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https://mathoverflow.net/questions/132191 | 17 | Let $m,n$ be nonnegative integers.
The sequence $\{a\_{m,n}\}$ satisfies the following three conditions.
1. For any $m$, $a\_{m,0}=a\_{m,1}=1$
2. For any $n$, $a\_{0,n}=1$
3. For any $m\ge0, n\ge1$, $a\_{m+1,n+1}a\_{m,n-1}=a\_{m+1,n-1}a\_{m,n+1}+a\_{m+1,n}a\_{m,n}$
Prove that $a\_{m,n}$ is an integer for any $m\g... | https://mathoverflow.net/users/34490 | Proving that every term of the sequence is an integer | This is the knight recurrence (Example 4.1) in Fomin and Zelevinsky's paper on the [Laurent phenomenon](http://arxiv.org/abs/math/0104241). They show more generally that if you replace the boundary entries with indeterminates, then all matrix entries are Laurent polynomials.
| 23 | https://mathoverflow.net/users/3106 | 132223 | 73,212 |
https://mathoverflow.net/questions/132244 | 31 | Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in this direction?
| https://mathoverflow.net/users/26250 | Can one recover a metric from geodesics? | The short answer is **no**, the geodesics do not determine the metric. For example, in the Cayley-Klein model of hyperbolic geometry the geodesics are straight lines. It is however rather rare for
two Riemannian metrics to have the same geodesics. In two dimensions these metrics were studied by Liouville (see Livre VI ... | 29 | https://mathoverflow.net/users/21123 | 132248 | 73,220 |
https://mathoverflow.net/questions/132247 | 1 | I have a question about some quotient groups of the lower central series of a free group.
When there's a free group $F = \langle x\_1,\cdots, x\_n, y\_1, \cdots, y\_m\rangle $,
let $A$ be the subgroup generated by elements of the forms $[x\_i, a]$ in $F\_{k+1}/F\_{k+2}$, with $1\leq i\leq n$ and $a\in F\_{k}/F\_{k+... | https://mathoverflow.net/users/15728 | quotient groups of the lower central series of a free group | I don't believe so.
I will associate my commutators on the left, so that $[a,b,c]=[[a,b],c]$; and use $[x,y]=x^{-1}y^{-1}xy$; I'm not sure it matters, but that's what I'm used to.
Say $k=2$; in $F/F\_{4}$, the Hall-Witt identity becomes
$$ [r,s,t][s,t,r][t,r,s]\equiv 1\pmod{F\_{4}}.$$
Now consider the case where... | 6 | https://mathoverflow.net/users/3959 | 132252 | 73,222 |
https://mathoverflow.net/questions/132251 | 8 | Let $A$ an abelian variety over a field $k$ and $A^{\*}$ the dual abelian variety.
How can we relate the étale cohomology of $A$ with etale cohomology of $A^{\*}$?
| https://mathoverflow.net/users/27398 | etale cohomology of an abelian variety and its dual | The Weil pairing induces a perfect pairing $T\_\ell A \times T\_\ell A^\vee \to \mathbf{Z}\_\ell(1)$, and $T\_\ell A = H^1\_{et}(\bar{A},\mathbf{Z}\_\ell)^\vee$ (here, $\bar{A}$ denotes $A \times\_k \bar{k}$). Further, $H^i\_{et}(\bar{A},\mathbf{Z}\_\ell) = \wedge^iH^1\_{et}(\bar{A},\mathbf{Z}\_\ell)$. One also has $A^... | 6 | https://mathoverflow.net/users/nan | 132256 | 73,225 |
https://mathoverflow.net/questions/132205 | 1 | Let $X\_1,X\_2,\ldots,X\_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X\_{i\_1}+X\_{i\_2}+\ldots+X\_{i\_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i\_1< i\_2 < \ldots < i\_k \leq n $.
Consider the following illustrative example. We are given constraints
$$
D=\{X\... | https://mathoverflow.net/users/21277 | Deducing Linear Inequalities | Consider the inequalities:
$X\_1+X\_2+X\_3 \geq 0$
$X\_3 + X\_4 + X\_5 \geq 0$
$X\_2+X\_4+X\_6 \geq 0$
$X\_1 + X\_5+X\_6 \geq 0$
By adding all and dividing by $2$, we can deduce $X\_1+X\_2+X\_3+X\_4+X\_5+X\_6\geq 0$.
But I don't think you can get it by this sort of logic:
These inequalities are very symme... | 2 | https://mathoverflow.net/users/18060 | 132267 | 73,229 |
https://mathoverflow.net/questions/132272 | 12 | This is a typical lazy mathematician question, so do not hesitate to close it and recommend me to do my homeworks...
Let $H$ be a subgroup of a finite group $G$, and let $Res\_H^G$ and $Ind\_H^G$ the restriction and induction functors between the categories of linear representations of $G$ and $H$, respectively. I'm... | https://mathoverflow.net/users/8320 | Wrong-way Frobenius reciprocity for finite groups representations | The point is that there are two ways to describe the restriction functor. The first way is as $\text{Hom}\_{\mathbb{C}[G]}(\mathbb{C}[G], -)$, thinking of $\mathbb{C}[G]$ as a $(\mathbb{C}[G], \mathbb{C}[H])$-bimodule, which by the tensor-hom adjunction means that it has left adjoint $\mathbb{C}[G] \otimes\_{\mathbb{C}... | 18 | https://mathoverflow.net/users/290 | 132273 | 73,231 |
https://mathoverflow.net/questions/132269 | 14 | Consider the following series:
$$S=\sum\_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
$$S=-\frac{5}{16}\Gamma\left(\frac{1}{4}\right)\ { \_3F\_2}\left(1,1,\frac{9}{4};2,2;-1\right).$$
I tried to find an expression of $S$ using ... | https://mathoverflow.net/users/34433 | An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$ | Use formula 16.5.2 from [DLMF](http://dlmf.nist.gov/16.5):
$$\_3F\_2\left(1,1,\frac94;2,2;-1\right)=\int\_0^1{\_2F\_1}\left(1,\frac94;2;-t\right)dt=\frac45\int\_0^1\frac{1-(1+t)^{-5/4}}tdt=\\\\\frac{2}{5}\left(8-4\sqrt[4]{8}+\pi-6\ln2+4 \ln
(1+\sqrt[4]{2})-4 \arctan\sqrt[4]{2}+2\ln(1+\sqrt2)\right).$$
Some elementary ... | 22 | https://mathoverflow.net/users/9550 | 132275 | 73,232 |
https://mathoverflow.net/questions/132233 | 6 | What is the (Phi,Gamma)-module of an elliptic curve over Z\_p, expressed by a direct construction ?
| https://mathoverflow.net/users/34503 | Examples of (Phi,Gamma)-modules | (By an elliptic curve over $\mathbb Z\_p$, I assume you mean an elliptic scheme over spec $\mathbb Z\_p$, or which amount to the same, an elliptic curve over $\mathbb Q\_p$ which has good reduction.) The Galois representation on $V\_p(E)$ is then crystalline, so its $(\phi,\Gamma)$-module $D$ will be triangulate -- in ... | 11 | https://mathoverflow.net/users/9317 | 132276 | 73,233 |
https://mathoverflow.net/questions/132281 | 3 | In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus\_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective iff $\operatorname{Ext}^1\_\mathbb{Z}(A,B)=0$.
Does this hold if $\mathbb{Z}$ is replaced with an arbitrary (unital) ring $R$? If not, w... | https://mathoverflow.net/users/36720 | Is there an $A$ such that $B$ injective iff 1st Ext functor vanishes? | Yes, such a module exists for each unital ring $R$: Take $A= \bigoplus\_I R/I$ where $I$ runs through the left ideals of $R$. This is because $B$ is injective iff $\text{Ext}^1\_R(R/I,B)=0$ for each $I$ (Weibel, Lemma 4.1.11). Also note that this is a straightforward generalization of your example $R=\mathbb{Z}$.
**... | 9 | https://mathoverflow.net/users/10194 | 132282 | 73,236 |
https://mathoverflow.net/questions/113867 | 4 | Hello,
I have the following question: Is it possible for a complete $L\_{\omega\_1,\omega}$ sentence $\phi$ to satisfy (a) the (unique) countable model of $\phi$ has $2^{\aleph\_0}$ many automorphisms and (b) there is a model $N$ of $\phi$, $N$ of size $\aleph\_\omega$ and $N$ has $\le\aleph\_\omega$ many automorhphism... | https://mathoverflow.net/users/13694 | $L_{\omega_1,\omega}$ sentence with many automorphism in $\aleph_0$ and few automorphism in $\aleph_\omega$ | It appears that we can already do this in first-order logic, without making any use of the infinitary language.
Namely, the unique countable endless dense linear order $\langle\mathbb{Q},\lt\rangle$ has continuum many order automorphisms, but Shelah has reportedly proved that for every uncountable cardinality $\lambd... | 2 | https://mathoverflow.net/users/1946 | 132285 | 73,237 |
https://mathoverflow.net/questions/132262 | 4 | Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube *covers* $P$ if $\operatorname{conv}(Q) \supseteq P$ where $\operatorname{conv}(Q)$ denotes the convex hull of $Q$. How can one compute a minimal set $Q$ that covers $P$? Trivial... | https://mathoverflow.net/users/34513 | convex polyhedron in the unit cube | This is not a direct answer, but a closely related problem is known to be NP-hard:
>
> Das, Goodrich.
> "On the Complexity of Approximating and Illuminating Three-Dimensional Convex Polyhedra."
> 1995. ([ACM link](http://dl.acm.org/citation.cfm?id=645930.672852); [PDF download](http://ranger.uta.edu/~gdas/website... | 6 | https://mathoverflow.net/users/6094 | 132292 | 73,239 |
https://mathoverflow.net/questions/132268 | 16 | Is there an axiomatic system where the deduction theorem does not hold?
| https://mathoverflow.net/users/34515 | Deduction theorem | Failures of the deduction theorem are one of the more mysterious topics in logic, in my experience. The motto is that *axioms are stronger than rules*.
Here is the simplest nontrivial example that I know. Start with propositional logic with two variables $A$ and $B$. Add the single new rule of inference $A \vdash B$... | 27 | https://mathoverflow.net/users/5442 | 132295 | 73,242 |
https://mathoverflow.net/questions/26978 | 12 | In Bill Goldman's paper "The Symplectic Nature of the Fundamental Groups of Surfaces" (Advances, 54, 200-225, '84) it is stated that the "Zariski tangent space" to a representation space Hom$(\pi, G)/G$ at a representation $\rho$ is the cohomology group $H^1 (\pi; Ad\ \rho)$. Here G is a Lie group, and Ad $\rho$ is the... | https://mathoverflow.net/users/4042 | Zariski tangent spaces to representation varieties | So I don't think I am going to answer your question here, and I think you probably know most of what I am going to write at this point. But perhaps this information will be of use to others that happen to pass by this thread.
Let $G$ be algebraic (including compact, and complex reductive) and denote $R(G)=\mathrm{Hom... | 10 | https://mathoverflow.net/users/12218 | 132301 | 73,246 |
https://mathoverflow.net/questions/132297 | 21 | Consider a country with $n$ families, each of which continues having children until they have a boy and then stop. In the end, there are $G$ girls and $B=n$ boys.
Douglas Zare's highly upvoted answer to [this question](https://mathoverflow.net/questions/17960/google-question-in-a-country-in-which-people-only-want-boy... | https://mathoverflow.net/users/10503 | Boys and Girls Revisited | Symmetry. Put them all together and tell them to multiply forever. The question then becomes whether the $n$-th boy was born at the $2n$-th birth or later, or not, i.e., who is the majority among the first $2n-1$ births: boys or girls.
| 31 | https://mathoverflow.net/users/1131 | 132303 | 73,247 |
https://mathoverflow.net/questions/132317 | 0 | Given a total function $f \colon A \to B$, by grouping the "points" on $A$ which have the same image on $B$, namely $a,b \in A$ located in the same equivalence class $G\_{i}$ iff $f(a) = f(b)$.
By that way, I get a partition $G = \{ G\_{1},G\_{2},\dotsc \}$ on $A$ derived from $f$, and concurrently I can represent $... | https://mathoverflow.net/users/12818 | Equivalence relation derived from function | Actually $G$ is the image of $f$. This can be stated categorically, see [the entry in the nLab](http://ncatlab.org/nlab/show/image), but it depends on the definition of “subobjects”.
If you are interested in the relation between $f$ and the equivalence relation you mentioned, have a look at universal algebra and its ... | 1 | https://mathoverflow.net/users/33842 | 132318 | 73,252 |
https://mathoverflow.net/questions/131004 | 4 | I have a very stupid question: I often see that the volume entropy of a compact Riemmannian manifold with negative curvature coincide with the Hausdorff dim of the limit set or Patterson sullivan measure on the boundary.
But the volume entropy is not invariant under scaling of the measure, and I assume the dimension ... | https://mathoverflow.net/users/21929 | Relation between volume entropy and Hausdorff dim of limit set? | For the geodesic flow on the unit tangent bundle of a compact manifold with nonpositive curvature, a classical result of Manning asserts that the topological entropy of the geodesic flow equals the volume entropy of the universal cover. This is true in variable (nonpositive) curvature.
For the geodesic flow on the u... | 5 | https://mathoverflow.net/users/30691 | 132319 | 73,253 |
https://mathoverflow.net/questions/132132 | 4 | I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, [give a construction [link]](http://arxiv.org/abs/math/0701916) of 2-colimits of *topological* groupoids, but I haven't seen any definitive statement on whether there is a const... | https://mathoverflow.net/users/34471 | Co-completeness of differential stacks? | The answer is no.
Claim: If $\mathfrak{DiffSt}$ were cocomplete, it would be reflective in $St\left(Mfd\right).$
Proof:
Let $i:Mfd \hookrightarrow \mathfrak{DiffSt}$ be the full and faithful inclusion of manifolds into differentiable stacks. Take the $(2,1)$-categorical left Kan extension of $i$ along the Yoneda ... | 4 | https://mathoverflow.net/users/4528 | 132329 | 73,258 |
https://mathoverflow.net/questions/130133 | 0 | Hi,
The PDE in question is: $A P\_{yy}(y,z) + B P\_{zz}(y,z) + ( [ C y -D z] P(y,z) )\_y + ( [ D y + C z ] P(y,z) )\_z=0,$
where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is the stationary bit of a Fokker-Planck equation derived from a generally complex Langevin equation, i.e. $x = y+iz$... | https://mathoverflow.net/users/33835 | I have this linear PDE... | If you are looking for solutions in the whole plane, take the Fourier transform. You get a first order PDE which you can then analyze by the method of characteristics.
| 1 | https://mathoverflow.net/users/12120 | 132336 | 73,260 |
https://mathoverflow.net/questions/132062 | 26 | Recently, when writing a review for MathSciNet, the following question arose:
Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true just because polynomials are holomorphic.
I was mainly interested in the affine case since that was the context of the... | https://mathoverflow.net/users/12218 | Biholomophic non-Algebraically Isomorphic Varieties | Question 1: The first example published seems to be the following (Corollary 4 in "Embeddings of Danielewski surfaces", G. Freudenburg and L. Moser-Jauslin, Math.Z. 2003 )
For any $a\in \mathbb{C}^\*$, the surfaces in $\mathbb{C}^3$ given by
$x^2z-y^2-a$ and $x^2z-(1+x)y^2-a$
are algebraically not isomorphic, but holo... | 11 | https://mathoverflow.net/users/23758 | 132337 | 73,261 |
https://mathoverflow.net/questions/132339 | 27 | This should be such an elementary problem in algebraic topology that I'm almost too embarrassed to ask, but here goes.
Let $f: X\to Z$ be a surjective fibration, and let $g: Y\to Z$ be any map. Assume all spaces are path-connected, and base points $x,y,z$ chosen so that $f(x)=g(y)=z$. Form the pullback in the topolog... | https://mathoverflow.net/users/8103 | Fundamental group of a topological pullback | There is a ``Mayer--Vietoris" sequence
$$\cdots \to \pi\_2(Z, z) \to \pi\_1(E, e) \to \pi\_1(X, x) \times \pi\_1(Y, y) \to \pi\_1(Z,z) \to \pi\_0(E) \to \cdots$$
that can be developed by fitting together the 4 long exact sequences of homotopy groups obtained from the 4 maps in your diagram.
With $X \simeq Y \simeq \*... | 31 | https://mathoverflow.net/users/318 | 132340 | 73,262 |
https://mathoverflow.net/questions/131932 | 3 | Let $L/\mathbf{Q}\_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}\_p/\mathbf{Q}\_p)$. Assume that $V$ is crystalline with distinct Hodge-Tate weights $k\_1 < k\_2 < \dotsb < k\_n$. We *do not* assume any condition ... | https://mathoverflow.net/users/17549 | Galois deformations with Panchiskin condition | Let $\pi\_1, \dots, \pi\_r$ be the irreducible representations appearing as a sub quotient in $V^-$. Note that none of these irreducible representation appear in $V/V^-$ because
these rep. have negative HT weights and $V/V^-$ non-negative HT weights.
Now consider the functor $F$ from the category of $L$-representatio... | 4 | https://mathoverflow.net/users/9317 | 132341 | 73,263 |
https://mathoverflow.net/questions/132335 | 2 | Helly's Theorem states the following. Suppose that $X\_1,X\_2,...,X\_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap\_{i \in I} X\_i \neq \varnothing$. Then $\bigcap\_{i=1}^N X\_i \neq \varnothing$.
If instead $X\_1,X\_2,...,X\_N$ are biconvex sets i... | https://mathoverflow.net/users/22627 | Helly's Theorem for Biconvex Sets | $\infty$.
Let $n=m=1$. We will form our sets with a braid-like construction. Each set will be a curve, and the curves will move back and forth in a fixed strip such that any $h+1$ of them intersect but all $N$ never do. We then stretch the strip out until the curves never move at an angle of more than $\pi/4$ from th... | 5 | https://mathoverflow.net/users/18060 | 132343 | 73,264 |
https://mathoverflow.net/questions/132305 | 2 | it seems that a module over a noetherian ring R is finitely generated if and only if it has finite length (sorry, it turns out to be **false**! i must have had a misunderstanding!)
but why in the following two cases, we have some extra assumptions?
**Example (i)** A module M over a (commutative) noetherian ring R h... | https://mathoverflow.net/users/34133 | finite length and finitely generated | A (long) comment:
the definition of an associated prime for $M$ you are mentioning is for the case of commutative rings. There are several problems with this definition, if $R$ is noncommutative. This is discussed in the following thesis: <http://math.fullerton.edu/sannin/Research/thesis2.pdf>.
The author uses the fo... | 2 | https://mathoverflow.net/users/32332 | 132346 | 73,267 |
https://mathoverflow.net/questions/132359 | 10 | Hi All,
Let $F$ be a finite field, $\lambda\in F$, and $$p\_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ^2z \end{array}\right|.$$
Can one choose $\lambda$ so that $(0,0,0)$ is the only zero in $F^3$ for $p\_\lambda (x,y,z)$?
Also, I'm wondering if ... | https://mathoverflow.net/users/34548 | zeros of a homogeneous polynomial | Let $t$ be a root of $t^3 + \lambda t + 1 = 0$. Suppose $\lambda$ can be chosen so this polynomial is irreducible. Then the cubic decomposes into three lines, one of which is given by $x = ty + z/t$ (and the other two are the Galois conjugates). Now it's easy to see this line has no $\mathbb{F}\_p$-rational points (els... | 8 | https://mathoverflow.net/users/2698 | 132373 | 73,278 |
https://mathoverflow.net/questions/132355 | 2 | 1. Let $X=(X\_{t},\,t \in T)$ be a non-homogeneous, continuous time Markov process with a finite state space $S=\{1,...,K\}$.
2. Let $\alpha\_{i,j}(t)$ be the hazard rates of some $\varGamma$-distributed random variables.
>
> **My question:** How can I simulate random paths of the Markov process with a transition i... | https://mathoverflow.net/users/34545 | How to simulate random paths of a non-homogeneous continuous-time Markov process with discrete state space for a given infinitesimal generator matrix? | Suppose you're in state $i$. For each $j$, let $X\_j$ be an independent Exponential random variable with mean 1.
Now solve
$\int\_{0}^{T\_j}\alpha\_{i,j}(t)\ dt=X\_j$ for each $i$. Whichever of the $T\_j$'s is smallest, you jump from state $i$ to state $j$ at time $T\_j$.
| 1 | https://mathoverflow.net/users/11054 | 132376 | 73,281 |
https://mathoverflow.net/questions/132372 | 3 | There is a paper by Gy\H{o}ry and Lemons "Hypergraphs with No Cycle of a Given Length", which bounds number of hyperedges for hypergraphs avoiding cycles of length $2k$ (unfortunately, I have access only to the abstract: <http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8514699>). Particularly, ... | https://mathoverflow.net/users/25905 | Uniform 4-hypergraph avoiding 2-cycles | No. You can't improve it to $o(n^2)$.
Let $\operatorname{ex}\_{C\_2}(n)$ be the largest possible number of edges of a $4$-uniform hypergraph on $n$ vertices that contains no cycle of length $2$. You can prove that there exists constant $c$ and $n'$ such that for all $n > n'$, $\operatorname{ex}\_{C\_2}(n) \geq cn^2$.... | 3 | https://mathoverflow.net/users/27829 | 132390 | 73,288 |
https://mathoverflow.net/questions/132326 | 13 | There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many unpublished results.
Does any one know some of them to state here.
Is there any way to get a copy of Woodin's unpu... | https://mathoverflow.net/users/11115 | Unpublished works of Woodin on SCH and Radin forcing | (Below, I refer to the Handbook. This is the **Handbook of Set Theory**, Foreman, Kanamori, eds., Springer, 2010.)
The bulk of these results appears in notes by [James Cummings](http://www.math.cmu.edu/~jcumming/). You may want to ask him about them. Originally these notes were intended for a book on Radin forcing to... | 12 | https://mathoverflow.net/users/6085 | 132391 | 73,289 |
https://mathoverflow.net/questions/132407 | 5 | **If $\kappa \rightarrow (\alpha)^r\_2$ holds for every $r\in \omega$, then is $\kappa$ an $\alpha$-Erdős cardinal? (or rather, does $\kappa \rightarrow (\alpha)^{<\omega}\_2$ hold?)**
$\kappa \rightarrow (\alpha)^r\_2$ means that $\forall f: [\kappa]^r \rightarrow 2$, $\exists A\in[\kappa]^\alpha$ such that $|f"[A]^... | https://mathoverflow.net/users/33625 | If $\kappa \rightarrow (\alpha)^r_2$ holds for every $r\in \omega$, then is $\kappa$ an $\alpha$-Erdős cardinal? | The two properties are not the same.
If $\kappa$ is weakly compact, then $\kappa\rightarrow(\kappa)^r\_\lambda$ for any $r<\omega$ and $\lambda<\kappa$. These cardinals are compatible with $V=L$.
A cardinal $\kappa$ for which $\kappa\rightarrow(\kappa)^{<\omega}\_2$ is called a Ramsey cardinal, and these cardinals ... | 8 | https://mathoverflow.net/users/18128 | 132412 | 73,293 |
https://mathoverflow.net/questions/132398 | 5 | Hello all,
The lemma 3.3 page 214 in Hartshorne Algebraic Geometry book states:" If $I$ is an injective module over a Noetherian ring $A$. Then for any $f\in A$, the natural map of $I$ to its localization $I\_f$ is surjective." I want to ask that is the lemma still true if $A$ is not Noetherian, are there any counter... | https://mathoverflow.net/users/34466 | Surjectivity of the natural map of injective module to its localization | A ring $A$ is said to have the ITI property with respect to an ideal $\mathfrak{a}$ if $\mathfrak{a}$-torsion submodules of injective $A$-modules are injective.
Noetherian rings have ITI with respect to every ideal, but a ring with ITI with respect to every ideal is not necessarily noetherian (e.g. absolutely flat ri... | 1 | https://mathoverflow.net/users/11025 | 132431 | 73,303 |
https://mathoverflow.net/questions/132385 | 3 | (Edited)
Both notions seem to measure "largeness" of an ideal in a ring. how are they related? does one imply the other and vice versa?
i am reading this book "Hereditary Noetherian Prime Rings and Idealizers" by Levy and Robson.
On Page 58, Lemma 13.6, it says:
(Here $R$ is a Hereditary Noetherian Prime (HNP)... | https://mathoverflow.net/users/34133 | Maximal ideals and Essential ideals | Here is a proof that the authors probably had in mind: if $M$ is not essential in $R$ then there is a non-zero right ideal $J$ such that $M \cap J = 0$. But then $J$ embeds into the simple module $R/M$ and so $J$ must be simple itself, and then $J$ is contained in the socle of $R$. However this socle is zero because $R... | 1 | https://mathoverflow.net/users/6827 | 132432 | 73,304 |
https://mathoverflow.net/questions/132410 | 10 | Let $G$ be a group with complementary subgroups $A$ and $B$ (meaning $A\cap B=1$ and $AB=G$).
If $A$ and $B$ are both normal in $G$, then $G\cong A\times B$ is a direct product. If $A$ is normal, then $G\cong A\rtimes\_{\varphi} B$ is a semi-direct product. More generally, $G\cong A\bowtie B$ is a *knit product*, or ... | https://mathoverflow.net/users/8103 | Cohomological dimension of knit products | First of all, there are examples for semi-direct products, but maybe this is not what you want.
1) Let $F\_m$ be the free group on $m$ generators. Given a surjection $\varphi: F\_m \rightarrow F\_n$, we obtain a short exact sequence
$$ 1 \rightarrow N \rightarrow F\_m \xrightarrow{\varphi} F\_n \rightarrow 1. $$
Here... | 7 | https://mathoverflow.net/users/31670 | 132437 | 73,308 |
https://mathoverflow.net/questions/131543 | 14 | $\DeclareMathOperator\Frob{Frob}$Iwaniec and Kowalski, in their famous book *Analytic Number Theory* states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's $L$-function and Artin's conjecture
(that the Artin $L$-function have no pole except maybe at $s=1$).... | https://mathoverflow.net/users/9317 | Effective Chebotarev without Artin's conjecture | Since no one answered my question, I have asked the author of the book. Emmanuel Kowalski
told me that this remark they make (namely that the form (1) or (2) they give of Chebotarev can be proved using GRH alone, without Artin) is mistaken. In the state of our knowledge,
Artin is necessary to get such a precise form.
... | 14 | https://mathoverflow.net/users/9317 | 132439 | 73,309 |
https://mathoverflow.net/questions/132111 | 2 | It is well known that given a short exact sequence $1\to H \to G \to G/H \to 1$ the transgression map
$$
H^{p-1}(G/H, H^1(H,A)) \to H^{p+1}(G/H,A^H)
$$
in the inflation-restriction sequence is in fact the cup product with the opposite of the class of the extension $1 \to H^{ab} \to G/H' \to G/H \to 1$ in $H^2(G/H, H^{a... | https://mathoverflow.net/users/2234 | transgression in terms of cup product in case of non-trivial action of the group on the coeffecients module | I can offer the following generalization for non-trivial coefficients $A$: Write $Q = G/H$. The cap product $$H\_1(H;\mathbb{Z}) \otimes H^1(H;A) \to \mathbb{Z}\otimes\_H A = A\_H$$ is $G$-linear with trivial $H$-action and induces a cup product
$$\cup: H^p(Q;H\_1(H;\mathbb{Z})) \otimes H^q(Q;H^1(H;A))\xrightarrow{} H... | 2 | https://mathoverflow.net/users/10194 | 132450 | 73,313 |
https://mathoverflow.net/questions/132324 | 3 | In ["Operator algebras with a faithful weakly-closed representation"](http://www.jstor.org/stable/1969954) (1955), Kadison describes a countable W\*-algebra as a C\*-algebra which has a faithful representation as a countably decomposable ring of operators. Unfortunately I was not able to find any other occurence of thi... | https://mathoverflow.net/users/33223 | References on countable W*-algebras | If I'm reading it correctly, "countably decomposable" means that there is no uncountable family of mutually orthogonal nonzero projections. Such von Neumann algebras are now called "$\sigma$-finite".
| 4 | https://mathoverflow.net/users/23141 | 132453 | 73,314 |
https://mathoverflow.net/questions/132445 | 7 | Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product
$AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are self-similar?
| https://mathoverflow.net/users/23542 | Arithmetic products of Cantor sets. | The results [my paper with M. Hochman](http://arxiv.org/abs/0910.1956) yield the following result on the products of self-similar sets:
---
Let $A=\bigcup\_{i=1}^m r\_i A + t\_i$, $B=\bigcup\_{i=1}^n s\_i B+u\_i$ be two self-similar sets on $\mathbb{R}$. Suppose that $\log r\_i/\log s\_j$ is irrational for some $... | 12 | https://mathoverflow.net/users/11009 | 132462 | 73,319 |
https://mathoverflow.net/questions/132442 | 10 | Suppose I have a metric $g\_0$ on the $\mathbb S^3$, and let $g\_t$ be the solution to Ricci flow (with surgery) with initial metric $g\_0$. What are some general results which give upper bounds on the extinction time of this flow?
Really, I want to flow for some (hopefully short) time $t$ so that all the pieces of t... | https://mathoverflow.net/users/35353 | How fast does Ricci flow converge on the three-sphere? | I'm not sure how to answer your question - I believe the Perelman or Colding-Minicozzi widths are the only known way to estimate extinction time in general.
At the time of extinction of a component, the sphere might not be round. Consider a dumbbell rotationally-symmetric metric on $S^3$ which [develops a neck singu... | 11 | https://mathoverflow.net/users/1345 | 132463 | 73,320 |
https://mathoverflow.net/questions/132446 | 15 | In Analysis Situs, Poincaré studies the following question:
which sequences of integers $b\_0,\ldots,b\_n$ are the Betti numbers of an orientable compact manifold of dimension $n$?.
He knows that necessary conditions are $b\_k=b\_{n-k}$ and if $n=4k+2$, $b\_{2k+1}$ is even.
Then he computes the homology of a produ... | https://mathoverflow.net/users/14547 | Realization problem for Betti numbers | For $M^{4k}$, the intersection form on middle-dimensional homology $H\_{2k}$ is non-degenerate and symmetric, so the signature equals $b\_{2k} \pmod 2$. Therefore in your case the signature theorem implies $b\_6$ must be even if $b\_4=0$.
See also [Torsion in cohomology of smooth manifolds](https://mathoverflow.net/q... | 4 | https://mathoverflow.net/users/34576 | 132464 | 73,321 |
https://mathoverflow.net/questions/132452 | 22 | I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.
Today I found a detail in the proof that seems outright wrong, and I am starting to worry that it is a serious problem (in particular since I promised to g... | https://mathoverflow.net/users/10811 | How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum | That also puzzled me first, but I think it is ok. By Cauchy-Schwarz,
$$ \mathcal E\_i \leq \left(\sum\_{ d < D^2, d | \mathcal P }\sum\_{ c \in \mathcal C\_i ( d ) } \tau\_3^2(d) \rho\_2^2(d) | \Delta( \theta,d,c) |\right)^{1/2}\left(\sum\_{ d < D^2 , d | \mathcal P } \sum\_{ c \in \mathcal C\_i(d) } | \Delta (\theta... | 21 | https://mathoverflow.net/users/11919 | 132466 | 73,322 |
https://mathoverflow.net/questions/132451 | 6 | Let $U,U'\subseteq\mathbf{C}$ be two connected open sets such that $\pi\_1(U)\simeq\pi\_1(U')$.
Q: Does this imply that $U$ is homeomorphic to $U'$?
In the case where the $\pi\_1$'s are trivial then the answer is yes, this is a consequence of
Riemann's mapping theorem. May be one should try to prove it when the $\... | https://mathoverflow.net/users/11765 | Is the homeomorphism class of a connected open set of C determined by its fundamental group? | The answer is indeed no as David Cohen has pointed out, and more generally the answer is determined by the complements of the sets U and U'.
The complete solution is effectively due to R.L. Moore (1925), the key fact being every nondegenerate monotone upper semicontinuous decomposition of the 2-sphere yields a 2-sphe... | 10 | https://mathoverflow.net/users/17029 | 132467 | 73,323 |
https://mathoverflow.net/questions/132475 | 18 | Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory provide us with a $\mathrm{K}$-theory space. Certain constructions also give us explicit deloopings, and if the ring is comm... | https://mathoverflow.net/users/23122 | Algebraic K-theory and Homotopy Sheaves | For the Zariski topology, one has cohomological descent if $R$ is regular. (This yields the Brown-Gersten spectral sequence.)
For the etale topology, still assuming $R$ regular, descent fails for $K$-theory but one has descent for the theory $K/(p^\nu)[\beta^{-1}]$ where $p^\nu$ is a prime power and $\beta$ is the B... | 17 | https://mathoverflow.net/users/10503 | 132477 | 73,328 |
https://mathoverflow.net/questions/132099 | 9 | Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition?
Even if we replace the existential quantifiers in the assertion of non-normality by explicit functions of the universally quantified variables, I don't see how an oracle in possession of those function... | https://mathoverflow.net/users/3621 | Normality of Chaitin's constant | Consider a program $P\_n$ which first unpacks $\Omega\_n$, the first $n$ digits of $\Omega$, then runs all finite programs tallying $2^{-k}$ each time a program of length $k$ halts. $P\_n$ continues until it gets within $2^{-n}$ of $\Omega\_n$, and then halts. If you run all finite programs forever, then you see all co... | 7 | https://mathoverflow.net/users/2954 | 132483 | 73,330 |
https://mathoverflow.net/questions/132484 | 1 | Suppose $X$ and $Y$ are two smooth surfaces (over the complex numbers), and $f: X \to Y$ is a finite flat morphism of degree two. Is it necessarily true that the locus where $f$ is not a smooth morphism (i.e. the ramification locus) always one dimensional (if not empty)?
| https://mathoverflow.net/users/32337 | Is the branching locus of the double cover of surfaces always one dimensional? | Yes. Otherwise there would be an isolated ramification point in $Y$. A link of that point is $S^3$, which does not have a nontrivial double cover. So upstairs, $X$ looks near that point like two ${\mathbb C}^2$s glued at a point, which isn't smooth.
| 5 | https://mathoverflow.net/users/391 | 132486 | 73,331 |
https://mathoverflow.net/questions/132444 | 9 | In $\mathbb R^3$ there are 3 natural foliations given by the lines parallel to each axis, which intersect transversally. Let $M^n$ a manifold with $n$ foliations by lines or circles that intersect transversally, is there a way to define a metric such that this fibrations are geodesics?
| https://mathoverflow.net/users/34575 | Geometry defined by foliation. | **I'm editing my original answer to add information about the $n=2$ and $n=3$ cases.**
The first question to answer is whether such a metric exists locally, and the answer to this is *yes* for $n=2$, *probably* (see below) for $n=3$, and *no* for $n>3$ (in general; the exact answer will depend on the foliations you s... | 10 | https://mathoverflow.net/users/13972 | 132488 | 73,333 |
https://mathoverflow.net/questions/62356 | 20 | Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^\* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle construction, the map $X \mapsto \mathrm T^\* X$ is not an endofunctor on the category of manifolds: whereas tangent vectors push f... | https://mathoverflow.net/users/78 | Functoriality of the cotangent bundle | I do not know a larger class of smooth mappings; and I considered this question intensively when co-writing the book "Natural operations in differential geometry, Springer-Verlag, 1993"[(pdf)](http://www.mat.univie.ac.at/~michor/kmsbookh.pdf).
See also 26.11 -- 26.16 in this book for a determination of all natural tr... | 4 | https://mathoverflow.net/users/26935 | 132501 | 73,338 |
https://mathoverflow.net/questions/132237 | 8 | does anyone know of a recreational maths venue which publishes, announces, or distributes mathematical results (at university level) which would be of interest to amateur mathematicians also ?
Thank you.
| https://mathoverflow.net/users/34133 | recreational maths venue for amateur mathematicians | Recreational Mathematics Colloquium mentioned in the comments above : <http://ludicum.org/ev/rm/13>
A mixed bag of (possibly) relevant journals, some of them well-known:
* [Convergence](http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=3334&bodyId=3703)
* [Forum Geometricorum](http://forumgeom.fau.... | 6 | https://mathoverflow.net/users/1223 | 132504 | 73,340 |
https://mathoverflow.net/questions/132471 | 4 | For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is:
---
There exist $C\_{H} > 0$ and $\delta \in (0,1)$ such that for any $B(x,r)$ in $\mathbb{R}$, if $u \geq 0$ satisfies $Hu \equiv 0$ in $B(x,r)$, then
... | https://mathoverflow.net/users/12968 | Elliptic Harnack inequality for 1D Schrodinger operator? | The constant $C\_H$ must depend on $x$ and $r$. Consider for example $u = e^{\frac{1}{2}|x|^2}$, which satisfies
$$\Delta u = (1+|x|^2)u.$$
Then $u$ has a minimum of $1$ at $0$, which clearly doesn't control the maximum of $u$ on $B\_{\delta r}$ times any constant independent of $r$.
Furthermore, $C\_H$ must depend o... | 4 | https://mathoverflow.net/users/16659 | 132507 | 73,343 |
https://mathoverflow.net/questions/132511 | 11 | Finite groups of Lie type include those obtained as rational points of a connected simple algebrraic group over a finite field $k = \mathbb{F}\_q$ of characteristic $p$: these are split or quasi-split. There are also several families obtained less directly from algebraic groups: the Suzuki groups $^2\!B\_2(q)$ and the ... | https://mathoverflow.net/users/4231 | Counting conjugacy classes in simple groups of Lie type | According to
>
> Fleischmann, Peter; Janiszczak, Ingo.
> [On the computation of conjugacy classes of Chevalley groups](http://link.springer.com/article/10.1007%2FBF01190331).
> *Appl. Alg. in Eng., Comm. and Comp.* 1996, 7(3), 221--234
>
>
>
the class number of ${\rm F}\_4(q)$ is
* $q^4 + 2q^3 + 6q^2 + 10q... | 8 | https://mathoverflow.net/users/28104 | 132519 | 73,347 |
https://mathoverflow.net/questions/132524 | 2 | I would like to know the sign of the following term in general. I tried approximation for $\log$ function and it had negative sign. Is there any $m\_0$ such that for all $n>m>m\_0$, the following function is positive or as I get it is always negative in its domain.
$$ f(m,n)=\frac{\log\log\log m}{\log\log\log n} \cen... | https://mathoverflow.net/users/11733 | determining sign of function containing logarithm. | This is my second response, after the clarifying remark from the asker (see below).
Let us assume that $m>e^e$ so that $\log\log m>0$. Then there is $m\_0>m$ such that $f(m,n)<0$ for any $n\in(m,m\_0)$. To see this claim, let $\epsilon>0$ be small, and consider $n$ such that
$$ \log n=(\log m)^{1+\epsilon}. $$ Then
... | 3 | https://mathoverflow.net/users/11919 | 132525 | 73,349 |
https://mathoverflow.net/questions/132521 | 1 | A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex and of class $C^2$ on the space of symmetric matrices, and uniformly elliptic at $u$, the solution is of class $C^{2,\al... | https://mathoverflow.net/users/12019 | Regularity of the right hand side (the source term) in Evans-Krylov theory | The equation $F(D^2u) = g$ for $g \in C^{\alpha}$ should have a $C^{2,\alpha}$ estimate by perturbation theory. See for instance Caffarelli-Cabre, Ch. 8. The idea is that the constant-coefficient equation $F(D^2w) = f(x\_0)$ has a $C^{2,\alpha}$ estimate (Evans-Krylov), so by using the quadratic approximation to $w$, A... | 2 | https://mathoverflow.net/users/16659 | 132530 | 73,351 |
https://mathoverflow.net/questions/132528 | 2 | Let $F$ be a local field of characteristic 0, and $f:F\rightarrow \mathbb{C}$ be an integrable function. Is the following formulation valid?
$
\int\_{F^\times}f(x^2) d^\times x=\int\_{F^{\times 2}}f(x) d^\times x
$
where $d^\times x$ is a chosen multiplicative measure on $F^\times$.
It seems if $F=\mathbb{R}$, th... | https://mathoverflow.net/users/32746 | A question on Haar measure on local field. | Since the subgroup of squares is open, the restriction of Haar measure from the full multiplicative group to the squares is a Haar measure on the subgroup of squares, and Haar measure is unique up to scalars.
| 2 | https://mathoverflow.net/users/15629 | 132535 | 73,353 |
https://mathoverflow.net/questions/132494 | 6 | This question regards counting the cusps of $Γ\_0 (N)$.
I’ve been trying for a while to understand the following argument to count the cusps of $Γ\_0(N)$ given in *A First Course in Modular Forms* by Diamond & Shurman, so here is the full argument for reference (you can find it in chapter 3.8. page 103):
>
> To c... | https://mathoverflow.net/users/26710 | How does this argument to count the cusps of $Γ_0 (N)$ work? | I don't have time to analyze the proof in Diamond-Shurman, but for what it is worth, here is my own treatment as I teach it to my students.
**Theorem.** The cusps $\frac{u\_1}{v\_1},\frac{u\_2}{v\_2}\in\mathbb{Q}\cup\{\infty\}$, given in lowest terms,
are equivalent under $\Gamma\_0(q)$ if and only if there exists $v... | 10 | https://mathoverflow.net/users/11919 | 132541 | 73,356 |
https://mathoverflow.net/questions/132527 | 5 | Let $X$ be a smooth projective algebraic variety over a field of characteristic zero and let $D$ be a simple normal crossing divisor on $D$. Put
$j: U \hookrightarrow X$
for the inclusion of the complement $U=X-D$ on $X$.
Are the functors $j\_\ast$ and $j\_!$ exact?
I think the answer is yes.
I would be ve... | https://mathoverflow.net/users/34593 | are these functors exact? | I assume you're asking about quasicoherent sheaves.
Since the inclusion of $U$ into $X$ is an open immersion, $j\_!$ is exact. You can find this in [Tag 03DJ](http://stacks.math.columbia.edu/tag/03DJ).
Since $j$ is an affine morphism, $j\_\*$ is exact (EGA 2 Corollary 5.2.2). $j$ is affine, because the affine prope... | 5 | https://mathoverflow.net/users/121 | 132545 | 73,359 |
https://mathoverflow.net/questions/132338 | 17 | I've come across a function from the set of integer partitions to the natural numbers which I don't recognise but which probably ought to be familiar; it arises in the homogeneous Garnir relations for graded Specht modules (see Kleshchev, Mathas & Ram: "Universal graded Specht modules", Proc. LMS 105). I hope someone w... | https://mathoverflow.net/users/6771 | A function from partitions to natural numbers - is it familiar? | Find a large enough $N$ that $i+j \leq N$ for all boxes $(i,j)$ of the partition. So your partition is contained in the staircase partition $(N,N-1,N-2, \ldots, 2,1)$. Turn your partition into a planar rooted forest on $N+1$ vertices using the bijection I'll list below. Your statistic is the number of ways to order the... | 10 | https://mathoverflow.net/users/297 | 132548 | 73,360 |
https://mathoverflow.net/questions/132538 | 3 | Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the measure $\mathbb P$ is the smallest closed set of full measure.
Is the support necessarily separable? If so, why? If not... | https://mathoverflow.net/users/238 | When is the support of a Radon measure separable? | Let $I$ be a set of cardinality larger than the continuum. Then the product topology $[0,1]^{I}$ is compact but not separable. Give the interval $[0,1]$ the Lebesgue measure, then give $[0,1]^{I}$ the product measure $\mu$. If $U$ is a non-empty open subset of $[0,1]^{I}$, then $U$ contains a basic open set $\prod\_{i\... | 4 | https://mathoverflow.net/users/22277 | 132550 | 73,362 |
https://mathoverflow.net/questions/132515 | 1 | A [*unital quantale*](http://ncatlab.org/nlab/show/quantale) is a suplattice with a compatible monoid structure. A quantale is called *idempotent* if it is idempotent as a monoid (every element is idempotent) (analogously for *commutativity*). Every suplattice can be regarded as a unital, commutative, idempotent suplat... | https://mathoverflow.net/users/33842 | Commutative, idempotent partially ordered monoids | The answer is yes. Here is a proof.
**Lemma 1**. Let $\leq$ be the suplattice order.
If $a\leq b$ the $c+a\leq c+b$.
Proof. $c+b=c+(a\vee b)=(c+a)\vee (c+b)$.
By the standard equivalence betwee join semilattices and idempotent commutative semigroups it suffices to prove:
**Lemma 2.** $a\leq b\iff a+b=b$.
... | 1 | https://mathoverflow.net/users/15934 | 132555 | 73,364 |
https://mathoverflow.net/questions/132553 | 2 | I am wondering about generalisations of the concept of equivalence relations to suplattices.
Here is my motivation: Given a set $X$. The powerset $\mathcal{P}(X)$ is a suplattice. For suplattices there is a tensor product, giving $\mathcal{P}(X)\otimes\mathcal{P}(X)=\mathcal{P}(X\times X)$. Now we can define the diag... | https://mathoverflow.net/users/33842 | Equivalence relations in suplattices | It seems like a more natural way to generalize equivalence relations from $P(X)$ to lattices is to look at partitions. One defines a partition of a lattice L(or Boolean algebra, frame, or whatever) to be a subset $p\subseteq L\setminus\{0\}$ such that $\bigvee p=1$ and where $a\wedge b=0$ whenever $a,b\in p,a\neq b$. T... | 1 | https://mathoverflow.net/users/22277 | 132557 | 73,366 |
https://mathoverflow.net/questions/98989 | 6 | [The Wikipedia page on vector space frames](http://en.wikipedia.org/wiki/Frame_of_a_vector_space) gives a construction to find a dual frame for a given frame. Specifically, given a set of vectors $\{ e\_k \}$ in a Hilbert space $\mathcal{H}$ such that for all $v\in\mathcal{H}$ and some $ A \leq B<\infty$ we have
$$A||v... | https://mathoverflow.net/users/20729 | Unicity of a vector space frame's dual frame | Consider the finite-dimensional case. Let $d$ be the dimension and $n$ the number of frame elements. Take $\Phi$ to denote the $d\times n$ matrix whose columns are the frame elements. Then $\Psi$ is a dual frame for $\Phi$ precisely when $\Phi\Psi^\*=I$. In particular, the $i$th row of $\Psi$ must have inner product $1... | 7 | https://mathoverflow.net/users/29873 | 132558 | 73,367 |
https://mathoverflow.net/questions/132544 | 4 | Peano is typically credited with giving the first abstract definition of a vector space (1888):
<http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_linear_spaces.html>
Apparently, Peano credits Möbius, Grassmann, and Hamilton for inspiring the idea. However, I recall reading somewhere that there was an earlier... | https://mathoverflow.net/users/33757 | Historical precursor for Peano's axioms of a linear space? | Before Peano (1888), a more limited notion of vector space over the reals was axiomatized by
Gaston Darboux (1875), in
*[Sur la composition des forces en statiques](http://archive.numdam.org/ARCHIVE/BSMA/BSMA_1875__9_/BSMA_1875__9__281_1/BSMA_1875__9__281_1.pdf)*.
This early history is discussed by Gregory Moore, [T... | 4 | https://mathoverflow.net/users/11260 | 132566 | 73,372 |
https://mathoverflow.net/questions/132564 | 2 | Consider the lens space $L\_{p,q}$, which we can describe using its standard heegaard splitting, i.e. define $L\_{p,q}$ as a quotient of two solid tori, identifying meridians on the boundary of one with $(p,q)$-curves on the other. Call $H\_1$ and $H\_2$ the two solid tori, and $T$ the torus with identified points. I a... | https://mathoverflow.net/users/47757 | A self-homeomorphism of $L_{p,q}$ is isotopic to one which preserves heegaard splitting | Suppose that $L = L(p,q)$ is a lens space. Let $T$ be the standard genus one Heegaard splitting. Then $T$ is unique up to isotopy. It follows that every self-homeomorphism of $L$ preserves $T$ up to isotopy.
The uniqueness result was first proved by Bonahon and Otal in their paper "Scindements de Heegaard des espace... | 5 | https://mathoverflow.net/users/1650 | 132569 | 73,374 |
https://mathoverflow.net/questions/132539 | 0 | The Hopf fibration $S^1\rightarrow S^3\rightarrow S^2$ gives a decomposition of $S^3$ into 2-tori and to circles, so that the tori are foliated by circles of slope 1. If you take the region between two of this tori and identified by an homeomorphism that preserves the foliation you get a foliation by circles on the 3 t... | https://mathoverflow.net/users/34575 | Fibration in the 3 torus. |
>
> If you take the region between two of
> this tori and identified by an
> homeomorphism that preserves the
> foliation...
>
>
>
There is not a single isotopy class of (doubly orientation-preserving) homeomorphism that does this. Instead, there is a $\mathbb{Z}$-family of homeomorphisms, generated by a Dehn... | 4 | https://mathoverflow.net/users/1650 | 132570 | 73,375 |
https://mathoverflow.net/questions/132532 | 5 | Recently, I stumbled upon an interesting statement regarding Quadratic Forms. It is quite well-known and, as I will describe briefly, equivalent to Goldbach's conjecture.
Let $p,q$ be odd primes and consider $Q(x,y) = x^2 - y^2$. A straightforward argument shows that there are only two non-negative integer solutions ... | https://mathoverflow.net/users/29971 | Goldbach's conjecture and Euler's idoneal numbers | There should be only finitely many such $n$. The reason is that there should be too many pairs of primes $(p\_i,q\_i)$ such that $p\_i=a\_i^2+b\_i^2 D\_1$ and $q\_i=c\_i ^2+d\_i ^2 D\_2$, and $n=p\_i+q\_i$ and if we have just two different such representations then we do not get the uniqueness that you want.
If a pri... | 4 | https://mathoverflow.net/users/10811 | 132571 | 73,376 |
https://mathoverflow.net/questions/132536 | 9 | I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight contact structures are less common. Also a tight contact structure gives a genus bound on an embedded closed surface and the... | https://mathoverflow.net/users/31475 | Tight vs. overtwisted contact structure | As mentioned in the previous answer, the first use of the overtwisted/tight dichotomy is most certainly Bennequin's Theorem that there are non-isomorphic contact structures on $\mathbb{R}^3$ and $\mathbb{S}^3$, a landmark result.
However, the relevance of this dichotomy goes now far beyond this. As you probably know... | 12 | https://mathoverflow.net/users/4961 | 132583 | 73,381 |
https://mathoverflow.net/questions/132561 | 5 | In their famous paper "On the differentiation of De Rham cohomology classes..." Katz and Oda construct the spectral sequence for de Rham cohomology for the situation of a smooth morphism
$\pi: X \rightarrow S$
of smooth $k$-schemes ($k$ a field), where $S$ is assumed affine.
There is a step which is not clear to ... | https://mathoverflow.net/users/18183 | Construction of the spectral sequence of Katz/Oda | Let me try to say the same thing as Will but in a different way. Let $f \colon A \to B$ be left exact between abelian categories and assume there are enough injectives. Then one can distinguish between *derived functors*
$$\newcommand{\R}{\mathrm R} \R^i f \colon A \to B,$$
as well as *hyper-derived functors*
$$\newc... | 7 | https://mathoverflow.net/users/1310 | 132584 | 73,382 |
https://mathoverflow.net/questions/132580 | 5 | Let $S$ be a finite nonabelian simple group such that the exact sequence
$$1 \to S \to {\rm Aut}(S) \to {\rm Out}(S) \to 1$$
is nonsplit, where $S$ is identified with ${\rm Inn}(S)$. Then there is always a (not necessarily unique) minimal subgroup $A$ in ${\rm Out}(S)$ with respect to the condition that
$$1 \to... | https://mathoverflow.net/users/12961 | Finite simple groups: smallest nonsplit automorphic extensions | It is not always true that $|A|=2$. The outer automorphism of $A\_6 \cong {\rm PSL}\_2(9)$ induced in $M\_{10}$ is the product of a field automorphism and a diagonal automorphism of ${\rm PSL}\_2(9)$ and I believe that it is true in general that the product of a field and a diagonal automorphism of the same order of a ... | 4 | https://mathoverflow.net/users/35840 | 132596 | 73,387 |
https://mathoverflow.net/questions/132595 | 10 | Given a p-divisible group over $\mathbb{F}\_p$, Grothendieck-Messing theory tells us that deforming the group to $\mathbb{Z}\_p$ is the same as finding an admissible filtration of the Dieudonne-module evaluated on $\mathbb{Z}\_p$. (I think the way this is done is that given the group on $\mathbb{F}\_p$, we pick some li... | https://mathoverflow.net/users/34612 | Deformations of p-divisible groups | The theory can be summarized as follows: First, to every $p$-divisible group $G$ over a scheme $S$ in characteristic $p$, you can attach a Dieudonne crystal $\mathbb{D}(G)$ over $S$. What this means is that, for any $S$-scheme $U$, and any divided power thickening of $U$--that is, a closed immersion of $\mathbb{Z}\_p$-... | 13 | https://mathoverflow.net/users/7868 | 132601 | 73,390 |
https://mathoverflow.net/questions/132605 | 6 | Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.
(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?
I suspect that this may be pos... | https://mathoverflow.net/users/2481 | Congruences between CM and non-CM modular forms | $\newcommand\rhobar{\overline{\rho}}$
$\newcommand\Q{\mathbf{Q}}$
$\newcommand\GL{\mathrm{GL}}$
$\newcommand\F{\mathbf{F}}$
$\newcommand\Frob{\mathrm{Frob}}$
If $\rhobar: G\_{\Q} \rightarrow \GL\_2(\F) = \GL(V)$ is a (projectively) dihedral representation, then congruences between $\overline{\rho}$ and non-projective... | 6 | https://mathoverflow.net/users/34616 | 132611 | 73,393 |
https://mathoverflow.net/questions/132603 | 4 | Sorry for the crossing-posting: original post is [here](https://math.stackexchange.com/questions/409645/algorithms-for-covering-a-rectilinear-polygon-using-rectangles-of-the-same-size)
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing... | https://mathoverflow.net/users/34611 | Algorithms for covering a rectilinear polygon using the same multiple rectangles | Suppose the rectangle is $a \times b$. If all rectangles in the cover must be oriented identically,
say, with the $a$-side horizontal, then by scaling the original polygon, the problem becomes covering
by squares of a given size. This problem, is, I believe, NP-complete for polygons with holes,
but *might* be polynomia... | 1 | https://mathoverflow.net/users/6094 | 132615 | 73,395 |
https://mathoverflow.net/questions/132551 | 2 | I am reading notes of Michel Brion on spherical varieties.
Consider a reductive group $G$, a Borel $B$ in $G$, a finite dimensional $G$-module $M$ and a closed orbit $Y$ of $G$ in $\mathbb{P}(M)$. The stabilisator of every point of $Y$ is a parabolic subgroup of $G$, so we can fix a $y \in Y$ such that $By$ is open i... | https://mathoverflow.net/users/34596 | Why are these parabolic subgroups opposed? | Let's start with $x \in Y$ a $B$-fixed vector. Then it's a high weight line in $M$, so the linear span $N \leq M$ of $Y$ is an irreducible subrepresentation of $M$.
Do you mind if I assume we're in characteristic $0$, as Brion does? Then we can split $M = N \oplus N'$, and $M^\* = N^\* \oplus N'^\*$.`
(Which is to ... | 5 | https://mathoverflow.net/users/391 | 132616 | 73,396 |
https://mathoverflow.net/questions/132622 | 2 | Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with
$$\mathcal D=\left< \bigcup\_{d\in P} \mathcal D\_d\right> \tag1 $$
where the angles show the filter generated by the subbace $\bigcup\_{d\in P} \mat... | https://mathoverflow.net/users/nan | Uniformities generated by metrics. | Here is a result that completely characterizes the uniformities that are generated by metrics as opposed to pseudometrics.
$\mathbf{Theorem}$ Let $(X,\mathcal{U})$ be a uniform space. Then the following are equivalent.
1. There are $R\_{n}\in\mathcal{U}$ for all natural numbers $n$ such that $\bigcap\_{n}R\_{n}:=1\... | 3 | https://mathoverflow.net/users/22277 | 132627 | 73,400 |
https://mathoverflow.net/questions/132629 | 0 | Let $E$ be an elliptic curve. I want to consider its degeneration to the union of two projective lines $C:=\mathbb{P}^1 \cup\_{x,y} \mathbb{P}^1$ attaching at two points $x,y$. The involution $-1$ on $E$ can "degenerates" to an involution on $C$ that preserve each component $\mathbb{P}^1$ and swaps $x$ and $y$.
My q... | https://mathoverflow.net/users/34620 | A question on degeneration of elliptic curves with actions. | Think of the group $E$ as degenerating to the group $C\_{reg} \cong {\mathbb G}\_m \times {\mathbb Z}/2$. In that group, the $2$-torsion is ${\mathbb Z}/2 \times {\mathbb Z}/2$, where one generator rotates each component of $C$ by $-1$, and the other one switches the two components.
EDIT: I'm very disturbed by the fa... | 1 | https://mathoverflow.net/users/391 | 132637 | 73,403 |
https://mathoverflow.net/questions/132625 | 9 | First-order order classical logic with standard semantics has a proof theory: it is complete, sound and effective.
In higher order logic with standard semantics one cannot obtain a proof theory - it cannot be simulateously complete, sound and effective.
Now, the internal language of a topos is higher order typed i... | https://mathoverflow.net/users/22002 | Is there a nice characterisation of topoi with nice meta-logical properties? | The undesirable properties of higher-order logic are created by an insufficient notion of model. That is, we cannot have all three, soundness, completeness and effectivness (decidability of proof checking), if we insist that formulas be interpreted in the "standard" set-theoretic way. Henkin semantics does not suffer f... | 19 | https://mathoverflow.net/users/1176 | 132639 | 73,404 |
https://mathoverflow.net/questions/129706 | 19 | Got numerical support that for odd $n$, $\zeta(n)$ might be
expressed in terms of the derivatives of $\zeta(\frac12)$.
Based on *More Zeta Functions for the Riemann Zeros*, Andre Voros, p.12, Table 3:
Conjecture: For odd $n$,
$$\zeta(n) = \left(\frac{2}{(n-1)!} (\log(|\zeta|)^{(n)} (\frac12) - 2^n \beta(n))\right)... | https://mathoverflow.net/users/12481 | zeta(3) in terms of derivatives of zeta at 1/2 and pi | In fact for odd $n\ge3$ we have
$$\Bigl.\frac{d^{n}}{ds^n}\log\zeta(s)\Bigr|\_{s=\frac12}=
\frac{(n-1)!}{2}\Bigl(2^n L(n,\chi)+(2^n-1)\zeta(n)\Bigr)$$
The proof (due to Voros) is the following:
It is well known that
$$\frac{\zeta'(s)}{\zeta(s)}=-\frac{1}{s-1}+\sum\_\rho\Bigl(\frac{1}{s-\rho}-\frac{1}{\rho}\Bigr)
... | 9 | https://mathoverflow.net/users/7402 | 132658 | 73,410 |
https://mathoverflow.net/questions/132657 | 2 | Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).
Let $\eta$ be the generic point of $S$, $K = S(\eta)$ and $ 0\_\eta \in X(K)$ denote the zero element of the generic fiber. Suppose $\alpha$ is a section of $\pi$ and ... | https://mathoverflow.net/users/34631 | Specialization of sections in an elliptic fibration | Your question is local, so we may take $S$ to be the ring of integers in a local field $K$. If the point $P\in E(K)$ is torsion or not does not matter in my answer here. I think here of an ellpitic curve over a number field.
I think the best theoretical tool to study fibres of type $I\_n$ is by using Tate's uniformis... | 2 | https://mathoverflow.net/users/5015 | 132660 | 73,412 |
https://mathoverflow.net/questions/132648 | 31 | This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
>
> **Conjecture:** If $A\subset \mathbb{N}$ and $$\sum\_{a\in A} \frac{1}{a}$$ diverges, then $A$ contains arbitrarily long arithmetic progressions.
>
>
>
T... | https://mathoverflow.net/users/12176 | The Erdős-Turán conjecture or the Erdős' conjecture? | One of the best places to track these things down is **The mathematical coloring book**, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about.
As far as I can tell, the question is sometimes ... | 23 | https://mathoverflow.net/users/6085 | 132677 | 73,421 |
https://mathoverflow.net/questions/112348 | 1 | How can I show that for a given volume, a convex orthogonal polyhedron will have a smaller surface area than a nonconvex orthogonal polyhedron?
If this is not possible to show, can it be shown that a cube will have a smaller surface area than any other orthogonal polyhedron, convex or nonconvex, of the same volume? H... | https://mathoverflow.net/users/28085 | Surface area of convex vs. non-convex polyhedra with same volume | Both questions have affirmative answer.
Let $S⊂\mathbb R^3$, and $S\_1,S\_2,S\_3⊂\mathbb R^2$ the projections planes $xy$, $yz$ and $xz$ of $S$ it is known that
$$V(S)^2≤A(S\_1)A(S\_2)A(S\_3)$$
This also follows from the first inequality in this text
<http://www.ma.huji.ac.il/~ehudf/docs/KKLBKKKL.pdf> using the ind... | 1 | https://mathoverflow.net/users/34575 | 132680 | 73,422 |
https://mathoverflow.net/questions/132635 | 3 | Set $X=\mathbb{R}^n$ and let $X^{I}$, the space of maps from the (bounded or unbounded) interval $I$ to $X$, be endowed with the locally convex topology of pointwise convergence.
**Is it true that the subspace $C(I, X)$ of continuous maps is *not* in the Borel algebra of $X^I$?**
It is clear from the proof of the ... | https://mathoverflow.net/users/16702 | Measurability of subspace of set of all functions | For $(x\_j)\_{j\in\Bbb N}\subset I$ and $B\in\mathcal B((\Bbb R^n)^{\Bbb N})$, define $S((x\_j)\_{j\in\Bbb N},B):=\{f\in X^I,(f(x\_j))\_{j\in\Bbb N}\in B\}$. The collection of sets of this form contains the $\sigma$-algebra generated by the semi-norms $f\mapsto |f(x)|$, $x\in I$, which is the same as the $\sigma$-algeb... | 2 | https://mathoverflow.net/users/17118 | 132684 | 73,424 |
https://mathoverflow.net/questions/132675 | 14 | I am interested in the following:
Let $G$ be a finite group of order $n$. Is there an explicit function $f$ such that
$|s(G)| \leq f(n)$ for all $G$ and for all natural numbers $n$, where $s(G)$ denotes
the set of subgroups of $G$?
| https://mathoverflow.net/users/nan | General bound for the number of subgroups of a finite group | A non-identity subgroup of a group of order $n$ can be generated by $\log\_{2}(n)$ or fewer elements. There is quite a lot of duplication, but if you count the number of subsets of $G$ of cardinality at most $\log\_{2}(n),$ you will have an upper bound (which could be made more precise with more care) and the case of e... | 23 | https://mathoverflow.net/users/14450 | 132702 | 73,433 |
https://mathoverflow.net/questions/132692 | 1 | Let V be a finitely generated vector space with dimension(V) = $n \in \mathbb{N}>1$. Now let T: $ V \to V$ be a map such that $\forall \hat{v},\hat{w} \in V$, $\; T(\hat{v}+\hat{w}) \neq T(\hat{v})+T(\hat{w})$.So if $\chi(V,V)$ be the collection of mappings with the property that T has, then what I'm looking for is a s... | https://mathoverflow.net/users/31611 | Nonlinear Operators(with the group property?) | Fix $a\in V, a\ne 0$ and for $L\in {\rm Hom}(V,V)$ denote $L\_a:x\to L(x)+a$. Let $G$ to be the set of all $L\_a (L\in {\rm Hom}(V,V)$ and $$\lambda (L\_a,M\_a)(x) =L\_a(x)+M\_a(x)-a=(L+M)(x)+a.$$
| 0 | https://mathoverflow.net/users/18814 | 132703 | 73,434 |
https://mathoverflow.net/questions/132489 | 7 | [EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that *I do not know what the right definition of a fractal in the following question should be*. But a nice answer might contain such a definition. Moreover, a nice comment by Joël correctly points out that choosing which theory of rigid analytic spaces ... | https://mathoverflow.net/users/18238 | Is Gouvêa-Mazur's "Infinite Fern" a fractal? | I post this as an auto-answer mainly not to leave the question open.
After googling a bit better, I discovered two recent works by M. Lapidus and L. Hung (both available on [Lapidus' webpage](http://math.ucr.edu/~lapidus/#FFIRG) )
* “Nonarchimedean Cantor Set and
String”, Journal of Fixed Point
Theory and Applicat... | 1 | https://mathoverflow.net/users/18238 | 132712 | 73,436 |
https://mathoverflow.net/questions/132708 | 0 | Problem Formulation
-------------------
under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$.
Related Art
-----------
This occurs to me when $\mathbf{A}$ was a Positive semidefinite matrix,... | https://mathoverflow.net/users/11273 | solving trace norm equality | Here is a result, which I think is interesting in this context (for reference see books on Lie algebras):
Theorem: Let $V$ be a finite-dimensional vector space over a field $K$ of characteristic zero. Let $E\subseteq F$ be
two linear subspaces of $End(V)$ and $M=\lbrace A \in End(V) \mid [A,F] \subseteq E\rbrace $.
A... | 1 | https://mathoverflow.net/users/32332 | 132719 | 73,438 |
https://mathoverflow.net/questions/132284 | 9 | I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles, so one either has to use the dynamics of the map, find some meaningful approximations, or try to generalize the theorems of s... | https://mathoverflow.net/users/34518 | C*-algebras, foliations and dynamical systems | A very good place to start is Connes' book "Noncommutative Geometry", available for free on his website. It's a huge book, but it's possible to skip around quite a bit to get what you need.
To begin, I'll remark that the foliations which are accessible to C\*-algebraic techniques are generally smooth in the horizonta... | 10 | https://mathoverflow.net/users/4362 | 132725 | 73,441 |
https://mathoverflow.net/questions/132723 | 9 | Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide a lot of constructive examples of compact hyperbolic manifolds, it is not that obvious that in higher dimensions, such ... | https://mathoverflow.net/users/25511 | Fundamental group of an hyperbolic $4$-manifold | For general $p$, the only known method is to construct a Dirichlet fundamental domain and read off the group presentation from it. The procedure for computation of a fundamental domain is called "Jorgenesen's algorithm: List elements of $\Gamma$ (using the embedding to $GL(2(n+1),Z)$ via restriction of scalars). For ea... | 14 | https://mathoverflow.net/users/21684 | 132726 | 73,442 |
https://mathoverflow.net/questions/132715 | 3 | Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees.
>
> **Question:** If $\mathfrak{g}$ is not free, must there exist linearly independent elements $a,b\in\mathfrak{g}$ such that $[a,b]=0$?
>
>
>
| https://mathoverflow.net/users/8103 | Zero-divisors in a graded Lie algebra | Consider the complex Lie algebra $\mathfrak{h}=\mathbb{C}[t,t^{-1}]\partial\_t$ with Lie bracket $$[t^m\partial\_t,t^n\partial\_t]=(n-m)t^{n+m-1}\partial\_t$$ ie the natural bracket that arises considering the natural action of $\mathfrak{h}$ on $\mathbb{C}[t,t^{-1}]$ by derivations. This is known as the [Witt algebra]... | 4 | https://mathoverflow.net/users/345 | 132727 | 73,443 |
https://mathoverflow.net/questions/132633 | 3 | I am working on a paper of R.P Langlands called "Representations of abelian algebraic groups", available here: <http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/AbelianAlg-ps.pdf>
Now on page 13-14 he makes the following claim, which I have summarized.
Let $G$ be a group, $L$ a $G$-module and let $G$ act ... | https://mathoverflow.net/users/15566 | Non-degeneracy of cup products on Tate-cohomology groups | This is explained completely in Ken Brown's **Cohomology of Groups** (always check here!), with a section devoted to it: *A Duality Theorem*, Section VI.7.
Ultimately, we have ordinary composition products from homological algebra, and using the evaluation map $Hom(M,\mathbb{Q}/\mathbb{Z})\otimes M\to \mathbb{Q}/\mat... | 3 | https://mathoverflow.net/users/12310 | 132735 | 73,445 |
https://mathoverflow.net/questions/132730 | 5 | Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear or nonlinear elliptic system. Further more, may I say that we can parallelly extened the $L^2$-theory of the second-order... | https://mathoverflow.net/users/32140 | Please recommend some literature on the systematical theory of the elliptic systems! | All you mentioned can be extended to elliptic systems. The only things you should be careful about are the maximum principles and DeGiorgi-Nash-Moser type regularity results. This means for linear systems the theory is very satisfactory. You can start with *strongly elliptic* systems and the variational approach to the... | 3 | https://mathoverflow.net/users/824 | 132736 | 73,446 |
https://mathoverflow.net/questions/132498 | 6 | Let $A \in M\_n(\mathbb Z)$ and $\|A\| = \max |a\_{ij}|$.
Denote $$ S(r) = \sum\_{\substack{\|A\| \leq r \\\ \det{A} \neq 0}} \dfrac{1}{|\det{A}|} $$
- the sum over **all** matrices $A \in M\_n(\mathbb Z)$ with $\|A\| \leq r$ and $\det{A} \neq 0$.
I'm interested about the asymptotic behavior of the function $S(r)$:
... | https://mathoverflow.net/users/30005 | Sums of inverse determinants over matrices | RV above links to a paper of [Duke, Rudnick and Sarnak](http://www.ams.org/mathscinet-getitem?mr=1230289) where they establish the following (see Example 1.6 and Theorem 1.10):
The number of $n \times n$ matrices with $||A|| \leq r$ and $\det A = k$ is
$$c\_n r^{n^2-n} \sum\_{d\_1 d\_2 \cdots d\_n=k} \frac{1}{d\_2 d\... | 1 | https://mathoverflow.net/users/297 | 132740 | 73,448 |
https://mathoverflow.net/questions/132699 | 2 | Suppose we have a Brownian motion(or transition density) on a Lie group G and a Riemmanian manifold H on which G acts transitively and isometrically. Can we construct a Brownian motion( or transition density) on H?
Any help will be useful.
| https://mathoverflow.net/users/32796 | Brownian motion on Homogeneous spaces | This question is not as trivial as some people think.
One point is that a quotient of a Markov chain is not Markov in general; there is a certain condition that has to be satisfied. More precisely, if $\pi\_x$ are the transition probabilities of a Markov chain on a space $X$, and $X\to \overline X,\; x\mapsto \overl... | 1 | https://mathoverflow.net/users/8588 | 132742 | 73,450 |
https://mathoverflow.net/questions/132737 | 1 | Sorry for the vague title. This question is about the Albanese map from the variety $M$ of canonically polarized varieties to the set of abelian varieties. (The variety $M$ is not of finite type...)
Let $A$ be an abelian variety of dimension $g$. For which integers $N$ does there exist an $N$-dimensional subvariety $... | https://mathoverflow.net/users/34658 | Can the Albanese map be anything? | Smooth hypersurfaces of fixed degree $d$ in $\mathbb P^n$ ($n\ge 3$) are simply connected, so they have trivial Albanese, and are of general type for $d>n+1$. The Hilbert polynomial is determined by $d$.
To get families of polarized varieties with fixed Albanese, just take products $X\times Y$, where $X$ varies in a... | 8 | https://mathoverflow.net/users/10610 | 132748 | 73,453 |
https://mathoverflow.net/questions/132746 | 2 | A link is called **quasipositive** if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form $\sigma\_{i,j} = (\sigma\_i \cdots \sigma\_{j-2})\sigma\_{j-1} ( \sigma\_i \cdots \sigma\_{j-2})^{-1}$ then we call... | https://mathoverflow.net/users/14006 | Is every quasipositive knot strongly quasipositive? | As pointed out by [Hedden](http://arxiv.org/abs/math/0509499), [Livingston](http://arxiv.org/abs/math/0311036) showed that strongly quasipositive knots have $g(K) = g\_4(K) = \tau(K)$, where $g\_4$ is the smooth slice genus and $\tau$ is the Ozsváth-Szabó concordance invariant, so this is an easy way to show that some ... | 11 | https://mathoverflow.net/users/428 | 132749 | 73,454 |
https://mathoverflow.net/questions/131744 | 4 | It is often used implicitly that the maps which associate to metrics curvature quantities (Riemann, Ricci, scalar curvature) and Differential operators like the Laplacian are analytic maps between certain $C^{k,\alpha}$ spaces.
This seems quite clear to me from local formulas. Even so, I would like to ask if there is... | https://mathoverflow.net/users/20823 | Analytic dependence on the metric | I find it strange that Besse does not discuss this! Here's my understanding of the issue:
Sun and Wang care about analyticity because they want to apply the Łojasiewicz--Simon inequality. This inequality was discovered in the finite dimensional setting by Łojasiewicz. Leon Simon later figured out how to use it in an ... | 4 | https://mathoverflow.net/users/1540 | 132754 | 73,456 |
https://mathoverflow.net/questions/132716 | 6 | I want to determine some structures of matrices that can be transformed into a symmetric matrices using similarity transformation, i.e.,
$B=T^{-1}AT$
where $T$ is the similarity transformation matrix.
Here, $A$ is a non-negative matrix and the diagonal elements of $A$ is supposed to be the same $(diag(A)=[a, a, \l... | https://mathoverflow.net/users/34651 | similarity transformation into symmetric matrices | We can say something on such matrices $B$ by characterizing its eigenvalues, which coincide with the eigenvalues of $A$.
Since $A$ is a real symmetric matrix, it has real eigenvalues. Hence a necessary condition on $B$ is that it has real eigenvalues. But one can say more. There are necessary and sufficient conditions... | 3 | https://mathoverflow.net/users/32332 | 132758 | 73,457 |
https://mathoverflow.net/questions/132756 | 3 | In a recent preprint [1302.1929](http://arxiv.org/abs/1302.1929) my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by people on MO, and playing around by ourselves.
Because the paper is probably going to be... | https://mathoverflow.net/users/763 | reference request for character theory of p-extraspecial groups | I'm sure you'll correct me quickly if I'm off the mark, but isn't this proven in the appendix of [Threads Through Group Theory](http://www-stat.stanford.edu/~cgates/PERSI/papers/threads_Diaconis.pdf)? (article by Persi Diaconis, appendix by Marty Isaacs, published in "Character Theory of Finite Groups".)
Quite apart ... | 1 | https://mathoverflow.net/users/11260 | 132766 | 73,460 |
https://mathoverflow.net/questions/132775 | 4 | For graphs $G$ and $H$, let $h(G,H)$ denote the number of graph homomorphisms from $G$ to $H$.
Fix some enumeration $G\_1,G\_2,\ldots$ of (isomorphism classes of) the set $\mathbf{D}$ of finite graphs, which contains only one representative from each isomorphism class.
The *right $\mathbf{D}$-profile* of a graph $G$ is... | https://mathoverflow.net/users/7252 | Do right-profiles determine graphs up to isomorphism? | It is true. It's Exercise 11 on page 128 of "Algebraic Graph Theory", by Royle and somebody.
If I recall correctly (and I might not) the trick is to show that the number of homomorphisms determines the number of injective homomorphisms. [EDIT: as Andras notes, I should have said surjections.]
The result is quite old,... | 5 | https://mathoverflow.net/users/1266 | 132780 | 73,468 |
https://mathoverflow.net/questions/132778 | 9 | Hi Mathoverflow. This question is about building intuition for the Proj construction. When I first started learning about schemes, I found the construction of the structure sheaves on Spec and Proj very confusing. However, after enough time had passed I began to understand the construction for Spec: It is sort of an al... | https://mathoverflow.net/users/4002 | Intuition for coordinate patches on Proj of a graded ring. | You in fact already found out the most natural way to think about the Proj construction, namely in terms of the projective space. Consider $X := \mathbb{P}^n$ with homogeneous coordinates $[x\_0: \cdots : x\_n]$. Then what are the basic open sets in $\mathbb{P}^n$? These are simply the complement of zero sets $V(f)$ of... | 4 | https://mathoverflow.net/users/1508 | 132783 | 73,470 |
https://mathoverflow.net/questions/132694 | 3 | Given a bounded decreasing sequence of rational number $a\_0,a\_1,\cdots$, then we know that it has limit $a$. Suppose this sequence satisfy that $|a\_k-a|<1/2^k$ for any $k$.
The sequence is given as an oracle: One can obtain $a\_k$ by querying $k$, and of course the limit $a$ is not known.
We are given another ra... | https://mathoverflow.net/users/4987 | Decide the limit of a decreasing sequence of real number | No, in the sense that if we had a decision procedure for this then it could also be used to decide the halting problem, for example.
For simplicity and without loss of generality, I will take $b = 0$. Then $a \neq 0$ corresponds to one of two $\Sigma^0\_1$-statements: $$(\exists k)(a\_k \leq 0) \quad\text{or}\quad (\... | 4 | https://mathoverflow.net/users/2000 | 132788 | 73,472 |
https://mathoverflow.net/questions/132763 | 4 | It is a famous open problem to determine the chromatic number of the graph $G=G(\mathbb{E}^2,\{1\})$ whose vertices are the point in the plane and two vertices are linked by an edge when they are at Euclidean distance $1$ one from the other. This number is known to be between $4$ and $7$.
It is a theorem of De Bruijn... | https://mathoverflow.net/users/4961 | Maximal minimum degree of unit-distance finite graphs | For arbitrary $m$ there is an $m$-regular unit-distance graph isomorphic as a graph to an $m$-cube (of course these all have chromatic number $2$): Suppose you have a unit graph isomorphic to a $k$-cube ( so $2^k$ vertices and regular of degree $k$ with a total of $k$ slopes for edges.) To get one isomorphic to a $k+1$... | 7 | https://mathoverflow.net/users/8008 | 132789 | 73,473 |
https://mathoverflow.net/questions/132761 | 1 | Let $\phi$ be an undecidable statement of ZFC set theory, for example let's take continuum hypothesis.
What is the ontological status of the "set" $X=\bigl\{x\in\{1,2\}:x=1\text{ or }(x=2\text{ and }\phi)\bigr\}$ in ZFC?
| https://mathoverflow.net/users/26007 | Ontological status of some "sets" in ZFC | Although you've been given a hard time in the comments, I think that this is actually a serious question in the philosophy of mathematics, whose answer depends on one's philosophical position concerning the nature of mathematical truth.
*What is the nature of existence for the mathematical objects that we define?*
... | 13 | https://mathoverflow.net/users/1946 | 132790 | 73,474 |
https://mathoverflow.net/questions/132266 | 6 | I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here.
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> Does there exist software to compute symmetric powers of Schur polynomials?
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I am seeking such software in the hopes of computing the characters of representations of the simple Lie al... | https://mathoverflow.net/users/25794 | Symmetric powers of Schur polynomials | This can be done with LiE:
<http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/>
(In fact it will compute the Schur functor of any irreducible representation.) There is a form interface so you can try LiE on the web: <http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html>
Here is an example of calculating $Sym^3$ of... | 6 | https://mathoverflow.net/users/321 | 132798 | 73,477 |
https://mathoverflow.net/questions/132800 | 5 | It is known that the set of natural numbers with the operation ab = {a}b, where {a} represents the index of a recursive function, forms a partial combinatory algebra (pca). All the references I have so far consulted mention the set of natural numbers as an example of a pca with no proof. If possible, I would appreciate... | https://mathoverflow.net/users/33090 | A Proof that the Natural Numbers Form a pca? | For a formalization of Turing machines you could look at
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> Andrea Asperti and Wilmer Ricciotti: [*Formalizing Turing Machines*](http://www.cs.unibo.it/~asperti/PAPERS/turing.pdf). Lecture Notes in Computer Science Volume 7456, 2012, pp 1-25.
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The book
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> Martin Davis. *Computability and Unsolvabili... | 11 | https://mathoverflow.net/users/1176 | 132819 | 73,485 |
https://mathoverflow.net/questions/132802 | 8 | Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f : \mathbb{C}^n \to \mathbb{C}$ defines a natural transformation $U^n \to U$ (in a way which is compatible with compositi... | https://mathoverflow.net/users/290 | Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras | I'll just consider a natural map $\phi:U\to U$; the multivariate case should be similar.
Let $D\_n$ be the closed disk of radius $n$ centred at the origin, and let $A\_n$ be the Banach algebra of functions that are continuous on $D\_n$ and holomorphic on the interior. Let $\iota\_n\in A\_n$ be the identity map. By na... | 5 | https://mathoverflow.net/users/10366 | 132823 | 73,489 |
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