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https://mathoverflow.net/questions/136303 | 8 | [Directed sets](http://ncatlab.org/nlab/show/direction) are defined to be sets equipped with a preorder that admit (finitary) upper bounds e.g. pairs $(D, \preceq)$ such that $\forall p,q \in D$ there exists $r \in D$ such that $p \preceq r$ and $q \preceq r$. Equivalently, they may be defined as thin categories in whi... | https://mathoverflow.net/users/31420 | What should the morphisms in the Category of Directed Sets be? | Here is a partial answer.
Given a directed set $D$, we say that a subset $A\subseteq D$ is cofinal if for each $d\in D$ there is some $a\in A$ with $d\leq a$. We say that $A\subseteq D$ is bounded if $A\subseteq\downarrow d=\{x\in D|x\leq d\}$ for some $d\in D$ and we say that $A$ is unbounded if it is not bounded. L... | 5 | https://mathoverflow.net/users/22277 | 136346 | 74,962 |
https://mathoverflow.net/questions/135759 | 21 | **Edit:** according to Narutaka Ozawa, question 3) is still open in the type $\mathrm{II}\_1$ case. In other terms, it is not known whether every topologically complemented type $\mathrm{II}\_1$ factor in $B(H)$ is injective.
Let $M$ be a von Neumann algebra sitting in $B(H)$. I will say that $M$ is B-complemented (r... | https://mathoverflow.net/users/35324 | On complemented von Neumann algebras | (i) Obviously, a vN subalgebra $N$ of a B-complemented vN algebra $M$ is also B-complemented if there exists a conditional expectation from $M$ onto $N$. (ii) Haagerup and Pisier's proof actually says if $N$ is a B-complemented vN algebra which contains a (possibly non-unital) copy of $M\_2(N)$ with a conditional expec... | 14 | https://mathoverflow.net/users/7591 | 136353 | 74,964 |
https://mathoverflow.net/questions/136345 | 7 | Let x be a positive element in the spatial tensor product of two non unital C\* algebras
A and B. Is there a single element $a \otimes b \geq x$?
How can we noncommutativize the following proof, in the commutative case:
Let $F^2$ be a positive function on $X\times Y$. Define $f(x)=\sup\_{y\in Y} F(x,y)$ and $g(y)=\sup\... | https://mathoverflow.net/users/36688 | positive elements in tensor product | Yes. For a self-adjoint element $y$, denote $s(y)=\sup{\rm Sp}(y)$. Then for $\gamma \geq s(y)$, one has
$$\inf \lbrace s(y - \gamma(e\otimes f)) : 0\le e\le 1,\ 0\le f\le 1\rbrace \le 0.$$
Indeed, if $e\_n$ and $f\_n$ are approximate units, then so is $g\_n:=e\_n\otimes f\_n$ and
$y - \gamma g\_n \le y-g\_n^{1/2} y g... | 12 | https://mathoverflow.net/users/7591 | 136354 | 74,965 |
https://mathoverflow.net/questions/136352 | 5 | Is there an algorithm to detect the Non-Haken Manifold?
Or, is there a sufficient condition for a manifold to be
a non-Haken manifold? (off course, I hope that condition is not the ones in its
definition.)
Note: a Non-Haken manifold means that either it is reducible or it contains no 2-sided properly embedded
incom... | https://mathoverflow.net/users/18496 | sufficient conditions on Non-Haken manifolds | Yes, there is an [algorithm due to Jaco-Oertel](http://www.ams.org/mathscinet-getitem?mr=744850) to detect if an irreducible manifold is Haken, and therefore to detect a non-Haken manifold (if it is given to be irreducible). The phrasing is a bit confusing (are you assuming the given manifold is irreducible?), but give... | 10 | https://mathoverflow.net/users/1345 | 136355 | 74,966 |
https://mathoverflow.net/questions/136351 | 9 | It is easy to prove that
(\*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.
It is also easy to see that statement (\*) remains true if the constant 2 is replaced with a somewhat smaller one. Contest: Find such a constant, the smaller the better.
### Update:
Reaching $\sqrt 2$ and ... | https://mathoverflow.net/users/36904 | Small quadrilaterals containing a given convex region | G. D. Chakerian, Minimum area of circumscribed polygons, Elem. Math. 28 (1973), 108–111, MR0322682 (48 #1044) proved that if $K$ is a convex body of area 1 in the plane then $K$ is contained in a quadrilateral of area at most $\sqrt2$.
W. Kuperberg, On minimum area quadrilaterals and triangles circumscribed about co... | 5 | https://mathoverflow.net/users/3684 | 136356 | 74,967 |
https://mathoverflow.net/questions/136358 | 2 | Let $R=k[x\_1,..,x\_n]/I$ and let $X=Spec(R)$ be it's associated affine scheme. Suppose that $X$ has only one isolated singularity, say at the origin $\mathfrak{m}=\langle x\_1,...,x\_n\rangle$. Now, let $R\_{\mathfrak{m}}$ be the localization at $\mathfrak{m}$, which is a local ring with maximal ideal $\mathfrak{m}\_{... | https://mathoverflow.net/users/36661 | Local blowup versus global blowup | Sure. This is true.
Indeed, the formation of the blowup is easily seen to commute with localization (your Rees algebra $R\_{\mathfrak{m}}[\mathfrak{m}\_{\mathfrak m} t]$s formation certainly commutes with localization of $R$. This generalizes to any sort of situation you'd like.
In particular, if $\mathfrak{m}$ is... | 3 | https://mathoverflow.net/users/3521 | 136378 | 74,977 |
https://mathoverflow.net/questions/136110 | 4 | Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f\_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the article of Serre and Tate on good reduction of abelian varieties).
Questions (a): Let $B$ be a non-zero abelian variety... | https://mathoverflow.net/users/36759 | Conductor of abelian varieties | The answer to (a) is yes. The conductor is given by the representation of an inertia group $I\_v$ in the Tate module. As $T\_\ell(A\times B)=T\_\ell(A)\times T\_\ell(B)$, the additivity is easy to see from definition (Serre: Facteurs locaux des fonctions zêta des variétés algébriques, §2. The definition you cite is the... | 4 | https://mathoverflow.net/users/nan | 136390 | 74,983 |
https://mathoverflow.net/questions/136380 | 3 | Given a finite dimensional (real) vector space $V$, and two non-degenerate bilinear forms $(\cdot,\cdot)\_1$ and $(\cdot,\cdot)\_2$, one can use a basic linear algebra argument to show that there exists a unique invertible linear map $F:V \to V$ such that
$$
(F(\cdot),\cdot)\_1 = (\cdot,\cdot)\_2.
$$
Now if we also as... | https://mathoverflow.net/users/36750 | Classifying all Equivariant Bilinear Forms on a Finite-Dimensional Module | This is true: Write $(v,w)\_i = \langle A\_i(v),\rangle$ where $A\_i:V\to V^\ast$ is invertible and $\langle\quad,\quad\rangle:V^\ast\times V \to K$ is the duality pairing. $(\quad,\quad)\_i$ is $G$-invariant iff $A\_i$ is $G$-equivariant, and $A\_1\circ F=A\_2$, or $F=A\_1^{-1}\circ A\_2$ is then $G$-equivariant too. ... | 2 | https://mathoverflow.net/users/26935 | 136397 | 74,986 |
https://mathoverflow.net/questions/136350 | 5 | **Background**
This is a follow-up question to:
[What (classes of) Banach spaces are known to have Schauder basis?](https://mathoverflow.net/questions/134727/what-classes-of-banach-spaces-are-known-to-have-schauder-basis)
In the previous question, I asked about what spaces are known to have Schauder basis. It seem... | https://mathoverflow.net/users/31548 | What Approximation Property does the space of Schatten-p class operators have? | It has a Schauder basis, namely $e\_i\otimes e\_j$, where $(e\_i)$ is an orthonormal basis of the Hilbert space. This holds for $1\le p<\infty$. For the Schatten ideal of compact operators, I do not know the answer.
| 8 | https://mathoverflow.net/users/26935 | 136398 | 74,987 |
https://mathoverflow.net/questions/136396 | 3 | Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p\_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two measurable functions $f, g:X\to Y$ such that $f = g$ holds $p\_X$-a.e. Is that true that
$$
p\left(\mathrm{Gr}[f]\,\D... | https://mathoverflow.net/users/11768 | Maps that are a.e. equal have almost the same graphs | The answer is easily yes, because we have
$$ \operatorname{Gr}(f)\Delta \operatorname{Gr}(g)\subset N\times Y$$
where $N:=\{x\in X\, :\, f(x)\neq g(x) \}$ by assumption has null measure
$$p\_X(N):= p(\operatorname{Pr}\_X^{-1}(N))=p( N\times Y )=0.$$
| 3 | https://mathoverflow.net/users/6101 | 136400 | 74,989 |
https://mathoverflow.net/questions/136275 | 2 | Given a spherical reference frame $\left(\rho,\phi,\theta\right)$, the reduced wave equation can be written as:
$$\nabla^2U=k^2n^2U$$
in which:
$U=U(\rho,\phi,\theta)$
The solutions of this equation can be found using classical methods of PDE knowing the initial and boundary conditions.
Suppose we have, $k$ constant wi... | https://mathoverflow.net/users/21258 | Solutions of a stochastic reduced wave equation | I refer to [this paper](https://www.researchgate.net/publication/228448963_Stochastic_Partial_Differential_Equations_as_priors_in_ensemble_methods_for_solving_inverse_problems). There is a straightforward approach in this case. You can separate the refraction index in the following way
$$
n^2=n\_0^2+\xi
$$
where $\xi$... | 1 | https://mathoverflow.net/users/19520 | 136403 | 74,990 |
https://mathoverflow.net/questions/136015 | 7 | In comprehension categories, dependent sums are defined as a choice of left adjoints for all reindexing functors along display maps, satisfying a Beck-Chevalley condition. Dependent products are right adjoints of the same functors, and identity types are left adjoints of reindexing functors along diagonal maps.
These... | https://mathoverflow.net/users/4913 | $\Pi$, $\Sigma$, and identity types without $\eta$ in comprehension categories | I'm not sure exactly what you're asking, but the general pattern for "positive" types is to assert that every "structured" display map over the type has a specified structured section. This matches the induction principle of a positive type former in type theory. One then needs to require a sort of "pullback-stability"... | 4 | https://mathoverflow.net/users/49 | 136404 | 74,991 |
https://mathoverflow.net/questions/136394 | 5 | Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set $A\subseteq X$ there exists a Borel set $B\subseteq X$ such that $p(A\Delta B) = 0$. Moreover, if $f:X\to\Bbb R$ is univers... | https://mathoverflow.net/users/11768 | Universally measurable map coincides a.e. with a Borel map | The proof is correct. The following result of a stronger assertion is taken from Hans Crauel's book *Random Probability Measures on Polish Spaces*, where it is Lemma 1.2.:
**Proposition:** Let $(\Omega,\Sigma,\mu)$ be a probability space, $Y$ a separable metric space and $f\_0:\Omega\to Y$ be measurable with respect ... | 5 | https://mathoverflow.net/users/35357 | 136407 | 74,992 |
https://mathoverflow.net/questions/136414 | 67 | There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
| https://mathoverflow.net/users/7089 | What if the Riemann Hypothesis were false? | An explicit zero $\rho$ for $\zeta(s)$, off the critical line, would give an explicit lower bound on the class number $h(-d)$ for $\mathbb Q(\sqrt{-d})$, for a range of $-d$ in terms of $\text{Im}(\rho)$. This is the 'Deuring-Heilbronn phenomenon,' with results due to these two and others beginning in the 1930's. For a... | 47 | https://mathoverflow.net/users/6756 | 136416 | 74,995 |
https://mathoverflow.net/questions/136401 | 8 | For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an abelian category one can define a *categorically finitely generated* object as being an object M such that for any directed f... | https://mathoverflow.net/users/36922 | Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks) | Note that on any qcqs algebraic space or Artin stack, a quasi-coherent sheaf is finite type if its pullback to any fppf scheme cover is finite type in the usual sense on schemes. We can similarly define the notion of "finite type" for a quasi-coherent sheaf of modules over any quasi-coherent sheaf of algebras on such a... | 8 | https://mathoverflow.net/users/36938 | 136423 | 74,999 |
https://mathoverflow.net/questions/136420 | 9 | I am looking for references on the following: **Spin structures on surfaces, and particularly the spin mapping class group.**
**What is known about generating the spin mapping class group? Has anybody found a finite set of generators?**
| https://mathoverflow.net/users/23204 | Reference request: Spin structures on surfaces and the spin mapping class group | The answer of course depends on the spin structure chosen. The paper
Johnson, Dennis,
Spin structures and quadratic forms on surfaces.
J. London Math. Soc. (2) 22 (1980), no. 2, 365–373.
proves that the set of spin structures on a closed surface $\Sigma\_g$ of genus $g$ can be identified with the set of quadrati... | 14 | https://mathoverflow.net/users/317 | 136425 | 75,001 |
https://mathoverflow.net/questions/136277 | 4 | The only simple groups having an irreducible character of degree 3 are $A\_5$ and $PSL\_2(7)$. What is it know for degrees 4 or 5? That is, do we know what simple groups posses an irreducible charcter of degree 4? of degree 5?
| https://mathoverflow.net/users/36868 | What simple groups have irreducible characters of degree 4 or 5? | I think the best modern source for this information is the paper:
G. Hiss and G. Malle.
Low-dimensional representations of quasi-simple groups.
LMS J. Comput. Math. 4 (2001), 22-63.
Corrigenda: LMS J. Comput. Math. 5 (2002), 95-126.
This lists all irreducible representations (in characteristic 0 and coprime to the ... | 4 | https://mathoverflow.net/users/35840 | 136427 | 75,003 |
https://mathoverflow.net/questions/136371 | 7 | This question came out of this other [MO question of mine](https://mathoverflow.net/questions/136236/prospects-for-reverse-mathematics-in-homotopy-type-theory). My question is
>
> Is there a formal comparison between $\mathsf{RCA}\_0$ and $\mathsf{BISH}$ (Bishop style constructive mathematics as used in constructiv... | https://mathoverflow.net/users/12978 | Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$ | BISH famously includes the full axiom of choice scheme (in the functional language of second-order arithmetic), which is utterly weak in that context but very strong when combined with the law of the excluded middle. This is precisely the context in which Bishop wrote that the axiom of choice follows from "the very mea... | 11 | https://mathoverflow.net/users/5442 | 136428 | 75,004 |
https://mathoverflow.net/questions/136368 | 1 | As far as I know, the April 2011 version of #143 on [this page](http://math.dartmouth.edu/~carlp) has not been improved upon.
On page 10 of that paper, the authors give an algorithm that uses a constant $\:c\_{\hspace{.01 in}5}\:$.
According to page 2 and page 6 of that paper, $\:c\_{\hspace{.01 in}5}\:$ should... | https://mathoverflow.net/users/nan | computing $c_5$ in "Primality testing with Gaussian periods" | Your question is about the result of Deshouillers–Iwaniec in the paper *"On the Brun-Titchmarsh theorem on average"*,
$$
\pi(x,q,a)\le \frac{(\frac{4}{3}+\epsilon c\_5)x}{\phi(x)\log (\frac{x}{q})},
$$
where the notations are explained in the above paper of Carl Pomerance. It is indeed mentioned
that the constants are... | 1 | https://mathoverflow.net/users/32332 | 136433 | 75,005 |
https://mathoverflow.net/questions/136431 | 18 | Recall that a group $G$ is called *Hopfian* if every surjective endomorphism $G\to G$ is injective. Malcev observed that all finitely-generated (f.g.) residually finite groups are Hopfian. It is well-known that residual finiteness is not a *coarse invariant*, i.e. a residually finite f.g. group can be quasi-isometric t... | https://mathoverflow.net/users/21684 | Is Hopf property a quasi-isometry invariant? | The answer to Question 1 is "no". The group $\langle x, y \mid x^{12}y = yx^{18}\rangle$
is Hopfian but contains a non-Hopfian subgroup of finite index (see Baumslag, Gilbert; Solitar, Donald Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 1962 199–201.).
**Edit.** Since the paper of Baums... | 24 | https://mathoverflow.net/users/nan | 136436 | 75,006 |
https://mathoverflow.net/questions/136432 | 1 | Unless I am very wrong, the following seems to be true:
>
> If the angle between two vectors in $\mathbb{R}^{n}\_{++}$ is small, then the
> value of the Hilbert projective metric between them is also small.
>
>
>
I am looking for a reference to a precise statement of this notion.
P.S.
What I mean by the ang... | https://mathoverflow.net/users/22051 | Angles and projective metric | This is equivalent to the following: in the Klein model of hyperbolic space, if two points are close together, then their hyperbolic distane is small. This is true locally (e.g., fix one point), but not true globally (in other words, no bound on Euclidean distance implies a fixed bound on hyperbolic distance -- the dis... | 2 | https://mathoverflow.net/users/11142 | 136449 | 75,013 |
https://mathoverflow.net/questions/136445 | 4 | Does there exist a category $\mathcal{C}$ and contravariant functors $F:\mathcal{C}\to\mathcal{C}^{op}, G:\mathcal{C}^{op}\to\mathcal{C}$ such that $F$ is left ajoint to $G$ and $F\neq G^{op}$? In the case of $\mathcal{C}=Set$, I proved that there exist no such functors, because if exist $G$ preserve limits, and so is ... | https://mathoverflow.net/users/36961 | Adjoint functors between dual categories | Here's a very simple example. Let $\mathcal{C}$ be any set with more than two elements, considered as a discrete category. Then $\mathcal{C}=\mathcal{C}^{op}$, and any permutation $F:\mathcal{C}\to\mathcal{C}^{op}=\mathcal{C}$ has an adjoint $G$ (on both sides!) given by the inverse. Any permutation that is not its own... | 7 | https://mathoverflow.net/users/75 | 136450 | 75,014 |
https://mathoverflow.net/questions/136454 | 4 | Let $X$ be a variety over $\mathbb{C}$, and $X^{an}$ be the analytic space associated to $X$. $X$ has Gorenstein singularity at $x \in X$ iff the local ring $\mathcal{O}\_{X,x}$ is a Gorenstein ring. Is this result true:
>
> $\mathcal{O}\_{X,x}$ is a Gorenstein $\iff$ $\mathcal{O}\_{X^{an},x}$ is a Gorenstein.
>
... | https://mathoverflow.net/users/29730 | Is Gorenstein singularity a locally analytic property | The key point is that if $X$ is a locally finite type $\mathbf{C}$-scheme and $x \in X(\mathbf{C})$ then $O\_{X,x}$ and $O\_{X^{\rm{an}},x}$ are both local *noetherian* rings with the *same* completion (induced by the evident canonical map from the algebraic local ring to the analytic one). So for any property of local... | 10 | https://mathoverflow.net/users/36938 | 136466 | 75,019 |
https://mathoverflow.net/questions/132779 | 18 | (Apologies if this is too obscure.)
In joint work with Izzet Coskun we came across the following kind of combinatorial identity, but we weren't able to prove it, or to identify what kind of identity it is. (We looked in some references, but to the outsider it can be difficult to distinguish one insanely complicated s... | https://mathoverflow.net/users/nan | Is this similar to a known combinatorial identity? | Here is a solution to your warm up problem. It uses a few known elementary identities, and some short inductions for identities I didn't recognize. I also changed your notation slightly from $l$ to $k$ in the internal summations.
Setting $x=2$, the first term vanishes, and we combine the second and third terms as $(... | 8 | https://mathoverflow.net/users/18086 | 136467 | 75,020 |
https://mathoverflow.net/questions/136463 | 0 | Suppose $\omega$ is a real closed $(1,1)$ form on a compact Kähler manifold. If we have a real $d$-closed two form $\sigma$ such that $[\sigma]=[\omega] \in H^2(M)$, can we claim that this two form $\sigma$ is also a $(1,1)$ form?
| https://mathoverflow.net/users/36974 | A question about a two form and a $(1,1)$ form on a compact Kähler manifold | I think not. Note that replacing $\sigma$ by $\sigma - \omega$ reduces us to the case $\omega=0$. Your question in that case is whether every real $d$-exact 2-form is a $(1,1)$-form.
Now unless I'm mistaken, on projective space $\mathbf{CP}^2$ with homogeneous coordinates $(a:b:c)$, we get a real $d$-exact 2-form wit... | 4 | https://mathoverflow.net/users/19276 | 136477 | 75,023 |
https://mathoverflow.net/questions/136472 | 1 | It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a coboundary. Does anyone know of a similar theorem for polynomial 4-cocycles? The proof of the theorem for 3-cocycles doesn... | https://mathoverflow.net/users/11546 | "Symmetric" Polynomial 4-cocycles | Heaton showed that if the field $K$ has characteristic zero, then all polynomial cocycles are coboundaries. In characteristic $p$ this is not always the case, but the following result holds (which you may know already).
**THEOREM:** For $n> 2$ and $K$ a field of characteristic $p$, every polynomial
$n$-cocycle over $... | 2 | https://mathoverflow.net/users/32332 | 136482 | 75,025 |
https://mathoverflow.net/questions/136453 | 2 | How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by Liebeck and Kleidman but I think this mostly deals with the maximal subgroups (please correct me if I'm wrong) and I'd lik... | https://mathoverflow.net/users/34884 | Subgroup structure of orthogonal groups of small dimension over finite fields | Despite the the fact that it deals only with maximal subgroups, I am afraid that I am unable to restrain myself from mentioning (i.e. advertising) the forthcoming book
The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. John N. Bray, Derek F. Holt,Colva M. Roney-Dougal, London Mathematical Society L... | 3 | https://mathoverflow.net/users/35840 | 136493 | 75,026 |
https://mathoverflow.net/questions/136506 | 14 | Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int\_a^b \| \dot\gamma(t) \| \, dt\,,$$ where the infinum is taken over all $C^1$-curves connecting $x$ and $y$.
I'm interested in the basic properties of ... | https://mathoverflow.net/users/17047 | How metric is Riemannian geometry | If you forget about the Riemannian metric, you should also forget about the tangent bundle and manifold structure. Then you end up with metric spaces of inner type or Aleksandrov spaces, where you can find a continuous curves with arc-length the distance.
Look at "M. Gromov: Metric structures for Riemannian and Non-Rie... | 17 | https://mathoverflow.net/users/26935 | 136510 | 75,029 |
https://mathoverflow.net/questions/136502 | 3 | [The ncatlab says:](http://ncatlab.org/nlab/show/Bousfield+localization+of+model+categories#localization_of_triangulated_categories_93)
>
>
> >
> > Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of... | https://mathoverflow.net/users/1291 | Bousfield localization before and after taking homotopy | Let $\tilde C$ be the left Bousfield localization of $C$. As categories $C=\tilde C$, but $\tilde C$ has more weak equivalences. In particular, the identity functor induces a functor $\varphi\colon \operatorname{Ho}(C)\rightarrow \operatorname{Ho}(\tilde C)$. Let $\mathcal L=\ker \varphi$, i.e. $\mathcal L\subset \oper... | 5 | https://mathoverflow.net/users/12166 | 136517 | 75,030 |
https://mathoverflow.net/questions/136503 | 1 | Where to find a proof of theorem which says that:
if a funcion $f: \mathbb R \rightarrow \mathbb R$ is bounded on a set of positive Lebesgue measure or on the set of second category with Baire property and
$$
\frac{\Delta\_h^nf(x)}{h^n} \rightrightarrows g(x) \textrm{ as } h \rightarrow 0
$$
on every compact interval $... | https://mathoverflow.net/users/36992 | Where find proof of such theorem about uniform convergence of differences | MR1245559 (94k:26024)
Frölicher, Alfred(CH-GENV-SM); Kriegl, Andreas(A-WIEN)
Differentiable extensions of functions. (English summary)
Differential Geom. Appl. 3 (1993), no. 1, 71–90.
The review:
A condition is given for functions defined on an arbitrary subset of the real line with values in a Fréchet space (or ev... | 2 | https://mathoverflow.net/users/26935 | 136518 | 75,031 |
https://mathoverflow.net/questions/136516 | 4 | Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works),
$f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic.
By the Lefschetz fixed point theorem (or by counting cells of $X$ and $Y$ with the induced cell structure) one obtains
$$\ch... | https://mathoverflow.net/users/32972 | Euler Characteristic of Coverings via Sheaf Theory | Generalization: Euler characteristic with coefficients in a rank $n$ local system equals $n$ times ordinary Euler characteristic. Then using Leray (as in Gunnar Magnusson's comment) gives the case of a finite covering (and more generally that of a fibre bundle).
The following `sheafy' proof of the generalization shou... | 5 | https://mathoverflow.net/users/23907 | 136519 | 75,032 |
https://mathoverflow.net/questions/136483 | 6 | I am wondering whether the following SDE can be solved explicitly?
$$
d X\_t = X\_t^2 d t + X\_t d B\_t
$$
where $B\_t$ is a standard Brownian motion. If not, can we say some thing about the moments of the solution, i.e., $E(|X\_t|^n)$?
Thank you very much for any hints!
Anand
| https://mathoverflow.net/users/36814 | Solving a SDE with quadratic drift | Solutions do exist locally. Globally they MAY blow up as you already know.
The blowup will be dominated by the deterministic system. You did not write what is your initial condition - note that $0$ is a perfectly fine solution to your equation. If your initial condition is say $X\_0=1$, then it may blow up, but with po... | 6 | https://mathoverflow.net/users/35520 | 136532 | 75,037 |
https://mathoverflow.net/questions/136451 | 12 | A metric space $M$ is *metrically homogeneous* if for every pair of points $x, y \in M$ there is an isometry $f$ of $M$ onto $M$ such that $f(x)=y$. What is known about metrically homogeneous spaces? Any references? In particular, is there an explicit description of all metrically homogeneous subspaces (under the Eucli... | https://mathoverflow.net/users/36904 | Metrically homogeneous subsets of the plane | Here is an description of homogeneous subsets of ${\mathbb R}^2$ using group theory, similar to the one you have in dimension 1. Let $M\subset {\mathbb R}^2$ be homogeneous and let $G< I=Isom({\mathbb R}^2)$ be its group of isometries. Then $G$ fits into short exact sequence
$$
1\to T\to G\to S \to 1
$$
where $T$ is a ... | 9 | https://mathoverflow.net/users/21684 | 136536 | 75,038 |
https://mathoverflow.net/questions/126986 | 22 | Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in operation in their writings a conception of mathematics which is quite extraneous to that of Euler." Ferraro concludes that "t... | https://mathoverflow.net/users/28128 | Euler's mathematics in terms of modern theories? | [Converted from comment to answer per Yemon Choi's suggestion.]
From a casual run-through of the Ferraro paper, it seems like Euler's ideas about infinitesimals were, unsurprisingly, not formalized to modern standards and therefore don't map exactly onto modern concepts. He apparently didn't think of a line segment a... | 5 | https://mathoverflow.net/users/nan | 136546 | 75,043 |
https://mathoverflow.net/questions/136294 | 4 | For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has:
* $G = K C\_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$)
* $G \neq K C\_G(K/L)$ and $H \cap K \leq L$ (so $K/L$ is non-inner and avoided by $H$)
If $G$ is solvable, then all such $H$ are conjugate... | https://mathoverflow.net/users/3710 | Generalized system normalizer for insoluble finite groups | I think a nonsplit extension of a simple group by an outer automorphism would provide a counterexample.
The smallest example of this is the group of order 720 sometimes denoted $M\_{10}$, the extension $A\_6.2\_3$ in ATLAS notation. The chief factor $A\_6$ is not inner, whereas the one of order 2 is inner, so a subgr... | 2 | https://mathoverflow.net/users/35840 | 136559 | 75,048 |
https://mathoverflow.net/questions/136550 | 7 | Can anyone help me to find a simple and good reference (a book, lecture notes or a website) for learning the [surgery theory](https://en.wikipedia.org/wiki/Surgery_theory) and its applications? I seek a reference together with many examples and figures for more intuition.
Thanks.
| https://mathoverflow.net/users/32817 | A simple and good reference on surgery theory | Well, there's surgery and there's Surgery Theory. By the former, I mean the basic geometric and homotopy-theoretic ideas that went into building a big machine (the latter) to systematically study problems in high-dimensional manifolds. Although you can dive right into Surgery Theory, I think it's much better to invest ... | 14 | https://mathoverflow.net/users/3460 | 136560 | 75,049 |
https://mathoverflow.net/questions/136554 | 15 | I am reasking a year-old [math.stackexchange.com question](https://math.stackexchange.com/questions/155138/generalizations-for-tietzes-extension-theorem) asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
[The Tietze extension theorem](htt... | https://mathoverflow.net/users/12978 | Generalizations of the Tietze extension theorem (and Lusin's theorem) | There is a nice characterization of the spaces $X$ where the Tietze extension theorem holds for all complete separable metric spaces $Y$. We say that a Hausdorff space $X$ is ultranormal if whenever $R,S$ are disjoint closed subsets of $X$, then there is some clopen set $C$ with $R\subseteq C$ and $S\subseteq C^{c}$. A... | 13 | https://mathoverflow.net/users/22277 | 136562 | 75,051 |
https://mathoverflow.net/questions/136545 | 5 | Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book *Model categories*, there is a model category structure on the diagram category $C^B$ such that weak equivalences and fibrations are defined pointwise.
Assume that $C\leftrightarrows D$ is a Quillen equivalence. By pointwise ... | https://mathoverflow.net/users/1291 | Quillen equivalence of diagram categories | Edit: As Fernando pointed out, the claim can be found in Hirschhorn's book as Proposition 15.4.1.
Here is my own attempt. We write $(F,G)\colon C\leftrightarrows D$ for the given Quillen equivalence and $(F^X,G^X)\colon C^X\leftrightarrows D^X$ for the pointwise induced adjunction. First we check that $(F^X,G^X)$ is ... | 1 | https://mathoverflow.net/users/1291 | 136563 | 75,052 |
https://mathoverflow.net/questions/134671 | 8 | Consider a complex symmetric matrix $$ C= C\_R + i C\_I $$ with $C\_R,C\_I \in \text{Mat}\_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C\_R$ are all strictly positive. Then, $C$ is invertible, and the real part of its inverse is $$ (C^{-1})\_R = \Big(C\_R + C\_I (C\_R)^{-1} C\_I\Big)^{-1}. $$... | https://mathoverflow.net/users/35281 | Inequalities for Hadamard products of complex symmetric matrices | The first inequality is **true** while the second one is **false**. The proof below suggests a version of the second inequality that does hold.
To ease notation, let $A = C\_R$, and $B=C\_I$. We wish to show that
\begin{equation\*}
(BA^{-1}B)\_{ii} \ge [(S\circ B)(S\circ A)^{-1}(S\circ B)]\_{ii}.
\end{equation\*}
... | 4 | https://mathoverflow.net/users/8430 | 136565 | 75,053 |
https://mathoverflow.net/questions/136515 | 12 | If $X$ is a compact Kahler manifold then it's well-known that $X$ can be embedded into a projective space if and only if it admits an ample line bundle. Suppose now that we look for other things to embed $X$ into, like complex tori, and ask for necessary and sufficient conditions for this to happen.
If $X$ is a subm... | https://mathoverflow.net/users/4054 | Which complex manifolds embed into tori? | See:
@article {MR0350057,
AUTHOR = {Matsushima, Yozo},
TITLE = {Holomorphic immersions of a compact {K}\"ahler manifold into
complex tori},
JOURNAL = {J. Differential Geometry},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {9},
YEAR = {1974},
PAGES = {309--328},
ISSN = {0022-040X},
MRCLASS = {32C... | 3 | https://mathoverflow.net/users/11142 | 136566 | 75,054 |
https://mathoverflow.net/questions/136573 | 2 | Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves under a covering map. In particular, I want to know which curves are null-homologous. The automorphisms give rise to auto... | https://mathoverflow.net/users/13832 | Finding null-homologous curves via the matrix equation $AB^iC^jx=0$ | I am pretty sure your general question is hopeless. However, your motivating question is not. Notice that a Dehn twist gives rise not to some random invertible matrix but to a *transvection*, which is to say $B(v) = v + a <v, u> u$ for some fixed scalar $a$ and vector $u$ (in your case the scalar is $1$ but that is not... | 1 | https://mathoverflow.net/users/11142 | 136575 | 75,057 |
https://mathoverflow.net/questions/136572 | 12 | I've recently been studying Euler's theories on music, and I came across Euler's concept of gradus suavitatis or 'degree of pleasure' of a rational number representing the ratio of two tones. (I found this on <http://www.mathematik.com/Piano/>)
The formula is $G(p/q)=1+\Sigma e\_i (p\_i-1)$ where $p,q$ are relatively... | https://mathoverflow.net/users/27933 | Number theory underlying Euler's theory of music | This is not directly an answer to your question (graph theoretic relation, rather than number theoretic), but it's too long for a comment.
Euler's formula is a special case of a disharmonicity function
$$D(x)=\sum\_i |e\_i| g(p\_i)$$
with $g(p\_i)>0$ a function of the prime factors $p\_i$ of a rational number $x=... | 7 | https://mathoverflow.net/users/11260 | 136584 | 75,061 |
https://mathoverflow.net/questions/136555 | 4 | Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a [distribution](http://en.wikipedia.org/wiki/Distribution_%28differential_geometry%29) $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of $\mathcal{D}$ generate the full tangent bundle... | https://mathoverflow.net/users/1049 | Converse to Chow's theorem in sub-riemannian geometry | Velimir Jurdjevic, **Geometric Control Theory**, p. 48, theorem 6: the orbits of any Lie algebra of real analytic vector fields have tangent space at each point given by the values of those vector fields at that point, so a more general result which immediately implies what you are looking for.
| 5 | https://mathoverflow.net/users/13268 | 136589 | 75,062 |
https://mathoverflow.net/questions/133520 | 8 | It is a well-known theorem first proved by Bourgain that any map $\varphi:T\_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a universal constant. Distortion is the least $D$ such that for some $s$ and all $x,y\in T\_n$, we have
$$ s d(x,y) \leq d(\var... | https://mathoverflow.net/users/4961 | Distortion of tree embedding in Alexandrov spaces | See the direct proof of markov convexity in proposition 2.1 of the following paper
Mendel, Manor; Naor, Assaf Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 287–337.
the only thing that is used there is lemma 2.3 of the above references, which holds with p=2 f... | 3 | https://mathoverflow.net/users/37045 | 136593 | 75,065 |
https://mathoverflow.net/questions/136576 | 3 | Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim linear utility:
$$
u\_i(k(\hat{\theta}), t(\hat{\theta}));\theta\_i)
= \theta\_i k\_i(\hat{\theta}) + t\_i(\hat{\theta}))... | https://mathoverflow.net/users/33290 | Optimal auction for risk-averse seller | Because all auction rules with $U\_i(0)=0$ yield the same expected revenue, the optimal auction will be one that returns that revenue to the seller with certainty.
Another way to say this is that each buyer shares the seller's uncertainty about all the other buyers, so each risk-neutral buyer should fully insure the ... | 3 | https://mathoverflow.net/users/10503 | 136595 | 75,066 |
https://mathoverflow.net/questions/135448 | 5 | I've been interested in abstract polyhedral decompositions of 3-manifolds for a long time. One thing I've tried to do a lot is to get nice polyhedral decompositions of manifolds with Nil geometry. It has been difficult, however, to find ones without some kind of irregularity (e.g. the boundary graph contains vertices o... | https://mathoverflow.net/users/27933 | Can the integer Heisenberg group be cubulated? | From the comments:
A discrete group with a proper action on an arbitrary CAT(0) cube complex has no distorted cyclic subgroups. Thus Heisenberg (which has a quadratically distorted Z) has no such proper action.
It follows from Pansu's results that if G is a discrete f.g. group QI to Heisenberg, then it has a finite i... | 1 | https://mathoverflow.net/users/27933 | 136601 | 75,068 |
https://mathoverflow.net/questions/136602 | 2 | Is the moduli space of Prym curves (curves $C$ with square root of $\mathcal{O}\_C$, compactified via admissible covers - by Beauville) of a given genus $g$ normal? Why?
| https://mathoverflow.net/users/4096 | normality of moduli of prym curves | There are various references. The one I like is the following, particularly Remark 1.3.3.
MR2007376 (2005b:14049) Reviewed
Abramovich, Dan(1-BOST); Corti, Alessio(4-CAMB); Vistoli, Angelo(I-BOLO)
Twisted bundles and admissible covers. (English summary)
Special issue in honor of Steven L. Kleiman.
Comm. Algebra 31 (20... | 3 | https://mathoverflow.net/users/13265 | 136606 | 75,069 |
https://mathoverflow.net/questions/136618 | 3 | are there any invariants of matrices, that are not affected by row- and/or column permutations?
To me it seems that the sequence of singular values could be such an invariant; am I right, resp. are there other invariants?
My guess for the sequence of singular values comes from the observation that a row permutation... | https://mathoverflow.net/users/31310 | Invariants of Matrix Reordering | As said in the comment, the *permanent* of a square matrix is such an invariant.
It is interesting for many topics in combinatorics.
For $A=(a\_{ij})$ it is defined as $\Sigma\_{\sigma \in S\_n}\prod\_{i=1}^na\_{i,\sigma(i)}$.
For $n=3$, as an example,
$Perm(A)=a\_{11}a\_{22}a\_{33} + a\_{11}a\_{23}a\_{32} + a\_{12}a\... | 5 | https://mathoverflow.net/users/32332 | 136622 | 75,076 |
https://mathoverflow.net/questions/136273 | 26 | When exactly were $\ell\_p$ norms first defined and used?
(Here is what I know, or think I know: Lebesgue and/or Riesz had something to do with them, but in some sense they go back to Minkowski, since Minkowski's inequality is (in essence) the statement that an $\ell\_p$ norm is a norm.)
Here is what is really my m... | https://mathoverflow.net/users/398 | Origin and first uses of $\ell_p$ norms? | Toeplitz in his [review](http://www.emis.de/cgi-bin/JFM-item?44.0401.01) of Riesz [[1913](http://archive.org/stream/lessystmesdq00riesuoft)] laments the lack of explicit motivation in generalizing from $\ell\_2$ to $\ell\_p$:
>
> The considerations of the 3rd Chapter require not the convergence of the sum of square... | 12 | https://mathoverflow.net/users/19276 | 136630 | 75,079 |
https://mathoverflow.net/questions/136632 | 3 | This might be a ridiculous question, but please bear with me.
Let $E$ be an elliptic curve over a $p$-adic field $K$. Denote by $K(E\_{p^∞}):=\bigcup\_{n∈Z≥1} K(E[p^n])$ the field extension obtained by adjoining to $K$ the coordinates of all $p$-power torsion points of $E$. By the Weil pairing, we know that all the $... | https://mathoverflow.net/users/37040 | Division Field of a nonCM elliptic curve | $E[p^n]$ is, over the maximal unramified extension of $K$ (obtained by adding all prime to $p$ roots of unity), an extension of $\mathbb{Z}/p^n$ by $\mu\_{p^n}$. The extension class is a principal unit, called the Serre-Tate parameter. To trivialize the extension you need to take the $p^n$-th root of this parameter. Th... | 6 | https://mathoverflow.net/users/2290 | 136634 | 75,081 |
https://mathoverflow.net/questions/136627 | 37 | ([This question](https://math.stackexchange.com/questions/415966/is-pi-definable-in-bbb-r-0-1-exp) is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields together with the unary function symbol ... | https://mathoverflow.net/users/37059 | Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$? | It seems to me that [Schanuel's conjecture](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture) (which is a kind of article of faith in transcendental number theory, but of course very far from proven itself) ought to imply that $\pi$ is not definable in this structure. The linked Wikipedia article contains the asser... | 40 | https://mathoverflow.net/users/2926 | 136642 | 75,086 |
https://mathoverflow.net/questions/136375 | 6 | What is the best known asymptotic formula for the number of graphs with a given degree sequence $(d\_1, ... ,d\_n)$, when the degrees are bounded by a constant and the number of vertices $n$ goes to infinity?
There are several papers of McKay et al. that provide bounds when the degrees are all $O(n^{\frac{1}{2}})$, b... | https://mathoverflow.net/users/17599 | Enumeration of graphs with a given and bounded degree sequence | In my comment I was misreading the question, sorry. The situation for the real question is as follows. For very low degrees (say, at most 3) it isn't hard to get the exact number as a single or double summation. For the more general case of bounded maximum degree, the best asymptotic formula is [my result with Wormald]... | 5 | https://mathoverflow.net/users/9025 | 136647 | 75,088 |
https://mathoverflow.net/questions/136652 | 3 | I am a beginner in [surgery theory](https://en.wikipedia.org/wiki/Surgery_theory). I have started learning with ALGEBRAIC AND GEOMETRIC SURGERY by Andrew Ranicki.
On page 4 of the book he defines surgery :
**Denition 1.2** A surgery on an $m$-dimensional manifold $M^m$ is the procedure
of constructing a new $m$-dim... | https://mathoverflow.net/users/32817 | On definition of surgery | 1. The "killing" terminology in surgery is the manifold version of the killing of homotopy classes by attaching cells: for any space $X$ the space $Y=X\cup\_fD^{n+1}$ obtained from $X$ by attaching an $(n+1)$-cell along a map $f:S^n \to X$ has $n$th homotopy group $\pi\_n(Y)=\pi\_n(X)/\langle [f] \rangle$, so the group... | 13 | https://mathoverflow.net/users/732 | 136659 | 75,093 |
https://mathoverflow.net/questions/136651 | 0 | I would like to ask if you may know how to prove this claim or any theorem related:
Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. Let vector a = [1, x, y, z]. Form the 16x9 matrix A = [a1⊗a1, a2⊗a2,.. ,a9⊗a9]. Prove rank of A is 9. (where is the ten... | https://mathoverflow.net/users/37078 | rank of outer product | I don't have a counterexample, but I do have a proof of something slightly weaker. (This uses the basic argument from [a previous answer of mine](https://mathoverflow.net/questions/136251/on-tensor-products-of-generic-vectors/136256#136256).)
Consider all possible ensembles of $9$ points: $\{(x\_i,y\_i,z\_i)\}\_{i=1}... | 2 | https://mathoverflow.net/users/29873 | 136663 | 75,094 |
https://mathoverflow.net/questions/136662 | 2 | Let me ask some basic question on the zeta function.
As you know well, the Riemann zeta function $\sum\_{n=1}^{\infty}\frac{1}{n^s}$ for $Re(s)>1$ has a meromorphic continuation to the whole complex plane and has a simple pole at $s=1$. We denote it by $\zeta(s)$. On the other hand, the zeta function has an Euler pro... | https://mathoverflow.net/users/29422 | On the analytic continuation of zeta function | The product $\prod\_p \zeta\_p(s)$ diverges at every point $s$ with $\Re(s)<1$, meaning that the partial products do not converge to a nonzero complex number. This is relatively straightforward to prove for $\Re(s)\leq 0$, while for $0<\Re(s)<1$ a proof can be found [here](https://mathoverflow.net/questions/63714/is-th... | 7 | https://mathoverflow.net/users/11919 | 136665 | 75,095 |
https://mathoverflow.net/questions/136590 | 9 | In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I work in the category of differential non-negatively graded algebras over some field $k$ of characteristic zero (denote this... | https://mathoverflow.net/users/32741 | Where should I search for resolutions? | One can try to reduce the computation to another easier computation in an abelian setting (e.g. Hochschild (co)homology). This is described in many places including Quillen's paper: On the (co-)homology of commutative rings.
In the abelian setting one often uses Koszul resolutions to compute $\mathrm{Tor}$ and $\mat... | 5 | https://mathoverflow.net/users/8818 | 136677 | 75,100 |
https://mathoverflow.net/questions/136672 | 10 | I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner (full proofs are not necessary, though sketches of the main arguments would be nice). Specifically covering the followin... | https://mathoverflow.net/users/14379 | Reference request: expository text on the structure of reductive groups over non-archimedean local fields | <http://www.math.umn.edu/~garrett/m/buildings/book.pdf>
was derived from seminar notes on structure of split classical p-adic groups, intending to circumvent the larger apparatus of algebraic groups and buildings. It became clear in the original project that it was necessary to develop some aspects of buildings, sinc... | 12 | https://mathoverflow.net/users/15629 | 136682 | 75,102 |
https://mathoverflow.net/questions/136681 | 10 | The question is simple but I still can't prove it or contradict it. Here it goes:
>
> Suppose $f$ and $g$ are defined on the circle
> (or, equivalently, $2\pi$ periodic functions) and Lebesgue integrable,
> is their convolution $(f\*g)(x) = \frac{1}{2\pi}\int\_{-\pi}^{\pi} f(x-y) g(y) dy $ continuous?
>
>
>
... | https://mathoverflow.net/users/37087 | the convolution of integrable functions is continuous? | Edit: sorry, there was a sign error. It should just be:
$$f(x) = g(x) = \begin{cases}
x^{-3/4}&x > 0\cr
0&x \leq 0.
\end{cases}$$
Then $\lim\_{x \to 0^+}f\*g(x) = \infty$ and $\lim\_{x \to 0^-} f\*g(x) = 0$.
| 13 | https://mathoverflow.net/users/23141 | 136686 | 75,103 |
https://mathoverflow.net/questions/136675 | 3 | In a comment to the top answer of [this](https://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/7952) question Darij Grinberg says that
>
> the problem with the dynamical perspective is that it is way harder to
> grasp for algebraic/combinatorial-minded pe... | https://mathoverflow.net/users/36246 | Explaining a comment: Difference between a transformation of points and a transformation of coordinates | Let $V$ be a finite-dimensional real vector space. Choosing a basis of $V$ amounts to giving an isomorphism $\phi : \mathbb{R}^n \to V$. Changing basis amounts to hitting $\mathbb{R}^n$ with an automorphism, but transforming points amounts to hitting $V$ with an automorphism.
| 3 | https://mathoverflow.net/users/290 | 136689 | 75,105 |
https://mathoverflow.net/questions/136683 | 6 | I think there aren't any discrete paradoxical subsets in $\mathbb{R}^2$ (any isometry that mapped a discrete subset into itself would have to either be a glide-reflection, a translation or a rotation by $2\pi/n$, and hence the subgroup of the isometry group on $\mathbb{R}^2$ generated by isometries that map a discrete ... | https://mathoverflow.net/users/26809 | Are there uniformly discrete paradoxical subsets in $\mathbb{R}^3$? | I think it is not possible for a uniformly discrete set $E$ in any ${\bf R}^d$ to be paradoxical, because one can create an invariant (or almost invariant) mean on such a set. Indeed, for any $\varepsilon>0$ and $C>0$, one can use the pigeonhole principle to find a large radius $R$ such that $|E \cap (B(0,R+C) \backsla... | 10 | https://mathoverflow.net/users/766 | 136698 | 75,109 |
https://mathoverflow.net/questions/136693 | -1 | I want to model the following situation: there is one production site (modelled by the source), a collection of depots (modelled by nodes without demand) and, of course many more customers (modelled by sinks). The production equals the demand and the transport of goods costs money; the costs are proportional to the amo... | https://mathoverflow.net/users/31310 | Effect of a Specific Restriction on the Integrality of Min-cost Flow Solutions | I don't see how your restriction captures the requirement you stated. In any case, it
does not preserve integrality. Consider a case where customers A and B (each with 1 unit demand) can both be served by depots 1 and 2, but you require the flow from 1 to A be the maximum of the flows out of 1. There are two extreme p... | 1 | https://mathoverflow.net/users/13650 | 136701 | 75,110 |
https://mathoverflow.net/questions/136105 | 8 | On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique up to simplicial Quillen equivalence.
On the other hand, we know that every model category can be simplicially enriche... | https://mathoverflow.net/users/36625 | Unicity up to homotopy of simplicial enrichments | If, given any fixed cofibrant object $A$, there is a funtor $map(A,-)$ from $M$ to simplicial sets which preserves weak equivalences between fibrant objects and commutes with homotopy limits up to canonical weak equivalences (e.g. $map(A,-)$ is a right Quillen functor), and such that $\pi\_0(map(A,X))=[A,X]$ (functoria... | 6 | https://mathoverflow.net/users/1017 | 136705 | 75,112 |
https://mathoverflow.net/questions/136700 | 6 | I just started reading Lambek and Scott's book "Introduction to higher-order categorical logic".
Right now I am reading Part I, section 5 (Polynomial categories). They explain two ways of adjoining an inderterminate arrow $x : A\_0 \to A$ to a category $\mathcal{A}$ (called $\mathcal{A}[x]$):
1. Take the underlying... | https://mathoverflow.net/users/37100 | Adjoining an arrow to a CCC | They mean this: given a cartesian closed $\mathcal{A}$ and objects $A, B$ of $\mathcal{A}$, the inclusion $i: \mathcal{A} \to \mathcal{A}[x]$ is universal with respect to strict cartesian closed functors $F: \mathcal{A} \to \mathcal{C}$ to cartesian closed categories $\mathcal{C}$ that come equipped with a specified ar... | 7 | https://mathoverflow.net/users/2926 | 136709 | 75,113 |
https://mathoverflow.net/questions/136655 | 5 | It is excerpt from "Algebraic Geometry Codes Basic Notions"(<https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCoQFjAA&url=http%3A%2F%2Fwww.math.umass.edu%2F~hajir%2Fm499c%2Ftvn-book.pdf&ei=CtLiUc2aAums4ATDloGoBA&usg=AFQjCNH8m6i46UGeRvF8J0nV_cMriSYSww&sig2=-doWN37rrQ2BMFnyUj3c1g&bvm=bv.48705608,d.bGE... | https://mathoverflow.net/users/31356 | Analogy between Jacobian of curve and Ideal class group | I think this excerpt from Tsfasman-Vladut-Nogin should not be taken as a literal statement--rather it is an explication of (part of) the number field/function field dictionary. Most of this dictionary comes from the fact that these fields are the function fields of Dedekind schemes.
In particular, suppose $X$ is a De... | 4 | https://mathoverflow.net/users/6950 | 136715 | 75,117 |
https://mathoverflow.net/questions/136708 | 4 | Among the different generalizations of the CLT available on the web, I found these
* CLT for the sum of non-identical (and independent) random variables
* CLT for the sum of identical (and independent) multivariate random variables.
However, I can't find any for the sum of non-identical (and independent) multivari... | https://mathoverflow.net/users/37101 | Multivariate Central Limit Theorem For Non-Identical Distribution | I always prefer to have error bounds for the CLT, so my favorite reference for your question is the paper "A Lyapunov type bound in $\mathbb{R}^d$" by Vidmantas Bentkus (Theory of Probability & Its Applications 49(2), 311--323, 2005).
From the abstract: Let $X\_1, \dots, X\_n$ be independent, mean-zero, $\mathbb{R}^d... | 2 | https://mathoverflow.net/users/658 | 136722 | 75,119 |
https://mathoverflow.net/questions/120546 | 14 | The question is whether, when you add a Cohen subset to a cardinal
$\kappa$, that cardinal becomes a characteristic of the resulting forcing extension $V[G]$. Or can there be strange instances in which the very
same model is realized as a Cohen subset forcing extension over
different ground models with different cardin... | https://mathoverflow.net/users/1946 | Can a model of set theory be realized as a Cohen-subset forcing extension in two different ways, with different grounds and different cardinals? | The main part of this question is answered by theorem 10 of
my joint paper with Bagaria, Tasprounis and Usuba, below.
>
>
> >
> > From [J. Bagaria, J. D. Hamkins, K. Tsaprounis, T. Usuba, Superstrong
> > and other large cardinals are never Laver
> > indestructible](http://jdh.hamkins.org/superstrong-never-indes... | 7 | https://mathoverflow.net/users/1946 | 136725 | 75,121 |
https://mathoverflow.net/questions/136726 | 1 | A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda\_1,\lambda\_2,\lambda\_3,....\lambda\_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands $\lambda\_i$ all like in a set $S\subset \mathbb{N}$ and we denote this number $p\_S(n)$.
If $S$ is ``dense" enough, one can... | https://mathoverflow.net/users/12337 | Does the asymptotic formula for Partitions into parts <c exist? | Parts greater than $c$ is complementary to parts smaller than $c.$ (as pointed out in the comments, parts greater than $c$ is the same as "at least $c$ parts") for the latter, there is a theorem of Szekeres from 1951, see the [math review.](https://dl.dropboxusercontent.com/u/5188175/szekeres.pdf) (of. Course this does... | 1 | https://mathoverflow.net/users/11142 | 136731 | 75,123 |
https://mathoverflow.net/questions/136740 | 6 | Let $\Phi (z,t)$ be a polynomial given by
$$ \Phi(z,t) := z^n + A\_{n-1}(t) z^{n-1} + \ldots + A\_1(t) z + A\_0(t).$$
Assume that $\Phi(0,0) =0$. It is a fact that a solution $z(t)$ of the equation
$$ \Phi(z(t), t) =0 $$
that is close to zero, can be expressed as a formal power series in $t^{1/r}$ for
some positive... | https://mathoverflow.net/users/4463 | Is there an algorithm to find out the number of small solutions to a polynomial equation, when we vary all the coefficients? | This algorithm is called the Newton Polygon, and it was really invented and carefuly
described by Newton, with examples, see for example,
MR1836037 Fischer, Gerd Plane algebraic curves. Translated from the 1994 German original by Leslie Kay. Student Mathematical Library, 15. American Mathematical Society, Providence,... | 9 | https://mathoverflow.net/users/25510 | 136741 | 75,128 |
https://mathoverflow.net/questions/136744 | 6 | Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer sum. Following the principle of [categorification](http://ncatlab.org/nlab/show/vertical+categorification), a finer and ... | https://mathoverflow.net/users/2841 | The 2-group of extensions | A version of this, but one categorical level higher (i.e. a 3-group of (central) extensions of 2-groups) appeared explicitly in this paper of mine:
[Central Extensions of smooth 2-groups and a finite dimensional string 2-group](http://arxiv.org/abs/0911.2483) Geometry & Topology 15 (2011) 609-676.
Of course this is... | 7 | https://mathoverflow.net/users/184 | 136747 | 75,130 |
https://mathoverflow.net/questions/135143 | 4 | I am consulting some papers (references below) about the *Carleson's problem* for the pointwise convergence of the Schrödinger group
\begin{equation}
S\_t=e^{i t \Delta}.
\end{equation}
In this context it is important to obtain $L^p$-estimates for the *maximal function*
\begin{equation}
S^\star f(x)=\sup\_{t\in(0,1)... | https://mathoverflow.net/users/13042 | To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function | I have had the opportunity to have a chat with some of the authors, so that I am now able to answer this question myself.
For the sake of clarity let us formulate the answer as a theorem.
**Theorem** Notation for $S\_t$ and $S^\star$ being as above, and $\mathscr{S}$ denoting the Schwartz class, the following prop... | 4 | https://mathoverflow.net/users/13042 | 136758 | 75,134 |
https://mathoverflow.net/questions/136733 | 4 | Let $X$ be a finite-dimensional complex manifold (possibly non-compact). Let $\mathcal{H}$ be a union of codimension-$1$ submanifolds such that the local picture is that of intersecting hyperplanes. I am interested in $M$ the complement of $\mathcal{H}$ inside $X$.
The well-known case, that I am aware of, is when $M... | https://mathoverflow.net/users/7494 | Cohomology of submanifold complements | If one moves from complements of hyperplanes arrangements to complements of hypersurfaces in $\mathbb{C}^n$, some of the above properties may fail. For instance, if $C$ is a plane curve, then the rational cohomology ring of its complement is not necessarily generated in degree $1$, see for instance José Ignacio Cogollu... | 3 | https://mathoverflow.net/users/17846 | 136761 | 75,136 |
https://mathoverflow.net/questions/136727 | 8 | I am not sure if this is the right venue to ask this. Apologies in advance.
I would like to clarify the following. When people give as reference:
J.-P. SERRE and J. TATE.-Mimeographed notes from the 1964 A.M.S. Summer Institute in Algebraic Geometry at Woods Hole;
which notes do they exactly refer to? I download... | https://mathoverflow.net/users/37040 | Serre-Tate 1964 Woods Hole notes | They are probably referring to the sections "Serre discussed..." and "Tate discussed..." in the Seminar report by Lubin, Serre, Tate, which outline what has become known as Serre-Tate theory. But without knowing the context, I can only guess. It is quite likely the author didn't have access to the notes of conference, ... | 7 | https://mathoverflow.net/users/36469 | 136762 | 75,137 |
https://mathoverflow.net/questions/136049 | 20 | Consider the problem of trying to identify which $n$-dimensional compact complex manifolds can be endowed with a Kähler metric.
$\underline{n = 1}:$ Any hermitian metric on a Riemann surface is a Kähler metric as the Kähler form is a two-form, and all two-forms are closed. (This is also true of non-compact Riemann su... | https://mathoverflow.net/users/21564 | Three-dimensional compact Kähler manifolds | The main obstruction to existence of Kahler metric (in addition to Lefschetz
SL(2)-action and Riemann-Hodge relations in cohomology)
is homotopy formality: the cohomology ring of a Kahler manifold is related to its de Rham algebra by a chain of homomorphisms of differential graded algebras inducing isomorphisms on coho... | 19 | https://mathoverflow.net/users/3377 | 136764 | 75,138 |
https://mathoverflow.net/questions/136479 | 9 | Let $D$ be a fixed positive squarefree integer. For a positive integer $x$, define
$S(D,x) = \{ q < x : D \text{is a quadratic residue} \pmod q \}$.
Here $q$ can be any integer, not necessarily a prime. Are elements of $S(D,x)$ evenly distributed? In other words,
let $0 < a < b < 1$ be constants and consider an int... | https://mathoverflow.net/users/36982 | Distribution of moduli of quadratic residues | For $D=-1$, Landau [proved that](http://mathworld.wolfram.com/Landau-RamanujanConstant.html)
$$\# S(-1, x) \sim K \frac{x}{\sqrt{\log x}}$$
where $K = \frac{1}{\sqrt{2}} \prod\_{p \equiv 3 \bmod 4} \frac{1}{\sqrt{1-p^{-2}}}$.
This shows that, for fixed $0<a<b$,
$$\# S(-1,x) \cap [ax,bx] \sim K \left( \frac{bx}{\sqrt{\... | 8 | https://mathoverflow.net/users/297 | 136775 | 75,140 |
https://mathoverflow.net/questions/136766 | 0 | According to a [conjecture p.4](http://dml.cz/dmlcz/136881)
$|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$
for $0 < \Delta < \frac12$ and $|t| > 2 \pi +1$.
Since $\zeta(\overline{s}) = \overline{\zeta(s)}$, this is equvalent to
$|\zeta(s)| > |\zeta(1-s)| $ for $0 < \sigma < \frac12$ and $t$ large eno... | https://mathoverflow.net/users/12481 | Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple | For fixed $t>12$, let us consider for $0\leq\sigma\leq \frac{1}{2}$ the function
$$ G(\sigma):=|\pi^{-(\sigma-it)/2}\Gamma(\sigma+it)|^2 = \pi^{-\sigma}|\Gamma(\sigma+it)|^2. $$
Following the accepted answer [here](https://mathoverflow.net/questions/89324/are-all-zeros-of-gammas-pm-gamma1-s-on-a-line-with-real-part... | 4 | https://mathoverflow.net/users/11919 | 136783 | 75,142 |
https://mathoverflow.net/questions/136787 | 0 | Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ is finite extension of fields?
| https://mathoverflow.net/users/37142 | Finite extension of a field | Suppose $k\subset Quot(A/p)$ is a finite extension of fields. Then every nonzero element $x$ of $A/p$ is algebraic over $k$ and so satisfies a minimal polynomial with non-zero constant term $a\_0\in k$. Therefore $x$ divides $a\_0$, so $x$ is a unit.
Because every nonzero element of $A/p$ is a unit, $A/p$ is a field... | 1 | https://mathoverflow.net/users/10503 | 136795 | 75,149 |
https://mathoverflow.net/questions/136752 | 2 | What would be the closed-form expression defining number of all possible labelled connected bipartite graphs given $\mid X \mid = m, \mid Y \mid = n - m $?
| https://mathoverflow.net/users/37134 | Enumeration of labeled connected bipartite graphs given partite sets | The number of connected labeled bipartite graphs with bipartition $(X,Y)$ where $|X|=m$ and $|Y|=n$ is the coefficient of $x^my^n/m!\,n!$ in
$$\log\biggl(\sum\_{m,n=0}^\infty 2^{mn} \frac{x^m}{m!}\frac{y^n}{n!}\biggr).$$
They are sequence [A123260](http://oeis.org/A123260) in the OEIS.
| 2 | https://mathoverflow.net/users/10744 | 136799 | 75,150 |
https://mathoverflow.net/questions/136797 | 7 | There's a well-known correspondence (traditionally called Hartshorne-Serre) between codimension 2 smooth subvarieties $S\subset X$ of a smooth algebraic variety $X$ and certain rank two vector bundles on $X$. Is there anything similar for higher codimensional subvarieties of $X$, i.e. codimension 3 and more?
| https://mathoverflow.net/users/4096 | Hartshorne-Serre's correspondence in higher codimension | To get some idea of the difficulty in generalizing Serre's correspondence to higher codimension, bear in mind that what makes a resolution $0 \rightarrow \mathcal{F} \rightarrow \mathcal{E} \rightarrow \mathcal{I}\_{Z|X} \rightarrow 0$ plausible when $\mathcal{F},\mathcal{E}$ are vector bundles on $X$ and $Z \subset X$... | 6 | https://mathoverflow.net/users/5496 | 136808 | 75,156 |
https://mathoverflow.net/questions/136796 | 10 | Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally defined? References would be lovely.
Best regards
| https://mathoverflow.net/users/37150 | History of profinite groups, when was it first mentioned? What was the original definition? | Profinite groups were first called "Groups of Galois type", see J.P. Serre's book "Cohomologie Galoisienne" of $1964$. The term "profinite" comes from Serre (if I am not mistaken).
Of course, some profinite groups have a much older history, e.g.,
already Hensel defined in $1910$ the $p$-adic integers during his studie... | 11 | https://mathoverflow.net/users/32332 | 136811 | 75,158 |
https://mathoverflow.net/questions/136750 | 1 | Let $\boldsymbol{X}\in\mathbb{R}^n$ be a random variable with positive entries ($X\_i\geq a>0$). I want to characterize the relation between the second moment matrix $\boldsymbol{M}$, defined as
$$
\boldsymbol{M} = \mathbb{E}\{\boldsymbol{X}\boldsymbol{X}^\*\},
$$
and that we can assume to be positive definite, a... | https://mathoverflow.net/users/37129 | Estimating the relation between the covariance of a vector and a monotone function of the same vector | I think you are probably dealing with random vectors with some symmetry. For a generic distribution the property you are trying to prove seems not to be true. Take, e.g., a random vector taking values $(1,1,1)$, $(1,2,1)$, $(1,1,2)$ with equal probabilities, and you will see that the property is violated. However, if t... | 0 | https://mathoverflow.net/users/2968 | 136816 | 75,161 |
https://mathoverflow.net/questions/136798 | 6 | If I want to learn about existence of weak solutions to PDEs of the form
$$u\_t + Au = f$$
or
$$Au = f$$
where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for parabolic PDEs with right hand side in $L^2(0,T;V')$ or $L^2(0,T;H)$ (where $V \subset H$). But now I want to a step deepe... | https://mathoverflow.net/users/35613 | PDEs involving measures; where to begin? | Weak solutions for PDEs with Radon measures as right-hand sides can be obtained by a duality technique, which roughly proceeds as follows. Assume that the adjoint operator $A^\*$ acts as isomorphism from $W^{1,q}\_0(\Omega)$ to $W^{-1,q}(\Omega)$ for some $q>n$ with $\Omega\subset\mathbb{R}^n$. (This is the case if the... | 5 | https://mathoverflow.net/users/30516 | 136835 | 75,166 |
https://mathoverflow.net/questions/129735 | 4 | One variant of Fulton's K-saturation conjecture is as follows:
$K\_{\lambda/\mu,w} > 0 \Leftrightarrow K\_{n\lambda/n\mu,n w} > 0$ for any integer $n>0.$
Here $K\_{\lambda/\mu,w}$ denotes the Kostka numbers (number of skew SSYT of shape $\lambda/\mu$ and weight $w.$
This has been proved in various ways, (Knutson,... | https://mathoverflow.net/users/1056 | Elementary proof of K-saturation conjecture | I answer this question [here.](http://arxiv.org/abs/1307.3999)
The proof is of combinatorial nature and quite short, using quite elementary
arguments.
| 1 | https://mathoverflow.net/users/1056 | 136837 | 75,167 |
https://mathoverflow.net/questions/136826 | 7 | It seems that considerable care is taken in the literature to ensure that a countable support (CS) iteration of proper forcings preserves $\aleph\_2$. Can you give an example, assuming CH, of a CS iteration of countably closed forcings such that each iterand is forced to have the $\aleph\_2$-c.c. but the full iteration... | https://mathoverflow.net/users/11145 | preservation of $\aleph_2$-c.c. in CS iterations | Assume CH. We start with a model with a function $F:[\omega\_2]^2\to\omega$
such that the following holds: if $\{A\_\alpha:\alpha<\omega\_2\}$
are pairwise disjoint countable subsets of $\omega\_2$ and $i<\omega$ then
there are $\alpha<\beta$ such that all values of $F$ between $A\_\alpha$
and $A\_\beta$ are $>i$. ... | 8 | https://mathoverflow.net/users/6647 | 136838 | 75,168 |
https://mathoverflow.net/questions/136836 | 2 | Suppose $A$ is a positive definite matrix such that$$ I \preceq A \preceq 1.01I.$$ Is it possible that $\sum\limits\_{i=1}^n A\_{1i}$ can be arbitrarily large?
| https://mathoverflow.net/users/36081 | Possible pathological properties of positive definite matrix | Outline of one possible answer (but am doing this in a rush so have not written anything down to check).
Consider the matrix $\Gamma$ which has 0 in top-left corner, 1 in rest of top row and rest of 1st column, and 0 in all other entries. (Note to hovering LaTeX hawks: please don't feel the need to enclose all those ... | 4 | https://mathoverflow.net/users/763 | 136840 | 75,170 |
https://mathoverflow.net/questions/136841 | 4 | The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics.
I am curious whether there exists an equivalent definition of such classes via $\mu$-recursive functions. What about register ... | https://mathoverflow.net/users/37176 | $\mu$-recursive definitions for the complexity classes P, NP, etc | Yes, there is. See [Cob64].
The idea is to replace primitive recursion
in the definition of primitive recursive functions with bounded recursion on notion.
Another more delicate approach is taken in [BC92].
* Alan Cobham, "[The Intrinsic Computational Difficulty of Functions](http://dx.doi.org/10.1016/j.tcs.2003.... | 5 | https://mathoverflow.net/users/7507 | 136843 | 75,172 |
https://mathoverflow.net/questions/136529 | 5 | This has been posted on SE, but I haven't gotten a reply, so I thought I'll try my luck here.
I would like to refer you to 2.30 & 2.32 in Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves.
2.30(b)[(c) in errata]: Suppose $\mathfrak{P}$ remains inert in $L'$, say $\mathfrak{P}R\_{L'}=\mathfrak{P}... | https://mathoverflow.net/users/36133 | Hecke $L$-series exercise in Silverman's Advanced Topics in Arithmetic of EC | There is nothing wrong with the question. The subtlety is that $T$ is not constant with respect to the fields $L'$ and $L$. My mistake was that I took $T$ to be the same in both fields when in reality it is not.
\begin{align\*}
L(s,E/L)&=\prod\_{\text{$\mathfrak{P}$ in $L$}}\left(1-a\_{\mathfrak{P}}q\_{\mathfrak{P}}^... | 3 | https://mathoverflow.net/users/36133 | 136844 | 75,173 |
https://mathoverflow.net/questions/136671 | 2 | A simple question that I was pondering on while examining some algorithms that work similarly for positive definite and nonnegative matrices.
Let $\mathcal{H}$ be the space of (let's say for now $2\times 2$) Hermitian positive-definite matrices. Let $\mathcal{P}$ be the space of entrywise nonnegative matrices of the ... | https://mathoverflow.net/users/1898 | Positive definite to nonnegative | This is not an answer, but a collection of observations and ideas, in hopes that it helps someone to get more.
Let us ask a bit more of our function:
>
> 1. If $AB$ is positive semidefinite, then $\Phi(AB) = \Phi(A)\Phi(B)$.
> 2. $\Phi(H)$ is nonsingular for at least one $H \in \mathcal{H}$. (I think this is quit... | 2 | https://mathoverflow.net/users/36450 | 136862 | 75,179 |
https://mathoverflow.net/questions/130075 | 4 | This is a [cross-post from math.se](https://math.stackexchange.com/questions/383358/expression-of-basis-vectors-of-permutation-modules-in-different-bases?noredirect=1#comment824882_383358), because I did not get any answer there:
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$... | https://mathoverflow.net/users/9947 | Expression of basis vectors of permutation modules in different bases. | Maybe I'm mistaken but I think the answer is quite easy if you use the language of Specht module, that is realize all these in polynomials. More precisely, you are working in the subspace of $C[x\_1,\dots,x\_n]$, where no variables are squared. In this space any standard tableau is realized as the product of the Vander... | 2 | https://mathoverflow.net/users/37190 | 136866 | 75,182 |
https://mathoverflow.net/questions/136820 | 2 | Suppose $f(n)$ is a periodic function with period $q$. Now from [this paper](http://www.sciencedirect.com/science/article/pii/S0022314X06002927) we get that if $\displaystyle\sum\_{n=1}^{q}f(n)=0$ then $\displaystyle\sum\_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\displaystyle\sum\_{a=1}^{q}f(a)\psi(\frac{a}{q})$, where... | https://mathoverflow.net/users/36735 | Summation of certain series | Thanks to everyone whoever thought over this problem. I have asked Professor Murty (one of the authors of the paper mentioned in the question) about this question. He told me that, of course such generalization exists. In fact, those same authors had another paper which encounters this generalization problem.
It is v... | 0 | https://mathoverflow.net/users/36735 | 136875 | 75,186 |
https://mathoverflow.net/questions/136847 | 0 | Let's consider a family of varieties defined by two equations in $\operatorname{Spec}(\mathbb{C}[x\_1,\dots,x\_n])$.
First, Let $\Delta\_i, i=1,2$ be a finite set of monomials in $\mathbb{C}[x\_1,\dots,x\_n]$, for simplicity, I use $X^v$ to denote an element of $\Delta\_i$. Then $$F\_i=\{\sum\_{X^v \in \Delta\_i}c\_v... | https://mathoverflow.net/users/29730 | Is dimension an open condition? | Probably this was downvoted because it is follows from standard material in most algebraic geometry textbooks. It might be more appropriate for Math Stack Exchange.
At any rate, with the question as formulated, this locus is not an open subset. For instance, consider a copy of $\mathbb{C}$ with coordinate $t$ inside... | 6 | https://mathoverflow.net/users/13265 | 136878 | 75,187 |
https://mathoverflow.net/questions/136882 | 4 | Is the field of real numbers $\mathbb{R}$ a finite extension of some subfield $k\subset \mathbb{R}$?
| https://mathoverflow.net/users/37142 | Field extension of fields | there is no such sub-field. [Edited] It is a theorem of Emil Artin that the only automorphism of the field of complex numbers of finite order is of order two. If such a sub-field $F$ existed, then $Aut ({\mathbb C}/F)$ would have only elements of order two and hence abelian. In particular, ${\mathbb R}/F$ would be abel... | 4 | https://mathoverflow.net/users/23291 | 136883 | 75,189 |
https://mathoverflow.net/questions/67275 | 3 | Let $X$ be a compact in the Polish space (metric, complete, separable) and $G\subseteq X\times X$ is open. For $x\in X$ we define the section of $G$:
$$
s(x) = (y\in X|\langle x,y \rangle \in \bar{G}).
$$
Here $\bar{G}$ is a closure of $G$.
The set $A'\subseteq X$ is invariant if for all $x\in A'$ holds $s(x)\subse... | https://mathoverflow.net/users/11768 | Sets invariant under sections | Your setup defines *set-valued dynamics* on $X$. More precisely, you have
$$X \stackrel{p}{\leftarrow} \overline{G} \stackrel{q}{\rightarrow} X$$
where $p$ and $q$ are the obvious projection maps. The set-valued transformation in question sends $x \in X$ to the subset $qp^{-1}(x) \subset X$. I'll write $F = qp^{-1}$ fo... | 1 | https://mathoverflow.net/users/18263 | 136890 | 75,191 |
https://mathoverflow.net/questions/136888 | 10 | For a sequence of positive integers $a\_0, \ldots, a\_n$ and a ring $R$, there is a graded ring $R[x\_0,\ldots, x\_n]$ where $x\_i$ is in degree $a\_i$. There is a corresponding $\mathbb{G}\_m$-action on $Spec R[x\_0,\dots, x\_n] = \mathbb{A}^{n+1}\_R$ induced by
$R[x\_0,\dots, x\_n] \to R[x\_0,\dots, x\_n] \otimes \... | https://mathoverflow.net/users/2039 | Reference for Weighted Projective Stacks | I don't know a reference, but I think you can make the following argument.
The weighted projective stack $X$ is a $\mathbb{G}\_m$-quotient of a scheme smooth over $R$, so it must be smooth over $R$ too: smoothness descends under flat morphisms, in the sense that if $Z \to W$ is faithfully flat and $Z$ is smooth, the... | 10 | https://mathoverflow.net/users/344 | 136893 | 75,193 |
https://mathoverflow.net/questions/136867 | 10 | When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have arguments strong enough to convince that measurability is indeed the must.
Let me focus on a particular case of stochasti... | https://mathoverflow.net/users/11768 | Why do we want maps to be measurable (in countably-additive setting) | You can do many more things with measurable functions than with nonmeasurable ones, so the theory is richer and has more applications. There's much less trouble with integrals, for example, right?
In fact, if in the beginning the measurability seems a necessary evil, in the end it becomes a blessing and helps interpret... | 11 | https://mathoverflow.net/users/2968 | 136905 | 75,200 |
https://mathoverflow.net/questions/136023 | 9 | **Definition**: Let be $f:M\to M$ a diffeomorphism of a compact manifold. We say that $A\subset M$ is an attractor when there exists a neighborhood $U\supset A$ such that $f( \overline{U})\subset int (U)$ and
$$
A=\bigcap\_{n\geq 0}f^n(U)
$$
$U$ is called an basin of attraction of $f$.
**Theorem**: Let $M$ be a comp... | https://mathoverflow.net/users/nan | Relationship between basic sets and attractors | Ok, here is the answer assuming that the basic set $\Lambda$ has local product structure.
Step 1. We will prove that $\forall x\in\Lambda$ $W^u(x)\subset \Lambda$. Then any point $y$ sufficiently close to $\Lambda$ belongs to a local stable manifold of some point in $\Lambda$, hence $\Lambda$ is an attractor.
Step ... | 2 | https://mathoverflow.net/users/2029 | 136907 | 75,202 |
https://mathoverflow.net/questions/132966 | 13 | I posted this question on Math.SE (<https://math.stackexchange.com/questions/409444/>), but got no answer. So I repost it here.
Let M be a closed manifold. Then there is a cap product $K^\ast(M) \times K\_\ast(M) \to K\_\ast(M)$ between the K-theory of M and its K-homology. For a definition of it one could see my pri... | https://mathoverflow.net/users/13356 | Duality between K-theory and K-homology in the non-spin^c case. | This is nicely understood by duality (in the sense of [dualizable objects](http://ncatlab.org/nlab/show/dualizable%20object) in monoidal categories) in the monoidal category [KK](http://ncatlab.org/nlab/show/KK-theory). The relevant results are collected and referenced on the nLab at *[Poincaré duality algebra](http://... | 8 | https://mathoverflow.net/users/381 | 136908 | 75,203 |
https://mathoverflow.net/questions/134381 | 8 | Is the solvability of finite groups of order coprime to 15
essentially easier to prove than the entire Classification of Finite Simple Groups?
| https://mathoverflow.net/users/28104 | Solvability of finite groups of order coprime to 15 -- proof without using CFSG? | As Derek Holt has pointed out, the answer to the question is *yes*. --
Thompson proved that the only finite simple groups of order coprime to 3 are the
Suzuki groups, and Glauberman later extended this to a classification of
simple groups that do not have ${\rm S}\_3$ as a subgroup.
Both of these results are pre-classi... | 4 | https://mathoverflow.net/users/28104 | 136922 | 75,212 |
https://mathoverflow.net/questions/136926 | 2 | I'm starting to study hyperKähler manifolds and I have a question for you. I hope it's not too stupid...
I know that given a Kähler metric $g$ on an hyperKähler manifold $X=(M,I)$ ($M$ is the differential structure and $I$ the complex structure) there are $J$, $K$ complex structures on $M$ with quaternionic relations... | https://mathoverflow.net/users/37224 | Holomorphic (2,0)-form on HyperKähler manifolds | Not sure which of the following is giving you trouble, so here is both:
* $\sigma\_I$ is **nondegenerate** (i.e. $\sigma\_I(u,\cdot) = 0$ implies $u=0$) for the simple reason that its real and imaginary parts are nondegenerate, since $g$ is.
* $\sigma\_I$ is a **(2,0)-form** relative to the complex structure $I$ beca... | 2 | https://mathoverflow.net/users/19276 | 136934 | 75,217 |
https://mathoverflow.net/questions/136827 | 6 | This question has two parts: A calculation that is giving me a lot of troubles, and a theoretical one on weighted projective spaces.
1) I want to find the genus of the curve $C\_7 \subset \mathbb{P}(1,2,3)$. Naive adjunction does not work, because this curve is not well-formed in the sense of [1]. To calculate the ge... | https://mathoverflow.net/users/17602 | On well-formedness of weighted projective spaces and a Hurwitz theorem calculation | Question 1: Take $F$ to be the polynomial that defines $C\_7$ as a hypersurface in your projective space. If you are looking for the cohomology to this (possibly singular) variety, then you should look at Dolgachev's Weighted Projective Varieties paper. In that paper, you can find a result (Theorem 4.3.2) where you can... | 5 | https://mathoverflow.net/users/14128 | 136936 | 75,219 |
https://mathoverflow.net/questions/136918 | 0 | If we have in certain norm
1). $g\_j(x) \rightarrow h(x), j\rightarrow\infty$ and
2). $ f\_{ij}(x)\rightarrow g\_j(x), i\rightarrow \infty, \forall j$ ,
then we can choose a subsequence $\{f\_{ij\_{(i)}}, i=1,2,...\}$ from $\{f\_{ij}, i=1,2,...;j=1,2,...\}$ such that $f\_{ij\_{(i)}}\rightarrow h, i\rightarrow\i... | https://mathoverflow.net/users/17708 | approximating functions pointwise | Let $h$ be the indicator function of the rationals. This is of Baire class 2, i.e. it is not the pointwise limit of a sequence of continuous functions, but it is the pointwise limit of a sequence of pointwise limits of continuous functions.
For example, let $r\_1, r\_2, \ldots$ be an enumeration of the rationals,
$g\_... | 3 | https://mathoverflow.net/users/13650 | 136938 | 75,221 |
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