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https://mathoverflow.net/questions/117774 | 4 | Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the orbital digraph $Graph(\Delta)$ is a digraph with vertex set $\Omega$ and edge set $\Delta$. Clearly, $G$ is a subgroup ... | https://mathoverflow.net/users/27831 | automorphism group of orbital graphs | The property of $G$ you are looking at is called *2-closure*, i.e., you ask for a classification of 2-closed permutation groups. See the paper by Liebeck, Praeger, Saxl,
On the 2-closures of finite permutation groups.
*J. London Math. Soc.* (2) **37** (1988), no. 2, 241–252, where this question is investigated for a p... | 1 | https://mathoverflow.net/users/11100 | 117788 | 66,518 |
https://mathoverflow.net/questions/117787 | 2 | Hello,
Does anybody know a reference for the following result: $d\ge 5$ points of $\mathbb P^2$ fail to impose independent conditions on curves of degree $d-3$ if and only if at least $d-1$ of these points are collinear. As usual, "fail to impose independent conditions" means $h^0(\mathcal I\_D(d-3))>h^0(\mathcal O\_... | https://mathoverflow.net/users/29992 | Points in the plane imposing independent conditions: reference request | You mean "curves of degree $d-3$". A reference (for a more general result) is: D. Eisenbud, M. Green, and J. Harris, CayleyBacharach theorems and conjectures, Bull. Amer.
Math. Soc. 33 (1996), 295–324.
| 4 | https://mathoverflow.net/users/30339 | 117792 | 66,520 |
https://mathoverflow.net/questions/117791 | 5 | What other axioms in set theory are stronger than AC ? I mean what are those axioms that will imply AC ?
| https://mathoverflow.net/users/nan | What axioms are stronger than the Axiom of choice? | The axiom "every set is constructible" (denoted **V** = **L**), and the axiom "every set is definable from an ordinal parameter" (denoted often as **V**= **HOD**, and sometimes as **V**= **OD**) each implies AC, and each is provably stronger than AC.
>
>
> >
> > More specifically, it is well-known that:
> >
> > ... | 14 | https://mathoverflow.net/users/9269 | 117795 | 66,522 |
https://mathoverflow.net/questions/117794 | 13 | I'm confused about a seeming contradiction that is probably just a reflection of ignorance on my part. Let's try to compute the Morava E-theory of $B \mathbb{Z}/p$ in two different ways.
1. First, following [Ravenel-Wilson](http://www.math.rochester.edu/people/faculty/doug/mypapers/emspace.pdf) (see also [Hopkins-Kuh... | https://mathoverflow.net/users/4649 | Rational Morava E-theory of cyclic groups | Your guess is correct, and I think this is a confusion pretty much everyone has when they first see these computations. What's going on here is essentially just the simple algebraic fact that $$\mathbb{Z}[p^{-1}][[x]]\neq\mathbb{Z}[[x]][p^{-1}].$$
In computation (1), you can work directly with a Gysin sequence for $E... | 15 | https://mathoverflow.net/users/75 | 117803 | 66,526 |
https://mathoverflow.net/questions/117806 | 5 | A cardinal $\kappa$ is weakly inaccessible iff $\kappa > \omega$, $\kappa$ is regular, and $\forall\lambda<\kappa(\lambda^+<\kappa)$
(here $\lambda^+$ is the successor cardinal)
A cardinal $\kappa$ is strongly inaccessible iff $\kappa > \omega$, $\kappa$ is regular, and $\forall\lambda<\kappa(2^\lambda<\kappa)$
M... | https://mathoverflow.net/users/30341 | If $\kappa$ is weakly inaccessible, then is it the $\kappa$-th aleph fixed point | If $\kappa$ is weakly inaccessible, then it is a limit cardinal and hence $\kappa=\aleph\_\lambda$
for some limit ordinal $\lambda$. Since the cofinality of $\aleph\_\lambda$ is
the same as the cofinality of $\lambda$, it follows by the regularity of $\kappa$ that
$\lambda=\kappa$, and so $\kappa=\aleph\_\kappa$, an $\... | 7 | https://mathoverflow.net/users/1946 | 117809 | 66,529 |
https://mathoverflow.net/questions/117808 | 0 | Hello all,
I am attempting to understand the proof of Lemma III.8 of Beauville's Complex Algebraic Surfaces:
Let $S$ be a minimal surface, $C$ a smooth curve, $p:S\rightarrow C$ a morphism with generic fibre isomorphic to $\mathbb{P}^1$.
Then $S$ is geometrically ruled by $p$.
The proof begins by selecting an arbit... | https://mathoverflow.net/users/21663 | The intersection multiplicity of the canonical divisor of a surface with a fibre of a map to a curve | You use that S is a minimal surface. Thus relatively minimal. Thus, it has no minus one curves. Also, the arithmetic genus is constant in the fibers. The latter is a fairly general fact about fibered surfaces.
| -1 | https://mathoverflow.net/users/30342 | 117811 | 66,531 |
https://mathoverflow.net/questions/117770 | 6 | In "stack project", there is a lemma on finite locally free morphisms, saying that a finite locally free morphism of schemes is equivalent to a morphism which is finite, flat, and locally of finite presentation.
For the proof, they refer to the commutative algebra fact that a module is finite locally free iff it is f... | https://mathoverflow.net/users/26460 | Does 'finite + finitely presented as an algebra' equal 'finitely presented as a module'? | EGA IV$\_1$, 1.4.7.
| 14 | https://mathoverflow.net/users/30180 | 117812 | 66,532 |
https://mathoverflow.net/questions/16833 | 20 | Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples.
Using his definition it is unclear how to separate the smooth structure from the metric.
How can we define a noncommutative smooth manifold without the additional Riemannian and spin^c structures?
Any re... | https://mathoverflow.net/users/402 | Noncommutative smooth manifolds | I'm a bit wary of resurrecting such an old question, but given that the precise content of the reconstruction theorem doesn't seem to be terribly well disseminated, please permit me to [cross-post from math.SE](https://math.stackexchange.com/a/268334/49610) and then make some extra comments:
"To be absolutely clear a... | 16 | https://mathoverflow.net/users/6999 | 117813 | 66,533 |
https://mathoverflow.net/questions/117690 | 6 | [This question](https://math.stackexchange.com/questions/34522/cubic-polynomials-that-generate-the-same-extension) asks how to tell whether two cubic polynomials with coefficients in $\mathbb{Q}$ have the same splitting field. There are several answers to the question, but they don't include proofs. Also, it's not clea... | https://mathoverflow.net/users/683 | Algorithm for determining whether two polynomials have the same splitting field | In practice, you can reduce f and g mod the first 1000 primes (deleting those where f and g are ramified) and see if the primes all have the same splitting behavior. If they don't, the splitting fields are not the same, and if they do, you can be pretty darn sure that they are. If you want to replace "be pretty darn su... | 10 | https://mathoverflow.net/users/431 | 117814 | 66,534 |
https://mathoverflow.net/questions/117786 | 7 | Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M people.
* What is the distribution $\mu$ on the positive integers that you get?
This is a serious question, obviously ... | https://mathoverflow.net/users/29333 | What is a random number? (poll experiment) | Cognitive sychologists study this kind of question, as you might expect. [Here's a paper](http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00250/abstract) (behind a paywall, sorry) where they asked people to name random digits. You don't get uniform distribution on 0,..,9.
I learned a little about this stuff when... | 14 | https://mathoverflow.net/users/431 | 117822 | 66,539 |
https://mathoverflow.net/questions/117815 | 1 | There are some facts that can be found by the spectrum of adjacency matrix of graph.For example, the number of edges and vertices, is bipartite or not, is complete multipartite or not and so on. Can we say anything about the clique number of two cospectral graphs?
We can construct the graphs $G\_1$ and $G\_2$ that th... | https://mathoverflow.net/users/19885 | The cliques of cospectral graphs | Let $X$ and $Y$ be two cospectral graphs with maximum clique size $a$ and $b$ respectively. Then their $k$-fold strong powers $X(k)$ and $Y(k)$ are cospectral and the
maximum size of a clique is $a^k$ and $b^k$ respectively. (The cliques of maximum size in a strong product are strong products of maximum sized cliques i... | 3 | https://mathoverflow.net/users/1266 | 117829 | 66,543 |
https://mathoverflow.net/questions/117827 | 0 | I want to know a proof of this fact: "every simple group has a minimal simple group as a subquotient." (If $H$ and $K$ are two subgroups of $G$ s.t. $H\lhd K$, then $\frac{K}{H}$ is called a subquotient of $G$) many thanks.
| https://mathoverflow.net/users/27962 | A question about minimal simple group | If every proper subgroup of G is soluble, then it is done. So there is a non-soluble subgroup H with every subgroup soluble. Let K be maximal normal subgroup of H, then H/K is a minimal simple group.
| 3 | https://mathoverflow.net/users/22049 | 117835 | 66,548 |
https://mathoverflow.net/questions/117840 | 1 | Hi,
as a continuation to the fully answered question:
[Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$](https://mathoverflow.net/questions/117609/injectiveintregrable-mapping-from-mathbb-r3-to-mathbb-r)
Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only a finite number ... | https://mathoverflow.net/users/28037 | Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$ | Assume $f:\mathbb{R}^n\rightarrow[0,1]$ is an injective map. Take any circle $C\subset \mathbb{R}^n$. If $f$ is continuous on $C$, then the restriction of $f$ to $C$ is a homeomorphism $C\to f(C)$ (because $C$ is compact), which is a contradiction, because $C$ minus any point is connected, while $f(C)$ minus any point ... | 7 | https://mathoverflow.net/users/11919 | 117844 | 66,550 |
https://mathoverflow.net/questions/117848 | 2 | What are the necessary and sufficient conditions for a group $H$ to be a central extension of the quaternions $Q\_8$ ?
| https://mathoverflow.net/users/nan | Extensions of Groups | So you mean "central extension". Let $H/K \cong Q\_8$ with $K \le Z(H)$. Since $Q\_8$ has trivial Schur Multiplier, $K \cap H' = 1$, so $|H'| = 2$, and $H$ is a subdirect product of an abelian group $L$ with $Q\_8$, where $L$ and $Q\_8$ have the common quotient $V = C\_2^2$ (the Klein 4-group).
More precisely, let $L... | 6 | https://mathoverflow.net/users/35840 | 117856 | 66,554 |
https://mathoverflow.net/questions/117857 | 0 | This question might be trivial but I cann't see.
Let $A$ and $B$ be two modules. is it always possible to have an exact sequence which begins with $A$, ends with $B$ with all modules in the sequence (other than $A$ and $B$) projective !?
| https://mathoverflow.net/users/30356 | Exact sequences | Not in general. The keyword is stable module category, the quotient of the module category by the ideal of morphisms which factor through a projective. The leftmost term is functorial on the rightmost term in this category if all intermediate modules are projective. This imposes some restrictions. If you take a heredit... | 1 | https://mathoverflow.net/users/12166 | 117858 | 66,555 |
https://mathoverflow.net/questions/117861 | 1 | I suspect the Granville-Langevin conjecture implies this.
Let $F(x,y)=0$ be a curve with infinitely many integral points $(F\_n,F\_{n+1})$.
Is it true that either $x^2+x y - y^2 -1$ or $x^2+ x y -y ^2 +1$ divides $F(x,y)$?
Probably this can be generalized to integral points $(F\_{f(n)},F\_{g(n)})$.
| https://mathoverflow.net/users/12481 | Curves with infinitely many integral points consecutive Fibonacci numbers | You don't need any conjectures for this. If two curves have infinitely many points in common then they have a common component (this is a basic fact of the Zariski topology). As $x^2+xy-y^2-1$ and $x^2+xy-y^2+1$ are irreducible, one of them must divide $F$.
| 7 | https://mathoverflow.net/users/4140 | 117864 | 66,559 |
https://mathoverflow.net/questions/117847 | 2 | Hello everyone
thanks to all of you
I have two questions and I hope to get some guide:
1. One of the Lie groups in the Berger's list of holonomy groups of locally irreducible Riemannian manifolds is $Sp(n).Sp(1)\cong Sp(n) \times Sp(1)/(\pm Identity)$, which can be considered as a Lie subgroup of $SO(4n)$. what... | https://mathoverflow.net/users/26148 | A lie Subgroup of SO(4n) | One way to make the subgroup $\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)\subset\mathrm{SO}(4n)$ explicit is to think of $\mathbb{R}^{4n}$ as $\mathbb{H}^n$, i.e. as columns of quaternions of height $n$, where $\mathbb{H}\simeq\mathbb{R}^4$ is the ring of quaternions. Then $\mathrm{Sp}(n)$ can be thought of as the group of $n$-b... | 2 | https://mathoverflow.net/users/13972 | 117869 | 66,560 |
https://mathoverflow.net/questions/117843 | 1 | Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
$\sum\_{i=0}^{n} (\frac{\partial^{2}f}{\partial t^{2}})^{i} K\_{i}(x,t,f,\frac{\partial f}{\partial t}, \frac{\partial^{2} ... | https://mathoverflow.net/users/22073 | Solution of a PDE and its uniqueness | Here is what you should try: Consider the function
$$
p(x,\lambda) = K\_0(x,0,\ldots,0)+\lambda\ K\_1(x,0,\ldots,0) + \cdots + \lambda^n\ K\_n(x,0,\ldots,0)
$$
For any analytic solution $\lambda=L(x)$ to this polynomial equation that is a simple root of $p(x,\lambda)=0$, then, by the Cauchy-Kovalevskya Theorem, there i... | 7 | https://mathoverflow.net/users/13972 | 117871 | 66,562 |
https://mathoverflow.net/questions/117879 | 1 | Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.
Question 1. Whether that $f$ is surjective implies that $f$ is injective?
Question 2. Whether that $f$ is injective implies that $f$ is surjective?
Question 3. $g: x\to x^b$ is a multiplication hom... | https://mathoverflow.net/users/30259 | Questions about multiplicative homomorphism of $\mathbb{R}$ | There is an isomorphism $\mathbb{Z}/2\mathbb{Z}\times (\mathbb{R}^+,\cdot )\to K$ given by $(x,y)\mapsto (-1)^xy$ (considering $\mathbb{Z}/2\mathbb{Z}$ to contain 0 and 1).
There is also an isomorphism $(\mathbb R,+)\to(\mathbb R^+,\cdot)$ given by $x\mapsto exp(x)$.
So for all your questions it is enough to consider... | 5 | https://mathoverflow.net/users/30363 | 117880 | 66,566 |
https://mathoverflow.net/questions/117878 | 3 | Assume $X$, $E$ and $G$ are topological groups and $1\to X\to E\to G\to 1$ a short exact sequence of continuous group homomorphisms.
Under which of these conditions is $E$ a profinite group?
(i) $G$ profinite, $X$ finite
(ii) $G$ profinite, $X$ pro-$p$ for some prime $p$
(iii) $G$ profinite, $X$ profinite
In th... | https://mathoverflow.net/users/30363 | Are extensions of profinite groups profinite? | The answer to all three questions is no in general. You need to assume in addition that $G$ carries the quotient topology from $E$. Otherwise, starting from any such exact sequence with $E$ compact and $G$ infinite, you can endow the compact group $E$ with the topology inherited from the embedding into $E\times G\_d$ w... | 8 | https://mathoverflow.net/users/14094 | 117885 | 66,568 |
https://mathoverflow.net/questions/117884 | 3 | Given the identity
$$ \int^\infty\_0 K\_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} K\_{v-\mu-1}(\alpha z), \quad \alpha>0,\quad \Re[\mu]>-1$$
how can I find a closed form for the integral:
$$ \int^\infty\_0... | https://mathoverflow.net/users/19493 | Integral of Modified Bessel Function of the Second Type | Here is one situation when you can give a closed-form answer. Re-write the integral as
$$ I= e^{\beta z^2}\int\_0^\infty e^{-\beta(x^2+z^2)} K\_\nu(\alpha\sqrt{x^2+z^2}) x^{2\mu+1}(x^2+z^2)^{-\frac{v}{2}} dx $$
( $x:=zy$
$$ =z^{2\mu+2-v} e^{\beta z^2}\underbrace{\int\_0^\infty e^{-\beta z^2(y^2+1)} K\_\nu(\alpha ... | 5 | https://mathoverflow.net/users/20302 | 117897 | 66,572 |
https://mathoverflow.net/questions/117900 | 5 | **Theorem (Michel).** *A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero.*
Is there a simple proof of this result due, I think, to R. Michel ?
I'm guessing there must be a representation theoretic proof (everything is $SL(3;\mathbb{R})$ equivariant) an... | https://mathoverflow.net/users/21123 | Forms satisfying the zero-energy condition on the projective plane | There is a simple proof along the following lines: Because the first deRham cohomology group of $\mathbb{RP}^2$ is trivial, a $1$-form on $\mathbb{RP}^2$ is exact if and only if it is closed. Let $\alpha$ be a $1$-form on $\mathbb{RP}^2$ whose integral over every line vanishes, and let $\beta = \pi^\*\alpha$ where $\pi... | 11 | https://mathoverflow.net/users/13972 | 117903 | 66,575 |
https://mathoverflow.net/questions/117820 | 2 | I do not know how to correctly interpret Hilbert's Irreducibility theorem
with Galois group as my aim.
Here $K$ is a number field (or simply $\mathbf{Q}$).
Scenario 1: Take a field $L$ that is a finite Galois extension of $K(t)$ ($t$ an indeterminate) with Galois group $G$. Writing $L=K(t)[X]/(f(t,X))$ for
an irred... | https://mathoverflow.net/users/22878 | How to apply Hilbert's Irreducibilty theorem? | In ``Scenario 2'', you have to take a minimal polynomial $g(t,X)$ of a primitive element of the splitting field of $f(t,X)$ over $K(t)$. Then $g(a,X)$ is irreducible for infinitely many $a\in K$ by Hilbert's irreducibility theorem, and $g(a,X)$ and $g(t,X)$ have the same Galois group over $K$ and $K(t)$, respectively.
... | 3 | https://mathoverflow.net/users/18739 | 117906 | 66,577 |
https://mathoverflow.net/questions/117912 | 5 | This arose from a question Gil Kalai asked about a problem I posed involving [the Fourier transform on the discrete cube](https://mathoverflow.net/questions/117754/fourier-transform-on-the-discrete-cube). Maybe it is more tractable. I'm afraid I'm not sure how to do this kind of computation.
A *$k$-dimensional face* ... | https://mathoverflow.net/users/23141 | faces in the discrete cube | No. The typical set doesn't contain anything like a 0.6n face. Some similar questions are considered in "The Probabilistic Method" by Alon and Spencer (which I thoroughly recommend).
Here is the calculation. Let's just deal with 0.5n faces. I want to make a crude estimate of the probability that a subset (chosen unif... | 9 | https://mathoverflow.net/users/11054 | 117916 | 66,582 |
https://mathoverflow.net/questions/117930 | 3 | Let $X=X\_{1}\times X\_{2}$ is locally compact space, and define $$E=\{E\_{1}\times E\_{2}\mid E\_{i}\text{ is a Borel set in }X\_{i}\;,\text{ for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in the $\sigma$-algebra generated by $E$? Of course that every Baire set is Borel too so all Baires of $X\_{i}$ is a Borel of ... | https://mathoverflow.net/users/27896 | Baire sets of $X$ possess the required Cartesian product property | The idea of the proof is to make use of compactness to show that sets of the form $f^{-1}(0)^{c}$ are countable unions of boxes in $E$.
For a proof, assume that $f:X^{2}\rightarrow[0,1]$ is a continuous function with compact support. Let $U=f^{-1}(0,1]$. Then since $f$ has compact support, the sets $f^{-1}[a,1]$ are ... | 3 | https://mathoverflow.net/users/22277 | 117932 | 66,588 |
https://mathoverflow.net/questions/117874 | 5 | Building on this question [scaling the imaginary part of $\rho$s in infinite products](https://mathoverflow.net/questions/115128/what-happens-to-zetas-when-all-its-im-rho-n-are-scaled-linearly), I like to conjecture that:
$$\displaystyle \prod\_{n=1}^\infty \left(1- \frac{s}{\mu\_n} \right) \left(1- \frac{s}{1-\mu\_n... | https://mathoverflow.net/users/12489 | A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH? | The proposed formula is not true if RH is not true.
Let $\Theta$ be the upper bound of the real parts of the zeros of $\zeta(s)$.
Your product has zeros at $\mu$, $1-\mu$, $\overline{\mu}$ and $\overline{1-\mu}$
with $\mu=a+\gamma x i$ where $\rho=\beta+i\gamma$ runs through the non trivial zeros of $\zeta(s)$
so ... | 6 | https://mathoverflow.net/users/7402 | 117945 | 66,593 |
https://mathoverflow.net/questions/117939 | 16 | It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite sets. It is a symmetric monoidal category with tensor product $\coprod$ and unit $\emptyset$. It contains a commutative mon... | https://mathoverflow.net/users/2841 | The symmetric monoidal category of finite sets | Marco Grandis has done some work on this, and you can extract answers for 1-3 from these papers
* *Finite Sets and Symmetric Simplicial Sets* - M Grandis - TAC ([pdf](http://www.emis.ams.org/journals/TAC/volumes/8/n8/n8.pdf))
* *Higher Fundamental Functors for Simplicial Sets* - M Grandis - CTGDC ([pdf](http://archiv... | 15 | https://mathoverflow.net/users/4315 | 117949 | 66,595 |
https://mathoverflow.net/questions/117928 | 2 | Let $k$ be a finite field (I care about $\mathbb F\_p$, especially $\mathbb F\_2$) and let $V\_1,...,V\_N\subset k^n$ be subspaces.
I want to find a subspace $S\subset k^n$ such that $S\cap V\_i=0$ for each $i$, with $dim(S)$ as large as possible, and I want to do this in polynomial time (in exponential time we can ... | https://mathoverflow.net/users/24021 | Finding a subspace disjoint from a union of subspaces | Let $B$ be the oriented vertex-edge incidence matrix of a graph, viewed as a matrix over $GF(p)$. Let the subspaces $V\_i$ be the 1-dimensional subspaces spanned by the columns of $B$.
There is a subspace of codimension 1 disjoint from these subspaces if and only if there is a non-zero vector $a$ such that no entry of ... | 2 | https://mathoverflow.net/users/1266 | 117950 | 66,596 |
https://mathoverflow.net/questions/117952 | 4 | Among all groups $G$ of order $n$ which one will maximum the value: $\frac 1n\sum\_{g \in G}O(g)$ ? (Where $o(g)$ is the order of $g$).
| https://mathoverflow.net/users/30356 | The Average of Orders | As quid has stated the maximum attained for $\mathbb{Z}\_n$. The problem goes back to 1991. See Americam Mathematical Monthly 1991, page 970.
| 4 | https://mathoverflow.net/users/nan | 117955 | 66,599 |
https://mathoverflow.net/questions/117959 | 11 | Let $R$ be a commutative ring with identity. Is there any characterization for invertible elements of $R[x,x^{-1}]$ ?
| https://mathoverflow.net/users/30356 | Laurent Polynomials | I have read about this somewhere. I think it was as follows: $\sum\_{i=-n}^n a\_ix^i \in R[x,x^{-1}]$ is invertible iff $\sum a\_i^2$ is invertible in $R$ and for all $i \not = j$, $a\_ia\_j$ is nilpotent.
| 10 | https://mathoverflow.net/users/nan | 117960 | 66,602 |
https://mathoverflow.net/questions/117963 | 5 | I know of Choi's theorem and some related problems, but not a solution to this exact problem:
>
> Characterize the linear maps from the space $S\_n$ of symmetric $n \times n $ matrices to itself that preserve positive semidefiniteness.
>
>
>
It looks a natural question; has a simple characterization been found... | https://mathoverflow.net/users/1898 | Linear maps preserving positive semidefiniteness | Somewhat surprisingly, this seems to be still open. It is a linear preserver problem, about which there is a nice overview [here](http://people.wm.edu/~cklixx/lpp.pdf). But your specific problem seems to be open, according to this [recent preprint](http://arxiv.org/pdf/1011.3739.pdf). They also say that in an [earlier ... | 4 | https://mathoverflow.net/users/22051 | 117966 | 66,604 |
https://mathoverflow.net/questions/117933 | 12 | I couldn't find similar question being asked here. The closest one I can find is [When to split/merge papers?](https://mathoverflow.net/questions/11366/when-to-split-merge-papers). Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it into two parts... | https://mathoverflow.net/users/30375 | submit the second part of a paper | Since the two papers together prove one main theorem (if I correctly understand the first few lines of the question), it seems reasonable to submit them to the same journal. I can imagine a referee or editor being unhappy about being asked to publish part 1, which builds up to a big theorem that will appear in a differ... | 12 | https://mathoverflow.net/users/6794 | 117969 | 66,606 |
https://mathoverflow.net/questions/117971 | 40 | First let me state two known theorems.
Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then
$$ \int \frac{K}{2 \pi} dA = \chi (M) $$
where $K$ is the Gaussian curvature, $dA$ is the area form and $\chi(M)$ is the Euler characteristic.
Theorem 2 (combinatorial v... | https://mathoverflow.net/users/4463 | Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit? | The answer is yes, to both questions.
First question first. For any geodesic $n$-gon $P$ on $M$, i.e., a simply connected region of $M$ whose boundary consists of $n$-geodesic arcs, define
$$ \delta(P)= \mbox{sum of the angles of $T$}-(n-2)\pi. $$
Note that if $M$ were flat, then the defect $\delta(P)$ would be $... | 52 | https://mathoverflow.net/users/20302 | 117976 | 66,609 |
https://mathoverflow.net/questions/117972 | 3 | I am doing something with the curve given parametrically by
$y = (-ar+b) r$, $x = \sqrt{r^2-y^2}$
for $r\in \lbrack (b-1)/a,b/a\rbrack$. It is nice enough (and of low enough degree) that I suspect it has been studied before - and, in particular, that it has a name. Does it?
| https://mathoverflow.net/users/398 | Name for curve? | I'm assuming that $a$ is nonzero, in which case, this curve of degree $4$ is a rational curve: Just set $x=r\cos\theta$ and $y=r\sin\theta$, then you get $\sin\theta = b-ar$, so setting
$$
\cos\theta = \frac{1-t^2}{1+t^2}\qquad\qquad{\text{and}}\qquad\qquad
\sin\theta = \frac{2t}{1+t^2} = b - a r,
$$
you can now solve... | 11 | https://mathoverflow.net/users/13972 | 117977 | 66,610 |
https://mathoverflow.net/questions/117544 | 1 | a) How to solve, or at least to prove the existence of a solution to differential equation for given initial condition $y(s)=y\_0>0$ and $y'(s)=y\_1$, $s<0$,
$$y''+(2-n)\coth(t) y'=\frac{(n-1)\sinh(2y)}{2}, t<0.$$ Here $n$ is an integer $>2$.
b) Can the previous equation have two different solution (with different in... | https://mathoverflow.net/users/26543 | Solution to differential equation | $\coth(t)$ has a singularity at $t=0$, so the hypotheses of the existence and uniqueness theorems are not satisfied there. In fact if $\lim\_{t \to 0} y(t) = y\_0$ and $\lim\_{t \to 0} y'(t) = y\_1$ exist, $y''(t) \sim (n-2) y\_1 t^{-1}$ as $t \to 0$. If $y\_1 \ne 0$, this is impossible, as $t^{-1}$ is not integrable a... | 0 | https://mathoverflow.net/users/13650 | 117980 | 66,612 |
https://mathoverflow.net/questions/117970 | 9 | Let $V \rightarrow M$ be an oriented vector bundle over a compact
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$).
Is there some ``natural'' differential $\omega\_T$ form
representing the Thom Class of $V$? In particular I want the
foll... | https://mathoverflow.net/users/4463 | Is there a natural form representing the Thom class of a vector bundle, which when pulled back via the zero section represents the Euler class on the level of forms? | Here is another construction which goes back to Chern's proof of the Gauss-Bonnet theorem.
Suppose that $\pi: E\to M$ is an oriented rank $2k$ real vector bundle over the manifold $M$. Assume additionally that $E$ is equipped with a metric $g$, and a connection $\nabla$ compatible with the metric.
In your case $2k... | 7 | https://mathoverflow.net/users/20302 | 117982 | 66,614 |
https://mathoverflow.net/questions/117891 | 11 | Note: This is an update formulation since many people misunderstood the question before.
Of course it is easy to make a statement like "Every n is a prime or at most 1000", which is true for every prime $n$ and every small $n$ but fails for $n=1002$. What are "real" conjectures that were known to hold for primes and ... | https://mathoverflow.net/users/955 | What are conjectures that are true for primes but then turned out to be false for some composite number? | I'll elevate my comment to an answer and give two more related ones. One seems less trivial for primes but has first exception at $30$, the other seems more obvious for primes but has first exception at $900$.
The [cyclotomic polynomials](http://en.wikipedia.org/wiki/Cyclotomic_polynomial) $\Phi\_d$ can be specified ... | 19 | https://mathoverflow.net/users/8008 | 117985 | 66,615 |
https://mathoverflow.net/questions/117915 | 7 | I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across with work only for $C^3$ Morse functions.
A Google search was inconclusive about the existence of a Morse lemma for ... | https://mathoverflow.net/users/17965 | Morse lemma with least amount of regularity. | You only need $C^2$. See Nirenberg's book *Topics in Nonlinear Functional Analysis*, Theorem 3.1.1. He attributes this version of the Morse lemma to the late great Lars Hormander, *Fourier Integral Operators I*.
| 9 | https://mathoverflow.net/users/20302 | 117993 | 66,619 |
https://mathoverflow.net/questions/117958 | 8 | Skandalis, Tu and Yu in "The coarse Baum-Connes conjecture and groupoids" proved that:
---
Let $\Gamma$ be a countable group with a proper left-invariant metric $d$. If $\Gamma$ admits a uniform embedding into Hilbert space, then Baum-Connes assembly map with coefficients is split injective.
---
My questio... | https://mathoverflow.net/users/9401 | Injectivity of the Baum-Connes assembly map for locally compact groups | I believe that, in full generality, this is open. However, the point of the Skandalis-Tu-Yu paper is to construct a locally compact groupoid of the form $X\rtimes\Gamma$ (with $X$ a compact $\Gamma$-space), which admits a proper isometric action on a continuous field of Hilbert spaces (then previous results by J.-L. Tu... | 7 | https://mathoverflow.net/users/14497 | 117996 | 66,621 |
https://mathoverflow.net/questions/118007 | 2 | My motivation to the following question stems from the discussion at [Zeros of "exponential" function](https://mathoverflow.net/questions/83999/zeros-of-exponential-function) about the real zeroes of Stirling numbers of the second kind, I am curious in exploring the complex zeroes of Stirling functions of the second ki... | https://mathoverflow.net/users/30277 | Complex Zeroes of Stirling functions of the second kind | Your function has infinitely many complex zeros.
Indeed, it is of the form
$$\sum\_{k=1}^n a\_k e^{\lambda\_k z},$$
where $a\_k$ are constants, and $\lambda\_k=\log k$ are all distinct.
A function of this form always has infinitely many zeros, unless
$n=1$.
Proof. This is an entire function of order $1$, normal type.... | 4 | https://mathoverflow.net/users/25510 | 118013 | 66,630 |
https://mathoverflow.net/questions/118008 | 16 | Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the *superposition property* (my terminology) if, for every pair of isometric subsets $A$, $B$, there is an isometry of $X$, $F: X \to X$, that superimposes $A$ onto ... | https://mathoverflow.net/users/6094 | Which metric spaces have this superposition property? | If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces.
[See *Metric methods in Finsler...* by Busemann and *Sur certaines classes d'espaces...* by Tits (1955); thanks to [Linus](https://mathoverflow.net/users/37911/linus) for [the reference](https://math... | 20 | https://mathoverflow.net/users/1441 | 118022 | 66,636 |
https://mathoverflow.net/questions/118021 | 3 | I am doing some calculation involving elliptic integrals/functions, and find the notations confusing.
In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" is not the Latin letter eitch but the Greek capital Eta. Similarly the function Z(u), defined as $\mathrm{Z}(u) = \Thet... | https://mathoverflow.net/users/30390 | Are traditional notations for elliptic integrals/functions in Latin or Greek letters? | According to Whittaker and Watson, these notations were introduced in Jacobi's paper
De functionibus ellipticis commentario, J. fur Math., 1829, IV, p. 371.
This paper is easily available on line. In it, $E$ is slanted (italic), while $\Pi, \Theta$
and $\Delta$ are not. I conclude that Jacobi either meant Latin $E$, or... | 5 | https://mathoverflow.net/users/25510 | 118024 | 66,638 |
https://mathoverflow.net/questions/118029 | 0 | It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their symmetries:
$$\sum\_{i=1}^{n+1}x\_i^{2n-3}=0.$$
Fermat hypersurfaces have a group of automorphisms of order $(2n-3)^n(n... | https://mathoverflow.net/users/13441 | Lines on degree 2n-3 Fermat hypersufaces | Regarding question 1, any line in the Fermat cubic $C = \{X\_0^3 + X\_1^3 + X\_2^3 + X\_3^3 = 0\}$ must meet the coordinate hyperplane $H\_0 = \{X\_0 = 0\}$. So which points $x \in (C \cap H\_0)$ can lie on lines? If $Y, Z$ are homogenous coordinates on $T\_x(C \cap H\_0) \cong \mathbb{P}^1$, then the restriction of $X... | 3 | https://mathoverflow.net/users/13061 | 118038 | 66,642 |
https://mathoverflow.net/questions/113811 | 13 | Classically, we can attach $L$-functions (with properties like, analytic continuation, functional equation) to Dirichlet characters, Hecke eigenforms, etc...
My question is: can one attach $L$-functions (properties similar to the above) to Half-Integral weight modular eigenforms as well? If they do exist, can someon... | https://mathoverflow.net/users/1816 | Do L-functions exist for Half-integral weight modular forms? | Since the original question asked only for analytic continuation, functional equation, I'd like to add that the Mellin transform $\sum a\_n n^{-s}$ of the half integral weight modular form $\sum a\_n \exp(2 \pi i n z)$ has
these two properties, but it lacks an Euler product decomposition even if the modular form is a H... | 12 | https://mathoverflow.net/users/8099 | 118040 | 66,643 |
https://mathoverflow.net/questions/118031 | 1 | First off: I'm not an expert in order theory, so some of my terms might be off; correct them if you wish.
Let me call a subset $A$ of a lattice $(S,\le)$ *upwards convex* (not sure if that's actually good terminology) if it holds that for all $a,b\in A$ and all $x\in S$ with $a\le x\le a\vee b$ (where $\vee$ is join)... | https://mathoverflow.net/users/30392 | Covering of a partial order by upwards convex sets | The answer to (1) is $2^{n-1}$. If $P$ is the collection of all subsets of even cardinality, then each convex subset of $P$ is a singleton. Hence $k\geq 2^{n-1}$.
On the other hand, one can fix $a\in M$ divide all the subsets into pairs differing only in $a$. Then $P$ is the union of its intersections with the pairs,... | 2 | https://mathoverflow.net/users/17581 | 118042 | 66,645 |
https://mathoverflow.net/questions/118044 | 5 | Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1\_{Ab}(G,M)$.
On the other hand $H^2$ is $\mathrm{Ext}^2\_{G-mod}(\mathbb Z,M)$ (by the definition of group cohomology), where $\mathbb Z$ is given the tr... | https://mathoverflow.net/users/1355 | An isomorphism between different Ext's coming from group cohomology | The group $H^2(G,M)$ classifies different types of extensions than $\operatorname{Ext}^1(G,M)$. On the one hand, $H^2(G,M)$ classifies extensions
$$M\hookrightarrow H\twoheadrightarrow G$$
where $H$ may be non-abelian, and the action of $H$ on $M$ by conjugation is encoded in the $G$-module structure of $M$. On the... | 15 | https://mathoverflow.net/users/12166 | 118049 | 66,646 |
https://mathoverflow.net/questions/118060 | 7 | Question 1: Does there exist models of the Zermelo-Fraenkel set theory without the axiom of choice, but such that every indexed family of non-void sets whose index set has a well-orderable cardinal admits a choice function ?
Question 2: The same as question 1, with "a well-orderable cardinal" replaced by "a linearly or... | https://mathoverflow.net/users/30395 | Existence of model of ZF without AC, but with many choice function | For the first question:
Yes. It is known that the axiom "Every well-orderable family of non-empty set has a choice function" implies $DC$ but not $DC\_{\aleph\_1}$.
You can find the proofs in Jech's "The Axiom of Choice" and in Felgner's "Models of ZF-Set Theory".
I am not sure about the second question, but I be... | 6 | https://mathoverflow.net/users/7206 | 118062 | 66,651 |
https://mathoverflow.net/questions/118052 | 15 | First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
**Problem S.** Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following properties.
**1.** $V$ is a finite dimensional complex vector space equipped with a Hermitian metric. $\DeclareMathOperat... | https://mathoverflow.net/users/20302 | Symbols of elliptic operators | Maybe I'm misunderstanding something, but it seems that the answer is probably 'no', at least if $d = \dim\_\mathbb{C} V$ is large enough.
What really matters is the $n$-dimensional real subspace $\mathsf{S}=\mathrm{Im}(S)\subset\mathrm{Sym}(V)$. What you need is that this space not meet the cone of singular matrice... | 12 | https://mathoverflow.net/users/13972 | 118064 | 66,652 |
https://mathoverflow.net/questions/118053 | 3 | Hello,
My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc}
A & B\\\
B^T & C \end{array}\right)
$.
According to the lemma $M\geq0$ iff $C>0$ and $A-BC^{-1}B^T\geq 0$.
In my current research I'm working on an optimization problem over a domain of matrices; I'm trying t... | https://mathoverflow.net/users/29228 | Schur complement and negative definite matrices | This should be a comment, but I can not yet post comments.
You got Schur's complement lemma wrong, the matrix
$$\left(\begin{array}{cc} 1 & 1 \newline 1 & -1 \end{array}\right)$$
satisfies you conditions, $A=1\ge 0$ and $A-BC^{-1}B^T=1-(-1)\ge 0$, but it is clearly not positive semi-definite. Replace the first condit... | 3 | https://mathoverflow.net/users/30364 | 118065 | 66,653 |
https://mathoverflow.net/questions/118046 | 1 | Is there a name and common technique for such equations, where $A$ and $B$ are matrices and $x$ a vector?
$Ax+f(\lambda)Bx=g(\lambda)x$.
| https://mathoverflow.net/users/22051 | What is such an equation called? | Akin to my comment, this equation can be called a nonlinear **generalized** eigenvalue problem. Usually, $f$ and $g$ are polynomials in $\lambda$, but more general nonlinearities might be allowed. In general, I doubt there will be robust, globally convergent method for this equation that gets all the solutions. The [ta... | 5 | https://mathoverflow.net/users/8430 | 118066 | 66,654 |
https://mathoverflow.net/questions/118058 | 4 | Given two cohomology theories $h^{\bullet}$ and $k^{\bullet}$, we define their direct sum $(h \oplus k)^{\bullet}$ as the cohomology theory $(h \oplus k)^{n}(X) := h^{n}(X) \oplus k^{n}(X)$. If I'm not wrong, this is another cohomology theory, whose spectrum is the wedge product of the corresponding spectra. For exampl... | https://mathoverflow.net/users/10758 | Irreducible cohomology theories | The sphere spectrum, representing stable cohomotopy, is irreducible (you can see this, for instance, from the fact that its homology is irreducible and any summand would be connective and thus would have to have nontrivial homology). But the associated cohomology of a point is the stable homotopy groups of spheres, whi... | 8 | https://mathoverflow.net/users/75 | 118067 | 66,655 |
https://mathoverflow.net/questions/118037 | 12 | In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
**QUESTION:**
Let $m,n\in\mathbb{N}$ and let $A\_1,\ldots, A\_m\in M\_n(\mathbb{R})$ be real, symmetric, indefinite matrices. I'm interested in conditions on $A\_1,\ldots,A\_m$ which ensures that the set
... | https://mathoverflow.net/users/27832 | On the positive definiteness of a linear combination of matrices | The following recent paper: ["An exact duality theory for semidefinite programming based on sums of squares"](http://www.math.uni-konstanz.de/~schweigh/publications/sosdualsdp.pdf) by I. Klep, and M. Schweighofer (both are on MO I think) addresses exactly your question: When is there a $\lambda \in \mathbb{R}^m$ such t... | 14 | https://mathoverflow.net/users/8430 | 118070 | 66,657 |
https://mathoverflow.net/questions/117723 | 5 | In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, but I don't have a feeling about the meaning of ext.
Is there a informal/intuitive interpretation of ext-groups?
I'm mostly interested in the case of $\mathcal{O}\_X$-Modules for (toric) varieties or $\ma... | https://mathoverflow.net/users/13488 | (geometric/intuitive) interpretation of ext | I got an answer on stackexchange: <https://math.stackexchange.com/questions/270228/geometric-intuitive-interpretation-of-ext>.
thank you guys.
| 1 | https://mathoverflow.net/users/13488 | 118073 | 66,660 |
https://mathoverflow.net/questions/117193 | 0 | Let $^{2}B\_{2}(q)$ be Suzuki simple group where $q=2^{2n+1}$. I want to know order of two Suzuki simple groups can divide each other? In other words, suppose that $|^{2}B\_{2}(q\_{1})|\mid |^{2}B\_{2}(q\_{2})|$, if this implies $q\_{1}=q\_{2}$ or not?
| https://mathoverflow.net/users/30203 | Suzuki group order | Geoff is on the right track about the way Suzuki groups occur naturally as subgroups of others. But in view of the incomplete formulation of the original question, and the string of comments following (with some references not directly relevant to Suzuki groups), it's worth pointing out sources in the literature.
1)... | 2 | https://mathoverflow.net/users/4231 | 118078 | 66,662 |
https://mathoverflow.net/questions/118081 | 21 | Background and motivation
-------------------------
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1\_\mathbb{Z}(A,\mathbb{Z})$. If $A$ is free then this abelian group is trivial. Is the converse true? The converse ... | https://mathoverflow.net/users/1437 | Nice algebraic statements independent from ZF + V=L (constructibility) | Let me address the part of your question seeking algebraic statements independent of ZFC+V=L.
The basic situation is that in set theory our tools are not so flexible for finding statements
independent of ZFC+V=L, as opposed to finding statements independent of ZFC.
The main reason for this is that one cannot directly... | 21 | https://mathoverflow.net/users/1946 | 118084 | 66,663 |
https://mathoverflow.net/questions/118083 | 2 | Hi All. Need some information. We all know axiom of choice (AC) and countable choice. Which axioms are between these two. I mean weaker that Axiom of choice but stronger than countable choice ?
| https://mathoverflow.net/users/nan | what axioms are between AC and Countable choice ! | Dependent choice, for example. Or choice for well-ordered families, see [Existence of model of ZF without AC, but with many choice function](https://mathoverflow.net/questions/118060)
| 2 | https://mathoverflow.net/users/14915 | 118086 | 66,665 |
https://mathoverflow.net/questions/118080 | 6 | Suppose $X$ is a reflexive space (possibly non-separable) which is not super-reflexive. Then (by definition) there exists a non-reflexive Banach space $Y$ which is non-reflexive but is finitely representable in $X$, meaning that for each $\lambda >1$, every finite dimensional subspace of $Y$ is $\lambda$-isomorphic to ... | https://mathoverflow.net/users/29433 | Non-super reflexive space | The first question is easy: Every non reflexive space has a separable non reflexive subspace (e.g. by the Eberlein-Smulian theorem or by R. C. James' characterization of non reflexivity).
The second question was a longstanding open problem that was solved by James in the 1970s. Pisier and Xu gave another proof--you ... | 6 | https://mathoverflow.net/users/2554 | 118087 | 66,666 |
https://mathoverflow.net/questions/118074 | 5 | It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and others.
Let’s however look at the spiral defined by the equation:
$r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$
I ... | https://mathoverflow.net/users/10903 | How to draw Archimedean-Galileo spiral? | This is not a complete solution, but a suggestion on what you can try. First, note that the ["planimeter"](https://en.wikipedia.org/wiki/Planimeter) will "compute" for you integrals
$$
F(x)=\int ydx,
$$
where $y=f(x)$ is the given curve. This "computation" is in the form of a numerical output measured by a rotating whe... | 10 | https://mathoverflow.net/users/21684 | 118089 | 66,667 |
https://mathoverflow.net/questions/118103 | 7 | In order to test the monadicity of a functor, there is a *precise monadicity theorem* (PM) as well as a *crude monadicity theorem* (CM), see the [nlab](http://ncatlab.org/nlab/show/monadicity+theorem). In CM, the forgetful functor should create [reflexive coequalizers](http://ncatlab.org/nlab/show/reflexive+coequalizer... | https://mathoverflow.net/users/2841 | The crude monadicity theorem | For $X$ an infinite set, the monad $(-)^X$ (induced from the comonoid structure on $X$ with respect to cartesian product) does not preserve reflexive coequalizers. See page 538 of [this paper](http://www.emis.de/journals/CMUC/pdf/cmuc0003/adamvele.pdf) by Adámek, Koubek, and Velebil. Correspondingly, the forgetful func... | 14 | https://mathoverflow.net/users/2926 | 118107 | 66,672 |
https://mathoverflow.net/questions/118099 | 4 | Background
----------
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all $X \in C$. Recall that the internal hom $\underline{\mathrm{hom}}(X,-)$ is defined, if it exists, as a right ... | https://mathoverflow.net/users/2841 | Example of a non-closed cocomplete symmetric monoidal category | Here is an amusing example which addresses Q2: take the universe $V$ of sets in a model of ZFC, as a class partially ordered by inclusion of sets. Consider a partially ordered class to be a category in the usual way. Of course, by Cantor's theorem, there is no terminal object in this category, but anyway we can freely ... | 6 | https://mathoverflow.net/users/2926 | 118108 | 66,673 |
https://mathoverflow.net/questions/118090 | 1 | As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book *DG, Lie groups and symmetric spaces* and Knapp's *Beyond an introduction*.
$\newcommand{\Real}{{\mathbb R}}\newcommand{\fa}{{\sf a}}\newcommand{\fg}{{\sf g}}\newcommand{... | https://mathoverflow.net/users/763 | Copies of ax+b inside the AN part of an Iwasawa decomposition? | Here is an elementary argument addressing Q1. The adjoint action of the subalgebra $\mathfrak{a}$ on $\mathfrak{g}$ has the following two key properties:
1. It is diagonalizable (the operators $ad(H)$ for $H\in\mathfrak{a}$ are symmetric with respect to the Killing form);
2. It stabilizes $\mathfrak{n}$.
It follows... | 2 | https://mathoverflow.net/users/5740 | 118109 | 66,674 |
https://mathoverflow.net/questions/118101 | 10 | According to:
Dirac, P. A. M. (1927). "The physical interpretation of the quantum dynamics."
Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 113(765), pp.621–641.
For any $y \in \mathbb{C}$, if $f$ is analytic, then $\int\_{-\infty}^\infty f(x) \delta... | https://mathoverflow.net/users/12983 | Dirac Delta function with a complex argument | I am afraid this is due to a misunderstanding of what Dirac meant. He does not write "for any $y\in \mathbb{C}$" but he refers to a "c-number". The c stands for "classical" as opposed to quantum, and what Dirac means is that $y$ is a real number and not a Hermitian operator. Dirac never considered the delta function of... | 17 | https://mathoverflow.net/users/11260 | 118114 | 66,677 |
https://mathoverflow.net/questions/118131 | 0 | I am looking for an example showingthat
a function $f$ which is $C^\infty$ on a submanifold $N$ of $M$, but it cannot be written as
the restriction of a $C^\infty$-function on $M$.
| https://mathoverflow.net/users/30415 | A $C^\infty$-function on a submanifold which is not the restriction of a a $C^\infty$ on $M$ | You can take $M = \{ z \in \mathbb{C} \colon |z|=1\}$, $N = M \setminus \{1\}$ and $f(e^{i\varphi}) = \varphi$.
| 3 | https://mathoverflow.net/users/nan | 118133 | 66,683 |
https://mathoverflow.net/questions/118095 | 0 | I 'm searching for an algorithm (and except the naive brute force solution had no luck) that efficiently ($O(n^2)$preferably) does the following:
Supposing I’m playing a game and in this game I’ll have to answer n questions (each question from a different category). For each category $i$, $i=1,...,n$ I’ve calculated ... | https://mathoverflow.net/users/30403 | An Algorithm to Determine a probable profit | It's best, I think, to think of the questions as being asked in *backwards* order: $n$ first, then $n-1$, etc., down to question $1$. And just for fun, let's tack on one final question for which the probability of answering correctly is $p\_0=0$. The relevant recursion then is
\begin{array}{rcl}
E(p\_0,p\_1,\ldots,p\... | 0 | https://mathoverflow.net/users/15837 | 118140 | 66,685 |
https://mathoverflow.net/questions/118039 | 2 | This might be a stupid question for expert in this area.
I am considering automorphic representation of algebraic group.
In studying it, local tempered, local square integrable representations occurs in the P-adic group case. So, I am wondering that what global automorphic representation gives rise to such represen... | https://mathoverflow.net/users/29334 | Global square integrability ensures local sq. integrability? | The naive form of Ramanujan-Peterssen-Selberg conjectures is that all the local repns attached to automorphic repns are tempered. This is not quite true, at least because of liftings, as observed by Roger Howe and I.I. Piatetski-Shapiro in the 1970s (maybe in the Corvallis proceedings). More recent conjectures of J. Ar... | 5 | https://mathoverflow.net/users/15629 | 118145 | 66,686 |
https://mathoverflow.net/questions/118142 | 3 | This is a question that comes from my (biological) research. I'm very weak in topology, so I'm not able to assure myself of the answer. The problem is this: I'm watching an animal move in two dimensions. At three successive points in time I have three positions, (x1,y1), (x2,y2), (x3,y3). But there are three uninterest... | https://mathoverflow.net/users/30419 | continuous R^2xR^2xR^2/E^+(2) -> R^3 injection? | By translation, fix the first point to be at the origin. Consider the other two points as complex numbers, and take their quotient. As long as the other two points are not both at the origin, this continuously gives an element of $\mathbb{CP}^1\cong S^2$. Up to rotation, the other two points are determined by this quot... | 3 | https://mathoverflow.net/users/75 | 118147 | 66,687 |
https://mathoverflow.net/questions/118117 | 23 | Does every irreducible curve admit an equation of the form $f(x)=g(y)$, where $f$ and $g$ are polynomials? What if we allow $f$ and $g$ to be rational functions?
Actually, I'd like to understand this in the presence of an additional constraint: if we're given a finite cover of curves $\pi\colon C\to\mathbb{P}^1$, do ... | https://mathoverflow.net/users/30412 | Can every curve be written as $f(x)=g(y)$? | This would contradict the Harris-Mumford(-Eisenbud) theorem that $M\_g$ is non-uniruled for $g$ at least $23$. Let $C$ be a general curve of genus $g$. If $C$ is in "Zieve form", then it is the normalization of the (almost certainly) singular curve in $\mathbb{CP}^1 \times \mathbb{CP}^1$,
$$D = \{ ([x\_0,x\_1],[y\_0,y\... | 24 | https://mathoverflow.net/users/13265 | 118148 | 66,688 |
https://mathoverflow.net/questions/118118 | 13 | Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, $A\cup \lbrace x\rbrace $ contains an arithmetic progression of length 3.
So there exists such a set (Using AC) but I f... | https://mathoverflow.net/users/nan | Is it necessary to use AC to solve this problem ? | I can't find the error in this argument, but everybody seems convinced (in the comments) that such a simply definable set with the desired property doesn't exist so there must be an error somewhere.
Let $A$ be the set of reals whose nonterminating base three expansion avoid the digit $1$. As usual, there is some hass... | 24 | https://mathoverflow.net/users/14913 | 118154 | 66,691 |
https://mathoverflow.net/questions/118138 | 12 | In a 1975 paper `Not Every Number is the Sum or Difference of Two Prime Powers', Cohen and Selfridge use covering congruences to prove their Theorem 1,
which states that there exists an arithmetic progression of odd numbers which are
neither the sum nor difference of a power of two and a prime.
The paper is availab... | https://mathoverflow.net/users/24963 | Cohen and Selfridge's proof about odd numbers which are neither the sum nor difference of a power of two and a prime | Selfridge liked to be terse in his papers. Indeed, he enjoyed being cryptic enough to puzzle the reader into doing some work. All the heavy lifting is there but some details are left unsaid. For a minor example "two prime powers" of the title is briefer than what that implies after some thought: " a power of $2$ and a ... | 14 | https://mathoverflow.net/users/8008 | 118163 | 66,693 |
https://mathoverflow.net/questions/117998 | 2 | In the course of reading a paper , I've encountered the following property of interest.
If $R$ is a ring, say it satisfies (\*) if: For any smooth, irreducible $R$-algebra $B$ of finite type such that all the fibers of $Spec B$ over points of codimension one in $Spec R$ are irreducible, then $(B \otimes\_R K)^\* = B... | https://mathoverflow.net/users/27736 | Rings satisfying a certain property | Here is a sketch. The hypothesis, when $R$ is a ufd says that any prime $p\in R$ remains a prime in $B$. So, if $a,b\in B\otimes K$ are units such that $ab=1$, then clearing denominators (from $R$), we get an equation $a'b'=f$ where $a',b'\in B, 0\neq f\in R$. Further we may assume that no prime in $R$ divides $a'$ or ... | 4 | https://mathoverflow.net/users/9502 | 118164 | 66,694 |
https://mathoverflow.net/questions/118143 | 3 | recursively extracting a MIS from an undirected simple graph $G$ does produce a minimal coloring for $G$ ?
I searched extensively the internet and found a paper [1] which answer partially to this question.
In this paper is shown a counterexample which is the graph depicted at the end of this post. In this graph if ... | https://mathoverflow.net/users/26449 | Coloring a graph by Maximum Independent Set extraction | this counterexample really confirms that both answers to my questions are negative.
In this instance is easy to see that $\chi(G)=\alpha(G)=4$, $\{1,2,3,4\}$ is a complete graph and the maximum independent set is $\{5,6,7,8\}$.
A minimum coloring for $G$ put necessarily each vertex of the complete graph in a diff... | 1 | https://mathoverflow.net/users/26449 | 118168 | 66,697 |
https://mathoverflow.net/questions/118129 | 3 | I have had the following problem on several occasions and I was wondering whether there is a general technique to solve this problem.
Given a list of graphs with property $P$. Is there a general technique to get a list of common substructures for the given list? Any approximations or heuristics are fine as well.
Th... | https://mathoverflow.net/users/15684 | Search for common substructures in list of graphs | I don't think there is a general technique. The only related situation I'm aware of is "motif detection" in biological networks, which involves exhaustively counting small subgraphs in very large sparse graphs. But I don't think it will be useful for finding missing subgraphs in lists of small graphs.
| 3 | https://mathoverflow.net/users/9025 | 118171 | 66,699 |
https://mathoverflow.net/questions/118177 | 6 | Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R\_k=R\_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ arithmetic progression
$$
\{a, a+r, a+2r, \dots, a+(k-1)r\}.
$$
Thus, the finiteness of $R\_k$ is Szemeredi's theo... | https://mathoverflow.net/users/1375 | Minimal period of arithmetic progressions occurring in sets of positive density. | Let
$$
S=\mathbb N\setminus\bigcup\_{n\ge 5}\bigcup\_k\lbrace 2^n(2k+1),2^n(2k+1)+1,\ldots,2^n(2k+1)+(n-1)\rbrace.
$$
This has positive upper density (in fact positive density), because what you're removing has density $\sum\_{n\ge 5}n/2^{n+1}<1$.
If you want to find an arithmetic progression with length $2^{n+1}$... | 6 | https://mathoverflow.net/users/11054 | 118186 | 66,704 |
https://mathoverflow.net/questions/118183 | 37 | I understand [Godel's Incompleteness Theorems](http://en.wikipedia.org/wiki/G%25C3%25B6del%27s_incompleteness_theorems) to be statements *about* effectively generated [formal systems](http://en.wikipedia.org/wiki/Formal_system), which basically makes them theorems *about* algorithms. This is cool, because despite being... | https://mathoverflow.net/users/84526 | What axioms are used to prove Godel's Incompleteness Theorems? | As Andres Caicedo points out in his comment (to the Question), the modest fragment $\sf{PRA}$ (Primitive Recursive Arithmetic) of $\sf{PA}$ (Peano arithmetic) is already is able to verify the incompleteness theorems.
>
>
> >
> > Indeed, the proof of the Gödel-Rosser incompleteness proof is entirely syntactic and ... | 39 | https://mathoverflow.net/users/9269 | 118189 | 66,706 |
https://mathoverflow.net/questions/118185 | 9 | I'd like to learn the basics of Hirsch-Smale immersion theory. What sources are best for this? My background is mostly topological; however, many of the sources I've found on the internet focused on later work of Gromov on the h-principle which seems more analytic than I would like.
| https://mathoverflow.net/users/30427 | Where should I learn about immersion theory? | M. Weiss has a very good survey on his homepage:
<http://wwwmath.uni-muenster.de/u/mweis_02/papers.html>
called "Immersion theory for homotopy theorists". I also like very much M. Adachi's book "Embeddings and immersions". I think both are very good starters if you have a topological background.
J. Francis has also ... | 7 | https://mathoverflow.net/users/27816 | 118191 | 66,708 |
https://mathoverflow.net/questions/117805 | 2 | Hello,
Let $A\_n = (a\_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda\_{n,i};\;i = 0,1,\ldots,n-1)$.
If $A\_n$ were a sequence of Hermitian Toeplitz matrices, and if $\sum\_k|a\_k|<\infty$, then Szego theorem states that for any continues function $F(\cdot... | https://mathoverflow.net/users/25265 | Asymptotic Behavior of Non-Analytic Function of the Eigenvalues | To create a partial answer,
eigenvalues of banded Toeplitz matrices accumulate on some real algebraic curves in $\mathbb{C}$, (this has been proved by Schmidt and Spitzer around 1970.).
Now, if we also attach a point-mass at each eigenvalue, (for fixed matrix size) with equal mass at each point, and with total mass on... | 0 | https://mathoverflow.net/users/1056 | 118192 | 66,709 |
https://mathoverflow.net/questions/117719 | 3 | Let $X$ be a smooth degree $d$ ($d >5$) surface in $\mathbb{P}^3$. Let $\pi:\tilde{X} \to X$ be a blow-up of $X$ at a point. When is it possible to embed $\tilde{X}$ into $\mathbb{P}^3$?
In general, when can we embed a projective surface in $\mathbb{P}^3$? When is the resulting surface smooth?
| https://mathoverflow.net/users/9164 | Embedding a surface in a projective space | Let $Y\subset \mathbb{P}^3$ be a smooth projective surface of degree $l$. If $l\geq 4$, then $K\_Y=(l-4)H|\_Y$ is nef (really either $0$ or very ample), and hence $Y$ cannot contain a $(-1)$ curve. If $l\leq 2$, then $Y$ is either a plane or a quadric, neither of which contain $(-1)$ curves. This leaves $l=3$. The cubi... | 4 | https://mathoverflow.net/users/10076 | 118194 | 66,711 |
https://mathoverflow.net/questions/118184 | 1 | Dear all,
According to the notation of Wilson's book "The finite simple groups" ( books.google.com/books?isbn=1848009879, page 9) $ A\_{.} B $ denotes an unspecified extension. Now, I want to know what does "unspecified extension of $A$ and $B$" mean?
| https://mathoverflow.net/users/27962 | Notation of Wilson's book "The finite simple groups" | I don't have the book at hand, but I think the usual meaning of $G = A.B$ for groups $A$ and $B$ is that $G$ has a normal subgroup isomorphic to $A$ such that the quotient $G/A$ is isomorphic to $B$. Some authors only write non-split extensions in this way, but if the book states that $A.B$ denotes an "unspecified" ext... | 8 | https://mathoverflow.net/users/28104 | 118195 | 66,712 |
https://mathoverflow.net/questions/118199 | 0 | There are many algebraic equivalences of AC in the literature. A famous one is "every ring with identity has a maximal ideal".
Where can I find this equivalences, specially those in rings theory !? Can you name some others ? Or a good references would be okay.
| https://mathoverflow.net/users/nan | Equivalent Forms of AC | One good source would be Rubin & Rubin **Equivalents of the Axiom of Choice**, where there is a section devoted for algebraic forms. There are two editions to this book, the second one is from the 1980's and contains a lot more information (naturally).
The book is a bit out dated and you can try Howard & Rubin's **Co... | 5 | https://mathoverflow.net/users/7206 | 118200 | 66,716 |
https://mathoverflow.net/questions/118201 | 5 | i'm trying to define modular forms for the full modular group for a general audience and want to make it as simple as possible. please tell me if this is correct:
definition of a modular form $f$:
1. holomorphic function $f$ from upper half plane to $\mathbb{C}$.
2. $f(z)$ bounded as Im($z$) tends to infinity.
3. $... | https://mathoverflow.net/users/30430 | simpler way to define modular forms | (Edited/corrected/amplified) "Bounded at infinity" is a condition that includes not only cuspforms but also Eisenstein series. If by "holomorphic at infinity" one means exactly holomorphy of $f(z)$ on the quotient by translations $z\rightarrow z+1$, then this is the same as "bounded at infinity".
On another hand, if ... | 2 | https://mathoverflow.net/users/15629 | 118207 | 66,719 |
https://mathoverflow.net/questions/118208 | 10 | Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the case of the Gelfand-Naimark Theorem. <http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem>
| https://mathoverflow.net/users/30081 | Intuitive meaning of Double Commutant Theorem | Okay, here's an explanation in terms of quantum mechanics. Let ${\cal A}$ be a family of observables, modeled as self-adjoint operators on some Hilbert space, and let ${\cal U}$ be the group of all unitary transformations that leave every observable in ${\cal A}$ invariant. You can consider ${\cal U}$ to be a kind of s... | 12 | https://mathoverflow.net/users/23141 | 118209 | 66,720 |
https://mathoverflow.net/questions/118139 | 4 | My question is just as in the box. Is every smooth projective toric variety diffeomorphic to a quotient of $\prod\_i S^{n\_i} \times T^k$ (I know torus is a one-sphere but I just wanted to make clear I allow this) by a free torus action? If not which ones can be realized like this? Maybe it can be proven using the Geom... | https://mathoverflow.net/users/9275 | Smooth projective toric varieties which are quotients of product of spheres and torii by a free torus action? | Let us consider the case of toric varieties of real dimension $4$ and prove they
can not be represented as such a quotient unless they have second Betti
number $1$ or $2$.
Proof.
Let us introduce some notations. Let $B$ be the toric manifold of real dimension $4$, $n=b\_2(B)$. Denote by $E$ the product $\Pi\_i S^... | 5 | https://mathoverflow.net/users/943 | 118217 | 66,724 |
https://mathoverflow.net/questions/118216 | 0 | **Absolute norm**
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|\_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|\_N=N((\|x\|, \|y\|)) \qquad \text{ for all } z=(x,y)\in Z.
$$
For example, the $\ell\_p$-norms are absolute norms.
**1-unco... | https://mathoverflow.net/users/30433 | Absolute norms and 1-unconditional sums | What Yemon says is correct. The "right" space to use is the space $Y$ of all elements in $Z$ such that only finitely many terms are non zero--you can always complete at the end.
Unconditional sums can be much more complicated than absolute sums. In an absolute sum, if you have linear operators $T\_n$ on $X\_n$ s.t. ... | 1 | https://mathoverflow.net/users/2554 | 118218 | 66,725 |
https://mathoverflow.net/questions/118215 | 6 | I would guess it was well known by the time of Cauchy. But are there earlier references to it?
| https://mathoverflow.net/users/23064 | Who discovered the winding number? | [Grünbaum and Shephard](http://www.ams.org/journals/tran/1990-322-01/S0002-9947-1990-1024774-2/S0002-9947-1990-1024774-2.pdf) suggest that the winding numbers (for closed polygons) have been discussed in the literature at least since 1769.
See
A.L.F. Meister, *Generalia de genesi figurarum planarum et inde pendenti... | 13 | https://mathoverflow.net/users/5371 | 118221 | 66,727 |
https://mathoverflow.net/questions/118222 | 5 | I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: [2008](http://arxiv.org/abs/0806.0349) and [2011](http://arxiv.org/abs/1005.2656)).
There, they work on a Hilbert space $H$ and on the bounded operators algebra $B(H)$ using some operato... | https://mathoverflow.net/users/21864 | definition of operator valued integral with spectral measure | Well, I would make sense of expressions like this by inserting $|e\_n\rangle\langle e\_n|$ between the operator and the measure and then summing over $n$.
Say we want to give a meaning to $A = \int A(x)d\mu$. If we know how to integrate scalar-valued functions against a spectral measure then we can define $$\langle v... | 6 | https://mathoverflow.net/users/23141 | 118244 | 66,739 |
https://mathoverflow.net/questions/117304 | 9 | This question is a follow up to: [Model for the (infinity,1)-category of (homotopy-)limit preserving functors](https://mathoverflow.net/questions/117267/model-for-the-infinity-1-category-of-homotopy-limit-preserving-functors).
>
> Warm-up Question: Given a simplicial model category $M$, what model category models t... | https://mathoverflow.net/users/2536 | Model for the (infinity,1)-category of functors preserving certain homotopy limits | The $(\infty,1)$-category of presheaves on any *small* $(\infty,1)$-category $C$ is presented by the model structure of simplicial presheaves on any simplicial category which incarnates $C$. So if $M$ is small, then simplicial presheaves on the simplicial category $M^{cf}$ would do it, or on the hammock localization of... | 4 | https://mathoverflow.net/users/49 | 118250 | 66,741 |
https://mathoverflow.net/questions/118251 | 5 | Let $X$ be a compact Kahler manifold of complex dimension $n$. Fix a nonzero class $u \in H^1(X,T\_X)$. This gives a linear morphism
$$
\phi\_u : H^0(X,\Omega^n) \to H^{1}(X,\Omega^{n-1}),
\quad \sigma \mapsto u \cup \sigma.
$$
Is $\phi\_u$ injective?
It is so for manifolds with $\Omega^n\_X = \mathcal O\_X$; the pro... | https://mathoverflow.net/users/4054 | Is the cup product of holomorphic $n$-forms with a fixed class injective? | The answer to this question is negative in dimensions $\ge 3$. For example, take a quintic in $\mathbb CP^4$ and consider its blow up $X$ in $10^{100}$ points (just to be safe). Then the space $H^1(X, T\_X)$ will be huge, since it parametrises deformations of the blown up variety and you can move points as you wish. So... | 12 | https://mathoverflow.net/users/943 | 118256 | 66,745 |
https://mathoverflow.net/questions/118245 | 4 | Let $e\_k$ be the $k$th-degree elementary symmetric polynomial in $\tan\theta\_1,\tan\theta\_2,\tan\theta\_3,\ldots$ (and if the sequence of $\theta$s is finite remember that the $k$th-degree elementary symmetric polynomial in $n<k$ variables is $0$). Then
$$
(e\_0-e\_2+e\_4-\cdots)^2 + (e\_1-e\_3+e\_5-\cdots)^2 = \sec... | https://mathoverflow.net/users/6316 | "Known" Pythagorean identity? (reference request) | I don't have enough reputation to comment, but for what it's worth, if we replace $\tan(\theta\_j)$ with $\lambda\_j$ and set $P(x)=\prod (x-\lambda\_j)$, then the left hand side will be $P(i)P(-i)$ by difference of squares and Vieta's formula, and the right hand side $\prod (1+\lambda\_j^2)$. The equality then follows... | 11 | https://mathoverflow.net/users/3404 | 118267 | 66,752 |
https://mathoverflow.net/questions/118246 | 17 | Consider the category $FdVect\_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only one up to isomorphism), and also compact closed with simple tensor unit which is a progenerator.
Can we characterise... | https://mathoverflow.net/users/4177 | Characterising categories of vector spaces | Let $C$ be an abelian monoidal category such that $1 \in C$ is simple, and each object is a finite sum of copies of $1$, i.e. isomorphic to $1^{\oplus n}$ for some $n \in \mathbb{N}$. It is well-known that $k:=\mathrm{End}(1)$ is a *commutative* ring and that $C$ is $k$-linear. By Schur's Lemma $k$ is even a field. Now... | 21 | https://mathoverflow.net/users/2841 | 118269 | 66,753 |
https://mathoverflow.net/questions/118275 | 5 | In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with *split* Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objects $\langle ... | https://mathoverflow.net/users/4231 | Origin of notion of "split Grothendieck group"? | These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K\_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K\_0(\mathcal{A})$ ... | 8 | https://mathoverflow.net/users/28011 | 118285 | 66,761 |
https://mathoverflow.net/questions/118043 | 1 | More specifically, if we only know that a complete Boolean algebra, $\mathbf{B}$, is $\kappa$-c.c., can we give a (reasonably tight) upper bound to the size of $\mathbf{B}$ in terms of $\kappa$?
Thanks in advance.
| https://mathoverflow.net/users/29231 | What can we infer about the size of a complete Boolen algebra, given it is $\kappa$-c.c.? | The $\kappa$-cc condition by itself does not put any bound on the cardinality of the algebra. For example, for each $\lambda$, the Cohen algebra of regular open subsets of $2^\lambda$ is ccc, but it has cardinality $\lambda^\omega$. However, one can bound the size of $B$ using additional cardinal characteristics: for a... | 3 | https://mathoverflow.net/users/12705 | 118288 | 66,763 |
https://mathoverflow.net/questions/118284 | 6 | Hi!
Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and $f\in C^{0,\alpha}(M)$ such that
$$L(u)=f$$
Do i have Schauder estimates of type
$$\left\|u\right\|\_{C^{2k,\alpha}\lef... | https://mathoverflow.net/users/4971 | Schauder estimates for higher order linear elliptic operator on manifold | The result is true with some caveats. Under your assumptions we have the following results.
**1.** If $u\in W^{2k,2}(M)$ and $Lu\in C^{j,\alpha}(M)$, then $u\in C^{2k+j,\alpha}(M)$.
**2.** There exists $C>0$ depending only on $M$, $L$, $j$, and $\alpha$ such that, for any $u\in C^{2k+j,\alpha}(M)$ we have
$$\Vert... | 4 | https://mathoverflow.net/users/20302 | 118295 | 66,767 |
https://mathoverflow.net/questions/118258 | 8 | Let $R$ be a non-zero ring with identity. Is it possible for $R[x]$ to have only a finite number of maximal left ideals !?
| https://mathoverflow.net/users/nan | Maximal Ideals in $R[x]$ | Lets try something like Euclid's proof of infiniteness of prime numbers. Let $f\_1 = x, f\_2 = xf\_1 + 1, f\_3=xf\_1f\_2+1, f\_4=xf\_1f\_2f\_3 + 1$ and so on. The left ideal generated by each $f\_i$ is proper. Thus each $f\_i$ is contained in a maximal left ideal. On the other hand for any $i < j$ we have $ f\_i | f\_j... | 22 | https://mathoverflow.net/users/nan | 118311 | 66,779 |
https://mathoverflow.net/questions/118178 | 7 | Does the following group have a name? Is it amenable?
Fix $p$ and $q$
>
> $\langle g,h: hg^qh^{-1}=gh^pg^{-1}\rangle$
>
>
>
| https://mathoverflow.net/users/8699 | Certain finitely presented group | For 1-related groups, over a 2-letter alphabet, draw the relator on the plane grid: $g$'s are horizontal, $h$'s are vertical, starting at the point $O=(0,0)$. Connect $O$ with the endpoint $М$ of the resulting path $P$ by a vector $\vec{v}=\vec{OM}$. Consider the two support lines of the path $P$ that are parallel to $... | 6 | https://mathoverflow.net/users/nan | 118312 | 66,780 |
https://mathoverflow.net/questions/118319 | 2 | It is a theorem of Neil Strickland's that the category of harmonic spectra (i.e. the category of $p$-localized spectra localized at the infinite wedge of Morava K-theories) has no small objects. That is to say that there is not a single spectrum $X$ in the category of harmonic spectra such that $[X,\bigvee X\_i]=\bigop... | https://mathoverflow.net/users/11546 | Counterexamples to Smallness of Harmonic Spectra | I believe that any example of a direct sum of harmonic spectra which isn't itself harmonic should answer the question. Namely, if the $X\_i, i \in \mathbb{N}$ are harmonic and $\bigoplus X\_i$ isn't, then the map
$$ \bigoplus [S, X\_i] \to [S, L\_{\mathrm{harm}}\bigoplus X\_i]$$
isn't an isomorphism, because $\bigoplu... | 5 | https://mathoverflow.net/users/344 | 118321 | 66,782 |
https://mathoverflow.net/questions/118324 | 11 | Let $N\subset M$ be an inclusion of ${\rm II}\_1$ factors of finite index, $[M:N]<\infty$. I would be mostly interested in the hyperfinite case, $N\simeq M\simeq R$, but let us just take them arbitrary.
There is an evolving theory about "what can be said about $N\subset M$, in the general case'', which started with J... | https://mathoverflow.net/users/29333 | What is known about arbitrary subfactors of integer index? | If you also assume finite depth, then there's a hope (it's too vague to call it a conjecture) that all integer index subfactors can be classified "using only finite group theory." That is, if you had a black box which could answer all questions about finite groups and their cohomology you'd be able to understand all fi... | 9 | https://mathoverflow.net/users/22 | 118330 | 66,787 |
https://mathoverflow.net/questions/118333 | 7 | I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat it as an L2 norm problem.
The two approaches I've seen ( <http://www.ee.ucla.edu/~vandenbe/236C/lectures/smoothing.pd... | https://mathoverflow.net/users/28115 | Smoothing L1 norm, Huber vs Conjugate | Following the suggestion of András Bátkai I post my comment as an answer:
Smoothing the dual or the primal problem are quite different things: Smoothing the dual will not give you a smooth primal. However, you get a strongly convex primal by dual smoothing (as opposed to merely a strictly convex primal by Huber smoot... | 4 | https://mathoverflow.net/users/9652 | 118346 | 66,794 |
https://mathoverflow.net/questions/118206 | 8 | Consider the usual sphere $S^{n-1}\subset\mathbb R^n$. By Stone-Weierstrass $C(S^{n-1})$ is generated by the standard coordinates $x\_1,\ldots,x\_n:\mathbb R^n\to\mathbb R$, and in fact we have the presentation result $C(S^{n-1})=C^\*\_{comm}(x\_1,\ldots,x\_n|x\_i=x\_i^\*,\sum x\_i^2=1)$.
The Riemannian structure of... | https://mathoverflow.net/users/29333 | Eigenvalues of the free sphere | How about
$\lambda\_k = \frac{U'\_k(n)}{U\_k(n)}$
where the $U\_k$ denote the Chebyshev polynomials of the second kind, $U\_0(x)=1$, $U\_1(x)=x$, and $U\_k(x)=xU\_{k-1}(x)-U\_{k-2}(x)$ for $k\ge 2$.
In Section 10 of <http://arxiv.org/abs/1210.6768> (See in particular Remark 10.4) we try to classify "Brownian moti... | 10 | https://mathoverflow.net/users/30364 | 118348 | 66,795 |
https://mathoverflow.net/questions/118104 | 0 | Definition: Let $(V,\Omega)$ be a symplectic vector space, we define
$\perp:\Lambda ^k(V^\*)\to\Lambda ^{k-2}(V^{\ast})$
by $\perp(\omega)=i\_{X\_{\Omega}}(\omega)$
here if $(e\_1,e\_2,...e\_n,f\_1,f\_2,...f\_n)$ is basis of symplectic vector space then $\Omega=e\_{1}^{\*}\wedge f\_{1}^{\*}+...+e\_{n}^{\*}\wedge f\_... | https://mathoverflow.net/users/nan | $q_{S^*\omega}(X)=S^{\ast}q_{\omega}(X)$ ? | [Edit: I repeat the question.
$X\in V$.
$X\_{\Omega} = \Omega^{-1}$, the dual symplectic structure on $V^\*$.
$\Lambda^2S(X\_\Omega)=X\_{\Omega}$ for all $S\in Sp(V)$.
$q\_\omega(X) = i\_{X\_{\Omega}}i\_{X\_{\Omega}}(i\_X\omega\wedge i\_X\omega).$]
Then
$S^\star(q\_\omega(X)) = q\_{S^\star\omega}(S^{-1}X)$
si... | 2 | https://mathoverflow.net/users/26935 | 118355 | 66,799 |
https://mathoverflow.net/questions/118352 | 11 | As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter.
Suppose $u\_n\in H^1(B\_1)$ (actually, $u\_n$ are smooth), where $B\_1$ is the unit ball in $\mathbb{R}^N$, and that... | https://mathoverflow.net/users/28090 | Boundedness of the derivative of the trace of an H^1 function | Clearly, what you call $\newcommand{\bn}{\boldsymbol{n}}$ $\nabla u\cdot \bn $ is the normal derivative $\frac{\partial u}{\partial \bn}$. The trace theorem (see e.g. Lions and Magenes, **Non-Homogeneous Boundary Value Problems and Applications. I**, Thm. 9.4, Chap 1) shows that for $s> \frac{3}{2}$ the restriction map... | 17 | https://mathoverflow.net/users/20302 | 118360 | 66,801 |
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