parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/130932 | 4 | Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is an upper bound for the distance between two points, for example it could be compact), is contractible (there is a homotopy of continuous self... | https://mathoverflow.net/users/32210 | Is there a contractible bounded homogeneous space? | According to [this](https://mathoverflow.net/questions/56524/example-of-a-compact-homogeneous-metric-space-which-is-not-a-manifold) MO question,
>
> A compact isometrically homogeneous metric space is a finite-dimensional manifold if and only if it is locally contractible.
>
>
>
So a compact isometrically hom... | 10 | https://mathoverflow.net/users/17836 | 131001 | 72,645 |
https://mathoverflow.net/questions/131000 | 3 | Is there a known definition of vector fields on a simplicial manifold?
For me, it seems natural that the definition should be something along the lines: Let $M\_{\bullet}$ be a simplicial manifold with degeneracy maps $s\_i: M\_n \rightarrow M\_{n+1}$ and face maps $\partial\_i: M\_{n+1} \rightarrow M\_n$. A vector ... | https://mathoverflow.net/users/34104 | Vector fields on a simplicial manifold. | How about this? Apply the tangent functor $T$ to $M\_\bullet$ to get a new simplicial manifold $TM\_\bullet,$ that is take the composite
$$\Delta^{op} \stackrel{M\_\bullet}{\longrightarrow} Mfd \stackrel{T}{\longrightarrow} VectBun \to Mfd,$$ where the last functor is the forgetful functor. There is an obvious map $\... | 4 | https://mathoverflow.net/users/4528 | 131003 | 72,646 |
https://mathoverflow.net/questions/131016 | 1 | What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?
| https://mathoverflow.net/users/35353 | Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero | One can call it pure object.
| 1 | https://mathoverflow.net/users/4428 | 131017 | 72,651 |
https://mathoverflow.net/questions/95137 | 18 | I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative question is easier to answer.
---
**First way**
1. Assume I have a real symmetric matrix $\mathbf{C} \in \mathbb{R}... | https://mathoverflow.net/users/23197 | Minimum off-diagonal elements of a matrix with fixed eigenvalues | I have a bound that will be of use to you. First, note that we can use the fact that the diagonal entries are all $1$s to relate $c\_\mathrm{max}$ to the Frobenius norm of $C$:
$$
\|C\|\_F^2\leq M+M(M-1)c\_\mathrm{max}^2.
$$
This Frobenius norm is easy to work with, since it's just the 2-norm of the spectrum:
$$
\|C\|\... | 7 | https://mathoverflow.net/users/29873 | 131020 | 72,653 |
https://mathoverflow.net/questions/131010 | 3 | I would like to understand the accounts of P. Gabriel ([link text](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1962__90_/BSMF_1962__90__323_0/BSMF_1962__90__323_0.pdf)), pag 365, when he shows that the composition of this category is well defined.
Definition: Given a Serre subcategory $\mathcal{C}$ of $\mathcal{A}$, ... | https://mathoverflow.net/users/34106 | Composition in the category quotient | First, you actually want $((N'' + N')/N')$ every time you have $(N'' + N')/N$. If this was the source of your confusion, sweet. If not, here goes. (Also, the double underscores in the NUMDAM url got messed up by MO's processing; [here's a working link](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1962__90_/BSMF_1962__90... | 5 | https://mathoverflow.net/users/27153 | 131023 | 72,654 |
https://mathoverflow.net/questions/130797 | 2 | I'm wondering if a compact set $A\subset\mathbb{C}$ satisfying the properties that
• $A$ and its complement have finitely many connected components
• every connected component of $\partial A$ is the image of a path (say piecewise analytic, or $C^1$)
has a standard name? I'm not sure "set with rectifiable boundary... | https://mathoverflow.net/users/24309 | What are these compact sets called? | I'll finaly settle for "a cutout compact set" for want of a better term. But I think this word expresses well the "finitely many" (connected components, non-smooth points) side of the object, which is what I wanted to highlight.
| 1 | https://mathoverflow.net/users/24309 | 131028 | 72,656 |
https://mathoverflow.net/questions/130999 | 13 | By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, I am more interested in the corresponding smooth site but the question may be posed for both.
Daniel Dugger states th... | https://mathoverflow.net/users/26470 | Is the site of (smooth) manifolds hypercomplete? | I think your idea to reduce the question to small slice topoi works perfectly. I will use it to show that every sheaf on $Man$ (either the continuous or the smooth version) is the limit of its Postnikov tower. This implies that the topos is hypercomplete since truncated objects are hypercomplete.
Let $F$ be a sheaf. ... | 12 | https://mathoverflow.net/users/20233 | 131032 | 72,657 |
https://mathoverflow.net/questions/131033 | 3 | I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated from maps in $S$ via directed colimits. This means every $f\in C$ can be written as a directed colimit of $f\_i \in S$,... | https://mathoverflow.net/users/11540 | Directed colimits of maps in a combinatorial model category | The answer is 'no' (sorry I didn't realize this before!)
I remembered someone showing me a weird counterexample in model categories a couple weeks ago (elaborated on from this paper by Rosicky: <http://www.math.muni.cz/~rosicky/papers/comb2.pdf>)
and it seems to do the trick here:
Consider the category ${\bf Pos}... | 5 | https://mathoverflow.net/users/6936 | 131041 | 72,661 |
https://mathoverflow.net/questions/130969 | 2 | I am writing a summary on a work on Fluid Dynamics that develops irrotational flow states that appear to interact amongst each other according to the equations of Electromagnetism <http://arxiv.org/abs/1301.7540>
So it begins with Euler Equations of inviscid compressible fluid. Apply some constraints and then find a ... | https://mathoverflow.net/users/34091 | Derivation of Bessel functions | I'll make an attempt at providing the steps you are seeking to go "from Euler equation to Bessel function".
You start from the Euler equation, describing conservation of momentum,
$$\rho\frac{\partial \vec{u}}{\partial t}+\rho\vec{u}\cdot\nabla\vec{u}=-\nabla p$$
and the continuity equation, describing conservati... | 3 | https://mathoverflow.net/users/11260 | 131055 | 72,667 |
https://mathoverflow.net/questions/131070 | 9 | I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon \partial\_x) f(x) = f(x + \varepsilon)$
$\exp(\varepsilon x \partial\_x) f(x) = f(x \exp(\varepsilon) )$
$\exp(\varepsil... | https://mathoverflow.net/users/34129 | Algorithm to find exponential map of differential operators acting on function | A partial answer: What you call the "exponential function" is the so-called flow semigroup, see [Engel-Nagel](http://www.fa.uni-tuebingen.de/research/publications/1999/one-parameter-semigroups-for-linear-evolution-equations/), Section II.3.28.
Another reference on the [Lie derivative](http://en.wikipedia.org/wiki/Lie... | 3 | https://mathoverflow.net/users/12898 | 131075 | 72,674 |
https://mathoverflow.net/questions/131078 | 3 | A real closed field can be ordered in one and only one way, and is therefore provided with a unique
order topology. Given any infinite cardinal number k, does there always exist a real closed field F
(whose cardinal number is greater than k), such that no non-empty subset of F having a cardinal number
not greater than ... | https://mathoverflow.net/users/4423 | A question about large real closed fields | If $\delta$ is the cofinality of an ordered field $F$, that is, the size of the smallest unbounded subset of $F$, then every point of $F$ fills a cut of type $(\delta,\delta)$. In other words, every point in $F$ is the limit of an increasing $\delta$ sequence from below and a decreasing $\delta$ sequence from above. On... | 5 | https://mathoverflow.net/users/1946 | 131081 | 72,676 |
https://mathoverflow.net/questions/131064 | 7 | What are the necessary and sufficient conditions for two finite groups $G$ and $H$
to have same complex-valued character table?
Is there any criterion for which one could know about the character table similarity of two finite groups without direct computations of each table?
Obviously two isomorphic groups have same... | https://mathoverflow.net/users/13525 | On finite groups with same complex-valued character table | Finite groups have the same complex character tables if and only if their group algebras
are isomorphic as quasi-Hopf algebras (if and only if the group algebras
are twisted forms of each other as Drinfel'd quasi-bialgebras, if and only if there is non-associative bi-Galois algebra over these groups).
For details see ... | 13 | https://mathoverflow.net/users/32332 | 131082 | 72,677 |
https://mathoverflow.net/questions/131084 | 3 | I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length $3-\varepsilon$?
(Inspired by [this question](https://mathoverflow.net/questions/131047/how-to-determine-the-number-of-a-cube-w... | https://mathoverflow.net/users/29873 | Is this cube packing possible? | Yes, since one can pack 10 unit cubes in a cube of side length $\:\:2+\left(\hspace{-0.023 in}\frac12\hspace{-0.05 in}\cdot\hspace{-0.03 in}\sqrt2\right) \;\;$.
See <https://erich-friedman.github.io/packing/cubincub> .
| 10 | https://mathoverflow.net/users/nan | 131090 | 72,680 |
https://mathoverflow.net/questions/131096 | 4 | I am reading Beauville's chapter IX on Elliptic surfaces.
Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the *elliptic fibration*.
We know $K^2=0$. Suppose the $m$-canonical system is non-empty and let $D\in \lvert m K \rvert$.
Then $D.F=0$, where $F$ is a fiber of $p$, by the g... | https://mathoverflow.net/users/34136 | Basics of minimal Elliptic Surfaces [following Beauville] | You are right, this follows from VIII.4.
The point is that you do not know if $D \in |mK|$ is connected.
If it is, then by VIII.4 and $D^2=0$ one deduces $D=rF$ with $r \in \mathbb{Q}$.
Otherwise, write $$D=D\_1+D\_2 + \ldots +D\_k,$$ where the $D\_i$ are the connected components. Then $D\_i D\_j=0$ for $i \neq... | 6 | https://mathoverflow.net/users/7460 | 131098 | 72,685 |
https://mathoverflow.net/questions/131100 | 3 | Is there a L-Lipschitz homeomorphism of the Elipse $x^2/4+y^2=1$ onto the unit circle $x^2+y^2=1$ such that $L<1$?
| https://mathoverflow.net/users/26543 | Lipschitz map of the ellipse | Assuming you mean Lipschitz with respect to the plane's Euclidean metric, as suggested by Noam D. Elkies, then no such homeomorphism exists.
The first thing to worry about is where $(0,\pm1)$ are mapped. Note that they have distance $2$, so a mapping $f$ with $L<1$ will not send these to opposite points on the unit c... | 8 | https://mathoverflow.net/users/29873 | 131113 | 72,693 |
https://mathoverflow.net/questions/131114 | 0 | What is a reference for the subject of "free resolutions for Lie algebras"?
Does the term "standard resolutions" means "free resolutions"?
What is a "bar resolution"?
Is there only one way to talk about this or it is always by considering the Lie algebra as a module over its universal enveloping algebra?
These ... | https://mathoverflow.net/users/40886 | Free resolution for Lie algebras (reference) | Here are some references (which are not mentioned in [Resolutions of Lie algebras](https://mathoverflow.net/questions/130376/resolutions-of-lie-algebras)).
First I can recommend the book of Charles A. Weibel, An Introduction to Homological Algebra.
It answers your questions and has a chapter where the general theory ... | 3 | https://mathoverflow.net/users/32332 | 131127 | 72,699 |
https://mathoverflow.net/questions/131128 | 4 | I've heard from others about the WO($\kappa$) as a counterpart of AC($\kappa$), but I cannot find a suitable way to express it in ZF since "every set of cardnality $\kappa$ can be well-ordered" is meaningless. Does WO($\kappa$) really exist?
---
I've found one. "Every set in $V\_\kappa$ can be well-ordered." Is t... | https://mathoverflow.net/users/34144 | Well-Ordering theorem of cardinal$\kappa$ | The counterpart you may intend to use is that every cardinal is comparable with $\kappa$. This means that every infinite set which is not smaller than $\kappa$ has a subset of cardinality $\kappa$.
This is the restriction of the well-ordering principle to $\kappa$, whereas $\sf AC\_\kappa$ is the restriction of the a... | 4 | https://mathoverflow.net/users/7206 | 131133 | 72,703 |
https://mathoverflow.net/questions/131131 | 7 | Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0\_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no quantifiers (Matiyasevich's theorem). The complement of $A$ is r.e. so $A$ may be defined by a $\Pi^0\_1$ formula $\for... | https://mathoverflow.net/users/34145 | Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ? | No, in general, a true $\Delta^0\_1$ assertion may not necessarily be provably $\Delta^0\_1$ in a given theory. For example, assume $\text{Con}(\text{PA})$ is true, and consider the formula $\phi(a)$ asserting that $a=a$ and the formula $\psi(a)$ asserting that "$a$ is not the code of a proof of a contradiction in $\te... | 8 | https://mathoverflow.net/users/1946 | 131136 | 72,705 |
https://mathoverflow.net/questions/131139 | -1 | Let $M$ and $N$ be finite dimensional smooth manifolds.
A smooth map $f: M \to N$ is an embedding if and only if there is an
open neighborhood $U$ of $f(M)$ in $N$ and a smooth mapping
$r : U \to M$ with $r \circ f = Id\_M$.
Does this mean we can pull back a vector-field $X$ on $N$ to
a vector field on $M$, like ... | https://mathoverflow.net/users/21302 | Vector field pull back from embedding | At each point $x\in M$ the differential $df\_x: T\_x M \to T\_{f(x)}N$ is a monomorphism. However, if $X$ is a vector field on $N$ the vector $X\_{f(x)}$ need not be in the image of $df\_x$. Hence to associate a tangent vector to $M$ at $x$, you need a procedure which associates to a vector in the vector space $T\_{f(x... | 3 | https://mathoverflow.net/users/8032 | 131141 | 72,707 |
https://mathoverflow.net/questions/131144 | 3 | It is well-known that if $\widetilde M\to M$ is a Galois cover of a compact Riemannian manifold $M$ with deck-transformation group $G$, then the growth of $G$ equals the volume growth of $\widetilde M$ (in the pullback metric).
**Question.** Is the same true when $M$ is a finite volume complete Riemannian manifold
an... | https://mathoverflow.net/users/1573 | Volume growth of covers and growth of deck-transformation groups | No, it's not true. A cusp is a counterexample (see below for an example with no boundary). Here by cusp I mean the subset $\{\text{Im}(z)\ge 1\}$ of the upper half-plane with hyperbolic metric modulo $z\mapsto z+1$. This has finite volume and the fundamental group has linear $\mathbf{Z}$ growth, but the universal cover... | 8 | https://mathoverflow.net/users/14094 | 131146 | 72,708 |
https://mathoverflow.net/questions/131107 | 20 | Hi. I was wondering if someone could explain why we call affine Lie algebras affine. Thanks!
Oliver
| https://mathoverflow.net/users/34139 | Why are affine Lie algebras called affine? | It's not easy to separate out the purely mathematical from the historical question here: What is the mathematical justification for use of the label "affine" and how did this label get attached to certain Lie algebras? The history in this case would be challenging to sort out, partly because some of the people involved... | 24 | https://mathoverflow.net/users/4231 | 131150 | 72,710 |
https://mathoverflow.net/questions/131134 | 4 | I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" $\theta\_{spheres}$ is achieved s.t. no further spheres can be placed inside of the cube. Is there an accepted range of values for $\the... | https://mathoverflow.net/users/34146 | Is there an "accepted" jamming limit for hard spheres placed in the unit cube by random sequential adsorption? | The process you describe is usually called Random Sequential Addition (RSA). In [this paper](http://arxiv.org/abs/cond-mat/0608402), Torquato, Uche, and Stillinger compute the saturation density up to $d=6$ (see Table I).
For $d=3$ they have $\phi\_s\approx 0.38278$ as the saturation density.
| 6 | https://mathoverflow.net/users/20186 | 131153 | 72,711 |
https://mathoverflow.net/questions/131085 | 1 | Let $R$ be a ring, $\Sigma$ be a multiplicatively closed subset of $R$. $M$ is an $R$-module. Denote the injective hull of $M$ by $E(M)$.
$M$ is $\Sigma$-torsion if for any $m$ in $M$, there is $\sigma \in \Sigma$ such that $\sigma m = 0$. we say that $\Sigma$ operates regularly on $M$ if $\sigma m =0$ implies $m =0$... | https://mathoverflow.net/users/34133 | decomposition of the injective hull of a torsion free module | Here is an answer as I understood the question (it was changed in the meantime). The $R$-module $A$ has an injective hull (or envelope) $E(A)$ containing $A$. Then we ask about conditions ensuring that the
injective module $E(A)$ has a decomposition (as a direct sum of indecomposable submodules).
One possibility to ens... | 1 | https://mathoverflow.net/users/32332 | 131162 | 72,715 |
https://mathoverflow.net/questions/131151 | 3 | Let $f \in C^{\gamma}\_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere except at $\vec{0}$
2) K homogeneous of degree $-n$, in particular $\|K(x)\| \leq \frac{c}{\|x \|^{n}}$
3) K has mean ... | https://mathoverflow.net/users/34152 | Is a Cauchy principal value invariant under a "change of variables"? | Changing variables, we have
$$(pv(K)\ast f)(x)=\lim\_{\delta\searrow 0}
\int\_{\vert x-y\vert>\delta}K(x-y) f(y) dy=\lim\_{\delta\searrow 0}
\int\_{\vert x-G(w)\vert>\delta}K(x-G(w)) f(G(w))\vert \nabla G(w)\vert dw,
$$
so that
$$
(pv(K)\ast f)(G(\nu))=\lim\_{\delta\searrow 0}\int\_{\vert G(\nu)-G(w)\vert>\delta}K(G(\... | 2 | https://mathoverflow.net/users/21907 | 131163 | 72,716 |
https://mathoverflow.net/questions/131164 | 4 | I find myself unable to solve question 24.1 of T. Jech's *Set Theory*:
>
> If $\beta<\omega\_1$ and if
> $2^{\aleph\_{\alpha}}\leq\aleph\_{\alpha+\beta}$
> for a stationary set of $\alpha$'s,
> then
> $2^{\aleph\_{\omega\_1}}\leq\aleph\_{\omega\_1+\beta}$.
>
>
> [By induction on $\beta$: If
> $\varphi(\alpha... | https://mathoverflow.net/users/22934 | Set Theory exercise. | Recall the definitions (24.1) and the consequence of the proof of Lemma 24.2, which states that $\|\varphi\|=\sup\lbrace\|\psi\|+1\mid\psi<\varphi\rbrace$.
Then we have that $\|\varphi\|=0$ if and only if $\varphi(\alpha)=0$ on a stationary set. Proceed by induction; suppose this claim is true for all $\delta<\beta$ ... | 7 | https://mathoverflow.net/users/7206 | 131166 | 72,717 |
https://mathoverflow.net/questions/131175 | 4 | Since in a compact Riemannian manifold $M$ the only totally convex subset is the whole manifold itself, see [Closed manifold has no nontrivial totally convex subset?](https://mathoverflow.net/questions/106169/closed-manifold-has-no-nontrivial-totally-convex-subset), it should follow that for every point $p\in M$ there ... | https://mathoverflow.net/users/24152 | Closed geodesic loops around points in compact manifolds | Here is a standard argument, I learned it from *"Comparision Theorems in Riemannian Geometry"* by Cheeger and Ebin. It avoids use of infinite dimensional spaces.
Choose the smallest $k>0$ such that $\pi\_kM\ne 0$.
Choose a spheroid which represents a nontricial element of $\pi\_kM$.
We can assume that the spheroid is... | 3 | https://mathoverflow.net/users/1441 | 131181 | 72,720 |
https://mathoverflow.net/questions/131184 | 5 | Let's define a net and subnet in this way:
* A net is any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set.
* A net $m:(P',\le)\to X$ is a subnet of the net $n:(P,\le)\to X$ iff there is a function:
$$\theta:(P',\le)\to (P,\le)$$
which is increasing:
$$x'\le y' \to \theta(x')\le\th... | https://mathoverflow.net/users/nan | Connection between subnet and superfilter | Yes. Let Z be the integers, let X consist of a single point and let m : {0} -> X and n : Z -> X be constant functions. Then n and m give the same filter but m cannot be
a subnet of n since no single integer in Z is cofinal.
| 6 | https://mathoverflow.net/users/34170 | 131187 | 72,722 |
https://mathoverflow.net/questions/131185 | 102 | Yitang Zhang recently published a [new attack on the Twin Primes Conjecture](https://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/). Quoting Andre Granville :
>
> “The big experts in the field had
> already tried to make this approach
> work,” Granville said. “He’s not... | https://mathoverflow.net/users/7126 | Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture | My understanding of this, which is essentially cobbled together from the various news accounts, is as follows:
Let $\pi(x;q,a)$ denote the number of primes less than $x$ congruent to $a\bmod q$, and $\pi(x)$ the number of primes less than $x$. $\phi(n)$ is the number of positive integers less than or equal to n that ... | 98 | https://mathoverflow.net/users/630 | 131188 | 72,723 |
https://mathoverflow.net/questions/131180 | 3 | I recently started reading about hyperbolic dynamics in the notes of L. Wen,
>
> <http://www6.cityu.edu.hk/rcms/publications/ln5.pdf>
>
>
>
and in this (page 8) there is the following statement: If the definition of hyperbolicity we allow $E^{s}= \{0\}$ or $E^u=\{0\}$ in this case the hyperbolic set $\Lambda$... | https://mathoverflow.net/users/nan | Hyperbolic sets | Let's suppose $E^u=0$ on $\Lambda$ and $\lambda\in(0,1)$ be the contracting constant of $Df$ on $\Lambda$. Pick an open neighborhood $U\supset \Lambda$ (small enough) such that $\|Df|\_{U}\|<\lambda<1$. Then pick $N$ large with $\lambda^N<1/6$.
First step is to show that, every nonwandering point $x\in\Omega(f,\Lambd... | 2 | https://mathoverflow.net/users/11028 | 131191 | 72,725 |
https://mathoverflow.net/questions/131173 | 18 | Building off of Qiaochu's comment on my answer to [a previous mathoverflow question](https://mathoverflow.net/questions/130883/is-there-any-proof-that-you-feel-you-do-not-understand/130901#130901), I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi\_e\text{ is total }\implies \Phi\_{\Phi\_e(k)}\... | https://mathoverflow.net/users/8133 | Lawvere's fixed point theorem and the Recursion Theorem | There is indeed a very close connection between Lawvere's fixed-point theorem and Recursion theorem, but one has to look at it the right way. Namely, it all becomes clear once we do it in the effective topos.
Let us start by recalling Lawvere's theorem. (I use $X \to Y$ and $Y^X$ as synonyms for the set of all functi... | 17 | https://mathoverflow.net/users/1176 | 131198 | 72,730 |
https://mathoverflow.net/questions/131196 | 3 | Hello,
I consider a compact and connected (smooth) Riemannian manofold $(M,g)$. I'm interested in the eigenfunctions of the Schrödinger Operator $L=-\Delta+ V$ acting on (smooth) functions. Do you know for what kind of potentials $V:M\rightarrow \mathbb{R}$, the eigenfunctions will be smooth?
Explicitly:
* Are th... | https://mathoverflow.net/users/21870 | regularity of eigenfunctions of Schrödinger Operator | If the first (lowest) eigenfunction $f\_0$ is smooth, then $V$ is smooth. Indeed, assuming $M$ connected, it is a classical fact that $f\_0$ doesn't vanish (it is the first case of Courant's nodal theorem for instance), and obviously $V=\lambda\_0 +\Delta f\_0/f\_0$.
With boundary and Neumann condition, the same argu... | 5 | https://mathoverflow.net/users/6451 | 131201 | 72,732 |
https://mathoverflow.net/questions/131202 | 1 | Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ''nice'' functions.
Can we characterize nice functions?
For example;
$f(x)=x$ is a nice function.
$f(x)=2x$ is no... | https://mathoverflow.net/users/34096 | A question about "nice" functions | Just a quick counterexample to your last question:
Let $C \subset \mathbb{R}$ be a fat Cantor set that is symmetric w.r.t. $0$ and define
$f(x) = \begin{cases}x,& \text{if }x \in C\\\ -x,& \text{if }x \notin C \end{cases}.$
(More trivial counterexample would be $f(x) = \begin{cases}x,& \text{if }x \notin \mathbb{Q}... | 1 | https://mathoverflow.net/users/11716 | 131207 | 72,735 |
https://mathoverflow.net/questions/131206 | 13 | According to the wiki of [Kakutani's fixed-point theorem](http://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem), A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W \rbrace$ is an... | https://mathoverflow.net/users/32564 | What is the definition of continuity of set-valued functions? | $\phi$ is upper semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\subset W\rbrace $ is open in $X$.
$\phi$ is lower semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\cap W\neq \emptyset\rbrace$ is open in $X$.
$\phi$ is continuous if it is both upper semincontin... | 14 | https://mathoverflow.net/users/10503 | 131209 | 72,736 |
https://mathoverflow.net/questions/131211 | 6 | Hello,
let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset L^2(M)\rightarrow L^2(M)$ with bounded potential $V:M\rightarrow \mathbb{R}$ (and Dirichlet/Neumann boundary conditions f... | https://mathoverflow.net/users/21870 | The first eigenvalue of the Schrödinger operator is simple. | Roughly, the trick is not to view $L$ as an operator on $L^2$, but on $C^0$
I will use the following version of Krein-Rutmann which is proven in "Du, Yihong: Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1: Maximum Principles and Applications.":
*Let $X$ be a Banach space... | 3 | https://mathoverflow.net/users/16702 | 131216 | 72,740 |
https://mathoverflow.net/questions/131200 | 1 | Hello!
I come across the word "vertex solution" in the context
" We can also assume that x and y are **vertex solutions**,so that the sequence {x,y} remains in a finite set."
Could anybody know any definition for "Vertex Solution"?
Thanks,
Tendow
| https://mathoverflow.net/users/33250 | What does "Vertex Solution" mean? | In the context of linear programming, and assuming that you're using the simplex method to solve your LP's rather than an interior point method, it's most likely that the author means "basic feasible solution" (BFS) here. In geometrical terms, the basic feasible solutions of an LP are vertices of the polytope of feasib... | 1 | https://mathoverflow.net/users/9022 | 131224 | 72,742 |
https://mathoverflow.net/questions/131208 | 6 | What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $k$, ie $f\in\mathbb{C}[k]$. References?
| https://mathoverflow.net/users/34179 | Algebraic closure of a polynomial ring | $\def\CC{\mathbb{C}}$A necessary and sufficient condition is that $k$ cannot be written as $h(\ell(x,y,z))$ for $h \in \CC[t]$ of degree $>1$ and $\ell \in \CC[x,y,z]$. Clearly, this is a necessary condition since, if $k = h(\ell)$, then $k$ and $\ell$ are integrally dependent. We now prove sufficiency.
Let $A = \CC[... | 11 | https://mathoverflow.net/users/297 | 131226 | 72,743 |
https://mathoverflow.net/questions/131223 | 6 | Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve and consider the ruled surface
$$ S=\frac{E\times\Bbb{P}^1}{G} $$
where $G$ is a group of translations of $E$, acting ... | https://mathoverflow.net/users/34136 | Surfaces ruled over elliptic curves | **EDIT** We show that the answer to the OP's question is *yes*. Thanks to Will Sawin for his comments.
I use the notation of [Hartshorne, *Algebraic Geometry*, Chapter V Section 2].
Since $S$ is a ruled surface, there exists a section $C\_0$ of minimal self-intersection; set $C\_0^2 = -e$. If we write $S=\mathbb{P}... | 6 | https://mathoverflow.net/users/7460 | 131236 | 72,745 |
https://mathoverflow.net/questions/131219 | 2 | Does a connected finite locally free group scheme G over a scheme S of characteristic p>0 has degree a power of p? I know that when S is the spectrum of a field k, it is true. Someone told me that this is true by considering a generic point of $, but I don't know how to do it?
| https://mathoverflow.net/users/5813 | Degree of a finite locally free group scheme over a base scheme of characteristic p | If $f:X\rightarrow S$ is finite locally free, then formation of $f\_\*\mathscr{O}\_X$ is compatible with arbitrary change of base on $S$. So if $X$ has constant rank $r$, i.e., $f\_\*\mathscr{O}\_X$ is finite locally free on $S$ of constant rank $r$, the same will be true for $f^\prime:X^\prime=X\times\_SS^\prime\right... | 2 | https://mathoverflow.net/users/4351 | 131239 | 72,746 |
https://mathoverflow.net/questions/131235 | 3 | Let $X$ be a normal complex affine algebraic variety. Suppose that $Y$ is an open subvariety of $X$, and that the codimension of $X\setminus Y$ in $X$ is at least $2$. One version of the Hartogs Theorem is that the restriction map $\mathbb{C}[X]\rightarrow\mathbb{C}[Y]$ is surjective. I am curious about whether there i... | https://mathoverflow.net/users/25358 | Hartogs Theorem and Canonical Bundles | I think the property you want is that the canonical sheaf $\omega\_X$ is S2. Note that on a normal affine variety, $\omega\_X$ is *not* necessarily a line bundle (it is if $X$ is a complete intersection though).
For simplicity, let's assume $X \subseteq A^{n}$ is of dimension $d$. Then
$$
\omega\_X = Ext^{n-d}(O\_X... | 7 | https://mathoverflow.net/users/3521 | 131240 | 72,747 |
https://mathoverflow.net/questions/131243 | 0 | Hi all,
Suppose that $\mathbf{f}=[f\_1, f\_2,\ldots,f\_m]$ and $\mathbf{g}=[g\_1,g\_2,\ldots,g\_m]$ are two $m$-dimensional vectors. All $f\_i$'s are chosen uniformly randomly from a finite field $\mathbb{F}\_q$, where $q$ is the finite field size. For convenience, we denote the elements of $\mathbb{F}\_q$ as $\{0, 1... | https://mathoverflow.net/users/34191 | Determine the probability that two random vectors over a finite field are orthogonal | If $\vec g = \vec 0$, then all vectors $\vec f$ are orthogonal to $\vec g$.
If $\vec g \ne \vec 0$, then $1/q$ of the vectors $\vec f$ are orthogonal to $\vec g$. Given the previous coordinates of $\vec f$, there is a unique choice for the last coordinate of $\vec f$ paired with a nonzero coordinate of $\vec g$ so t... | 3 | https://mathoverflow.net/users/2954 | 131251 | 72,750 |
https://mathoverflow.net/questions/131238 | 1 | I have a function f(x,n) can be expressed as a cubic function of x with coefficients that are functions of n. For example x^3 + (n-2)x^2 + (3n-6)x + n.
I want to prove that for every positive value of n, there exists a real, positive value of x such that f(x,n)=0.
I know this is true for the function I have in min... | https://mathoverflow.net/users/34190 | Real root of a cubic equation | For convenience, write the cubic function as
$$f(x) = x^3 + 3ax^2 + 3bx + c,$$
where $a$, $b$, and $c$ are (polynomial?) functions of $n$. As has been noted in comments, if $c<0$, you're guaranteed a positive real zero $x$, so the only question is what to do for values of $n$ for which $c\ge0$.
The only way you ... | 2 | https://mathoverflow.net/users/15837 | 131260 | 72,756 |
https://mathoverflow.net/questions/131265 | 4 | Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-singular everywhere else).
Suppose that this variety $A$ is normal.
In order to study singularities it often seems like a good idea to study ... | https://mathoverflow.net/users/34197 | When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal? | I don't think so. There are examples of isolated normal threefold singularities that are not Cohen-Macaulay. A hyperplane section is not Cohen-Macaulay, hence it can not be normal, because a normal surface is Cohen-Macaulay.
| 9 | https://mathoverflow.net/users/4790 | 131267 | 72,760 |
https://mathoverflow.net/questions/131126 | 17 | In Brian Conrad's notes
[here](http://math.stanford.edu/~conrad/papers/aws.pdf) for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped with a non-zero section such that all stalks vanish (at all the "usual" points correspon... | https://mathoverflow.net/users/34143 | Why do rigid spaces have "not enough points"? | If you are familiar with Berkovich spaces, you can do the following construction. Let $X$ be an affinoid space of positive dimension and pick a point $x$ in $X$ that is not a rigid point. Consider the inclusion map $i\colon x \to X$. Then the sheaf $F = i\_\*\mathbb{Z}$ does the job. Since the space $X$ is Hausdorff, t... | 10 | https://mathoverflow.net/users/4069 | 131274 | 72,764 |
https://mathoverflow.net/questions/131037 | 2 | Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$.
Let $X\_1,X$ be two Banach spaces such that $X\_1\subset X$ with compact embedding. Let $x\_n(t)\in X\_1$ be a bounded sequence in $X\_1$ (this sequence converge strongly to $x(t)\in X$ for almost every $t\in [0,\infty)$). M... | https://mathoverflow.net/users/33135 | Strong convergence in the Bochner space L^p([0,T],X) | Rafa: if I'm not mistaken your dominated convergence argument is wrong.
I agree that for a.e. fixed $t\in [0,T]$ you can extract a subsequence $n\_k$ such that $x\_{n\_k}(t)\to x(t)$ strongly in $X$. The mistake you made is that the extraction procedure depends on the time $t$ you fix. Of course you can always use di... | 4 | https://mathoverflow.net/users/33741 | 131275 | 72,765 |
https://mathoverflow.net/questions/131297 | 1 | $$\begin{array}{ll} \text{minimize} & \beta^{T} A \beta\\ \text{subject to} & \beta^{T} C \beta=1\\ & \beta \geqslant 0\end{array}$$
where $A, C\in \mathbb{R}^{M\times M}$ and $\beta \in \mathbb{R}^{M}$. I saw in one paper that it could be solved via its semidefinite programming relaxation by adding an auxiliary vari... | https://mathoverflow.net/users/34206 | A non-convex quadratically constrained quadratic program | It's helpful if you cite the paper in which you saw something that you're asking a question about- we could provide a better answer if we knew where the question came from.
First, assume without loss of generality that $A$ and $C$ are symmetric matrices. It's easy to take these quadratic forms and write them in term... | 2 | https://mathoverflow.net/users/9022 | 131302 | 72,779 |
https://mathoverflow.net/questions/131311 | 4 | **PART I** (Initial version)
Let $P$ be the set of all primes $2\ 3\ \ldots$. Let
$$P\_d\ \ :=\ \ \{\ p\in P\ :\ \exists\_{q\in P}\ \ 0 < |p-q|\le d\ \}$$
and
$$S\_d\ :=\ \sum\_{p\in P\_d}\ \frac 1p$$
for every real $d>0$. Thus $d\mapsto S\_d$ is non-decreasing, $S\_2 < \infty$, and $\li... | https://mathoverflow.net/users/8385 | Are sums of the inverses of prime siblings finite? | It turns out that $S\_d$ is finite for every $d$. See my answer to this [very similar question](https://math.stackexchange.com/questions/397391/convergence-of-the-sum-of-reciprocals-of-fake-twin-primes/397466#397466).
| 14 | https://mathoverflow.net/users/5091 | 131313 | 72,784 |
https://mathoverflow.net/questions/131317 | 0 | I have a convex optimization problem of finding a function Q(x,y) as below:
Minimize $\int{k(x,y)Q(x,y)dxdy}$ subject to a list of constraints which are not relevant to the question, so I'm skipping them. The function $k(x,y)$ is known and my aim is to find the form of $Q(x,y)$. I was thinking if I could represent th... | https://mathoverflow.net/users/34213 | Interpreting numerical double integration as a matrix multiplication | Okay found it out myself. It could be done the following way:
Compute $k(x,y)$ for each incrementation and make a matrix $K$ which contain the values of $k(x,y)$ all through the interval for specified x,y limits. $Q(x,y)$ becomes a matrix itself which is to the optimization variable. Now the objective is:
Minimize $\... | 0 | https://mathoverflow.net/users/34213 | 131325 | 72,786 |
https://mathoverflow.net/questions/131295 | 1 | I came across this problem when trying to solve the following integral equations arising in direct scattering:
$$
\begin{align}
n\_{11}(x,z)=1+\int\_{-\infty}^xe^{-izy}u(y)n\_{21}(y,z)dy, \quad
n\_{21}(x,z)=\int\_{-\infty}^xe^{izy}\bar{u}(y)n\_{11}(y,z)dy
\end{align}
$$
I was suggested to iterate thoses two equations... | https://mathoverflow.net/users/27040 | Solving systems of integral equations using Volterra series | In order to iterate, you have to substitute the second equation into the first one. So,
$$
n\_{11}(x,z)=1+\int\_{-\infty}^xdye^{-izy}u(y)\int\_{-\infty}^ydy\_1e^{izy\_1}{\bar u}(y\_1)n\_{11}(y\_1,z).
$$
This equation is generally the starting pointing for an iterative procedure, the main tool of perturbation technique... | 0 | https://mathoverflow.net/users/19520 | 131333 | 72,790 |
https://mathoverflow.net/questions/131340 | 2 | This is a question I asked at Math.SE but got no answers: <https://math.stackexchange.com/q/396217/7110/>
The tautological vector bundle $\gamma\_k(\mathbb{K}^N)$ over the Grassmann manifold $G\_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$) is defin... | https://mathoverflow.net/users/13356 | Non-(stable)-triviality of the tautological bundles | For simplicity, let's take $\Bbb K = \Bbb R$.
By the bundle classification theorem, your question amounts to understanding whether the
inclusion map
$$
G\_k(\Bbb R^N) \to \underset j{\text{colim }} \, G\_{k+j}(\Bbb R^{N+j}) = BO
$$
is null homotopic.
First consider the inclusion
$$
i: G\_k(\Bbb R^N) \to G\_k(\Bb... | 4 | https://mathoverflow.net/users/8032 | 131342 | 72,793 |
https://mathoverflow.net/questions/131334 | -2 | Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality:
$$\sum\_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$
| https://mathoverflow.net/users/21258 | Upper bound of a series | Since $\displaystyle \frac{k^a}{(k+1)^a+(k+2)^a}<\frac{k^a}{k^a+k^a}=\frac{1}{2}$ then $\displaystyle\sum\_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}<\sum\_{k=1}^N\frac{1}{2}=\frac{N}{2}$.
| 2 | https://mathoverflow.net/users/19642 | 131344 | 72,794 |
https://mathoverflow.net/questions/131345 | 3 | Let $\scr A$ be an abelian category with exact products and a cogenerator (e.g. $\scr A$ is a category of modules). Let ${\mathbf K}(\scr A)$ be the homotopy category of cochain complexes over $\scr A$. We call homotopcally injective a complex $X\in{\mathbf K}(\scr A)$ with the property that every map $N\to X$, startin... | https://mathoverflow.net/users/15541 | Why every complex of injectives is homotopically injective (provided that, the injective dimension is finite)? | Let $J^\bullet$ be an acyclic complex of injective objects in an abelian category $\mathcal A$. Consider its finite subquotient complexes of canonical truncation $0\to Z^m\to J^m\to J^{m+1}\to \dotsb\to J^{n-1}\to Z^n\to 0$, where $Z^i$ denotes the kernel of the differential $J^i\to J^{i+1}$. This finite complex is a r... | 3 | https://mathoverflow.net/users/2106 | 131350 | 72,797 |
https://mathoverflow.net/questions/131318 | 8 | Suppose we are given a commutative ring $R$ with a unit. Suppose that $R$ is the direct product of two rings $R\cong R\_1\times R\_2$. It's straightforward to show that any ideal $I\subset R$ maps to an ideal $I\_1\times I\_2\subset R\_1\times R\_2$ by the above isomorphism. It is, however, not straightforward at all t... | https://mathoverflow.net/users/34217 | Quotients in Sums of Rings | There are several issues to address here, but let me first point out that the formula for $B\_n$ at the top of p. 71 of our Memoir has a typo: it should say
$B\_n = (Z/p^{j+1})^s \oplus (Z/p^j)^{p^2-p-s} $ where $0 < s \leq p^2-p$ and
$n = 2j(p^2-p)+2s+2p-3$. The point is that there should be $(p^2-1) - (p-1) = p^2-p$... | 7 | https://mathoverflow.net/users/6872 | 131362 | 72,806 |
https://mathoverflow.net/questions/131353 | 2 | A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm trying to solve:-
For a research problem that I'm working on, I need to solve the following system of Diophantine equations:-
$ a^3 + 400... | https://mathoverflow.net/users/23202 | Help with this system of Diophantine equations | The first two equations amount to $b^3 - a^3 = 721$.
Now since for any solution $b^3 - a^3 \ge (a+1)^3 - a^3 = 3a^2 + 3a +1$ this directly gives an upper bound on $a$ namely $15$.
Now checking for which of $a=1, \dots, 15$ one has that $a^3 + 721$ is the third power of an integer, by calculating its third root for... | 10 | https://mathoverflow.net/users/nan | 131363 | 72,807 |
https://mathoverflow.net/questions/131086 | 7 | It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In [these](http://mat.uab.es/~kock/RLN/rcv.pdf) notes, at the top of page 6, it is said that the problem is that showing a variety is not unirat... | https://mathoverflow.net/users/24268 | Proving a variety is not unirational | Just so there is an answer: the problem of equivalence / non-equivalence of rational connectedness and unirationality is still open.
| 4 | https://mathoverflow.net/users/13265 | 131369 | 72,809 |
https://mathoverflow.net/questions/131176 | 9 | Are there tetrahedra which can be subdivided into three non-overlapping parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedra for which this decomposition is impossible. In 2d, for right triangles you get a decomposition into two similar parts b... | https://mathoverflow.net/users/34162 | Question about tetrahedron decomposition | In the case where the three parts are each congruent to one another, the answer to your question is **no**: there is no such decomposition of a tetrahedron.
The terminology needed to find such an answer in the literature is "reptile" or "$k$-reptile simplices."
**Citation for proof:**
Safernová, Z.: Perfect tilin... | 7 | https://mathoverflow.net/users/22971 | 131372 | 72,811 |
https://mathoverflow.net/questions/131338 | 2 | Let $\Gamma\subseteq \Gamma'\subset SL\_2(\mathbb Z)$ be congruence subgroups, and
$X(\Gamma)$, $X(\Gamma')$ be the associated smooth projective modular curves over $\mathbb C$. The inclusion $\Gamma\subseteq \Gamma'$ induces a (canonical) non-constant morphism $p:X(\Gamma)\to X(\Gamma')$ of curves over $\mathbb C$.
... | https://mathoverflow.net/users/32209 | Field of definition of canonical morphism between (congruence) modular curves | Yes. Please see Theorem 7.1.3 of Katz-Mazur.
| 4 | https://mathoverflow.net/users/3384 | 131377 | 72,815 |
https://mathoverflow.net/questions/131383 | 1 | Let $X$ be a sympletic manifold and $A\in H\_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers.
Assume that $$\mathcal{M}\_{g,k}(X;A)$$
is dense in
$$\overline{\mathcal{M}}\_{g,k}(X;A).$$
Are the primary Gromov-Witten invariants corresponding to $X$ and $A$ enumerative?
If not, when does the condition $... | https://mathoverflow.net/users/15512 | enumerative Gromov-Witten invariants | I am not sure what "primary" means. However, I believe the answer to your first question is "no". For a sufficiently general quintic hypersurface $X$ in $\mathbb{C}P^4$, for sufficiently small curve classes $A$, all genus $0$ curves in $X$ of class $A$ are pairwise disjoint and smooth with normal bundle $\mathcal{O}\_{... | 3 | https://mathoverflow.net/users/13265 | 131385 | 72,818 |
https://mathoverflow.net/questions/131062 | 11 | For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which can be made for the importance of separability.
Let me subsume the situation: Both notions are intended to guarantee smallness known from cl... | https://mathoverflow.net/users/33842 | Importance of separability vs. second-countability | An arbitrary product of separable spaces satisfies Suslin´s condition (i.e. any disjoint family of open sets is countable). I find this result remarkable since separability is not preserved under (large) products while Suslin´s condition might or might not be preserved under (even finite) products, depending on the und... | 10 | https://mathoverflow.net/users/17836 | 131390 | 72,821 |
https://mathoverflow.net/questions/131225 | 1 | What is the relation between $H^i\_I(-)$ and $H^i\_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
| https://mathoverflow.net/users/21992 | Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$ | There's a map between them, and these do fit into a long exact sequence together. This is explained in the book on local cohomology by Hartshorne, see Lemma 1.8: You can even download this book **[if your institution has access...](http://link.springer.com/book/10.1007/BFb0073971/page/1)**
I will sketch it briefly.
L... | 0 | https://mathoverflow.net/users/3521 | 131391 | 72,822 |
https://mathoverflow.net/questions/131386 | 4 | Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric) realization of a simplicial group (re-topologize this using the compactly generated topology).
Let $R^G(\ast)$ be th... | https://mathoverflow.net/users/8032 | Equivariant versus retractive spaces: a reference request | Appendix A in this paper seems to do this, unless I've misunderstood:
<http://arxiv.org/abs/0810.4535>
| 3 | https://mathoverflow.net/users/6936 | 131397 | 72,825 |
https://mathoverflow.net/questions/131392 | 7 | I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$:
1. Are all reductive algebraic groups strongly polynomial-count?
2. Are products of strongly polynomial-count varieties also strongly polynomial-count? What about (disjoint) unions?
3. If X is strongly polynomial-count variety, and... | https://mathoverflow.net/users/12218 | Constructing Polynomial Count Varieties | A combination of easy and hard questions here. The easy ones:
(1) No. For example, the group scheme $\{ (x,y) : x^2+y^2=1 \}$, with multiplication $(x\_1, y\_1) (x\_2, y\_2) = (x\_1 x\_2 - y\_1 y\_2, x\_1 y\_2 + x\_2 y\_1)$ has $q - (-1)^{(q-1)/2}$ points over a field with $q$ elements. Or, similarly, the group schem... | 7 | https://mathoverflow.net/users/297 | 131401 | 72,826 |
https://mathoverflow.net/questions/131400 | 3 | Let $G$ be an Abelian group. Let $A \subseteq G$. In additive combinatorics, one of the primary measures of the additive structure of $A$ is its *additive energy*, defined as $E(A) = |\lbrace(a\_1,a\_2,a\_3,a\_4) \in A^4 : a\_1 + a\_2 = a\_3 + a\_4 \rbrace|$.
A related quantity that I'm interested in is: $F(A) = |\l... | https://mathoverflow.net/users/5534 | A measure of closure under sumset? | On the one hand there are some notions that seem related that are studied (see at the end), but on the other hand the precise defintion you give does not have some, at least from a cetain point of view, desirable features.
First, on this second aspect an example (there are various other 'good' properties of the addit... | 4 | https://mathoverflow.net/users/nan | 131402 | 72,827 |
https://mathoverflow.net/questions/131407 | 25 | Assume for this question that ZF set theory is sound.
Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF proof that M runs forever on a blank input.
It's clear that PROVELOOP is recursively-enumerable, and hence reducible to the halting proble... | https://mathoverflow.net/users/2575 | Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem? | The first thing to notice is that if ZF is consistent, then it is
consistent with ZFC that what you call ProveLoop is actually decidable. The
reason is that if ZF is consistent, then by the incompleteness
theorem, it is consistent with ZFC that $\neg$Con(ZF), in which case everything is provable in
ZF, in which case ev... | 22 | https://mathoverflow.net/users/1946 | 131410 | 72,833 |
https://mathoverflow.net/questions/131413 | 33 | I understand that one can give a proof of each of these propositions assuming the truth of the other. But this seems a bit squishy to me, since there is a trivial sense in which any two true theorems are equivalent (to any proof of Theorem A, prepend "Assume Theorem B", and vice versa; the objection "But the proof of T... | https://mathoverflow.net/users/3621 | In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent? | Sperner's lemma is not equivalent to Brouwer's Fixed Point Theorem. All that one can prove directly from Sperner's Lemma is the following weaker statement.
**Approximate Fixed Point Theorem.** Let $K$ be the standard $n$-dimensional simplex and let $f:K \to K$ be a continuous function. For every $\varepsilon \gt 0$ t... | 48 | https://mathoverflow.net/users/2000 | 131414 | 72,836 |
https://mathoverflow.net/questions/131420 | 14 | Let $I$ be a compact interval and $\mathcal{M}(I)$ the space of (signed) Borel measures. We equip it with the weak topology, i.e. a sequence $\mu\_n$ converges to zero if and only if
$$ \left|\int\_I f(x) \mathrm{d}\mu\_n(x)\right| \longrightarrow 0$$
for all $f \in C(I)$.
**Now the question is the following: Let $V ... | https://mathoverflow.net/users/16702 | Dirac measures dense in space of measures? | Equipped with the mentionned weak($-\star$) topology, am I wrong or the set of continuous linear forms on $\mathcal{M}(I)$ is **precisely** given by $C(I)$ ?
Then by the classical use of Hahn-Banach theorem, your vectorspace $V$ if dense if and only if the only continuous linear form of $\mathcal{M}(I)-w\star$ vanish... | 17 | https://mathoverflow.net/users/27767 | 131427 | 72,841 |
https://mathoverflow.net/questions/131424 | 3 | I am unsure which is the right spelling (if there even is a ‘right’ spelling), but maybe native speakers can enlighten me: When should I use
* *fixed point*
* *fixed-point*
* *fixedpoint*
when I refer to the point itself, but also in composite works (“fixed point equation”, “fixed-point juggling”, “fixed-point oper... | https://mathoverflow.net/users/28027 | fixedpoint or fixed point or fixed-point | When it is a phrasal adjective you use a hyphen.
So when it modifies a noun uses a hyphen: fixed−point equation, fixed−point operator, fixed-point theory.
But, on the other hand, take a fixed point of the operator, consider the fixed point in X. We have found our fixed point.
When the phrasal adjective ends wit... | 15 | https://mathoverflow.net/users/26674 | 131437 | 72,844 |
https://mathoverflow.net/questions/131417 | 4 | The nonlinear pde
$$
\partial\_t^2\phi-\partial\_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\_0t-p\cdot x+\varphi,i)
$$
with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the dispersion relation hol... | https://mathoverflow.net/users/19520 | Exact solutions to nonlinear Klein-Gordon equation | regularization of the Klein-Gordon equation proceeds in the same way as for the Schrödinger equation; you restrict $x$ to the interval $(0,L)$ and impose periodic boundary conditions $\phi(0,t)=\phi(L,t)$; this quantizes the wave vector $p=p\_n(t)$, $n\in\mathbb{Z}$ --- for the linear Schrödinger equation the quantizat... | 2 | https://mathoverflow.net/users/11260 | 131447 | 72,850 |
https://mathoverflow.net/questions/131453 | 13 | Let $(X,\mathcal{O}\_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}\_X$-modules. Can we invoke the fact that $X$ is contractible to conclude, in some cases, that $\mathcal{F}$ is isomorphic to $\mathcal{O}\_X^{\oplus n}$ for some $n$? If you like, you may ta... | https://mathoverflow.net/users/25358 | Sheaves on Contractible Analytic Spaces | The so-called Oka-Grauert principle states that for any Stein space $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide.
The original reference is
*Hans Grauert*, [**Analytische Faserungen über holomorph-vollständigen Räumen**](http://www.ams.org/mathscinet-getitem?mr=98... | 30 | https://mathoverflow.net/users/7460 | 131458 | 72,854 |
https://mathoverflow.net/questions/131145 | 6 | Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings.
As far as I understand, in various generalizations of this situation, such a map is called smooth if $B$ is a perfect object in $D(B\otimes\_A B)$. (See for example Definition 2.2 of <http://arxiv.org/pdf/1006.4721v2.pdf>).
... | https://mathoverflow.net/users/3759 | Homological characterization of smooth maps | Yes to the second question. More generally, if $f:A \to B$ is a flat homomorphism of noetherian commutative rings such that the flat dimension of $B$ over $B\otimes\_A B$ is finite, then $f$ is regular. See Rodicio: Smooth algebras and vanishing of Hochschild homology, Comm. math. Helv. 65 (1990) 474-477. When $f$ is o... | 6 | https://mathoverflow.net/users/34259 | 131472 | 72,856 |
https://mathoverflow.net/questions/131449 | 8 | [Frankl's conjecture](http://en.wikipedia.org/wiki/Union-closed_sets_conjecture), open since 1979, says that if $F$ is a union-closed family of subsets of $X$, then there is some $x \in X$ such that $x$ appears in at least half the sets in $F$.
What was the motivation for this conjecture? Is it a generalization of so... | https://mathoverflow.net/users/22051 | Motivation for Frankl's conjecture? | Frankl originally stated the dual of the problem as written here, i.e., in terms of intersections instead of unions. This seems to have been in the 1979 edition of the Handbook of Combinatorics [edit: No such edition exists, and I'm not sure of the original source. See comments below], which isn't that easy to find, bu... | 17 | https://mathoverflow.net/users/19729 | 131474 | 72,857 |
https://mathoverflow.net/questions/131477 | 0 | Is there a connections between the number of vertices and the number of lattice points of $P\_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine?
Or if I have a bound on the number of vertices, can I also bound the number of lattice points?
Is counting the number of vertices or latti... | https://mathoverflow.net/users/34261 | Connection between the number of vertices and the number of lattice points of the integer hull of a polytope? | In response to your first question the number of vertices does not control the number of lattice points. Consider the polytopes $P=conv[(1,1),(1,-1),(-1,-1),(-1,1)]$, and $Q=conv[(0,1),(2,-1),(-2,-1)]$. They have the same number of lattice points but different number of vertices.
In regard to your second question it... | 1 | https://mathoverflow.net/users/19642 | 131482 | 72,860 |
https://mathoverflow.net/questions/66661 | 5 | I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, such as locally finite covers and subordinate partitions of unity, a tubular neighborhood theorem, smoothing of submanifol... | https://mathoverflow.net/users/3928 | Basic results in bounded geometry | Many results, in particular about Sobolev spaces, for Riemannian manifolds with bounded geometry, are in:
* J. Eichhorn.
Global Analysis on Open Manifolds.
Nova Science Publishers Inc., New York, 2007.
* H. Triebel.
Theory of Function Spaces. II, Volume 84 of Monographs
in Mathematics.
Birkhauser Verlag, Basel, 1992.... | 2 | https://mathoverflow.net/users/26935 | 131496 | 72,865 |
https://mathoverflow.net/questions/131499 | 1 | I know that exist a Lie Group Called the Orthogonal Group $O(n)$.
That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for $\mathbb{R}^n$. Is posible to construct a "General Orthogonal Group" over a field $k$ of characteristic zero?
It won't be a Lie Group, but ... | https://mathoverflow.net/users/31524 | General Orthogonal Group and its properties | Yes, the orthogonal group makes sense over any field $k$. It is an linear algebraic group.
In fact the theory of linear algebraic groups generalizes that of linear Lie groups over the real or complex numbers to give something that makes sense over an arbitrary field $k$.
| 4 | https://mathoverflow.net/users/32332 | 131503 | 72,868 |
https://mathoverflow.net/questions/131468 | 0 | In the [8 queen puzzle](http://en.wikipedia.org/wiki/Eight_queens_puzzle), if we use the incremental approach, i.e. put the queen one by one on the board, the number of possible sequences would be 2057. How is that number calculated?
(This number is taken from the book AI by Peter Norvig)
| https://mathoverflow.net/users/32768 | 8 queens puzzle | This is an exercise in recursion and programming, so in THIS particular instance,
id say use bruteforce recursion, (for example the mathematica code below).
However, for $n=8$, there are only 92 solutions, without removing symmetric solutions.
Thus, there must be something strange where you see this number.
Your numb... | 0 | https://mathoverflow.net/users/1056 | 131504 | 72,869 |
https://mathoverflow.net/questions/82331 | 14 | Let $V\_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V\_n$.
What is known about $V\_n$? Is there a known general formula? If not, then what are the known best bounds for $V\_n$?
| https://mathoverflow.net/users/9550 | Smallest containing simplex | The problem seems to be still open even for $n=3$:
*Weisstein, Eric W. ["Tetrahedron Circumscribing."](http://mathworld.wolfram.com/TetrahedronCircumscribing.html)*
| 14 | https://mathoverflow.net/users/34258 | 131505 | 72,870 |
https://mathoverflow.net/questions/131490 | 7 | In his 'Märchen' Langlands considers for a local field $F$ a certain abelian category $\Pi(F)$ whose objects are given by isomorphisms classes of irreducible admissible representations of $GL\_n(F)$, where $n \in \mathbf{N}$ runs over all natural numbers. For $[\pi],[\pi']$ represented by cuspidal reps $\pi,\pi'$ of $G... | https://mathoverflow.net/users/3824 | Langlands product | Basically, no. Marc Palm's answer addresses L-functions, but that is a long long way from determining the irrep -- you'd need L-functions and epsilon-factors of twists, plus an impressive ability to translate such information into a construction of a smooth irrep (if you want more than existence).
To me, your questio... | 7 | https://mathoverflow.net/users/3545 | 131508 | 72,871 |
https://mathoverflow.net/questions/131484 | 4 | A partition of $n$ is a weakly decreasing sequence of natural numbers $\lambda = (\lambda\_1, \lambda\_2, \dots)$ such that $\sum \lambda\_i = n$. Its length $l(\lambda)$ is the number of positive summands $\lambda\_i$.
In exercise 3.19 of his book "The $q, t$-Catalan Numbers and the Space of Diagonal Harmonics" Jame... | https://mathoverflow.net/users/34263 | Why are the dinv-statistic and the partition length equidistributed? | This formula appears in Exercise 1.103 of *Enumerative Combinatorics*, vol. 1, second ed. It was first proved by K. Liu, C. H. F. Yan, and J. Zhou, *Sci. China, Ser. A* **45** (2002), 420-431. A combinatorial proof was given by G. Warrington, *J. Combinatorial Theory Ser. A* **116** (2009), 379-403.
| 6 | https://mathoverflow.net/users/2807 | 131516 | 72,875 |
https://mathoverflow.net/questions/131509 | 0 | What is the Bahadur-Anderson Algorithm, and which book could one read to learn it?
| https://mathoverflow.net/users/34267 | What is the Bahadur-Anderson Algorithm? | M. I. Schlesinger,Václav Hlavác̆, Ten lectures on statistical and structural pattern recognition, Springer, 2002.
| -1 | https://mathoverflow.net/users/18814 | 131517 | 72,876 |
https://mathoverflow.net/questions/121444 | 2 | I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, there exist a minimal log semistable model?
Thanks.
| https://mathoverflow.net/users/17495 | Minimal semistable model for K3-surfaces. | The answer is yes when p>3. Look at Kawamata's paper
Semistable minimal models of threefolds in positive or mixed characteristic.
J. Algebraic Geom. 3 (1994), no. 3, 463–491.
and a correction in
Index 1 covers of log terminal surface singularities.
J. Algebraic Geom. 8 (1999), no. 3, 519–527.
| 3 | https://mathoverflow.net/users/10083 | 131525 | 72,880 |
https://mathoverflow.net/questions/131514 | 7 | EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with no conjectural answer. END OF EDIT
EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^... | https://mathoverflow.net/users/33828 | Growth of Thompson's group $F$ | These questions have been studied (perhaps except the third one). See Section 5.8.7 in [my book](http://www.math.vanderbilt.edu/~msapir/book/book11513.pdf) and the references there.
| 11 | https://mathoverflow.net/users/nan | 131529 | 72,881 |
https://mathoverflow.net/questions/131520 | 16 | Is the following statement true?
>
> For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points on its surface.
>
>
>
| https://mathoverflow.net/users/33698 | Integer lattice points on a hypersphere | There are several related and very interesting problems and theorems:
* [Schinzel's theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) - solves the problem in $\mathbb{R}^2$ using so-called [Schinzel circles](http://mathworld.wolfram.com/SchinzelCircle.html). It seems intuitively clear that it generalizes ... | 17 | https://mathoverflow.net/users/33829 | 131533 | 72,883 |
https://mathoverflow.net/questions/71681 | 14 | Are there any interesting examples of semisimple algebras in nonsemisimple categories which don't "come from" a semisimple algebra in a semisimple category? That is, if you want to study semisimple algebra objects can you assume wlog that the underlying category is semisimple?
Here's one way of trying to make this qu... | https://mathoverflow.net/users/22 | Are there interesting semisimple algebras in non-semisimple categories? | If $H$ is a finite dimensional Hopf algebra and $\mathcal C=\mathcal M^H$ is the category of corepresentation then $H$ is an algebra in $\mathcal C$ and $\mathcal{C}\_H=\mathcal M\_H^H= Vec$, where the last equivalence follows by the fundamental theorem of Hopf modules. Then if $H$ is not semisimple (for example the Ta... | 3 | https://mathoverflow.net/users/6517 | 131535 | 72,885 |
https://mathoverflow.net/questions/130456 | 4 |
>
> Let $(\Omega,\Sigma)$ be a measurable space and $K$ be a compact
> metrizable space endowed with its Borel $\sigma$-algebra
> $\mathcal{B}(K)$. Let $A\subseteq\Omega\times K$ be universally
> measurable and such that $$C\_\omega=\{x\in K:(\omega,x)\in A\}$$ is
> closed for all $\omega\in\Omega$. Let $\Sigma\_... | https://mathoverflow.net/users/35357 | From universal measurability to measurability | I found a solution that suffices for what I do. It is based on strengthening the assumption that the graph $A$ is universally measurable to it being analytic. The notion of analyticity being used is that a subset $S$ of a measurable space $(M,\mathcal{M})$ is analytic if there is a compact metric space $T$ with Borel $... | 3 | https://mathoverflow.net/users/35357 | 131539 | 72,887 |
https://mathoverflow.net/questions/131511 | 24 | Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$, i.e. $E=\{n \mapsto n, n \mapsto n^n, n \mapsto n^{n^n}, n \mapsto (n^n)^n, n \mapsto (n^n)^{n^n},\ \dots\}$. Let $E$ ... | https://mathoverflow.net/users/34258 | Order type of the smallest set containing the identity function and closed under exponentiation | As Joel showed, the set $E$ is well-ordered with order type no more than the Cantor ordinal $\epsilon\_0$. In fact, its order type is exactly $\epsilon\_0$. This can be proved by constructing the order isomorphism between $\epsilon\_0$ and $E$.
First, note that if $F,G\in E$ are of the form $F(n)=n^{n^{f(n)}}$, $G(n)... | 18 | https://mathoverflow.net/users/1004 | 131542 | 72,889 |
https://mathoverflow.net/questions/131547 | 2 | In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surjection from $B$ to $A$, or $A=\emptyset$. In topos theory people consider the relation '$A$ is a subquotient of $B$' (usually in the non-bool... | https://mathoverflow.net/users/4177 | Subquotients in ZF | In $\sf ZF$ injections can be split, so if $A\leq B$ then we have $A\leq^\ast B$ as well. For this reason I prefer to use a slightly modified (but equivalent) definition for $\leq^\ast$:
>
> $A\leq^\ast B$ if there is some $C\subseteq B$ such that there is a surjection from $C$ onto $A$.
>
>
>
This is dual to ... | 3 | https://mathoverflow.net/users/7206 | 131553 | 72,895 |
https://mathoverflow.net/questions/131440 | 0 | How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x\_1,...,x\_n)\in C^n$ ?
suppose that $\lVert Ax \rVert^2= x^T(A^TA)x,\ \lVert\Sigma\rVert\_p$ denotes the spectral (operator) norm of a matrix $\Sigma$
Pro... | https://mathoverflow.net/users/34221 | What is the corresponding version in the complex space of this proposition got in the real space real | The answer should be
$$\text{Pr}\left\{\lVert Ax \rVert^2 > \text{tr}(\Sigma) +\sqrt{2\text{tr}(\Sigma^2)t}+\lVert\Sigma\rVert\_p t \right\}\leq\exp(-t).
$$
| 0 | https://mathoverflow.net/users/34221 | 131556 | 72,897 |
https://mathoverflow.net/questions/131528 | 59 | Recently I posted [a conjecture at Math.SE](https://math.stackexchange.com/questions/395818/closed-form-for-int-0-infty-ln-fracj-mux2y-mux2j-nux2y-nux):
$$\int\_0^\infty\ln\frac{J\_\mu(x)^2+Y\_\mu(x)^2}{J\_\nu(x)^2+Y\_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J\_\mu(x)$ and $Y\_\mu(x)$ are ... | https://mathoverflow.net/users/9550 | How closed-form conjectures are made? | Part of what makes this question subtle is that what's intuitive depends on your background knowledge. In particular, the question of what counts as an "explicit tiny quantity" is hard to pin down. For example, Bailey, Borwein, and Borwein pointed out the example
$$
\left(\frac{1}{10^5} \sum\_{n=-\infty}^\infty e^{-n^2... | 60 | https://mathoverflow.net/users/4720 | 131557 | 72,898 |
https://mathoverflow.net/questions/97711 | 3 | I am interested in finding a canonical general expression for the area of a spherical polygon in $\mathbb{S}^2$ knowing the side lengths of the polygon and a bound on the internal angles (we can assume a radius of $R=1$).
For what I am researching (I will not go into the background) I need the following conditions to... | https://mathoverflow.net/users/20343 | The Area of Spherical Polygons | This excellent paper collects many useful formulae for spherical calculations, including (but hardly limited to) polygon area. Explanations are clear and well developed.
Some Algorithms for Polygons on a Sphere
Robert.G.Chamberlain
William.H.Duquette
Jet Propulsion Laboratory
<http://hdl.handle.net/2014/40409>
... | 3 | https://mathoverflow.net/users/34284 | 131564 | 72,902 |
https://mathoverflow.net/questions/131527 | 17 | Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b\_{1} & 0 & ... & 0 \\\
b\_{1} & a & b\_{2} & & ... \\\
0 & b\_{2} & a & ... & 0 \\\
... & & ... & & b\_{n-1} \\\
0 & ... & 0 & b\_{n-1} & a
\end{pmatrix} $
All of the entries can be taken to be positive real numbers and all of the $a\_{i}$ ... | https://mathoverflow.net/users/34275 | Eigenvalues of Symmetric Tridiagonal Matrices | The type of matrix you have written down is called Jacobi matrix and people are still discovering new things about them basically their properties fill entire bookcases at a mathematics library. One of the reasons is the connection to orthogonal polynomials. Basically, if $\{p\_n(x)\}\_{n\geq 0}$ is a family of orthogo... | 27 | https://mathoverflow.net/users/3983 | 131568 | 72,904 |
https://mathoverflow.net/questions/102503 | 7 | Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume that $\pi\_n(X/G)\cong \pi\_n(Y/G)$ for all $n\geq 0$ and are induced by the cellular inclusion $Y/G\hookrightarrow X/G$.
... | https://mathoverflow.net/users/12218 | G-equivariant Whitehead's Theorem | The usual statement is that if $X\to Y$ is an equivariant map of $G$-CW complexes and if for every closed subgroup $H$ the induced map of fixed-point spaces $X^H\to Y^H$ is a homotopy equivalence then in fact the map has an inverse up to equivariant homotopy.
This is part of the following picture: The category of $G... | 10 | https://mathoverflow.net/users/6666 | 131590 | 72,914 |
https://mathoverflow.net/questions/131583 | 25 | My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any **algorithm**) that we can use to rewrite it using language of differential forms, tensor, exterior calculus, Hodge star and other... | https://mathoverflow.net/users/2391 | Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc? | This is rather standard, though, unfortunately this knowledge is confined to a rather narrow segment of mathematicians working on PDEs. The main reason is that, most of the time, this is not necessary for practical problems. In fact there are multiple ways converting a PDE into invariant form.
In mathematical physics... | 36 | https://mathoverflow.net/users/2622 | 131591 | 72,915 |
https://mathoverflow.net/questions/131538 | 10 | I've read [@TauMu's question](https://mathoverflow.net/questions/131511/order-type-of-the-smallest-set-containing-the-identity-function-and-closed-under) about the set of functions $\mathbb N\rightarrow\mathbb N$ generated from the identity map by repeatedly applying exponentiation of two already accepted funct... | https://mathoverflow.net/users/8385 | Do operations generate well-ordered sets only? | If we further require that the operation is nondecreasing in all arguments, the answer is negative. In fact, something more general holds:
**Theorem:** Let $L$ be a finite set of operations $f\colon\mathbb N^k\to\mathbb N$ (where the arity $k$ is finite, but not necessarily the same for all operations). Assume that e... | 10 | https://mathoverflow.net/users/12705 | 131596 | 72,918 |
https://mathoverflow.net/questions/131570 | 3 | Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime.
Can we say that $(p^2+1)/2$ is not equal to the square of a prime number?
Many thanks for your help
BHZ
| https://mathoverflow.net/users/31045 | can we say that $(p^2+1)/2\ne p_0^2$ where $p$ is a Mersenne prime | Suppose, $p^2-2p\_1^2=-1.$ Substituting $p=2^a-1,$ we arrive at
$$2^a(2^{a-1}-1)=(p\_1-1)(p\_1+1).$$ Observe, $(p\_1-1,p\_1+1)=2,$ so we must have the following options: $p\_1-1=2^{a-1}k$ and $p\_1+1=2l$ and $kl=2^{a-1}-1.$ This is impossible unless $k,l$ and thus $a$ are small. Indeed, if $k\ge 2,$ then $p\_1\ge 2^a+... | 4 | https://mathoverflow.net/users/17503 | 131611 | 72,926 |
https://mathoverflow.net/questions/131602 | 4 | There is an elementary statement that I *believe* I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I can find a proof of it.
The statement is about the prime-counting function $\psi(x) = \sum\_{ p^n < x } \log p$. I... | https://mathoverflow.net/users/9317 | Estimate on the prime-counting function $\psi(x)$. | The reference for this (given in the 3rd edition of Davenport's *Multiplicative Number Theory*) is:
Grosswald, Émile. "Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-\ell i(x)$." (French)
C. R. Acad. Sci. Paris **260** (1965), 3813–3816.
It seems (according to the Math Review) that for $\alpha$ fix... | 6 | https://mathoverflow.net/users/3659 | 131612 | 72,927 |
https://mathoverflow.net/questions/131604 | 7 | Let $f = \sum\_n a\_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma\_0(N)$ for some $N$. Let $K\_f$ be the number field generated by the $a\_q$ as $q$ runs over all primes.
My question: if we consider the field generated by all but one of these $a\_q$, can this field be smaller than $K\_f$?
(edited based on ... | https://mathoverflow.net/users/27315 | Field generated by the Fourier coefficients of a modular form | No, strong multiplicity one says that all-but-finitely-many of the $a\_p$'s determine all the others.
Edit in response to comment/query, and further in response to subsequent comments: ... and, once all the other coefficients are determined by strong multiplicity one (for newforms), invoke Shimura's results (arguably... | 12 | https://mathoverflow.net/users/15629 | 131615 | 72,928 |
https://mathoverflow.net/questions/131603 | 2 | Assume we have a homomorphism $\phi: C(S^{1},M\_{n}(\mathbb{C}))\rightarrow C(S^{1},M\_{m}(\mathbb{C}))$ where $n$ divides $m$. Under what conditions does $\phi$ send constant functions to constant functions?
| https://mathoverflow.net/users/33724 | Homomorphisms preserving constant functions | This is perhaps just a partial answer; but I feel that Will's answer is incomplete, and this is a more interesting question that he is suggesting.
Given a \*-homomorphism $\phi$, composing with point-evaluation at any point $x \in S^1$ gives us a representation
$$\phi\_x: C(S^1, M\_n) \to M\_m, $$
which is therefore ... | 0 | https://mathoverflow.net/users/22052 | 131620 | 72,930 |
https://mathoverflow.net/questions/131619 | 4 | I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations.
`http://en.wikipedia.org/wiki/Ostrowski's_theorem`
Wikipedia refers to Koblitz (p-adic Numbers, p-adic Analysis, and Zeta-Functions), but I ca... | https://mathoverflow.net/users/33757 | Reference for Ostrowski's 1916 Theorem? | Barry Cipra is right. See <http://link.springer.com/article/10.1007/BF02422947>
I've changed the wikipedia's entry.
| 4 | https://mathoverflow.net/users/34310 | 131623 | 72,933 |
https://mathoverflow.net/questions/131618 | 29 | If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular
$$ (a-b)^2 \geq 0. \qquad (1)$$
Combining this fact with the identity
$$ ab + ba + (a-b)^2 = a^2 + b^2 \qquad (2)$$
we obtain the inequality
$$ ab + ba \leq a^2 + b^2 \qquad (3)$$
whic... | https://mathoverflow.net/users/766 | Can the fact that the square of an integer is a natural number be categorified? | Probably the most natural thing to ask for Theorem 1 is as follows. Let $\mathcal{A}$ be the category whose objects are pairs of finite sets, and whose morphisms are pairs of bijections. Let $\mathcal{B}$ be the category of finite sets and functions. (Allowing more morphisms in $\mathcal{A}$ or fewer morphisms in $\mat... | 18 | https://mathoverflow.net/users/10366 | 131627 | 72,935 |
https://mathoverflow.net/questions/131625 | 2 | Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?
| https://mathoverflow.net/users/34312 | Determining A valid Convex Hexagon given The length of Six sides | A chain of edges can close iff the longest edge is not longer than the sum of the lengths of all the other edges. This is Theorem 8.6.3 (p.326) in [*Computational Geometry in C*](http://cs.smith.edu/~jorourke/books/compgeom.html) and Theorem 5.1.2 (p.61) in [*Geometric Folding Algorithms*](http://gfalop.org/). You can ... | 6 | https://mathoverflow.net/users/6094 | 131629 | 72,936 |
https://mathoverflow.net/questions/131531 | 9 | Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: \mathrm{Ho}(\mathcal{C}) \leftrightarrows \mathrm{Ho}(\mathcal{D}) :\mathbb{R}G
$$
It is then a well-known fact that $\mathbb{L}F$ pr... | https://mathoverflow.net/users/1797 | Homotopy left-exactness of a left derived functor | I do not know the answer for a general Quillen adjunction, but I will attempt to give a complete answer in the case you're interested in, when the adjunction $(F,G)$ is of the form $(f\_!,f^\ast)$ for $f\colon C\to D$ a continuous functor between sites:
**Claim.** The functor $\mathbb{L}f\_!$ preserves finite homotop... | 10 | https://mathoverflow.net/users/20233 | 131633 | 72,938 |
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