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https://mathoverflow.net/questions/138351 | 9 | I am writing an academic paper for submission to a journal. One of my co-authors wrote the following:
>
> **Theorem** Statement of the theorem
>
>
> **Proof of theorem** We first show the following result
>
>
> **Lemma** Statement of lemma used to prove the theorem
>
>
>
However, I think that it is more na... | https://mathoverflow.net/users/38053 | What are good ways to present proofs of theorems requiring auxiliary lemmas? | The guiding principle should be: think of your readers. Nesting results inside the proofs of other results can make things confusing. I am looking right now at a paper that has three instances of that, and one of them is truly egregious: the bracket diagram is like this: $((((()))(())))$.
Maybe one nested lemma is ok... | 6 | https://mathoverflow.net/users/703 | 138354 | 75,833 |
https://mathoverflow.net/questions/138339 | 3 | Thanks to responses in a previous question I asked, I was able to show that if $p$ and $q$ are two distinct polynomials with equal degree greater than 2 and coefficients in $\mathbb{N}$ that the number of integer solutions to $p(x)=q(y)$ is finite.
This is enough for a part of the research problem I am working on, bu... | https://mathoverflow.net/users/37316 | Bounds on solutions to Diophantine equations of the form p(x)=q(y) | Tengely's thesis is a good place to learn the state of the art for bounding these solutions in the irreducible case. All the main methods appear there. In the reducible case there aren't any known bounds, but there are Galois-theoretic results about reducibility of $p(x)-q(y)$ (for instance Theorem 8.1 in the Bilu-Tich... | 3 | https://mathoverflow.net/users/30412 | 138358 | 75,837 |
https://mathoverflow.net/questions/130155 | 12 | I have been trying to hunt down the original reference for the construction of the projective plane of order $4$ from the complete graph on $6$ vertices.
The reference I have at hand are [Cameron and van Lint](http://books.google.ie/books/about/Designs_Graphs_Codes_and_Their_Links.html?id=j1CZeuHI7q0C&redir_esc=y) (C... | https://mathoverflow.net/users/262 | Who constructed the projective plane of order $4$ from $K_6$? | All references converge on W.L. Edge, Some implications of the geometry of the 21-point plane, Math.Zeitschr. 87, 348--362 (1965)
Available at <http://www.maths.ed.ac.uk/cheltsov/edge2013/pdf/1965a.pdf>
Section 6 lays out all the necessary ideas.
| 6 | https://mathoverflow.net/users/262 | 138362 | 75,838 |
https://mathoverflow.net/questions/138361 | 5 | I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", <http://arxiv.org/pdf/math/0311202v1.pdf>, and I have a question regarding the nature of the function field for $X0(N)$: if I have a modular function for $Γ\_0(N)$ that has a rational $q$-expansion at $\infty$, does it foll... | https://mathoverflow.net/users/38058 | Modular Functions with Rational Fourier Expansions | The statement about being in ${\bf Q}(j,j\_N)$ if the $q$-expansion is rational is true. See Prop 12.7 (a) in Cox's book "Primes of the form $x^2 + Ny^2$". Moreover, part (b) of that proposition tells you that you can specialize when the partial derivative $\partial \Phi\_N(x,y)/ \partial(x)$ does not vanish at that po... | 6 | https://mathoverflow.net/users/2698 | 138364 | 75,840 |
https://mathoverflow.net/questions/138345 | 4 | *Are there non-trivial diffeomorphisms (i.e., different from isometries) of the complex projective plane that map geodesics (for the canonical Riemannian metric) to geodesics?*
Same question for all other rank one symmetric spaces different from spheres and real projective spaces.
| https://mathoverflow.net/users/21123 | Geodesic transformations of the complex projective plane | The answer is no. The explanation of Anton is of course correct but there exist stronder statements in the literature: for example by Sinjukov (Dokl. Akad. Nauk SSSR (N.S.) 98, (1954) 21--23) any symmetric space is locally \emph{geodesically rigid} is the sense that any
other metric having the same (unparameterized) g... | 5 | https://mathoverflow.net/users/14515 | 138370 | 75,843 |
https://mathoverflow.net/questions/138284 | 9 | ZFC-, which is ZFC minus power set, is modelled by $ L\_{\delta}$ where $\delta$ is an admissible ordinal larger than
any least $\Sigma\_{n}$-admissible ordinal for n a natural number. Can some provide more information? Does it have a name? What is its most natural omega sequence?
| https://mathoverflow.net/users/37385 | A question on an ordinal for ZFC- | The least such ordinal $\beta$, often written $\beta\_0$, for which $L\_\beta$ is a $ZF^-$ is also characterised as the "ordinal of ramified analysis". This is because the ramified analytical hierarchy, which builds up cumulative second order number theoretic structures of the form $$\underline{P\_\alpha}=( P\_\alpha,\... | 14 | https://mathoverflow.net/users/6942 | 138372 | 75,844 |
https://mathoverflow.net/questions/138374 | 17 | Let $G$ be a finite group acting linearly on a finite dimensional vector space $V$ over a finite field. By Burnside's lemma,
$$
|V/G| = \frac 1{|G|} \sum\_{g\in G} q^{\dim(ker(g - I))}.
$$
Since $g-I$ and its dual map $g^\*-I$ have kernels of the same dimension, it follows that $|V/G|=|V^\*/G|$.
The above argument sh... | https://mathoverflow.net/users/9672 | Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action? | Yes. In particular, it can happen that $V$ has non-zero fixed points, but $V^\*$ doesn't.
For example, let $G$ be the symmetric group of degree 3 acting in the obvious way on the set $\{e\_1,e\_2,e\_3\}$, and let $W$ be the corresponding permutation module over the field of 3 elements. Let $V$ be the submodule spanne... | 21 | https://mathoverflow.net/users/22989 | 138378 | 75,845 |
https://mathoverflow.net/questions/138258 | 4 | This is a question I asked on math.stackexchange without success.
Let $p$ be a prime number and denote by $\mathbb{F}\_p(1)$ the one dimensional vector space over $\mathbb{F}\_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}\_p / \mathbb{Q}\_p)$ via the mod $p$ cyclotomic character.
We have a nice description ... | https://mathoverflow.net/users/38010 | Generalization of Kummer isomorphism? | It depends what you mean by "nice". There is, to the best of my knowledge, no such easy explicit description of $H^1(G,F\_p(n))$ if $n \neq 0,1$. If $n=1$, it's Kummer theory as you have described. If $n=0$, the $H^1$ is the group of homomorphisms of $G$ into $F\_p$, so it boils down to local class field theory.
In g... | 6 | https://mathoverflow.net/users/5743 | 138381 | 75,847 |
https://mathoverflow.net/questions/138387 | 5 | I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns..." After stating the following proposition, they make the statement: " a Koszul ring is a positively graded ring that i... | https://mathoverflow.net/users/38075 | "as close to being semisimple as it can possibly be." | In the semisimple case it is really easy to calculate $ext\_A^i(M,N)$, $i\ge 1$, with the above assumptions. It is zero. For Koszul rings this is almost
true, i.e., $ext^i(M,N)$ is concentrated in degree $i$ (that is, if $S$ is the direct sum of all simples, then the two natural gradings on the algebra $A^{!}=ext^{\bul... | 4 | https://mathoverflow.net/users/32332 | 138389 | 75,851 |
https://mathoverflow.net/questions/138106 | 33 | Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x\_1, \dots, x\_n]$,
decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective,
respectively, injective? --
And what is the answer if $\mathbb{Q}$ is replaced by $\mathbb{Z}$?
The motivation for this question is Jonas Meyer's c... | https://mathoverflow.net/users/28104 | Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable? | We treat all four problems in turn. In all that follows $n>1$.
Surjectivity over $\mathbb{Q}$:
If there is an algorithm to test whether an arbitrary polynomial with rational coefficients is surjective as a map from $\mathbb{Q}^n$ into $\mathbb{Q}$ then Hilbert's Tenth Problem for $\mathbb{Q}$ is effectively decidab... | 28 | https://mathoverflow.net/users/5229 | 138410 | 75,860 |
https://mathoverflow.net/questions/138377 | 7 | Take the equation $y^{d}=\Pi\_{1}^{n}(x-t\_{i})^{m\_{i}}$ over $\mathbb{C}$. This affine equation gives a cyclic cover of $\mathbb{P}^{1}$. Now it is usually said without explanation that if the sum $\sum m\_{i}$ is not divisible by $d$, then the family is ramified over $\infty$. My question is how to see explicitly--i... | https://mathoverflow.net/users/37808 | How to explicitly see the ramification over infinity | One way to see this is the following. Let $U$ be the complement of the points $t\_1,\ldots,t\_n$ in $\mathbb{P}^1(\mathbb{C})$. The fundamental group $\pi\_1(U)$ is generated by elements $\gamma\_1,\ldots,\gamma\_n,\gamma\_\infty$ with relation
$\gamma\_\infty^{-1} = \gamma\_1\cdots \gamma\_n$. Consider the character ... | 2 | https://mathoverflow.net/users/37622 | 138411 | 75,861 |
https://mathoverflow.net/questions/138416 | 4 | Suppose I have two lists $\alpha\_1,\dots,\alpha\_k$ and $\beta\_1,\dots,\beta\_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,\dots,k\}$, let $\phi\_k$ be the affine linear map $x \mapsto \alpha\_i x + \beta\_i$.
Let $I$ be the image of th... | https://mathoverflow.net/users/35981 | Algebraically Independent Numbers and Affine Linear Maps | The sequence of maps $\phi\_{i\_1}^{e\_1},\dots,\phi\_{i\_{\ell}}^{e\_{\ell}}$ is not uniquely determined by the image set $\phi\_{i\_{\ell}}^{e\_{\ell}}\circ\dots\circ\phi\_{i\_1}^{e\_1}(\mathbb{Z})$. One reason for this is that all commutators $\phi\_i\phi\_j\phi\_i^{-1}\phi\_j^{-1}$ are translations $x\mapsto x+b$, ... | 5 | https://mathoverflow.net/users/30412 | 138427 | 75,863 |
https://mathoverflow.net/questions/138448 | 2 | I have a set of lattice points S in R^n (listed in memory in a computer for n=8 say). I want to computationally certify that they do *not* form the lattice points of a convex polytope P in R^n. (Ex. S={-1,1} in R^1.) Is there an easy (and hopefully efficient enough) way to do this?
Remarks:
* I don't have much fee... | https://mathoverflow.net/users/38119 | Proving non-convexity of a set of lattice points | You can compute the convex hull $P$ of these points, and then apply, say, [Barvinok's algorithm](http://math.uchicago.edu/~shmuel/AAT-readings/Combinatorial%20Geometry,%20Concentration,%20Real%20Algebraic%20Geometry/barvinok%20MSRI.pdf "An Algorithmic Theory of Lattice Points in Polyhedra") for counting $|P\cap\mathbb{... | 2 | https://mathoverflow.net/users/11100 | 138450 | 75,872 |
https://mathoverflow.net/questions/138454 | 28 | Let $(X,d)$ be a compact connected metric space with the property that for any distinct points $a,b$, $X\backslash \lbrace a,b\rbrace$ is disconnected. Clearly the unit circle has this property. Is there any other example (up to isomorphism) ??
| https://mathoverflow.net/users/nan | A property of the unit circle | This property indeed characterizes the circle, but this is not obvious.
This was shown by R. L. Moore, according to Sam Nadler's *Continuum Theory* p. 156.
Added: the precise reference is [522] in [this historical survey](http://web.mst.edu/~wjcharat/JJC/publications/p138.pdf) of continuum theory.
| 23 | https://mathoverflow.net/users/6451 | 138458 | 75,875 |
https://mathoverflow.net/questions/138457 | 4 | If ZF has a standard model there is a least ordinal $\sigma$ such that $L\_{\sigma}$ is a model of ZFC. What is $\sigma$ called?
| https://mathoverflow.net/users/37385 | A question on the minimal ordinal for ZFC | It does not have any particular name or notation: it is just ''the height of the minimal model of set theory''
| 7 | https://mathoverflow.net/users/6942 | 138461 | 75,877 |
https://mathoverflow.net/questions/138456 | 5 | Kripke Platek set theory has collection instead of replacement, and it is a weakening of KP if one has replacement instead of collection. Call KP minus collection plus replacement KF for *Kripke Fraenkel*. Is KF weaker than KP in that some transfinite recursion can be done by KP which cannot be done by KF? If so, is th... | https://mathoverflow.net/users/37385 | Transfinite recursion, collection and replacement in KP and KF | The main deficit of KF over KP is that one cannot prove that the set of formulae equivalent to a $\Sigma\_1$ or $\Pi\_1$ formula is closed under bounded quantification. (And this persists up through the $KF\_n/KP\_n$ you define.)
Therefore recursions involving schemes defined by a formula of the form $\forall u \in v... | 7 | https://mathoverflow.net/users/6942 | 138465 | 75,880 |
https://mathoverflow.net/questions/138245 | 2 | Let $F\_n$ denote the free group on $n$ generators $x\_1,\ldots , x\_n$.
Recall that an element $\varphi\in\mathrm{Aut}(F\_n)$ is an *IA automorphism* if it induces the identity on the abelianization $F\_n/[F\_n,F\_n]=\mathbb{Z}^n$. In other words, $\varphi$ is in the kernel of the induced map $\mathrm{Aut}(F\_n)\to ... | https://mathoverflow.net/users/8103 | Fixed points of IA automorphisms | Here's a compilation of some of the answers given in comments.
As pointed out by Mark Sapir and Misha, if $S$ is a compact surface with one boundary component whose fundamental group is equipped with an isomorphism $\alpha : \pi\_1(S) \to F\_n$, and if $f : S \to S$ is any pseudo-Anosov homeomorphism, then the outer ... | 3 | https://mathoverflow.net/users/20787 | 138468 | 75,882 |
https://mathoverflow.net/questions/138435 | 13 | **Short version**: what can we say about the place of idempotent ultrafilters in the Rudin-Keisler ordering?
**Longer version**:
If $U$, $V$ are (nonprincipal) ultrafilters on $\omega$, then we write $U\ge\_{RK}V$ in case there is some function $f:\omega\rightarrow\omega$ such that $$ \forall X\subseteq\omega,\quad... | https://mathoverflow.net/users/8133 | Idempotent ultrafilters and the Rudin-Keisler ordering | The idempotent ultrafilters closest to being Ramsey are the stable ordered-union ultrafilters. These are officially defined as certain ultrafilters on the set $\mathbb F$ of finite subsets of $\omega$, but they can be transferred to $\omega$ via the "binary expansion" map $\mathbb F\to\omega: s\mapsto\sum\_{n\in s}2^n$... | 12 | https://mathoverflow.net/users/6794 | 138477 | 75,885 |
https://mathoverflow.net/questions/138478 | 11 | Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is there any characterization for idempotent elements in $R[x]$ ?
| https://mathoverflow.net/users/nan | Idempotent polynomials | Let $f = a\_0 + a\_1x + ... + a\_nx^n$ be idempotent. Then $a\_0^2 = a\_0$. Also
$a\_0a\_1 + a\_1a\_0 = a\_1$. Multiply by $a\_0$ to get $a\_0a\_1 = 0$ which
means that $a\_1 = 0$ and by induction it is easy to show that $a\_2 = ... = a\_n = 0$
Therefore **$f$ is idempotent iff its constant term is idempotent and ot... | 24 | https://mathoverflow.net/users/nan | 138479 | 75,886 |
https://mathoverflow.net/questions/138471 | 4 | Background of my reference request is an observation that I made, while I was still in school: there are two ways to calculate $x\*999$: either do it directly, by applying the multiplication algorithm that is taught in school or, calculate it as $x\*1000-x\*1$ of which the second way is much easier.
But, I had no cl... | https://mathoverflow.net/users/31310 | Reference Request: Representing Positive Integers as Differences with Minimal Hamming Weight | Subtraction will produce an improvement in total weight if and only if there is an interval of $k$ bits where then number $r$ of ones satisfies $2r > k+2$.
*Proof:* We may assume the interval is preceded by a zero. Subtraction will transform it into the difference between the length $k+1$ string $10\cdots0$ (with 1 o... | 4 | https://mathoverflow.net/users/121 | 138493 | 75,892 |
https://mathoverflow.net/questions/138494 | 6 | I'm new to computational geometry and advanced mathematics in general here so bear with me. I've spent a decent amount of time attempting to figure out this problem and I just can't find a solution.
My problem is to find the vertices that make up each face of a convex polyhedron. At my disposal are a set of planes, w... | https://mathoverflow.net/users/38144 | Finding the vertices of a convex polyhedron from a set of planes | For the description of a pretty good algorithm see [Avis and Fukuda.](http://cgm.cs.mcgill.ca/~avis/doc/avis/AF92b.pdf) For an efficient implementation (in any dimension), and much additional discussion and references, see [Komei Fukuda's page.](http://www.inf.ethz.ch/personal/fukudak/soft/soft.html)
| 5 | https://mathoverflow.net/users/11142 | 138497 | 75,893 |
https://mathoverflow.net/questions/138273 | 1 | As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and
NOT on existence/uniqueness etc. which is usually the focus of most theoretical PDE texts.
NOT on numerical methods to solve the PDE, which is usually the focus of engineering/p... | https://mathoverflow.net/users/30684 | Reference request: Spectral analysis of advection diffusion PDE | I am not sure whether there has been a comprehensive study of spectral properties of such equations, but you may be interested in the paper ``Diffusion and Mixing in Fluid Flow'' by Constantin, Kiselev, Ryzhik and Zlatos and the references therein.
<http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.6407>
... | 1 | https://mathoverflow.net/users/7193 | 138500 | 75,895 |
https://mathoverflow.net/questions/138475 | 5 | Math people:
A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form
$a\_{i,j} = \frac{1}{x\_i-y\_j}$, with $x\_i \neq y\_j$ for all $(i, j)$, and the $x\_i$'s and $y\_i$'s belong to a field (<http://en.wikipedia.org/wiki/Cauchy_matrix>). Also it seems to be part of the definition that the $x\_i$'s ... | https://mathoverflow.net/users/31107 | What is known about the spectrum of a Cauchy matrix? | Suppose $x\_i > 0$ and $y\_j -x\_j$, then $c\_{ij} = 1/(x\_i+x\_j)$. These matrices are infinitely divisible, i.e., $[c\_{ij}^r]$ is also positive definite for all $r > 0$.
Spectral properties of Cauchy-like matrices and kernels are studied [here.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.81.3678&rep... | 3 | https://mathoverflow.net/users/8430 | 138502 | 75,896 |
https://mathoverflow.net/questions/138498 | 2 | I am a theoretical physics major student working on string theory. I want to understand the work of Nakajima, "Lectures on Hilbert Schemes of Points on Surfaces" . What kinds of mathematical background does it need? Hartshorne or Griffiths & Harris? More precisely, which chapters are necessary? (I only learned Nakahara... | https://mathoverflow.net/users/25715 | About "Lectures on Hilbert Schemes of Points on Surfaces" of Nakajima | If by a book of Nakahara you mean "Geometry, Topology and Physics", you are going to need alot from Hartshorone, as based on this book, you don't even know what a sheaf or a scheme precisely is. So at least the first 2 chapters of Hartshorne are needed. But I think reading that book of Nakahara, you almost have all the... | 4 | https://mathoverflow.net/users/37808 | 138506 | 75,898 |
https://mathoverflow.net/questions/138473 | 3 | Let $f : X \to Y$ be a finite morphism between compact complex manifolds of the same dimension $n$. We denote by $R\_f \subset X$ the ramification divisor of $f$ and by $J\_f \subset X$ the set of points where $\operatorname{Jac} f = 0$.
It is easy to see that $R\_f = J\_f$ as *sets* ($J\_f \subset R\_f$ is trivial;... | https://mathoverflow.net/users/4054 | Ramification divisor and degenerate locus of jacobian | I don't know it is exactly what you want or not. But a reference not in the context of complex manifolds but smooth schemes is for example "Neron models" by Bosch-Lütkebomert, setion 2.2: If $f:X\rightarrow Y$ is a morphism of smooth schemes, then $f$ is etale at $x\in X$ if and only if the canonical homomorphism $(f^{... | 5 | https://mathoverflow.net/users/37808 | 138522 | 75,901 |
https://mathoverflow.net/questions/138480 | 1 | Let $A$ be a $\*$-algebra, $E,$ and $F$ two $A$-modules, and a map $f:E \to F$ such that
$$
f(ae) = a^\*f(e), ~~~~~~~ a \in A.
$$
This seems to me to be the natural generalisation of a conjugate linear map. However, it seems to be too restrictive. Since for, $b$ also in $A$, we must have
$$
f(abe)=a^\*f(be) = a^\*b^\*f... | https://mathoverflow.net/users/2612 | Conjugate linear maps between $*$-algebra modules | Yes, the problem here is with your definition.
It helps to try to keep yourself in the category of modules and module maps, which have to be linear.
You don't want to work with conjugate-linear maps.
The basic idea is this: given a complex vector space $V$, define the complex conjugate vector space $\overline{V}$ to ... | 2 | https://mathoverflow.net/users/703 | 138529 | 75,904 |
https://mathoverflow.net/questions/138525 | 16 | Related to the questions [mathoverflow.net question No. 137850](https://mathoverflow.net/questions/137850/A-question-about-a-question-about-3-dimensional-convex-bodies) and [mathoverflow.net question No. 39127](https://mathoverflow.net/questions/39127/is-the-sphere-the-only-surface-all-of-whose-projections-are-circles-... | https://mathoverflow.net/users/36904 | (A question about)${}^3$ 3-dimensional convex bodies | A convex body $K\subset\mathbb{R}^n$ all of whose $(n-1)$-dimensional projections have the same $n-1$-content is known as a body of constant brightness, by analogy with bodies of constant width. The theory is very similar to that of bodies of constant width. The surface area measure $dS\_K(\mathbf{x})$ takes the place ... | 16 | https://mathoverflow.net/users/20186 | 138534 | 75,907 |
https://mathoverflow.net/questions/138535 | 14 | I know the article of Hamilton on the inverse function theorem of Nash and Moser (with the same title) where he proves that $C^\infty(M)$ is a tame Fréchet space, when $M$ is closed or compact with boundary.
However, as far as I see no word is lost on the case of non-compact $M$. In particular, what if $M$ is an open... | https://mathoverflow.net/users/16702 | Are smooth functions tame? | The idea behind a tame Frechet space is that you have smoothing operators with controlled estimates by which you can counteract loss of derivatives in a nonlinear operator so that a Banach fixed point argument works. You look at a formal version of this. Of course you can have smoothing operators on noncompact manifold... | 12 | https://mathoverflow.net/users/26935 | 138536 | 75,908 |
https://mathoverflow.net/questions/86959 | 7 | Consider a dynamical system $(T,X)$ that admits a Markov partition $\mathcal{M}$ (e.g., an Anosov map), and consider the corresponding 0-1 transition matrix $A$. It is commonplace to study information about the growth of the number of closed orbits as a function of iterates through a zeta function.
However, it seems... | https://mathoverflow.net/users/1847 | Persistent homology of Markovian dynamical systems | The answer to the question as stated seems to be "no".
Consider a three-element Markov partition $\mathcal{M} = \{A, B, C\}$ with directed edges $(A,B)$, $(B,C)$ and $(C,A)$. There is an obvious periodic orbit $A \to B \to C \to A$ but the Vietoris-Rips complex never has non-trivial persistent homology in dimension $... | 5 | https://mathoverflow.net/users/18263 | 138537 | 75,909 |
https://mathoverflow.net/questions/138540 | 4 | For what prime numbers $p$ there exists a ring with identity and exactly $p$ invertible elements ?
**REMARK** It can be shown that for $p=5$ there is no such ring, so I am wondering for what values of $p$ such a ring exists.
| https://mathoverflow.net/users/nan | Rings with group of units cyclic of prime order | For $p=2$, $\mathbb F\_3$ is an example.
Otherwise, $p$ is odd, so $-1$ is a unit of order $2$ unless it is equal to $1$, so $1=-1$, so $2=0$. We can form the ring:
$\mathbb Z[x]/(2,x^p-1)= \mathbb F\_2[x]/(x^p-1)$. Any ring with exactly $p$ invertible elements admits a map from this universal ring whose image is t... | 17 | https://mathoverflow.net/users/18060 | 138541 | 75,910 |
https://mathoverflow.net/questions/138552 | 3 | [*Commutative* diagrams](https://en.wikipedia.org/wiki/Commutative_diagram) usually express path equivalences in a category and thus involve pairs of paths in a category with the same source and target.
[*General* diagrams](https://en.wikipedia.org/wiki/Diagram_%28category_theory%29) - in categories resp. category th... | https://mathoverflow.net/users/2672 | Pairs of paths with the same source and target | Pairs of paths with the same source and target but with no other nodes in common are called *parallel paths*, at least on the computer science side of things in graph theory -- you can google the term to get some relevant CS papers, but I couldn't find a wikipedia article on parallel paths.
If your paths are allowed ... | 4 | https://mathoverflow.net/users/18263 | 138554 | 75,916 |
https://mathoverflow.net/questions/138563 | 12 | A group $G$ is called locally indicable if for any finitely generated subgroup $H \subset G$, there is a non-trivial homomorphism from $H$ to the real additive group $(\mathbb{R},+)$.
Is the Thompson group $F$ locally indicable?
| https://mathoverflow.net/users/38190 | Is the Thompson group F locally indicable? | Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log\_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$.
A more fancy (... | 20 | https://mathoverflow.net/users/nan | 138564 | 75,920 |
https://mathoverflow.net/questions/135047 | 6 | Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $\tilde{\mathfrak{g}}=\mathfrak{g}((t))\oplus \mathbf{C}K\oplus \mathbf{C}d$ its Kac-Moody extension ($K$ is the level and $d$ is the energy operator). $\tilde{\mathfrak{h}}=\mathfrak{h}\oplus \mathbf{C}K\oplus \mathbf{C}d$ is the Cartan subalgeb... | https://mathoverflow.net/users/18512 | Restriction of highest-weight representations to Heisenberg subalgebras | The answer to your second question is no. Take for instance $\Lambda=k \Lambda\_0$, where the level $k$ is two or more. I'll consider only the homogeneous Heisenberg subalgebra. In this case, the space of invariants you're interested is very important in 2-dimensional conformal field theory and is known as the $parafer... | 2 | https://mathoverflow.net/users/nan | 138585 | 75,928 |
https://mathoverflow.net/questions/138547 | 5 | Let $p>3$ be a prime number and $G$ be a finite group of order $2p(p^2+1)$.
Is it true that always the Sylow $p$- subgroup of $G$ is a normal subgroup of $G$?
As I checked by Gap it seems true.
Thanks for your answer.
| https://mathoverflow.net/users/31045 | the number of Sylow subgroups | As $p^2+1 \equiv 2 \bmod 4$, a Sylow $2$-subgroup of $G$ has order four. As $p^2+1 \equiv 2 \bmod 3$ and $p>3$, $G$ has no element of order three. It follows now that a Sylow $2$-subgroup $X$ of $G$ is central in its normalizer in $G$. By Burnside's normal $p$-complement theorem, $G$ contains a normal complement $N$ to... | 14 | https://mathoverflow.net/users/36466 | 138591 | 75,932 |
https://mathoverflow.net/questions/138466 | 4 | Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows.
For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular matrix of the same size $\mathbf{R}$, we have
\begin{align}
\det(\mathbf{D}\mathbf{D}^H+\mathbf{R}\mathbf{R}^H) \geq \pr... | https://mathoverflow.net/users/38129 | Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix | I have found the proof by mathematical induction. Thanks to Sean Shih and Chu-Lan Kao for fruitful discussions.
For $n=1$, the inequality is obvious.
Suppose that for $n=m\in\mathcal{N}$, $\det(\mathbf{D}\mathbf{D}^H+\mathbf{R}\mathbf{R}^H)\geq \prod\_{i=1}^m(|\mathbf{D}\_{ii}|^2+|\mathbf{R}\_{ii}|^2)$ holds.
Fo... | 1 | https://mathoverflow.net/users/38129 | 138594 | 75,934 |
https://mathoverflow.net/questions/138539 | 4 | I am studying about Mock modular forms and Mock theta functions. I wonder how Zwegers connected mock theta functions with Harmonic Maass Forms? I mean, what was the philosophy/idea of Mock Theta function that motivated him to make a connection with the holomorphic projection of weight 1/2 harmonic Maass forms?
Any refe... | https://mathoverflow.net/users/36735 | Mock Theta Functions | In addition to Zagier's excellent Bourbaki seminar I would also recommend some notes by Ken Ono that include both a summary of the history of mock theta functions and mock modular forms and a survey of applications. They are available at <http://swc.math.arizona.edu/aws/2013/2013OnoNotes.pdf>.
You might also want to ... | 8 | https://mathoverflow.net/users/10475 | 138603 | 75,937 |
https://mathoverflow.net/questions/138418 | 14 | A theory $T$ has the *existence property* (EP) if the following holds:
Let $\phi(x)$ be a formula with one free variable (and no parameters) such that $T \vdash (\exists x) \phi(x)$. Then there is another formula $\psi(x)$ (again no parameters) such that $T \vdash (\exists ! x)\psi(x)$ (ie there is a unique $x$ such ... | https://mathoverflow.net/users/30790 | Existence property for ordered fields | The answer is negative.
For any model $M$, let $M\_d$ denote the submodel of its parameter-free definable elements. We have the following characterization for classical theories.
**Lemma:** $T$ has EP iff $M\_d\preceq M$ for every $M\models T$.
**Proof:** $\leftarrow$ is left as an exercise. $\to$: By Tarski’s te... | 8 | https://mathoverflow.net/users/12705 | 138604 | 75,938 |
https://mathoverflow.net/questions/138610 | 2 | Suppose that I have a category $C$ which has all finite limits and colimits. I have read that in general the canonical functor $C \to Ind(C)$ does not behave nicely with respect to filtered colimits.
Suppose that $I$ is a filtered category and a $F$ functor from $I$ to $C$. I would like to know that if the colimit (... | https://mathoverflow.net/users/3396 | The functor $C\to Ind(C)$ and filtered colimits | Since $\mathrm{Ind}(C)$ is the free cocompletion of $C$ under filtered colimits, this is only true if the colimit of $F$ is absolute, i.e. preserved by every functor whatsoever.
In more detail, suppose $C\to \mathrm{Ind}(C)$ preserves the colimit of $F$. Then any functor $G:C\to D$ to a category $D$ with filtered col... | 4 | https://mathoverflow.net/users/49 | 138618 | 75,941 |
https://mathoverflow.net/questions/138624 | 2 | I recently read this article [*Syntactic semigroups*](http://www.liafa.jussieu.fr/~jep/PDF/HandBook.pdf). In page $8$, he speaks about a J class having an idempotent is called *regular*:
>
> A $\mathcal J$-class containing an idempotent is called *regular*. One can show that in a regular $\mathcal J$-class, every $... | https://mathoverflow.net/users/13898 | Idempotents in Green J classes | Any regular $J$-class with a joined zero is a 0-simple semigroup. So your question is reduced to the following: who many idempotents has a 0-simple semigroup? In particular, if $S$ is finite, a $J$-class (with 0) is completely 0-simple semigroup, so it has just one idempotent $\ne 0$ iff it is a group.
Moreover, if a... | 1 | https://mathoverflow.net/users/18814 | 138628 | 75,945 |
https://mathoverflow.net/questions/138633 | 3 | I asked a mixed-up version of this question earlier.
The Lie algebras I have in mind are the homotopy Lie algebras of wedges of finitely many spheres (in dimensions greater than $1$). Thus each element has a degree and the bracket is the Whitehead product, which satisfies a graded version of the Jacobi identity, etc.... | https://mathoverflow.net/users/3634 | Linear independence in (graded) Lie algebras | Let me say that since you are interested in square-free elements where the weight is equal to the number of generators, you actually are asking questions about *multilinear* elements, that is elements of degree one in each generator, or in other terms, about the operad Lie of Lie algebras. (This actually implies that i... | 2 | https://mathoverflow.net/users/1306 | 138640 | 75,951 |
https://mathoverflow.net/questions/138620 | 8 | Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral mixed Hodge structures.
Is $\Phi$ expected to be faithful? If not, what kind of information does one expect to lose?
... | https://mathoverflow.net/users/1310 | Motives over the complex numbers versus mixed Hodge structures | Yes, this functor is expected to be faithful, but this has very little to do with Hodge theory. The reason for this is that, if $\Psi$ denotes the forgetful functor from integral mixed Hodge structures to abelian groups, then the composed functor $\Psi\circ\Phi$ simply is the Betti realization functor. As $\Psi$ itself... | 10 | https://mathoverflow.net/users/1017 | 138645 | 75,952 |
https://mathoverflow.net/questions/138643 | 2 | Suppose $Y\_1,Y\_2,\ldots, Y\_n$ are independent, where $Y\_i$ is a continuous valued random variable with a density $p\_{Y\_i}(y\_i)$ on its domain $\mathcal{D}\_i\subseteq \mathbb{R}.$ Can one show that for any continuous function $f:\mathbb{R}^n\to\mathbb{R},$ that is non-constant on $\mathcal{D}\_1\times \mathcal{D... | https://mathoverflow.net/users/7576 | Function of independent random variables cannot be independent of each variable? | First define $f:[0,1]\rightarrow [0,1]$ as follows:
$$f(x,y)=2(y-x)-1 \hbox{ if } 1\ge y-x \ge 1/2$$
$$f(x,y)=-2(y-x)+1 \hbox{ if } 1/2\ge y-x \ge 0$$
$$f(x,y)=2(y-x)+1 \hbox{ if } 0\ge y-x \ge -1/2$$
$$f(x,y)=-2(y-x)-1 \hbox{ if } -1/2\ge y-x \ge -1$$
Then let $X$ and $Y$ be uniformly (and independently) distribut... | 4 | https://mathoverflow.net/users/10503 | 138650 | 75,955 |
https://mathoverflow.net/questions/138576 | 2 | Assuming the axiom DC($\omega\_1$), is there a definition of the rank of a group ?
Another related question: assuming DC($\omega\_1$), if we have two groups $A$ and $B$ of the same infinite rank, is there necessarily a surjective homomorphism from $A$ onto $B$?
Edit: in the second question , we assume $A$ and $B$ ... | https://mathoverflow.net/users/38200 | Axiom of dependent choice (up to $\omega_1$) and group rank | Without sitting to verify the details in full, here is a sketch of a proof:
Consider Lauchli's construction of a vector space with two bases of different cardinality, as outlined in Jech **The Axiom of Choice** in problem 10.5 (p. 149). The construction is to create two infinite sets which span isomorphic vector spac... | 1 | https://mathoverflow.net/users/7206 | 138655 | 75,957 |
https://mathoverflow.net/questions/138651 | 7 | It is known that Ramsey property is a kind of generalizition of pigeon hole principle, and some kinds of Ramsey properties have lots of equivalent forms.
We often deal with the case $a\rightarrow (b)^r\_c $,when $a,b,c$ are cardinals; however, if we consider the ordertype of the homogeneous set, the question becomes mu... | https://mathoverflow.net/users/38228 | A problem about Ramsey Property | For $b=\omega+2$, see
>
> András Hajnal. *Some results and problems on set theory*, Acta Math. Acad. Sci. Hungar., **11**, (1960), 277–298. [MR0150044 (27 #47)](http://www.ams.org/mathscinet-getitem?mr=150044).
>
>
>
In this paper, András shows that $\omega\_1\to(\omega\cdot n,\omega\cdot 2)^2\_2$ for all $n... | 7 | https://mathoverflow.net/users/6085 | 138659 | 75,958 |
https://mathoverflow.net/questions/138658 | 10 | Suppose $X$ is a topological space, and $\mu$ is a Borel measure on $X$. Also suppose we have an $n$-dimensional vector bundle $E \to X$, with an inner product $\langle \cdot,\cdot \rangle\_x$ on the fibre $E\_x$ for all $x \in X$, in such a way that each $E\_x$ is complete and such that there exists a vector bundle tr... | https://mathoverflow.net/users/nan | Duality relations for Lebesgue spaces of sections of vector bundles | Maybe the following helps:
Theorem 3.12 (page 20) in the following source has such a related result, albeit for higher Sobolev spaces.
There are quite subtle requirements for the trivialising atlas and the partition of unity which are used in the proof.
* [MR2343536](http://www.ams.org/mathscinet-getitem?mr=2343536);... | 5 | https://mathoverflow.net/users/26935 | 138661 | 75,959 |
https://mathoverflow.net/questions/138582 | 3 | Is there a location where one can access generator matrices (not just bounds) of best known linear codes?
| https://mathoverflow.net/users/10035 | Generator Matrices of Best Known Linear Codes | There is a Magma BKLC (Best Known Linear Codes, i.e., linear $[n,k,d]\_q$-codes, which have the highest minimum weight among all known linear $[n, k,d]\_q$-codes)
database. It contains also generator matrices.
The construction of this Magma BKLC database has been undertaken by John Cannon (Sydney), Markus Grassl (Karls... | 5 | https://mathoverflow.net/users/32332 | 138664 | 75,962 |
https://mathoverflow.net/questions/138412 | 9 | Harvey Friedman's "Concrete Mathematical Incompleteness" at <http://www.math.osu.edu/~friedman.8/pdf/0.Intro061311.pdf> cites the Hasse Minkowski theorem saying quadratic forms over a number field are equivalent if and only if they are equivalent over every completion of the field (real, complex, or $p$-adic). He says ... | https://mathoverflow.net/users/38783 | The Hasse Minkowski theorem in Peano arithmetic | The community has spoken by silence. No one has worked on this.
| 1 | https://mathoverflow.net/users/38783 | 138666 | 75,964 |
https://mathoverflow.net/questions/123491 | 3 | wikipedia has an entry on the [Collatz conjecture](http://en.wikipedia.org/wiki/Collatz_conjecture) with a section on [As an abstract machine that computes in base two](http://en.wikipedia.org/wiki/Collatz_conjecture#As_an_abstract_machine_that_computes_in_base_two). this apparently describes a construction of a [FSM t... | https://mathoverflow.net/users/20793 | Collatz conjecture— finite state machine transducer construction, origination? | I gave a sequential machine computing the 3n+1/n:2 function in base 2 in several courses
since 1990, but of course I am not claiming any originality here, since it is just an easy
exercise.
Anyway, if you want to see in more details how this sequential machine can be computed in a systematic way, you can look at <ht... | 8 | https://mathoverflow.net/users/38236 | 138675 | 75,967 |
https://mathoverflow.net/questions/138638 | 1 | The question is about the family of tensors that are naturally associated to any nice Lie group.
Take the Mauer-Cartan form, $\omega=g^{-1} dg$ and I would like to make the covariant index of this one-form explicit, $\omega\_a =g^{-1}\partial\_a g$. Then the Riemannian, left/right invariant, metric is just $g\_{ab}=tr(... | https://mathoverflow.net/users/2052 | tensor hierarchy for Lie groups from Maurer-Cartan form | Sounds like you are in physics; particle physicists often assume that all Lie groups are compact. Compact Lie groups admit biinvariant metrics. The tensors you are considering are essentially the characteristic polynomials in the adjoint representation, so if you insert wedge product signs (work with the associated alt... | 5 | https://mathoverflow.net/users/13268 | 138686 | 75,970 |
https://mathoverflow.net/questions/138641 | 4 | Let $G$ be a finite abelian $p$-group. What is known about the minimal number of generators of a $p$-sylow of $Aut(G)$? is it bounded in terms of $d(G)$ the minimal number of generators of $G$ (and perhaps $p$)?
| https://mathoverflow.net/users/31883 | Number of generators of the automorphism group of an abelian group | For the special case that $G=(\mathbb{Z}/p\mathbb{Z})^n$, this is true.
Then we have $d(G)=n$ and $Aut (G)=GL(n,p)$. In the article of A. Patterson, "The minimal number of generators for $p$-subgroups of
$GL(n, p)$" of $1974$ it is shown that any $p$-subgroup of $\text{GL}(n,p)$, where $p$ is an odd prime, can be gener... | 3 | https://mathoverflow.net/users/32332 | 138688 | 75,971 |
https://mathoverflow.net/questions/131061 | 5 |
>
> I would like to know the ring structure of $K(Q\_n)$ explicitly where $Q\_n \subset \mathbb{P}^{n+1}$ is the non-singular $n$-dimensional complex quadric and $K(Q\_n) = K^0(Q\_n)$ is the complex topological $K$-theory of $Q\_n$ (with analytic topology). What is it? Can anyone provide a reference?
>
Ideally, I ... | https://mathoverflow.net/users/6801 | What is the ring structure of the complex topological K-theory of a non-singular complex quadric? | I guess I might as well answer my own question as it might help somebody in the future. I ended up working this out using the methods of Hodgkin. (The Atiyah-Hirzebruch SS leaves one with a series of extension problems and so only gives the Abelian group structure.)
In fact I have written up a careful proof of this i... | 3 | https://mathoverflow.net/users/6801 | 138699 | 75,977 |
https://mathoverflow.net/questions/138694 | 14 | Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb P=\left\{(p,A)\mid p\in[\kappa]^{<\omega}, A\in\mathcal U, \max p<\min A\right\}.$$
We also define the order, $(q,B)$ is s... | https://mathoverflow.net/users/7206 | Is Prikry forcing minimal? | There is a theorem of Gitik, Kanovei and Koepke that characterises the degrees of constructibility in $M[G]$ where $G$ is prikry generic over the model $M$: they are isomorphic to the $P(\omega)/Fin$ of $M[G]$.
That is they show (letting $M, G$ be as above with $x=G\_C$ the actual prikry sequence):
$$∀Z ∈ M[x] ∃y ⊆... | 14 | https://mathoverflow.net/users/6942 | 138701 | 75,978 |
https://mathoverflow.net/questions/137805 | 7 | The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are "easier"?
| https://mathoverflow.net/users/30081 | Riemann hypothesis and Kakeya needle problem | The Riemann hypothesis is proved over function fields (like the fraction field of F\_q[t]), not finite fields, and the "real version" is a question about the integers. Kakeya is proved over finite fields, and the "real version" is a question about, well, the reals. So the situations are quite different. I'd say that th... | 24 | https://mathoverflow.net/users/431 | 138707 | 75,982 |
https://mathoverflow.net/questions/138705 | 3 | Consider the left regular representation of $\mathbb{R}$ in $\mathrm{L}^2(\mathbb{R})$. Let us denote by $\mathrm{L}^2(\mathbb{R})^\infty$ the algebraic subspace of smooth vectors, or equivalently, the Gårding subspace (the theorem of Dixmier-Malliavin assumed to be at our disposal). I want to prove
$$\mathrm{L}^2(\m... | https://mathoverflow.net/users/19142 | On a characterization of the Gårding subspace of the left regular representation of reals | Yes, your desired equality is true: regarding the left regular representation as
$$
\operatorname{Ind}\_{\{0\}}^{\mathbf R}1,
$$
it becomes a special case of the characterization of smooth vectors in induced representations by N. S. Poulsen, [*On $C^\infty$-vectors and intertwining bilinear forms for representations of... | 3 | https://mathoverflow.net/users/19276 | 138710 | 75,983 |
https://mathoverflow.net/questions/138704 | 2 | Let $X = \textrm{Spec} A$ be a reasonable scheme and $I\subset A$ an ideal generated by a regular sequence. Then we have a full set of generators/relations for the blow-up of $X$ along $V(I)$.
Are there other situations where one can compute a presentation for the Rees algebra by hand? The schemes I'm interested are ... | https://mathoverflow.net/users/25854 | Equations for blow-ups non-regular centers | I believe that you are looking for a description of the defining equations of Rees algebras in commutative algebra. In general, this is a hard question. The complete intersection case can be generalized to ideals of linear type.
>
> Definition: Let $R$ be a ring. An ideal $I$ is call of linear type if $\operatorna... | 4 | https://mathoverflow.net/users/22388 | 138720 | 75,989 |
https://mathoverflow.net/questions/138693 | 9 | In homotopy type theory, homotopy types can be viewed as logical types and it is possible to prove some theorems about them without using any underlying space (no simplicial set, no topological space). It is a kind of synthetic algebraic topology. Is it just a coincidence that the word "type" may have these two meaning... | https://mathoverflow.net/users/24563 | A (very naive) question about Homotopy Type Theory | Maybe I should repost my comment above as a genuine answer:
It's a coincidence, but a very fortunate one (for a change).
* When people started saying "[homotopy type](http://ncatlab.org/nlab/show/homotopy%20type)" $X$ they meant "the type of $X$" as in "what kind of space is X?".
* When people said "[type](http://... | 17 | https://mathoverflow.net/users/381 | 138721 | 75,990 |
https://mathoverflow.net/questions/138310 | 50 | Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been considered almost certainly true and have guided a lot of the research in these areas for a long time. A new preprint by Geordie... | https://mathoverflow.net/users/19113 | What to do now that Lusztig's and James' conjectures have been shown to be false? | The questions raised here will probably need some substantial research papers to answer, inventing new approaches and methods. In any case, the question of what to do about "small" primes has been around for decades without any clear program emerging. Three logical outcomes are possible: these range from least satisfac... | 23 | https://mathoverflow.net/users/4231 | 138723 | 75,991 |
https://mathoverflow.net/questions/138732 | 5 | I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a question about some affirmations:
**Definition** A normal crossing variety is a reduced complex analytic space which is... | https://mathoverflow.net/users/17495 | Existence of logarithmic structures and d-semistability | It is true that if $X\subseteq Y$ is a normal crossings divisor, then $Y$ has a log structure whose sheaf of monoids is the sheaf of regular functions invertible outside of $X$. It is also true that this log structure can then be restricted to $X$, giving a log structure on $X$.
On the other hand, if we start with $X... | 6 | https://mathoverflow.net/users/23917 | 138737 | 75,994 |
https://mathoverflow.net/questions/138722 | 1 | I want to know what the above is. I would also like to know what H^5(B[SO(3)xU(1)],U(1)) is, where B[SO(3)xU(1)] is the classifying space of SO(3)xU(1). I don't know how to do the calculation myself
| https://mathoverflow.net/users/38269 | H^4(SO(3)xU(1),U(1)) with Borel cohomology | If I'm understanding your notation right, then you're asking for the fourth cohomology of the Lie group $SO(3) \times U(1)$ with coefficients in the abelian group $U(1) = S^1$. Using the exponential sequence $1 \to \mathbb{Z} \to \mathbb{R} \to S^1 \to 1$, we have
$$H^4(SO(3) \times U(1), U(1)) = H^5(SO(3) \times U(1),... | 3 | https://mathoverflow.net/users/4649 | 138741 | 75,996 |
https://mathoverflow.net/questions/138740 | 1 | I'm trying to understand a proof of the following theorem (from section II of Hall's paper [*An Isomorphism Between Linear Recurring Sequences and Algebraic Rings*](http://www.ams.org/journals/tran/1938-044-02/S0002-9947-1938-1501967-8/S0002-9947-1938-1501967-8.pdf)):
>
> If $F(a\_1, \ldots, a\_k)$ is a polynomial ... | https://mathoverflow.net/users/38282 | If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$ | Pick two distinct primes $p$ and $q$. Now choose any monic irreducible polynomial $f\_q$ of degree $k$ modulo $q$ (i.e. in ${\bf F}\_q$); these exist. Also choose $a\_1, \dots, a\_k$ arbitrary mod $p$ and let $f\_p = x^k - a\_1 x^{k-1} + \dots + (-1)^k a\_k$. Then by CRT, there is a common lift $f(x) \in {\bf Z}[x]$ wh... | 3 | https://mathoverflow.net/users/2698 | 138742 | 75,997 |
https://mathoverflow.net/questions/138739 | 2 | I heard the statement "the blowup of a toric variety corresponding to a subdivison of fan" many times, but could not find reference in the literature. What is the precised statement (blowup a point? or certain subscheme?)? And how does blowup related to the fan?
I think one can just consider an affine toric variety d... | https://mathoverflow.net/users/29730 | Why the blowup of a toric variety corresponding to a subdivison of fan? | If $\Delta$ is a fan in a lattice $N$, and $\sigma \in \Delta$ is a cone, the star of $\sigma$ $-$ call it $\Delta'$ $-$ is a refinement of $\Delta$. Then the morphism $X(\Delta') \to X(\Delta)$ of toric varieties induced by identity map of $N$ exhibits $X(\Delta')$ as the blowup of $X(\Delta)$ at the distinguished poi... | 4 | https://mathoverflow.net/users/18289 | 138755 | 76,002 |
https://mathoverflow.net/questions/138750 | 4 | $U(1)$ seems to lead a dual life. On one hand it is the group we know and love, and on the other, it is the classifying space of the integers. Thinking about $n$-groups says that we should also think about this delooping $B\mathbb{Z}$ as the 2-group presented by the crossed module $\mathbb{Z} \to 1$. What is the relati... | https://mathoverflow.net/users/36591 | U(1) vs. BZ and representations of 2-groups | This is about the difference between and relation of "bare" groups and groupoids and higher groupoids (bare = no extra geometry) and *[smooth](http://ncatlab.org/nlab/show/smooth%20infinity-groupoid)* groups and groupoids and higher groupoids.
The main fact is that there is an $\infty$-functor
$$
\Pi : Smooth \inf... | 3 | https://mathoverflow.net/users/381 | 138758 | 76,003 |
https://mathoverflow.net/questions/111448 | 7 | Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms:
\begin{align}
\neg Sx&=0\tag{1}\\\
Sx=Sy&\rightarrow x=y\tag{2}\\\
x+0&=x\tag{3}\\\
x+S(y)&=S(x+y)\tag{4}\\\
x\times 0&=0\tag{5}\\\
x\times S(y)&=(x\times y)+x\tag{6}
\end{alig... | https://mathoverflow.net/users/22019 | (Finite) Models of two subtheories of Peano Arithmetic | Infinite models are partly classified by two theorems of $PA^{(-1,2)}$ the subtheory of $PA$ without axioms 1 or 2. Then I will describe the finite models completely.
In $PA^{(-1,2)}$ if axiom 1 fails then axiom 2 holds. For proof, express failure of axiom 1 by a constant $c$ with $S(c)=0$. Then $PA^{(-1,2)}$ proves... | 6 | https://mathoverflow.net/users/38783 | 138772 | 76,008 |
https://mathoverflow.net/questions/138767 | -2 | For a scheme $X$ of finite type over $k$, and a coherent sheaf $\mathcal{F}$ on $X$, the Hilbert polynomial of $\mathcal{F}$ is defined by $\Phi(n)=\chi(\mathcal{F}(n))$.
And for a scheme $X$ over $S$ (with some suitable conditions), I was told that we can define Hilbert polynomials by defining on each fibers. That i... | https://mathoverflow.net/users/38244 | Hilbert polynomials on a scheme | Seconding Jack, it is not clear what question you are asking. However, my guess is that you want to know why, or rather when, the Hilbert polynomial is well-defined independently of the geometric point $s$ of $S$. With the hypotheses that Jack listed, if you also assume that $\mathcal{F}$ is an $S$-flat, locally finite... | 3 | https://mathoverflow.net/users/13265 | 138774 | 76,009 |
https://mathoverflow.net/questions/138677 | 5 | Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A\_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil pairing, it is known that $K(A\_{p^\infty})$ contains the field $K(\mu\_{p^\infty})$, the field obtained by adjoining to... | https://mathoverflow.net/users/37040 | Weil pairing, fixed field of a $p$-adic Galois representation | The answer to question 1 as states is no. For example if $i=0$, $V$ will be the trivial Galois representation (assuming your variety to be geometrically connected).
But the answer to question 2 is yes (which implies that some corrected version of question 1 holds as well): a "generalized Weil pairing" is given by the... | 10 | https://mathoverflow.net/users/9317 | 138776 | 76,011 |
https://mathoverflow.net/questions/138759 | 12 | What are invisible sets? In order to illustrate what we mean here let me to explain some examples:
Consider you want to find the biggest treasure of the world and you have found a magic map of some hidden treasures. If you look at it in "sun light", you will see some objects, signs and sentences which guide you ... | https://mathoverflow.net/users/nan | Can we see invisible sets? | I think the question loses some of its appeal once we note that the preorder $T\sqsubseteq T'$ defined by the OP coincides with the preorder defined by the inclusion $\mathtt{Th}(T)\subseteq\mathtt{Th}(T')$ when restricted to theories which are strong enough to prove say $\exists!x\forall y(y\not\in x)$ (a formula sugg... | 4 | https://mathoverflow.net/users/37548 | 138780 | 76,013 |
https://mathoverflow.net/questions/137058 | 7 | Suppose I am given a subset of $2^\omega\times\omega^\omega$ of some bounded Borel rank. Can I get an **analytic** uniformization of this set?
| https://mathoverflow.net/users/25700 | Analytic uniformization | There is an arithmetical set $A\subseteq 2^{<\omega}\times \omega^{<\omega}$ so that for any $x\in 2^{\omega}$, $A(x)=\{\sigma\mid \exists n(x|n,\sigma)\in A\}$ is an $x$-recursive tree which has an infinite path but no infinite path hyperarithmetic in $x$.
Now let $B$ be an arithmetical set so that $(x,y)\in B$ if a... | 5 | https://mathoverflow.net/users/14340 | 138783 | 76,016 |
https://mathoverflow.net/questions/138765 | 2 | The type of booleans, denoted by ${\mathbf{2}}$, has two terms $0\_{\mathbf{2}}:{\mathbf{2}}$ and $1\_{\mathbf{2}}:{\mathbf{2}}$. The induction principle of ${\mathbf{2}}$ states that, given a dependent family $C:2\to {\mathcal{U}}$ and terms $c\_0:C(0\_{\mathbf{2}})$, $c\_1:C(1\_{\mathbf{2}})$, there exists a dependen... | https://mathoverflow.net/users/nan | Does the induction principle of the type of booleans imply its recursion principle? | Recursion principles are usually special cases of induction principles in dependent type theory. For example, for the boolean type $\mathbf{2}$, given $a\_0, a\_1 : A$, we may form (using the induction principle) the function $f : \prod\_{x : \mathbf{2}} A$ where $f (0\_{\mathbf{2}}) \equiv a\_0$ and $f(0\_{\mathbf{1}}... | 4 | https://mathoverflow.net/users/11640 | 138787 | 76,018 |
https://mathoverflow.net/questions/138785 | 5 | It is quite easy to show that if $\mu$ is positive finite Borel measure on, say $[0,1]$, and for all $n \in \mathbb{N}$
$$\int\_{[0,1]} e^{-nx}\mu(dx)=0$$
holds true, then $\mu=0$. Does this still hold if $\mu$ is signed finite Borel measure?
For positive measure, by applying Holder's inequality it can be shown t... | https://mathoverflow.net/users/38288 | On vanishing signed measure | This is an old Theorem of M. Lerch, 1903. For more info see Theorem 6.2, Chap.II, $\S 6$ of
>
> D.V. Widder: The Laplace Transform, Princeton University Press 1941 (or Dover 2010).
>
>
>
| 5 | https://mathoverflow.net/users/20302 | 138794 | 76,020 |
https://mathoverflow.net/questions/134317 | 0 | Have they been studied? In particular, what is the analogue of the Schmidt theorem for compact operators in Hilbert spaces?
[Helemskii A. Ya., Lectures and Exercises on Functional Analysis](http://books.google.ru/books?id=wjzZCLzx6hUC&printsec=frontcover#v=onepage&q&f=false), Ch. 3, $\S4$,
>
> **Theorem** (Schmi... | https://mathoverflow.net/users/14551 | Reference for compact operators in quaternionic Hilbert spaces | Quaternionic functional analysis is reviewed in <http://arxiv.org/abs/math/0609160> (Chi-Keung Ng, On quaternionic functional analysis). In fact "Hilbert spaces of any one of the three kinds - real, complex and quaternionic - can be seen as Hilbert spaces of the other kinds, equipped with extra structure" (John C. Baez... | 2 | https://mathoverflow.net/users/32389 | 138796 | 76,022 |
https://mathoverflow.net/questions/138791 | 10 | In his paper *A finitely generated infinite simple group* (J. London Math. Soc., 1951), Higman introduced the following finitely presented group:
$$
H = \langle x,y,z,w \mid [x,y]=y, \, [y,z]=z, \, [z,w]=w, \, [w,x]=x \rangle .
$$
This group has many remarkable properties, including being acyclic.
Let $H\_{x,y} \... | https://mathoverflow.net/users/8103 | A question about conjugacy in Higman's group | No, there are no such element. Indeed $H$ is amalgam of the subgroups $\langle x,y,z\rangle$ and $\langle z,w,x\rangle$ over the intersection $\langle x,z\rangle$, which is free (as we see by viewing $\langle x,y,z\rangle$ itself as amalgam of the Baumslag-Solitar $\langle x,y\rangle$ and $\langle y,z\rangle$ over $\la... | 13 | https://mathoverflow.net/users/14094 | 138798 | 76,023 |
https://mathoverflow.net/questions/138793 | 4 | Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial?
I suspect that the answer is "yes", for the following reason. Some neighbourhood of $x$ is analytically isomorphic to the 'conifold' geometry
$$
xy - wz = 0 ~~\mathrm{in}~... | https://mathoverflow.net/users/22975 | Factoriality of one-nodal Calabi-Yau threefolds | Let $X$ be a Calabi-Yau three-fold with only ordinary double points. One can always find a small resolution $Y\rightarrow X$ where $Y$ is a (not necessarily Kaehler) complex manifold. Let $C\_1,\ldots,C\_n$ be the exceptional curves. Friedman proved in his paper "Simultaneous Resolution of Threefold Double Points" that... | 6 | https://mathoverflow.net/users/23917 | 138813 | 76,029 |
https://mathoverflow.net/questions/138764 | 5 | In this Mathoverflow question, [Examples of Eigensheaves outside of langlands](https://mathoverflow.net/questions/100943/examples-of-eigensheaves-outside-of-langlands), David Ben-Zvi says
" Given a G -space X you can recover quasicoherent sheaves on X from sheaves on X/G (ie equivariant sheaves) as eigenobjects for t... | https://mathoverflow.net/users/36931 | A naive question on eigensheaves for group actions on derived categories | Given a group $G$ acting on a category $QC(X)$, you get a sheaf of quasicoherent categories over $BG$, whose fiber at $pt\rightarrow BG$ is $QC(X)$. Global sections of this sheaf are exactly invariants $QC(X/G)$ with an action of "global functions" $QC(BG)$. See <http://arxiv.org/abs/1306.4304> for a reference on quasi... | 2 | https://mathoverflow.net/users/18512 | 138824 | 76,033 |
https://mathoverflow.net/questions/138667 | 9 | A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose velocity vectors are dense in the unit tangent bundle).
*Are there Riemannian metrics on closed surfaces of genus $g > ... | https://mathoverflow.net/users/21123 | Transitive geodesics on closed surfaces of genus greater than one | This is both: answer on the new version of the question and comment on comment of Andrey Gogolev, who asked whether one can make the question more complicated assuming
additionally that the set of non-periodic geodesics is dense.
Hier is an example that answers both: it is not much different from the answer of And... | 4 | https://mathoverflow.net/users/14515 | 138826 | 76,034 |
https://mathoverflow.net/questions/138823 | 7 | I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this stability condition is well-determined or still conjectural. More precisely, is there any precise statement of this conjectur... | https://mathoverflow.net/users/30081 | Yau's conjecture for positive Chern class | A precise statement and proof of the relationship between stability and the existence of Calabi-Yau metrics is in:
1. arXiv:1302.0282, Xiuxiong Chen, Simon Donaldson, Song Sun,
*Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof*
2. arXiv:1212.4714, Xi... | 7 | https://mathoverflow.net/users/13268 | 138831 | 76,035 |
https://mathoverflow.net/questions/138810 | 3 | I have the following question regarding group actions on trees to which I suspect the answer to be "yes", but it could very well be that extra conditions are required (it is certainly true for free actions on trees):
Let $G$ be a finitely generated group acting by simplicial automorphisms on a minimal simplicial tree... | https://mathoverflow.net/users/12996 | Group actions on trees and translates under hyperbolic elements | The answer is yes. For the proof, the ray $R$ contains three distinct points in order, $y\_0,y\_1,y\_2$, such that $y\_1 = g\_1 y\_0$ and $y\_2 = g\_2 y\_1$ for some elements $g\_1,g\_2 \in G$. If either of $g\_1,g\_2$ is hyperbolic, you are done. Suppose that both of $g\_1,g\_2$ are elliptic. The midpoint $p\_1$ of $[... | 6 | https://mathoverflow.net/users/20787 | 138835 | 76,036 |
https://mathoverflow.net/questions/138812 | 4 | I came across the Fujita conjecture which is perhaps very widely known. I want to know what are the supporting facts to the truth of the conjecture.
<http://en.wikipedia.org/wiki/Fujita_conjecture>
| https://mathoverflow.net/users/30081 | supporting facts to fujita conjecture | There are a number of things known.
(1). As Libli answered, if $L$ is globally generated and ample (for example, very ample) then
* $K\_X \otimes L^{\dim X + 1}$ is globally generated
* $K\_X \otimes L^{\dim X+ 2}$ is very ample.
These follow from Castelnuovo-Mumford regularity and Kodaira vanishing (see Positiv... | 7 | https://mathoverflow.net/users/3521 | 138843 | 76,041 |
https://mathoverflow.net/questions/133921 | 12 | There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to [this question](https://mathoverflow.net/questions/51863/does-riemann-map-depend-continuously-on-the-domain). Given two non-empty simply-connected domains $U$ and $V$ in the complex plane which are not equal to ... | https://mathoverflow.net/users/674 | Constructing Riemann maps using Brownian motion? | Let $U$ be a Jordan domain and $b\_0,b\_1,b\_2$ be points appearing in order on $\partial U$. Let $f : U \to \mathbb{H}$ be the Riemann map normalized so that $f(b\_0)=0$, $f(b\_1)=1$ and $f(b\_2)=\infty$. Chris Bishop taught me a poor man's way to construct $f$ using Brownian motion.
Let $A\_k$ be the arc of $\parti... | 7 | https://mathoverflow.net/users/38319 | 138846 | 76,043 |
https://mathoverflow.net/questions/138819 | 2 | How can I explicitly calculate all the orbits of the action of $SO(3)$ on $\mathbb C\mathbb P\_2$?
For example I know that one of the orbits is the quadric $\{[z\_0:z\_1:z\_2]\in\mathbb C\mathbb P\_2: z\_0^2 + z\_1^2 + z\_2^2= 0\}$ but I dont know how to calculate it. Other orbits are copies of $\mathbb P\_1$ and $\mat... | https://mathoverflow.net/users/13559 | Orbits of an action | $SO\_3$ acts with cohomogeneity one on $CP^2$, where we view $SO\_3$ inside $SU\_3$ (the identity component of the isometry group of $CP^2$, up to a $Z\_3$-kernel) by the standard inclusion. If the metric in $CP^2$ is normalized to be the quotient metric given by the Hopf fibration
$S^5(1)\to CP^2$, then the orbit spa... | 5 | https://mathoverflow.net/users/15155 | 138850 | 76,044 |
https://mathoverflow.net/questions/138860 | 1 | I apologize in advance if this question is not considered research-level.
I am reading material on Teichmüller theory and I am getting confused as to the nature of the space $Q(R)$ of all integrable, holomorphic quadratic differentials in terms of the complex structure of $\mathrm{Teich}(R)$, for a Riemann surface $R... | https://mathoverflow.net/users/38325 | Confusion about the dual/predual to the tangent plane to a Teichmüller space | For any Riemann Surface $R$, the tangent space $T\_0{\rm Teich}$ is the topological dual of $Q(R)$ - whence the dual of the tangent space is the double dual of $Q(R)$. All this and more is discussed in the texts of Hubbard and Gardiner-Lakic.
| 1 | https://mathoverflow.net/users/15819 | 138865 | 76,051 |
https://mathoverflow.net/questions/138815 | 6 | [Oriented matroids](http://www2.lirmm.fr/~sol/Rapports/References/orientedMatroids.pdf) are abstractions of hyperplane arrangements, or equivalently vector configurations. Let me recall the definition in terms of covectors.
Let $R=\lbrace 0,+,-\rbrace$ with the monoid structure given by the table
$$\begin{array}{c|cc... | https://mathoverflow.net/users/15934 | Functionals on oriented matroids | I think the answer to your first question is "yes".
Oriented matroids can be realized topologically, and I am going to use that language. (This means is that I can pretend the oriented matroid is a hyperplane arrangement, provided I am a little careful.)
Let $\phi$ be a functional. Let the rank of the oriented mat... | 5 | https://mathoverflow.net/users/468 | 138873 | 76,054 |
https://mathoverflow.net/questions/138867 | 9 | In introductory knot theory books, authors usually make a choice of smooth knots or piecewise-linear knots. I often find myself wanting to work in the larger setting of piecewise-smooth knots which subsumes both smooth and PL knots. To do this I would need to prove a "piecewise smooth isotopy-extension theorem" in orde... | https://mathoverflow.net/users/38347 | Piecewise Smooth Knot Theory | Everything that you wish for is true.
You can approximate "ambient" (in the sense explained below)
isotopies by smooth isotopies and keep the ends (the knots) fixed or not. This is spelled out in great detail in:
* MR0674117, Bröcker, Theodor; Jänich, Klaus: Introduction to differential topology. Translated from the ... | 6 | https://mathoverflow.net/users/26935 | 138883 | 76,060 |
https://mathoverflow.net/questions/138844 | 0 | For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric condition is relaxed, do there exist a large number of non-symmetric equivariant Riemannian metrics. Also, what is a spe... | https://mathoverflow.net/users/38254 | Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces | I assume that, by 'equivariant metric', you mean 'invariant metric', i.e., you are looking for $G$-invariant metrics on $M=G/H$ and wondering how one describes the non-symmetric ones (when they exist). Also, I assume that you are using 'symmetric' in its standard sense, i.e., that the metric is locally symmetric, i.e.,... | 7 | https://mathoverflow.net/users/13972 | 138894 | 76,065 |
https://mathoverflow.net/questions/138900 | 1 | Assume DC($\aleph\_1$).
Can we define the following:
1. Basis for a vector space $V$ over a field $K$ such that $\operatorname{card}(K) \leq \aleph\_1$ and we happen to find a generating set of $V$ of cardinality $\leq \aleph\_1$.
2. Linear dimension making the same assumption as above.
3. Transcendence degree of ... | https://mathoverflow.net/users/38200 | Some definitions without full choice | Without any appeal to the axiom of choice, if you have a vector space which is well-ordered then it has a basis. Moreover, if you have a generating set which is well-ordered then the vector space has a basis. The latter one allows you to have that the field itself is not necessarily well-ordered (as in your first point... | 2 | https://mathoverflow.net/users/7206 | 138901 | 76,067 |
https://mathoverflow.net/questions/138911 | 3 | Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y\_0$ we have
$$
\left| {f\left( {x + iy} \right)} \right| \le f\left( x \right)
$$
for any $x>0$ and $y>y\_0$. Does it follow that this inequality holds for any $y\geq 0$? This question is related to my earlier q... | https://mathoverflow.net/users/35433 | Inequality for certain analytic functions | The answer is no. For example, consider $f(z)=1/(z^2+1)$. Then for every $y \geq \sqrt{2}$, we have
$$
|f(x+iy)|\leq f(x)
$$
for all $x>0$. Indeed, if you write everything out, you get that this inequality is equivalent to
$$
(x^2+1)^2 \leq (x^2-y^2+1)^2 + 4x^2y^2,
$$
which after simplification is seen to be equivalent... | 9 | https://mathoverflow.net/users/38319 | 138920 | 76,074 |
https://mathoverflow.net/questions/138676 | 7 | **Definition**: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$ be a function.
**Definition** : $\omega$ is **algebraically normal** if for all $h$ polynomial and $\forall r \geq ... | https://mathoverflow.net/users/34538 | Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$? | $\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}$I suspect this is false! At least, I'll show that a similar statement is false for $\ZZ^4$ and I'll sketch how I think a similar construction should work for $\ZZ^2$.
Fix a coloring $\omega : \ZZ^4 \to \{ 0, 1 \}$. Define a directed graph whose vertices are quadruples $(p,q,p',... | 2 | https://mathoverflow.net/users/297 | 138927 | 76,077 |
https://mathoverflow.net/questions/138930 | 6 | Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that
A) The integral of $F$ over the hemisphere $(x-x\_0)^2 + (y-y\_0)^2 + z^2 = R^2$ vanishes for all $x\_0$, $y\_0$, and $R$.
implies
B) $F = 0$ everywhere
My question is whether this is known to be... | https://mathoverflow.net/users/38386 | Vanishing of integral on hemispheres implies vanishing of function? | As @alvarezpaiva notes, this is a question about hyperbolic hyperplane (Radon) transform. This has been studied. See, e.g., Kurusa, the Radon transform in hyperbolic space [Geometriae Dedicata, 1991] (available for free, thanks to Springer). and references therein (Kurusa inverts it on fairly natural subspaces of $L^2,... | 3 | https://mathoverflow.net/users/11142 | 138943 | 76,085 |
https://mathoverflow.net/questions/138905 | 12 | There are examples of functions $f \colon [0,1] \longrightarrow [0,1]$ such that
for any $\alpha $, $f^{-1}(\lbrace \alpha \rbrace)$ is uncountable. My favorite example is $$f(r) = \limsup\_n \frac{a\_1 + a\_2 +\cdots + a\_n}{n}$$ where $0.a\_1a\_2\cdots$ is the (non-terminated) binary expansion of $r$.
Is there a ... | https://mathoverflow.net/users/nan | Is it possible to have the set $f^{-1}(\lbrace x \rbrace)$ perfect for every $x$? | Existence of such a function goes back to 1939:
**J. Gillies, Note on a conjecture of Erdos, Quart. J. Math. Oxford 10, 1939, 151-154**
Also, it can be shown that there is a residual set (a set whose complement is of first category) of continuous functions on $[0,1]$ such that for any $f$ in that set $f^{-1}(\lbrac... | 6 | https://mathoverflow.net/users/nan | 138948 | 76,089 |
https://mathoverflow.net/questions/138884 | 2 | A well known theorem about abelian subgroups of index $p^2$ in $p$-groups is that
(\*) *if a $p$-group contains abelian subgroup $A$ of index $p^2$, then it contains an abelian normal subgroup $A\_1$ of index $p^2$* ([see this](http://summit.sfu.ca/system/files/iritems1/4186/b13538421.pdf)).
A non-trivial theorem ... | https://mathoverflow.net/users/6761 | Abelian subgroups of maximum order in $p$-Groups | First I note that your assertion about Alperin's theorem is not true for $p=2$. I think the answer to your second question is negative. There is $p$-groups ($p>3$) having abelian subgroups of index $p^{\frac{p+3}{2}}$, and no normal abelian subgroup of that index. In particular, for $p=5$ the analogue of Alperin's theo... | 3 | https://mathoverflow.net/users/31883 | 138955 | 76,092 |
https://mathoverflow.net/questions/138736 | 9 | **Background**: A [preorder](http://en.wikipedia.org/wiki/Preorder) is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms that give us partial orders etc., so plenty of everyday "order" concepts ca... | https://mathoverflow.net/users/37405 | What kind of category is a cyclically ordered set? | It seems like cyclically ordered sets ought to be regarded as cousins of categories rather than as categories themselves. More precisely, a cyclically ordered set $C$ has a nerve $N(C)$ analogous to the [nerve](http://ncatlab.org/nlab/show/nerve) of a (higher) category or (higher) groupoid. The nerve is first of all a ... | 8 | https://mathoverflow.net/users/290 | 138961 | 76,095 |
https://mathoverflow.net/questions/138959 | 9 | It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ *there exists* a sequence $(k\_i)$ such that $\lim\sin k\_i = \theta,$ but there appear to be no explicit such (infinite) sequences, even for $\theta=0.$ Does anyone know of such?
| https://mathoverflow.net/users/38393 | Convergent subsequence of $\sin n$ | As to the convergenge to zero: note that the convergents of the [continuous fraction for $\pi$](http://oeis.org/A001203) provide a rational approximation $|\pi - p\_n/q\_n| < 1/q\_nq\_{n+1}$ so that $\sin p\_n\to 0$.
The sequence of numerators $0, 1, 3, 22, 333, 355,\dots$ is OEIS' [A002485](http://oeis.org/A002485).... | 15 | https://mathoverflow.net/users/6101 | 138963 | 76,096 |
https://mathoverflow.net/questions/136030 | 3 | [Parikh and Parnes](http://link.springer.com/chapter/10.1007/BFb0066012) showed that one can define a isometry-invariant finitely-additive conditional probability for all pairs of subsets of the interval. Can one extend this result to bounded subsets of $\mathbb R^2$? Of course, due to the non-amenability of SO($n$), $... | https://mathoverflow.net/users/26809 | Conditional probabilities for all pairs of subsets in $\mathbb R^2$? | The answer seems to be negative. Here's a quick and rough sketch.
Say that $C:2^\Omega \times (2^\Omega \backslash \{\varnothing\})\to [0,\infty]$ is a comparative probability on $\Omega$ iff $C(-,Y)$ is a finitely additive measure with $C(Y,Y)=1$, and $C(X,Y)C(Y,Z)=C(X,Z)$ whenever the product on the lhs is defined ... | 0 | https://mathoverflow.net/users/26809 | 138968 | 76,099 |
https://mathoverflow.net/questions/138969 | 3 | Just as the zero section in $T^\*\mathbb{R}^N$ (equipped with the standard symplectic form) is the "model" / "quintessential" Lagrangian submanifold, does $T^\*\mathbb{R}^N$ have a "model" co-isotropic submanifold?
| https://mathoverflow.net/users/38394 | model co-isotropic submanifold | Let $(M, \Omega)$ be a presymplectic manifold and let $E \longrightarrow M$ be the **characteristic bundle** of $(M, \Omega)$, defined fiberwise by
$$E\_x = \{v \in T\_x M : \Omega\_x(v, -) = 0\}.$$
If $E^\ast$ denotes the dual bundle to $E$, then we have the following classification theorem.
>
> **Theorem.** There... | 6 | https://mathoverflow.net/users/21375 | 138974 | 76,101 |
https://mathoverflow.net/questions/138971 | 6 | Let $G\_0$ be the $\mathbb{R}$-points of a real reductive group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. What is the precise relation between the category of admissible representations of $G\_0$ and the category of admissible $(\mathfrak{g},K)$-modules? If we restrict to unitary (a... | https://mathoverflow.net/users/3544 | Infinitesimal equivalence of admissible representations | Probably the relevant work is that of Casselman and Wallach (independently) on (in effect) adjoints (right? left?) to the forgetful functor taking Lie group repns to Lie algebra repns. The keyword is "globalization". It turns out that a right adjoint is not the left adjoint (which could be anticipated by observing that... | 6 | https://mathoverflow.net/users/15629 | 138975 | 76,102 |
https://mathoverflow.net/questions/137632 | 6 | Let $a\_n$ denote the Fibonacci numbers, for a prime $p$ let $\alpha(p)$ denote the first index $n$ such that $p|a\_n$ and let $r$ denote the golden ratio.
Q: Is there a proof of $\lim\_{x\rightarrow \infty}\frac{1}{x^2}\sum\_{\alpha(p)\leq x} \log p = \frac{3 \log r}{\pi^2}$ ?
The sum is taken over all primes $p$ ... | https://mathoverflow.net/users/17879 | Sum of the log of all primes dividing at least one Fibonacci number up to index x | As far as I understand the answer to my question is no. That seems plausible since, differently from what I thought first, the truth of the conjecture apparently does not have serious consequences.
| 1 | https://mathoverflow.net/users/17879 | 139002 | 76,113 |
https://mathoverflow.net/questions/138790 | 9 | We are forced to use forcing for almost all "hard" independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. The question simply is:
**Primary Question:** Is there any "forcing free" proof for $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $ or $Con(ZF)\longrightarrow Con (ZF+\neg AC) $ or any other "ha... | https://mathoverflow.net/users/nan | Is there any forcing free proof for hard independence results? | Krivine realizability can be used to obtain independence results over $\mathsf{ZF}$. For instance, in Krivine's paper *[Realizability algebras II : new models of ZF + DC](http://www.pps.univ-paris-diderot.fr/~krivine/articles/R_ZF.pdf)*, Logical Methods in Computer Science 8 (1:10) p. 1-28 (2012), he constructs a reali... | 6 | https://mathoverflow.net/users/30790 | 139017 | 76,119 |
https://mathoverflow.net/questions/139020 | 8 | Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) \times \operatorname{Ho} (\cal D)$, and that this is true for any (finite I assume) number of model categories.
Is this t... | https://mathoverflow.net/users/38418 | Is the localisation of a product of categories the product of the localisation? | It is true for arbitrary products of model categories (or just cofibration categories) as proven in Theorem 7.1.1 of <http://arxiv.org/abs/math/0610009v4>.
It is also true for finite products of arbitrary relative categories as discussed in this answser: [Localizing an arbitrary additive category](https://mathoverflo... | 12 | https://mathoverflow.net/users/12547 | 139023 | 76,120 |
https://mathoverflow.net/questions/138962 | 8 | This fact is an easy consequence of results of the [paper](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-90/issue-2/Classes-of-Banach-spaces-with-unique-isometric-preduals/pjm/1102778987.full) by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?
| https://mathoverflow.net/users/19593 | Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces | In this particular case, you can proceed similarly as in Example 2.1 [here](https://web.archive.org/web/20100610042345/http://math.uchicago.edu/%7Eamwright/DavidsonWright2.pdf). For more general results please consult a fantastic survey on unique preduals by G. Godefroy:
>
> G. Godefroy. Existence and uniqueness of... | 6 | https://mathoverflow.net/users/15129 | 139024 | 76,121 |
https://mathoverflow.net/questions/139025 | 5 | Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?
| https://mathoverflow.net/users/38423 | Fixed points on boundary of hyperbolic group | Let $G$ be a hyperbolic group with the Cayley graph $X$.
Let $F<G$ be a finite subgroup and $L\subset \partial G$ be the fixed-point set of $F$.
I will assume that $F$ is the maximal finite subgroup with the fixed-point set $L$.
Lemma. The normalizer $H$ of $F$ in $G$ has the property that:
1. $H$ is quasiconvex... | 6 | https://mathoverflow.net/users/21684 | 139032 | 76,125 |
https://mathoverflow.net/questions/139030 | 3 | A codimension one foliation $\cal F$ on a smooth manifold $M$ is *taut* if every leaf of $\cal F$ meets a closed transversal (i.e., a simple closed curve that is everywhere transversal to the leaves of the foliation). Is it true that a taut foliation admits a closed transversal $\gamma$ that meets *all* leafs of the fo... | https://mathoverflow.net/users/38433 | taut foliations and the existence of total transversals | Yes, this is true. You can find a proof in Calegari's book, "Foliations and the Geometry of
3–Manifolds", lemma 4.26.
| 6 | https://mathoverflow.net/users/3460 | 139034 | 76,126 |
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