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https://mathoverflow.net/questions/275168 | 4 | In homotopy type theory, or dependent type theories more generally, there is a "top-level" type called the universe, generally denoted $\newcommand{\type}{\mathtt{Type}}\type$. So for a concrete example, I could describe having 3 types: $\mathtt{Nat}$, $\mathtt{Real}$, $\mathtt{Complex}$ and the type of those types wou... | https://mathoverflow.net/users/112089 | Homotopy type theory: Are the hierarchy of Type_k universes isomorphic? | This question is about type theory in general and is not specific to homotopy type theory. $\newcommand{\Type}{\mathtt{Type}}$
The thing you are missing is that **a universe $\Type\_k$ contains very many types**, not just one as you claim. Each $\mathtt{Type}\_k$ is closed under type forming operations $\times$, $+$,... | 15 | https://mathoverflow.net/users/1176 | 275203 | 122,616 |
https://mathoverflow.net/questions/273834 | 26 | I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.
I could try to do it myself but I really lack expertise, hence am afraid to miss something or do it wrong.
Let me just provide some glimpses, and maybe somebody can nicely tie them together.
At the "initial end... | https://mathoverflow.net/users/41291 | Has anyone seen a nice map of multiplicative cohomology theories? | I'm not sure I understand what "the" map is here, but I'll attempt to answer
the questions that were asked in the body of the question. Sorry if I'm just
saying things that you already know.
$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\Mfg}{\mathscr{M}\_{\textbf{fg}}}\newcommand{\QCoh}{\mathrm{QCoh}}\newcommand{\Eoo}{\ma... | 24 | https://mathoverflow.net/users/102390 | 275211 | 122,618 |
https://mathoverflow.net/questions/275222 | 4 | In complex analysis one learns Hartogs' theorem:
>
> Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$
> $$ z \mapsto f(z\_1, \dots, z\_{i-1}, z, z\_{i+1}, \dots, z\_n) $$
> is analytic.
>
>
>
Can we generalize this theorem to t... | https://mathoverflow.net/users/91098 | Hartogs' theorem in Banach spaces | Yes this is true and stated as Theorem 14.27 in the book [Holomorphy and Calculus in Normed Spaces](https://books.google.com/books?id=wtrvhCdbsd8C&printsec=frontcover#v=onepage&q&f=false) by Soo Bong Chae. If you need even more sophisticated generalizations, you can also look at the notes related to Section 2.76 on Pag... | 3 | https://mathoverflow.net/users/7410 | 275225 | 122,622 |
https://mathoverflow.net/questions/275217 | 3 | Assume $X\_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X\_t = -\alpha X\_t dt + \sigma dB\_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max\_{t\le T} |X\_t| \ge z
\right)
\le
?
$$
P.S. when $\alpha = 0$ above $X\_t$ is a Brownian motion, and hence
$$
P\left( \sup\_{t\le T} |\... | https://mathoverflow.net/users/88033 | Large deviation bound for O-U process | [Fernique's theorem](https://en.wikipedia.org/wiki/Fernique%27s_theorem), valid for all continuous Gaussian processes, implies that there are constants $C,c$ (depending on $T$) such that $P(\max\_{0 \le t \le T} |X\_t| \ge z) \le C e^{-c z^2}$.
See also [Deviation bound for the maximum of the norm of Wiener process]... | 3 | https://mathoverflow.net/users/4832 | 275229 | 122,625 |
https://mathoverflow.net/questions/275213 | 3 | Dear Colleagues and Friends,
Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive.
I believe that it's known that if a function $f(z)$ satisfies an equation $P(z, f(z)) = 0$ with $P(z, \xi) \in \mathbb{C}[[z, \xi]]$ a non-zero analytic f... | https://mathoverflow.net/users/39331 | Analytic solutions to algebraic differential equation | Your belief is not correct, as Robert Israel pointed in his comment. Same for
differential equations: $yy'-(3/2)z^2=0$ has a solution $y(z)=z^{3/2}$ which is not
analytic at $0$. The correct theorem is:
If $P(z\_0,y\_0,y\_1,\ldots,y\_{n})=0$, where $P$ is a polynomial,
(or an analytic function in a neighborhood of $(z\... | 5 | https://mathoverflow.net/users/25510 | 275233 | 122,626 |
https://mathoverflow.net/questions/275234 | 2 | I am considering the following question.
>
> **Question 1.** Let $G$ be a reductive algebraic group over $\mathbb{R}$, can we find finitely many reductive $\mathbb{R}$-subgroups $H\_1,...,H\_m$ such that for any reductive $\mathbb{R}$-subgroup $H\subset G$, there is a $g\in G(\mathbb{R})$ such that $H=gH\_ig^{-1}$ ... | https://mathoverflow.net/users/90978 | About the conjugation of reductive subgroups | No. Again the 2-torus $T$ is a counterexample. Indeed, first an embedding into $\mathrm{GL}\_n$ is given by $n$ characters, hence $n$ pairs $(k\_1,m\_1)$,... $(k\_n,m\_n)$ of integers such that the corresponding matrix defines a surjective endomorphism $\mathbf{Z}^n\to\mathbf{Z}^2$. Namely it's conjugate to the embeddi... | 4 | https://mathoverflow.net/users/14094 | 275236 | 122,628 |
https://mathoverflow.net/questions/258265 | 6 | Let $k$ be a field and $A$ be an ordinary $k$-algebra. Let $M$ and $N$ be two left $A$-modules then we could consider the Ext group $\mathrm{Ext}^i\_A(M,N)$ which is actually a $k$-vector space. Let $l/k$ be a field extension and we consider $A\_l$, $M\_l$, and $N\_l$ be the base-change algebras and modules, i.e. $A\_l... | https://mathoverflow.net/users/24965 | Is the hom in derived category of a dg-algebra compatible with base field extension? | Yes.
The full subcategory consisting of those $M$ for which this natural map is an isomorphism (for all $N$) is a thick subcategory by a five lemma argument, and the thick subcategory generated by $A$ is precisely the category of perfect complexes, so it is sufficient to prove it for $M=A$.
But $\text{Hom}\_{D(A)}(... | 3 | https://mathoverflow.net/users/22989 | 275259 | 122,633 |
https://mathoverflow.net/questions/275251 | 27 | The title of the question more or less says it all.
The question asks for precise and scientific descriptions (submission-rules, editor-behavior, referee-recruitment, *anonimity issues in an age where handwriting was still the norm*, all the way to rejection or acceptance-and-concomitant-galley-proof-process), example... | https://mathoverflow.net/users/108556 | How did the refereeing system of Gösta Mittag-Leffler's Acta Mathematica function from 1882 to (about) 1918? | Looking over the following references, I believe that they contain some of what is being asked after:
>
> Nickerson, Sylvia (2012). Referees, publisher’s readers and the image of mathematics in nineteenth century England. *Publishing History, 71*, 27-67. [**Link**](http://sts.gradstudies.yorku.ca/files/2015/12/Nick... | 27 | https://mathoverflow.net/users/22971 | 275261 | 122,635 |
https://mathoverflow.net/questions/275169 | 17 | **Komlos Conjecture**: the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v\_1,\ldots, v\_n\in \mathbb{R}^d$ with $\left\lVert v\_i\right\rVert \_2=1$ we can find weights $w\_i\in\{-1,1\}$ such that $\left\lVert w\_1v\_1+\cdots+w\_nv\_n\right\rVert \_{\infty}<K$.
I would like t... | https://mathoverflow.net/users/24494 | Current state of the Komlos conjecture on vector balancing | For fixed $d$, one can actually achieve a bound independent of $n$. More precisely, $K=K(d)=O(\sqrt{d})$ is fine, uniformly in $n$.
Proof : the unit ball of $\mathbb{R}^d$ can be covered by $2C^d$ balls of radius $\frac{1}{2}$, for some absolute constant $C$. Let $v\_1,\dots,v\_n$ be $n$ vectors in $\mathbb{R}^d$ of ... | 9 | https://mathoverflow.net/users/21724 | 275262 | 122,636 |
https://mathoverflow.net/questions/275238 | 5 | An elliptic curve defined over a field $k$ is a smooth projective curve of genus $1$, plus a $k$-rational point. Every elliptic curve can be written in a Weierstrass form, i.e. as a plane cubic curve of the form $y^2 + a\_1 x y + a\_3 y = x^3 + a\_2 x^2 + a\_4 x + a\_6$.
Now let $A$ and $B$ be two symmetric $4 \times... | https://mathoverflow.net/users/76332 | Intersections of quadratic planes as elliptic curves | Regarding your main question, this is done in Cassels, *Lectures on elliptic curves*, $\S$ 8 (iv) p. 36. We may assume that the common rational point of the quadrics is $(X:Y:Z:T)=(0:0:0:1)$. Then the quadrics have the shape
\begin{align\*}
Q\_1 & = TL + R\\
Q\_2 & = TM + S
\end{align\*}
where $L,M$ (resp. $R,S$) are l... | 7 | https://mathoverflow.net/users/6506 | 275268 | 122,638 |
https://mathoverflow.net/questions/275230 | 5 | Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that:
* $F$ separates points of $X$, i.e. for any two distinct points $x,y\in X$ there is $\varphi\in F$ such that $\varphi(x)\ne\varphi(y)$;
* for every $x\in X$ there is an open neigh... | https://mathoverflow.net/users/53155 | Generating the topology of a manifold | This is just a variation of Corbennick's comment.
The statement in the question is wrong for any non-compact manifold: take $x\in X$ and consider all maps into $\mathbf{R}^{d}$, such that their limit at infinity equals their value at $x$. This collection generates the topology of the one point compactification of $X$... | 2 | https://mathoverflow.net/users/53155 | 275275 | 122,639 |
https://mathoverflow.net/questions/275271 | 3 | A truth predicate for first order set theory would allow you to determine the truth of statements in first order set theory. A definition is given [here](https://mathoverflow.net/a/273121/65915).
My question is, can you formulate a statement or axiom asserting that such a truth predicate does *not* exist.
In partic... | https://mathoverflow.net/users/65915 | Can you formulate a theory stating that a truth predicate does not exist for first order set theory? | The assertion that there is (or is not) a truth predicate is expressible in the second-order language of set theory, but assuming consistency, not by any first-order assertion.
**Second-order.** In the second-order case, one simply says that there is a class $T$ satisfying the Tarskian recursion $$\exists T\ (T\text... | 9 | https://mathoverflow.net/users/1946 | 275276 | 122,640 |
https://mathoverflow.net/questions/113989 | 6 | Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory ([Borger, Wieland](http://arxiv.org/pdf/math/0407227.pdf), 2.5). The corresponding comonad on $\mathrm{CAlg}(k)$ is simply $\mathrm{Hom}\_{\math... | https://mathoverflow.net/users/2841 | Classification of plethories over $\mathbb{Q}$ | The preprint <https://arxiv.org/abs/1701.01314> of Magnus Carlson, "Classification of plethories in characteristic zero" answers the question about plethories over $\mathbb{Q}$ in the affirmative.
| 10 | https://mathoverflow.net/users/437 | 275295 | 122,645 |
https://mathoverflow.net/questions/275224 | 1 | I know that for standard Brownian motion, the total variation sampled at intervals of length $\Delta$ converges to $V(\Delta) = C \Delta^{-1/2}$ for some constant $C$. I wish to use this fact to study whether or not my data behaves as Brownian motion by calculating $V(\Delta)$ for many $\Delta$ in order to estimate $\b... | https://mathoverflow.net/users/112146 | Literature on the total variation of fractal graphs/fractal Brownian motion? | I presume that you are considering sample paths on the time interval $[0,1]$, so that the number of samples is $n=1/\Delta$. The fractional Brownian motion has stationary increments, whence
$$
\mathbf E V(\Delta) = n \mathbf E |Z(\Delta)| ;.
$$
Since $Z(\Delta)$ is Gaussian with variance $\Delta^{2\alpha}$,
$$
\mathbf... | 1 | https://mathoverflow.net/users/8588 | 275307 | 122,651 |
https://mathoverflow.net/questions/275299 | 6 | We define $\mathsf{BWKL}$ as follows:
Every infinite binary tree of bounded width has an infinite path.
This obviously follows from $\mathsf{WKL}$. Is this principle true in $\mathsf{RCA}\_0$? If not, what does it need?
I've found [a paper](https://www.mimuw.edu.pl/~lak/buchi_strength.pdf) that addresses a relat... | https://mathoverflow.net/users/38989 | Bounded-width Konig's lemma in reverse math | If I understand your question correctly, this principle is indeed provable from either WKL or I$\Sigma^0\_2$, but not in RCA$\_0$, or even WWKL$\_0$. See slides 12 and 13 in [this talk](https://www.birs.ca/cmo-workshops/2016/16w5072/files/Yokoyama.pdf) by Yokoyama. Based on [this paper](https://arxiv.org/pdf/1704.00931... | 5 | https://mathoverflow.net/users/47312 | 275310 | 122,654 |
https://mathoverflow.net/questions/275280 | 6 | In [this question](https://mathoverflow.net/questions/273331/behaviour-at-natural-boundary) some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging *to* the unit circle. Well, nothing is new under the ... | https://mathoverflow.net/users/11142 | zeros on the circle of convergence | **Disclaimer:** I learned the trickery below from N.K.Nikolskii, who was giving us a special topics course in complex analysis when I was a fourth year undegraduate student. I have no idea whether it can also be traced back 100 years but, certainly, it has been used by many people on many occasions and is worth teachin... | 3 | https://mathoverflow.net/users/1131 | 275315 | 122,656 |
https://mathoverflow.net/questions/275292 | 16 | I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.
Let $n>1$ be an integer. If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) \cong GL\_2(\mathbb{Z}/n\mathbb{Z})$ (edit) and containing the $n$-th roots of unity (/edit), must there exist an elliptic c... | https://mathoverflow.net/users/37644 | Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field? | I have two things to add to this discussion.
$\bullet$ For $n = 2$ and $n = 3$, every Galois extension of $\mathbb{Q}$ with Galois group ${\rm GL}\_{2}(\mathbb{Z}/n\mathbb{Z})$ *does* arise from an elliptic curve (by a result of Shepard-Barron and Taylor from 1997 - see the reference in the paper of Dieulefait linked... | 12 | https://mathoverflow.net/users/48142 | 275320 | 122,659 |
https://mathoverflow.net/questions/275207 | 8 | **Definition.** A function $f:X\to Y$ between topological spaces is called
$\bullet$ *$G\_\delta$-measurable* if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G\_\delta$ in $X$;
$\bullet$ *$\sigma$-continuous* if $X$ has a countable cover $\mathcal C$ such that $f|C$ is continuous for every $C... | https://mathoverflow.net/users/61536 | Is each $G_\delta$-measurable map $\sigma$-continuous? | I looked to my own [old paper](https://arxiv.org/pdf/0801.2131) with Bokalo and have found there Example 9.3 answering this problem:
**Example.** Under Martin's Axiom (more precisely, $\mathrm{add}(\mathcal M)=\mathrm{cof}(\mathcal M)$) there exists a bijective function $f:X\to Y$ between zero-dimensional separable m... | 1 | https://mathoverflow.net/users/61536 | 275322 | 122,660 |
https://mathoverflow.net/questions/275161 | 5 | Let $E \to X$ be a vector bundle. We can associate to $E$ several invariants: among them are the *Stiefel-Whitney* classes $w\_i(E) \in H^i(X;\mathbb{Z}\_2)$. These classes may be defined using the axioms:
0. $w\_0(E)=1$ and $w\_i(E) \in H^i(X;\mathbb{Z}\_2)$.
1. $w(f^\*E)=f^\*w(E)$ for continuous maps $f$ (here ... | https://mathoverflow.net/users/24078 | Two set of axioms for Stiefel-Whitney classes | The naturality condition 1'' reduces the question of whether $v\_1=w\_1$ to the special case of the tautological bundles $\gamma\_n$. In this case both $v\_1$ and $w\_1$ lie in $H^1(BO(n);{\mathbb Z}\_2)$. This group is just ${\mathbb Z}\_2$, as shown in the book of Milnor and Stasheff for example, or just from the fac... | 6 | https://mathoverflow.net/users/23571 | 275325 | 122,662 |
https://mathoverflow.net/questions/273336 | 4 | When defining the $A\_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic discs with incidence conditions. To acheive a virtual fundemental class, a Kuranishi structure is built on these moduli s... | https://mathoverflow.net/users/14105 | How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space? | You are right that the solutions of the perturbed equation do not satisfy the $\bar\partial\_J$ equation for any $J$ anymore. Please note that this is a feature: if they would still be properly $J$-holomorphic, you could not in general achieve transversality. Not even with multisections.
If you don't know such exampl... | 3 | https://mathoverflow.net/users/13514 | 275328 | 122,664 |
https://mathoverflow.net/questions/275342 | 1 | A Lax-Friedrichs (LF) type of flux for a conservation law $\partial\_tU+\partial\_xf(U)=0$ is given by
\begin{align}
F(U^-, U^+) = \frac{1}{2} \Big(f(U^-) + f(U^+)\Big)\cdot \nu - \frac{1}{2} \lambda(U^+ - U^-)
\end{align}
where for a classical LF flux in one dimension, $\lambda = \Delta t / \Delta x$. I understand th... | https://mathoverflow.net/users/17223 | A mathematical motivation for Lax-Friedrich type of Numerical Fluxes | The most natural way to derive Lax-Friedrichs's scheme is to consider the discretisation where the approximate state $U$ is constant ($\equiv U\_i^n$) in cells $((i-1)\Delta x,(i+1)\Delta x)\times((n-1)\Delta t,n\Delta t)$ where $n+i\in2{\mathbb Z}$. To pass from the array $(U\_i^n)\_i$ to the next one $(U\_i^{n+1})\_i... | 2 | https://mathoverflow.net/users/8799 | 275344 | 122,667 |
https://mathoverflow.net/questions/275352 | 0 | Let ${\cal U}$ be a non-principal [ultrafilter](https://en.wikipedia.org/wiki/Ultrafilter) on $\omega$. If $\kappa>0$ is a cardinal, we say that a function $c:\omega \to \kappa$ is a *coloring for ${\cal U}$* if for all $U\in{\cal U}$ the restriction $c|\_U$ is not constant. The *coloring number* of ${\cal U}$ is the l... | https://mathoverflow.net/users/8628 | Coloring non-principal ultrafilters on $\omega$ | If a function $c$ has finite image then $\omega$ decomposes into a disjoint union of finitely many subsets according to the value of $c$. By the property of the ultrafilter exactly one of those sets will be in $\mathcal U$. Thus $c$ is constant on a member of the ultrafilter. Therefore there is no ultrafilter with fini... | 6 | https://mathoverflow.net/users/28128 | 275353 | 122,670 |
https://mathoverflow.net/questions/275300 | 6 | Let $G$ be a semisimple, simply-connected, complex algebraic group. Fix a Borel subgroup $B$ and let $P$ be a parabolic subgroup properly containing $B$. If $M$ is a $B$-module, then we have the Leray-Serre spectral sequence corresponding to the fibration $P/B\to G/B\to G/P$
$E^{p,q}\_2=H^p(G/P, H^q(P/B,M))\implies H... | https://mathoverflow.net/users/31556 | Leray-Serre spectral sequence for algebraic groups | In the vein of @MarkGrant's comment: Since this is a first quadrant spectral sequence, for a given pair $(p,q)$ the individual terms $E\_2^{p,q},E\_3^{p,q},E\_4^{p,q},\ldots$ occurring on each subsequent page of the spectral sequence will eventually reach a stable value $E\_\infty^{p,q}$. (The larger $p+q$ is, the more... | 2 | https://mathoverflow.net/users/7932 | 275363 | 122,672 |
https://mathoverflow.net/questions/275372 | 3 | Suppose $X$ is a complete separable metric space, and there is a continuous map $x \mapsto \mu\_x$ associating to each point in $X$ a probability measure on $X$ (where we use the weak topology on the space of probabilities).
For each probability $\mu$ on $X$, let $T\mu = \int \mu\_x \mathrm{d}\mu(x)$. A probability $... | https://mathoverflow.net/users/7631 | Markov chain dichotomy | No - it's not true. In particular, you are asking whether, under your conditions, if the state space is compact then there is a unique stationary measure. However, already deterministic Markov chains provide a counterexample, because there are minimal not uniquely ergodic homeomorphisms of compact sets.
| 1 | https://mathoverflow.net/users/8588 | 275377 | 122,676 |
https://mathoverflow.net/questions/275355 | 7 | Let $A,B$ be positive dimensional Abelian varieties over a finite field and $p$ be an arbritrary prime. By Zarhin, *Homomorphisms of abelian varieties over finite fields* <http://www.math.nyu.edu/~tschinke/books/finite-fields/final/10_zarhin.pdf>, Theorem 10.2, one has an isomorphism $$\mathrm{Hom}(A,B) \otimes \mathbf... | https://mathoverflow.net/users/nan | Galois action on $p$-adic Tate module of Abelian variety over finite field semisimple? | You're just asking whether Frobenius acts semsimply on the $p$-adic Tate module.
We know from Tate's theorem that Frobenius acts semisimply on the $\ell$-adic Tate module, and hence satisfies some squarefree polynomial (its minimal polynomial). Now because the map from the ring of endomorphisms to the endomorphisms o... | 11 | https://mathoverflow.net/users/18060 | 275379 | 122,677 |
https://mathoverflow.net/questions/275190 | 3 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). If $\kappa>0$ is a cardinal, we say the hypergraph $H$ is $\kappa$-*chromatic* if there is a function $c:V\to\kappa$ such that for all $e\in E$ the restriction $c|\_e$ is not constant (that is, the vertices of every edge are colored with at leas... | https://mathoverflow.net/users/8628 | Coloring hypergraphs with no singleton intersections | To answer Jon Noel's question in the comments, there is no such example for finite hypergraphs.
**Claim.** Let $H=(V,E)$ be a finite hypergraph such that $|e| > 1$ for all $e \in E$ and for all distinct $e\_1, e\_2 \in E$, $|e\_1 \cap e\_2| \neq 1$. Then $H$ is $2$-chromatic.
*Proof*. We proceed by induction on $... | 3 | https://mathoverflow.net/users/2233 | 275382 | 122,679 |
https://mathoverflow.net/questions/272820 | 7 | While working on some computations on Hilbert schemes, I came across the following combinatorial problem.
Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ internal nodes (internal node = node with children), with each node labeled by an integer (labels need not be unique, negativ... | https://mathoverflow.net/users/19088 | Counting some binary trees with lots of extra stucture | I was able to find an interesting generalization of your formula, but I'm having trouble finding a reference in the literature.
Let's call a 0-1-2 tree, a rooted tree where every vertex can have no child, a left child, a right child, or both. Let's call such a tree increasing if we also have a total ordering on the v... | 6 | https://mathoverflow.net/users/2384 | 275383 | 122,680 |
https://mathoverflow.net/questions/275286 | 0 | I have a polynomial $f\in\mathbb{C}[X\_1,...X\_n]$ and a bounded (non-empty) compact region $\Omega \subset \mathbb{C}^n$. Let's say additionally that $f$ is not zero on the boundary of $\Omega$.
Does there exist a certificate/criterion, which tells me whether there exists an $a\in\Omega$ such that $f(a)=0$?
I think ... | https://mathoverflow.net/users/64330 | Certificate/Criterion for the existence of zero of a complex multivariate polynomial in a bounded region | So, I think Zach is right, though you need a little more care. Since $\partial\Omega$ is not algebraic, it is not entirely straightforward that any hypersurface should intersect it, so one would need a version of the Jordan–Brouwer separation theorem to conclude that if there is a zero inside, then there has to be one ... | 0 | https://mathoverflow.net/users/10076 | 275385 | 122,681 |
https://mathoverflow.net/questions/275393 | 2 | I was curious to see whether the following conjecture of Morton-Silverman is (known to be) a consequence of Lang (or Lang-Vojta's) conjecture.
**Conjecture.** *Let $D$, $N$, and $d$ be positive integers. Then, there is an integer $C=C(D,N,d)$ such that, for all number fields $K$ of degree $D$ and all endomorphisms $... | https://mathoverflow.net/users/4333 | Does Lang's conjecture imply Morton-Silverman's Uniform Boundedness conjecture? | Uniform boundedness of torsion for elliptic curves (Mazur-Merel) and for abelian varieties (conjectured) is analogous to the dynamical conjecture on uniform boundedness of preperiodic points that Morton and I made and you have stated. An interesting and non-trivial result of Fakhruddin says that our conjecture for $\ma... | 3 | https://mathoverflow.net/users/11926 | 275395 | 122,685 |
https://mathoverflow.net/questions/275386 | 1 | Given a continuous and deriviable function of many variables, how do I know when this function is equals to zero on all the corners (or vertices) of a unit hypercube, i.e. all points, where each coordinate is equals either to 1 or 0.
I think that if I explore the function inside or/and around the unit hypercube, I wi... | https://mathoverflow.net/users/93648 | Given a continuous function of many variables, how do I know when this function is equals to zero on all the corners of a unit hypercube? | Since $x^m = x$ for $x \in \{0,1\}$ and $m \ge 1$, we may assume wlog all exponents in your polynomial are $0$ or $1$. Thus your polynomial can be written as $$P(x) = \sum\_{\alpha \in A} c\_\alpha x^\alpha$$
where $A$ is a collection of not too many subsets of $\{1,\ldots, n\}$ and
$x^\alpha = \prod\_{j \in \alpha} ... | 3 | https://mathoverflow.net/users/13650 | 275397 | 122,687 |
https://mathoverflow.net/questions/275348 | 10 | Let $M$ be a closed oriented manifold. Chas and Sullivan (<https://arxiv.org/abs/math/0212358>) introduced a Lie bialgebra structure on $H\_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop space $LM=\mathrm{Map}(S^1, M)$ relative to constant loops $M\rightarrow LM$.
It seems an "explanation" of the strin... | https://mathoverflow.net/users/18512 | String cobracket from TFT | The string cobracket you are referring to is not part of the TQFT structure of string topology given by the smooth Calabi Yau algebra structure on $C\_\*(\Omega M)$ but rather associated to an action of the chains of certain *compactification* of the moduli space of Riemann surfaces as explained in Sullivan's survey "S... | 7 | https://mathoverflow.net/users/5450 | 275401 | 122,689 |
https://mathoverflow.net/questions/275378 | 1 | Suppose $M$ is an $C^\infty$ surface in $\mathbb{R}^3$ and let
$$
K\_-=\{p\in M:\ K(p)<0\}\neq \emptyset,
$$
where $K$ denotes Gaussian curvature. Consider the following statement:
>
> Let $p\in K\_-$. Then, there exists a neighborhood $U$ of $p$ and a diffeomorphism $T:U\to U$ such that if $J$ is an open inte... | https://mathoverflow.net/users/53175 | Transformation which sends asymptotic lines to principal lines over a surface | I don't know a reference, but, as you've stated it, this is a trivial result:
If $p$ is a point on a smooth surface $S\subset\mathbb{R}^3$ at which the Gauss curvature is negative, then $p$ is non-umbilic, so, in an open $p$-neighborhood in $S$, both the principal curves and the asymptotic curves define transverse f... | 2 | https://mathoverflow.net/users/13972 | 275410 | 122,691 |
https://mathoverflow.net/questions/275404 | 3 | I have the following problem: in the free group $F\_2=\langle a,b\rangle$, we define the sequence
$\begin{cases}
w\_0=a, \\
w\_1=b, \\
w\_{n+2}=[w\_{n+1},w\_{n}] & \text{for }n\ge 0.
\end{cases}$
So $w\_2$ is the classical commutator $[b, a]$ (I take $[a,b]=aba^{-1}b^{-1}$ but this doesn't really matter), and then ... | https://mathoverflow.net/users/47274 | Maximal power in a sequence of iterated commutators in the rank two free group | A word in $F\_2$ can be represented by a path on the unit square grid on the plane. Now, $w\_0$ is a horizontal unit interval, $w\_1$ is a vertical unit interval, $w\_2$ is a unit square and the image of $w\_3$ is a union of two adjacent squares (on on the top of the other; it looks like a figure 8 from an old calculat... | 10 | https://mathoverflow.net/users/23500 | 275416 | 122,693 |
https://mathoverflow.net/questions/275376 | 5 | Let $T$ be a compact symmetric operator on $\ell^2$ and $T\vert\_{\ell^1}$ be bounded on $\ell^1$. Are there any non-trivial conditions that $T\vert\_{\ell^1}$ is compact as well (for example would $T$ belonging to some Schatten-class on $\ell^2$ be sufficient)?
The obvious proof estimating
$$\left\lVert \sum\_{i=0... | https://mathoverflow.net/users/57155 | Compact operators on $\ell^1$ | Here's an example showing that $T$ can be trace-class but $T|\_{\ell^1}$ is not compact.
Let $(x\_n)$ be a sequence of vectors in $\ell^2$ with disjoint supports, $\sum\_n \|x\_n\|\_2 \leq 1$ and $\|x\_n\|\_1=1$ for all $n$. Define
$$ T(\xi) = \sum\_n \xi\_n x\_n \qquad (\xi\in\ell^2). $$
Then $\| T(\xi) \|\_1 \leq \... | 5 | https://mathoverflow.net/users/406 | 275423 | 122,695 |
https://mathoverflow.net/questions/275428 | 0 | Given a positive integer $ n $ , let $ S\_{b}(n) $ the set of functions $ f $ fulfilling the following conditions :
1) $ f $ is continuous, positive and increasing on $(n,+\infty) $
2) for all $ x>n $ , the interval $I\_{f}(x) : =[x, x+f(x)] $ contains at least one prime number
3) the number of primes in $ I\_{... | https://mathoverflow.net/users/13625 | Space of functions f such that the number of primes in $ [x, x+f(x)] $ remains bounded | There is no such function $f(x)$. First, the prime number theorem implies that the average gap between two primes $p\_{n}$ and $p\_{n+1}$ is about $\log p\_{n}$ and for this reason, if $f(x)$ is smaller than $\log(x)$, then your condition (2) will fail.
On the other hand, Theorem 3.2 from Maynard's paper "[Dense clus... | 10 | https://mathoverflow.net/users/48142 | 275430 | 122,698 |
https://mathoverflow.net/questions/201049 | 3 | Sions minimax theorem ([wiki](http://en.wikipedia.org/wiki/Sion%27s_minimax_theorem), [paper](http://projecteuclid.org/download/pdf_1/euclid.pjm/1103040253)) can be stated as follows:
>
> Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex
> subset of a linear topological space. Let $... | https://mathoverflow.net/users/36356 | On compactness in Sion's minimax theorem | Here is a counterexample where neither $X$ nor $Y$ are compact. Consider $f(x,y)=y/(x+y)$ on $X\times Y$, where $X=Y=[1,\infty)$. Then
$$ 1=\inf\_x\sup\_y f(x,y)>\sup\_y\inf\_x f(x,y)=0.$$
| 3 | https://mathoverflow.net/users/12518 | 275432 | 122,699 |
https://mathoverflow.net/questions/275424 | 7 | Is there a connected $T\_2$-space $(X,\tau)$ with more than 1 point and with the following property?
>
> Whenever $D\subseteq X$ is dense, $X\setminus D$ is not dense.
>
>
>
| https://mathoverflow.net/users/8628 | Dense and co-dense subsets in connected $T_2$-spaces | A topological space is called *irresolvable* if it is not the disjoint union of two dense subsets. So you are asking whether there is a connected, $T\_2$, irresolvable space with more than one point. The answer is yes! You can find a proof, along with a few references to relevant literature, in
>
> D. Anderson, "O... | 12 | https://mathoverflow.net/users/70618 | 275433 | 122,700 |
https://mathoverflow.net/questions/275394 | 6 | The answer is "yes" if $\bar K = \overline{k(x)}$ (in fact, *every* $\phi \in Gal(\overline{k(x)}/\bar k)$ preserves all algebraically closed subfields -- there's only two of them, so it's not that hard!), but I'm not sure about higher transcendence degree. I suspect that for large enough $\bar K$ the answer must be "n... | https://mathoverflow.net/users/2362 | Can a nontrivial $\phi \in Gal(\bar K /\bar k )$ preserve all algebraically closed subfields? | Let me rephrase your question using a little bit of model theory (actually combinatorial geometry). Since the theory of algebraically closed fields is strongly minimal, the algebraic closure induces a pregeometry. More precisely given $\bar k \subseteq \bar K$ there is a pregeometry $G(\bar K/\bar k) = (\bar K, cl\_{\b... | 9 | https://mathoverflow.net/users/57712 | 275440 | 122,703 |
https://mathoverflow.net/questions/275434 | 4 |
>
> Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
> categories. $\mathcal{A}$ reflective and closed under filtered
> colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
> $R:\mathcal{K}\to \mathcal{A}$ be the reflection. Let $G$ be a dense
> generator of $\mathcal{K}$ consisting of... | https://mathoverflow.net/users/24563 | About small-orthogonality classes of a locally presentable category | It seems to me that a negative answer follows immediately from the possibility, in many cases, of choosing $G\subset \mathcal A$, so that $G$ doesn't know anything about $R$ until one closes it under some colimits. Given such a $G$, every object of $\mathcal K$ is right orthogonal to each $\eta\_g$, since these are iso... | 3 | https://mathoverflow.net/users/43000 | 275443 | 122,704 |
https://mathoverflow.net/questions/275422 | 0 | Given the 2D wave equation in polar coordinates:
$$u\_{\rho\rho}+\dfrac{1}{\rho}u\_{\rho}+\dfrac{1}{\rho^2}u\_{\theta\theta}=\dfrac{1}{a^2}u\_{tt}$$
with $u=u(\rho,\theta,t),(\rho,\theta,t)\in [0,c]\times(-\pi,+\pi]\times [0,+\infty)$
and boundary conditions:
$$u(0,\theta,t)=\nu(t),\forall \theta\in(-\pi,+\pi], t\in [0... | https://mathoverflow.net/users/21258 | 2D wave equation with gaussian boundary condition | You have then
$
\rho^2u\_{\rho\rho}+\rho u\_{\rho}=\rho^2 a^{-2} u\_{tt}.
$
Looking for a solution of the form $\phi(t) w(\rho)$, we obtain
$$
\bigl(\rho^2w''(\rho)+\rho w'(\rho)\bigr)\phi(t)=\rho^2 a^{-2} \phi''(t)w(\rho).
$$
If $\phi''(t)=-\lambda^2\phi(t)$, i.e. $\phi(t)= \alpha e^{i\lambda t}+\beta e^{-i\lambda t}$... | -1 | https://mathoverflow.net/users/21907 | 275448 | 122,706 |
https://mathoverflow.net/questions/275452 | 9 | What kind of upper bound can one get for
$$
\sum\_{d|n}\frac{1}{d^{\sigma}},
$$
where $0<\sigma<1$? The best I can do is $\ll n^{(1-\sigma)/2}$ by breaking the sum at $\sqrt{n}$, using the symmetry of the divisors, and replacing the sum on $d|n$, $d<\sqrt{n}$, with the sum over all $d<\sqrt{n}$. Since the hyperbola met... | https://mathoverflow.net/users/6756 | Divisor sum estimate | This is the topic of an [article fragment by Ramanujan](http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf). The story as I understood it was that there were wartime paper shortages, and the original article was shortened. Nicolas and Robin seem to be giving a different reason.
I also asked about aspects of this on ... | 9 | https://mathoverflow.net/users/3324 | 275457 | 122,707 |
https://mathoverflow.net/questions/275460 | 1 | Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits\_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c \ln \ln n}{\ln^\beta n}$ holds for infinitely many $n$.
Statement 2 : if the R.H. is false then there exist constants $0<\bet... | https://mathoverflow.net/users/106239 | On Robin's criterion for the Riemann Hypothesis | Statement 2 was proved by Jean-Louis Nicolas: see Theorem 3 in [*Petites valeurs de la fonction d'Euler*, Journal of Number Theory, 17 (1983), 375-388](http://www.sciencedirect.com/science/article/pii/0022314X83900550). More precisely, Statement 2 follows from the bound $\varliminf x^b\log f(x)<0$ in part (c) of the qu... | 7 | https://mathoverflow.net/users/11919 | 275462 | 122,711 |
https://mathoverflow.net/questions/275474 | 0 | Yesterday I needed to do some calculations with circles and "ventured" to calculate the arc length via the $\int{\sqrt{1+\left(f'(x)\right)^2}}$ formula and was baffled to see that in the case of unit half-circles that amounts to $\ ds := \sqrt{1+\left(f'(x)\right)^2} = \frac{1}{f(x)}$
Stuffing that differential equa... | https://mathoverflow.net/users/31310 | Has this Peculiar Property of Unit Circles Already been Noticed? | You are asking whether there are functions $f$ such that $(f')^2=h(f)$. (I denote $h=g^2-1$). The answer is yes: these functions are solutions of the differential equation
$f'=\sqrt{h(f)}$. The general solution of this equation is: $$\int\frac{df}{\sqrt{h(f)}}=x+c.$$
In your case, $h$ is a quadratic rational function. ... | 9 | https://mathoverflow.net/users/25510 | 275477 | 122,719 |
https://mathoverflow.net/questions/275454 | 2 | Let $X$, $Y$ be reflexive Banach spaces, and let $\imath:X\hookrightarrow Y$ be a bounded inclusion with dense image. Then for any domain $\Omega\subset\Bbb R^n$ we may define $C\_{\mathrm c}^\infty(\Omega;X)$ to be the (Fréchet)-smooth functions $\Omega\to X$ with compact support, with the usual inductive limit topolo... | https://mathoverflow.net/users/94022 | Let $X\subset Y$ be a dense inclusion of reflexive Banach spaces. Then is $C_c^\infty(\Omega;X)\subset C_c^\infty(\Omega;Y)$ dense? | $C^\infty\_c(\Omega, X) = C^\infty\_c(\Omega,\mathbb R){\bar\otimes} X$ completed inductive tensor product which agrees with the projective tensor product since $C^\infty\_c(\Omega,\mathbb R)$ is nuclear. Do the same for $Y$. Finite rank tensors are dense in $C^\infty\_c(\Omega,\mathbb R){\bar\otimes} Y$ and these can ... | 4 | https://mathoverflow.net/users/26935 | 275478 | 122,720 |
https://mathoverflow.net/questions/275451 | 10 | Let $X$ be a smooth complex projective variety. Let $M\_{Higgs}(X, P)$ be the coarse moduli which universally corepresents the functor:
$$M^{\#}\_{Higgs}(X, P): Sch/\mathbb{C}\longrightarrow \mathcal{Set}$$
which assigns to any scheme $S$ the set of isomorphism classes of semi-stable Higgs bundles $(E, \theta)$ over ... | https://mathoverflow.net/users/63996 | Interesting geometric application of Hitchin Fibration | You may know this already, but I'm not sure if you'll get many other answers. A nice result of Brunebarbe, Klingler and Totaro [Symmetric differentials and the fundamental group, Duke 2013] is that if the fundamental group of a smooth complex projective variety $X$ has a finite dimensional representation with infinite ... | 10 | https://mathoverflow.net/users/4144 | 275481 | 122,722 |
https://mathoverflow.net/questions/275351 | 2 | For G-spaces one has the property that fixed points of a join of two spaces is the join of the fixed points. I want to prove an analogue for homotopy fixed points. Is it true that
$$
X^{hG} \ast Y^{hG} \simeq (X\ast Y)^{hG}
$$
where $(-)^{hG}$ means homotopy fixed points.
| https://mathoverflow.net/users/112196 | Homotopy fixed points of a join | It is indeed not true.
Let $G = \mathbb{Z}$ and let $X = Y$ be the set of two elements with the action of $\sigma$, the generator of $\mathbb{Z}$ switching the two points.
For a space $Z$ with a $<\sigma> = \mathbb{Z}$ action,
a point $z \in Z^{h\mathbb{Z}}$ is the same as a point $z\_0 \in Z$ and a path between $z\_0... | 3 | https://mathoverflow.net/users/43850 | 275486 | 122,723 |
https://mathoverflow.net/questions/275495 | 4 | On $L^2(\mathbb{R}^n)$ it is true that $\Delta$ has $\sigma(\Delta)=(-\infty,0].$ Also, there are no eigenfunction. Yet, even if one would not know this, negativity $\langle \Delta u,u \rangle \le 0$ does immediately imply that there could only be such functions satisfying $\Delta u = \lambda u$ for $\lambda \le 0.$
... | https://mathoverflow.net/users/57155 | Eigenfunction of Laplacian | We can find all the tempered distributions $u$ such that $\Delta u=\lambda u$ (thus, including continuous functions going to $0$ at infinity, since these are locally integrable): taking Fourier transform, $(r^2+\lambda)\widehat{u}=0$. Unless $\lambda$ is real and non-positive, this implies that the support of $\widehat... | 11 | https://mathoverflow.net/users/15629 | 275510 | 122,733 |
https://mathoverflow.net/questions/275484 | 1 | Let $P:\ell^1(\mathbb{Z}^d) \rightarrow \ell^1(\mathbb{Z}^d)$
be given by
$$(Pz)(x)=\sum\_{y \tilde \ x} \frac{1}{2d} z(y)$$
where the tilde indicates that $y$ is a neighboured vertex of $x.$
I would like to know: Let $\lambda \notin [-1,1]$ is it true that
$\operatorname{ran}(P-\lambda)$ is dense in $\ell^1.$ I... | https://mathoverflow.net/users/57155 | Stochastic operator on $\ell^1$ has dense range | Yes, it is true (if we replace $2^d$ to $2d$). If the range of $P-\lambda$ is not dense, there exists a bounded linear functional $\eta\in \ell^\infty=(\ell^1)^\*$ which vanishes on this range. In other words, there exists a not identically zero bounded function $\eta$ on $\mathbb{Z}^d$ which satisfies $P^\*\eta=P\eta=... | 2 | https://mathoverflow.net/users/4312 | 275511 | 122,734 |
https://mathoverflow.net/questions/275499 | 3 | Kunen came up with the idea of, starting with a large cardinal notion A, kill the weak compactness of A and then resurrect some property without resurrecting weak compactness. Here we can for instance start with a weakly compact and find a generic extension in which the cardinal reflects stationary subsets but isn't we... | https://mathoverflow.net/users/38602 | Separation of large cardinal notions | Let me argue that Kunen's argument actually shows the best possible thing here.
First, let's think about consistency results. The "ideal" result here would be:
>
> (i) Con(ZFC) implies Con(ZFC + "There is a stationary-reflecting, non-weakly-compact cardinal.")
>
>
>
Obviously we can't prove this in any reaso... | 7 | https://mathoverflow.net/users/8133 | 275521 | 122,736 |
https://mathoverflow.net/questions/275524 | 5 | **Main idea shortly:** As we discussed recently [MO272045](https://mathoverflow.net/q/272045/10446), there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S\_n$.
Let me give ["field with one element"](https://en.wikipedia.org/wiki/Field_with_one_element) interpretation of that formula,
... | https://mathoverflow.net/users/10446 | Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$? | Yes this is known. The paper ["On Wohlfahrt series and wreath products"](http://www.sciencedirect.com/science/article/pii/S0001870806001642) by Y. Takegahara contains slightly more general results and builds on previous work (Muller's paper). Corollary 2.7 says:
$$\sum\_{n=0}^{\infty}\frac{|\operatorname{Hom}(A,G\wr ... | 7 | https://mathoverflow.net/users/2384 | 275526 | 122,740 |
https://mathoverflow.net/questions/275498 | 2 | Let $B$ be a $C^{\*}$-subalgebra of a $C^{\*}$-algebra $A$ with a faithful conditional expectation $P: A\rightarrow B$. [Kumjian](https://cms.math.ca/openaccess/cjm/v38/cjm1986v38.0969-1008.pdf) suggests on page 15 that since $P$ is faithful we have
$\left\Vert a\right\Vert =\text{sup}\left\{ \left\Vert P\left(c^{\*}... | https://mathoverflow.net/users/64444 | Characterization of $C^{*}$-algebra norms via conditional expectations | I think this can be answered with GNS representation theory.
Note that all is needed is that $P$ is a faithful positive map between C\*-algebras.
Let $S(B)$ be the state space of $B$. Consider the representation
$$
\pi = \bigoplus\big\{ \pi\_{\phi\circ P} \mid \phi\in S(B) \big\}
$$
of $A$ on some (huge) Hilbert spac... | 5 | https://mathoverflow.net/users/29404 | 275530 | 122,741 |
https://mathoverflow.net/questions/275468 | 5 | I have the following inequality:
$$ \left(\frac{1}{3}
\left(
\left(\frac{a+b}{2}\right)^3 +
\left(\frac{a+c}{2}\right)^3 +
\left(\frac{b+c}{2}\right)^3 \right) \right)^\frac{1}{3}
\leq
\left(\frac{a^p+b^p+c^p}{3}\right)^\frac{1}{p}$$
which is true $\forall a,b,c \in \mathbb{R}^+\cup\{0\} $.
I would like to ... | https://mathoverflow.net/users/112257 | Conditions under which inequality holds for all triplets of non-negative reals | *it would be quite a nice result, but unfortunately it's proven quite difficult.*
I'm not sure about "nice" (after all, one can invent infinitely many inequalities for 3 positive numbers) but it is certainly not difficult.
WLOG, $a+b+c=3$.
Write $a=1+A, b=1+B, c=1+C$ with $A,B,C\ge -1, A+B+C=0$.
Then the LHS equal... | 5 | https://mathoverflow.net/users/1131 | 275531 | 122,742 |
https://mathoverflow.net/questions/273839 | 11 | Suppose that you want to look at the left Kan extension of a functor $F : \mathcal{C} \to \mathcal{A}$ along a functor $K : \mathcal{C} \to \mathcal{B}$. It is widely known that if the colimit of the canonical diagram
$$ \mathrm{colim}(K \downarrow d \to \mathcal{C} \overset{F}{\rightarrow} \mathcal{A})$$
exists in $\... | https://mathoverflow.net/users/111321 | About pointwise Kan extension | The answer is no (I think -- non-pointwise Kan extensions are a pain and I may have messed something up!). I wouldn't lose too much sleep over this, though -- in practice, you never know that some functor is a Kan extension without knowing it's a pointwise one.
The non-pointwise Kan extension I gave [here](https://ma... | 2 | https://mathoverflow.net/users/2362 | 275535 | 122,744 |
https://mathoverflow.net/questions/275528 | 6 | Given an algebra $A$ with a right $A$-module $M$ with $End\_A(M) \cong A$.
Then we can view $M$ as a natural $A$-bimodule. When is $M$ as a bimodule indecomposable and what is its endomorphism ring as a bimodule?
In fact in my examples $M$ was always indecomposable and the endomorphism ring was isomorphic to the cent... | https://mathoverflow.net/users/61949 | Endomorphism ring of bimodules | The endomorphism ring of such a bimodule is indeed always the center: If $f$ is a bimodule-endomorphism of $M$, then in particular $f\in End({\_A M})$ so that by assumption $f(m)=ma$ for some $a\in A$. If this is also a right-module homomorphism, then $\forall b,m: mab = f(m)b = f(mb) = mba$. Because the right action i... | 8 | https://mathoverflow.net/users/3041 | 275536 | 122,745 |
https://mathoverflow.net/questions/273613 | 2 | Let $G=(V,E)$ be an infinite simple, undirected graph with $\chi(G) \geq \aleph\_0$. Is there a minor $M$ of $G$ such that
1. $M\not\cong G$, and
2. $\chi(M)=\chi(G)$
?
| https://mathoverflow.net/users/8628 | Minors of graphs with infinite chromatic number | For every such $G$ there is an $M$ satisfies your requirement.
It is enough to show that for every graph $G$ with infinite chromatic number has two minors $G\_{0}$ and $G\_{1}$ such that
$(i)$ $G\_{0}$ has no isolated vertex and $G\_{1}$ has exact one isolated vertex, and
$(ii)$ $\chi(G\_{0}) = \chi(G\_{1}) = \c... | 4 | https://mathoverflow.net/users/38228 | 275543 | 122,747 |
https://mathoverflow.net/questions/275553 | 1 | Let $\lambda=(\lambda\_1,\lambda\_2)$, $\mu=(\mu\_1,\mu\_2)$ be two compositions of $n$. Just to remind that $\lambda,\mu$ are not necessarily partitions. Denote $S\_{\lambda}$ and $S\_{\mu}$ the Young subgroups of $S\_n$. Say, $S\_{\lambda}=S\_{\{1,2,\ldots,\lambda\_1\}}\times S\_{\{\lambda\_1+1,\lambda\_1+2,\ldots,\l... | https://mathoverflow.net/users/106987 | Is there a way to find a representative set for double cosets of groups? | $S\_n$ is a Coxeter group (w.r.t. the neighbour-transpositions) and the Young subgroups are its parabolic subgroup. Therefore there is a canonical way to describe coset and double-coset representatives: There is a unique element of minimal length in each of these (double)cosets.
This also provides algorithmic ways to... | 4 | https://mathoverflow.net/users/3041 | 275555 | 122,751 |
https://mathoverflow.net/questions/275556 | 1 | Let $G$ be a simple group which has a maximal subgroup of the form $p:q$ (semidirect product of ${\Bbb Z}\_{p}$ and ${\Bbb Z}\_{q}$) where $p$ and $q$ are some primes and $q\vert(p-1)$. Also suppose that every Sylow $p$-subgroup of $G$ is ${\Bbb Z}\_{p}$ and every Sylow $p$-subgroup is contained properly in exactly one... | https://mathoverflow.net/users/97247 | Characterizing simple groups by their maximal subgroups | The Mathieu group $M\_{23}$ is a counterexample with $p=23$ and $q=11$.
But there are many (almost certainly infinitely many) counterexamples that are semilinear maximal subgroups of classical groups.
As John Shareshian pointed out, for $p=2^k+1$ a Fermat prime, there is an example with $G={\rm PSL}(2,2^k)$ and $q=... | 3 | https://mathoverflow.net/users/35840 | 275559 | 122,752 |
https://mathoverflow.net/questions/275449 | 2 | Let $B = \{ 0, 1 \}$. For two points $\textbf{x}, \textbf{y} \in B^n$ we will write $\textbf{x} \preceq \textbf{y}$ iff $\textbf{x}\_i \leq \textbf{y}\_i$ for every $i \in \{ 1, \ldots, n \}$.
A boolean function $f: B^n \rightarrow B$ is called *monotone* if $f(\textbf{x}) \leq f(\textbf{y})$, whenever $\textbf{x} \p... | https://mathoverflow.net/users/83519 | Characterization of monotone boolean functions with minimum number of extremal points | Lemma
-----
The set of functions with the minimum number of extremal points is closed under fixing variables.
That is, if $f\colon B^n\to B$ essentially depends on $k$ variables and has exactly $k+1$ extremal points,
then for each index $i$ and each $a\in\{0,1\}$, the function
$f\_{i\mapsto a}\colon B^n\to B$ defin... | 4 | https://mathoverflow.net/users/112284 | 275563 | 122,754 |
https://mathoverflow.net/questions/275475 | 7 | Let $X=\{x\_1,\dots,x\_N\}$ and $F=F(X)$ be a free group generated by $X$. Let $\phi\colon F\to F$ be an automorphism of $F$. Define a growth function of $\phi$ as:
$$
\operatorname{gr}\_{\phi,X}(n)=\operatorname{max}\_{1\le i\le N}\{\|\phi^n(x\_i)\|\_X\},
$$
where $\|.\|\_X$ denotes the word length with respect to $X$... | https://mathoverflow.net/users/2164 | growth of a free group automorphism is same for finite index subgroups? | While I cannot point to any place in the literature with this particular statement, I would say that it is an exercise in relative train track theory, which is the very nice normal form theory for $\text{Out}(F\_n)$ and $\text{Aut}(F\_n)$ found in papers of Bestvina, Feighn, and Handel (in various subgroupings). Usuall... | 8 | https://mathoverflow.net/users/20787 | 275564 | 122,755 |
https://mathoverflow.net/questions/275389 | 4 | [Kneser's conjecture](https://en.wikipedia.org/wiki/Kneser_graph) states that the chromatic number of the Kneser graph $KG(n,k)$ is $n-2k+2$. A [simple proof](http://www.renyi.hu/~barany/cikkek/1.pdf) using topological methods was given by Bárány, and an involved [combinatorial proof](https://link.springer.com/article/... | https://mathoverflow.net/users/83212 | Simpler combinatorial proof for special case of Kneser's conjecture | We have found such a proof together with Gábor Tardos while working on the local chromatic number of Kneser graphs and their relaitives. I try to give a detailed sketch.
Let me use the notation KG(2k+2,k) (instead of KG(2n,n-1), I use n for the size of the basic set, that is, n=2k+2). We will actually argue for Schri... | 4 | https://mathoverflow.net/users/112299 | 275577 | 122,757 |
https://mathoverflow.net/questions/275568 | 2 | Given the historical development of modern mathematics, everything is ultimately encoded as a set (possibly with some additional structure, also encoded as set(s) ). For example, a topological space is an ordered pair $(X, \tau)$ where $X$ is the underlying point-set and $\tau$ is a set of subsets on $X$ called it's to... | https://mathoverflow.net/users/112089 | Topological space (or math structure more generally) without encoding as set | There are several ways to understand this question, leading to different answers.
1. If the question is about studying *topological spaces*, in the sense of points equipped with open sets, independent of a particular axiom system for set theory, then the commentators are right: ordinary point-set topology is (like ne... | 13 | https://mathoverflow.net/users/49 | 275582 | 122,759 |
https://mathoverflow.net/questions/275554 | 21 | Seen $(\Bbb N,+,\cdot)$ as a [semiring](https://en.wikipedia.org/wiki/Semiring), is it possible to extend it to a semiring $(R,+,\cdot)$ so that the additive and multiplicative monoids become isomorphic? This means there is some monoid-isomorphism
$$\varphi:(R,\cdot)\cong(R,+)$$
and $\Bbb N$ is a sub-semiring of $R... | https://mathoverflow.net/users/108884 | Extending $\Bbb N$ to a semiring with isomorphic additive and multiplicative structure | There is an extension $R$: take the closure of $\mathbb N$ by the operations $\text{L}$ (or $\varphi$ in the OP) and its inverse $\text{E}$, which are the logarithm and exponential in base $1.2$. Notice that the choice of base (see below) implies that all ***new*** numbers generated by repeated applications of the 4 op... | 18 | https://mathoverflow.net/users/2480 | 275584 | 122,761 |
https://mathoverflow.net/questions/275548 | 16 | It is well known that locales are much more well behaved in a constructive setting than topological spaces. Nevertheless, many authors develop the theory of locales in classical mathematics. Are there any textbooks in which the theory is treated constructively?
I'm actually interested in specific questions. First, le... | https://mathoverflow.net/users/62782 | Locales in constructive mathematics | For this type of question the first reference that comes to my mind is P.T.Johnstone Sketches of an elephant, part C.
Most of the results in this book are constructively valid: If a result is proved over an arbitrary base (either a topos or a locale), it means that it is constructive, the few non-constructive result... | 12 | https://mathoverflow.net/users/22131 | 275605 | 122,769 |
https://mathoverflow.net/questions/275609 | 2 | Does there exist a finitely presented, torsion-free group $G$ which has conjugacy classes of finite size greater than one?
This condition came up in a research project, and we would like to rule out the existence of such examples.
**Edit**: Sorry for the confusion. I meant to ask for an example of such $G$ such tha... | https://mathoverflow.net/users/8103 | Torsion-free groups with finite conjugacy classes | Such groups do not exist by Schur's theorem <https://chiasme.wordpress.com/2015/01/07/a-theorem-of-schur-on-commutator-subgroup/>
| 6 | https://mathoverflow.net/users/nan | 275612 | 122,773 |
https://mathoverflow.net/questions/275529 | 8 | Ribet's paper on the Herbrand-Ribet theorem constructs a representation $\rho: Gal(\overline{\Bbb Q}/\Bbb Q) \to GL\_2(\mathbb F\_q)$ where $q = p^r$ of the specific form:
$
\begin{bmatrix}
1 & \*\\
0 & \chi
\end{bmatrix}$ where $\chi$ is a power of the cyclotomic character mod $p$.
In particular, if we let $K$ be th... | https://mathoverflow.net/users/58001 | Does Ribet's construction of class fields give us eigenspaces of rank 1? | I don't think we know how to prove this directly. Indeed, recent works by [Wake](https://arxiv.org/pdf/1401.3764.pdf) and [Wake–Erickson](https://arxiv.org/abs/1505.05128) show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the effect that a suitable localization of the Hecke a... | 3 | https://mathoverflow.net/users/18238 | 275617 | 122,774 |
https://mathoverflow.net/questions/194749 | 4 |
>
> A pernicious number is a positive integer such that the Hamming weight of its binary representation is prime.
>
> [[Wikipedia](https://en.wikipedia.org/wiki/Pernicious_number)]
>
>
>
The meaning of ‘pernicious’:
>
> **pernicious (adj.)**: highly injurious or destructive, deadly
>
> [[Merriam Webs... | https://mathoverflow.net/users/50685 | Why are they called ‘pernicious’ numbers? |
>
> Let $t\,(n)$ be the digit sum of the binary representation of $n$. Then Google will tell you that $n$ is called odious if $t\,(n)$ is odd and evil if $t\,(n)$ is even. Thus every number is either odious or evil, and therefore the words "odious" and "evil" cannot be pejorative in this context. It seems very likely... | 3 | https://mathoverflow.net/users/50685 | 275618 | 122,775 |
https://mathoverflow.net/questions/275589 | 3 | I searched so many articles about Bloch-Kato $p$-selmer Groups defined by $p$-adic representation but it seems that $p$ is not necessary to be odd.
Hence, I am wondering if it is worth considering Bloch-Kato 2-selmer Groups (2-primary) or Bloch-Kato conjecture for $p=2$ (2 part of Bloch-Kato Conjecture). Besides, there... | https://mathoverflow.net/users/112303 | Selmer $p$-Groups | You seem to be asking what the reason is for many papers on $p$-adic Selmer groups to assume throughout that $p > 2$: is because the case $p = 2$ is less interesting, or because it is more difficult?
The answer is definitely the latter. For many theorems concerning p-adic Selmer groups, there is an argument that work... | 5 | https://mathoverflow.net/users/2481 | 275622 | 122,776 |
https://mathoverflow.net/questions/275601 | 21 | **Title edited** I thank მამუკა ჯიბლაძე and Corbennick for their suggestion on the title of this question. I changed the title based on the suggestion of Corbennick.
What is an example of a manifold $M$ which does not admit an atlas $\mathcal{A}$ with the following property?:
For every two charts $(\phi,U)$ and $(\... | https://mathoverflow.net/users/36688 | Manifolds with polynomial transition maps | If I remember correctly, this is impossible for any (nonempty) simply-connected compact manifold of positive dimension. In particular, $S^2$ cannot have such an atlas. Off the top of my head, I don't remember where I saw this statement, but I'll try to find a reference.
Followup: I still don't remember the reference,... | 25 | https://mathoverflow.net/users/13972 | 275627 | 122,777 |
https://mathoverflow.net/questions/275623 | 0 | 1. Given $k\in\Bbb N$ is there coprime $1<a,1<b$ with $(k,a^2-b^2)=1$ and coprime $1<c,1<d$ such that $k|(ac-bd)$ and $k|(ad-bc)$?
2. What is the smallest $\max(|a|,|b|,|c|,|d|)=\max(a,b,c,d)$ among such $a,b,c,d$?
| https://mathoverflow.net/users/10035 | On symmetric linear diophantine equations | From your assumptions follows that $$k\mid (ac-bd\pm(ad-bc))=(a\mp b)(c\pm d).$$ So $k\mid(c\pm d)$, i.e. $k\mid(c+d,c-d)\in\{1,2\}.$
| 3 | https://mathoverflow.net/users/5712 | 275630 | 122,778 |
https://mathoverflow.net/questions/275613 | 1 | The following situation is given: Let $A$ be a unital, separable, nuclear $C^\*$-Algebra, $i:\mathbb{C}\to A$ the unital embedding. All $C^\*$-algebras are considered as trivially graded. Consider the induced map of $i$ in $KK$-theory: $i^0:KK^0(A,\mathbb{C})\to KK^0(\mathbb{C},\mathbb{C}),\; [E,\phi, T]\mapsto [E,\phi... | https://mathoverflow.net/users/nan | description of a map in KK-theory | I don't think that there is anything special about $\mathcal O\_\infty$ being used here. One observation that might help is that the map $KK(A, \mathcal O\_\infty)\to KK(\mathbb C, \mathcal O\_\infty)$ given by Kasparov product with the $KK$ class of $i\_0$ is the same as the map induced by $i\_0$ using the functoriali... | 2 | https://mathoverflow.net/users/85913 | 275632 | 122,780 |
https://mathoverflow.net/questions/275638 | 0 | For a line bundle $L$ on a curve $C$ the base locus of $L$, or equivalently the locus where the evaluation map $$H^0(L)\otimes \mathcal{O}\_C\to L$$ fails to be surjective is a proper closed subset of $C$. That is, the above map is always (at least) generically surjective. I am wondering if that holds for a vector bund... | https://mathoverflow.net/users/48522 | Is the evaluation map always generically surjective? | First, I assume that you meant $h^0(L)\geq 1$ in the first statement, else it is clearly false. Second one is false too, since you can take $E$ to be the direct sum of a line bundle $L$ with $h^0(L\geq 2$ and another one $M$, with $h^0(M)=0$. Then the above map is not generically surjective.
| 2 | https://mathoverflow.net/users/9502 | 275640 | 122,781 |
https://mathoverflow.net/questions/275643 | 5 | I am looking for a reference regarding the maximal proper connected algebraic subgroups of $PGL\_3$ and $PGL\_2 \times PGL\_2$ respectively when the base field is any algebraically closed field (of characteristic $\neq 2$).
For instance in the case $G=PGL\_2 \times PGL\_2$, if $H$ is a maximal proper connected algeb... | https://mathoverflow.net/users/23236 | Connected algebraic subgroup of $PGL_3$ and $PGL_2 \times PGL_2$ | Gary Seitz (and various collaborators over the years) have worked out lots of concrete information about maximal closed subgroups of classical groups and exceptional algebraic groups over an algebraically closed field of prime characteristic. Much of this is used in the study of maximal subgroups of corresponding finit... | 8 | https://mathoverflow.net/users/4231 | 275647 | 122,784 |
https://mathoverflow.net/questions/275633 | 7 | Let $\kappa\geq\aleph\_0$ be an infinite cardinal, and suppose that ${\cal A}$ is a collection of subsets of $\kappa$ such that for all $A\in {\cal A}$ we have $|A| = \kappa$ and for $A,B\in {\cal A}$ with $A\neq B$ we have $|A\cap B|<\kappa$. Is there $D\subseteq \kappa$ such that for all $A\in {\cal A}$ we have
1. ... | https://mathoverflow.net/users/8628 | Existence of a "diagonal" set in certain set systems | This is false in general when $\kappa=\omega$. Let $\mathcal{A}=\langle A\_\alpha:\alpha<\mathfrak{c}\rangle$ be an almost disjoint family of size continuum, and let $\langle D\_\alpha:\alpha<\mathfrak{c}\rangle$ list all subsets of $\omega$.
For each $\alpha<\mathfrak{c}$, one of $A\_\alpha\cap D\_\alpha$ and $A\_\a... | 8 | https://mathoverflow.net/users/18128 | 275653 | 122,786 |
https://mathoverflow.net/questions/275596 | 2 | Let $P\subset k[x\_0,\dots, x\_n]=:S$ be a prime ideal such that
$\dim( \text{Proj}(S/P))\geq 2$.
Is it true that for a general linear form $H\in S\_1$ we have that
the ideal $\langle P, H \rangle$ is again a prime ideal?
If not, for what kind of projective schemes $\text{Proj}(S/P)$ we can expect this?
| https://mathoverflow.net/users/38983 | Generic Algebraic Hyperplane Section of a Prime Ideal | This type of theorem is called a *Bertini theorem*, and there are versions for smoothness, (geometric) irreducibility, and geometric reducedness. Over a perfect field, geometric reducedness is equivalent to reducedness [[Tag 035X](http://stacks.math.columbia.edu/tag/035X)]. However, my last example below shows that thi... | 2 | https://mathoverflow.net/users/82179 | 275656 | 122,787 |
https://mathoverflow.net/questions/275614 | 2 | My purpose is to understand if in a graph $G = \langle V, E\rangle$ given 4 vertices in input (a, b ,c and d) they belong to an infinite path. With infinite path I mean a vertex succession that has a known head without ending.
In other words what I'm trying to do is to find and algorithm that could identify this kind... | https://mathoverflow.net/users/112312 | Algorithm to check if vertex belong to infinite path in Graph theory | If there is an infinite path then there is also a finite path using two of $a,b,c,d$ as endpoints. However the converse is not true, so the question makes sense. There is some ambiguity about how explicitly one knows the graph.
I will give a graph $H$ which is a disjoint union of paths and has an infinite path exact... | 0 | https://mathoverflow.net/users/8008 | 275667 | 122,792 |
https://mathoverflow.net/questions/275672 | 2 | What are the names of the research journals that focus on convex geometry? I know of "Advances in Geometry" and "Discrete & Computational Geometry" but no others.
Context of question: I am a graduate student who recently submitted a paper to a geometry research journal. They did not accept my paper and suggested that... | https://mathoverflow.net/users/112337 | Names of convex geometry journals | I am not aware of such a journal (and as someone who has been the editor in charge of convex geometry in a couple of good journals, I would probably know). My suggestion is to do one of the following:
Meditate on your paper, and try to discover more connections to other fields.
Go down the list of journals and see ... | 5 | https://mathoverflow.net/users/11142 | 275679 | 122,797 |
https://mathoverflow.net/questions/275670 | 1 | Given a graph $G(V,E)$ where every edge $e\in E$ has some positive weight $c(e)$. The graph can be directed/undirected/mixed. The graph is assumed to be strongly connected. Moreover, we define a subset of vertices $V'\subset V$.
Let $P\_{u,v}$ be the edges on the least cost path from $u$ to $v$, and $EP=\cup\_{u\in V',... | https://mathoverflow.net/users/106690 | Find all edges not covered by a shortest path in an all-pairs shortest path over a subset of vertices | I'll assume for simplicity the graph is undirected.
Run Dijkstra $|V'|$ times to find the distance $D(v,w)$ from each vertex $v$ of $V'$ to each vertex $w$ of $V$. Then an edge $E = (a,b)$ is in $EP$ if and only if there exist
$u,v \in V'$ such that $D(u,v) = D(u,a) + D(v,b) + c(a,b)$.
| 1 | https://mathoverflow.net/users/13650 | 275681 | 122,798 |
https://mathoverflow.net/questions/275658 | 6 | Schauder's Lemma in functional analysis states the following:
>
> Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\to F$ is *nearly open*, that is to say, for any neighborhood of the origin $U\subset E$ we have that $\emptyset\ne\... | https://mathoverflow.net/users/94022 | Proof of the Schauder Lemma | The result has not much to do with the linear structure. In the book *Introduction to Functional Analysis* of Meise and Vogt you find a version for metric spaces (Lemma 3.9):
>
> Let $X$ and $Y$ be metric spaces; $X$ be complete. Let $f:X\to Y$ be continuous and assume that
> for every $\varepsilon>0$ there exist... | 6 | https://mathoverflow.net/users/21051 | 275695 | 122,801 |
https://mathoverflow.net/questions/275685 | 3 | If $x\_n$ is a normalized, weakly-null sequence in a Banach space, and $\epsilon\_n\to 0$, does there exists a non-zero functional $f$ such that $|f(x\_n)|<\epsilon\_n$ for all $n$?
| https://mathoverflow.net/users/69275 | Rate of convergence of weakly null sequences | No, in fact in the Banach space $c\_0$, for any sequence $\epsilon\_n$ of positive numbers decreasing to $0$, we can choose a normalized weakly-null sequence $x\_n$ such that for every nonzero bounded linear functional $f$,
$\limsup\_{n \to \infty} |f(x\_n)|/\epsilon\_n = \infty$.
Let $e\_j$ be the $j$'th unit vecto... | 4 | https://mathoverflow.net/users/13650 | 275700 | 122,804 |
https://mathoverflow.net/questions/275592 | 3 | I asked the following question at Math Stackexchange a while ago [here](https://math.stackexchange.com/questions/2212682/for-an-arbitrary-gx-t-does-f-t-2g-xfgf-x-fx-0-0-have-a-unique-solut) but did not get a correct answer.
>
> Let $f(x,t)$ and $G(x,t)$ be smooth functions from
> $\mathbb R^2\to\mathbb R$.
>
> ... | https://mathoverflow.net/users/20838 | For an arbitrary $G(x,t)$, does $f_t=2G_xf+Gf_x$, $f(x,0)=0$ have a unique solution for $f$? | The answer is "No, it is not necessarily true that $f(x,t)=0$ for all $(x,t)\in\mathbb{R}^2$".
Here is a counterexample: Let $G(x,t) = x^2$ and let $h:\mathbb{R}\to\mathbb{R}$ be *any* smooth function that vanishes to infinite order at $0\in\mathbb{R}$. Now define $f:\mathbb{R}^2\to\mathbb{R}$ by requiring that $f(x,... | 6 | https://mathoverflow.net/users/13972 | 275702 | 122,806 |
https://mathoverflow.net/questions/275710 | 1 | Let $$I\_{N} = \int\_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N}(2\pi x)dx = \frac{2N-1}{2N} \int\_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N-2}(2\pi x)dx $$ therefore (I think)
$$ I\_N = \frac{(2N-1)!!}{(2N)!!}$$ hence $$ \frac{(2N)!!}{(2N-1)!!} I\_N = 1$$
Now let for $n \in \mathbb{N}$
$$J\_{3,N}(n) = \frac{(2N)... | https://mathoverflow.net/users/110322 | On Riemann zeta function and Dirac delta function/distribution | Let $$C\_n = \int\_{-1}^1 (1-\frac{x^2}{2})^n dx \sim \int\_{-\pi/2}^{\pi/2} |\cos(x)|^n dx$$
If $f$ is continuous
$$\lim\_{n \to \infty} \int\_{-1}^{1} \frac{(1-\frac{x^2}{2})^{n} }{C\_n} f(x)=\lim\_{n \to \infty} \int\_{-\pi/2}^{\pi/2} \frac{|\cos(x)|^{n} }{C\_n} f(x) = f(0)$$
(So that $\frac{|\cos(x)|^n }{C\_n}1\_{... | 2 | https://mathoverflow.net/users/84768 | 275714 | 122,809 |
https://mathoverflow.net/questions/275706 | 11 | The product
$$
F(s)=\prod\_{p}\frac1{(1-p^{-s})^p},
$$
converges for $\mathrm{Re}(s)>2$, when $p$ runs over all primes. Does it admit analytic continuation beyond the line $\mathrm{Re}(s)=2$? Any papers where it has been studied?
| https://mathoverflow.net/users/nan | Does this product have analytic continuation? | $$P(s) = \sum\_p p^{-s}, \qquad \log F(s) = \sum\_{p^k} \frac{p^{1-sk}}{k} = \sum\_{k\ge 1} \frac{P(sk-1)}{k}$$
* $P(s) = \sum\_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(ns)$ and $P\_N(s) = \sum\_{n=N+1}^\infty \frac{\mu(n)}{n} \log \zeta(ns)$ is analytic for $\Re(s) > \frac{1}{N+1}$
so that $$e^{N! P(s)} = e^{N! P\_... | 8 | https://mathoverflow.net/users/84768 | 275723 | 122,812 |
https://mathoverflow.net/questions/275731 | 0 | For any $n\in\mathbb{N}$ let $S\_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S\_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \pi(x) = x\}.$$
For any $n\in \mathbb{N}$ the expected value of the number of fixed points of a randomly chosen elemen... | https://mathoverflow.net/users/8628 | Asymptotic behaviour of fixed points in permutations | $E\_n=1$, it is well known and easy to prove by double counting: for any specific point $x$, it is fixed for $(n-1)!$ permutations, now sum up by $x$.
| 0 | https://mathoverflow.net/users/4312 | 275734 | 122,815 |
https://mathoverflow.net/questions/275704 | 5 | Take the straight forward Fibonacci equation
$$F\_0 = F\_1 = 1$$
$$F\_{n-2} + F\_{n-1} = F\_n$$
Let's consider a holomorphic function $F: \mathbb{C} \to \mathbb{C}$ such that
$$F(z)\Big{|}\_{\mathbb{N}} = F\_n$$
$$F(z-2) + F(z-1) = F(z)$$
Let's call such $F$, those that satisfy the Fibonacci equation in the com... | https://mathoverflow.net/users/nan | How do we classify all possible extensions of the Fibonacci recursion to the complex plane? | Yes, these are all entire solutions. This follows from a general theorem which says that linear combinations of exponential solutions of such equations (linear homogeneous with constant coefficients) are dense
in the set of all solutions, see, for example Gelfond, Calculus of finite differences, Chap V, sect. 7.
| 5 | https://mathoverflow.net/users/25510 | 275735 | 122,816 |
https://mathoverflow.net/questions/275743 | 2 | Does there exist any polynomial $p(x) \in \mathbb{R}[x]$ such that $p(n)$ is a prime number if and only if $n$ is a palindrome number ?
($n$ must be a positive palindrome number to give $p(n)$ a prime number).
| https://mathoverflow.net/users/109471 | Existence of polynomial p with real coefficients such that p(n) is prime if and only if n is palindrome | There is no such non-constant polynomial, even if we only assume that $p(10^m+1)$ is prime for $m=0,1,\dots$. Using this weaker condition for $m=0,1,\dots,\deg p$, we infer that $p(x)$ has rational coefficients. Let $N$ be a common denominator of the coefficients, and write it as $N=2^r5^sM$ with $(M,10)=1$. Fix $k\geq... | 8 | https://mathoverflow.net/users/11919 | 275753 | 122,819 |
https://mathoverflow.net/questions/275673 | 2 | When will the value of automorphic function $f(x)$ satisfy an algebraic equation? Or what is the value of $x$ such that the value of automorphic function $f(x)$ is algebraic?
If the question is too broad, may anyone familiar with such a topics give example or class of examples?
Any reference and answer is welcome. ... | https://mathoverflow.net/users/14024 | When will the value of automorphic function $f(x)$ satisify an algebraic equation? | Again, I think there is too much that could be said in response to this question to fit into this format, but I can give some indications.
Again, thinking that exponential functions are a sort of automorphic function, certainly special values of them (at "division points", hence, roots of unity) generate many abelian... | 3 | https://mathoverflow.net/users/15629 | 275756 | 122,821 |
https://mathoverflow.net/questions/275660 | 6 | I have a task:
Find all $n\ \epsilon \ N, \ n > 1$ for which a permutation $a\_1,\ a\_2,\ ...,\ an\ $ of numbers $ 0,1, ..., n - 1$ exists such that $a\_1,\ a\_1+a\_2,\ ...,\ a\_1+a\_2+\ ...\ +an\ $ form a CRS $mod\ n$.
So far I've come to the conclusion that $a\_1$ must be $0$ because otherwise there would be tw... | https://mathoverflow.net/users/112298 | Complete residue system modulo n (permutation of numbers 0 to n-1) such that | Your question is about *sequenceable groups*, [introduced in 1961](http://msp.org/pjm/1961/11-4/pjm-v11-n4-p14-p.pdf) by Basil Gordon.
A finite group is called *sequenceable* if its elements can be written as a sequence $(g\_1,g\_2,\dotsc,g\_n)$ so that all the partial products $g\_1,g\_1g\_2,\dotsc,g\_1g\_2\dotsb g... | 11 | https://mathoverflow.net/users/9924 | 275759 | 122,823 |
https://mathoverflow.net/questions/275742 | 4 | Given an unclasped necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the same number of beads of type $i$ where $i \in \{1, \dots, d\}$. We also assume that the number of beads of type $i$ is div... | https://mathoverflow.net/users/112374 | Which necklaces require maximal cuts? | I think the answer is "no".
Let's consider $p=2,d=3$. Suppose that we have a necklace which can be fairly divided using only 2 cuts (one less than the maximum number that may be required).
Let the types of the beads be 0, 1, and 2, and let the number of beads of type $i$ be $2a\_i$.
Choose three vectors in the... | 2 | https://mathoverflow.net/users/468 | 275763 | 122,824 |
https://mathoverflow.net/questions/275762 | 19 | The number of commuting pairs of elements in finite group G is equal to the product $k(G)\*|G|$ (see [MO271757](https://mathoverflow.net/a/271757/10446) ) where $k(G)$ is the number of conjugacy classes. Thus it is is divisible by $|G|$ (the number of elements of $G$).
That divisibility also follows from a theorem by ... | https://mathoverflow.net/users/10446 | The number of commuting m-tuples is divisible by order of group: Improvements? | The answer to questions 0 and 1 is yes. Here is a generalization.
>
> **Claim:** Let $\pi$ be a finitely generated group and $G$ be a finite group. Then
>
>
> $$\frac{|\text{Hom}(\pi \times \mathbb{Z}, G)|}{|G|}$$
>
>
> is equal to the number of conjugacy classes of homomorphisms $\pi \to G$.
>
>
>
We get ... | 16 | https://mathoverflow.net/users/290 | 275769 | 122,825 |
https://mathoverflow.net/questions/275493 | 24 |
>
> **Definition 1** A subset $B$ of a metric space $(M,d)$ is called a **metric basis** for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$
>
>
> **Definition 2** A metric space $(M,d)$ has "**metric dimension**" $n \in \mathbb{N}$ if there exists a *minimal* (in terms of cardinality) m... | https://mathoverflow.net/users/93694 | When does a metric space have "infinite metric dimension"? (Definition of metric dimension) |
>
> I'll try to make a couple of remarks. it'll be easier for me to write only a bit at a time.
>
>
>
*The metric category of metric spaces and metric maps (i.e. Lipschitz with constant $1$, i.e. never stretching) has a certain rigidity to it so that the idea of a metric base seems attractive. However, I feel th... | 5 | https://mathoverflow.net/users/110389 | 275771 | 122,826 |
https://mathoverflow.net/questions/275776 | 1 | Let $X$ be a finite set ordered by $R$, where $R$ is a transitive, reflexive, and antisymmetric relation on $X$. We define, for all $x\in X$, $C\_R(x)=(m\_R(x),M\_R(x))\in \mathbb N^2$, such that $m\_R(x)$ is the cardinality of the set of all the minorants of $x$, and $M\_R(x)$ is the cardinality of the set of all majo... | https://mathoverflow.net/users/112382 | Characterisation of a poset | If I understand correctly the meaning of minorant and majorant, then the answer is negative. Let $P$ be the poset with vertices $1,\dots,8$ defined by $1<5,6$; $2<6,7$; $3<7,8$; $4<5,8$. Let $Q$ have the same vertices and be defined by $1<5,6$; $2<5,6$; $3<7,8$; $4<7,8$.
| 4 | https://mathoverflow.net/users/2807 | 275777 | 122,829 |
https://mathoverflow.net/questions/275784 | 3 | Suppose $X\neq \emptyset$ is a set. Let $\tau\_1, \tau\_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau\_1,\subseteq)$ and $(\tau\_2,\subseteq)$ are order-isomorphic.
Does this imply that $(X,\tau\_1)\cong (X,\tau\_2)$ as topological spaces?
---
If Hausdorffness is dro... | https://mathoverflow.net/users/8628 | $T_2$-spaces with order-isomorphic topologies | If $h:(\tau\_1, \subseteq) \to (\tau\_2, \subseteq)$ is an order isomorphism.
Define $M(\tau\_1) = \{O \in \tau\_1: O \neq X \land \forall O' \in \tau\_1: O \subseteq O' \implies O' = X\}$, the "submaximal" elements.
Because the definition is purely order theoretical, $h[M(\tau\_1)] = M(\tau\_2)$.
Then all $O \in... | 3 | https://mathoverflow.net/users/2060 | 275790 | 122,837 |
https://mathoverflow.net/questions/275783 | 1 | It seems that there may be example of a transcendental entire function with finite (but positive) planar area of the Fatou set in Eremenko-Lyubich class. However, I can't not find it in the literature for Eremenko-Lyubich class. I remember that I may see a related title without any further details in the lecture notes ... | https://mathoverflow.net/users/11966 | Is there an example with Area $0<F(f)<\infty$ for some transcendental entire function | Yes there are such examples, the simplest one is $\sin(z^3)$.
It has a stronger property that the area of non-escaping set is finite.
MR2213937
Hemke, Jan-Martin,
Recurrence of entire transcendental functions with simple post-singular
sets.
Fund. Math. 187 (2005), no. 3, 255–289.
| 6 | https://mathoverflow.net/users/25510 | 275795 | 122,840 |
https://mathoverflow.net/questions/275785 | 7 | **Question.**
What are examples (preferably documented and explicitly commented on from this perspective in the literature, preferably in an article dedicated to this aspect alone) of the following well-known aspect of the usual completeness theorem for first-order logic (summarizing it here slightly flippantly, for ... | https://mathoverflow.net/users/108556 | Notable examples of syntactic proofs whose existence is guaranteed by completeness, but having been found later than a semantic proof? | There's a good reason that concrete examples are going to be rare. Existence of a first-order proof (at least in a countable language) is a $\Sigma\_1$ property in the language of arithmetic, and it's a general metamathematical principle that proofs of $\Sigma\_1$ statements should be constructive.
More precisely, if... | 9 | https://mathoverflow.net/users/8991 | 275813 | 122,849 |
https://mathoverflow.net/questions/275811 | 4 | It is well known that there are $2^{2^{\aleph\_0}}$ many non-principal ultrafilters on $\omega$. Is there a set ${\frak U}$ of non-principal ultrafilters on $\omega$ with $|{\frak U}| = 2^{2^{\aleph\_0}}$ such that for ${\cal U}\_1\neq {\cal U}\_2\in{\frak U}$ the partially ordered sets $({\cal U}\_1,\subseteq)$ and $(... | https://mathoverflow.net/users/8628 | Cardinality of a set of pairwise non-order-isomorphic ultrafilters on $\omega$ | Let me prove that a two ultrafilters $\mathcal{U}\_{1},\mathcal{U}\_{2}$ on $\omega$ are isomorphic as posets if and only if they are Rudin-Kielser equivalent. Suppose that $\phi:\mathcal{U}\_{1}\rightarrow\mathcal{U}\_{2}$ is a poset isomorphism. Then whenever $x\in\omega$, the element $\{x\}^{c}$ is a coatom in the p... | 10 | https://mathoverflow.net/users/22277 | 275836 | 122,860 |
https://mathoverflow.net/questions/275792 | 7 | Consider the following condition on a bounded operator $T$ on a Hilbert space:
$\ \ \ \ \ $(A) there exists an orthonormal basis $(e\_j)$ with $\sum\_j\parallel Te\_j\parallel<\infty$.
We have the implications
$\ \ \ \ \ $(trace class) $\ \Rightarrow\ $ (A) $\ \Rightarrow\ $ (Hilbert-Schmidt)
But what about the... | https://mathoverflow.net/users/nan | Property between trace class and Hilbert-Schmidt | Condition (A) is equivalent to $T$ being trace class. I will use the usual notation $|T|:=(T^\ast T)^{1/2}$.
**1.** Assume that condition (A) holds. Then
$$\sum\_j\langle|T|e\_j,e\_j\rangle\leq\sum\_j\||T|e\_j\|=\sum\_j\|Te\_j\|<\infty.$$
The left hand side is finite, hence $T$ is trace class by definition.
**2.**... | 5 | https://mathoverflow.net/users/11919 | 275837 | 122,861 |
https://mathoverflow.net/questions/275831 | 7 | I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-1}\\
r & 1 & \cdots & r^{d-2}\\
\vdots & \vdots & \ddots & \vdots\\
r^{d-1} & r^{d-2} & \cdots & 1
\end{pmatrix}$$
| https://mathoverflow.net/users/88033 | Determinant of correlation matrix of autoregressive model | The determinant of this matrix is $(1-r^2)^{d-1}$. This can be proven by induction on $d$ using the block matrix identity $\det \pmatrix{A & B\cr C & D} = \det(A) \det(D - C A^{-1} B)$, where $A$ is the top left $(d-1) \times (d-1)$ submatrix. Note that $$A \pmatrix{0\cr \ldots\cr 0\cr r\cr} = B$$
so $D - C A^{-1} B = ... | 7 | https://mathoverflow.net/users/13650 | 275844 | 122,862 |
https://mathoverflow.net/questions/275843 | 4 | Although [this problem](https://mathoverflow.net/questions/275633/existence-of-a-diagonal-set-in-certain-set-systems) got an accepted answer, I still feel there are more questions to ask in this direction.
>
> Suppose that to every $\alpha\in\mathbb R$ there corresponds a subset $A\_\alpha\subseteq\mathbb R$ with $... | https://mathoverflow.net/users/9924 | Choosing *two* representatives | I assume by $\aleph$ you [mean](https://math.stackexchange.com/questions/444273/how-many-infinite-cardinals-are-smaller-than-aleph?noredirect=1#comment4871997_444273) $\mathfrak c$, the cardinality of the continuum. You can build $D$ by transfinite recursion: Well-order the continuum in type $\mathfrak c$. At stage $\a... | 5 | https://mathoverflow.net/users/6085 | 275849 | 122,866 |
https://mathoverflow.net/questions/275782 | 7 | Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}\_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure on $\mathrm{Set}\_\Delta$ to define this). It has been proved that the $(\infty,1)$-category 2-$\mathrm{Cat}$ of $(\inft... | https://mathoverflow.net/users/37400 | Internal hom in $(\infty,2)$-categories | Yes because the model structure on Fun(Δ^op, sSet)
is a cartesian model structure.
This follows from the fact that the Joyal model structure is cartesian,
and §10, §11 in Rezk's paper “A model for the homotopy theory of homotopy theory”.
| 5 | https://mathoverflow.net/users/402 | 275870 | 122,870 |
https://mathoverflow.net/questions/275877 | 6 | A lecture I heard had a remark - "There is a rich class of pseudohoplomorphic curves to a symplectic manifold with an almost complex structure (tamed by the symplectic structure). On the other hand, there are very few complex valued functions on a symplectic manifold". Can someone explain why? Or point me to some refer... | https://mathoverflow.net/users/6822 | More pseudoholomorphic curves than complex valued functions | I think that, instead of complex-valued functions $f:M\to\mathbb{C}$ you mean *$J$-holomorphic functions*, i.e., complex-valued functions $f:M\to\mathbb{C}$ that satisfy $f'(x)(Jv) = i\,f'(x)(v)$ for all $v\in T\_xM$. The complex-valued functions don't have anything to do with the almost-complex structure $J$, while th... | 10 | https://mathoverflow.net/users/13972 | 275878 | 122,871 |
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