Quasi-dense subsets of boolean algebras | Here is a way to construct an atomless version of Joel´s counterexample:
Let $A\_0 \subseteq A\_1 \subseteq A\_2$ be the algebras in Joel´s example (in his notation $A \subseteq B \subseteq C$) and let $X\_0$,$X\_1$ and $X\_2$ be the corresponding Stone spaces. So $X\_0$ and $X\_1$ are both just a converging sequenc... | 2 | https://mathoverflow.net/users/17836 | 88459 | 52,377 |
https://mathoverflow.net/questions/88460 | 9 | Let G and H be two groups. There is a one-to-one correspondence between:
(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;
(ii) an (isomorphism class of) action of the group G on the category of H-sets.
Actually, this can even be made an equivale... | https://mathoverflow.net/users/21405 | Group extensions and actions on categories | The beginning of SGA 7, Exposé VII, essentially proves an equivalence between extensions of $G$ by $H$ and monoidal functors from $G$ (thought of as a discrete monoidal category) to the monoidal category of $H$-bitorsors. The correspondence here takes an extension $1 \to H \to E \stackrel{\pi}{\to} G \to 1$ to the func... | 9 | https://mathoverflow.net/users/396 | 88462 | 52,379 |
https://mathoverflow.net/questions/88440 | 4 | Hi all,
do you know how to compute (as a function of n) the largest eigenvalue of this matrix (or at least to bound it)?
$$
\left(\begin{array}{cccccc}
0 & 1 & & & & \cr
1 & 0 & \sqrt 2 & & & \cr
& \sqrt 2 & 0 & & & \cr
& & & \ddots & & & \cr
& & & & 0 & \sqrt n & \cr
& & & & \sqrt n & 0 &
\end{array}\right)
$... | https://mathoverflow.net/users/16292 | Bounding largest eigenvalue | If you denote $A\_n$ your tri-diagonal matrix of order $n$, and $H\_n(x):= \det(x+A\_n)$, the sequence $H\_n$ satisfies the two-term linear recurrence $H\_{n+1}=xH\_n - nH\_{n-1}$ with initial conditions $H\_0=1$ and $H\_1=x$. Thus, they are the [Hermite polynomials](http://en.wikipedia.org/wiki/Hermite_polynomials) (h... | 11 | https://mathoverflow.net/users/6101 | 88464 | 52,380 |
https://mathoverflow.net/questions/87855 | 2 | Hi Guys,
Just wondering if you could suggest applications of distribution of the supremum of a fractional Brownian motion process with a drift ?
Also if you could possibly recommend how to approach this problem, that would be much appreciated.
Cheers
| https://mathoverflow.net/users/18929 | Applications of this project | An application that springs to mind immediately is *option evaluation*.
Suppose I offer you to buy a contract to me: after three months I pay you the maximum value of the price of an asset.
How much are you willing to pay this contract ?
If you model the price of the assert by a drifted brownian motion, then you'll pro... | 5 | https://mathoverflow.net/users/20997 | 88469 | 52,383 |
https://mathoverflow.net/questions/88448 | 5 | I'm studying a small category $A$ and diagrams of based spaces or spectra indexed by $A$ (so let's say diagrams in a category $C$ that's closed symmetric monoidal, has a compatible model structure, etc.). I'm told that in this setting there's a projective model structure on diagrams $A \rightarrow C$, where the fibrati... | https://mathoverflow.net/users/1874 | mapping spaces of diagrams | Let me mention yet another approach using the simplicial localization due to Dwyer and Kan.
Given any category $\mathcal{C}$ equipped with a class of morphisms $W$ one can form a simplicial category $L(\mathcal{C}, W)$ which depends only on $\mathcal{C}$ and $W$ and not on any auxiliary structure like model structure... | 6 | https://mathoverflow.net/users/12547 | 88473 | 52,386 |
https://mathoverflow.net/questions/88398 | 5 | **Intro**
The question is about [Game of Life](http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life).
Let us denote the set of points obtained from initial configuration $A$ after $m$ steps as $A(m)$ (we are only interested in finite initial configuration, i.e. those one which formed by finite number of marked cel... | https://mathoverflow.net/users/7095 | Decay of Relative Growth in Conway's Game of Life | *[expanded from the comment above]*
There cannot be such a result. The simplest aperiodic counterexample
is a "lightweight spaceship gun" of even period $p$, whose
$m$-th generation has population $9m/p + O(1)$ or $12m/p + O(1)$
according to the parity of $m$, whence
$i(A,m)/N(A(m)) \rightarrow 1/3$ in one congruence... | 9 | https://mathoverflow.net/users/14830 | 88477 | 52,387 |
https://mathoverflow.net/questions/88457 | 6 | Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of the elements of this set.
I am interested in figuring out the computational complexity (and algorithm, if available) of ... | https://mathoverflow.net/users/16571 | Finding the convex combination of vertices which yields an inner point of a polytope | Hopefully I am using the right notion of convex combination. The following requires at most n+1 steps, however I do not know how complicated a step is.
Take the given point x and a vertex v visible from x. Thus the line through v and x passes through the polygon from v to x to a point p on a face or facet on the othe... | 7 | https://mathoverflow.net/users/3568 | 88485 | 52,390 |
https://mathoverflow.net/questions/88406 | 42 | Having now seen and "understood" (quotes necessary) the Seiberg-Witten equations on a closed oriented Riemannian 4-manifold $X$, I have no real understanding of **where they came from**.
We take an orthogonal frame bundle $P$ of $TX$, a $\textrm{spin}^\mathbb{C}$ structure $\tilde{P}$ with determinant line bundle $\... | https://mathoverflow.net/users/12310 | Meaning/origin of Seiberg-Witten equations/invariants | After thinking, and reading other references and re-reading the papers I mentioned, I may have found a sufficient explanation (at least to my care): Both instantons/monopoles are solutions to corresponding equations of motions from associated actions, and they "bloom" from an overarching SUSY action.
Witten formulate... | 12 | https://mathoverflow.net/users/12310 | 88489 | 52,392 |
https://mathoverflow.net/questions/87700 | 3 | A theorem in [Geoghean's](http://books.google.com/books/about/Topological_methods_in_group_theory.html?id=BwX6gblqV8MC) book is the following (theorem 18.3.18):
Let $G$ be a finitely presented group and let the rank of $G/G'$ (as
a $\mathbb{Z}$-module) be at least 2. If $G$ has no non-abelian free subgroup, then ther... | https://mathoverflow.net/users/7307 | Subgroups with Infinite cyclic quotients of the Thompons's group | The first answer is incomplete and moreover I suspect that it is incorrect!
Here is an answer submitted to me via email by Andrew Brunner, who asked me to post his answer for him since he is not signed up for MO.
Let $F = \langle a,b| [ab^{-1},a^{-1}ba],[ab^{-1},a^{-2}ba^2] \rangle$ be the usual
finite presentatio... | 1 | https://mathoverflow.net/users/7307 | 88493 | 52,393 |
https://mathoverflow.net/questions/88481 | 3 | Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it works under the condition that $\pi\_1(B)$ acts trivially on $H\_\*(F;G)$. If this condition ($\pi\_1(B)$ acts trivially on... | https://mathoverflow.net/users/15770 | Computing the homology groups of spaces in a fibration | Repeating Mark Grant's comment, the spectral sequence when all spaces are $K(\pi,1)$s goes under the name Lyndon-Hochschild-Serre spectral sequence.
Good references for this spectral sequence are:
D. Benson: Representations and Cohomology II: cohomology of groups and modules
L. Evens: The cohomology of groups
... | 8 | https://mathoverflow.net/users/6574 | 88502 | 52,397 |
https://mathoverflow.net/questions/88501 | 27 | Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a\_0,a\_1,\ldots,a\_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum\_{i=0}^k a\_i(c)t^i+o(t^k)$ (for any $t$ in a neighbourhood of $0$). Can we conclude that $f$ is of class $C^k$?
| https://mathoverflow.net/users/36952 | "Converse" of Taylor's theorem | Yes. It's a classical result that goes back to Marcinkiewicz and Zygmund (*On the differentiability of functions and summability of trigonometric series*, Fund.Math **26** (1936) ).
There is a sublety in the form of the remainder: a first and natural characterization of $C^k$ is obtained asking a remainder of the fo... | 21 | https://mathoverflow.net/users/6101 | 88508 | 52,398 |
https://mathoverflow.net/questions/88114 | 13 | Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the idea is to "average" left invariant forms on $G$ using a Haar measure.
Can we extend this result for non compact infinit... | https://mathoverflow.net/users/5450 | free loop space and invariant forms | Iterated integrals define a map
$$
\sigma: C(\Omega(M)) \to \Omega(LM)
$$
where $C(\Omega(M))$ is the cyclic bar complex of $\Omega(M)$. It has various nice properties; for instance, it induces an isomorphism in cohomology, when $M$ is simply-connected.
Unfortunately, the forms in the image of $\sigma$ are *not* inv... | 5 | https://mathoverflow.net/users/3473 | 88510 | 52,399 |
https://mathoverflow.net/questions/88511 | 5 | Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G\_p = \mathrm{Gal}(\overline{\mathbb{Q}\_p}: \mathbb{Q}\_p)$.
Abelian class field theory gives us for the abelinisation
$$ G^{ab} = G / [G, G] = \prod\limits\_{p} GL\_1( \mathbb{Z}\_p).$$
>
> How can we relate the group... | https://mathoverflow.net/users/10400 | What is the relation of the absolute Galois group and classical profinite groups? | This question has been solved by Paskunas in his PhD thesis, *Unicity of types for supercuspidals*, arXiv:[math/0306124](https://arxiv.org/abs/math/0306124).
Corollary 8.2 of this reference gives an "inertial Galois correspondence" between supercuspidal types of the form $({\rm GL}(n,{\mathbb Z}\_p), \lambda )$ and c... | 9 | https://mathoverflow.net/users/4767 | 88517 | 52,401 |
https://mathoverflow.net/questions/88520 | 5 | Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P\_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group.
Is the following assertion true:
"The Abelian variety $A$ is simple over the finite field with $q$ elements if and onl... | https://mathoverflow.net/users/21030 | Can we decide if an abelian variety is simple by knowing its Zeta function ? | The following result follows from Tate-Honda theory
>
> Let $A$ be an abelian variety over a finite field $k$, and let $f\_A$ be the characteristic polynomial of $A$. Then $A$ is isogenous to a power of a simple abelian variety if and only if $f\_A$ is a power of an irreducible polynomial.
>
>
>
I can't find a... | 14 | https://mathoverflow.net/users/297 | 88523 | 52,404 |
https://mathoverflow.net/questions/88512 | 5 | Hi all,
it is well known that the complex projective space with the fubini study metric is Einstein, but what is the explicit value, i.e. for which $\mu$ does $Ric=\mu g$ hold?
Moreover, I would like to know how to calculate the sectional cuvature explicitly, because I would like to calculate the number $\sqrt{\sum... | https://mathoverflow.net/users/20823 | Fubini Study Metric and Einstein constant | As suggested by Anton, you can use the O'Neill formulas in the Riemannian submersion $\mathbb C^{n+1}\to \mathbb{C} P^n$ that defines the Fubini-Study metric on $\mathbb C P^n$. This gives the following: suppose $X,Y$ are orthonormal tangent vectors at some point in $\mathbb C P^n$, and denote by $\overline X,\overline... | 6 | https://mathoverflow.net/users/15743 | 88525 | 52,405 |
https://mathoverflow.net/questions/88296 | 2 | Let $A$ be a $m \times n$ matrix and let Y be the set of paths "from left to right through the matrix"
\begin{equation}
Y=\lbrace 1 \ldots m \rbrace ^N
\end{equation}
Let $f(y;A)$ be the "sum along the path $y$"
\begin{equation}
f(y;A) = \sum\_{i=1}^n A\_{y\_i,i}
\end{equation}
Let $Z\_k = \lbrace y \in Y : f(y... | https://mathoverflow.net/users/19899 | Computing the sum over paths through a matrix satisfying constraints | Here is an example calculation using generating functions. We begin with a matrix
$$ A = \left( \begin{array}{cccc}
1 & -1 & 0 & 2\\
2 & 2 & -3 & 0\\
0 & 2 & 1 & 2 \end{array} \right). $$
Each column contributes a factor of $\sum x^e$ where $e$ ranges over the column:
$$f(x)=(x+x^2+x^0)(x^{-1}+2x^2)(x^0 + x^{-3}+... | 4 | https://mathoverflow.net/users/9068 | 88533 | 52,409 |
https://mathoverflow.net/questions/88518 | 2 | I take a random but practical example direct from "R-matrices and the magic square"
by Bruce Westbury: The adjoint irrep $A$ of the $E\_7$ family has quantum dimension
$qi[2\*m+3]\*qi[3\*m/2+2]\*qi[3\*m/2]/qi[m/2]/qi[m/2+2]$
where $qi[m]$ (at the value $q$) denotes the quantum integer function and $m$ is "mostly... | https://mathoverflow.net/users/11504 | Divisibility Rules for Quantum Integers | Factor a fraction in the $q$-integers as a product of cyclotomic polynomials $\Phi\_d$, using that $q^n-1 = \prod\_{d|n} \Phi\_d$, and look at the multiplicity of each polynomial $\Phi\_d$.
| 1 | https://mathoverflow.net/users/10881 | 88546 | 52,414 |
https://mathoverflow.net/questions/88539 | 20 | It is a well-known fact that if an integer is a sum of two rational squares then it is a sum of two integer squares. For example, Cohen vol. 1 page 314 prop. 5.4.9. Cohen gives a short proof that relies on Hasse-Minkowski, but he attributes the theorem (without reference) to Fermat, who didn't have Hasse-Minkowski avai... | https://mathoverflow.net/users/12669 | sums of rational squares | This result is pretty shy of needing the full Hasse-Minkowski Theorem. Indeed, since Fermat already knew which integers were a sum of two integer squares, it would suffice for him to show that those that weren't (i.e., those with an odd power of some prime congruent to 3 mod 4 showing up in its prime factorization) cou... | 11 | https://mathoverflow.net/users/35575 | 88549 | 52,415 |
https://mathoverflow.net/questions/88513 | 31 | According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs)".
I'd like to know if h-principle and theory from M. Gromov's "Partial Differential Relations" is a useful too... | https://mathoverflow.net/users/19379 | H-principle and PDE's |
>
> I'd like to know if h-principle and theory from M. Gromov's "Partial Differential Relations" is a useful tool in the field of nonlinear PDE's.
>
>
>
Useful is a relative word.
>
> What type of problems can be attacked using h-principle? What type of results can be obtained?
>
>
>
This is easier to a... | 27 | https://mathoverflow.net/users/10839 | 88557 | 52,420 |
https://mathoverflow.net/questions/88542 | 4 | Consider a semisimple Lie group or a $p$ adic reductive group $G$.
To what extent can the character of a representation as a distribution on $C\_c^\infty(G)$ determine the representation?
| https://mathoverflow.net/users/10400 | Character determines the representation? | For a reductive Lie group, the character characterizes an irreducible admissible representation up to infinitesimal equivalence. Referring to Knapp's "Representation Theory, etc", Proposition 10.5 says that two infinitesimally-equivalent irreducible admissible representations have the same character, and Theorem 10.6 s... | 8 | https://mathoverflow.net/users/6753 | 88558 | 52,421 |
https://mathoverflow.net/questions/88544 | 2 | Fix $\epsilon, 0\leq \epsilon\leq 1/2.$ Let $Z\_1,Z\_2$ be zero mean, unit variance Gaussian random variables which are jointly Gaussian with $\mathbb{E}Z\_1Z\_2=-(1-2\epsilon)\leq 0.$
Then,
$$P(Z\_1Z\_2>0)\geq \epsilon.$$
This curious fact popped out of some calculations I was doing using the Central Limit Theo... | https://mathoverflow.net/users/7576 | On two-dimensional Gaussian integrals | Since $P(Z\_1Z\_2>0)=\frac12-\frac1{\pi}\arcsin(1-2\epsilon)$, indeed $P(Z\_1Z\_2>0)\geqslant\epsilon$, and the inequality is strict except when $\epsilon=0$, $\epsilon=\frac12$ and $\epsilon=1$.
| 3 | https://mathoverflow.net/users/4661 | 88566 | 52,427 |
https://mathoverflow.net/questions/88567 | 4 | **Background**
Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be its unipotent radical. Consider $k[U]$ as a $U$-module under left multiplication in $U$; let's call this module $k... | https://mathoverflow.net/users/1528 | Decomposition of the ring of functions on the unipotent radical of a Borel | It is probably hard to give a complete description. A lot of partial information about this is contained in this paper of Kostant <https://arxiv.org/abs/1201.4494>
(in particular there is a description of the subring of invariant elements in $k[U]$;
in fact Kostant works with the Lie algebra of $U$ instead of $U$ itsel... | 2 | https://mathoverflow.net/users/3891 | 88572 | 52,430 |
https://mathoverflow.net/questions/88378 | 9 | Let $i : U \to X$ be a quasicompact open immersion of schemes. I would like to know whether the canonical morphism $i\_\* \mathcal{O}\_U \otimes\_{\mathcal{O}\_X} i\_\* \mathcal{O}\_U \to i\_\* \mathcal{O}\_U$ is an isomorphism.
Remark that this is trivial if $i$ is an affine morphism (base change formula), so we're ... | https://mathoverflow.net/users/2841 | Is the restriction map an epimorphism of commutative rings? | Martin, Mike is correct, $\Gamma(U)$ is indeed not free over $\Gamma(X)$, but most of what you are saying can be salvaged.
Here is what's happening:
Let $X=X\_1\cup X\_2$ be the union of the two copies of $\mathbb A^2$ glued at one point as you describe in your answer. Let $X'=X\_1\overset{\cdot}\cup X\_2$ the disj... | 2 | https://mathoverflow.net/users/10076 | 88574 | 52,431 |
https://mathoverflow.net/questions/88522 | 2 | A shift space $(X, \sigma)$ is a *coded system* if there exist a countable collection of finite words $(\omega^n)\_{n \in \mathbb{N}}$, called generators, such that $X$ is the closure of the set of sequences obtained by freely concatenating the generators.
In Lind and Marcus book *An introduction to symbolic dynamic... | https://mathoverflow.net/users/10518 | Coded Systems and dense subsets | In case the book referred to in Doug Lind's comment didn't have what you're looking for, a proof of this statement can be found in Section 2 of [this paper](http://www.ams.org/mathscinet-getitem?mr=1869067):
Doris Fiebig and Ulf-Rainer Fiebig, [Invariants for subshifts via nested sequences of shifts of finite type](h... | 2 | https://mathoverflow.net/users/5701 | 88578 | 52,434 |
https://mathoverflow.net/questions/88576 | 2 | What is the Hausdorff dimension of the subset
$$F := \{ x = \sum^\infty\_{n=1} \frac{2 x\_n}{3^n} \in [0,1] : x\_n \in \{ 0 , 1 \} , x\_n = 1 \Rightarrow x\_{n+1}=0 \}$$
of the Cantor set? Is it known already?
As far as I know, this set can be corresponded to a binary tree related to the Fibonacci sequence. (I don't ... | https://mathoverflow.net/users/21437 | Hausdorff dimension of a subset of Cantor set | The comments by Andreas and Anton give you the answer already to your specific question. Let me give a more general answer, since your question is very representative of a whole class of examples.
The condition that $x\_n = 1 \Rightarrow x\_{n+1} = 0$ is a *Markov* condition: the value of $x\_{n+1}$ is restricted by ... | 7 | https://mathoverflow.net/users/5701 | 88579 | 52,435 |
https://mathoverflow.net/questions/88582 | 11 | Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb R^M)=0$.
This is a very strong transition from having sets with a full measure being annihilated into nullity. Is it p... | https://mathoverflow.net/users/7206 | Can we change the Lebesgue measure by forcing? | The answer is no. If $\mathbb{R}^V$ is measurable in a forcing extension $V[G]$ having new reals, then the measure must be $0$. The point is that every new real $x$ in $V[G]$ but not in $V$ is transcendental over $\mathbb{R}^V$, since one cannot add algebraic numbers by forcing. It follows that the translates of $\math... | 13 | https://mathoverflow.net/users/1946 | 88583 | 52,437 |
https://mathoverflow.net/questions/88589 | 6 | A non-zero complex number can be uniquely written in polar form as $re^{i\theta}$. There is an analogous result for complex matrices: any invertible complex matrix can be uniquely written as $UP$, where $U$ is a unitary matrix and $P$ is a positive definite Hermitian matrix (see e.g. the [description on Wikipedia](http... | https://mathoverflow.net/users/8938 | Polar decomposition for quaternionic matrices? | A key wordphrase here is "[Cartan decomposition](http://en.wikipedia.org/wiki/Cartan_decomposition)". Since $G=SL\_n(\mathbb H)$ is a semisimple group, there is a diffeomorphism
$$K\times \mathfrak p\rightarrow G$$
taking $(k,X)$ to $k\cdot {\rm exp}(X)$, where $K$ is a maximal compact subgroup of $G$ (i.e. the compa... | 7 | https://mathoverflow.net/users/6753 | 88596 | 52,445 |
https://mathoverflow.net/questions/88529 | 3 | Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ *locally nilpotent* if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. Equivalently, for every finitely generated submodule $N \subseteq M$ there is some $n \in \mathbb{N}$ with $f^n |\_N = 0$. In pa... | https://mathoverflow.net/users/2841 | Infinite products of representations of the additive group | Question A: Demazure-Gabriel, Groupes Algébriques, II, §2, 2.6
Question B: It is easily checked that the maximal submodule of $\prod M\_i$ on which $\prod f\_i$ acts locally nilpotent is the categorical product. This may also be described as $\bigcup\_n \prod\_i \ker f\_i^n\subseteq\prod\_i M\_i$. Actually, since the... | 4 | https://mathoverflow.net/users/2035 | 88602 | 52,448 |
https://mathoverflow.net/questions/88555 | 6 | Here by stable homotopy category I mean the homotopy category of spectra, or maybe just some monogenic, Brown, algebraic, etc. stable homotopy category (in the language of Hovey, Palmieri and Strickland).
It is my understanding that it is still open as to whether or not a given localizing subcategory of the stable h... | https://mathoverflow.net/users/11546 | Coreflective Subcategories of the Stable Homotopy Category | I'm not sure if I constitute an expert or this constitutes a real answer but let me try.
If I understand correctly your first question is whether it is open that every localizing subcategory of an algebraic stable homotopy category $\mathcal{S}$ (so it should be triangulated, symmetric monoidal in a way compatible wi... | 6 | https://mathoverflow.net/users/310 | 88606 | 52,450 |
https://mathoverflow.net/questions/88608 | 3 | Let $R$ be a ring. Let $M$ be a left $R$-module.
Then: $M$ is not finitely generated <=> $M$ is the union of a set of proper submodules closed under binary sums. To recall why: (<=) If $M$ would be f.g., then chosen finitely many generators appear in certain members of the given set of proper submodules. Hence, this... | https://mathoverflow.net/users/9300 | Non-finitely-generated module is union of countable chain? | Counterexample: let $R$ be a valuation ring with value group $\Gamma=\mathbf Z^{\omega\_1}$ with the lexicographic ordering and $M$ the maximal ideal of $R$. Any proper submodule of $M$ is contained in an ideal of the form $\{r\in R\mid v(r)>\gamma\}$ for some $\gamma>0$, but every countable collection of positive elem... | 3 | https://mathoverflow.net/users/2035 | 88610 | 52,451 |
https://mathoverflow.net/questions/86683 | 4 | Let $\pi: V\to W$ be a resolution of singularities, let $E \subset V$ be the exceptional divisor, and let $F$ be a coherent sheaf such that $R^i\pi\_\*F=0$ for $i>0$.
>
> Can we conclude that $R^i\pi\_\*F(-E)=0$ for $i>0$?
>
>
>
The idea being to somehow use that$-E$ is nef on $E$.
EDIT:
the exceptional fib... | https://mathoverflow.net/users/1231 | Vanishing associated to a resolution of singularities | 1
-
The statement is still false with the most recently added conditions.
The problem is that $-E$ is **not** necessarily ($\pi$-)nef. I will discuss below what one can do to fix the statement so it will hold. First, here is an example:
Let $W$ be a two dimensional rational double point that's either a $D\_n$ or a... | 7 | https://mathoverflow.net/users/10076 | 88615 | 52,453 |
https://mathoverflow.net/questions/88612 | 1 | Hello,
I have the following series:
$$
\sum\_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0,
$$
where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum gives $t(e^t-1)$. When $a=1/2$, it gives $e^{t^2}t^2 (1+ Erf(t))$ where $Erf(\cdot)$ is the standard error function. I... | https://mathoverflow.net/users/36814 | About one series. Are there some related special functions? | It's a special case of the [Mittag-Leffler function](http://en.wikipedia.org/wiki/Mittag-Leffler_function).
| 9 | https://mathoverflow.net/users/4854 | 88616 | 52,454 |
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