parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/433819 | 6 | I am trying to generalize the Gödel sentence as follows.
Define a pair of sentence $A$ and $B$ such that:
\begin{gather\*}
A := \lnot \operatorname{Prov}(\hat B) \\
B := \operatorname{Prov}(\hat A)
\end{gather\*}
where $\hat A$ and $\hat B$ are the Gödel numbers of $A$ and $B$ respectively.
From the definition above ... | https://mathoverflow.net/users/480476 | Generalize the Gödel sentence requires a fixed point theorem | Yes, in fact every such finite self-referential system has a fixed-point solution, and this can be proved using the same methods usually used to prove the unary fixed-point lemma. Such systems were explored at great length by Raymond Smullyan in several of his books. (But your desired example requires only the ordinary... | 14 | https://mathoverflow.net/users/1946 | 433822 | 175,463 |
https://mathoverflow.net/questions/433812 | 2 | Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $\mathcal{U}$.
Instead, we call a cover $\mathcal{U}$ **finitely intersecting** if every member of $\mathcal{U}$ interse... | https://mathoverflow.net/users/121875 | A stronger version of paracompactness | Lemma: Let $(U\_{\alpha})\_{\alpha\in A}$ be a finitely intersecting open cover of a space $X$. Then there is some partition $P$ of $A$ where $(\bigcup\_{\alpha\in R}U\_\alpha)\_{R\in P}$ is a partition of $X$ into clopen sets and where each $R\in P$ is finite or countable.
Proof: Let $E$ be the smallest equivalence ... | 4 | https://mathoverflow.net/users/22277 | 433823 | 175,464 |
https://mathoverflow.net/questions/433815 | 6 | Suppose $X$ is a smooth closed oriented 4-manifold, and $\Sigma\_1,\Sigma\_2$ are smoothly embedded compact oriented surfaces in $X$. Suppose they intersect transversally at two points with different sings, so that their algebraic intersection is zero: $[\Sigma\_1]\cdot [\Sigma\_2]=0$. Then can we take other representa... | https://mathoverflow.net/users/164671 | Two surfaces in a 4-manifold whose algebraic intersection number is zero | Yes, this can be done by tubing one surface along the other.
Suppose that you have two intersection points $p\_+, p\_- \in \Sigma\_1 \cap \Sigma\_2$ of opposite signs. Suppose also that $\Sigma\_1$ and $\Sigma\_2$ are connected (which we can assume, without loss of generality).
Choose a path $\gamma \subset \Sigma\... | 10 | https://mathoverflow.net/users/13119 | 433824 | 175,465 |
https://mathoverflow.net/questions/433733 | 4 | I have been recently studying different methods to construct Hopf algebras.
In Theorem IX.2.3 of "*Quantum groups*" by Kassel the **bicrossed product** of a pair of matched bialgebras (or Hopf algebras) $X$ and $A$ is defined as the vector space $X \otimes A$ with unit $1 \otimes 1$ and the following structure:
$$ (x... | https://mathoverflow.net/users/493533 | Bicrossed and bismash product of Hopf algebras | The two construction are similar but different. In the former, we deform the algebra structure with the two actions but preserve the coalgebra structure. For instance, if $G$ and $H$ are groups we have that $\mathbb{C}[G \bowtie H] \cong \mathbb{C}[G] \bowtie \mathbb{C}[H]$. In the second construction, we deform both, ... | 2 | https://mathoverflow.net/users/493533 | 433825 | 175,466 |
https://mathoverflow.net/questions/433695 | 6 | In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-category whose objects are all compact. One argument for this is that a certain fundamental result (Theorem 2.14 of [this pa... | https://mathoverflow.net/users/158123 | How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects? | After some discussion with Brian and reading the papers referenced by Balmer's survey, I realized what's going on here. The problem is not that the theory fails for "big" categories. Rather, it's that there are two possible theories: taking the thick primes of $C^{\omega}$, and taking the smashing primes of $C$. While ... | 0 | https://mathoverflow.net/users/158123 | 433848 | 175,472 |
https://mathoverflow.net/questions/433847 | 5 | Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. See for example
<https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0250340-1/S0002-9947-1969-0250340-1.pdf>
... | https://mathoverflow.net/users/387190 | Maximum symmetry metric on $ \mathbb{C}P^n $ | There's an easy counterexample to your guess: Let $M^6 = \mathrm{SU}(3)/\mathbb{T}^2$, where $\mathbb{T}^2\subset\mathrm{SU}(3)$ is the maximal torus (for example, the diagonal subgroup). In that case, there is a 3-parameter family of non-isometric metrics on $M^6$ that are invariant under $\mathrm{SU}(3)$, so they are... | 13 | https://mathoverflow.net/users/13972 | 433850 | 175,473 |
https://mathoverflow.net/questions/433593 | 2 | Consider a union-closed family $\mathcal{F}$ of $n=\vert \mathcal{F} \vert$ sets, $n$ odd, and its family $\mathit{J}(\mathcal{F})$ of $m = \vert\mathit{J}(\mathcal{F})\vert$ basis sets (or $\cup$-irreducible sets, also called generators), and define $\mathcal{F\_a} = \{A \in \mathcal{F}: a \in A \}$, $\mathit{J\_a}(\m... | https://mathoverflow.net/users/136218 | How to find an example of a union-closed family with two given properties | Not a good example, but it might hint for a better one.
Take two integers $N=2k+1,M$. Let the basis sets be $\{x,x+1,...,x+k\},1\leq x\leq N$, taken modulo $N$ (identify $N\equiv 0$) and $\{1,2,...,N,a\_1,a\_2,...,a\_M\}$.
The union-closed family $\mathcal{F}$ contains all set of the form $\{x,x+1,...,x+l\},1\leq x... | 2 | https://mathoverflow.net/users/432274 | 433861 | 175,478 |
https://mathoverflow.net/questions/433854 | 2 | I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All of the integrals I am interested in have the following form
$$ \int\_0^\infty f(x) \sin(x) dx \text{ or } \int\_0^\infty ... | https://mathoverflow.net/users/146528 | Assigning values to divergent oscillating integrals | There is indeed such a method which is elementary and allows one to rigorously give a numerical value to such integrals, in particular, the value $1$ to $\int\_0^\infty \cos x\, dx$. This uses two facts:
1. Every continuous function has a primitive. In your case this is $\cos x$.
2. About 60 years ago, a notion of th... | 1 | https://mathoverflow.net/users/494111 | 433868 | 175,481 |
https://mathoverflow.net/questions/433865 | 10 | I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one.
Let $M$ be a smooth $n$-manifold with boundary $N:= \partial M$, and assume that there exists a *continuous* retraction of $M$ onto $N$, name... | https://mathoverflow.net/users/7460 | Is every retraction homotopic to a smooth retraction? | Using a collar of the boundary we resort to the case when $r\in C^0(M, \partial M)$ is given by the projection $\partial M\times I \to \partial M$ over the collar .
Since $r$ is smooth on an open neighbourhood of $\partial M$, for any $\epsilon>0$ we can find a smooth $g\in C^\infty(M,\partial M)$ such that $g=r$ in ... | 10 | https://mathoverflow.net/users/158806 | 433875 | 175,483 |
https://mathoverflow.net/questions/433876 | 1 | Let
* $X$ be a metric space,
* $\mathcal M(X)$ the space of all finite **signed** Borel measures on $X$, and
* $\mathcal C\_b(X)$ be the space of real-valued bounded continuous functions on $X$.
Then $\mathcal C\_b(X)$ is a real Banach space with supremum norm $\|\cdot\|\_\infty$. We endow $\mathcal M(X)$ with the ... | https://mathoverflow.net/users/99469 | Are there some conditions on a metric space $X$ such that these two types of weak converge of finite signed Borel measures on $X$ are related? | An example. $X = [0,1]$ with the usual metric. $\mathcal C[0,1] = \mathcal C\_b[0,1] = \mathcal C\_0[0,1]$. $\mathcal C[0,1]^\* = \mathcal M[0,1]$.
Let $\mu\_n$ be the unit point-mass at $1/n$ and $\mu$ the unit point-mass at $0$. Show $\mu\_n \overset{1}{\rightharpoonup} \mu $ is true but $\mu\_n \overset{2}{\righthar... | 4 | https://mathoverflow.net/users/454 | 433880 | 175,485 |
https://mathoverflow.net/questions/433874 | 2 | Let $G$ be a compact (connected) semisimple Lie group. Let $G\_\mathbb{C}$ be the complexification of $G$.
Is $G$ a maximal compact subgroup of $G\_\mathbb{C}$?
| https://mathoverflow.net/users/172459 | Question about maximal compact subgroups of Lie groups | $\DeclareMathOperator\Lie{Lie}\newcommand\g{\mathfrak g}\newcommand\C{{\mathbb C}}$$\g = \Lie(G)$ is maximal among subalgebras of $\g\_\C = \Lie(G\_\C)$ on which the Killing form is negative definite, so $G$ is a maximal connected, compact subgroup of $G\_\C$. According to [@YCor](https://mathoverflow.net/questions/433... | 5 | https://mathoverflow.net/users/2383 | 433885 | 175,487 |
https://mathoverflow.net/questions/433839 | 8 | EDIT: in this question, I was proposing a conjecture, Prop. 1. Fedor Pakhomov showed a counter-example. [In this new question](https://mathoverflow.net/questions/433954/computational-complexity-and-commuting-functions-improved-conjecture) I propose a slightly weaker conjecture that holds even for that example and seems... | https://mathoverflow.net/users/138060 | Computational complexity and commuting functions | There is a counterexample to Proposition 1 iff $\mathsf{P}\ne\mathsf{PSPACE}$. The idea is to make a pair $f,g$ such that on certain inputs iterations of them individually are trivial, but their combination performs computation of a deciding algorithm for some $\mathsf{PSPACE}$-complete problem.
If $\mathsf{P}=\maths... | 5 | https://mathoverflow.net/users/36385 | 433887 | 175,488 |
https://mathoverflow.net/questions/433835 | 6 | Let $X$ be a metrizable topological space, $A\subseteq X\times X$ a nonempty closed subset which is reflexive, symmetric and transitive, $d:A\to \mathbb{R}\_+$ a continuous function that satisfies the metric axioms. Does there exist a metric on $X$ which extends $d$ and defines the topology on $X$?
If it makes any di... | https://mathoverflow.net/users/105656 | Extending a partially defined metric on a metrizable space | Here is a counterexample to Q2, with your stated extra condition.
Let $X$ consist of the half-open unit interval $(0,1]$ on the $x$-axis in the plane, together with the full unit interval $[0,1]$ at height $1$. That is,
$$X=B\sqcup T$$
where $B$ is the bottom line segment, having points of the form $(x,0)$ for $0<x\l... | 3 | https://mathoverflow.net/users/1946 | 433888 | 175,489 |
https://mathoverflow.net/questions/433881 | 4 | There is a somewhat forgotten sieve-theoretic approach to the Goldbach conjecture, due to Buchstab et al, see e.g. pp.247-248 of R.D. James.
On p.247, James defines some function $F$ such that for any fixed $a \in \mathbb{N}$ and even $x \geq 6$:
1. $F(x ; 2, a, 1) = F(x; 2)$ with $a=1$ is the number of positive in... | https://mathoverflow.net/users/493772 | On Buchstab et al's "forgotten" sieve and the Goldbach conjecture for certain integers | The result (aka Buchstab's identity) you mentioned is not forgotten. In modern sieve theory texts such as Halberstam & Richert's *Sieve Methods* and Friedlander & Iwaniec's *Opera de Cribro*, the identity is written as
$$
S(\mathcal A,z)=S(\mathcal A,w)-\sum\_{w\le p<z}S(\mathcal A\_p,p)
$$
where $S(\mathcal A,z)$ ... | 13 | https://mathoverflow.net/users/449628 | 433890 | 175,490 |
https://mathoverflow.net/questions/433561 | 1 | Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) such that category of conilpotent reps of UC(T) is equivalent to the category of conilpotent reps of T. It can be construct... | https://mathoverflow.net/users/81055 | Two (or less) filtrations on coenveloping coalgebra | Yes, both of those filtrations should be equal, at least when the ground field has characteristic $0$. I don't know if there is a good standard reference in the literature, but I did have to grapple with the dual version of this problem in a paper I wrote a couple of years back (see the appendix of arXiv:1909.05734, es... | 1 | https://mathoverflow.net/users/126183 | 433892 | 175,491 |
https://mathoverflow.net/questions/433889 | 1 | Setup
-----
To clarify, let constants $0 < a < b < \infty$, and $p \in \mathbb{N}$ be fixed. Further let $B \subset \mathbb{R}^{p}$ be a fixed compact support. We then define the space of bounded (probability) densities on $B$, as follows:
$$
\mathcal{F}\_{B}^{[a, b]} := \left\{f \colon B \to [a, b] \mid \int\_{B}{... | https://mathoverflow.net/users/486152 | Lower bound $L_{1}$-metric with $L_{2}$-metric for bounded pdfs, on common support | This is false. If (let's say) $f-g=1$ on a set of measure $\epsilon$ and $|f-g|\simeq \epsilon$ otherwise (and note that we can still normalize them both), then $\|f-g\|\_1^2\simeq \epsilon^2$, $\|f-g\|\_2^2\simeq\epsilon$.
For a concrete example, you can take $\mu$ as Lebesgue measure on $B=[0,1]$ and then
$$
f(x) =... | 3 | https://mathoverflow.net/users/48839 | 433899 | 175,493 |
https://mathoverflow.net/questions/433898 | 7 | The sequence of polynomials
$$P\_n=\sum\_{k=0}^{\lfloor(2n-1)/3\rfloor}
\frac{(2n-2k-1)!(2n-2k-2)!}{k!(n-k)!(n-k-1)!(2n-3k-1)!}x^k$$
satisfies apparently the identities
$$0=\sum\_{j=0}^nP\_{n-j}(P\_j-(-x)^j)$$
for all $n\geq 2$. (The previous condition
$n\geq 1$ was incorrect, as pointed out in the answer of Ira Gessel... | https://mathoverflow.net/users/4556 | A sequence of polynomials related to Catalan numbers | I find a slightly different initial condition for the recurrence:
$$0=\sum\_{j=0}^nP\_{n-j}(P\_j-(-x)^j)$$
for $n\ne 1$; for $n=1$ the sum is $-1$. It's easy to derive a formula for the generating function
$\sum\_{n=0}^\infty P\_n(x) z^n$ from this recurrence. We find that, as noted by Tewodros,
$$\sum\_{n=0}^\infty P\... | 9 | https://mathoverflow.net/users/10744 | 433913 | 175,499 |
https://mathoverflow.net/questions/433911 | 1 | I apologize if this question is not suited for MathOverflow. This has been crossposted in MathStackExchange [here](https://math.stackexchange.com/questions/4568587/vector-subbundles-of-a-given-one-in-mathbbcp1) and it is related to some open questions on that site that remain unsolved.
I would like to understand and ... | https://mathoverflow.net/users/494147 | Vector subbundles of a given one in $\mathbb{CP}^1$ | $\sum\_{i=1}^r \mathcal O(a\_i)$ is a sub-bundle of $\sum\_{i=1}^s \mathcal O( b\_i)$ if and only if, for all $c$, (1) $\# \{ i \mid a\_i \geq c \} \leq \# \{i \mid b\_i\geq c\}$ and (2) if equality holds for one $c$ it holds for all greater $c$.
Proof of "only if": By twisting, we may assume $c=0$. Then $\sum\_{i, a... | 1 | https://mathoverflow.net/users/18060 | 433915 | 175,500 |
https://mathoverflow.net/questions/433866 | 1 | I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e\_m, f\_1$ and action of ${\mathbb C}^\*$ by $t \cdot e\_m = t^m e\_m$ and $t \cdot f\_1 = f\_1$ and I have the projective line ${\mathbb P}^1$ which is the set of lines through the origin in this ${\mathbb C}^2$. I have the tautological line bundle $\L... | https://mathoverflow.net/users/198061 | Short exact sequence of equivariant line bundles on $\mathbb P^1$ | This is equivariant. The map $\Lambda \to \mathcal O$ explicitly on sections sends a section $(e\_m, f\_1) $ valued in $ L \subseteq \mathbb C^2$ to the section $f\_1$ of the trivial line bundle. This map is an isomorphism everywhere but the point corresponding to the line spanned by $e\_m$, and vanishes to order $1$ t... | 3 | https://mathoverflow.net/users/18060 | 433919 | 175,502 |
https://mathoverflow.net/questions/433860 | 6 | $\DeclareMathOperator\Ho{Ho}\DeclareMathOperator\Hom{Hom}$There are (at least) seven kinds of morphism spaces for a simplicial set $X$:
1. The [left-pinched morphism space](https://kerodon.net/tag/01KW) $\Hom^L\_X(x,y)$,
2. The [right-pinched morphism space](https://kerodon.net/tag/01KW) $\Hom^R\_X(x,y)$,
3. The (non... | https://mathoverflow.net/users/130058 | Do the various notions of morphism spaces of simplicial sets agree on the underived level? | There is no hope of comparisons between $\pi\_0\operatorname{Hom}^L\_X(x,y)$ and $\pi\_0\operatorname{Hom}^R\_X(x,y)$ in general, even when $x=y$. (Note that the statement you cited says $\operatorname{Hom}^L\_{X^{\text{op}}}(x,x) \cong \operatorname{Hom}\_X^R(x,x)^{\text{op}}$, so it computes Homs in a different simpl... | 7 | https://mathoverflow.net/users/82179 | 433929 | 175,506 |
https://mathoverflow.net/questions/433923 | 2 | Find the function-constant pairs $\langle f(x),c\rangle$ that satisfy the differential equation below:
$$f'(x)=f(x+c),$$
where $c \in \mathbb{C}$ and $f(x) \in \mathbb{C}$.
I found two families of functions (i.e. exponential functions and sine/cosine functions) that satisfy this differential equation. For example, $f... | https://mathoverflow.net/users/369335 | One question about a specific first-order differential equation | Following Euler, let us look at solutions of the form $f(z)=e^{sz}$, where $s$ is a complex number. We obtain a transcendental equation
$$s=e^{cs},$$
which for every complex $c\neq 0$ has infinitely many complex solutions $s\_n$. Now any linear combination
$$\sum\_{n}a\_ne^{s\_nz}$$
with complex $a\_n$, and any limit o... | 5 | https://mathoverflow.net/users/25510 | 433932 | 175,508 |
https://mathoverflow.net/questions/433928 | 2 | Let $G$ be a compact Lie group and Let $G\_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.
Consider $H$ to be a Lie subgroup of $G$ and denote by $H\_\mathbb{C}$ its complexification. Let $x \in X$, denote by $H\_x$ and ${(H\_{\mathbb{C}})}\_x$ the stabilizers... | https://mathoverflow.net/users/172459 | Generalization of $G/T \simeq G_\mathbb{C}/B$ | No.
Take $G = SU(2)$, $G\_{\mathbb C} = SL\_2(\mathbb C)$, $G/B$ the complex projective line alias the sphere, $H$ the diagonal $U(1)$, $x$ any point other than the two fixed points of $H$, so that the orbit $H/H\_x$ is a circular slice of the sphere, and $H\_{\mathbb C}$ the diagonal $GL\_1(\mathbb C)$, so that the ... | 4 | https://mathoverflow.net/users/18060 | 433933 | 175,509 |
https://mathoverflow.net/questions/433667 | 7 | The famous [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)) and its [hypergraphs generalization](https://mathoverflow.net/questions/432595/de-bruijn-erd%C5%91s-theorem-for-hypergraphs) states the following.
**Theorem.** *Let $V$ be a set, and $E\subset2^V$ b... | https://mathoverflow.net/users/482790 | Two questions on infinite hypergraphs | The answer to the first question is also no, by a minor modification of the proof of the Elekes-Hoffmann result cited in the answer to the second question. In fact, we get the following:
**Theorem** There is a set $V$ of cardinality $2^{\aleph\_0}$ and a collection $E$ of countably infinite subsets of $V$ such that, ... | 5 | https://mathoverflow.net/users/26002 | 433957 | 175,516 |
https://mathoverflow.net/questions/433951 | 4 | I'm reading a proof of below theorem from [this paper](https://arxiv.org/pdf/2205.13207.pdf).
>
> **Theorem A.3.** Let $\Omega$ be a locally compact normal Hausdorff space. Let $\left\{\mu\_n\right\} \cup\{\mu\} \subset \mathcal{M}(\Omega)$ and assume that $\underset{n \rightarrow \infty}{\operatorname{v-lim}} \mu\... | https://mathoverflow.net/users/477203 | Vague convergence: confusion about the regularity of a signed Radon measure and that of its variation | $\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\Om)$, any open subset $\Th$ of $\Om$, and any real $\ep>0$. Let $\de:=\ep/4$.
By the [Hahn decomposition theorem](https... | 4 | https://mathoverflow.net/users/36721 | 433987 | 175,523 |
https://mathoverflow.net/questions/431870 | 3 | Let $\Omega$ be a bounded smooth domain,
$Lu = D\_i \left( a^{ij} (x) D\_ju \right)$, and two constants
$\lambda, \Lambda > 0$. Suppose the coefficient $a$ is measurable,
symmetric, and satisfies
$$
a^{ij} \xi\_i \xi\_j \ge \lambda \vert{\xi} \vert^2 \quad \text{ and} \quad
\sum\_{i,j}^{} \vert{a^{ij}(x)}\vert \le \Lam... | https://mathoverflow.net/users/82772 | Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger | I recommend [Luigi Orsina's Lecture Notes](https://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf). They are beautifully written, and page 24 you will read Stampacchia's approach, which is (in my view) more elegant than Moser iterations and gives you the result you need in a jiffy.
| 2 | https://mathoverflow.net/users/40120 | 433995 | 175,526 |
https://mathoverflow.net/questions/433982 | 5 | $\newcommand{\U}{\mathcal{U}}$
$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\D}{\mathcal{D}}$
$\newcommand{\C}{\mathcal{C}}$
For any infinite $X \subseteq \omega$, we define:
$$
\D\_X := \{Y \in [\omega]^\omega : Y \subseteq X \vee Y \cap X = \emptyset\}
$$
It's easy to see that $\D\_X$ is a dense open subset of $([\ome... | https://mathoverflow.net/users/146831 | Minimum number of dense sets to make a filter an ultrafilter | No; $\mathfrak u'=\mathfrak c$.
To prove it, consider any $\mathcal C\subseteq[\omega]^\omega$ with cardinality $<\mathfrak c$. Working modulo finoite subsets of $\omega$ , and closing under (finitary) Boolean operations, we may assume that $\mathcal C$ is a Boolean subalgebra of $\mathcal P(\omega)/\text{fin}$, and ... | 7 | https://mathoverflow.net/users/6794 | 434012 | 175,531 |
https://mathoverflow.net/questions/434010 | 3 | Can we solve the follwing functional equation
$$f(xy)=g(x)h(y)+g(y)h(x)$$
on semigroups for unknown complex valued functions $f,g,h$ ?
| https://mathoverflow.net/users/494236 | A pexiderization of the sine addition law on semigroups | I corrected a typo on the right-hand-side of the equation in the OP, I'm unsure whether the left-hand-side has a typo, but the generalized sine addition law on semigroups is known in the form
$$g(xy)=g(x)h(y)+h(x)g(y),$$
so in terms of two unknown functions $g$ and $h$, generalizing
$$\sin(x+y)=\sin x\cos y+\cos x\sin ... | 5 | https://mathoverflow.net/users/11260 | 434015 | 175,533 |
https://mathoverflow.net/questions/433838 | 6 | Consider the $G(n,p)$ random graph model where $n$ is a ``large'' positive integer and $p\in (0,1)$. We may equip every realized random graph $G$ with its shortest path distance, making it into a (random) metric space $(G,d\_G)$. Since $G$ is finite then $(G,d\_G)$ is [doubling](https://en.wikipedia.org/wiki/Doubling_s... | https://mathoverflow.net/users/36886 | Expected doubling constant of a random Erdős–Rényi graph | Let me assume $p > (1 + \varepsilon)(2 \ln n/ n)^{1/2}$, this in particular includes the case of constant $p \in (0,1)$.
If $p > (1 + \varepsilon)(2 \ln n/ n)^{1/2}$ then w.h.p. the binomial random graph $G(n,p)$ has diameter at most $2$. Also, for $p = \omega( \ln n / n)$ then w.h.p. every vertex has degree $(1 \pm ... | 2 | https://mathoverflow.net/users/45545 | 434020 | 175,534 |
https://mathoverflow.net/questions/72201 | 1 | Is there an explicit formula for the number of fourth powers mod n?
Finch & Sebah [1] give theorems, partially folklore, for squares and cubes mod n, but I don't know of a similar formula for higher powers.
[1] S. R. Finch and Pascal Sebah, [Squares and Cubes Modulo n](http://arXiv.org/abs/math.NT/0604465)
| https://mathoverflow.net/users/6043 | Number of biquadrates mod n | As stated in the comments ([1](https://mathoverflow.net/questions/72201/number-of-biquadrates-mod-n#comment182932_72201) [2](https://mathoverflow.net/questions/72201/number-of-biquadrates-mod-n#comment182983_72201) [3](https://mathoverflow.net/questions/72201/number-of-biquadrates-mod-n#comment183408_72201)), the count... | 3 | https://mathoverflow.net/users/31084 | 434039 | 175,535 |
https://mathoverflow.net/questions/434035 | 9 | The following special cases are obvious:
1. A Grothendieck topos is concretizable ($F \mapsto \times\_i F(i)$)
2. A well-pointed topos is concretizable ($X \mapsto \rm{Hom}(1, X)$)
I looked at some more different examples and they are all obviously concretizable (but I'm just starting to learn topos theory). Is the... | https://mathoverflow.net/users/148161 | Is any elementary topos a concretizable category? | As Ivan Di Liberti suggests in the comments, according to Lemma 1.2 ([Freyd's paper, *Concretness*](https://doi.org/10.1016/0022-4049(73)90031-5), JPAA 1973) every regular well-powered category with equalizers is concretizable. In particular, every locally small elementary topos is concretizable.
| 12 | https://mathoverflow.net/users/148161 | 434043 | 175,538 |
https://mathoverflow.net/questions/434026 | 3 | For a given weighted graph $G = (V, E)$, there is a simple algorithm for finding the minimum weight circuit by running [Dijkstra's algorithm](https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Applications) $|E|$ times.
Also for a matroid $M = (E, I)$ one can use the greedy Rado-Edmonds algorithm to find a basis of... | https://mathoverflow.net/users/494271 | Algorithm for finding a minimum weight circuit in a weighted binary matroid | The problem is NP-hard (even in the unweighted case) via a well-known connection to coding theory. Namely, if $A$ is the parity check matrix of a binary linear code $C$, then the distance of $C$ is the size of a shortest circuit in the binary matroid represented by $A$. In [The intractability of computing the minimum d... | 2 | https://mathoverflow.net/users/2233 | 434045 | 175,539 |
https://mathoverflow.net/questions/434054 | 8 | Given the famous Littlewood-Richardson rule, in terms of Schur polynomials:
$$s\_\mu s\_\nu=\sum\_\lambda c^{\lambda}\_{\mu\nu} s\_\lambda,$$
is there a classification of the cases where the LR coefficients are equal to 1?
| https://mathoverflow.net/users/166314 | When the Littlewood-Richardson rule gives only irreducibles? | The answer is **Yes**, but this requires some elaboration.
1. [Knutson-Tao-Woodward](https://arxiv.org/abs/math/0107011) prove Fulton's conjecture in $\S$6.1. In principle, you can follow the approach by [De Loera-McAllister](https://arxiv.org/abs/math/0501446) or [Mulmuley-Narayanan-Sohoni](https://www.emis.de/journ... | 14 | https://mathoverflow.net/users/4040 | 434055 | 175,541 |
https://mathoverflow.net/questions/434041 | 4 | Let $K$ be a number field i.e. a finite extension of $\mathbb{Q}$, $\overline{K}$ a fixed separable closure of $K$, and $G\_K:=\mathrm{Gal}(\overline{K}/K)$ the absolute Galois group of $K$. Let $S$ be a finite set of non-Archimedean places of $K$, or equivalently, a finite set of non-zero prime ideals in the ring of i... | https://mathoverflow.net/users/492396 | Shafarevich's conjecture on Galois groups over fields ramified at finitely many places | In the footnote on Page 11 of [Fernando Q. Gouvea: Deformations of Galois Representations],
>
> The reference is
>
>
> [I.R. Shafarevich, *Algebraic number fields*, Proceedings of the
> international Congress of Mathematicians, Stockholm, 1962 (Djursholm),
> Inst. Mittag-Leffler, 1963, Translated version reprinte... | 4 | https://mathoverflow.net/users/492970 | 434057 | 175,542 |
https://mathoverflow.net/questions/434061 | 5 | As the title says. Which finite projective planes admit a symmetric incidence matrix?
I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F\_q)$ can always have a symmetric matrix, but he was unsure about the non-Desarguesian planes.
I am mostly asking for reference (both ... | https://mathoverflow.net/users/161063 | Which finite projective planes can have a symmetric incidence matrix? | The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, such that $\pi^2 = \operatorname{id}$ and such that a point $p$ is incident with a line $L$ if and only if the line $p^\pi$ is incid... | 10 | https://mathoverflow.net/users/12858 | 434062 | 175,543 |
https://mathoverflow.net/questions/433974 | 5 | It is [well-known](http://www.scholarpedia.org/article/Attractor) in dynamical systems that the concept of "attractor" differs in the literature.
*My question is whether attractors need to be defined as subsets of $\omega$-limit sets of some point on the phase space, or can be **any** compact set in the phase space?*... | https://mathoverflow.net/users/14101 | On the correct definition of attractors | This is just an extended comment, as I'm not even trying to recall all the uses of the term attractor in the literature, pointing out when equivalence holds or when there are some subtle differences depending, for instance, on the topology of the phase space or on properties of the map. Even the more basic definition o... | 3 | https://mathoverflow.net/users/167834 | 434066 | 175,545 |
https://mathoverflow.net/questions/434038 | 26 | There are several ways to generalize the notion of "algebraic closure" from fields to arbitrary commutative rings. A good overview is [On algebraic closures](https://www.jstor.org/stable/43737179) by R. Raphael. I am more interested in the notion suggested in [Stacks/0DCK](https://stacks.math.columbia.edu/tag/0DCK). In... | https://mathoverflow.net/users/2841 | Uniqueness of the "algebraic closure" of a commutative ring | Assume $R$ is reduced and $\mathrm{min}(R)$, the set of minimal prime ideals, is finite. Then $R$ has a *universal* aic, one in which every aic of $R$ can be embedded. We show this below.
Rings of this type have quite remarkable properties, and they include all Noetherian reduced rings. Domains are just the special c... | 11 | https://mathoverflow.net/users/31923 | 434072 | 175,546 |
https://mathoverflow.net/questions/412068 | 14 | Let's call an $E\_\infty$-algebra $A$ in spaces free if there is a space $A\_0$ and an equivalence of $E\_\infty$-algebras:
$
\coprod\_{n \ge 0} (A\_0)^n\_{h\Sigma\_n} \simeq A.
$
Consider a diagram of $E\_\infty$-algebras in spaces
$$
A \longrightarrow B \longleftarrow C
$$
and assume that each of them is free in th... | https://mathoverflow.net/users/91925 | Are free $E_\infty$-algebras in spaces closed under pullbacks? (as a full subcategory of all $E_\infty$-algebras in spaces) | ### Yes,
the full subcategory
$\mathrm{Mon}\_{\mathbb{E}\_\infty}(\mathcal{S})^{\rm free} \subset \mathrm{Mon}\_{\mathbb{E}\_\infty}(\mathcal{S})$
on those $\mathbb{E}\_\infty$-monoids that are equivalent to a free $\mathbb{E}\_\infty$-monoid, is **closed under all finite limits and retracts.**
This is Corollary 2.1.... | 7 | https://mathoverflow.net/users/91925 | 434074 | 175,547 |
https://mathoverflow.net/questions/434047 | 3 | Let $(x\_n)\_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property that
$x\_1<x\_2<...<x\_N$ and $$\frac1N \gtrsim \vert x\_j-x\_{j-1} \vert \gtrsim \frac1N.$$
We then define a function
$$f(x) = \sum\_{j=1}^{N} \frac{\alpha\_j}{x-x\_j},$$
where $\alpha\_j$ are positive numbers satisfying $1/N^2 \lesss... | https://mathoverflow.net/users/457901 | Bounds on zeros of rational function | Let us drop the assumption $x\_j\in[1,2]$, it is not needed.
Proving the result by contradiction, denote our function by $f\_N$, suppose that $f\_N(z\_N)=-i$, and $\mathrm{Im}\ z\_N= 1/(N^2R\_N)$ where $R\_N\to\infty$. Since nothing depends on a shift in horizontal direction, one may assume without loss of generality t... | 4 | https://mathoverflow.net/users/25510 | 434078 | 175,548 |
https://mathoverflow.net/questions/434076 | 12 | A *pseudo-Kähler manifold* is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a [pseudo-Riemannian metric](https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold).
>
> **Question.** What are e... | https://mathoverflow.net/users/392184 | Non-Kähler pseudo-Kähler manifolds | I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the [Kodaira-Thurston manifold](https://www.jstor.org/stable/2041749?origin=crossref#metadata_info_tab_contents). See for instance the introduction of
*Yamada, Takumi*, [**Ricci flatness of certain compact pse... | 13 | https://mathoverflow.net/users/7460 | 434079 | 175,549 |
https://mathoverflow.net/questions/434003 | 2 | Let
* $\Omega$ be a metric space,
* $C\_b(\Omega)$ the space of all real-valued bounded continuous functions on $\Omega$, and
* $\mathcal{M}(\Omega)$ the space of all finite signed Borel measures on $\Omega$.
For $\mu \in \mathcal{M}(\Omega)$, we denote by $|\mu|$ its associated variation measure. We say that a seq... | https://mathoverflow.net/users/477203 | Complex Borel measures: does $\mu_n \to \mu$ weakly imply $|\mu|(\Theta) \le \liminf_n |\mu_n|(\Theta)$ for every open subset $\Theta$? | $\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$By the [polar decomposition of complex measures](https://en.wikipedia.org/wiki/Complex_measure#Variation_of_a_complex_measure_and_polar_decom... | 4 | https://mathoverflow.net/users/36721 | 434082 | 175,550 |
https://mathoverflow.net/questions/371012 | 4 | If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1\_{et}(\mathrm{Spec}(K),\mu\_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, Corollary 4.4.3. I assume that this is connected, perhaps even implies the main theorem from Kummer theory stating tha... | https://mathoverflow.net/users/2841 | Etale cohomology and Kummer theory | The question was answered by R. van Dobben de Bruyn in the comments.
>
> "Serre's Galois cohomology states this relation briefly in a remark
> (p. 73 in §II.1.2 of the second English edition), and refers to
> Bourbaki's Algèbre (Chap. V) for a statement of Kummer theory (but
> Bourbaki doesn't use Galois cohomology... | 0 | https://mathoverflow.net/users/2841 | 434084 | 175,551 |
https://mathoverflow.net/questions/433954 | 2 | History of the question. I was proposing a conjecture [here](https://mathoverflow.net/questions/433839/computational-complexity-and-commuting-functions/433887#433887), called Prop. 1. Fedor Pakhomov showed a counter-example. Here I am proposing a slightly weaker version of the conjecture, Prop. 2, that holds for that c... | https://mathoverflow.net/users/138060 | Computational complexity and commuting functions, examples and conjectures | This is not a proper answer. I will give a construction of a pair of functions $f,g$ assuming the access to some cryptographic function $e$ that probably should form a counterexample for Proposition 2 under some reasonable computability-theoretic assumptions which I will not properly specify. Unfortunately my knowledge... | 1 | https://mathoverflow.net/users/36385 | 434101 | 175,559 |
https://mathoverflow.net/questions/434096 | 2 | We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started with simple cases on a smooth complex projective varieity $X$ of dimension $n$.
Taking $X$ itself to be the covering i... | https://mathoverflow.net/users/nan | Could certain closed covering determine a coherent sheaf? | It is not true even for $\mathbb{P}^n$. For instance, the tangent bundle $T\_{\mathbb{P}^n}$ restricts to each line as
$$
T\_{\mathbb{P}^n}\vert\_L \cong \mathcal{O}\_L(2) \oplus \mathcal{O}\_L(1)^{\oplus (n-1)},
$$
and on the other hand
$$
(\mathcal{O}\_{\mathbb{P}^n}(2) \oplus \mathcal{O}\_{\mathbb{P}^n}(1)^{\oplus (... | 2 | https://mathoverflow.net/users/4428 | 434102 | 175,560 |
https://mathoverflow.net/questions/225235 | 0 | It is well known that for an Stochastic differential equation (on the real line) of the form:
$dX\_t = \mu(X\_t)dt + \sigma(X\_t)dW$
where $W$ is the standard Wiener process, the transition probability densities of the process can be related to a PDE of the form
$\frac{\partial u}{\partial t} = \frac{\partial^2}{\parti... | https://mathoverflow.net/users/83631 | Weak solutions of linear parabolic PDEs and corresponding SDEs | One of the latest conditions on $\mu,\sigma$ are from ["A Numerical Method for SDEs with Discontinuous Drift"](https://arxiv.org/pdf/1503.08005.pdf):
>
> This result states that the SDE (1) ($dX\_t = \mu(X\_t)dt + \sigma(X\_t)dW$) admits a unique strong solution $X$ if the drift coefficient $\mu$ has finitely many ... | 0 | https://mathoverflow.net/users/99863 | 434120 | 175,569 |
https://mathoverflow.net/questions/434000 | 0 | Let $X$ be a metric space and $\mathcal B$ its Borel $\sigma$-algebra. For $B \in \mathcal B$ we denote by $\Pi(B)$ the collection of all finite measurable partitions of $B$, i.e.,
$$
\Pi(B)=\left\{\left(B\_{1}, \ldots, B\_{n}\right) \,\middle\vert\, n \in \mathbb{N^\*}, B\_{i} \in \mathcal B, B\_{i} \cap B\_{j}=\varno... | https://mathoverflow.net/users/477203 | Complex Borel measures: relation between the total variation norm of a measure and those of its real and imaginary parts | I represent below @NikWeaver's idea of improving $\frac{1}{2} [\cdot]' \le [\cdot]$ to get $\frac{1}{\sqrt 2} [\cdot]' \le [\cdot]$.
---
For complex number $z = x + iy$ with $x,y \in \mathbb R$, we have
$$
|z| \ge \frac{|x| +|y|}{\sqrt{2}} .
$$
Fix a Borel subset $B$ of $X$. Then
$$
\begin{align}
|\mu|(B) &= \s... | 0 | https://mathoverflow.net/users/477203 | 434123 | 175,571 |
https://mathoverflow.net/questions/428422 | 7 | Let $G$ be a countable amenable group with a (left) Følner sequence $(F\_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F\_n)$, in the sense that
$$
\lim\_{n \to \infty} \frac{\lvert\Gamma \cap F\_n\rvert}{\lvert F\_n\rvert} \ = \ 1.
$$
Now define
$$
\Gamma' := \{ (s,t) \in G^2 : st^{-1} \i... | https://mathoverflow.net/users/166445 | Density of “diagonal sets” in amenable groups | The answer to your question as stated is "no", but a variant of it is true (see the proposition below).
**Proof that the answer is "no":** Let $(F\_n)$ be the Følner sequence in $\mathbb{Z}$ given by $F\_n = [2^n, 2^n + n]$, and let $\Gamma = \bigcup\_{n \in \mathbb{N}}{F\_n}$ so that $\Gamma$ has full density along ... | 2 | https://mathoverflow.net/users/171304 | 434124 | 175,572 |
https://mathoverflow.net/questions/434127 | 3 | Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
1. If $r(G) > r(K)$ (here, $r$ means the rank of maximal torus), is it true that $G/K$ admits free $S^1$-action? If so, why? If not, what is a ... | https://mathoverflow.net/users/166199 | Free $S^1$-action on compact homogeneous spaces | Here is a counterexample where $K$ is connected.
Let $W = SU\_3/SO\_3$ be the Wu manifold. This 5-dimensional manifold is simply-connected and has $H\_2(W;\Bbb Z) = \Bbb Z/2$, and in particular, $\pi\_2(W) = \Bbb Z/2$ by the Hurewicz theorem. Note that $r(SU\_3) = 2$ and $r(SO\_3) = 1$.
Now I claim that there is no... | 2 | https://mathoverflow.net/users/40804 | 434162 | 175,580 |
https://mathoverflow.net/questions/434132 | 2 | Let $K\_0$ and $K\_1$ be two knots in $S^3$. We say $K\_0$ and $K\_1$ are *concordant* if there exists a smoothly embedded annulus $A \subset S^3 \times [0,1] $ such that $\partial A = -(K\_0) \cup K\_1$.
Given two **non-trivial** concordant knots $K\_0$ and $K\_1$, assume that one of them is hyperbolic, say $K\_0$. ... | https://mathoverflow.net/users/475366 | Knot concordance, hyperbolicity and amphichirality | Neither of these properties are preserved by concordance. As was pointed out in the comments, any hyperbolic (for the first question) or chiral (for the second question) knot which is concordant to the unknot will be a counter-example. For a specific example, the knot 6\_1 is slice (concordant to the unknot), but is hy... | 6 | https://mathoverflow.net/users/173304 | 434167 | 175,582 |
https://mathoverflow.net/questions/433907 | 26 | *Update.* It's now on the [arXiv](https://arxiv.org/abs/2211.07508).
---
Some time ago I found my "own" proof of the [fundamental theorem of Galois theory](https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory). You can find a pdf with the proof (link removed, see arXiv). It is quite short, self-cont... | https://mathoverflow.net/users/2841 | A simple proof of the fundamental theorem of Galois theory | At a first glance your approach reminds me of [Meinolf Geck's](https://www.jstor.org/stable/10.4169/amer.math.monthly.121.07.637#metadata_info_tab_contents) American Mathematical Monthly article, see also the [arxiv version](https://arxiv.org/abs/1306.3853) of his article.
| 9 | https://mathoverflow.net/users/18739 | 434176 | 175,586 |
https://mathoverflow.net/questions/434161 | 3 | Considering a quotient singularity $\mathbb{C}^n/G,$ its crepant resolution $Y$ (i.e. having $c\_1(Y)=0$) has rational cohomology supported in even degrees only. This holds for many other resolutions of singularities, in particular for symplectic resolutions of conic symplectic singularities.
I am wondering whether t... | https://mathoverflow.net/users/114985 | Resolution of conical singularities have even-only cohomology? | Thanks to @Yosemite Stan, we have a counterexample, and actually many of them:
Pick a projective variety $Z$ with some odd-cohomology such that
$$c\_1(\omega\_Z)=-m c\_1(H), \text{ for some } m>0,$$
where $H$ is the bundle by which $Z$ embeds to $\mathbb{P}^n,$
and $\omega\_Z$ is the canonical bundle.
In other words,... | 2 | https://mathoverflow.net/users/114985 | 434179 | 175,588 |
https://mathoverflow.net/questions/370484 | 0 | I have been tyring to understand the first condition given in the link <https://en.wikipedia.org/wiki/Regularity_structure> for quite some time now, at least a year. I have posted a similar question in <https://math.stackexchange.com/questions/3524946/how-to-think-of-a-set-that-has-no-accumulation-point> but did not ge... | https://mathoverflow.net/users/71105 | When and why do we require the condition that :"a subset bounded from below and has no accumulation points?" | From the references mentioned in the comments , (section 4.4,[*"Renormalisation of parabolic stochastic PDEs"*](https://arxiv.org/pdf/1803.03044.pdf)) (RP) and (section 6.1,*"Introduction to regularity structures"*) (IRS), we first start from the pde fixed point problem
$$\Phi = G \ast (\xi − \Phi^3 ) + G\Phi\_0 ,$$
... | 1 | https://mathoverflow.net/users/99863 | 434181 | 175,589 |
https://mathoverflow.net/questions/434148 | 4 | $\DeclareMathOperator\MCG{MCG}$Let $\Sigma$ be a compact oriented surface, with empty or connected boundary. Let $\mathcal{O}$ the space of orbits of nontrivial simple closed curves on $\Sigma$ under $\MCG(\Sigma)$-action. (so, $\mathcal{O}$ has finitely many elements: the sets of nonseparating curves and the sets of s... | https://mathoverflow.net/users/62201 | Action of noncentral mapping classes on curves or arcs on a surface | Yes, this is true - there are many ways to prove it, but I'll hit it with a hammer. Let $C(\Sigma)$ denote the [curve complex](https://en.wikipedia.org/wiki/Curve_complex) of $\Sigma$. Suppose not, then $f$ would map every curve $[c]$ in $\mathcal{O}\subset C(\Sigma)$ to $[f(c)]$ which has distance $\leq 1$ from $[c]$ ... | 5 | https://mathoverflow.net/users/1345 | 434187 | 175,592 |
https://mathoverflow.net/questions/434185 | 1 | Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz constant of $f-g$ is less than epsilon for any positive epsilon?
There should be some simple counterexample.
| https://mathoverflow.net/users/490101 | Construction of the Lipschitz function with a given Lipschitz constant and given two values | $\newcommand\ep\varepsilon$Yes, it is easy to construct a counterexample here.
Indeed, if $g\_\ep$ is such a function for each given real $\ep>0$ (so that $|g\_\ep| \geq c$, $g\_\ep(a)=f(a)$, $g\_\ep(b)=f(b)$, and the Lipschitz constant of $f-g\_\ep$ is less than $\ep$), then $g\_\ep\to f$ pointwise (as $\ep\downarro... | 2 | https://mathoverflow.net/users/36721 | 434189 | 175,594 |
https://mathoverflow.net/questions/434144 | 4 | Let $\mathcal{M}$ be a
>
> locally finitely presentable model category, cofibrantly generated by
> two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial
> cofibrations with presentable domain and codomain.
>
>
>
I know that weak equivalences and fibrations are stable by filtered colimits.
>
... | https://mathoverflow.net/users/139854 | When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations? | If both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy category (such as $\Delta$) with cofibrations the monomorphisms, whatever the weak equivalences are (see Cor. 3.4.41 & (t... | 9 | https://mathoverflow.net/users/1017 | 434194 | 175,597 |
https://mathoverflow.net/questions/434183 | 3 | The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In some sense, this is an analytic continuation result, as it says that if measure of points close to $0$ on which the fun... | https://mathoverflow.net/users/146531 | Quantitative analytic continuation estimate for a function small on a set of positive measure | Unfortunately, no, as requested:
Take any sequence $\delta\_j\in(0,1)$ decaying to $0$, choose small $\mu\_j>0$ such that $\prod\_j \delta\_j^{\mu\_j}=e^{-1}$ and put $f\_n(z)=e^n\prod\_j B\_{\delta\_j}(z)^{[\mu\_j n]}$ where $B\_\delta(z)=\frac{\delta-z}{1-\delta z}$ is the usual Blaschke factor. Then $|f\_n(0)|\ge ... | 11 | https://mathoverflow.net/users/1131 | 434203 | 175,598 |
https://mathoverflow.net/questions/433978 | 1 | Cross post with mse
For example, let's say I have the following equations.
\begin{gather\*}
a^{x-1}+b^{x-1}=337 \\
a^{x}+b^{x}=1267 \\
a^{x+1}+b^{x+1}=4825 \\
a^{x+2}+b^{x+2}=18751.
\end{gather\*}
What we can notice is that we can rewrite the left-hand sides of all the equations as
$a^{x+k}(1+(b/a)^{x+k})$
for ou... | https://mathoverflow.net/users/494220 | Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables | We can use linear algebra to make this easier.
Consider the space of sequences of the form $A a^n + B b^n$. We can use several dual bases for this space. One is the obvious basis of maps that send $A a^n + B b^n$ to its coefficients, $A$ and $B$. If neither of $a$ and $b$ are zero and the two are distinct, then anoth... | 2 | https://mathoverflow.net/users/44191 | 434208 | 175,601 |
https://mathoverflow.net/questions/434204 | 6 | Let $n\ge2$ be a given positive integer, and $z\_{1},z\_{2},\cdots,z\_{n}\in \mathbb{C}$,such
$$|z\_{1}|^2+|z\_{2}|^2+\cdots+|z\_{n}|^2\ge n.$$
Prove or disprove
$$f\_{n}=\sum\_{j=1}^{n}\left|\sum\_{I\subseteq \{1,2,3,\cdots,n\},|I|=j}\prod\_{k\in I}z\_{k}\right|^2\ge 1$$
In the particular case when $n=2$, it can be ... | https://mathoverflow.net/users/38620 | a problem in complex-variable inequality | For a polynomial $Q(x)=\sum\_i q\_ix^i$, define $N(Q)=\sum\_i|q\_i|^2$. We need to show that $N(R)\geq 2$, where $R(x)=\prod\_i (x-z\_i)$.
For a polynomial $Q(x)=\sum\_{i=0}^k q\_ix^i$, define $Q^\*(x)=\sum\_{i=0}^k\overline{q\_i}x^{k-i}=x^k\overline{Q(\bar x^{-1})}$. Here is the lemma which I definitely saw somewher... | 12 | https://mathoverflow.net/users/17581 | 434219 | 175,604 |
https://mathoverflow.net/questions/434200 | -1 | I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer...
---
I've been recently playing around with the linear recurrence sequences. Consider the following recurrence equation:
$$ a\_n = c\_1a\_{n-1} + \cdots + c\_ka\_{n-k}, \quad \forall n > k $$
i... | https://mathoverflow.net/users/198287 | Companion matrices must have geometric multiplicity one, linear recurrence sequence view | If $M$ had more than one Jordan block corresponding to some eigenvalue, then its minimal polynomial's degree would be smaller than $k$. This yields that *all* sequences satisfying your recurrence relation in fact satisfy a fixed smaller order linear recurrence relation (given by that minimal polynomial). But that is ab... | 1 | https://mathoverflow.net/users/17581 | 434220 | 175,605 |
https://mathoverflow.net/questions/433176 | 13 | I recently came across Kac algebra. They are roughly Hopf algebras and $C^\*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra can be seen as a $C^\*$-algebra structure, but I am unsure whether this structure will be compatible with the comultiplic... | https://mathoverflow.net/users/493533 | Hopf algebras vs. Kac algebras | As pointed out in the comments all semisimple Hopf algebras of dimension up to $23$ are Kac algebras. So it seems like dimension $24$ is the smallest one where this is open.
Corollary 9.7 of "Weakly group-theoretical and solvable fusion categories" by Etingof, Nikshych and Ostrik says that a semisimple Hopf algebra o... | 1 | https://mathoverflow.net/users/493533 | 434225 | 175,606 |
https://mathoverflow.net/questions/434210 | 8 | For a natural number not a perfect square, is there always at least a prime number for which it is a primitive root?
[Artin's conjecture on primitive roots](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) is that there are infinitely many such primes. Do we know for sure that there is at least ... | https://mathoverflow.net/users/494432 | For a non-square, is there a prime number for which it is a primitive root? | We do not know that. It is typical for questions about the infinitude of primes satisfying some property that even proving the existence of at least one of those primes in sufficient generality is unproved. And sometimes it turns out that proving there is at least one such prime in sufficient generality (the type of th... | 24 | https://mathoverflow.net/users/3272 | 434228 | 175,607 |
https://mathoverflow.net/questions/434173 | 2 | **Question.** Let $f: \mathbf{R} \to \mathbf{R}$ be an analytic function. Is there a harmonic function $u$ on the circular cylinder $D \times \mathbf{R} \subset \mathbf{R}^3$ so that $u = f$ along the axis $\{ (0,0) \} \times \mathbf{R}$?
* The problem is obviously ill-posed in the sense of Hadamard because it is ver... | https://mathoverflow.net/users/103792 | 'Dirichlet problem' along axis for harmonic functions | Assuming the Taylor series of $f$ has an infinite radius of convergence, the sum
$$
\sum\_{k=0}^\infty \left(x^2+y^2\right)^kf^{(2k)}(z)\cdot \frac{(-1)^k}{4^k k!^2}
$$
converges absolutely and locally uniformly on $\mathbb{R}^3$ to a function $u(x,y,z)$ which is harmonic and satisfies $u(0,0,z)=f(z)$
| 3 | https://mathoverflow.net/users/49822 | 434229 | 175,608 |
https://mathoverflow.net/questions/434021 | 0 | Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Denote by $\mathfrak{g}^\*$ the dual space of $\mathfrak{g}$. Let $r$ be an element of $\mathfrak{g}^\*$ such that $G\_r$ the stabilizer of $r$ under the coadjoint action is a maximal torus of $G$. Denote by $\mathcal{O}\_r$ the coadjoint orbit of $G$ whic... | https://mathoverflow.net/users/172459 | Question about coadjoint orbits of compact connected Lie groups | $\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Lie}{Lie}
\newcommand{\g}{{\mathfrak g}}
\newcommand{\z}{{\mathfrak z}}
\newcommand{\s}{{\mathfrak s}}
\newcommand{\O}{{\mathcal O}}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}$
Let $G$ be a connected compact Lie group, and l... | 2 | https://mathoverflow.net/users/4149 | 434234 | 175,610 |
https://mathoverflow.net/questions/433564 | 1 | Pair of sequences $\ v\_n\ $ and $\ U\_n\ $ of integers start as in the following table:
[\begin{array}{rrrrrrrrrr}
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\
v\_n= & 0 & 2 & 5 & 10 & 17 & 37 & 50 & 82 & \ldots \\
U\_n= & 0 & 2 & 3 & 6 & 8 & 12 & 14 & 18 & \ldots
\end{array}]
These two sequences are defined as... | https://mathoverflow.net/users/110389 | Consecutive non-powerful integers | Infinitely often (and with positive density, as proved by Shiu) there are no powerful numbers between $n^2+1$ and $(n+1)^2-1$. Hence the maximal gap below $x$ is infinitely often of size $\sim2\sqrt x$. (This is more than can be deduced from the Erdős & Szekeres result, or the Bateman & Grosswald result, since $\zeta(3... | 4 | https://mathoverflow.net/users/6043 | 434236 | 175,611 |
https://mathoverflow.net/questions/434231 | -1 | I have a polynomial, and I want to get the conditions for the number of positive roots
What are the different methods out there to determine these conditions?
this is the polynomial:
f(g)=A1*g^5 + A2*g^4 + A3*g^3 + A4*g^2 + A5\*g + A6
and A1 to A6 are constants
I will appreciate any help
Warm regards
| https://mathoverflow.net/users/493074 | finding positive roots for a polynomial | Abel-Ruffini Theorem postulates the existence of polynomials of every degree above $4$ that are not solvable by radicals (see also Galois' results on the same topic at the beginning of the XIX century, about the necessary and sufficient condition to let a polynomial equation be solvable by radicals).
A good (simple t... | -1 | https://mathoverflow.net/users/481829 | 434238 | 175,612 |
https://mathoverflow.net/questions/434242 | 1 | $\DeclareMathOperator\inc{inc}$Let $|X|=n$ and $\inc(X,\leq)=\{\{x,y\} : \neg (x\leq y)\wedge \neg (y\leq x)\}$, where $(X,\leq)$ is poset (possibly unconnected). Define the function:
$$\pi(n,m):=|\{(X,
\leq):\inc(X,\leq)=m\}/\cong|$$
, where $\cong$ is the relation of isomorphism of partial orders. It's obvious that i... | https://mathoverflow.net/users/175589 | The quantity of poset with a given number of pairs of incomparable elements | Yes. This is true. We shall prove this result by induction on $n$. Suppose that $n>0$. and $0\leq m\leq\binom{n}{2}$. If $m\leq\binom{n-1}{2}$, then there is some poset $X$ with $|X|=n-1$ and where $\text{inc}(X)=m$. Now attach a new element $1$ where $x\leq 1$ for $x\in X$ to obtain a poset $X\cup\{1\}$. Then $|X\cup\... | 1 | https://mathoverflow.net/users/22277 | 434248 | 175,614 |
https://mathoverflow.net/questions/433149 | 3 | Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a *local-isometry* where $H$ is either a Hilbert space or a Euclidean space?
* **Strengthened:** If $(X,d)$ is simply connected and is the length space induced by a Riemannian geometry and if we... | https://mathoverflow.net/users/491352 | Spaces satisfying a strong Cartan-Hadamard theorem | Note that Hilbert spaces (of all dimensions finite or infinite) are the only geodesic spaces with extendable geodesics which are flat in the sense of Alexandrov.
Therefore $X$ has to have extendable geodesics + it has to be locally flat (in the sense of Alexandrov).
By Cartan--Hadamard theorem, these two conditions are... | 2 | https://mathoverflow.net/users/1441 | 434258 | 175,618 |
https://mathoverflow.net/questions/434217 | 7 | In the very first chapter of *Elements of* $\infty$*-category theory*, E. Riehl and D. Verity define their notion of an $\infty$-cosmos, which should axiomatise a category in which $\infty$-categories live. (So, for example, the category of quasi-categories is an example of an $\infty$-cosmos.) An $\infty$-cosmos is a ... | https://mathoverflow.net/users/131975 | Intuition for isofibrations in $\infty$-categories | As has been mentioned, there's no homotopically meaningful content to the notion of an isofibration, since every map of $\infty$-categories is equivalent to an isofibration. So the point is really all in the definition of an $\infty$-cosmos: isofibrations are the kinds of maps between $\infty$-categories that you can t... | 6 | https://mathoverflow.net/users/43000 | 434259 | 175,619 |
https://mathoverflow.net/questions/434252 | 4 | Given a set $X$, by a *tree in $X$* I mean a set $T$ of finite sequences of elements of $X$ which is closed under initial segments. It is *pruned* of every element has a proper extension, and *finitely branching* if every element has at most finitely many immediate successors.
If $X$ is countable, then trees in $X$ a... | https://mathoverflow.net/users/16107 | Is there a standard Borel space of finitely branching real trees? | A natural way to represent a finitely branching tree over $\mathbb{R}$ is to separate the structure of the tree from the content (ie its labels from $\mathbb{R}$).
We can describe the structure of the tree by a function $d : \mathbb{N} \to \mathbb{N}$, where $d(n)$ is the number of children the $n$-th vertex in the t... | 5 | https://mathoverflow.net/users/15002 | 434262 | 175,620 |
https://mathoverflow.net/questions/434250 | 0 | Suppose you have a set of objects *X* and a scoring function *f* (in which order does not matter; *f(x,y) = f(y,x)*) which works in the following way.
* Passing a viable pair of these objects to the function will return a real number between, say, 0 and 100.
* Pairing an object to itself will return 101. Think of thi... | https://mathoverflow.net/users/494467 | Pairing optimisation w.r.t. a given function, or at least close to optimised | If you negate the function, this is a [maximum weight matching](https://en.wikipedia.org/wiki/Maximum_weight_matching) problem in an undirected graph where each object is a node, and there are polynomial algorithms for both exact and approximate solutions.
| 0 | https://mathoverflow.net/users/141766 | 434263 | 175,621 |
https://mathoverflow.net/questions/418619 | 25 | *[Cross-posted from MSE.](https://math.stackexchange.com/questions/4231158/is-there-an-infinite-topological-space-with-only-countably-many-continuous-maps?noredirect=1)*
Is there an infinite countable topological space $X$ with only countably many continuous functions to itself?
It cannot be a metrizable space. Ano... | https://mathoverflow.net/users/469928 | Is there an infinite topological space with only countably many continuous functions to itself? | A partial answer: the only place where Kannan and Rajagopalan use the inequality $(2^\kappa)^+<2^{2^\kappa}$ is in the application of the Theorem on page 121. That theorem is a consequence of Corollary 10.15 in Comfort and Negrepontis' *The Theory of Ultrafilters*. However the particular case that they use can be prove... | 7 | https://mathoverflow.net/users/5903 | 434266 | 175,622 |
https://mathoverflow.net/questions/424584 | 11 | *(Below I conflate quantifiers and quantifier **symbols** in a couple places for readability; I can change that if that actually makes things less readable.)*
For the purposes of this question, an **$n$-ary quantifier** is a (class) function $\mathscr{Q}$ assigning to each (nonempty) set $X$ a family $\mathscr{Q}X$ o... | https://mathoverflow.net/users/8133 | Are there quantifiers that require multiple "steps" to define? | The quantifier $\exists^\infty$ is not definable over $\mathcal{L}\_0$.
To prove it, we show that a sentence $S$ tautologised by $\exists^\infty$ is also a tautology for the "always false" quantifier $\mathsf{F}$.
Let $\mathfrak{M}$ be a model for $S$. From $\mathfrak{M}$, we build a finite model $\mathfrak{M}'$ wh... | 3 | https://mathoverflow.net/users/142409 | 434267 | 175,623 |
https://mathoverflow.net/questions/434260 | 1 | A *Fibonacci-type sequence* is a sequence with two seed-values, $F\_1$ and $F\_2$, and which, for all $n>2$, abides by the recurrence relation $F\_n = F\_{n-1} + F\_{n-2}$. If $F\_1 = F\_2 = s$, then the $n$th number is equal to the number of compositions of $n-1$, consisting only of $1$'s and $2$'s, multiplied by $s$:... | https://mathoverflow.net/users/493854 | Explicit formula for Fibonacci numbers; compositions of $n$ | Yes, this identity is well known. According to Singh's [The so-called Fibonacci numbers in ancient and medieval India](https://www.sciencedirect.com/science/article/pii/0315086085900217), the $s=1$ case has been known since at least the the 14th century. Since everything in the sequence with $F\_1 = F\_2 = s$ is a mult... | 1 | https://mathoverflow.net/users/14807 | 434268 | 175,624 |
https://mathoverflow.net/questions/434211 | 13 | Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have [a paper](https://arxiv.org/abs/2107.06308) where they use equivariant methods to compute the Picard group of the stable module category of representations for certain finite groups. I was wo... | https://mathoverflow.net/users/489806 | Applications of equivariant homotopy theory to representation theory | There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ...
(1) Quillen's work on ring theoretic aspects of the cohomology of finite groups [The spectrum of an equivariant cohomology ring. I, II. Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), ... | 13 | https://mathoverflow.net/users/102519 | 434271 | 175,625 |
https://mathoverflow.net/questions/434274 | 5 | For the past year and a half, I have been working my way through Diamond & Shurman's "A First Course in Modular Forms", and I have just finished it. I Have Some Questions.
1. What is so special about two dimensions? One can think about lattices/tori in N dimensions and their moduli space $$SL\_n(\mathbb{Z})\backslash... | https://mathoverflow.net/users/17496 | Lots of questions about modular forms | (1.) This is a very good question and shows you are thinking in the right directions, but it also is asking for a summary of multiple entire fields of mathematics. Some keywords are "automorphic forms", "locally symmetric spaces", "Shimura varieties".
In brief, you can exactly do that. You have to think about what yo... | 9 | https://mathoverflow.net/users/18060 | 434275 | 175,627 |
https://mathoverflow.net/questions/434291 | 5 | Consider a four-dimensional Lorentzian manifold $(\mathcal{M},g)$ and a $3$-dimensional compact manifold $\Sigma$, such that there exists a spacelike embedding $i:\Sigma\to\mathcal{M}$ so that $h:=i^{\ast}g$ becomes a Riemannian metric on $\Sigma$.
In a paper it was, without references, stated the space
$$\mathrm{E... | https://mathoverflow.net/users/259525 | Space of spacelike embeddings as infinite-dimensional manifold | A standard reference on infinite dimensional manifolds is
*Kriegl, Andreas; Michor, Peter W.*, [**The convenient setting of global analysis**](http://www.ams.org/online_bks/surv53/), Mathematical Surveys and Monographs. 53. Providence, RI: American Mathematical Society (AMS). x, 618 p. (1997). [ZBL0889.58001](https:/... | 6 | https://mathoverflow.net/users/2622 | 434294 | 175,631 |
https://mathoverflow.net/questions/434138 | 1 | Let $G =$ PGL$\_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}\_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-subgroups in $T$ of the same rank.
The Kronecker product $\otimes I\_{2}$ embeds $A$ and $B$ in $H$ = $\operatorname{P... | https://mathoverflow.net/users/488802 | Kronecker product preserves the conjugacy relation? | If $A$ and $B$ are elementary abelian $2$-subgroups of $\mathrm{PGL}\_n(\mathbf C)$ of rank $r$ then they lift uniquely to elementary abelian $2$-subgroups of $\mathrm{GL}\_n(\mathbf C)$ of rank $r+1$ (take all lifts $a$ of elements of $A$ such that $a^2 = 1$), so let us assume $A, B \le \mathrm {GL}\_n(\mathbf C)$ to ... | 0 | https://mathoverflow.net/users/20598 | 434296 | 175,632 |
https://mathoverflow.net/questions/434150 | 10 | [Zhang 2022](https://arxiv.org/abs/2211.02515) proves a somewhat suspicious formula:
$$L(1,\chi) \gg (\log D)^{-2022}$$
This raises the obvious-but-frivolous question: did he intentionally weaken the constant to get the current year?
| https://mathoverflow.net/users/22930 | Did Zhang weaken the constant in his Landau-Siegel zero paper to get the current year? | According to himself, yes. The following is a link to some of his comments that he posted on a Chinese forum.
<https://www.zhihu.com/question/564799818/answer/2752632822>
>
> Regarding the question of whether the fixed power of logD , which is taken for many parameters in the paper, is to get the number 2022, in te... | 19 | https://mathoverflow.net/users/494533 | 434308 | 175,636 |
https://mathoverflow.net/questions/434276 | 18 | Consider a function $h$ defined on real numbers, which is not of the form $kx+b$ i.e. a linear function. If $h$ maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers, could $h$ be analytic? If so, how to give an example?
| https://mathoverflow.net/users/494497 | Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers? | Answering a question of Erdos, Barth and Schneider proved that for every countable dense sets $A$ and $B$
in the complex plane, there exists an entire function such that $f(z)\in B$ if and only if $z\in A$.
K. Barth and W. Schneider, Entire functions mapping arbitrary countable dense sets and their complements to eac... | 22 | https://mathoverflow.net/users/25510 | 434311 | 175,637 |
https://mathoverflow.net/questions/434289 | 1 | Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L\_p := L\_p(X, \mu, E)$ be the [Bochner space](https://en.wikipedia.org/wiki/Bochner_space) of all $\mu$-integrable functions $f:X \to E$. Here we use [Bochner integrals](https://en.wikipedi... | https://mathoverflow.net/users/477203 | Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space? | A reference for this result would be [Some more uniformly convex spaces](https://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07499-9/S0002-9904-1941-07499-9.pdf) by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941).
(Alternative link at [Project Euclid](https://projecteuclid.org/journals/bulleti... | 2 | https://mathoverflow.net/users/85906 | 434314 | 175,639 |
https://mathoverflow.net/questions/434305 | 1 | For a multidimensional subshift $X$ over $\mathbb Z^d$, the topological full group $[X]$ is the set of homeomorphisms $f$ of $X$ that can be written as $f : x \mapsto \sigma\_{c(x)}(x)$ with $c : X \to \mathbb Z^d$ a continuous function (namely, a cocyle).
My questions would mainly be about embedability of those grou... | https://mathoverflow.net/users/494521 | Topological full groups of subshifts: differences between one-dimensional and multi-dimensional subshifts | I don't know that much literature on the multidimensional case (though I'm not sure I'm the one who would if there is literature, either), but I can collect the comments and try to add a few things. Sorry in advance if there are some mistakes in my claims, I never wrote some of these carefully before (and still didn't)... | 3 | https://mathoverflow.net/users/123634 | 434317 | 175,640 |
https://mathoverflow.net/questions/423541 | 7 | For various types of groups, there exist catalogues of those groups of the
particular type which are "small" in a certain sense. — For example:
* The [GAP Small Groups Library](https://www.gap-system.org/Packages/smallgrp.html) catalogizes groups of small order,
* The [GAP Transitive Groups Library](https://www.gap-s... | https://mathoverflow.net/users/28104 | Catalogue of groups with short finite presentations | I would very much like to have such a database and would like to contribute to its development. Prompted by this question, we talked about what such a database could look like (e.g. in terms of groups covered, functionality etc.) at a discussion session of a workshop in Manchester with Ian Leary, Marco Linton, Saul Sch... | 3 | https://mathoverflow.net/users/24447 | 434318 | 175,641 |
https://mathoverflow.net/questions/434239 | 2 | Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$
It is easy to see that if $\|f\|\_{\infty}< \frac{\varepsilon}{2} =: \delta (\varepsilon)$ then we can find $g$ with followning properties:
1. $$\|f-g\|\_{\infty}< \varepsilon$$
2. $$g(a)=f(a), \... | https://mathoverflow.net/users/490101 | Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm | I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$.
(1) because $C = \{t: |f(t)| \geq c\}$ is a closed subset of $[a,b]$, so its complement is a countable set of disjoint ope... | 2 | https://mathoverflow.net/users/23141 | 434323 | 175,643 |
https://mathoverflow.net/questions/434329 | 11 | Suppose $m$ is a positive integer.
I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-element subsets.
An example for $m=2$ is a projective space over a finite field.
| https://mathoverflow.net/users/1441 | Not very transitive actions | According to Theorem 4.11 of Peter Cameron's book `Permutation Groups' it follows from the classification of finite simple groups that the only finite 6-transitive groups are (some of the) symmetric and alternating groups in their natural actions, and the only finite 4-transitive groups are symmetric, alternating and t... | 11 | https://mathoverflow.net/users/124004 | 434333 | 175,646 |
https://mathoverflow.net/questions/427125 | 6 | A function $f:X\to Y$ between topological spaces is called
$\bullet$ *$\sigma$-continuous* if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\restriction}\_C$ is continuous;
$\bullet$ a *$\sigma$-homeomorphism* if $f$ is bijective and the maps $f$ and $f^{... | https://mathoverflow.net/users/61536 | Classification of Polish spaces up to a $\sigma$-homeomorphism | There are indeed continuum many. See:
Kihara, T., & Pauly, A. (2022). Point Degree Spectra of Represented Spaces. Forum of Mathematics, Sigma, 10, E31. doi:10.1017/fms.2022.7
| 3 | https://mathoverflow.net/users/94358 | 434334 | 175,647 |
https://mathoverflow.net/questions/434300 | 4 | $\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Proj{Proj}\DeclareMathOperator\Pic{Pic}$I have a question about an example for a line bundle not admitting a
$G$-linearization from Mumford's GIT, page 33:
We consider the action of $\PGL(n+1)$ on projecive space
$\mathbb{P}^n= \Proj k[X\_0,\dotsc, X\_n]$.
Observe th... | https://mathoverflow.net/users/108274 | Example of a line bundle not admitting a $\operatorname{PGL}(n+1)$-linearization in Mumford's GIT | $PGL(n+1)$-linearization of $\mathcal O\_{\mathbb P^n}(1)$ could be used to produce isomorphism $\gamma : p\_2^\*(\mathcal O\_{\mathbb P^n}(1)) \to \sigma^\*(\mathcal O\_{\mathbb P^n}(1))$, hence, taking to attention isomorphism
$$ \sigma^\*(\mathcal{O}\_{\mathbb{P}^n}(1)) \cong
p\_1^\*(\mathcal{O}\_{\mathbb{P}^{n^2+2... | 2 | https://mathoverflow.net/users/54337 | 434335 | 175,648 |
https://mathoverflow.net/questions/433421 | 4 | *Throughout I'm only interested in the **standard** semantics for second-order logic, and all structures/languages are relational for simplicity.*
If defined naively, second-order logic without equality is equivalent to second-order logic, since $x=y$ is equivalent to $\forall A(x\in A\leftrightarrow y\in A)$. Howeve... | https://mathoverflow.net/users/8133 | Source on equality-free second-order logic (nontrivially construed) | It currently appears that this logic is not already treated in the literature.
*(I'm posting this answer to move this question off the unanswered queue, but if someone does find a source on it of course please add it!)*
| 0 | https://mathoverflow.net/users/8133 | 434340 | 175,649 |
https://mathoverflow.net/questions/434347 | 3 | Consider a second order gradient-like system with linear damping
$$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x\_0,\quad\dot{x}(0)=0$$
Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf\_{x\in\mathbb{R}^n}f(x)>-\infty$. The solution $x:[0,\infty)\rightarrow \mathbb{R}^n$ is bounded, i.e., $\lVert x(t)\rVert\leq c$ for all $t\i... | https://mathoverflow.net/users/490600 | Does gravity constant affect boundedness of solution? | $\newcommand\la\lambda$No. E.g., if $n=1$, $x\_0\ne0$, and $f(u)=-u^2/2$ for all real $u$, then the solution
$$x(t)=\frac{x\_0}{2} \, \Big(\frac{e^{\la\_+ t}-e^{\la\_- t}}{\sqrt{4 g+1}}+e^{\la\_- t}+e^{\la\_+ t}\Big)$$
of the problem
$$\ddot{x}+\dot{x}+g\nabla f(x)=0, \quad x(0)=x\_0,\quad\dot{x}(0)=0 \tag{1}\label{1}$... | 4 | https://mathoverflow.net/users/36721 | 434351 | 175,653 |
https://mathoverflow.net/questions/434350 | 2 | Is there a simple proof that there is no Anosov flow on $S^2$? Where can I find it?
| https://mathoverflow.net/users/479780 | Anosov flow on the 2-sphere | The usual definition of Anosov flow requires three invariant sub-bundles, so I guess you are actually asking about the 3-sphere?
Plante and Thurston have proved in
*Plante, J. F.; Thurston, W. P.*, [**Anosov flows and the fundamental group**](https://doi.org/10.1016/0040-9383(72)90002-X), Topology 11, 147-150 (1972... | 6 | https://mathoverflow.net/users/47274 | 434352 | 175,654 |
https://mathoverflow.net/questions/434343 | 6 | For any commutative [Frobenius algebra](https://en.wikipedia.org/wiki/Frobenius_algebra) $A$ there is an associated *window element* $\omega \in A$. If $\mu: A \otimes A \to A$ denotes the multiplication, $1 \in A$ the unit, $b: A \otimes A \to k$ the non-degenerate pairing, and $c: k \to A \otimes A$ the copairing, th... | https://mathoverflow.net/users/184 | Commutative Frobenius algebra with non-invertible window element, but not square zero | Assume $A$ is a connected (not necessarily commutative) non-semisimple Frobenius algebra that is finite dimensional over a field of characteristic 0 and given by quiver and relations. (for the commutative case all this reduced to be a local commutative Frobenius algebra that is not a field).
We should have $c(1)= \su... | 2 | https://mathoverflow.net/users/61949 | 434356 | 175,656 |
https://mathoverflow.net/questions/434339 | 15 | The surreal numbers are sometimes called the "universally embedding" ordered field, in that every ordered field embeds into them. What "universally embedding" means seems to be [somewhat complicated](https://mathoverflow.net/questions/410437/what-does-it-mean-for-the-surreal-numbers-partizan-games-to-be-universally-emb... | https://mathoverflow.net/users/24611 | Can you build the surreal numbers as a simple direct limit of ordered fields? | Here is one way to get a positive answer to the title question.
**Theorem.** There is a definable class $\mathcal{F}$ of ordered fields, containing isomorphic copies of any given field, and a directed order $\unlhd$
on them, with a definable commutative system of embeddings between them $\pi\_{F,K}:F\to K$ for $F\unl... | 15 | https://mathoverflow.net/users/1946 | 434358 | 175,658 |
https://mathoverflow.net/questions/434360 | 3 | I am looking for non-trivial examples of flat $U(2)$ connections over the complement of a torus link $\mathcal{S}^3-L$ i.e.
$\mathcal{A}:\mathcal{S}^3-L \longrightarrow \mathfrak{U}(2)$ such that $F\_{\mathcal{A}} = 0$ and $Hol\_{\gamma}(\mathcal{A}) \neq 0$ with $\gamma$ a non-trivial element in $\pi\_1(\mathit{S}^3... | https://mathoverflow.net/users/494010 | Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy | For torus *knots*, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–828). The starting point is a rather simple presentation of the fundamental group of the complement of a torus knot; sin... | 4 | https://mathoverflow.net/users/3460 | 434362 | 175,660 |
https://mathoverflow.net/questions/434344 | 3 | Let $v\_1 =\lambda\_1 \zeta\_1$ and $v\_2 = \lambda\_2 \zeta\_2$ with $\zeta\_1 = \frac{4\pi i\omega}{3}$ and $\zeta\_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and $\lambda\_1,\lambda\_2$ some positive integers.
I would like to ask if there is an entire function $f$ such t... | https://mathoverflow.net/users/457901 | Entire function with almost periodic boundary condition? | The answer seems to be negative. Suppose that an entire function $f$
satisfies $f(z+v\_i)=e^{A\_iz}f(z)$, where $v\_1$ and $v\_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this lattice and integrate $f'/f$ over $\partial \Pi$. You obtain the ``Legendre's relation'':
$$v\_2A\_1-v\_1A\_2=2\pi in,$... | 4 | https://mathoverflow.net/users/25510 | 434364 | 175,661 |
https://mathoverflow.net/questions/434320 | 4 | In the setting of symbolic dynamics over $\mathbb{Z}^d$, one can define for the $n$-th pattern complexity of a given a subshift $\Omega\subseteq \mathcal{A}^{\mathbb{Z}^d}$ as
$$ c\_n(\Omega):= \Big\vert \{P\in \mathcal{A}^{Q\_n}: P= \omega\vert\_{Q\_n} \; \text{for some} \; \omega\in \Omega \} \Big\vert $$
with $Q... | https://mathoverflow.net/users/143153 | Lower bounds for pattern complexity of aperiodic subshifts | The answer is no in a very strong sense: there does not exist such $C\_d$ for $d \geq 3$ even for aperiodic minimal subshifts.
As far as finding lower bounds goes, complexities of subshifts containing aperiodic configurations are more or less the same as complexities of individual aperiodic configurations, namely a s... | 4 | https://mathoverflow.net/users/123634 | 434371 | 175,664 |
https://mathoverflow.net/questions/434195 | 1 | I am concerned with unweighted directed graphs where each node contains exactly one edge pointing to another node, which could be itself. In other words, each row of the adjacency matrix contains one entry equal to 1, and the rest are 0. Im not sure if these graphs have a name, but they could be called a deterministic ... | https://mathoverflow.net/users/94774 | Eigenvalues of directed graph with one outward edge for each vertex | Here is an alternative (more combinatorial) proof to the one linked to in my comment.
Suppose that the digraph $D$ has a vertex of in-degree zero, which we may assume is vertex $1$. Then letting $\varphi(D)$ denote the characteristic polynomial of the adjacency matrix $A(D)$, we have
$$
\varphi(D) = |xI - A(D)| = \... | 1 | https://mathoverflow.net/users/1492 | 434373 | 175,665 |
https://mathoverflow.net/questions/434378 | 0 | You are given a series of vectors $u\_0,\ldots,u\_k$ of non-negative entries, each of dimension $n$, for which the sum of the entries of each vector is $1$ (i.e., for all $i\in \{0,\ldots,k\}$, it holds that $\sum\_j u\_{i,j}=1$).
Is it always true that either there exists a vector $v\in R^{n}$, such that $\sum\_j (u... | https://mathoverflow.net/users/480123 | Non-convex combination of vectors | Yes, either point $u\_0$ belongs to the convex hull $H$ of the points $u\_1,\dots,u\_k$ (in which case $\alpha\_i$ exist), or there exists a hyperplane separating $u\_0$ and $H$, when we can take $v$ as a normal vector of this hyperplane "pointing" towards $u\_0$.
| 3 | https://mathoverflow.net/users/7076 | 434381 | 175,668 |
https://mathoverflow.net/questions/433707 | 3 | Consider a binary relation $R$ over a finite set $X$ of size $n$. Assume $R$ is antisymmetric and connected but not necessarily transitive. In essence, we are modeling an "option x beats option y" relation, which is not necessarily transitive. It might be the result of a voting process for instance.
It is sometimes p... | https://mathoverflow.net/users/8737 | Representing a binary relation | Every binary relation $R$ has a representation with $d=2$. Enumerate $X=\{x\_i:i\in[n]\}$, and define $g\colon X\to\mathbb R^2$ by $g(x\_i)=(i,-i)$. Since $\{(i,-i,j,-j):i,j\in[n]\}$ is a set of pairwise incomparable elements of $\mathbb R^4$, the Proposition below implies that there exists a continuous increasing func... | 2 | https://mathoverflow.net/users/12705 | 434387 | 175,670 |
https://mathoverflow.net/questions/434369 | 1 | We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int\_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial data $f$; that on cones (change $|\xi|^2$ to $|\xi|$) a solution to the wave equation with the same initial data; and on (... | https://mathoverflow.net/users/494230 | Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE? | Your question was basically answered by David Roberts in the comments, but let me write a few more words.
Given a **constant coefficient linear differential operator** of degree $N$
$$ L = \sum\_{|\alpha| \leq N} c\_\alpha \partial^\alpha $$
(here I use multi-index notation for $\alpha$), we can formally take the Fou... | 6 | https://mathoverflow.net/users/3948 | 434399 | 175,673 |
https://mathoverflow.net/questions/434328 | 0 | Let $X$ be a metric space, $\mu$ a $\sigma$-finite non-negative Borel measure on $X$, and $(E, |\cdot|)$ a Banach space. Let $\mathcal L\_p := \mathcal L\_p (X, \mu, E)$ and $\|\cdot\|\_{\mathcal L\_p}$ be its semi-norm. Here we use [Bochner integral](https://math.stackexchange.com/questions/4298588/dominated-convergen... | https://mathoverflow.net/users/477203 | A generalization about the density of $\mathcal C_c(X, E)$ in $\mathcal L_p (X, \mu, E)$ when $E$ is a Banach space | Below is a counter-example taken from [this thread](https://math.stackexchange.com/q/4574262/1019043). It works even when $\mathcal C\_c$ is replaced by $\mathcal C$, the space of all continuous functions from $X$ from $E$.
---
Let $X:=[0, 1]$, $E:=\mathbb R$, and $\mu$ the Lebesgue measure on $[0, 1]$. Let $C$ b... | 0 | https://mathoverflow.net/users/477203 | 434400 | 175,674 |
https://mathoverflow.net/questions/434368 | 9 | Let $n > 13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property:
$$AA^T=(n-1)I+J$$
where $I$ is the $n\times n$ identity matrix and $J$ is the $n\times n$ matrix of ones.
I conjecture that the answer is no. But I can't prove it.
| https://mathoverflow.net/users/369335 | One question on circulant $(-1,1)$-matrices | This is a question about a sequence $a(t)\in \{\pm 1\}$ of period $n$ with 2 level periodic autocorrelations, with the nontrivial autocorrelations identically equal to 1. All these problems have a design theoretic aspect as well. For the relationship to the $\{0,1\}$ alphabet see the question [here](https://mathoverflo... | 5 | https://mathoverflow.net/users/17773 | 434406 | 175,676 |
https://mathoverflow.net/questions/424965 | 5 | *(For simplicity, all languages are relational.)*
In analogy with *first-order* languages, say that a **second-order language** is a set of relation symbols of two kinds: *first-order* relation symbols and *second-order* relation symbols. Both types of symbols have a notion of arity; the arity of a first-order relati... | https://mathoverflow.net/users/8133 | Does second-order logic satisfy Craig interpolation for second-order languages? | *This is just an expansion of Emil Jerabek's comments above; I've made it CW to avoid reputation gain, and will delete this if he posts an answer of his own.*
Craig interpolation can be rephrased as a "syntactic separation" property: the statement $$\varphi\models\psi$$ can be rephrased as $$\emptyset\models\exists\m... | 2 | https://mathoverflow.net/users/8133 | 434414 | 175,678 |
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