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https://mathoverflow.net/questions/136920 | 5 | Let $T$ be a triangle-free graph on $n$ vertices with minimum degree $\delta$ (which can be $0$). How does one show that $n >2\delta -1$? It seems to be true for bipartite graphs, but I cannot see how to prove it for general triangle free graphs in general. The motivation behind this question is if this is solved, I wi... | https://mathoverflow.net/users/37221 | Relationship between triangle free graphs and their minimum degree | In addition to the literature mentioned in the other answers, one can try some
arguments based on counting.
Let $u$ and $v$ be two of $n$ vertices in a triangle-free graph, and further assume they
are distinct and connected by an edge, with degrees $c$ and $d$ and $c$ at most $d$.
Then their neighborhoods are disjoin... | 5 | https://mathoverflow.net/users/35626 | 136951 | 75,229 |
https://mathoverflow.net/questions/136948 | 15 | For some $g \geq 2$, let $\Gamma\_g$ be the fundamental group of a closed genus $g$ surface and let $S\_g=\{a\_1,b\_1,\ldots,a\_g,b\_g\}$ be the usual generating set for $\Gamma\_g$ satisfying the surface relation $[a\_1,b\_1]\cdots[a\_g,b\_g]=1$. Dehn proved the following famous theorem.
**Theorem** : Let $w$ be a r... | https://mathoverflow.net/users/37231 | Dehn's algorithm for word problem for surface groups | Dehn gave at least three solutions of the conjugacy problem for surface
groups, which can be found in my translation in the book *Papers on Group
Theory and Topology* (Springer 1986), Papers 2, 4, and 5.
The first is based on an idea of Poincaré: lifting a curve to the universal
cover, which is the disk model of t... | 17 | https://mathoverflow.net/users/1587 | 136954 | 75,230 |
https://mathoverflow.net/questions/136919 | 2 | I have a square S, and I want to convert it to the unit disc D.
The [Riemann mapping theorem](http://en.wikipedia.org/wiki/Riemann_mapping_theorem) says that I can do this with a conformal bijective map. But, any such mapping will cause some distortion.
Specifically, if S contains small sub-squares, they will be ma... | https://mathoverflow.net/users/34461 | What happens to small squares in Riemann mapping? | 1. Question: In general it seems difficult to find max and mean distortion as a function of $x$. For the square is looks doable.
The following papers describe numerical code to find the Riemann mapping for arbitrary simple curves (described by landmark points).
* Sergey Kushnarev: Teichons: Soliton-like Geodesics on... | 2 | https://mathoverflow.net/users/26935 | 136956 | 75,232 |
https://mathoverflow.net/questions/136959 | 4 | Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}\_p$. Put $K\_\infty = K(\mu\_{p^\infty})$, the field extension obtained by adjoining all $p$-power roots of unity to $K$.
I want to prove that: if $u \in {\mathcal{O}\_K^\times}$ is not a root of unity, then the field extension $K\_u$ of $K\_\infty$... | https://mathoverflow.net/users/37040 | Kummer theory isomorphism and Kummer extensions | Well, let me try in an elementary way. Pick any field $K$ of characteristic prime to $p$ and suppose it does not contain any $p^n$-th root of unity. Let $a\in K^\times\setminus (K^\times)^p$. Then, the extension $K(\sqrt[p^n]{a},\mu\_{p^n})/K$ is normal, not abelian and its Galois group is isomorphic to $\Delta\_n\ltim... | 2 | https://mathoverflow.net/users/18238 | 136964 | 75,234 |
https://mathoverflow.net/questions/136894 | 3 |
>
> **Question:** For which "interesting" classes of finitely generated groups is
> it known whether every infinite group in the class has an element of infinite order?
>
>
>
Some examples:
1. For finitely generated abelian groups the question trivially has a positive answer.
2. For recursively presented gro... | https://mathoverflow.net/users/28104 | Classes of finitely generated groups for which it is known whether they contain periodic groups | Since the comments are already answering the question I am copying here the answers so far as Community Wiki. Please add new answers here.
* Linear groups
* Groups with a regular language of unique normal forms. This includes hyperbolic groups, automatic groups and groups with a finite complete rewriting system.
(T... | 5 | https://mathoverflow.net/users/15934 | 136967 | 75,235 |
https://mathoverflow.net/questions/136903 | 7 | Is there a counter-example showing that Morley rank in the theory of compact complex spaces (as defined by Zilber, Pillay, Moosa, ...) is not definable in families?
Given the existence of such a counterexample in DCF (constructed by Hrushovski and Scanlon; it is a family of Abelian varieties) and the similarity betwe... | https://mathoverflow.net/users/2234 | Definability of Morley rank in the theory of Compact Complex Spaces(CCS) | Pillay and Scanlon gave an example showing Morley rank isn't definable in elementary extensions in: Compact complex manifolds with the DOP and other properties, J. Symbolic Logic, vol. 67 (2002), pp. 737–743.
In a different direction Dale Radin showed the you do have definability of dimension in the standard model in... | 6 | https://mathoverflow.net/users/5849 | 136975 | 75,239 |
https://mathoverflow.net/questions/135543 | 3 | Let me first say that my background is theoretical physics so I find it hard to look at some of the mathematical literature.
The kind of problem I am interested in is the following one. Consider a 4-dimensional smooth, simply connected orientable manifold M and suppose A and B are immersed surfaces. Denote with $A\#B... | https://mathoverflow.net/users/29850 | Self-intersection of immersed surfaces and connected sum | The answer is: **yes**, the formula holds.
For any $4n$-dimensional manifold and $2n$-dimensional immersed submanifolds $A$ and $B$. (Everything must be oriented if you want consider an integer formula, intersections with signs. If you drop the condition on orientability, then you still can count the points of inters... | 3 | https://mathoverflow.net/users/36950 | 136981 | 75,241 |
https://mathoverflow.net/questions/136993 | 3 | Let $F$ be a nonarchimedean local field. Does there exist a semisimple algebraic group over $F$ which is not split over a maximal unramified extension of $F$ ?
| https://mathoverflow.net/users/3824 | Semisimple group not split by an unramified extension? | The Galois group of $F$ maps onto $S\_3$, the symmetric group on three letters. The group $S\_3$ is also the group of diagram automorphisms of the Dynkin diagram of $SO(8)$. That means that there is a quasi-split group $G$ defined over $F$, which over an extension $E/F$ with Galois group $S\_3$, becomes the split group... | 3 | https://mathoverflow.net/users/23291 | 136996 | 75,244 |
https://mathoverflow.net/questions/136982 | 2 | Assume $\Gamma$ is a discrete subgroup of some $GL\_n$, and let $G$ be its Zariski closure.
Let $H$ an algebraic cocompact normal subgroup of $G$. Do we have that $H\cap \Gamma$ is of finite index in $\Gamma$.
I'm not sure this is true (actually i think it should be false), i'm far from being an expert in this field, ... | https://mathoverflow.net/users/37253 | on the zariski closure and cocompactness of discrete groups | This is not true. There are irreducible (arithmetic) lattices $\Gamma=G({\mathbb Z})$ in $$SL \_2({\mathbb R})\times SU(2)=G({\mathbb R})$$ such that $\Gamma$ projects densely into $SU(2)$ and its projection to $SL\_2({\mathbb R})$ is a cocompact lattice. $G$ is an algebraic $\mathbb Q$-subgroup of $GL\_n$ and the Zari... | 3 | https://mathoverflow.net/users/23291 | 136998 | 75,246 |
https://mathoverflow.net/questions/126123 | 13 | In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get access to Takagi's thesis, does anyone have any idea how he did this?
I know how to prove complex multiplication (e.g.... | https://mathoverflow.net/users/1355 | How did Takagi prove Kronecker's Jugendtraum for Q(i)? | Takagi's goal is the following:
1. show the existence of sufficiently many cyclic extensions defined
by division values of sn $u$ (the lemniscatic sine);
2. prove that each abelian extension of ${\mathbb Q}(i)$ is contained in
the compositum of these fields.
Step 1 is analogous to the construction of the field... | 23 | https://mathoverflow.net/users/3503 | 137003 | 75,248 |
https://mathoverflow.net/questions/136980 | 6 | An $\omega$-Turing machine is just a usual Turing machine $T=(Q,\Sigma,\Gamma,\delta,q\_0,F)$ where $Q$ is the finite set of states, $\Sigma$ is the input alphabet, $\Gamma\supset\Sigma$ is the tape alphabet, $\delta$ is is the transition relation, $q\_0$ is the initial state and $F\subset Q$ is the set of accepting st... | https://mathoverflow.net/users/33842 | Where does the deterministic simulation of non-deterministic ω-Turing machines fail? | You say we can remove the condition that a run read every input only finitely many times, but I don't think that's so. As you noted, König's lemma shows that acceptance is a $\Pi^0\_1$ property if we remove that condition. That means that the language of such a machine is a $\Pi^0\_1$-class.
On the other hand, if we ... | 5 | https://mathoverflow.net/users/32178 | 137004 | 75,249 |
https://mathoverflow.net/questions/137001 | 5 | Does every contractively complemented subspace of $c\_0(I)$ is isometric to $c\_0(J)$ for some $J\subseteq I$?
May be someone has a counterexample?
| https://mathoverflow.net/users/19593 | Contractively complemented subspaces of $c_0(I)$ | Yes, it is true, but I don't know where it is proved in the literature. The reason I say yes is that what you want is a corollary of a unpublished almost isometric theorem that Zippin proved in the 1970s! Here is either an outline of a proof or complete nonsense: Let $P$ be a contractive projection on $c\_0(I)$. Then t... | 6 | https://mathoverflow.net/users/2554 | 137005 | 75,250 |
https://mathoverflow.net/questions/136999 | 3 | I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.
Let $X$ be a complex manifold. Let $E$ be a smooth, complex vector bundle. Let $C^{\infty}(E)$ be the sheaf of smooth sections of $... | https://mathoverflow.net/users/297 | Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle | This is Theorem 2.1.53 in Donaldson--Kronheimer's *Geometry of 4-manifolds*. They prove it from scratch via the contraction mapping principle but there are arguments from the Newlander--Nirenberg theorem also.
| 3 | https://mathoverflow.net/users/1143 | 137008 | 75,253 |
https://mathoverflow.net/questions/136962 | 5 | I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.
Do you know where I can find a complete proof of such a result :
There exists a unique solution $u$ to the problem
$\left\{
\... | https://mathoverflow.net/users/37238 | Heat Equation on $[0,T] \times \mathbb{R}^n$ | Fourier transform in $x$ gets you there:
$
\dot v+\eta\vert\xi\vert^2 v=g(t,\xi),\quad v(0)=v\_0,
$
so that
$$
v(t,\xi)=e^{-t\eta \vert\xi\vert^2} v\_0(\xi)+\int\_0^te^{-(t-s)\eta \vert\xi\vert^2} g(s,\xi) ds.
$$
Since $\iint\_0^T\vert g(t,\xi)\vert^2 dtd\xi<+\infty$ and
$\int\vert v\_0(\xi)\vert^2 d\xi<+\infty$,
you ... | 7 | https://mathoverflow.net/users/21907 | 137014 | 75,256 |
https://mathoverflow.net/questions/137013 | 2 | If I have $n$ variables $x\_1, \cdots, x\_n$, and a set $S$ of inequalities of the form $p(x\_1, \cdots, x\_n) > 0$ where $p$ is a homogenous real polynomial, is it true that I need $|S| \geq n$ to prove that $x\_i > 0$ for each $i$?
| https://mathoverflow.net/users/32926 | Homogenous polynomial inequalities to prove positivity? | No. Take $n=3$ with the equations $x^2+y^2+z^2 < 2(xy+xz+yz)$ and $x+y+z > 0$.
The first inequality is a double cone tangent to all three coordinate planes; one component lies in the positive orthant and the other in the negative orthant.
The second inequality is a half space which makes sure we are in the first co... | 5 | https://mathoverflow.net/users/297 | 137015 | 75,257 |
https://mathoverflow.net/questions/133781 | 9 | **Now I understand the answers.**
I am trying to understand **Henselian Weierstrass Theorem** in Hironaka's **Idealistic exponents of singularity**, page 76 - 77. A glance at [the paper](https://docs.google.com/file/d/0B-vOsANdAoEJUUk3TnZhZENUNVU/edit?usp=sharing).
At some point there is a ring $R$, which is Noeth... | https://mathoverflow.net/users/5506 | Finite generation and Henselization | (Note: There is no new content in this answer; it is simply an explanation of parts of user's excellent response. I'm writing it as an answer, because that seems less perverse than splitting it across many comments.)
The key result that is being used here is the following consequence of Zariski's Main Theorem + Hense... | 4 | https://mathoverflow.net/users/7868 | 137018 | 75,260 |
https://mathoverflow.net/questions/137023 | 0 | EDIT: Let me modify the question then: for what submanifolds $N$ does the torsion $T$ preserve tangent vectors to $N$?
If $\nabla$ is a connection on a manifold $M$, then torsion is defined to be the map
$$
T(X,Y)=\nabla\_XY-\nabla\_YX-[X,Y]
$$
where $X$ and $Y$ are vector fields on $M$. It can be shown that $T$ is... | https://mathoverflow.net/users/14454 | Torsion and submanifolds | Edit: Answering Robert, moved to 3-space to give a general example:
A simple example in $M=\mathbb R^3$: Let $N=0\times \mathbb R^2$ and put
$$
\nabla\_XY = dY(X) + \begin{pmatrix}X^T\,A^1\,Y \\ X^T\,A^2\,Y \\ X^T\,A^3\,Y\end{pmatrix},
\quad
A=\begin{pmatrix} a^i\_{11} & a^i\_{12} & a^i\_{13}\\ \dots \\a^i\_{31} & a... | 3 | https://mathoverflow.net/users/26935 | 137040 | 75,266 |
https://mathoverflow.net/questions/137048 | 1 | Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the [Hewitt-Ross, Abstract harmonic analysis](http://rads.stackoverflow.com/amzn/click/3540048324), (27.7)), i.e. the algebra of functions $u:G\to {\mathbb C}$ which can be represented as complex linear c... | https://mathoverflow.net/users/18943 | Tangent vectors on the algebra of trigonometric polynomials | Yes, of course. The functions you have are representation functions on $G$ (the representations are finite dimensional as well); they are thus algebraic functions on the complexification of $G$. It is well known that the "algebraic Lie algebra is the true tangent space for algebraic groups.
| 2 | https://mathoverflow.net/users/23291 | 137053 | 75,272 |
https://mathoverflow.net/questions/137044 | 6 | I have two questions about [this](http://ncatlab.org/toddtrimble/published/Towards+a+doctrine+of+operads) note by T. Trimble, which I warmly invite to answer.
**First question**
I am trying to prove that [these](http://ncatlab.org/toddtrimble/published/Towards+a+doctrine+of+operads#concrete_description_of_cartesian... | https://mathoverflow.net/users/7952 | Trimble's theory of operads | For the first question, we have natural isomorphisms
$$U V(\Phi)(F) = \int^n F(n)\cdot \Phi(\hom(1,-))^n \cong \int^n F(n)\cdot \Phi(\hom(1,-)^n) \cong \int^n \Phi(F(n)\cdot \hom(1,-)^n)$$
$$\cong \Phi(\int^n F(n)\cdot \hom(1,-)^n) \cong \Phi(\int^n F(n) \cdot \hom(n, -)) \cong \Phi(F)$$
where the first isomor... | 11 | https://mathoverflow.net/users/2926 | 137062 | 75,274 |
https://mathoverflow.net/questions/137022 | 14 | What is known, and what is published, on the reverse mathematics of the nest of results called [Hilbert's Theorem 90](http://en.wikipedia.org/wiki/Hilbert%27s_Theorem_90)?
| https://mathoverflow.net/users/38783 | Reverse mathematics of Hilbert's Theorem 90 | It is provable in RCA0 that if $K$ is a finite Galois extension of $F$ with Galois group $G$ then $H^1(G,L^\times)$ is trivial (every cocycle $a:G \to L^\times$ is a coboundary).
The finite Galois theory in RCA0 was discussed by Friedman, Simpson and Smith in [*Countable algebra and set existence axioms*](http://dx.d... | 11 | https://mathoverflow.net/users/2000 | 137067 | 75,276 |
https://mathoverflow.net/questions/137080 | 4 | A cardinal $\kappa$ is huge iff there is $\lambda>\kappa$ and a $\kappa$-complete normal ultrafilter on
$P\_{\leq \kappa}(\lambda)$, or, equivalently, on the set of families of subsets of $\lambda$ of order-type $\kappa$. [1]
I want to know whether one can further assume that $U$ is concentrated on something smaller? F... | https://mathoverflow.net/users/4745 | ultrafilter characterisation of huge cardinals | The hugeness of $\kappa$ is witnessed by an embedding $j:V\to M$ for which $M^\lambda\subset M$, where $\lambda=j(\kappa)$. In particular, for such an embedding we have $j''\lambda\in M$, and one may accordingly consider the induced measure on $P\_{\leq\kappa}(\lambda)$ defined by $X\in U\iff j''\lambda\in j(X)$. The u... | 6 | https://mathoverflow.net/users/1946 | 137085 | 75,283 |
https://mathoverflow.net/questions/137076 | 4 | I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the length of a shortest path between $\gamma(-r)$ and $\gamma(r)$ in the complement $X \setminus B(\gamma(0), r)$ of a ball a... | https://mathoverflow.net/users/37302 | Divergence of geodesics in mapping class groups | As Sam Taylor told you, the paper to read is by Duchin and Rafi. They prove that the divergence rate is between linear and quadratic. The second question is about rate of divergence along axes of pA elements in the mapping class group. In this case, the rate is quadratic, see theorem 4.2 in the same paper of Duchin and... | 5 | https://mathoverflow.net/users/21684 | 137087 | 75,285 |
https://mathoverflow.net/questions/103098 | 16 |
>
> What is known about isometric actions on $\mathbb S^n$ such that the quotient space is homeomorphic to $\mathbb S^n$?
>
>
>
**Comments.**
* I am mostly interested in (maybe trivial) properties of such actions for large $n$.
* I see that the orientation preserving part of Coxeter's group has this property.
... | https://mathoverflow.net/users/1441 | Actions on Sⁿ with quotient Sⁿ | I think I have completely answered the question in the following form:
**Theorem**. *For a finite subgroup $\Gamma < O(n)$ the quotient space $S^{n-1}/\Gamma$ is homeomorphic to $S^{n-1}$ if and only if $\Gamma$ has the form
\begin{eqnarray\*}
\Gamma = \Gamma\_{ps} \times P\_1 \times \ldots \times P\_k
\end{eqnarray... | 12 | https://mathoverflow.net/users/37353 | 137134 | 75,310 |
https://mathoverflow.net/questions/137135 | 21 | One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity topos among locally Cartesian closed infinity categories. This is generally presented as a commonly accepted fact but no... | https://mathoverflow.net/users/22131 | $\infty$-categorical interpretation of type theory | Mike Shulman has a good discussion of what exactly needs to be shown in his
* Michael Shulman, "The univalence axiom for inverse diagrams" ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253))
The issue is that in the type theory the classification of objects is by strict pullbacks. So roughly the problem is that one... | 14 | https://mathoverflow.net/users/381 | 137142 | 75,313 |
https://mathoverflow.net/questions/137114 | 47 | First, I will explain my situation.
In my University most of the careers are doing videos to explain what we do and try to attract more people to our careers.
I am in a really bad position, because the people who are in charge of the video want me to explain what a pure mathematician does and how it helps society. Th... | https://mathoverflow.net/users/37338 | How does the work of a pure mathematician impact society? | Here are two recent speakers from our departments lecture series on applied mathematics (the Wing Lectures at U. Rochester). We've had a lot of great lecturers, but these two stick out to me as having an impact on society.
[Adrien Treuille](http://www.cs.cmu.edu/~treuille/) -- he works in computer graphics, and his r... | 26 | https://mathoverflow.net/users/36862 | 137148 | 75,316 |
https://mathoverflow.net/questions/137152 | 6 | I've been going back over some results from Munkres's *Topology*, and I'm curious about some things. (I originally posted this on [M.SE](https://math.stackexchange.com/q/425948/28900), but I think it is probably a better fit here.)
I know that Choice principles have *some* connection to the separation axioms (in ZF, ... | https://mathoverflow.net/users/22840 | A few standard results (on metrizability and relative separation strength) without Choice? | * Paul Howard, Kyriakos Keremedis, Jean E. Rubin, Adrienne Stanley,
[*Paracompactness of metric spaces and the axiom of multiple choice*](http://dx.doi.org/10.1002/%28SICI%291521-3870%28200005%2946:2%3c219%3a%3aAID-MALQ219%3e3.0.CO%3B2-2) [MLQ 46 (2000), 219–232; [MR1755811](http://www.ams.org/mathscinet-getitem?mr=175... | 4 | https://mathoverflow.net/users/2000 | 137160 | 75,319 |
https://mathoverflow.net/questions/137155 | 6 | Let $X$ be a either a projective scheme or a compact complex space. Then one has an exact sequence $$ (1) \quad 0 \to \textrm{Pic}(X) \to \textrm{Cl}(X) \to \bigoplus\_{x \in \textrm{Sing}(X)} \textrm{Cl}(\mathcal{O}\_{X,x}),$$
where $\textrm{Pic}(X)$ and $\textrm{Cl}(X)$ denote the groups of Cartier (resp. Weil) divi... | https://mathoverflow.net/users/7460 | Exact sequence for divisor class groups | The exponential sequence does not quite work that way for the *algebraic cohomology* $H^q(X,\mathcal{O}\_{X}^\*)$ for $q>1$. Of course the analytic cohomology, $H^q(X^{\text{an}},\mathcal{O}\_{X^{\text{an}}}^\*)$, does satisfy the exponential sequence. However, GAGA does not directly apply for $q>1$. Of course it does ... | 11 | https://mathoverflow.net/users/13265 | 137164 | 75,323 |
https://mathoverflow.net/questions/137115 | 7 | Say we have an AF C\* algebra $A$ described by some Bratteli diagram $E$. If $M\_\infty (A)=\displaystyle{\lim\_\rightarrow M\_n(A)}$ and $P(A)$ are the projections in this algebra, we know that $K\_0(A)^+=P(A)/\sim$ where $\sim$ is the von Neumann equivalence relation, and the Grothendieck group construction gives us ... | https://mathoverflow.net/users/37339 | $K_0$ group of graph underlying an approximately finite (AF) C* algebra | Kerov's thesis "Asymptotic representation theory of the symmetric group" has a nice discussion of these ideas. I think I can remember an answer to your question in that book, but I'll have to check later.
As you probably know, the ordered abelian group $K\_0(A)$ is a complete invariant of the algebra $A$. Here is a c... | 5 | https://mathoverflow.net/users/9068 | 137169 | 75,325 |
https://mathoverflow.net/questions/137168 | 3 | I've seen some estimates for the decay in $d$ of the probability a SRW makes a distance $d$ in time $n$, but is there any reference for the "speed" of a random walk in a group? I'm interested mostly for nilpotent groups (transient ones), though subexponential or polycyclic could be also nice.
Let $X\_n$ be the rando... | https://mathoverflow.net/users/18974 | Speed of random walks in groups | For nilpotent groups the speed exponent is 0.5. See this paper for the general picture <http://arxiv.org/abs/1203.6226> .
| 4 | https://mathoverflow.net/users/33828 | 137170 | 75,326 |
https://mathoverflow.net/questions/137156 | 2 | My current research project involves sampling a sequence along non-constant polynomials with non-negative integer coefficients defined over the natural numbers, and I'd like to be able to say when two such polynomials are "independent".
Say for two polynomials of this type $p$ and $q$, that $q$ is independent of $p$ ... | https://mathoverflow.net/users/37316 | Relative density of images of diophantine polynomials | Faltings theorem seems overkill here. As Gerry suggested, you should use Siegel's theorem since you are dealing with integer points and the hypotheses are easier to check. It states that an affine irreducible curve has finitely many integer points unless it is of genus zero and has at most two points at infinity.
In ... | 3 | https://mathoverflow.net/users/2290 | 137178 | 75,327 |
https://mathoverflow.net/questions/137186 | 2 | Does Sigma(n)-KP, i.e. Kripke-Platek set theory with Sigma(n)-collection and Sigma(n)-separation have Sigma(n)-replacement.
| https://mathoverflow.net/users/37385 | A question on collection and replacement in Sigma(n)-KP | Yes. If we have a set $A$ and for every $a\in A$ there is a unique $b$ such that $\varphi(a,b,z)$, where $\varphi$ has complexity $\Sigma\_n$, then by $\Sigma\_n$ collecton there is a set $B$ such that every $a\in A$ has a $b\in B$ with $\varphi(a,b,z)$. But perhaps there is extra stuff in $B$, so this set may be too l... | 1 | https://mathoverflow.net/users/1946 | 137187 | 75,329 |
https://mathoverflow.net/questions/137051 | 3 | I have a question about Faltings' paper "Crystalline cohomology and p-adic Galois representations". Suppose $R$ is a smooth $\mathbb{Z}\_p$-algebra, of relative dimension $1$, such that there is an etale map $\mathbb{Z}\_{p}[T,T^{-1}]\to R$.
By $\bar{R}$ we denote the maximal extension of $R$ which is etale in charac... | https://mathoverflow.net/users/4504 | Surjectivity of frobenius | One can prove this using the almost purity theorem. The argument is presented below (essentially extracted from Scholze's paper on perfectoid spaces, but is presumably also in Faltings' papers), and it works in any dimension. I do not know an elementary proof.
---
Note first that $\overline{R}$ is an algebra ove... | 2 | https://mathoverflow.net/users/37395 | 137203 | 75,336 |
https://mathoverflow.net/questions/137208 | 5 | Is there any evidence that axiomatic reasoning has been used prior to Thales of Milet (624-547BC), who is generally credited for the "invention" of axioms.
In this context I understand axioms in the original interpretation as checkable facts, whose truth is beyond doubt.
The motivation for my question is the follow... | https://mathoverflow.net/users/31310 | Origins of Axiomatic Reasoning | You'll find an extensive overview of early uses of the axiomatic method in [Axiomatic Philosophy](http://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=8122408117), by Pritam Sen (1996). One of the earliest sources is the *Upanishad* of the Hindu faith, written in the period 1000-500 BC.
| 5 | https://mathoverflow.net/users/11260 | 137212 | 75,341 |
https://mathoverflow.net/questions/137197 | 2 | Are Sigma(n)-admissible ordinals for n>0, i.e. ordinals such that Gödelś constructible hierarchy at that level is a model of Sigma(n)-KP, recursively inaccessible? According to Wikipedia on large countable ordinals it seems to follow that already Sigma(1)-admissible ordinals are *nonprojectible* and highly recursively ... | https://mathoverflow.net/users/37385 | A question on Sigma(n)-admissible ordinals | For $n\gt 1$, the answer is yes. An ordinal is *recursively inaccessible* if it is admissible and a limit of admissible ordinals. And every $\Sigma\_{n+1}$-admissible ordinal is a limit of $\Sigma\_n$-admissible ordinals, by a reflection type argument, and hence recursively inaccessible.
But for $n=1$, the answer is... | 2 | https://mathoverflow.net/users/1946 | 137213 | 75,342 |
https://mathoverflow.net/questions/137215 | 0 | Let $X$ and $Y$ be identically distributed bounded random variables defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$. I want to know if there always exists an invertible measure-preserving transformation $T: \Omega \to \Omega$ such that $ X \circ T= Y$ holds almost surely.
When $X$ and $Y$ are chara... | https://mathoverflow.net/users/3635 | identically distributed random variables and measure-preserving transformations | No. Take for example the case where $\Omega = [0,1]$ with Lebesgue measure, $X$ is the identity, and $Y = 2X\mod 1$. Then you ask for $T = X\circ T = Y$ almost surely, but changing $Y$ on a set of measure zero cannot make it invertible.
| 1 | https://mathoverflow.net/users/5963 | 137217 | 75,343 |
https://mathoverflow.net/questions/137108 | 7 | Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” begins by remarking that when X is the projective line over the complex numbers, minus three points: "every finite cover of X can be described by equations with algebraic number coefficients."
see <http://www.math.ias.edu/files/del... | https://mathoverflow.net/users/38783 | Algebraic numbers and the complex projective line minus three points | The comments are all correct. Deligne is citing one half of what is called Belyi's theorem, though experts feel this half was actually clear much earlier due to Weil. Weil used a version of the insight described above by JSE. B Kock gives a nice exposition in "<http://arxiv.org/abs/math/0108222>".
| 3 | https://mathoverflow.net/users/38783 | 137224 | 75,345 |
https://mathoverflow.net/questions/137219 | 8 | Let $M\subset\mathbb{N}$ be a finite set. For every positive integer $n$ set
$$D\_n(M)=\{W\subset \mathbb{N} \text{ finite }|\ \forall\ i=0,\ldots,n-1\ \exists\ w\in W: w\in M+i\},$$ where $M+i=\{m+i|\ m\in M\}$. I would like to understand
$$d\_n(M)=Min\{|W|\ | \ W\in D\_n(M)\}.$$
I am mainly interested in the asymptot... | https://mathoverflow.net/users/21885 | Is this a known combinatorial problem? | Yes this is a known combinatorial problem. You should look into the literature of covering or tiling groups by translates of a finite set. Your function $d(M)$ is commonly known as the "covering density". Newman called it "codensity" in his article "Complements of finite sets of integers", Michigan Math J 14 (1967), 48... | 14 | https://mathoverflow.net/users/2384 | 137225 | 75,346 |
https://mathoverflow.net/questions/137172 | 8 | Is there a standard name for order-preserving maps $f\colon P\to Q$ of posets with the property that the image of a lower set is a lower set, or equivalently if $q\leq f(p)$ then there exists $p'\leq p$ with $f(p')=q$?
If you view the poset as a category, then this condition says that the functor associated to $P$ is... | https://mathoverflow.net/users/15934 | Terminology question for poset maps | I think one name for this is a [simulation](http://ncatlab.org/nlab/show/simulation).
| 4 | https://mathoverflow.net/users/49 | 137226 | 75,347 |
https://mathoverflow.net/questions/137220 | 4 | Let $\mathcal{C}$ be a $c$-unital $A\_\infty$-category. If $\mathcal{A}$ is a $c$-unital and triangulated $A\_\infty$-category, then there is a $c$-unital $A\_\infty$- functor
$$Tw: fun(\mathcal{C},\mathcal{A})\rightarrow fun(Tw \mathcal{C}, Tw\mathcal{A})\rightarrow fun(Tw \mathcal{C}, \mathcal{A}),$$
See (3.26) in s... | https://mathoverflow.net/users/37400 | Is the functor $mod(Tw\mathcal{C})\rightarrow mod(\mathcal{C})$ cohomologically full and faithful? | This seems to me to be unproblematic (if messy to prove). Try if this does the job:
Proposition 12.15(2) in Bespalov-Lyubashenko-Manzyuk, Pretriangulated A\_infty-categories
| 2 | https://mathoverflow.net/users/37410 | 137229 | 75,349 |
https://mathoverflow.net/questions/137153 | 6 | Let $A$ be a real $n\times n$ matrix and let $\mu\_1, \dots, \mu\_n$ the (generalized, complex) eigenvalues of $A$. Assume that
$$ 0 < \alpha < \mathrm{Re}(\mu\_1) < \dots < \mathrm{Re}(\mu\_n).$$
I am interested in a lower bound on the eigenvalues of $A^t A$ in relation to $\alpha$.
If $A$ is symmetric, then the l... | https://mathoverflow.net/users/16702 | Lower bounds on matrix eigenvalues | No, there is no such $C(\alpha)$, even for fixed $n>1$.
Indeed, consider the matrix $A = \begin{pmatrix} \alpha & x \\ 0 & \alpha\end{pmatrix}$.
The product of the eigenvalues of $A^t A$ does not depend on $x$ (it is equal to $det(A)^2=\alpha^4$), whereas their sum goes to $\infty$ (it is $Tr(A^tA)=2\alpha^2+x^2$).... | 5 | https://mathoverflow.net/users/10265 | 137239 | 75,354 |
https://mathoverflow.net/questions/137241 | 3 | Suppose that $(M,\omega)$ is a (connected compact) symplectic manifold with a Hamiltonian $S^1$-action given by Hamiltonian $H$. I would like to find a reference for the fact that every level set of $H$ is connected. I tried to find this statement in McDuff Salamon, but could not.
| https://mathoverflow.net/users/13441 | Level sets of Hamiltonians of S^1 actions | Michael F. Atiyah, [Convexity and commuting Hamiltonians](http://www.ams.org/mathscinet-getitem?mr=642416) (1982), Lemma 2.3.
Dusa McDuff and Dietmar Salamon, [Introduction to symplectic topology](http://www.ams.org/mathscinet-getitem?mr=1698616) (2nd ed., 1998), Lemmas 5.51 and 5.54.
Michèle Audin, [Torus actions ... | 6 | https://mathoverflow.net/users/19276 | 137250 | 75,357 |
https://mathoverflow.net/questions/137244 | 0 | Let $\pi:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connexion $A$. Let $X$ be a tangent vector field on $M$ and let $Y$ be a (not-necessarily horizontal) lift of $X$ to $P$. In other words, for every $p\in P$ we have
$$\pi\_{\*,p}(Y\_p)=X\_{\pi(p)}\ .
$$
The vector field $X$ defines a one-parame... | https://mathoverflow.net/users/35227 | Invariant connection by local diffeomorphism | Let $\zeta:\mathfrak g \to \Gamma(VP)$ be the infinitesimal principal right action, where $VP=ker(\pi)$ is the vertical bundle, and describe the connection $A$ equivalently as $\Phi=\zeta\circ\omega:TP\to VP$ where $\omega\in\Omega^1(P,\mathfrak g)$ is the principal connection form. This is the vertical projection asso... | 2 | https://mathoverflow.net/users/26935 | 137253 | 75,359 |
https://mathoverflow.net/questions/137232 | 2 | I have asked this question previously at Math.stackexchange, but it seems to receive little attention there.
In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's properties) infini... | https://mathoverflow.net/users/37415 | Resource on Infinite Systems of Difference Equations | If your equations look like $x\_{i+1} = Ax\_i$, where each $x\_j$ is an element of a (possibly infinite dimensional) vector space and $A$ is a linear operator thereon, then the general solution is $x\_j = A^j x\_0 = \exp(j\log(A))x\_0$, with $x\_0$ specifying the initial conditions. The properties of $A^j$, or equivale... | 1 | https://mathoverflow.net/users/2622 | 137261 | 75,362 |
https://mathoverflow.net/questions/137273 | 3 | In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair $(G\_{1}, G\_{2})$ is called a matched pair of l.c. groups if there exists a l.c. group $G$ such that, $G\_{1}$, $G\_{2}$ are closed subgroups of $G$, intersection of $G\_{1}$ and $G\_{2}$... | https://mathoverflow.net/users/8934 | Matched pair of locally compact groups | I suppose you could ask that the complement of $G\_1G\_2$ is nowhere dense, or more generally a [meagre set](https://en.wikipedia.org/wiki/Meagre_set). But whether this notion is appropriate or not really depends on what application you have in mind.
Also, unless I am missing something, isn't every pair of locally co... | 0 | https://mathoverflow.net/users/703 | 137274 | 75,364 |
https://mathoverflow.net/questions/137227 | 6 | I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
* $E$ is a holomorphic vector bundle.
* There is a Dolbeault operator $\bar{\partial}\_E$, i.e. a $\mathbb{C}$ linear operator $\bar{\partial}\_E : \Omega^{0,0}(E) \to \Omega^{0,1}(E)$ which satisfie... | https://mathoverflow.net/users/21564 | $E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$ | A. Moroianu gives a detailed proof on pp. 72-74 of his [*Lectures on Kähler geometry*](http://www.ams.org/mathscinet-getitem?mr=2325093) (Theorem 9.2), available on the internet. (The [preprint](http://arxiv.org/abs/math/0402223v1) has it as Theorem 3.2.)
He attributes that proof to S. Kobayashi, [*Differential geome... | 4 | https://mathoverflow.net/users/19276 | 137279 | 75,366 |
https://mathoverflow.net/questions/136678 | 9 | What is the shape of the $n$-gon $P\_1P\_2\cdots P\_n$ which gives the maximum of $A\_n$? The quantity $A\_n$ is defined by
$$ A\_n = \frac{{\sum\_{i\lt{j}\le{n}}{\lvert P\_i P\_j\rvert}^2}-{\sum\_{i=1}^{n}{\lvert P\_i P\_{i+1}\rvert}^2}}{{\sum\_{i=1}^{n}{\lvert P\_i P\_{i+1}\rvert}^2}} $$
Here, $\lvert P\_i P\_j \rver... | https://mathoverflow.net/users/34490 | What is the shape of the $n$-gon which gives the maximum of a function? | The optimal shape is the regular $n$-gon and all its affine images.
I am going to optimize the ratio
$$
\frac{\sum\_{i<j} |P\_iP\_j|^2}{\sum\_i|P\_iP\_{i+1}|^2}
$$
which differs from $A\_n$ by 1. For the regular $n$-gon this ratio equals
$$
\frac{n}{2(1-\cos\frac{2\pi}n)} .
$$
Let us prove that this is the maximum.... | 13 | https://mathoverflow.net/users/4354 | 137284 | 75,368 |
https://mathoverflow.net/questions/137282 | 3 | Does Sigma(n)-KP with Sigma(n)-replacement instead of Sigma(n)-collection have Sigma(n)-collection as a theorem?
| https://mathoverflow.net/users/37385 | Another question on collection and replacement in Sigma(n)-KP | In your comment, you linked to my paper [V. Gitman, J. D. Hamkins, T. A. Johnstone, What is the theory ZFC without power set?](http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/) (see it also [at the arXiv](http://arxiv.org/pdf/1110.2430v1.pdf)). And although your theories are much weaker, in fact the mai... | 4 | https://mathoverflow.net/users/1946 | 137286 | 75,369 |
https://mathoverflow.net/questions/137289 | 3 | I would like some help with the proof of the preservation of measurable cardinals in mild extensions, as I am a bit new with forcing.
By mild extensions, I mean the generic extension produced from a forcing notion of size less than $\kappa$, where $\kappa$ is a large cardinal (in my question measurable). I am reading... | https://mathoverflow.net/users/35968 | Preservation of measurable cardinals in mild extensions | If $T$ is not in $U$, then the complement of $T$ is in $U$, which means that $U$ concentrates on ordinals that $p$ forces are not in $\dot X$, which means that $p$ forces that the complement of $\dot X$ is in $W$.
If $T$ is in $U$, then let $T\_q=\{\alpha\mid q\vdash\check\alpha\in \dot X\}$ be the ordinals that $q$... | 5 | https://mathoverflow.net/users/1946 | 137292 | 75,370 |
https://mathoverflow.net/questions/137295 | 8 | Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to $\mathbb R\cup\{\infty\}$ (the last set being the one-point compactification)?
(The fact that $\theta\mapsto\sec\theta+\tan\t... | https://mathoverflow.net/users/6316 | certain trigonometric homeomorphisms | The magic words are $\tan(\theta/2).$ That substitution reduces your question to asking which rational functions $\mathbb{R} \rightarrow \mathbb{R}$ are homeomorphisms. Those are precisely the functions whose derivative does not change sign, so differentiating our function we get a rational function which does not chan... | 17 | https://mathoverflow.net/users/11142 | 137297 | 75,372 |
https://mathoverflow.net/questions/137294 | 2 | In complex field, assume $$f(x)=\sum\_{i=1}^{\infty} a\_i x^i$$ where $a\_i \in {\bf N}$ or $a\_i = 0$, and $f(x)$ converges in an area.
**Question 1:** are there $$f(x)=p(x)+\sum\_{i=1}^{\infty}r\_i(x), $$
or $$f(x)=p(x)+\sum\_{i=1}^{n}r\_i(x),$$
where $p(x)$ is a polynomial with all coefficients which are natural n... | https://mathoverflow.net/users/14024 | Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$ | Consider the sequence $1,0,0,1,0,0,0,0,1,\dots$ given by $a\_n=1$ if $n$ is a square, otherwise $a\_n=0$. I claim that no nonzero sequence $b\_n$ of non-negative integers with $b\_n\le a\_n$ for all $n$ can satisfy a constant coefficient homogeneous linear recurrence relation. For any sequence satisfying such a relatio... | 3 | https://mathoverflow.net/users/3684 | 137302 | 75,373 |
https://mathoverflow.net/questions/137259 | 1 | Let $C$ be a category and for each object $X$ let us denote by ${\sf Mono}(X)$ the category of all monomorphisms in $C$ going to $X$ (i.e. monomorphisms which have $X$ as range -- I hope it is clear how morphisms in ${\sf Mono}(X)$ are defined).
$C$ is said to be *well-powered* (see [MacLane](http://www.maths.ed.ac.u... | https://mathoverflow.net/users/18943 | "Uniformly (co)well-powered" categories? | At Sergei Akbarov's suggestion, I'm copying my comment into this answer.
In the presence of the other usual axioms of set theory, including especially the axiom of regularity, the axiom of global choice gives you a well-ordering of the class of all sets. Given such a well-ordering, you can define $\mathcal S\_X$ to c... | 3 | https://mathoverflow.net/users/6794 | 137304 | 75,374 |
https://mathoverflow.net/questions/137315 | 2 | Let $\bf Top$ a convenient category of topological spaces, $G$ a group in $\bf Top$, ${}^G\bf Top$ the category of (left) $G$-spaces, and $Sgrp(G)$ the poset of subgroups of $G$.
Define two functors [and prepare yourself to a couple of slight abuses of notation]:
1. ${}^G{\bf Top}\times Sgrp(G)^\text{op}\to {\bf To... | https://mathoverflow.net/users/7952 | A $G$-space as a coend | It doesn't seem to be true. Suppose we take $G = \{-1, 1\}$ with the discrete topology, acting on $X = \mathbb{R}$ by usual multiplication. Here $X^G$ consists of a single point $0$. The coend amounts to a pushout in left $G$-spaces of the diagram
$$\{-1, 1\} \times \mathbb{R} \stackrel{id \times 0}{\leftarrow} \{-1... | 5 | https://mathoverflow.net/users/2926 | 137317 | 75,381 |
https://mathoverflow.net/questions/137243 | 2 | I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST
In the internet I only could find the LP formulations, but no motivation for the constraints.
The need for topological constraints seems contradictory to me, bec... | https://mathoverflow.net/users/31310 | Why does the LP Formulation of the MST Problem need Topology Constraints? | I'm not sure that I understand your question properly but the constraints you propose do not ensure connectivity. As far as I guess, you propose to use the following polytope to describe the spanning trees of a graph $G(V,E): \sum\_{e\in E} x\_e=|V|-1; \sum\_{v\in e\in E} x\_e\geq 1$ for all $v\in V$ and $x\_e\geq 0, e... | 3 | https://mathoverflow.net/users/37513 | 137339 | 75,388 |
https://mathoverflow.net/questions/137196 | 6 | A quick reminder of the definition of Schmidt's game:
>
> Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let
> ${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball
> ${B\_0\subset X}$ with radius ${\rho\_0}$. Then Alice chooses a ball
> ${B\_1\subset B\_0}$ with radius ${\rho\_1=\alpha\rho\_... | https://mathoverflow.net/users/35222 | Winning sets of full measure (Schmidt's game) | I'm not sure if you're more interested in general conditions for this or an example with a simple proof. I can give a simple example and explanation. Let $S$ be the set of numbers normal in base $b$. $S$ is a set of full measure (follows by SLLN, Birkhoff ergodic theorem, etc.) W. Schmidt proved that $S^c$ is $1/2$-win... | 3 | https://mathoverflow.net/users/32036 | 137343 | 75,389 |
https://mathoverflow.net/questions/137338 | 3 | Let there be an omega sequence of ordinals such that the first is the least $\Sigma\_1$-admissible ordinal and the $n+1$st is the least $\Sigma\_{n+1}$-admissible ordinal. What is the name, if any, of the union of all of these? Incidentally, $L$ at the level of this ordinal would be the minimal model of ZFC minus the p... | https://mathoverflow.net/users/37385 | Question on a limit of admissible ordinals | This ordinal is not as large as you claim. In particular, the constructible universe up to this ordinal does not satisfy ZFC minus power set, and the ordinal is not even admissible.
To see this, let $\gamma\_n$ be the least $\Sigma\_n$-admissible ordinal (which I understand you to mean that $L\_{\gamma\_n}$ satisfie... | 5 | https://mathoverflow.net/users/1946 | 137344 | 75,390 |
https://mathoverflow.net/questions/137356 | 5 | I came across the following function transformation:
$$
\sum\_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum\_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j \in \mathbb{Z}$ (i.e. integers).
Can anyone help me to understand why this relation is true? Thanks!
| https://mathoverflow.net/users/37526 | Function transformation of exponentials | This is the functional equation for the theta function. A nice proof (using Poisson summation) can be found [here.](http://math.mit.edu/~brubaker/Math784/thetafunctions.pdf)
| 5 | https://mathoverflow.net/users/11142 | 137358 | 75,395 |
https://mathoverflow.net/questions/137345 | 6 | Say $X \to Y$ is a surjective map of algebraic varieties, and $Z \subset Y$ is nonreduced. Then is the preimage $Z \times\_Y X$ also nonreduced?
| https://mathoverflow.net/users/4707 | Is the preimage of a nonreduced subscheme nonreduced? | In Allen's notation, take:
$R = k[t]$, $X = \operatorname{Spec} R$
$S = k[x,y]/(y^2 - x^2(x-1) )$, $Y= \operatorname{Spec} S$.
with the map defined by:
$y = t(t^2+1)$
$x=(t^2+1)$
$I=(x)$, $Z = \operatorname {Spec} S/I$.
$S/I$ contains a nilpotent, $y$ so $Z$ is non-reduced. $X \to Y$ is a surjective map o... | 11 | https://mathoverflow.net/users/18060 | 137360 | 75,396 |
https://mathoverflow.net/questions/137359 | 6 | Let $({\cal C}, \otimes, \cal I)$ be a monoidal (locally small) category and consider $\cal Hom : \cal C^{op} \times C \to Set$. The $\cal Hom$ functor is always a lax functor in the sense that $\forall A,B,C, D \in {\cal C}$ there is some natural transformation $\cal Hom(A,C) \times Hom(B,D) \to Hom(A \otimes B, C \ot... | https://mathoverflow.net/users/37532 | What are the conditions for the Hom-functor to be strong monoidal ? | It seems any such monoidal category $\mathcal{C}$ must be equivalent to the terminal category.
Let $I$ be the monoidal unit. First I claim there is exactly one morphism $I \to I$. For we have an isomorphism
$$\hom(A, B) \cong \hom(I \otimes A, I \otimes B) \cong \hom(I, I) \times \hom(A, B)$$
where the first i... | 12 | https://mathoverflow.net/users/2926 | 137361 | 75,397 |
https://mathoverflow.net/questions/130988 | 38 | Given a finite simple group $G$, we can consider the [quasisimple](http://en.wikipedia.org/wiki/Quasisimple_group) extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the standard trick of averaging a cocycle to try to make it into a coboundary) sh... | https://mathoverflow.net/users/766 | Why are Schur multipliers of finite simple groups so small? | The Schur multiplier $H^2(G;{\mathbb C}^\times) \cong H^3(G;{\mathbb Z})$ of a finite group is a product of its $p$-primary parts
$$H^3(G;{\mathbb Z}) = \oplus\_{ p | |G|} H^3(G;{\mathbb Z}\_{(p)})$$
as is seen using the transfer. The $p$-primary part $H^3(G;{\mathbb Z}\_{(p)})$ depends only of the $p$-local struct... | 32 | https://mathoverflow.net/users/6574 | 137394 | 75,405 |
https://mathoverflow.net/questions/137184 | 1 | In a paper I'm reading, it refers to Theorem 8, Page 217 of the book
"Introduction to Stochastic Control" H. J. Kushner, New York: Holt, Reinhart, and Winston 1971. Unfortunately I don't have it and the copy in our library was checked out.
Does anyone here happen to have that book at hand and let me know what the ... | https://mathoverflow.net/users/37386 | Anyone has Kushner's book "Introduction to stochastic control" 1971? I need a theorem from it | The Carleton College library has a copy of the Kushner book. Here's the theorem:
>
> **Theorem 8**
>
>
>
> >
> > Let\* $P\gt0, C\ge0$ and
> >
> >
> > $$EA\_n'PA\_n-P=-C.\ \ \ (8.24)$$
> >
> >
> > Then $EX\_n'CX\_n\rightarrow0$ and $X\_n'CX\_n\rightarrow0$ w.p.l. Also
> >
> >
> > $$P\_x(\sup\_{\infty\gt n\... | 10 | https://mathoverflow.net/users/15837 | 137410 | 75,410 |
https://mathoverflow.net/questions/137180 | 3 | Let $w\_1$ and $w\_2$ be two permutations of $\{1, \cdots , k\}$ such that for all $1\leq i \leq k$, $w\_1(i)\neq w\_2(i)$. Let $m$ and $n$ be two relatively prime integers. Then is there exist two diagonal matrices $D\_1, D\_2 \in M\_k(\mathbb{Z})$, with $\gcd(\text{det}D\_1, n)=1$, $\gcd(\text{det}D\_2, m)=1$ such th... | https://mathoverflow.net/users/19078 | A problem about Determinant of sum of permutation matrices | I think Igor Rivin's original answer is correct, and that no hypothesis of $w\_1(i) \neq w\_2(i)$ is needed, nor do I see the subtlety he refers to in his recent edit. I'm writing this out in full so people can point out whether I missed anything.
First of all
$$\det( m D\_1 w\_1 + n D\_2 w\_2 ) = \pm \det(m D\_1 + n... | 4 | https://mathoverflow.net/users/297 | 137413 | 75,411 |
https://mathoverflow.net/questions/137406 | 8 | This is a crosspost of <https://math.stackexchange.com/questions/446470/combinatorics-of-resultants> which received no answer. [**EDIT:** I deleted the initial copy of the question on MathSE].
Let $f(z)=\sum\_{i=0}^{D\_f}x\_iz^i$ and $g(z)=\sum\_{i=0}^{D\_g}y\_iz^i$ be two polynomials. I would like to know the number... | https://mathoverflow.net/users/37394 | Combinatorics of resultants | Yes, this (like everything else under the Sun) was studied, but not a lot. The reference is:
[M. Kalkbrener, An upper bound on the number of monomials in the Sylvester resultant.](https://www.evernote.com/shard/s24/sh/6c3eedc7-0e1e-483e-b5ca-baa54aa235a7/97920f99147371c0dc934d805c646704)
| 5 | https://mathoverflow.net/users/11142 | 137416 | 75,412 |
https://mathoverflow.net/questions/137409 | 10 | I have read it claimed in several places that in a stable $(\infty,1)$-category, the coequalizer of parallel maps $f,g:X\to Y$ can be identified with the cokernel of $f-g$ (i.e. the pushout of the map $X\to 0$ along $f-g$). How is this proven? Is it written down anywhere?
| https://mathoverflow.net/users/49 | Coequalizers in stable (infinity,1)-categories | It seems to me like the proof for Abelian categories works here too. Consider the following diagram where $d = (id,-id)$, the middle vertical map is the fold (or $+$) map, and the square on the right is a homotopy pushout:
$$ \begin{array}{ccccc}
X & \xrightarrow{d} & X \oplus X & \xrightarrow{(f,g)} & Y \\
\downar... | 6 | https://mathoverflow.net/users/644 | 137417 | 75,413 |
https://mathoverflow.net/questions/137415 | 7 | I am sure that this is well-known, but I looked around for the last half hour and couldn't see an answer. I just wondered whether it's possible to insist on taking all primes to be large in Vinogradov type results?
Thanks in advance for any help.
| https://mathoverflow.net/users/37573 | Is there a constant $c>0$, such that every natural number $n>1$ is the sum of primes, each with size at least $cn$? | The usual proof of Vinogradov's result can be modified to show that every sufficiently large odd $n$ has $\asymp n^2/(\log n)^3$ representations as a sum of three primes with each prime exceeding $cn$, provided $c>0$ is sufficiently small. This gives (easily) a positive answer to your original question.
The best unco... | 16 | https://mathoverflow.net/users/11919 | 137419 | 75,415 |
https://mathoverflow.net/questions/137430 | 3 | I apologize if this is something standard and/or elementary, but I was unable to find anything relevant via Google.
Consider a Dirichlet series
$$
f(s) = \sum\_{n=1}^\infty \frac{a\_n}{n^s}
$$
and assume that $s\_0$ is a real number such that $f(s)$ converges absolutely for all $s>s\_0$. In addition, assume that the ... | https://mathoverflow.net/users/4384 | Convergence of Dirichlet series ("at the boundary") | The answer is no. Take, for example, $a\_n:=n^{-it}$ with any fixed $t\neq 0$. Then $f(s)$ converges absolutely to $\zeta(s+it)$ for any $s>1$, and $\lim\_{s\downarrow 1}f(s)=\zeta(1+it)$ exists, but $f(1)$ diverges.
The answer would be yes under some stronger assumptions, see e.g. the main theorem in Chapter VII of ... | 5 | https://mathoverflow.net/users/11919 | 137434 | 75,421 |
https://mathoverflow.net/questions/137363 | 7 | My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called an angle) to each edge of a tetrahedron such that when we glue the tetrahedron up to obtain the $3$--manifold, around eac... | https://mathoverflow.net/users/37434 | Standard (special) spines and hyperbolic structure on 3-manifolds | If an ideal triangulation (or its dual spine) admits an angle structure (with *positive* angles), then the manifold must admit a complete hyperbolic metric of finite volume (see [Theorem 10.2](http://www.ams.org/mathscinet-getitem?mr=1985831), an observation of Casson). However, when one straightens these tetrahedra in... | 4 | https://mathoverflow.net/users/1345 | 137440 | 75,422 |
https://mathoverflow.net/questions/137378 | 0 | Let me ask you a question about $Z/pZ$ - acyclic affine toric varieties (for some $p$) i.e. toric varieties $X$ such that homologies $H\_{j}(X,Z/pZ) = 0$ for $j > 0$ and $H\_{0}(X,Z/pZ) = Z/pZ$. I am interested how big such class of $Z/pZ$ - acyclic affine varieties is? For example, it is known that the affine spaces a... | https://mathoverflow.net/users/37547 | $Z/pZ$ - acyclic affine toric varieties | It turns out not to be the same as every other notion of acyclic toric varieties.
If the associated cone of affine toric variety has nonempty interior, equivalently, if the toric variety contains a torus-invariant point, then it will be acyclic, because it will be homotopic to a point. We just choose a one-parameter ... | 0 | https://mathoverflow.net/users/18060 | 137445 | 75,424 |
https://mathoverflow.net/questions/137381 | 12 | The famous Quillen-Suslin theorem (formerly known as Serre's problem/conjecture) states that every projective module over $k[x\_1,\dots, x\_n]$ is free for $k$ a field. Replacing $k$ by a more general ring, we get the Bass-Quillen conjecture:
>
> Let $R$ be a regular ring and $P$ a projective module over $R[x\_1,\d... | https://mathoverflow.net/users/2039 | An analogue of the Bass-Quillen conjecture with power or Laurent series | Here is an attempt at the power series question. One can easily reduce to the one variable case. So, I will attempt to prove that if $R$ is any Noetherian ring and $A=R[[x]]$ and $P$ a projective $A$-module, then $P\cong P/xP\otimes\_R A =P'$. First, note that if $K$ is any finitely generated $A$-module, then it is com... | 12 | https://mathoverflow.net/users/9502 | 137447 | 75,425 |
https://mathoverflow.net/questions/137354 | 3 | Suppose $A$ is an abelian variety, $X, Y$ are subvarieties of $A$ of complementary dimension,
>
>
> >
> > Does every component of $X \cap Y$ contribute non-negatively to the intersection number?
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/4707 | Intersection multiplicity in abelian varieties | The following answer expands on my comment.
We use Fulton's definition of the intersection product. Consider the diagonal embedding $\Delta$ of $A$ in $A \times A$ (which is regular) and intersect this with $X \times Y$. Since $A$ is an abelian variety, the normal bundle $E$ of $\Delta$ is trivial. The normal cone $C... | 3 | https://mathoverflow.net/users/519 | 137460 | 75,431 |
https://mathoverflow.net/questions/137347 | 5 | In his comment to the question [Algebraic numbers and the complex projective line minus three points](https://mathoverflow.net/questions/137108/algebraic-numbers-and-the-complex-projective-line-minus-three-points#comment353577_137108) JSE says that for algebro-geometric objects defined over the complex numbers "in prac... | https://mathoverflow.net/users/38783 | Rigidity, moduli space, and moduli field | While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (<http://www.math.ias.edu/files/deligne/GaloisGroups.pdf>) and are not working with someone who already understands it, then you need to l... | 0 | https://mathoverflow.net/users/38783 | 137466 | 75,432 |
https://mathoverflow.net/questions/137458 | 1 | If $G$ is a finite $p$-group and $H$ is a proper subgroup of $G$, then, it is well known that the union of conjugates of $H$ in $G$ is proper subset of $G$. The problem I considering is the following:
**Question** Let $G$ be a finite $p$-group, $H$ be a non-normal subgroup, such that its normalizer $N\_G(H)$ is norma... | https://mathoverflow.net/users/6761 | Union of Conjugate Subsgroups in $p$-Groups | I believe the answer is yes, the union of the conjugates of $H$ is a proper subset of $N\_G(H)$.
Let $H < G$ be a counterexample with $|G|$ minimal. Since $N\_G(H) \lhd G$, the intersection $Z := N\_G(H) \cap Z(G)$ of $N\_G(H)$ with the centre of $G$ is nontrivial.
If $Z < H$, then $H/Z < G/Z$ is a smaller countere... | 4 | https://mathoverflow.net/users/35840 | 137467 | 75,433 |
https://mathoverflow.net/questions/137421 | 24 | There is well-known [Rado's "Busy Beaver" sequence](http://oeis.org/A028444) — the maximal number of marks which a halting Turing machine with *n* states, 2 symbols (blank, mark) can produce onto an initially blank two-way infinite tape. It begins:
>
> 1, 4, 6, 13, ...
>
>
>
The sequence grows faster than any... | https://mathoverflow.net/users/34258 | Busy Beaver modulo 2 | I will not answer the question since I think
it is not completely defined without completely specifying the machine model.
In the following I explain why.
Note that the exact running time of a machine is not natural mathematically,
it heavily depends on the particular model of the machine we use
(that is why in co... | 13 | https://mathoverflow.net/users/7507 | 137469 | 75,434 |
https://mathoverflow.net/questions/137429 | 8 | Shelah isolates the notion of "$\aleph\_1$-free iteration" in the first two sections of Chapter IX from *Proper and Improper Forcing*, and he proves there that properness is preserved by this sort of iteration. He mentions in his "On what I do not understand" paper that this was his third proof of the preservation of p... | https://mathoverflow.net/users/18128 | On free limits of iterations | This is more of a comment than an answer, but as I couldn't post a comment, I'm writing here.
If I understood correctly the definitions in chapter IX from Proper and Improper Forcing, it seems that you can find some variants of free limits (not $\aleph\_1$-free, though) in the following work of Shelah:
In the paper... | 4 | https://mathoverflow.net/users/37613 | 137474 | 75,436 |
https://mathoverflow.net/questions/137472 | 1 | Who proved it? When?
See also:
[Irreducible Degrees and the Order of a Finite Group](https://mathoverflow.net/questions/126828/irreducible-degrees-and-the-order-of-a-finite-group)
<http://planetmath.org/proofthatdimensionofcomplexirreduciblerepresentationdividesorderofgroup>
[Why would dim primitive irrep divi... | https://mathoverflow.net/users/4690 | What is the name of the following theorem: dimension of complex irreducible representation divides order of group | Georg Frobenius, *Über die Primfactoren der Gruppendeterminante*, Sitzungsber. Akad. Berlin (1896) 1343-1382. The theorem is announced at the beginning, p. 1344:
>
> Der Grad $f$ ist ein Divisor der Ordnung $h$
>
>
>
and proved at the [very end](http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schr... | 12 | https://mathoverflow.net/users/19276 | 137477 | 75,437 |
https://mathoverflow.net/questions/137386 | 8 | Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the integer ring ${\mathcal O}\_K$ is infinite cyclic up to the roots of unity in $K$ and one can pick a generator $\varepsil... | https://mathoverflow.net/users/32210 | Fundamental units of imaginary quartic fields | There's certainly *some* uniform bound, as a special case of the
theorem that for each $n$ and $M$ there are only finitely many
algebraic integers $\epsilon$ of degree $n$ each of whose conjugates
has absolute value at most $M$. Here $n=4$, and since $\epsilon$
is a unit conjugate to $\pm\epsilon^{-1}$ (once $D \lt -4$... | 8 | https://mathoverflow.net/users/14830 | 137480 | 75,438 |
https://mathoverflow.net/questions/137384 | 1 | I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial data, i.e;
$$ u\_t - \bigtriangleup u = f(u) \ \ \ \forall (x,t)\in\mathbb{R}^n\times (0,T] $$
$$ u(x,0)= 0 \ \ \ \foral... | https://mathoverflow.net/users/33849 | Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data | While not exactly the same problem that you solved, there has been previous results considering nonuniqueness of solutions (with zero initial data) for power-law type semilinear term. Interestingly, [contrary to what you wrote](https://mathoverflow.net/questions/137384/spatially-inhomogeneous-solutions-to-parabolic-pde... | 2 | https://mathoverflow.net/users/3948 | 137481 | 75,439 |
https://mathoverflow.net/questions/137479 | 1 | Let $M$ be a compact manifold with an $\mathbb S^1$-action that fixes a point on $M$.
Is it correct that $\pi\_1(M/S^1)=\pi\_1(M)$?
I believe this is correct and is a corollary of some well-known statement.
| https://mathoverflow.net/users/13441 | The fundamental group of an $S^1$-quotient | Suppose $S^1$ acts on $S^2$ by rotation around one axis. This action commutes with the antipodal map and hence gives an action on $\mathbb{R}P^2$. But $\mathbb{R}P^2/S^1\cong [0,1]$ and hence this cannot be true.
| 6 | https://mathoverflow.net/users/3969 | 137486 | 75,440 |
https://mathoverflow.net/questions/137482 | 7 | Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma\_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi\_1(M)\ne 0$.
2) $\pi\_1(M)\ne \pi\_1(\Sigma\_g)$.
2)' If 2) is too hard I would be happy just to know that the map $\pi\_1(\Sigma\_g)\t... | https://mathoverflow.net/users/13441 | The fundamental group of a $3$-manifold with a boundary of genus $>0$ | 1. The long exact sequence of the pair $(M, \partial M)$ combined with Poincare duality immediately imply that the natural map
$$
i\_\*: H\_1(\partial M)\to H\_1(M)
$$
cannot be an isomorphism, unless $H\_1(\partial M)=0$, which means that the boundary is a sphere. Thus, the map of fundamental groups cannot be an isom... | 11 | https://mathoverflow.net/users/21684 | 137492 | 75,443 |
https://mathoverflow.net/questions/137478 | 6 | Let G be the group with 4 generators, each of order 2, such that the product of any 2, say ab, has order 3 (i.e., ababab=e). That is, this is an infinite reflection group with Coxeter diagram a tetrahedron. I am looking for references for this group...
| https://mathoverflow.net/users/37616 | Reference for tetrahedral Coxeter group | In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.
Here are some:
Maclachlan and Reid's "[The Arithmetic of Hyperbolic 3-Manifolds](https://doi.org/10.1007/978-1-4757-6720-9)" has the hyperbolic te... | 7 | https://mathoverflow.net/users/27453 | 137494 | 75,445 |
https://mathoverflow.net/questions/137487 | 2 | I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no special qualities. From this, I first need to calculate $\mathrm{tr}(\textbf{H}^{-1})$, and then I solve $(\textbf{H}+\lambda... | https://mathoverflow.net/users/37618 | Is my use of the eigendecomposition correct here? | If $D = diag(H)$ is positive, let $K = D^{-1/2} H D^{-1/2}$ which again is positive semidefinite. Then $(H + \lambda D)^{-1} = D^{-1/2} (K + \lambda I)^{-1} D^{-1/2}$, so if you do the eigendecomposition $K = Q \Lambda Q'$ with $Q$ orthogonal you get
$(H+\lambda D)^{-1} = (D^{-1/2} Q) (\Lambda + \lambda I)^{-1} (D^{-1/... | 1 | https://mathoverflow.net/users/13650 | 137508 | 75,450 |
https://mathoverflow.net/questions/132508 | 3 | Let $S$ be an Enriques surface, i.e. a quotient of a K3 surface by a free involution. Enriques surfaces arise as elliptic fibrations $S\rightarrow \mathbb{P}^1$ with 12 singular fibers and 2 double points.
>
> **Claim** There exist elliptic surfaces $S\_1, S\_2$ such that
> we can degenerate the elliptic fibrati... | https://mathoverflow.net/users/50973 | A question on existence of degeneration of Enriques surface. | The surface $S\_1$ is rational and the surface is an elliptic ruled surface (induced by the projection $E\times \mathbb{P}^1\to E$). The glued surface is a standard Type II degeneration of Enriques surfaces (an elliptic ruled chain) classified by Vic. Kulikov and D. Morrison. I believe that the general member of the de... | 1 | https://mathoverflow.net/users/37622 | 137511 | 75,452 |
https://mathoverflow.net/questions/137183 | 6 | Let $H$ be an infinite dimensional separable Hilbert space.
**Definition**: The **commutant** $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \} $.
**Definitions** : An operator $A \in B(H)$ is :
* **Irreducible** ([Halmos 1968](https://projecteuclid... | https://mathoverflow.net/users/34538 | Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous? | Essentially I think weighted shifts should be a sufficiently rich class of operators. Consider, for instance, the following example.
Take the doubly infinite sequence
$$ w\_k=\left\{\begin{array}{ll} 2 & \text{if } k<0 \\ 1 & \text{if } k\geq 0 \end{array}\right. $$
and let $W$ be the weighted shift on $\ell^2(\mathb... | 6 | https://mathoverflow.net/users/10779 | 137522 | 75,456 |
https://mathoverflow.net/questions/132804 | 4 | This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.
Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ be a principal bundle with compact structure group $G$. Let $E$ be a finite dimensional complex vector space and $\rho... | https://mathoverflow.net/users/24965 | The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold | Your understanding is correct. All the identify is only for the $G$-invariant section.
| 1 | https://mathoverflow.net/users/16326 | 137523 | 75,457 |
https://mathoverflow.net/questions/137528 | 4 | Let T be a tree which satisfies the following conditions. (A) The set of vertices of T is denumerably infinite. (B) Each vertex of T is an end-point of at most finitely many edges of T. Does there always exist a planar graph G which is isomorphic to T and which satisfies the following conditions? (1) Every edge of G is... | https://mathoverflow.net/users/4423 | A question about trees and planar graphs | The answer is clearly yes.
Pick a vertex and choose it as a root. Put it at (0,0). Then put its children at (0,1), (1,1), ..., (n1,1). Then for each of these vertices put their children at (0,2), (1,2), ..., (n2,2) (in doing so make first appear the children of (0,1), then the one of (1,1) etc). Then repeat the same ... | 11 | https://mathoverflow.net/users/15342 | 137531 | 75,461 |
https://mathoverflow.net/questions/137504 | 2 | Let $G$ be a finite group and consider $R=K[V]^G$ be the invariant ring of the group G over the field K of char 0. Let $f\_1,\ldots f\_n$ be a set of primary invariants. Is there a nice geometric characterization of the points where the Jacobian of these polynomials vanishes? For finite reflection groups, this is the r... | https://mathoverflow.net/users/36892 | Jacobian of primary invariants | The points of constant rank of $f=(f\_1,\dots,f\_n)$ essentially are the strata of the stratification into orbit types. See [here](http://www.mat.univie.ac.at/~michor/rep-tens.pdf) or [here](http://www.mat.univie.ac.at/~michor/alg-tens.pdf) and references therein.
| 2 | https://mathoverflow.net/users/26935 | 137533 | 75,463 |
https://mathoverflow.net/questions/137539 | 9 | Let $p$ be a prime number and $h\_p^+$ the class number of $\mathbb{Q}(\zeta\_p + \zeta\_p^{-1})$. What is known about the values of $p$ for which $h\_p^+ = 1$?
Are there infinitely many? Finitely many? Something else?
(As usual, $\zeta\_p$ denotes a primitive $p$-th root of unity.)
| https://mathoverflow.net/users/7313 | Class number of real maximal subfield of cyclotomic fields | It is expected that there are infinitely many primes for which it is $1$; indeed one expects this to hold for more than 70 per cent of all primes. However, this is open.
The exact value is only know for very few primes; I think only up to $67$ (this was true until some years ago, but perhaps I missed something).
Th... | 13 | https://mathoverflow.net/users/nan | 137541 | 75,466 |
https://mathoverflow.net/questions/137545 | 7 | Let $X$ denote a topological affine space (with no additional assumptions). Let $X^\*$ denote its dual space of continuous affine functionals, equipped with the weak-$\*$ topology. It is easy to see that $X^\*$ is a topological vector space, since it has a zero functional; it is also locally convex.
Is the dual space... | https://mathoverflow.net/users/238 | Are dual spaces barreled? | No, it need not be barreled.
Let $X$ be an infinite-dimensional normed vector space. The closed unit ball $B\_{X^\ast}$ in the dual space $X^\ast$ is a barrel in the weak-$\ast$ topology: it is compact, convex, balanced and absorbing. It is not a weak-$\ast$ neighborhood of zero because its interior is empty: every b... | 10 | https://mathoverflow.net/users/29555 | 137549 | 75,471 |
https://mathoverflow.net/questions/137530 | 7 | Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z\_1$, $\bar{z\_1}$, $z\_2$, $\bar{z\_2}$, ..., $z\_d$, $\bar{z\_d}$ with $z\_i$ in the upper half plane. Let $g(x) = \prod(x-z\_i)$, a complex polynomial of degree $d$. Is there a way to compute the coefficients of $g$ which i... | https://mathoverflow.net/users/297 | Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system | As @Federico notes (but does not say explicitly), the magic words are "spectral factorization" -- for an algorithm see [here.](https://www.evernote.com/shard/s24/sh/ba0b0a31-e398-41c7-a5a0-66e50b9554cd/3a98723085e004097be0ed22a5dfdda9) I should say that it is very far from clear to what extent the fancy algorithms are ... | 1 | https://mathoverflow.net/users/11142 | 137552 | 75,472 |
https://mathoverflow.net/questions/137439 | 6 | I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ has significant applications, as do monomial symmetric functions but for the purposes of this question, we should ignore them. (*Edit: As Graham has ... | https://mathoverflow.net/users/7717 | Generating functions with all non-zero coefficients equal to one | Although you say you're not interested in examples with indicator functions, I think there are some examples with indicator functions of polytopes that fit your criteria. I'm writing this answer because I don't think they have the same flavor which you are trying to rule out.
To every lattice point one may associate ... | 7 | https://mathoverflow.net/users/2384 | 137558 | 75,475 |
https://mathoverflow.net/questions/137499 | 7 | Let $(X, x\_0)$ be a based topological space, and $\Omega X$ its based loop space. The group of path components of $\Omega X$ is $\pi\_0(\Omega X) = \pi\_1(X, x\_0)$. For brevity, let's call this group $G$. Since $\Omega X$ is a loop space, all of its components are homotopy equivalent, and so if we define $\Omega\_1 X... | https://mathoverflow.net/users/4649 | Components of a loop space, semidirect products, and multiplicativity | I think the answer is ' yes' for trivial reasons. If the loop space were a topological group the map $\varphi$ would be a homeomorphisms and the group structure on the target could be transferred to the source. Now, any space has models for the loop space which are honest topological groups, so you can transfer the res... | 1 | https://mathoverflow.net/users/12166 | 137568 | 75,478 |
https://mathoverflow.net/questions/137554 | 7 | I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more specifically, I care about the proof of Theorem 4.10, see below.
Here's the necessary background. For $0 < \delta \ll 1$, and $f$ ... | https://mathoverflow.net/users/37640 | Kakeya and Nikodym maximal functions | Yes, the argument is dyadic decomposition followed by rescaling. I think I forgot to mention in the paper one initial reduction, which is to only consider the portion of the Nikodym maximal function coming from tubes which make an angle of at most 1/10 (say) with the basis vector $e\_n$; note that one can reduce to thi... | 9 | https://mathoverflow.net/users/766 | 137570 | 75,480 |
https://mathoverflow.net/questions/137462 | 7 | In
>
> David Pincus, *Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods*,
> The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743
>
>
>
Pincus introduces the notion of *injectively bounded statements*, which he proves are sentences which can be transferred from a (permutatio... | https://mathoverflow.net/users/4177 | Intuition behind Pincus' "injectively bounded statements" | (Note: this isn't something I really know, so this might be wildly off base.)
To start with, let's look at a weaker transfer principle: the Jech-Sochor Embedding Theorem.
Jech-Sochor says that sentences depending only on a bounded amount of the cumulative hierarchy above the set $A$ of atoms can be "passed over" to... | 5 | https://mathoverflow.net/users/8133 | 137595 | 75,489 |
https://mathoverflow.net/questions/130896 | 3 | The number of triangulations of a convex $n$-gon is $C\_{n-2}$ the $n-2$nd Catalan number. What I am wondering, is if there is a way to enumerate the isomorphism types of these as graphs? I am currently working an a project where I want to only consider all possible unlabeled triangulations of $n$-gons as a way to clas... | https://mathoverflow.net/users/34027 | Number of isomorphism classes of triangulations of a convex polygon | If I understand correctly, these are [A001683](http://oeis.org/A001683) if turning over the $n$-gon is not allowed as an isomorphism, and [A000207](http://oeis.org/A000207) if it is. In both articles there are formulae.
You might also be interested that [plantri](http://cs.anu.edu.au/~bdm/plantri/) can compute these ... | 6 | https://mathoverflow.net/users/9025 | 137599 | 75,493 |
https://mathoverflow.net/questions/137581 | 2 | A "not well understood" proof, do you know if one knows by now the conceptual background?: <http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps>
| https://mathoverflow.net/users/451 | Unexplained techniques in Demjanenko's "Sums of 4 Cubes": | There is a paper by Philippe Revoy that analyses and expands on this result.
Its abstract reads (the paper itself is in French):
>
> In this note, we do a systematic study of first degree identities $\sum\_{i=1}^4 P\_i(x)^3 = P x + q$, $P\_i \in \mathbb{Z}[x]$, occuring in the four cube problem over $\mathbb{Z}... | 4 | https://mathoverflow.net/users/nan | 137601 | 75,494 |
https://mathoverflow.net/questions/137550 | 19 | Is there any convincing sense in which the standard trigonometric basis for
the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is
optimal among all the orthonormal bases?
(If this question strikes you as overly subjective, replace "convincing" by "described in the existing mathematical literatur... | https://mathoverflow.net/users/3621 | In what ways is the standard Fourier basis optimal? | Here's something similar to what you conjecture. Let's work instead with functions $\mathbb{R} / 2\pi \mathbb{Z} \to \mathbb{C}$; one can do similar things in the real-valued case but I find the complex case simpler to write down.
Say we want to construct an ONB $f\_0, f\_1, \ldots$ of $L^2(\mathbb{R} / 2\pi \mathbb... | 5 | https://mathoverflow.net/users/1044 | 137604 | 75,496 |
https://mathoverflow.net/questions/137612 | 9 | This question is related to this [MO question](https://mathoverflow.net/q/107168/21564) and this [MSE question](https://math.stackexchange.com/q/279957/39599).
---
Let $E$ be a hermitian holomorphic vector bundle over a hermitian manifold $X$. The bundle $\bigwedge^{\bullet,\bullet}X\otimes E$ has an induced herm... | https://mathoverflow.net/users/21564 | Weitzenböck Identity for $\Delta_{\bar{\partial}_E}$ | You can find all what you want to know in the chapter 1 of the book X.Ma and G. Marinescu:Holomorphic Morse Inequalities and Bergman Kernels.
In Kahler case, all the metric connection is the chern connection.
And for the general case(hermitian manifold), the connection is called Bismut connection (see section 1.2.3... | 5 | https://mathoverflow.net/users/16326 | 137619 | 75,504 |
https://mathoverflow.net/questions/137560 | 1 | Given a class $C$ of arithmetical sentences,
an arithmetical theory $T$ is said to be $C$-sound if
all the theorems of $T$ which are in $C$ are true.
For instance, $T$ is $\Sigma\_1$-sound if all the $\Sigma\_1$ theorems of $T$ are true.
Now, for some classes $C$, like the class of $\Sigma\_1$ sentences,
the sta... | https://mathoverflow.net/users/5017 | Can a class of arithmetical statements containing its own soundness condition be closed under negation? | The answer to both questions is negative.
**Theorem.** There is no class $C$ of formulas in the language of
arithmetic, such that
the assertion "$T$ is not $C$-sound" is uniformly expressible in $C$
for c.e. theories $T$ (regarded as an index of $T$ as a c.e. set).
Proof. Suppose $C$ is like that. By the Gödel fix... | 7 | https://mathoverflow.net/users/1946 | 137624 | 75,507 |
https://mathoverflow.net/questions/137556 | 3 | I'm interested in Buchberger's criterion for determining if G={g\_1,....,g\_n} is a Grobner basis for the ideal it generates. In the procedure, I consider the S-polynomial S(g\_i,g\_j) and check if it has remainder zero upon division by G *in any order*.
My question is if Buchberger's criterion holds if one use a dif... | https://mathoverflow.net/users/37645 | Buchberger algorithm question | Indeed, it is enough to check that each $S(g\_i,g\_j)$ can be reduced to zero in *some* way. Indeed, if you trace carefully the proof of Buchberger's criterion, you will see that one only needs the *existence* of a representation of $S(g\_i,g\_j)=\sum\_{k}a\_{ijk}g\_k$ with each leading term of $a\_{ijk}g\_k$ being les... | 5 | https://mathoverflow.net/users/1306 | 137633 | 75,511 |
https://mathoverflow.net/questions/137643 | 5 | My primary question is: given a cellular decomposition of a sphere is there any way to check if it can be embedded as the boundary of a polytope?
My question is motivated by the following problem. I began with a polyhedral cell complex $P$ homeomorphic to a ball. Then I had an unbounded polyhedral cone $C$ of the sa... | https://mathoverflow.net/users/19642 | Cellular decomposition of a sphere to polytope | The "primary question" is a well-known hard problem, and I think this reference:
MR0889977 (89b:52009) Reviewed
Bokowski, Jürgen(D-DARM); Sturmfels, Bernd(D-DARM)
Polytopal and nonpolytopal spheres: an algorithmic approach.
Israel J. Math. 57 (1987), no. 3, 257–271.
52A25 (05B35)
PDF Clipboard Journal Article Mak... | 5 | https://mathoverflow.net/users/11142 | 137645 | 75,516 |
https://mathoverflow.net/questions/137641 | 4 | An interval graph is an intersection graph of real intervals, that is, an undirected graph whose vertices can be labeled with real intervals so that there is an edge between two vertices iff their intervals intersect.
A comparability graph is an undirected graph that connects elements that are comparable in some part... | https://mathoverflow.net/users/37671 | For what classes of comparability graphs are their complements also comparability graphs? | One source I found over the Internet is [Information System on Graph Classes and their Inclusions](http://www.graphclasses.org/index.html). There you can find a page devoted to [comparability graphs](http://www.graphclasses.org/classes/gc_72.html), [complements of comperability graphs](http://www.graphclasses.org/class... | 3 | https://mathoverflow.net/users/1532 | 137647 | 75,518 |
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