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https://mathoverflow.net/questions/104117 | 1 | Let $S$ be a noetherian scheme (possibly nicer assumptions) and $G$ a smooth group scheme of finite type over $S$ with geometrically integral fibers with structure morphism $f: G\rightarrow S$ and $f\_\*\mathcal O\_G =\mathcal O\_S$ universally.
For $n\geq 0$ let $G\_n$ denote the $n$-th infinitesimal neighborhood of... | https://mathoverflow.net/users/18183 | Locally free sheaves on group schemes and thickenings | No. Let $G=\mathbb A^2$ with the projection map to $S=\mathbb A\_1$. It is a relative group scheme under addition of the $y$ coordinate. Take the module on $G\_2=\textrm{Spec} k[x,y]/y^2$ with two generators, $a$ and $b$, under the relations $ya-xb=0$, $yb=0$. This gives a coherent sheaf. The direct image is the free $... | 2 | https://mathoverflow.net/users/18060 | 104118 | 60,321 |
https://mathoverflow.net/questions/101042 | 5 | I am sure that there is an answer out there for the following question. If one is given an n dimensional Kahler manifold $X$ with Betti numbers that are the same as in the case of $\Bbb{CP}^n$, then is $X$ holomorphically diffeomorphic to $\Bbb{CP}^n$? This is of course true in the one dimensional case, but other then ... | https://mathoverflow.net/users/17965 | Same betti numbers as $\Bbb{CP}^n$ | Dear Common Crane,
let me list several results and question going in the direction of your intuition
(all the information below I learned from Sergey Galkin)
1) Theorem. Hirzebruch-Kodaira, Yau. A Kahler manifold homeomorphic to $\mathbb CP^n$ is biholomorphic to $\mathbb CP^n$. There is a nice exposition of this r... | 5 | https://mathoverflow.net/users/943 | 104122 | 60,324 |
https://mathoverflow.net/questions/104032 | 3 | It may be easy for the expert.
Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$.
My question is when this map is faithfully flat?
I known that for the faithfully part, (if already known it is flat) it is equivalent to the surjectivity ... | https://mathoverflow.net/users/16762 | When the affine quotient is faithfully flat? | The morphism $\phi$ is not flat for $n = m = 2$. Otherwise all fibers would be of dimension 1, namely the dimension of all 2 by 2 matrices minus the dimension of all symmetric 2 by 2 matrices. But the fiber over the zero matrix has dimension 2: matrices with coefficients $a$, $b = ia$, $c$, $d = ic$ for $i$ a square ro... | 3 | https://mathoverflow.net/users/13302 | 104139 | 60,332 |
https://mathoverflow.net/questions/104144 | 1 | Suppose we have a known group G and an unknown subgroup H. The permutation representation of G on the cosets of H gives a permutation group C, which is known. Is it possible to identify the generators of H using this information? If so, how? If not, what further information is needed?
(For background, my G is the mod... | https://mathoverflow.net/users/25565 | Identifying Subgroups of the Modular Group via Permutation Representations on Cosets | Choose a set of generators for $G$. Draw the Cayley graph of the action of $G$ on $G/H$. Choose a set of generators for the fundamental group of the graph. For each generator of the fundamental group, multiply the generators of $G$ corresponding to the edges together in order. This will be a set of generators for $H$.
... | 5 | https://mathoverflow.net/users/18060 | 104147 | 60,336 |
https://mathoverflow.net/questions/88107 | 9 | We have a finite dimensional representation of a finite quiver over, say, the rationals. Is there an algorithm known to decompose this representation into its irreducible components?
A related question: we have two (finite-dimensional) representations of a finite quiver. Is there an algorithm to check if one is a sum... | https://mathoverflow.net/users/21291 | Is there an algorithm known to decompose quiver representation? | If you want to decompose a finite-dimensional representation over, say, $\mathbb{Q}$ into its indecomposables over, say, $\overline{\mathbb{Q}}$, there is not only an algorithm but an efficient one (at least theoretically efficient: polynomial-time). See
Chistov, A., Ivanyos, G. and Karpkinski, M. [Polynomial Time A... | 8 | https://mathoverflow.net/users/38434 | 104154 | 60,338 |
https://mathoverflow.net/questions/104137 | 5 | Let $n\geq 1$ be an integer. The Friendship Graph (or Dutch windmill graph or $n$-Fan) $F\_n$ is a graph that can be constructed by coalescence $n$ copies of the cycle graph $C\_3$ with a common vertex. By construction, the friendship graph $F\_n$ is isomorphic to the windmill graph $Wd\left(3,n\right)$.
Can $F\_n$ b... | https://mathoverflow.net/users/19075 | Can the friendship graph be determind by its adjacency spectrum? | This isn't a real answer, but wolfram-alpha says that $F\_n$ is determined by its (adjacency) spectrum for $n \in \lbrace 2,3,4 \rbrace $. It doesn't say anything about $n=5$.
<http://www.wolframalpha.com/input/?i=%285%2C3%29-windmill+graph>
| 1 | https://mathoverflow.net/users/24848 | 104159 | 60,342 |
https://mathoverflow.net/questions/104162 | 7 | Recently I have been working with a certain subgroup of $GL\_{10}(\mathbb{F}\_2)$ and for various reasons was fairly sure it contained a normal subgroup isomorphic to $A\_5$. Today I was able to affirmatively show that this presumed copy of $A\_5$ does indeed exist in the bigger group. Now the matrices composing this c... | https://mathoverflow.net/users/12301 | Failure of a basic fact from Representation Theory | To reinforce Geoff's answer, I'd emphasize that the "fairly standard exercise" mentioned in the question is only standard when you consider characteristic 0 irreducible representations as is usually done in a first course (then it's also usual to start with a big field like $\mathbb{C}$ to avoid extra complications).
W... | 13 | https://mathoverflow.net/users/4231 | 104167 | 60,346 |
https://mathoverflow.net/questions/42234 | 20 | It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
>
> Is it still true for topological manifold?
>
>
>
**Comments:**
* To proof the smooth case, fix a Riemannian metric and consider exponential map up to cutlocus.
* The question was ask... | https://mathoverflow.net/users/1441 | Rugged manifold | The answer is yes -- Morton Brown's mapping theorem says that for every closed (connected) topological $n$-manifold $M$ there is a continuous map $f$ from the $n$-cube $I^n$ onto $M$ which is injective on the interior of the cube (for manifolds with boundary, see Remark 1 below). This was proved in early sixties and ca... | 22 | https://mathoverflow.net/users/25556 | 104175 | 60,350 |
https://mathoverflow.net/questions/104152 | 5 | Is it described somewhere in the literature how exponentials in functor categories other than presheaf categories can be constructed? So far I did not find anything in this regard. In the following, I sketch what I have developed myself. I would like to know whether this can be improved, and whether it is similar to so... | https://mathoverflow.net/users/25527 | Exponentials in functor categories | If $\mathcal{C}$ is small and $\mathcal{D}$ is complete and cartesian closed, then $\mathcal{D}^{\mathcal{C}}$ is also cartesian closed. We can compute exponentials in essentially the same way as for presheaves, motivated by $\mathcal{D}$-enriched category theory:
$$
G^F(x) = \int\_{y\in \mathcal{C}} \prod\_{\mathcal{C... | 18 | https://mathoverflow.net/users/49 | 104178 | 60,353 |
https://mathoverflow.net/questions/104171 | 3 | Let $k$ be a field. Given a (not necessarily commutative) $k$ graded ring $A$, M. Artin and J.J. Zhang introduced a notion of "noncommutative projective scheme" $Proj(A)$ in [this paper](http://www.sciencedirect.com/science/article/pii/S0001870884710875). It is defined as a $k$-linear abelian category
$$
Tails(A)=Gr(A... | https://mathoverflow.net/users/25571 | Global dimension of quantum $\mathbb{P}^{n}$ | The usual strategy is to take first the resolution of the diagonal --- that is a projective resolution of $A$ as $A\otimes A$-module up to $Tor\_2(A\otimes A)$, where $Tor\_2(A\otimes A)$ is the subcategory of bigraded $A\otimes A$-modules $M$ such that $M\_{i,j} = 0$ for all $i,j \ge N$ for some $N$. Then tensoring th... | 1 | https://mathoverflow.net/users/4428 | 104181 | 60,355 |
https://mathoverflow.net/questions/104172 | 11 | Short version: **Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams?**
Yes, I understand that the concept of knot has a natural geometrical significance, that one usually views knot diagrams as a tool to study... | https://mathoverflow.net/users/10909 | Knot diagrams, sets of moves and equivalence relations | Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves.
1. The study of [claspers](https://en.wikipedia.org/wiki/Clasper_(mathematics)). For example, $C\_k$-moves are a special type of clasper surgeries. MathSciNet indicates [123 c... | 14 | https://mathoverflow.net/users/2051 | 104184 | 60,356 |
https://mathoverflow.net/questions/104187 | 2 | Let $P(X) = a\_nX^n + \cdots + a\_1X + a\_0$ be a polynomial, $a\_i \in \mathbb{R}$ for all $i$. Set
$$S = \lbrace A \in \mathbb{M}\_n: P(A) = 0 \rbrace.$$
We consider the following relation $\sim$ on $S$: $A \sim B$ iff there exists an invertible matrix $C$ such that $B = C^{-1}AC$. It is easy to see that $\sim$ is a... | https://mathoverflow.net/users/17901 | The number of solutions of a matrix equation | The minimal polynomial $\mu\_A$ of each matrix $A\in M\_n(\mathbb R)$ with $P(A)=0$ divides $P$. Clearly, due to Jordan normal form, there are only finitely many conjugacy classes of such matrices. If you decompose $P$ into irreducible factors:
$$
P = P\_1^{k\_1}\cdots P\_r^{k\_r}
$$
then representatives for each equi... | 4 | https://mathoverflow.net/users/17498 | 104189 | 60,360 |
https://mathoverflow.net/questions/104182 | 1 | I apologize in advance if this question is not of sufficient level. Define a perfect magic hypercube of side length $k$ and dimension $n$ to be one in which the cells are filled with consecutive integers and the sum of numbers over cells in any geometric line is equal to the appropriate constant depending on $n$ and $k... | https://mathoverflow.net/users/22202 | Nonexistence of high dimensional perfect magic hypercubes of fixed side length | Here is an argument for $k=3$. For simplicity let the average value be $0$. Take any plane. Let the central element be $a$. The sum of the lines through the center minus the sum of the rows equals both $3a$ and $0$, so the central element of the plane must be $0$. Since there is only one $0$, there can't be more than o... | 2 | https://mathoverflow.net/users/2954 | 104191 | 60,362 |
https://mathoverflow.net/questions/104145 | 4 | I apologize if this question is too basic, but I haven't been able to work this out for myself.
Let $X$ and $Y$ be projective schemes, say over the complex numbers. There exists a scheme $Hom(X,Y)$ parameterizing morphisms $f : X \to Y$. In Kollar's "Rational curves on algebraic varieties" it is proved that the stalk... | https://mathoverflow.net/users/4054 | Tangent sheaf of a hom scheme | There are references for the statement with the sheaf of relative differentials. Although it is not the original reference, one reference is on p. 83, the discussion immediately preceding Proposition 6.3 of the following article.
MR1437495 (98e:14022) Reviewed
Behrend, K.(3-BC); Fantechi, B.(I-TRNT)
The intrinsic nor... | 3 | https://mathoverflow.net/users/13265 | 104193 | 60,364 |
https://mathoverflow.net/questions/104009 | 2 | Suppose that $A$ is a Hermitian matrix and that $u,v$ are two vectors. Is there some known function $\kappa(A)$ so that $||u-v|| \leq \kappa(A) |\frac{u^{\\*}Au}{u^{\\*}u}-\frac{v^{\\*}Av}{v^{\\*}v}|$?
UPDATE: Andrew T. Baker has shown that the answer is "no" in general (to take an even simpler counter-example, take ... | https://mathoverflow.net/users/22051 | 'Condition number' for Rayleigh-Ritz quotient | I think the answer for a general Hermitian matrix $ A $ is no.
Let $ u, v $ be distinct eigenvectors of $ A $ with the same eigenvalue $ \lambda $ and normalized so that $ u^\ast u = v^\ast v = 1 $. Then $ \| u - v \| > 0 $ and
\begin{equation\*}
| u^\ast Au - v^\ast A v | = | \lambda u^\ast u - \lambda v^\ast v | ... | 3 | https://mathoverflow.net/users/17113 | 104199 | 60,366 |
https://mathoverflow.net/questions/104198 | 3 | Is there any difference between the equivalent classes of $\mathbb R^n$ vector bundles and $\mathbb R^n$-fiber bundles? The first one is related to $K$ group. What is the second one? I am thinking they are the same for $\mathbb R^n$.
| https://mathoverflow.net/users/1190 | Classify $\mathbb{R}^n$-bundles | I assume that by $\mathbb{R}^n$-fiber bundles you mean differentiable bundles with fiber diffeomorphic to $\mathbb{R}^n$. So you are forgetting the linear structure of $\mathbb{R}^n$, keeping only the smooth structure. It is a fact that $\textit{Diff}(\mathbb{R}^n)$ deformation retracts to $\textit{GL}(n,\mathbb{R})$, ... | 11 | https://mathoverflow.net/users/25577 | 104203 | 60,367 |
https://mathoverflow.net/questions/104061 | 9 | Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in \mathbb{F}\_p[\mu]/(m(\mu))[x]$, I'd like the CAS to be able to compute things like $f(x + \mu) + g(x)$ where I specify the polynomial $m(\mu)$.
... | https://mathoverflow.net/users/8341 | computer algebra system for polynomial algebras over finite fields | You can do these things in Sage quite easily. Here is an example (using Sage 5.2):
```
sage: Fp.<mu>=GF(5)[]
sage: Fp
Univariate Polynomial Ring in mu over Finite Field of size 5
sage: m=mu^5-mu+1
sage: K.<y>=GF(5^5, name='y', modulus=m) # your mu becomes y
sage: A.<x>=K[]
sage: A
Univariate Polynomial Ring in x ove... | 18 | https://mathoverflow.net/users/11100 | 104204 | 60,368 |
https://mathoverflow.net/questions/104200 | 4 | I have the following functional
$$F(y)=\left[\int \frac{1}{y(x)+A}\cos(x)dx\right]^2$$
How do I find the functional derivative $dF$?
(I never encountered the square of an integral before when I did functional derivatives)
| https://mathoverflow.net/users/24208 | Functional derivative of the square of an integral | Replace $y$ by $y+t\delta y$ and then compute
$$\lim\_{t\to 0} \Bigl(\; F(y+t\delta y)-F(y)\;\Bigr). $$
Here are the details. Observe first that
$$\frac{1}{y+ A+ t\delta y}-\frac{1}{y+A} = -\frac{1}{(y+A)^2} t\delta y+ O(t^2),$$
so that
$$F(y+t\delta y)= \left(\int\;\Bigl(\; \frac{1}{y+A} -\frac{1}{(y+A)^2} t... | 4 | https://mathoverflow.net/users/20302 | 104205 | 60,369 |
https://mathoverflow.net/questions/104206 | 7 | I'm looking for, or hoping to inspire the creation of, a list of conventional names for categories that come up often.
For example, we have *the terminal category* $\fbox{$\bullet$}$, a nice name. I've heard this category $\fbox{$\bullet\to\bullet$}$ called *the free arrow category*. That's fine by me. What about the... | https://mathoverflow.net/users/2811 | Conventional names for finite categories | I notice that the categories considered for naming here are all the domains, or shapes, of basic *diagrams*; an object, an arrow, an endomorphism (n.b., my instinct was just to call that $\mathbb{N}$), a composable sequence, parallel arrows, equalized arrows... Not that diagrams *in* these categories aren't also intere... | 3 | https://mathoverflow.net/users/1631 | 104220 | 60,380 |
https://mathoverflow.net/questions/104212 | 5 | Suppose $X$ is a $T\_1$ space with an infinite set of isolated points. Show that if $X^\sharp = X \cup \lbrace \infty \rbrace$ is obtained by adding a single new isolated point, then $X$ and $X^\sharp$ are homeomorphic.
I am almost embarrased to raise this, which seems obvious. The proof must be simple, but it eludes... | https://mathoverflow.net/users/20300 | Adjoining a new isolated point without changing the space | In my answer to <https://mathoverflow.net/questions/26414> I descibed a somewhat simpler-looking example than Nik's, but proving that it works may be harder. Take two copies of $\beta\mathbb N$ and glue each non-isolated point of one copy to the corresponding point of the other copy. Any way of "absorbing" a new isolat... | 9 | https://mathoverflow.net/users/6794 | 104222 | 60,381 |
https://mathoverflow.net/questions/104213 | 6 | On a smooth algebraic variety X, every coherent sheaf F has a finite resolution by
locally free sheaves. Using such resolution, we can define the determinant
of F, det F, which is a line bundle on X.
My question is :
why if the support of F is of codimension greater or equal to 2
is the determinant of F trivial ?
... | https://mathoverflow.net/users/25309 | triviality of determinant sheaf | Outside the support of $F$, the resolution is an exact sequence, so the alternating tensor product of the determinants is trivial. On a smooth scheme, a line bundle trivial outside a codimension $2$ subset is trivial.
| 16 | https://mathoverflow.net/users/18060 | 104227 | 60,384 |
https://mathoverflow.net/questions/104223 | 42 | I have been chosen by my university to give a 1 hour public research lecture. Every year a researcher is chosen for this honour. Traditionally people explain their own research about designing improved airplanes or the life of a peasant in the middle ages or whatever. I don't feel that I can explain my research in an h... | https://mathoverflow.net/users/16684 | Justifying/Explaining math research in a public address | David, I'd suggest to use physics or biology as targets.
I mean, try to build a bridge between your research area and the applications... which ultimately could turn into a technological or daily application.
**Examples:**
* Differential geometry -> General relativity (gravitation) -> Fine corrections in GPS devi... | 14 | https://mathoverflow.net/users/25356 | 104234 | 60,390 |
https://mathoverflow.net/questions/101799 | 7 | **Background:** Let $k$ be a field and denote by $P = k[x\_1,\ldots,x\_n]$ the polynomial ring in $n$ (commuting) variables over $k$. A *resolution* of an ideal $I \lhd P$ is an exact sequence of $P$-modules $$\ldots \to F\_n \to F\_{n-1} \to \ldots \to F\_0 \to P/I \to 0.$$ This resolution if *minimal* if the rank of ... | https://mathoverflow.net/users/18263 | Why are minimal resolutions of polynomial ideals important? | As it has already been pointed out, the key concept is *uniqueness*.
We know that free (or, if you will, projective) resolutions exist in the case we are interested in. Yet, a module may possess many of them. However, each finitely generated module (over a polynomial algebra over a field) has a minimal free resolutio... | 4 | https://mathoverflow.net/users/nan | 104239 | 60,394 |
https://mathoverflow.net/questions/104148 | 4 | Hello,
In the differentiable case it is quite easy to prove that vector bundles are equivalent to smooth maps to the Grassmannian $G\_{k}(\mathbb{R}^N)$ for some integer $N>>0$. The proofs I have seen of this use a partition of unity on the base space to embed the bundle into a trivial bundle.
Now in the holomorphi... | https://mathoverflow.net/users/25566 | Holomorphic bundles and maps to the Grassmannian ? | Here are some remarks; hope they will help.
Let us consider the algebraic case first: then we shall see what one can hope for in the analytic case. Let $X$ be a smooth compact complex algebraic variety and let $\mathcal{E}$ be a vector bundle on $X$. As Youloush points out, in general it is not true that $\mathcal{E}... | 5 | https://mathoverflow.net/users/2349 | 104242 | 60,396 |
https://mathoverflow.net/questions/104221 | 5 | I recently came upon the following theorem which was attributed to J. Conway:
For each $A\subset \omega\_1$, let $\Phi(A)$ be a linear ordering of type $\sum\_{\alpha<\omega\_1} \tau\_\alpha$, where $\tau\_\alpha$ is $\eta$ (the order-type of the rationals) whenever $\alpha\notin A$, and $1+\eta$ whenever $\alpha\in ... | https://mathoverflow.net/users/13694 | Characterizing $\omega_1$-like dense linear orderings | Concerning question 2, the theorem that you mention in the case of
$\omega\_1$ is not actually difficult. I had briefly sketched a
proof of it at the conclusion of my answer to the math.SE question
[Linearly ordered sets somewhat
similar to $\mathbb{Q}$](https://math.stackexchange.com/a/174404/413), which is concerned ... | 7 | https://mathoverflow.net/users/1946 | 104247 | 60,400 |
https://mathoverflow.net/questions/104084 | 2 | Is every long exact sequence
$$\cdots\to\pi\_{d+1}(B)\to\pi\_d(F)\to\pi\_d(E)\to\pi\_d(B)\to\pi\_{d-1}(F)\to\cdots$$
with topological spaces $F,E$ and $B$, where $F$ is a subspace of $E$ with inclusion map $i$, induced by a Serre fibration $p:E\to B$ ? By "induced" I mean that the maps in the sequence are given by
$$p... | https://mathoverflow.net/users/25549 | Is every long exact sequence of homotopy groups induced by a fibration? | Ok, so one counterexample to the other interpretation of the question would be to take a fibration $F\to E\to B'$
And let $B$ be a space with the same homotopy groups as $B'$ but not homotopy equivalent to $B$. Note that $B$ must have at least two non-trivial homotopy groups.
| 0 | https://mathoverflow.net/users/3874 | 104249 | 60,402 |
https://mathoverflow.net/questions/104253 | 3 | Let $E\_{1},E\_{2}$ be elliptic curves over $\mathbb{C}$. We denote by $\iota\_{i}$ the translation by a 2-torsion point on $E\_{i}$. Then $G=\mathbb{Z}/2\mathbb{Z}$ acts freely on the the product $E\_{1}\times E\_{2}$ via the involution $\iota=(\iota\_{1},\iota\_{2})$. and the quotient
$$
X=(E\_{1}\times E\_{2})/G
$$... | https://mathoverflow.net/users/25571 | Intersection form on quotient manifold | Luckily, the $X$ thus described is a torus. This means that $H^2(X,\mathbb Z)$ has a nice explicit description: It is $\wedge^2 H^1(X,\mathbb Z)$. The intersection form is the the symmetric bilinear map to $\wedge^4 H^1(X,\mathbb Z)=\mathbb Z$. $H^1(X,\mathbb Z)$ is an index two sublattice of $H^1(E\_1 \times E\_2,\mat... | 4 | https://mathoverflow.net/users/18060 | 104257 | 60,407 |
https://mathoverflow.net/questions/104254 | 12 | I recently learned of an [interesting result](http://www.ams.org/mathscinet-getitem?mr=242185) of Narasimhan and Ramanan from 1969, which says that moduli space of rank two vector bundles with trivial determinant on a curve $X$ of genus two is naturally isomorphic to $\mathbb PH^0(\operatorname{Pic}^1X,\mathcal L\_\The... | https://mathoverflow.net/users/35353 | Rank two vector bundles on a curve of genus two | Dear unknown, by NR, there exists always a family of degree -1 line sub bundles for every holomorphic vector bundle of rank two with trivial determinant on a genus 2 surface. As they have degree -1, they do not admit flat connections. This means that when your representation $\rho$ has the upper triangular form you wro... | 7 | https://mathoverflow.net/users/4572 | 104260 | 60,409 |
https://mathoverflow.net/questions/104201 | 7 | Let $k$ be an algebraically closed field with char($k$)$= p > 0$.
Let $P$ be a finite $p$-group. For any homomorphism
$\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be
put inside the group of upper triangular matrices(standard borel)
after conjugation. The question I have is what can we say abou... | https://mathoverflow.net/users/25576 | homomorphism into reductive groups | This is a (rather) expanded version of Yves' answer, because for the specific focus of the question one can understand the proof without getting into many other technicalities of the Borel--Tits paper (which has bigger fish to fry, related to fields of definition and much more). We will explain the proof of the followi... | 4 | https://mathoverflow.net/users/22479 | 104266 | 60,412 |
https://mathoverflow.net/questions/104282 | 10 | Let $M$ be a square matrix with integer entries. Then **Fermat's little theorem for matrices** holds:
$$\text{tr}(M^p) \equiv \text{tr}(M) \bmod p.$$
This follows by an examination of the action of the Frobenius map on the roots of the characteristic polynomial of $M$ over a splitting field; there is also a combina... | https://mathoverflow.net/users/290 | Does this notion of pseudoprime relative to a matrix appear in the literature? | I think this is weaker than the notion of Frobenius pseudoprime. The trace of a matrix is the sum of the roots of its characteristic polynomial, so you're testing whether the sum of the roots remains invariant upon raising to the pth power. The Frobenius test is stronger than that.
| 8 | https://mathoverflow.net/users/11722 | 104290 | 60,425 |
https://mathoverflow.net/questions/104270 | 7 | Let $G$ be a finite group and let $p$ be a prime dividing the order of $G$. Let $\chi$ be a $\mathbb{C}\_p$-valued irreducible character of $G$ and let $e\_{\chi} = |G|^{-1}\chi(1)\sum\_{g \in G} \chi(g^{-1})g$ be the associated primitive central idempotent in $\mathbb{C}\_p[G]$. Let $\mathbb{Q}\_{p}(\chi)=\mathbb{Q}\_... | https://mathoverflow.net/users/7443 | Modular representation theory: central idempotents in $\mathbb{Z}_p[G]$ | EDIT: There is a MUCH simpler proof than my first one, which I found after looking up the proof of (90.4) in Curtis-Reiner (Rep. Th. of finite groups...) mentioned by Florian Eisele in the comments:
The central idempotent $e\in \mathbb{Z}\_p[G]$ is supported on $p$-regular elements. (By the way, a quite elementary p... | 7 | https://mathoverflow.net/users/10266 | 104292 | 60,426 |
https://mathoverflow.net/questions/104156 | 4 | In the book *Open Problems in Topology* by Jan Van Mill and George M. Reed, the following problem was presented:
108. *Is there a para- Lindelof Dowker space?*
Recall that a *para-Lindelof Dowker space* has a locally countable open refinement, satisfies Axiom T4, and is not countably paracompact.
Some results on this ... | https://mathoverflow.net/users/nan | Concerning Problem 108 in "Open Problems in Topology" | It is now problem 502 in "Open problems in topology II", I would guess it is still open.
| 6 | https://mathoverflow.net/users/17836 | 104295 | 60,429 |
https://mathoverflow.net/questions/104291 | 2 | Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even more, I think it is not possible to triangulate the euclidean plane with triangles that are $C$-bilipschitz with a unit ... | https://mathoverflow.net/users/7894 | Euclidean triangulation of the plane with degree 7 at each vertex. | Unless I am missing something: start with the image of a degree-seven triangulation of the hyperbolic plane (coming from the (2, 3, 7) triangle group -- that is the triangulation both you and @Will seem to be alluding to) in the Beltrami-Klein model (this will be a triangulation of the disk with Euclidean straight edge... | 6 | https://mathoverflow.net/users/11142 | 104299 | 60,431 |
https://mathoverflow.net/questions/104300 | 10 | Hello,
I asked this question already on StackExchange with no answer, maybe it will be better suited here. It is well known that $\operatorname{Ext}^1\_\mathbb{Z}(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Is there a generalization of this formula outside PIDs? I mean $\operatorname{Ext}^1\_R(R/I,M) \simeq M/IM$ for maxim... | https://mathoverflow.net/users/25602 | Ext of cyclic module | In general $Ext^1\_R(R/I,M) \not\cong M/IM$.
As an example take a finite abelian group $G$ and set $R := \mathbb{Z}G$ and let $I:= I\_G = \ker(\mathbb{Z}G \to \mathbb{Z},\; g \mapsto 1)$ be the augmentation ideal. $I\_G$ is a prime ideal since $\mathbb{Z}G/I\_G \cong \mathbb{Z}$. Then, with trivial coefficients
$$E... | 6 | https://mathoverflow.net/users/10194 | 104302 | 60,432 |
https://mathoverflow.net/questions/104297 | 3 | Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries?
Thanks!
| https://mathoverflow.net/users/24715 | Eigenvectors of asymmetric graphs | I think the answer is no. Take the [Frucht Graph](http://en.wikipedia.org/wiki/Frucht_graph), the simplest nontrivial asymmetric graph. Its adjacency matrix is
\begin{equation\*}
\left(
\begin{array}{cccccccccccc}
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\
... | 5 | https://mathoverflow.net/users/8430 | 104307 | 60,436 |
https://mathoverflow.net/questions/104308 | 7 |
>
> If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
>
>
>
<http://math.stanford.edu/~pardon/img/dehntwists.png>
| https://mathoverflow.net/users/35353 | Do the following set of Dehn twists generate the mapping class group? | No. If they did, then they would still generate the mapping class group of the closed surface that results from gluing a disc to the boundary component. However, in that surface they all commute with the hyperelliptic involution, which is not central for $g$ at least $3$.
In fact, Humphries proved that you need at le... | 20 | https://mathoverflow.net/users/317 | 104311 | 60,437 |
https://mathoverflow.net/questions/104309 | 0 | Given an alphabet of $q \ge 2$ letters, I want to construct a set $S$ of $x$ strings (of uniform length) such that the minimum Hamming distance between any two strings is $d$. What I need to figure out is the minimum string length $n$ that could produce such a set.
When $d = 1$, then $n = ceiling(log\_q(x))$, but I c... | https://mathoverflow.net/users/25603 | How to find the minimum string length to produce a set of a given size with a minimum pairwise Hamming distance | For $q$ the size of the alphabet and $n$ the length of the code it is costumary to denote by
$A\_q(n,d)$ the maximal size of a code with minimum distance $d$.
There are numerous investigations on this. tergi already mentioned tables of explicit values.
There are however also general bounds known. In particular a c... | 2 | https://mathoverflow.net/users/nan | 104314 | 60,439 |
https://mathoverflow.net/questions/104312 | 8 | The Steenrod algebra $\mathcal{A}^\*$ is the algebra of cohomology operations for the spectrum $\mathrm{H}\mathbb{F}\_2$, i.e. $\mathcal{A}^\*\simeq (\mathrm{H}\mathbb{F}\_2)^\*(\mathrm{H}\mathbb{F}\_2)$.
The dual Steenrod algebra $\mathcal{A}^\vee \_\*$ can be interpreted as the coalgebra of homology cooperations, i.e... | https://mathoverflow.net/users/23122 | Homology-Cohomology Pairing | Propositions 17.10 and 17.11 of [Switzer's book](http://books.google.co.uk/books/about/Algebraic_Topology_homotopy_and_Homology.html?id=f1xnOzmX2FYC) seem to be examples of things working nicely.
For those who don't have the book to hand: let $E$ be a commutative ring spectrum such that $E\_\ast(E)$ is flat as a righ... | 5 | https://mathoverflow.net/users/8103 | 104327 | 60,444 |
https://mathoverflow.net/questions/104318 | 3 | I have two related questions that I can't seem to find any literature on:
1) What can be said about $\tau(a+b)$ knowing $\tau(a)$ and $\tau(b)$ (where $\tau(n)$ is the number of positive divisors of $n$)?
2) I have a feeling the answer to the previous is "nothing" because of the same types of issues surrounding the... | https://mathoverflow.net/users/3400 | Number of divisors of a sum / ABC conjecture equivalent statement | I agree that the answer to the first question should be more or less nothing. However, I disagree that this is for 'the same types of issues surrounding the ABC-conjecture', and so I highly doubt there is much of a direct link.
The ABC-conjecture is very different in that (in a certain sense) it says something on th... | 5 | https://mathoverflow.net/users/nan | 104342 | 60,451 |
https://mathoverflow.net/questions/104324 | 0 | what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a discrete subset of the space of all continuous functions X--->X with compact-open topology.the converse is false,in general, ... | https://mathoverflow.net/users/25609 | G-structures and complete riemannian manifolds | Concerning the proof that in Riemannian setting proper discontinuity is equivalent to discreteness, even more is true: If $G$ is a subgroup of group of isometries of a Riemannian manifold $M$ so that (for some $x\in M$) $d(x,g\_i(x))$ is bounded for a sequence $g\_i\in G$, then the sequence $(g\_i)$ contains a converge... | 1 | https://mathoverflow.net/users/21684 | 104346 | 60,452 |
https://mathoverflow.net/questions/104344 | 9 | I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to arise on the big Zariski site.
Thanks!
| https://mathoverflow.net/users/12693 | Representable Presheaf | Since you ask specifically for an *easy* example (not a natural, useful, etc., one), I suggest taking any category in which there are two objects $A$ and $B$ for which Hom$(A,B)$ is not a one-element set (i.e., the category is equivalent to neither the initial nor the terminal category) and giving it the topology in wh... | 9 | https://mathoverflow.net/users/6794 | 104360 | 60,460 |
https://mathoverflow.net/questions/102779 | 11 | Among the basic algorithms of quantum computations Lov Grover's [result](http://en.wikipedia.org/wiki/Grover%27s_algorithm) on quantum search stands out, both in regards to its intrinsic interest, and for its undisputable elegance.
Grover's algorithm enables one to search an unsorted database of N elements in
$O(N^... | https://mathoverflow.net/users/15293 | How much does a quantum oracle to find a needle in a haystack really cost? | $U\_f$ is easy to create from a circuit that computes $f$. (I'm assuming you have a circuit that computes $f$. If you have a Turing machine, convert it to a circuit in the standard way.)
Now given a circuit (using AND, OR, NOT gates) of G gates that computes $f$, i.e., accepts a binary string $x$ as input and outputs... | 8 | https://mathoverflow.net/users/8075 | 104363 | 60,462 |
https://mathoverflow.net/questions/104332 | 1 | Let $D$ be a bounded Lipschitz domain and $f$ is continuous up to $\partial D$. Is it true that $$\int\_{\partial D}f(x)d\sigma(x) = \lim\_{\epsilon\to 0}\frac{1}{\epsilon}\int\_{D^{\epsilon}}f(x)dx$$
where $D^{\epsilon}=\{y\in D: d(y,\partial D)<\epsilon\}$?
When $\partial D$ is $C^2$, we can parametrize each point ... | https://mathoverflow.net/users/23381 | Surface integral approximation | This is really a question about [Minkowski content.](http://en.wikipedia.org/wiki/Minkowski_content) If you look at the cited article (under properties), this seems to indicate that under your Lipschitz condition, the answer to your question is YES (since the full-dimensional Hausdorff measure is the Lebesgue measure).... | 0 | https://mathoverflow.net/users/11142 | 104365 | 60,464 |
https://mathoverflow.net/questions/104362 | 7 | Some big picture questions -
1. What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with additional level structure.
(Edit: I know there must be tons, though I'm still having trouble finding good resourc... | https://mathoverflow.net/users/15242 | moduli interpretations for modular curves | Over $\mathbb{C}$, elliptic curves with, say, a point of order $N$ can be identified with the quotient of the upper half plane $\mathbb{H}$ by $\Gamma\_1(N)$ just by associating to the class of $\tau\in \mathbb{H}$ the isomorphism class of the air $(\mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau), 1/N)$. Under this ident... | 9 | https://mathoverflow.net/users/12107 | 104369 | 60,465 |
https://mathoverflow.net/questions/104371 | 0 | I am studying virtual knot theory.
A virtual knot is a knot diagram with real or virtual crossing information.
The equivalence relation includes generalized Reidemeister moves.
There are premitted virtual Reidemeister moves, and forbbiden moves.
Question
Do the forbidden moves have some validity?
How are generalize... | https://mathoverflow.net/users/16516 | Validity of generalized Reidemeister moves for a virtual knot | A virtual knot is the same thing as a Gauss diagram. The virtual crossing doesn't really exist. So moving something across it makes no sense. Another way of seeing this is that, given that the object of study is Gauss diagrams modulo Reidemeister 1, 2, and 3, the virtual Reidemeister moves don't change the Gauss diagra... | 4 | https://mathoverflow.net/users/2051 | 104375 | 60,466 |
https://mathoverflow.net/questions/104368 | 5 | I can show that there infinitely many solutions to this equation. Is it possible that the set
of rational solutions is dense?
| https://mathoverflow.net/users/25619 | Rational solutions to x^3 + y^3 + z^3 - 3xyz = 1 | I think this surface has a rational parameterization in terms of (a,b), given by:
$x = (1 + a + a^2)^2/(9 (3 + a (6 + (-1 + a)^2 a)) b^2) + ((-2 + (-2 + a) a) b)/(1 + a + a^2)$
$y = (1 + a + a^2)^2/(9 (3 + a (6 + (-1 + a)^2 a)) b^2) + (b + 2 a b)/(1 + a + a^2)$
$z = (1 + a + a^2)^2/(9 (3 + a (6 + (-1 + a)^2 a)) ... | 6 | https://mathoverflow.net/users/11841 | 104377 | 60,468 |
https://mathoverflow.net/questions/104339 | 2 | Hello,
Recently I've been studying the problem of quadratic form diagonalization. Suppose that we have a form $F(x,y,z)$ with corresponding symmetric matrix $M$. This form is equivalent to another form $F'(x',y',z')$ which is in [Smith normal form](http://en.wikipedia.org/wiki/Smith_normal_form). The matrix that corr... | https://mathoverflow.net/users/22733 | Diagonalization of a quadratic form in integers | The actual correct manipulation is
$$
\left( \begin{array}{rrr}
3 & -1 & 0 \\\
-5 & 2 & 0 \\\
-34 & 13 & 1
\end{array}
\right) \cdot
\left( \begin{array}{rrr}
5 & 13 & 1 \\\
13 & 34 & 0 \\\
1 & 0 & 35
\end{array}
\right) \cdot
\left( \begin{array}{rrr}
3 & -5 & -34 \\\
-1 & 2 & 13 \\\
0 & 0 & 1
\e... | 3 | https://mathoverflow.net/users/3324 | 104382 | 60,471 |
https://mathoverflow.net/questions/104386 | 9 | Is the property of having a bound on the dimensions of irreducible representations preserved by an extension?
For example $G\_1=\mathbb{Z}$ and $G\_2=\mathbb{Z}/2\mathbb{Z}$ are discrete abelian groups each of which only have 1 dimensional irreducible representations, while the group
$$\mathbb{Z}\rtimes\mathbb{Z}/2\... | https://mathoverflow.net/users/21007 | Dimensions of unitary representations of group extensions | A discrete group has the property you describe if and only if it is virtually abelian. I must admit to never learning the proof, but some foraging on MathSciNet indicates the result is due to Isaacs and Passman:
>
> Isaacs, I. M.; Passman, D. S.
> Groups with representations of bounded degree.
> [Canad. J. Math. ... | 13 | https://mathoverflow.net/users/763 | 104387 | 60,472 |
https://mathoverflow.net/questions/93262 | 49 | Given two categories $I$ and $J$ we say that *colimits of shape $I$ commute with limits of shape $J$* in the category of sets, if for any functor $F : I \times J \to \text{Set}$ the canonical map $$\textrm{colim}\_{i\in I} \text{lim}\_{j\in J} F(i,j) \to \textrm{lim}\_{j\in J} \text{colim}\_{i\in I} F(i,j)$$ is an isom... | https://mathoverflow.net/users/644 | Which colimits commute with which limits in the category of sets? | Some very involved necessary and sufficient conditions are found in a [paper of Foltz](http://archive.numdam.org/article/DIA_1981__5__A1_0.pdf) (in French). Some observations on his paper:
1. An elementary observation (Proposition 3, section 1, p. F 12): $I$-colimits commute in $\mathrm{Set}$ with $P$-limits iff $I$-... | 21 | https://mathoverflow.net/users/2362 | 104389 | 60,474 |
https://mathoverflow.net/questions/104380 | 5 | One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C\_\ast$, define
$D\_0 : C\_\ast\to C\_\ast\otimes C\_\ast$,
so that the cup product is given (on cocycles)
$(u\cup v)(\sigma) = (u\otimes v)(D\_... | https://mathoverflow.net/users/24021 | Why are cup-i products and Steenrod Squares often (always?) unary? | I use $\cup\_i$ products as binary products in my work on the algebraic theory of surgery
<http://www.maths.ed.ac.uk/~aar/papers/ats2.pdf>
They give the higher symmetry properties {$\phi\_s|s \geq 0$} of the Poincare duality chain equivalence
$$\phi\_0=[M] \cap - : C(M)^{m-\*} \to C(M)$$
of an $m$-dimensional mani... | 12 | https://mathoverflow.net/users/732 | 104391 | 60,475 |
https://mathoverflow.net/questions/104393 | 0 | We recall that if $f\_1\in L^p(\mathbb R)$ and if $f\_2\in L^q(\mathbb R)$ where
$1 \lt p \lt \infty$ and $\frac 1p+\frac1q=1$ then the function $f\_1\ast f\_2(x)=\int\_{\mathbb R} f\_1(x-y) f\_2(y)dy$ is a continuous function in $x$. Now if we take $f$ to be a $L^\infty$ function on $\mathbb R$ and $\mu$ an element... | https://mathoverflow.net/users/17822 | Convolution with an element in the dual space | This should be a comment on Yulia Kuznetsova's answer, but I didn't have the points and it's gone now anyway:
Can't you take $\mu$ as corresponding to a functional $f \mapsto f(0)$ if $f$ is an $L^\infty$ function continuous at $0$, extended to all of $L^\infty(\mathbb{R})$ by Hahn-Banach? Then if you take e.g. a ste... | 5 | https://mathoverflow.net/users/3210 | 104398 | 60,477 |
https://mathoverflow.net/questions/104277 | 18 | My colleagues and I are working on a project related to an [old paper of C. Borell](http://www.springerlink.com/content/bp92875n35324173/) and we have boiled it down to the following problem:
Show, for all integers $1 \leq i \leq k$, that the univariate real polynomial $P(x) = \frac12 \binom{2k}{2i} (1+x)^{2k-2i} + ... | https://mathoverflow.net/users/658 | Proving univariate polynomials (defined by sums, binomial coeffs, etc.) are nonnegative: is it 'routine'? | There are some techniques for proving positivity of sums using computer
algebra, but they are all heuristic approaches rather than solid algorithms.
This is in contrast to identities, which we understand very well
algorithmically.
One possibility to prove positivity of some quantity f[n] involving a
discrete paramete... | 9 | https://mathoverflow.net/users/25631 | 104401 | 60,478 |
https://mathoverflow.net/questions/104405 | 3 | Let $A$ be a non-singular matrix and let $s(A)$ be the sum of its entries. Under which conditions can it be assured that $s(A) \neq 0$?
if you like, you can assume that $A$ is symmetric.
Here is an example with $s(A)=0$:
$A=\begin{bmatrix}1 & 2 & 3 \\\\ 2 & -4 & -1\\\\ 3 & -1 & -5 \end{bmatrix}$
| https://mathoverflow.net/users/22051 | matrices whose entries sum to zero | It seems that, as quid suggested, very little can be said, at least if we want to say something invariant under rotations of coordinates. Specifically, the following are equivalent for a symmetric real matrix $M$:
(1) There is an orthogonal matrix $T$ such that $T^{-1}MT$ has entries summing to 0.
(2) The eigenval... | 15 | https://mathoverflow.net/users/6794 | 104417 | 60,483 |
https://mathoverflow.net/questions/104422 | 6 | I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,j\_! \mathcal{S})$ and $H^i(U,\mathcal{S})$? What if I add that I know that $H^i(U,\mathcal{S}) = 0$ for $i>0$. I'm ho... | https://mathoverflow.net/users/4542 | cohomology and $j_!$ | Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line.
Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \m... | 7 | https://mathoverflow.net/users/25309 | 104431 | 60,490 |
https://mathoverflow.net/questions/104394 | 4 | I would like to know why every non-degenerate irreducible projective curve has a three-dimensional secant variety. It is clear to me that the dimension can't be larger.
Thanks for your help!
| https://mathoverflow.net/users/25630 | Secant variety of an irreducible non-degenerate projective curve | Suppose that the secant variety of your curve is a surface $S$. This implies that $S$ is covered by lines in such a way that there is a $1$-dimensional family of such lines through every general point of $S$.
Now let $P$ be a general smooth point on $S$. The lines contained in $S$ through $P$ are necessarily included... | 7 | https://mathoverflow.net/users/2868 | 104437 | 60,491 |
https://mathoverflow.net/questions/104433 | 10 | Suppose the $n \times n$ matrix $A$ has eigenvalues $\lambda\_1, \ldots, \lambda\_n$ and singular values $\sigma\_1, \ldots, \sigma\_n$. It seems plausible that by comparing the singular values and eigenvalues we gets some sort of information about eigenvectors. Consider:
a. The singular values are equal to the absol... | https://mathoverflow.net/users/21162 | Eigenvalues, singular values, and the angles between eigenvectors | Suprprisingly, no, or at least not using any obvious measure of eigenvalue equality. Let $M$ be a matrix that is very non-normal, and consider the matrix:
$\left(\begin{array}{cc} I+\epsilon M & 0 \\ 0 & 2I + \epsilon M\end{array}\right)$
This matrix has a bunch of eigenvalues very close to $1$ and a bunch of eigen... | 12 | https://mathoverflow.net/users/18060 | 104445 | 60,496 |
https://mathoverflow.net/questions/104442 | 0 | Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?
| https://mathoverflow.net/users/25641 | Find both maximum and minimum values in linear programming problem | You can certainly construct problems where the active constraints for the maximum are disjoint from the active constraints for the minimum, so this is as hard as separately solving for max and min.
| 1 | https://mathoverflow.net/users/11142 | 104448 | 60,497 |
https://mathoverflow.net/questions/104451 | 14 | Some examples of irreducible homology 3-spheres that bound **smooth** contractible 4-manifolds are listed in the comment to problem 4.2 in [Kirby's problem list](http://math.berkeley.edu/~kirby/problems.ps.gz), and all of them happen to occur among the [Brieskorn spheres](http://www.maths.ed.ac.uk/~aar/papers/milnbri... | https://mathoverflow.net/users/1573 | Irreducible homology 3-spheres that bound smooth contractible manifolds | Yes, there's loads of other contractible 4-manifolds bounding homology spheres with other geometries.
For example $1$-surgery on the Stevedore knot is a hyperbolic manifold with volume $1.3985\cdots$. This bounds a contractible manifold by the same kind of arguments Casson and Harer used to cook up their big list of ... | 19 | https://mathoverflow.net/users/1465 | 104452 | 60,499 |
https://mathoverflow.net/questions/104450 | 8 | In the first semester of 2012 I took a course in General Topology and Set Theory, at undergraduate level. For topology, I was instructed to use Engelking's *[General Topology](http://rads.stackoverflow.com/amzn/click/3885380064)*; albeit I had a great difficult to approach it, I got used to the text and did (and I'm st... | https://mathoverflow.net/users/17760 | Road to Solovay's Land. | With no background in logic, it's a bit of a long road to Solovay's model; the good news is that every step of it is incredibly interesting!
(Some of this you may already know - I'm just listing a complete roadmap to Solovay.)
To start with, you need a good understanding of what models of ZFC look like. The last co... | 16 | https://mathoverflow.net/users/8133 | 104456 | 60,501 |
https://mathoverflow.net/questions/104439 | 6 | How to deduce a formula (see below) for number of conjugacy classes in $\operatorname{GL}\_n(F\_2)$? (More generally F\_q) ?
Is there some description of conjugacy classes or we just know how many of them but do not know how to describe them ?
Can someone send me a paper, please ?:
"Pairs of commuting matrices over a... | https://mathoverflow.net/users/10446 | Conjugcy classes in GL(F_2)? GL(F_q) | **Edit:** Unfortunately, the first version of this answer was wrong, so I had to rewrite it almost entirely.
So the formula you want to prove is:
$$
\sum\_{n=1}^\infty C\_{n,q} t^n =\prod\_{k=1}^\infty \frac{1-t^k}{1-qt^k}
$$
where $C\_{n,q}$ denotes the number of conjugacy classes in $\textrm{GL}\_n(q)$.
Note tha... | 6 | https://mathoverflow.net/users/17498 | 104457 | 60,502 |
https://mathoverflow.net/questions/104467 | 4 | In Hatcher's Algebraic Topology book it is noted after 4.61 that:
fiber preserving map + homotopy equivalence $\Rightarrow$ fiber homotopy equivalence.
Question:
---------
Could there be two fibrations over the **same** base space where the total spaces ***are*** homotopy equivalent, but there is **no** fiber ho... | https://mathoverflow.net/users/17083 | A homotopy equivalence between total spaces in a (Hurewicz) fibration which is not a fiber homotopy equivalence | You can fiber a circle over a circle in many ways.
| 14 | https://mathoverflow.net/users/6666 | 104470 | 60,506 |
https://mathoverflow.net/questions/104473 | 2 | A sequence of vector fields on an open subset of a manifold is linearly independent in module sense, i.e., if we multiply by smooth functions. Does it imply the pointwise independence of the vector fields?
| https://mathoverflow.net/users/16425 | Independent vector fields | No: consider the vector fields $\frac{\partial}{\partial x}$ and $x\frac{\partial}{\partial y}$ on $\mathbb R^2$.
| 6 | https://mathoverflow.net/users/4790 | 104475 | 60,507 |
https://mathoverflow.net/questions/104465 | 5 | Suppose that $P$ is a forcing poset in $V$. If $\pi$ is an automorphism of $P$ then $\pi$ extends to automorphisms of the names by induction: $$\pi\dot x = \lbrace(\pi p,\pi\dot y)\mid (p,\dot y)\in\dot x\rbrace$$
I've been stuck on the following proposition for quite some time now and I don't see an argument for, no... | https://mathoverflow.net/users/7206 | Equivalence of forcing automorphisms | It depends on the forcing notion, on the generic filter, on the
group of automorphisms and on the name $\dot x$.
First, there are some trivial cases where it is normal, such as
the case of a check name $\check x$, which is fixed by every
automorphism $\pi$, and so that subgroup is the whole group and hence normal. Si... | 5 | https://mathoverflow.net/users/1946 | 104478 | 60,510 |
https://mathoverflow.net/questions/104434 | 6 | This is a follow-up question to [another](https://mathoverflow.net/questions/101892/when-is-homotopy-orbit-space-weakly-equivalent-to-orbit-space-other-than-situati) question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Qu... | https://mathoverflow.net/users/11540 | State of knowledge on the Commutative W-spaces which appear in "Model Categories of Diagram Spectra" | There is a very illuminating paper I did not know about when I last answered a similar question:
Tyler Lawson. ``Commutative $\Gamma$ rings do not model all commutative ring spectra". Homology,
Homotopy and Applications Vol 11, 2009, 189-194. The title says it all. Together with the monoidal functor from $\Gamma$-space... | 5 | https://mathoverflow.net/users/14447 | 104483 | 60,514 |
https://mathoverflow.net/questions/104484 | 3 | Hi all. I have a linear program with the restriction that every variable can be zero or greater than or equal to a positive constant. That is:
minimize: $w^Tx$
subject to: $Ax=b$, $Cx \le d$ and for each $x\_i$, $x\_i = 0$ OR $x\_i \ge k\_i, k\_i > 0$.
EDIT: it is also known that $w \ge 0$ and $A$ and $C$ are binar... | https://mathoverflow.net/users/25641 | linear programming with OR restrictions | Your additional constraints make the feasible region for your problem nonconvex, and thus it cannot be represented as a linear programming problem.
If you can obtain an upper bound $M\_{i}$ on the maximum value of $x\_{i}$, then there is a standard approach to these problems in which the problem is formulated as a 0-... | 3 | https://mathoverflow.net/users/9022 | 104487 | 60,516 |
https://mathoverflow.net/questions/104413 | 11 | Is the following fact about Poisson random variables true?
>
> For any $\lambda \in (0,1)$ and integer $k > 0$, if $X$ is a Poisson random variable with mean $k \lambda$, then $\Pr(X < k) \geq e^{-\lambda}$.
>
>
>
It clearly holds for $k=1$, and for any fixed $\lambda$ it's easy to see that it holds for all su... | https://mathoverflow.net/users/8049 | Tail bound for Poisson random variable | The argument below is a bit weird because it is a mixture of an explicit computation and a general handwaving (rigorous handwaving, of course), but, when figuring it out at 70 mph
under medium strength rain, I could use neither pen and paper, nor the full power of my imagination, so I just took whatever came easily fro... | 7 | https://mathoverflow.net/users/1131 | 104495 | 60,522 |
https://mathoverflow.net/questions/104464 | 1 | I am building upon MO question 102846 concerning the Tarski-Grothendieck set theory (TG).
I have two questions;
1/ I think that it is possible that the axiom of pairing (axiom 4 of the TG theory presented in question 102846) could be deleted. So, my question is: Is axiom 4 independant of the others axioms of the TG set... | https://mathoverflow.net/users/30395 | Tarski-Grothendieck set theory, the axiom of pairing and the axiom of specification | Assume all the axioms of TG except for pairing. We will show that pairing and specification (a.k.a. separation) both follow, with the caveat in the following paragraph.
We will also assume that the empty set exists. This does not seem to follow from the TG axioms as defined in the linked question [What's the differen... | 1 | https://mathoverflow.net/users/1682 | 104497 | 60,524 |
https://mathoverflow.net/questions/104493 | 2 | Is the set of Godel numbers of $\Delta\_1$ formulae itself $\Delta\_1$ definable (i.e., computable)?
| https://mathoverflow.net/users/25648 | The set of $\Delta_1$ indices | The answer is no, in either of the cases Trevor outlined.
For example, for a fixed $e$, consider the formulas $p(x)\equiv$"There is some $n$ such that $\Phi\_e(x)[n]\downarrow=1$" and $q(x)\equiv$"There is some $n$ such that $\Phi\_e(x)[n]\downarrow=0$." These formulas are both $\Sigma^0\_1$, and they define compleme... | 6 | https://mathoverflow.net/users/8133 | 104501 | 60,526 |
https://mathoverflow.net/questions/104500 | 0 | Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. Therefore, I have a cascaded optmization problem. Is there anyway to combine them in a single optimization problem?
| https://mathoverflow.net/users/25641 | Cascading minimization problems | You can solve the first LP, obtain the optimal value, and then add a constraint that the first objective has that optimal value. Then you can add your second set of constraints to the original set of constraints and solve that LP. e.g. if your original LP is
$\max c^{T}x $
subject to
$Ax=b$
$x \geq 0$
Suppo... | 3 | https://mathoverflow.net/users/9022 | 104502 | 60,527 |
https://mathoverflow.net/questions/104350 | 28 | Let $R$ be commutative unital ring, and $M$ an $R$-module. $M$ is called invertible (a.k.a. projective module of rank one), if it is finitely generated, and $M\_{\mathfrak{p}} \cong R\_{\mathfrak{p}}$ for every $\mathfrak{p} \in \operatorname{Spec} R$. Equivalently, there are $a\_1, \dotsc, a\_n \in R$ such that $(a\_1... | https://mathoverflow.net/users/22804 | Equivalent definitions of invertible modules | I will happily incorporate this strengthening of the statement of Proposition 19.8 of my notes. Thanks for this, and in the future please feel free to contact me directly (as well).
As an aside, I well remember Rota's criticism of commutative algebra texts for their "hygienic theorems": see [this previous MO answer](... | 12 | https://mathoverflow.net/users/1149 | 104503 | 60,528 |
https://mathoverflow.net/questions/104494 | 6 | For every product in a category, there exists an operation $\langle\cdot, \cdot\rangle$ that turns morphisms $f : C \to A$ and $g : C \to B$ into morphisms $\langle f, g\rangle : C \to A \times B$. How are this operation and its results usually called? Calling $\langle f, g\rangle$ the product of $f$ and $g$ does not m... | https://mathoverflow.net/users/25527 | product operation: name and notation | Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the operation have called it the "pairing arrow."
I would write $\langle f\_i\rangle\_{i \in I} : C \to \prod\_{i \in I} A\_i... | 7 | https://mathoverflow.net/users/38783 | 104509 | 60,529 |
https://mathoverflow.net/questions/104505 | 2 | Let $\pi \colon X\to Y$ be a morphism, where $X$ is projective. Suppose the existence of an open subset $U\subset X$ such that $\pi^{-1}(\pi(U))=U$, that $\pi$ is injective on $U$, and that its derivative is injective.
I would like to say that $\pi$ restricts to an isomorphism between $U$ and its image. And I suspect... | https://mathoverflow.net/users/23758 | Birational maps and open subsets inducing isomorphisms | The useful theorem here is "a proper, universally injective, unramified morphism is a closed immersion." see: <http://math.columbia.edu/~dejong/wordpress/?p=1358>
$X$ being projective and $\pi^{-1}(\pi(U))=u$ ensures that $U$ is proper over $\pi(U)$, by the base change property of properness.
You need to assume tha... | 4 | https://mathoverflow.net/users/18060 | 104512 | 60,530 |
https://mathoverflow.net/questions/104519 | 10 | It's now well known that Continuum Hypothesis (CH) is independent from standard axioms of set theory: one can assume that $c=\aleph\_1$ or assume that $c \neq \aleph\_1$. Let us assume the second case-then the natural queation rise:
What 'values' $c$ can take? I know that it's impossible that $c=\aleph\_{\omega}$? Is... | https://mathoverflow.net/users/24078 | Around Continuum Hypothesis | Any cardinal of uncountable cofinality can consistently be the cardinal of the continuum. In particular, it is consistent that the cardinal of the continuum is weakly inaccessible (assuming, of course, that inaccessible cardinals are consistent.) The relevant paper is Solovay's "$2^{\aleph\_0}$ can be anything it ought... | 11 | https://mathoverflow.net/users/6794 | 104520 | 60,532 |
https://mathoverflow.net/questions/104513 | 11 | As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem:
>
> A weak homotopy equivalence induces isomorphisms of the corresponding integral singular homology. Conversely, a map between simply connected spaces which induces isomorphisms of the corresponding in... | https://mathoverflow.net/users/17083 | A map inducing isomorphisms on homology but not on homotopy | If the isomorphism in homology is meant with integral coefficients (or all constant coefficients), you can take the classifying space $BG$ of any non-trivial discrete acyclic group, for example Higman's four-generator four-relator group (see <http://www.encyclopediaofmath.org/index.php/Acyclic_group>).
By definition... | 12 | https://mathoverflow.net/users/10194 | 104522 | 60,533 |
https://mathoverflow.net/questions/104536 | 9 | Equivalently, is the free Lie algebra on finitely many generators over a fixed field $k$ (say of characteristic not equal to $2$) residually finite-dimensional in the sense that any nonzero element remains nonzero in some finite-dimensional quotient? Some quick Googling on my part was not successful here.
**Motivati... | https://mathoverflow.net/users/290 | Does any identity holding in all finite-dimensional Lie algebras hold in all Lie algebras? | Maybe I am missing something but let me try. The free Lie algebra is a subalgebra of the free associative algebra made Lie via the bracket. The free associative algebra is residually finite dimensional by truncating polynomials. Hence its Lie algebra is residually finite dimensional. Here I assume finitely generated bu... | 9 | https://mathoverflow.net/users/15934 | 104537 | 60,539 |
https://mathoverflow.net/questions/104540 | 5 | Let $R$ be a (not necessarily commutative) ring and $M$ a simple right $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. It is seems known that
$$
pdim\_{R}(M)=pdim\_{R\_{\mathfrak{m}}}(M\_{\mathfrak{m}})
$$
where $R\_{\mathfrak{m}}$ is the localization of $R$ with respect to $\mathfrak{m}$. "$\ge$"-par... | https://mathoverflow.net/users/25659 | Projective dimension of simple module | In general, it is not true that $Ann\_R(M)$ is a maximal two-sided ideal of $R$ if $M$ is a simple $R$-module. For example, let $k$ be a field of characteristic zero, let $\mathfrak{g} = \mathfrak{sl}\_2(k)$ and let $A = U(\mathfrak{g}) / \langle C \rangle$ where $C$ is the Casimir element. Then the Verma module $V$ of... | 8 | https://mathoverflow.net/users/6827 | 104546 | 60,542 |
https://mathoverflow.net/questions/104549 | 4 | Consider the following one-dimensional version of the game battleships. There is a battleship somewhere on $\mathbb N$, i.e., a interval $N,\ldots,N+k$. Your task is to find whether this battleship lies in the interval $1,\ldots,n$ using the minimal number of tests (on can ask if the battlship includes $i$ for any $1\l... | https://mathoverflow.net/users/5734 | battleship permutation | The version of the Battleship game doesn't seem to match either the question or the motivation.
Suppose $n=15$ and $k=2$ so the Battleship has length $3$. There are $5$ disjoint positions $1-3, 4-6, 7-9, 10-12, 13-15$ so you have to make at least $5$ guesses to conclude the Battleship is not on the interval, and you ... | 3 | https://mathoverflow.net/users/2954 | 104555 | 60,548 |
https://mathoverflow.net/questions/104482 | 0 | Edit: So, my original question (stated below) was to find an error in my "proof" that immediate parabolic basins for rational maps are always simply connected.
Since I have not received any answers as of yet I would ask alternatively if someone could point out an explicit example of a rational map with a parabolic fixe... | https://mathoverflow.net/users/25646 | parabolic immediate basins always simply connected? | The mistake is in the statement that $\partial B\subset F(f)$.
There can be points on $\partial B$ and $\partial V\_m$ which are in $J(f)$,
namely preimages of $z\_0$ :-)
An example is $f(z)=z+1-1/z$. There is one petal for the neutral point at infinity.
Let $A$ be the dmain of attraction of $\infty$.
Critical points... | 3 | https://mathoverflow.net/users/25510 | 104557 | 60,549 |
https://mathoverflow.net/questions/104458 | 11 | I've been reading Chen's original works on iterated integrals and in order to consider differential forms on the path space $PM$ of a smooth manifold $M$ he gives $PM$ the following "differentiable space" structure:
Let $N$ be a smooth manifold. A continuous map $\alpha: N \to PM$ (where $PM$ has the compact open top... | https://mathoverflow.net/users/5450 | Vector fields on path spaces | Good question!
There is a fairly well developed theory of *diffeological spaces*, with many recent references. The only difference between diffeological spaces and Chen's *smooth spaces* is that the plots of a diffeological space have *open* subsets $U \subset \mathbb{R}^n$ as their domains, while the ones of a smoot... | 6 | https://mathoverflow.net/users/3473 | 104564 | 60,551 |
https://mathoverflow.net/questions/104556 | 17 | Is there something well-known about which ring can be a cohomology ring of a manifold? More concretely, I would be interested in the following question: does there exists an $m$, s.t. for each $r,s\in\mathbb{N}$ and $\beta\_{i,j}^k\in\mathbb{Z}$, there exists a connected compact $m$-manifold $M$ s.t. $H^2(M)\simeq \mat... | https://mathoverflow.net/users/10072 | What can be the cohomology ring of a manifold? | You probably meant $\beta\_{ij}^k=\beta\_{ji}^k$ (the cup-product is commutative). With that condition, there is always a simply-connected compact $7$-manifold $M$ whose only non-trivial cohomology groups are $H^0(M)\cong\mathbb{Z}$, $H^2(M)\cong\mathbb{Z}^r$, and $H^4(M)\cong\mathbb{Z}^s$, and such that the cup-produc... | 28 | https://mathoverflow.net/users/12166 | 104568 | 60,552 |
https://mathoverflow.net/questions/104563 | 5 | What can one say about the relation between the Hodge decompositions
of $H^\\*(X,C)$ and $H^{\*}(X\_\sigma,C)$
for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism
of the field of complex numbers ? I am also interested in conjectural results, e.g.
whether Standard Conjectures imply someth... | https://mathoverflow.net/users/22247 | comparing Hodge structures on cohomology of conjugate varieties | As Jason & Will have already commented, the Hodge numbers are the same for
conjugate pairs because of GAGA and the fact algebraic coherent cohomology behaves well with respect to field extensions. In fact, there would be an isomorphism of filtered vector spaces
$$(\mathbb{H}^i(X,\Omega^\bullet), F^\bullet)\otimes\_\mat... | 8 | https://mathoverflow.net/users/4144 | 104569 | 60,553 |
https://mathoverflow.net/questions/104552 | 3 | What is the asymptotic behaviour or an upper bound for $\int\_0^{\infty} \exp(-c x^a+K x^b) \, dx$, for $a>b>0,$ as $K\rightarrow \infty$?
Or any good reference for tools to tackle this question?
I think the growth in $K$ should be polynomial because $-c x^a+K x^b=0$ yields $x=(K/C)^\frac{1}{a-b}$ on the range $[0,(K... | https://mathoverflow.net/users/19874 | Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $K\rightarrow \infty$? | One can get a full asymptotic expansion as an application of [Watson's Lemma](http://en.wikipedia.org/wiki/Watson%27s_lemma). One need only observe that the integrand is maximized at $x\_0 = \left(\frac{Kb}{ac}\right)^{1/(a-b)}.$
Substituting $x = x\_0 u,$ one gets(where $I$ is the original integral) $I = x\_0 \int\_... | 3 | https://mathoverflow.net/users/11142 | 104587 | 60,561 |
https://mathoverflow.net/questions/104598 | 1 | I have arrived to this conjecture in my work, I am not sure that is true or false. So I would appreciate if someone give a counterexample or prove it.
My question: Let n be a non-prime such that n-1 be a prime. Is it true that there exist a prime p such that n-p is a prime as well?
| https://mathoverflow.net/users/25674 | A question concerns prime numbers | The conjecture that any even number (greater than 4) is the sum of two odd primes is well-known to be open. You are asking about a special case that is also open.
| 6 | https://mathoverflow.net/users/121 | 104604 | 60,570 |
https://mathoverflow.net/questions/104592 | 4 | Hello everyone,
Suppose that I am defining a function which embeds a surface (manifold) in $\mathbb{R}^3$.
Is there a standard symbol or letter that is used for this function?
Additionally, is there any other classical or standard notation (such as the hooked arrow for inclusion maps) of which I ought to be aware... | https://mathoverflow.net/users/12069 | Standard notation/symbol for an embedding function | I usually use $\iota$ (\iota) for all kinds of embeddings.
| 8 | https://mathoverflow.net/users/15327 | 104612 | 60,573 |
https://mathoverflow.net/questions/104602 | 3 | I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x\_0$$ where the equation is over same domain $t \in (a,b)$, then via the implicit function theorem, there is a unique solution around neighborhoods of the initial value and the parameter, and the s... | https://mathoverflow.net/users/25650 | ODE continuous dependence on parameters to PDE | Yes, you can get some results on some PDE's directly with abstract ODE methods. However, in my experience with nonlinear PDE's it tends to be preferable to deal directly with the iteration/fixed point method, because the abstraction you get by using ODE method is very thin, and so it does not add much insight. Removing... | 2 | https://mathoverflow.net/users/824 | 104619 | 60,578 |
https://mathoverflow.net/questions/104628 | 5 | I'm looking at irreducible, real representations of $SO(3)$. The 5-dimensional irrep is isomorphic to the space of trace-free quadratic forms on $\mathbb{R}^3$, and we all know that any such quadratic form can be diagonalized by the standard $SO(3)$ action on $\mathbb{R}^3$. My question is: is there any analogous "norm... | https://mathoverflow.net/users/18048 | Normal form for trace-free real cubic forms in 3 variables under SO(3)-action? | One way to do this is to reduce this to understanding sets of points on the $2$-sphere up to a certain equivalence. If you let $\mathcal{H}\_k$ denote the homogeneous polynomials $p$ of degree $k$ on $\mathbb{R}^3$ that are harmonic, i.e., that satisfy $\Delta p = 0$, then $\mathcal{H}\_k$ is an irreducible $\mathrm{SO... | 6 | https://mathoverflow.net/users/13972 | 104631 | 60,582 |
https://mathoverflow.net/questions/102854 | 7 | If AD$\_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be weakened to "AD holds and there is a Suslin cardinal above $\kappa$." Is this correct? If so, does anyone know where a proof can ... | https://mathoverflow.net/users/1682 | Weakly homogeneous trees under AD | I hope it's okay to post an answer to my own question. I am essentially repeating Woodin's proof that he just showed me. Any errors were probably introduced by me.
Recall that under AD all measures on ordinals are countably complete and ordinal-definable.
By the coding of measures theorem of Kechris assuming AD and t... | 6 | https://mathoverflow.net/users/1682 | 104632 | 60,583 |
https://mathoverflow.net/questions/104630 | 8 | Suppose we have three $n \times n$ matrices $A$, $B$, $C$ with floating point entries. We would like to compute the polynomial $\det (xA+yB+zC)$. At least in Mathematica, and I think in all computer algebra systems, this will take $n!$ steps; Mathematica chokes around $n=15$.
What's the smart way to do this? I have ... | https://mathoverflow.net/users/297 | Computing determinants of matrices of linear forms | The smart way of doing this is by interpolation (this, by the way, is how Mathematica computes the characteristic polynomial), and in Mathematica you can use InterpolatingPolynomial[] to do this in a couple-of-line program.
**Remark** In fact, to compute the characteristic polynomial of an integer matrix, Mathematica... | 4 | https://mathoverflow.net/users/11142 | 104634 | 60,585 |
https://mathoverflow.net/questions/104627 | 25 | This question was motivated by a more general question raised by Jan Weidner [here](https://mathoverflow.net/questions/104102/). In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}\_q$ of characteristic $p$. Here there is a natural action of a Frobenius morphism... | https://mathoverflow.net/users/4231 | Example of non-projective variety with non-semisimple Frobenius action on etale cohomology? | Actually, the semisimplicity should hold with no hypotheses on X, so no example should exist. In fact it is generally expected that, with char. 0 coefficients and over a finite field (both hypotheses being necessary), every mixed motive is a direct sum of pure motives -- so the question for arbitrary varieties reduces ... | 27 | https://mathoverflow.net/users/3931 | 104635 | 60,586 |
https://mathoverflow.net/questions/104636 | 0 | Hello!
Let $K = G\ast\_U H$ be the free product of the groups $G$ and $H$ amalgamated along the common subgroup $U$. I have a homomorphism of groups $f:K\to K'$ defined using the universal property of free products (i.e., it is the unique map determined by two homomorphisms $j\_1 : G\to K'$ and $j\_2 : H\to K'$ that ag... | https://mathoverflow.net/users/25685 | Injectivity of a group homomorphism with domain a free product | If you have a homomorphism to $K'$ where $K'$ acts on a set $X$, then you could use ping-pong arguments to verify injectivity. In your setting, this is called 1-st Maskit combination theorem, you can find some examples in Maskit's book "Kleinian groups". If $K=G\_1\star G\_2$, then ping-poing setting is a pair of subse... | 2 | https://mathoverflow.net/users/21684 | 104639 | 60,588 |
https://mathoverflow.net/questions/104601 | 1 | Let $R$ be (not necessarily commutative) ring and $S$ a simple right $R$-module. Let $f\in Ann(S)$ be normalizng and a non-zero divisor. Is it always true that
$$
pdim\_{R}(S)=pdim\_{R/(f)}(S)+1?
$$
| https://mathoverflow.net/users/25659 | Projective dimension over hypersurface | With the definition of normalizing you give, it is not always the case that the projective dimension of $S$ as an $R$-module equals $1$ more than the projective dimension of $S$ as an $R/\langle f \rangle$-module. Let $R$ be $\mathbb{Z}$, let $f$ be $p^2$ for some prime $p$, and let $S$ be $\mathbb{Z}/p\mathbb{Z}$. The... | 1 | https://mathoverflow.net/users/13265 | 104644 | 60,590 |
https://mathoverflow.net/questions/104651 | 5 | Haken proved that an incompressible surface in a triangulated irreducible 3-manifold is isotopic to a surface which is *normal* with respect to the triangulation ([Theorie der Normalflächen. Acta Math. 105 1961 245–375](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=79845&vfpref=html&r=29&mx-pid=141106)).... | https://mathoverflow.net/users/4325 | When are incompressible surfaces isotopic into a two-skeleton? | Unfortunately, you can't always expect every incompressible surface to be isotoped into the $2$-skeleton. There are only finitely many subsets of the $2$-skeleton, but there can be infinitely many nonparallel incompressible surfaces. This is simplest in the $3$-torus, but there are many other examples. Jaco's stair con... | 10 | https://mathoverflow.net/users/2954 | 104653 | 60,593 |
https://mathoverflow.net/questions/104305 | 17 | In topology, given topological spaces $X$ and $Y$, the compact-open topology is considered, under the relatively mild requirement that $X$ be locally compact Hausdorff, to be the most "natural" topology on the set $\mathcal{C}(X,Y)$ of continuous functions $X\to Y$. (I'm not going to even attempt to define "natural" he... | https://mathoverflow.net/users/36720 | "Nice" sigma-algebra on set of measurable functions | There is an impossibility theorem: If you let $\mathcal{L}$ be the the space of Borel-measurable functions $f:[0,1]\to[0,1]$, and $e:\mathcal{L}\times [0,1]\to[0,1]$ the evaluation given by $e(f,x)\mapsto f(x)$, then there is no $\sigma$-algebra on $\mathcal{L}$ such that the evaluation is jointly measurable. The resul... | 16 | https://mathoverflow.net/users/35357 | 104656 | 60,595 |
https://mathoverflow.net/questions/104609 | 3 | I am wondering wether it exists a theorem that any continuous path on the plane one can
approximate with algebraic curve $P(x,y)=0$ ($P$- is a polynom)?
| https://mathoverflow.net/users/24233 | Algebraic curve approximation | Given any compact set $K$ in the plane (in particular the image of a compact interval under a continuous function) and $\epsilon > 0$, there is a finite set $\{(x\_j, y\_j)\}\_{j=1}^n \subseteq K$ such that $K$ is contained in the union of the disks of radius $\epsilon$ centred at $(x\_j, y\_j)$. Then $K$ is within dis... | 3 | https://mathoverflow.net/users/13650 | 104659 | 60,598 |
https://mathoverflow.net/questions/104655 | 5 | This is (quite obviously) inspired by [this question.](https://mathoverflow.net/questions/104645/descartes-rule-of-signs-for-noncommutative-polynomia) Let $C\_i$ be symmetric positive definite matrices. Then is it true that there is exactly one symmetric positive definite $X$ such that
$F(X) = X^n - \sum\_{i=0}^n C\_i... | https://mathoverflow.net/users/11142 | A conjecturally easier version of matrix Descartes rule of signs. | I suspect you mean "is it true that there is exactly one symmetric $X$ such that $F(X) = X^n - \sum\_{i=0}^{n-1} C\_i \circ X^i = 0$", which is more in line with the inspiration question. If you meant something else, I'll just delete this later today.
In that case, the answer is no. Take
$$
C\_2 =
\begin{bmatrix}
1... | 7 | https://mathoverflow.net/users/18086 | 104669 | 60,599 |
https://mathoverflow.net/questions/104594 | 1 | I'm trying to prove a conjecture and need some help.
Consider a continuous, twice differentiable function $p(a)$ such that $p(0) = 0$ and $\forall a$, $p'(a) > 0$ and $p''(a) < 0$ and $p$ is bounded above by 1. I believe it's the case that for $\gamma > 1$, $\exists a' | p'(a) = \gamma p'(\gamma a)$. Another way of sta... | https://mathoverflow.net/users/13456 | Conjecture that two nested convex curves have a point with the same slope | The differentiable function $x\mapsto p(\gamma x)−p(x)$ vanishes at zero and at infinity, and it is non-negative because p is increasing, so it has a maximum point a, which is what you want.
| 1 | https://mathoverflow.net/users/6101 | 104676 | 60,601 |
https://mathoverflow.net/questions/104672 | 0 | Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an infinite linear combination $\sum\_{i=1}^\infty\beta\_i a\_i$ of elements of $A$. Are there simple (non-trivial) conditions g... | https://mathoverflow.net/users/23297 | Infinite linear span vs closed linear span | In the case of linear $A$, which seems to be your case of interest, you can simply do it as follows. For $x$ in the closure of $A$ take a sequence $(x\_n)$ in $A$ converging to $x$ such that $\|x\_n-x\_{n+1}\|<2^{-n}$. Then set $a\_i=x\_i-x\_{i-1}$, $\beta\_i=1$ and the sum will converge absolutely, hence also uncondit... | 4 | https://mathoverflow.net/users/15327 | 104679 | 60,602 |
https://mathoverflow.net/questions/104670 | 4 | Let $f$ be a continuous function on $\mathbb R$ with compact support and $\mu$ a finitely additive measure which is in the dual space of $L^\infty(\mathbb R)$. Is the convolution
$f\ast \mu(x)=\int\_{\mathbb R} f(x-y)d\mu(y)$ a continuous function in $x$? This is really an update of a question I asked, where I took $f... | https://mathoverflow.net/users/17822 | Convolution of a continuous function and a finitely additive measure | Since $f$ has compact support, it is uniformly continuous. Let $h$ be a uniform modulus of continuity. If $|x'-x| < \varepsilon$, then $|f(x-y)-f(x'-y)| < h(\varepsilon)$ for all $y$, hence $|f\*\mu(x)-f\*\mu(x')| < h(\varepsilon)\|\mu\|$ (where $\|\cdot\|$ is variational norm) and $f\*\mu$ is uniformly continuous.
... | 5 | https://mathoverflow.net/users/15327 | 104682 | 60,603 |
https://mathoverflow.net/questions/104686 | 3 | Let $E$ be an elliptic curve in characteristic zero with $Aut(E)=\pm$. Is it possible to have two different isogenies to another elliptic curve $E'$ with distinct cyclic kernels? This isn't possible if $End(E)\cong\mathbb{Z}$ (see for example Lang's Elliptic functions Chapter 2, $\S 2$), but how about if $E$ does have ... | https://mathoverflow.net/users/16858 | Image of isogeny of elliptic curves determines kernel? | It is possible. Suppose $E$ has complex multiplication defined over a number field $F$ by the ring of integers $\mathcal{O}\_K$ in a quadratic imaginary field $K$ (and we take $K$ that only has $\pm 1$ as roots of unity, so that your $Aut(E) = \pm 1$ condition is satisfied). Then one can take $E^{\prime} = E$ and the f... | 6 | https://mathoverflow.net/users/5498 | 104688 | 60,606 |
https://mathoverflow.net/questions/104678 | 9 | If, with a bit abuse of notation, $U\notin R\bigotimes{R}\bigotimes{R}\bigotimes{...}$ (i.e., regardless how long you clebsch up $R$, $U$ won't appear in the expansion), I call $U$ unsociable with respect to $R$ (in the group $G$). Clearly everything else is unsociable with the $1$ irrep, since (duh) $1\bigotimes{1}\bi... | https://mathoverflow.net/users/11504 | Are there "unsociable" irreps? (Definition inside) | It follows from a theorem of Burnside/Steinberg that if G is a finite group then the tensor powers of a module V contain all irreps if and ony if V is faithful for G.
| 15 | https://mathoverflow.net/users/15934 | 104689 | 60,607 |
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