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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/405577 | 4 | I have been looking through Alan Pears' "Dimension theory of general spaces" recently. In this book Pears references a 1960 paper by Aleksandrov and Ponomarev called "Some classes of $n$-dimensional spaces" that contains a sufficient condition for the large inductive dimension and the covering dimension to coincide for... | https://mathoverflow.net/users/115694 | Sufficient conditions for the covering dimension and large inductive dimension of compact Hausdorff spaces to coincide | You can read a review of the paper [in Zentralblatt](https://www.zbmath.org/?q=an%3A0108.35605), it contains a short description in German.
The review on [MathSciNet](https://mathscinet.ams.org/mathscinet-getitem?mr=124031) is a bit more extensive (but requires a subscription). There is indeed the condition of having... | 1 | https://mathoverflow.net/users/5903 | 405732 | 166,327 |
https://mathoverflow.net/questions/405673 | 2 | Is there a name for a pseudo-Riemannian manifold that admits no nonzero null vectors? More precisely: For a pseudo-Riemannian manifold $(R,g)$, a **null vector** is a non-zero vector field $X:M \to TM$ such that
$$
g(X,X)(m) = 0, \forall m \in M.
$$
As this [question][1] shows - vector like this exist. But can there ex... | https://mathoverflow.net/users/378228 | A name for a pseudo-Riemannian manifold that admits no nonzero null vectors | Note first that every pseudo-Riemmanian manifold admits a null vector field which is not identically $0$ (just construct one locally and multiply it by a bump function). So by "non-zero vector field" I assume you mean "nowhere vanishing".
Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. The tangent b... | 5 | https://mathoverflow.net/users/173096 | 405740 | 166,330 |
https://mathoverflow.net/questions/403486 | 3 | After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have applications to classical mechanics as it appears to be generally useful for dimensionality reduction.
Motivation:
------... | https://mathoverflow.net/users/56328 | Has the von Neumann entropy ever been used in classical mechanics? | You are really asking just about the applications of the notion of entropy to an analysis of correlation matrices, so that this question is not necessarily related with classical mechanics. The entropy here is the plain Boltzmann - Planck - Shannon entropy rather than von Neumann's one, as the mere appearance of a posi... | 2 | https://mathoverflow.net/users/8588 | 405748 | 166,332 |
https://mathoverflow.net/questions/405752 | 0 | When I did some research, I have not found the analytical expression about the eigenvalues and eigenvectors of Laplacian matrix in a chain graph while I only found those in a cycle graph. The Laplcian matrix is as follows.
$$L=\begin{bmatrix}1 & -1 & 0 & 0 & 0\\-1 & 2 & -1 & 0 & 0\\0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 ... | https://mathoverflow.net/users/394256 | The eigenvectors and eigenvalues of Laplacian matrix in a chain graph | You can find it in [this paper](https://ieeexplore.ieee.org/abstract/document/5717507?casa_token=wzm-Ny2TJyYAAAAA:yoCoqQc-JPp7yPxO9AhtR43atzGbV0L61c4o1teX4A3QxPGlJEv0_traXxbEmajwKjBfgOstKZs).
In case that link doesn't work, search at Google Scholar for "On the observability of path and cycle graphs".
| 2 | https://mathoverflow.net/users/9025 | 405759 | 166,335 |
https://mathoverflow.net/questions/405731 | 3 | Let $X$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $X$ must be locally finite, outer regular and tight. Let $\mathcal{P}(X)$ be the set of all Borel probability measures on $X$.
A **rational preference relation** on $\mathcal{P}(X)$ is a binary relation $\precsim$ on $... | https://mathoverflow.net/users/49284 | Conditions for the existence of von Neumann-Morgenstern utility on a Polish space | There exists a continuous, bounded utility function if and only if the relation is continuous in the stronger sense of being closed in $P(X) \times P(X)$, using the weak${}^\*$ topology on each factor. See Section 3.3 of [this paper](https://arxiv.org/abs/2104.11205), for example. (The point of that paper is to find Li... | 4 | https://mathoverflow.net/users/23141 | 405764 | 166,337 |
https://mathoverflow.net/questions/405766 | 5 | Let $p: \mathcal{X} \rightarrow \text{Spec } \mathcal{O}\_K$ be a normal proper Artin stack with finite diagonal. A $K$-rational point is by definition a section $x: \text{Spec}(K) \rightarrow \mathcal{X}$ of $p$ over the generic point of $\text{Spec } \mathcal{O}\_K$, and an integral point is a section $\bar{x}: \text... | https://mathoverflow.net/users/96891 | Extending rational to integral points | $\DeclareMathOperator{Spec}{Spec}$ For 1. take $\mathcal{X} = B \mu\_2$ to be the classifying stack of $\mu\_2$ over $\mathbb{Z}$. For any $\mathbb{Z}$-scheme $S$ we have $\mathcal{X}(S) = \mathrm{H}^1(S, \mu\_2)$. Thus we have $\mathcal{X}(K) = K^\times/K^{\times 2}$, but this doesn't equal $\mathcal{X}(\mathcal{O}\_K... | 7 | https://mathoverflow.net/users/5101 | 405767 | 166,338 |
https://mathoverflow.net/questions/392060 | 8 | My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.
I would like to start with unrefined trace formula, but there are bunches of papers already. There is a paper on trace formula for ... | https://mathoverflow.net/users/159973 | How to read the paper of Arthur on trace formula on general reductive groups | One of the best places to learn about trace formula, other than David Whitehouse's wonderful notes, are the notes by Erez Lapid [Introductory notes on the trace formula](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.494.5118).
Arthur's trace formula relies on Langlands's work on Eisenstein series and Spect... | 7 | https://mathoverflow.net/users/303419 | 405772 | 166,340 |
https://mathoverflow.net/questions/405491 | 3 | In [Salamon's notes on Floer homology](https://people.math.ethz.ch/%7Esalamon/PREPRINTS/floer.pdf), it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial\_s+J\_0\partial\_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\rightarrow L^p(\mathbb{R}\times S^1,\mathbb{R}^{2n})$$ is Fred... | https://mathoverflow.net/users/155363 | Computing the Fredholm index in Floer theory | The idea is to show that the kernel & cokernel consist of smooth sections, and thus are independent of $p$. Since the Fredholm index is the difference between the dimensions of these, it doesn't depend on $p$.
This is proved, admittedly in the case without punctures, in an appendix of McDuff & Salamon's J-holomorphic... | 4 | https://mathoverflow.net/users/477 | 405782 | 166,346 |
https://mathoverflow.net/questions/405754 | 2 | It is a well-known fact that the range of a positive measure $\mu$ on a measure space $(X,\cal M)$ is the interval $[0,\mu(X)]$ provided $\mu$ is atomless (i.e., there is no measurable set $A\in \cal M$ such that $\mu(A)>0$ and for any measurable set $B\subset A$ we have $\mu(B)=0$ or $\mu(B)=\mu(A)$).
It seems that ... | https://mathoverflow.net/users/65954 | A question about the range of a positive measure | (1) Existence of sets of small measure.
Let $(\Omega,\Sigma,\mu)$ an atomless finite measure space with $\mu(\Omega)>0$.
It is easy to show that any measurable set $A$ with $\mu(A)>0$ contains a measurable subset $B\subset A$ with $0<\mu(B)\le \mu(A)/2$, and therefore a measurable set $M$ of measure $0<\mu(M)\le \var... | 3 | https://mathoverflow.net/users/7402 | 405783 | 166,347 |
https://mathoverflow.net/questions/405781 | 4 | Recall that a [door space](https://en.wikipedia.org/wiki/Door_space) is a topological space where every set is either open or closed (or both). A topological space is *finite* if it has finitely many points. I'm interested in learning about finite door spaces.
**Question 0:** What is an example of a finite topologica... | https://mathoverflow.net/users/2362 | How do finite door spaces work? | A door space $X$ is $T\_0$, for if $x,y\in X$ are not separated by the
$T\_0$ axiom then the set $\{x\}$ is neither open nor closed. A finite $T\_0$
space is equivalent to a finite poset $P$ (*Enumerative
Combinatorics*, vol. 1, second ed., Exercise 3.3). An open set
corresponds to an order ideal of $P$. Thus we want t... | 19 | https://mathoverflow.net/users/2807 | 405785 | 166,348 |
https://mathoverflow.net/questions/402822 | 5 | With the aid of the simple identity
\begin{equation\*}
\sum\_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^{k}}=2^n
\end{equation\*}
in Item (1.79) on page 35 of the monograph
R. Sprugnoli, *Riordan Array Proofs of Identities in Gould’s Book*, University of Florence, Italy, 2006. (Has this monograph been formally published somew... | https://mathoverflow.net/users/147732 | How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$? | For $k\in\mathbb{N}$, let $s\_k$ and $S\_k$ be two sequences independent of $n$ such that $n\ge k\in\mathbb{N}$. The inversion theorem, Theorem 4.4 on page 528 in the freely downloading paper [1] below, reads that
\begin{equation}\label{Qi-Zou-Guo-Inversion-thm}\tag{1}
s\_n=\sum\_{k=1}^{n}\binom{k}{n-k}S\_k
\quad\text{... | 0 | https://mathoverflow.net/users/147732 | 405812 | 166,356 |
https://mathoverflow.net/questions/405806 | 5 | Let $X$ be a compact Hausdorff space and $C(X)$ its algebra of continuous complex valued functions. The Gelfand-Naimark theorem tells us that we have a duality between commutative $C^\*$-algebras and compact Hausdorff spaces: Send $X$ to its $C^\*$-algebra of continuous functions.
The Swan-Serre theorem tells that th... | https://mathoverflow.net/users/153228 | Hermitian vector bundles and Hilbert $C^*$-modules | In addition to Nik Weaver's references, let me just sketch the proof which is in fact not very difficult:
1. A construction of Kaplansky (Rings of operators, Thm 26) shows that if $\mathcal{A}$ is a $\*$-algebra with the property that for every $A \in M\_n(\mathcal{A})$ the matrix $1 + A^\*A$ is invertible, then ever... | 4 | https://mathoverflow.net/users/12482 | 405817 | 166,359 |
https://mathoverflow.net/questions/405746 | 4 | If $R$ is a commutative ring with identity with a 'nice' action of a finite group $G$, the subring $R^G\subset R$ gives a Galois extension of rings. In this case, S.U. Chase, D.K. Harrison, A. Rosenberg have shown that there exsits the following exact sequence (also knonwn as Chase-Harrison-Rosenberg exact sequence):
... | https://mathoverflow.net/users/90911 | Regarding a 'global' version of Chase-Harrison-Rosenberg exact sequence for rings | I'm converting my comment to an answer. Let $\pi:X\to Y$ be a Galois étale cover, with Galois group $G$. One has a Hochschild-Serre spectral sequence
$$E\_2 = H^p(G, H^q(X\_{et},\mathbb{G}\_m))\Rightarrow H^{p+q}(Y\_{et}, \mathbb{G}\_m)$$
(The reference is given in "A User"'s comment.)
The associated exact sequence of ... | 3 | https://mathoverflow.net/users/4144 | 405818 | 166,360 |
https://mathoverflow.net/questions/401720 | 5 | $\newcommand\P{\mathbb P} \newcommand\C{\mathcal C}$I am a bit confused by a proof I am reading on the fact that a projective algebraic space-curve (i.e. an algebraic curve in $\P^3(k)$, where $k$ is an algebraically closed field) is the intersection of 3 hypersurfaces. I am trying to constructively create the hypersur... | https://mathoverflow.net/users/3949 | Question on a constructive proof that space projective curves are the intersections of three hypersurfaces | $\newcommand\C{\mathcal C} \newcommand\spn[1]{\langle #1\rangle}$Using cohomology is interesting when dealing with higher dimensions and more general results but I don't see why it would result into a *shorter* proof. In fact the generalization of this result of Kneser (Eisenbud and Evans) avoids any sheaf cohomology a... | 3 | https://mathoverflow.net/users/1245 | 405819 | 166,361 |
https://mathoverflow.net/questions/405834 | 2 | Suppose $F\subset V\subsetneq \mathbb {R}$ where $F$ is closed and $V$ is open. I want to show that $\exists$ an open set $U\subset \mathbb {R}$ satisfying $F\subset U\subset \overline {U}\subset V$. My proof is below, but I think it's problematic since I never use the condition that $F$ is closed.
For each $x\in F$,... | https://mathoverflow.net/users/197849 | Between an open set and its closed subset | $\newcommand{\R}{\mathbb{R}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\de}{\delta}\newcommand{\ol}{\overline}$The property of $\R$ that you want to prove is that the topological space $\R$ is [normal](https://en.wikipedia.org/wiki/Normal_space). [All metric spaces are perfectly normal](https://en.wikipedia.org/wiki/Normal... | 2 | https://mathoverflow.net/users/36721 | 405840 | 166,366 |
https://mathoverflow.net/questions/405133 | 11 | It is known that the Haefliger trefoil $S^3\hookrightarrow S^6$ is PL trivial but non-trivial smoothly. I wonder, where exactly does the problem come? Consider its tubular neighborhood $T\cong S^3\times D^3$. By the disc bundle theorem its closed complement $C$ is diffeomorphic to $S^2\times D^4$. Therefore, $S^6$ is p... | https://mathoverflow.net/users/9800 | Haefliger trefoil $S^3\hookrightarrow S^6$ | The problem comes from the fact that $S^3\subset S^6$ admits more $PL$-framings than smooth ones: $\pi\_3 SO\_3=\mathbb{Z}$, while $\pi\_3 PL\_{6,3}=\mathbb{Z}^2$. (The group $PL\_{6,3}$ is the group of PL self homeomorphisms of $\mathbb{R}^6$ preserving $\mathbb{R}^3\subset\mathbb{R}^6$ pointwise.) Thus, when we PL is... | 5 | https://mathoverflow.net/users/9800 | 405843 | 166,369 |
https://mathoverflow.net/questions/405829 | 2 | This question is related to a [question](https://mathoverflow.net/questions/405567/is-this-internalization-of-a-bijection-between-a-set-and-its-powerset-possible?) lately posted to $\cal MO$. Here, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$.
The question is about if we can add the followi... | https://mathoverflow.net/users/95347 | Can we internalize a bijection between a set and its powerset in this way? | The above theory is inconsistent. Since j is surjective, there is an x∈ with jx=. Then f[x]=. Therefore x=, since f is a bijection. But is not an element of .
| 5 | https://mathoverflow.net/users/133981 | 405851 | 166,374 |
https://mathoverflow.net/questions/405830 | 4 | Are there any cases of finite subgroups of $O\_n(\mathbb{Q})$ ~~not contained in~~ *not isomorphic to any subgroup of* $O\_n(\mathbb{Z})$?
| https://mathoverflow.net/users/172799 | Finite subgroups of $O_n(\mathbb{Z})$ versus $O_n(\mathbb{Q})$ | Let $n=m^2$ be a square. Then the vector $(1,\dots,1)\in\mathbf{Q}^n$ has norm $m$, as does $(0,\dots 0,m)$. Hence by Witt's theorem there exists an element $u$ of $\mathrm{O}\_n(\mathbf{Q})$ mapping $(0,\dots,0,m)$ to $(1,\dots,1)$. Hence $u$ maps orthogonals to orthogonals: it maps the hyperplane $\mathbf{Q}^{n-1}\ti... | 4 | https://mathoverflow.net/users/14094 | 405854 | 166,376 |
https://mathoverflow.net/questions/404378 | 0 | Let $Q=(I,\Omega)$ be the $D\_4$ affine quiver. We choose as dimension vector $(2,1,1,1,1)$ (where $2$ is on the central vertex). As this dimension vector is indivisible, we can choose a generic $\theta \in \mathbb{Z}^I$. We have the usual moment map $$\mu: R(\overline{Q},v) \to \mathbb{C}^I $$ from representations of ... | https://mathoverflow.net/users/146464 | Quiver varieties associated to D_4 | It is [Kronhimer's result](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-29/issue-3/The-construction-of-ALE-spaces-as-hyper-K%C3%A4hler-quotients/10.4310/jdg/1214443066.full) that $\mathfrak M\_{\zeta\_{\mathbb R},\zeta\_{\mathbb C}}(\mathbf v)$ is $\mathbb C^2/\Gamma$ ($\zeta\_{\mathbb R}=... | 2 | https://mathoverflow.net/users/3837 | 405865 | 166,380 |
https://mathoverflow.net/questions/405862 | 3 | Let $A$ be a finite semisimple algebra over $\mathbb{k}$, a **perfect** field. Is true that the second Hochschild cohomology group vanishes, i.e. $$HH^2(A) = 0?$$
In order to make this question a little bit more complete, it would also be interesting to discuss whether $$HH^{\*>0}(A) = 0?$$
| https://mathoverflow.net/users/105094 | Hochschild cohomology of finite semisimple algebras | By corollary 18 of <https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-12/issue-none/On-the-dimension-of-modules-and-algebras-VIII-Dimension-of/nmj/1118799929.full>
we have (using the perfect condition which ensures the seperability condition needed for corollary 18 to hold) that the enveloping alg... | 4 | https://mathoverflow.net/users/61949 | 405870 | 166,381 |
https://mathoverflow.net/questions/405793 | 5 | **Question:**
Consider the set of continuous, differentiable a.e. functions from $\mathbb R \to \mathbb R$. Can we characterise the subset of these that satisfy $f’(x) = f(x)$ for almost every $x \in \mathbb R$?
*Remarks:*
*1) The problem can be thought of as a weakening of the defining ODE for the exponential fu... | https://mathoverflow.net/users/173490 | Which continuous, differentiable a.e. functions have $f’(x) = f(x)$ a.e.? | Let $g(x) = e^{-x} f(x)$, so that $f(x) = e^x g(x)$. For a given $x$, $f'(x)$ exists if and only if $g'(x)$ exists, and $g'(x) = e^{-x} (f'(x) - f(x))$. In particular, $f'(x) = f(x)$ if and only if $g'(x) = 0$.
It follows that $f$ is necessarily of the form $f(x) = e^x g(x)$, where $g$ satisfies $g'(x) = 0$ almost ev... | 11 | https://mathoverflow.net/users/108637 | 405880 | 166,385 |
https://mathoverflow.net/questions/403249 | 3 | Let $Q$ be a finite quiver. As far as I know there's a great amount of work concerning the so-called quiver varieties one can associate to it. Loosely speaking, these are obtained by taking GIT quotients of semistable representations of the (deformed) preprojective algebra one get from $Q$.
These varieties turn out t... | https://mathoverflow.net/users/146464 | Moduli stack of quiver representations | There is a differential graded version of the preprojective algebra, called *Ginzburg dg-algebra*, that is a special case of the Calabi-Yau completion introduced by [Keller](https://arxiv.org/abs/0908.3499).
The derived moduli stack of object (as defined by [Toen-Vaquié](https://arxiv.org/abs/math/0503269)) of a Cala... | 2 | https://mathoverflow.net/users/7031 | 405888 | 166,388 |
https://mathoverflow.net/questions/405869 | 1 | I am studying von Neumann algebras. In the wiki article [abelian von Neumann algebras](https://en.wikipedia.org/wiki/Abelian_von_Neumann_algebra), it mentions that every abelian von Neumann algebras acting on a separable Hilbert space is \*-isomorphic to $L^{\infty}[0,1]$, $l^{\infty}(\mathbb{N})$ or their direct sum. ... | https://mathoverflow.net/users/172458 | Questions about Maharam's classification theorem | The spaces $[0,1]$, $[0,1]^2$, and $S^1$ are all isomorphic as measurable spaces, including their sets of measure 0, as required by the [Gelfand-type duality for measurable spaces](https://mathoverflow.net/questions/23408/reference-for-the-gelfand-duality-theorem-for-commutative-von-neumann-algebras/360088#360088).
F... | 5 | https://mathoverflow.net/users/402 | 405892 | 166,389 |
https://mathoverflow.net/questions/405895 | 2 | This question is related to: <https://math.stackexchange.com/q/4270522/168758>
---
Let $H\_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the following equivalent formulae (which ever helps)
$$
\begin{split}
H\_n(x) &= n!\sum\_{k=0}^{\lfloor n/2\rf... | https://mathoverflow.net/users/78539 | Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere | To find the dependence of $s\_{nm}$ on $t=a\cdot b$, we take $a=(t,\sqrt{1-t^2},0,0,\ldots 0)$, $b=(1,0,0,0,\ldots 0)$, so that
$$s\_{nm} = \mathbb E[H\_n(X^\top a)H\_m(X^\top b)]=\mathbb E[H\_n(X\_1 t+X\_2\sqrt{1-t^2})H\_m(X\_1)].$$
The marginal distribution $P(X\_1,X\_2)$ of two elements from a vector that is uniform... | 2 | https://mathoverflow.net/users/11260 | 405901 | 166,390 |
https://mathoverflow.net/questions/405751 | 5 | Lets take a closed four manifold $M:=\Sigma\_1\times \Sigma\_2,$ where $\Sigma\_i$s are compact Riemann surfaces. Now if $V$ and $W$ are Spin$^\mathbb{C}$ bundles on $\Sigma\_1$ and $\Sigma\_2$ respectively, then one can define Spin$^\mathbb{C}$ bundle on $M$ with positive Spin$^\mathbb{C}$ bundle defined as
\begin{equ... | https://mathoverflow.net/users/131004 | Understanding the quadratic part in Seiberg Witten equation | So your $\phi$ in this special case really means $(\phi,0)$ in the direct sum, while $\psi$ means $(0,\psi)$. Then $q(\phi)$ is the 2x2 matrix with vanishing off-diagonal entries and nontrivial diagonal entries ($\frac12|\phi|^2,-\frac12|\phi|^2)$, which is traceless. Particular values are $q(\phi)\phi=\frac12|\phi|^2\... | 0 | https://mathoverflow.net/users/12310 | 405903 | 166,391 |
https://mathoverflow.net/questions/405896 | 0 | I really want to prove the statement in the title but I'm struggling with it. Here my current state:
Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ and let $x \in \mathcal{H}$. Then:
\begin{align}
\|{k\_x - \hat{k}\_x}\|^2 &= \langle k\_x - \hat{k}\_x, k\_x - \ha... | https://mathoverflow.net/users/407441 | Proof: If a reproducing kernel exists for a Hilbert space, then it is unique | $\newcommand\tk{\tilde k}\newcommand\ip[2]{\langle #1,#2\rangle}$Let $k$ be a reproducing kernel of a [reproducing kernel Hilbert space (RKHS)](https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space#Definition) $H:=\mathcal H$ of real-valued functions on a set $X$. Then
$$\ip f{k\_x}=f(x)\tag{1}$$ and
$$k(x,y)=... | 3 | https://mathoverflow.net/users/36721 | 405905 | 166,392 |
https://mathoverflow.net/questions/405907 | 0 | $$\ \int\_0^{\pi} \bigl(\sin(x)\bigr)^{2n-2k+1} e^{a\cos(x)} dx , \qquad a,n,k\in\mathbb Z.$$
I tried to solve this integral by parts, but I didn't get any result. I look forward to your experience.
| https://mathoverflow.net/users/380650 | Integration of the product of sin and exponential with power | $$\int\_0^{\pi} (\sin x)^{2n-2k+1} e^{a\cos x} dx =\int\_{-1}^1 (1-\xi^2)^{n-k}e^{a\xi}\,d\xi$$
$$\qquad\qquad=\sqrt{\pi }\, 2^{-k+n+\frac{1}{2}} a^{\frac{1}{2} (2 k-2 n-1)} \Gamma (-k+n+1) I\_{\frac{1}{2} (-2 k+2 n+1)}(a),\;\;n-k+1>0,$$
with $I\_\alpha(x)$ the modified Bessel function of the first kind.
| 2 | https://mathoverflow.net/users/11260 | 405909 | 166,395 |
https://mathoverflow.net/questions/405791 | 3 | In Constructive Set Theory (CZF) the Power Set axiom is replaced with the Subset Collection axiom which I will state here:
$$ \exists c \forall u [\forall x \in a \exists y \in b (\psi(x,y,u)) \longrightarrow \exists d \in c (\forall x \in a \exists y \in d (\psi(x,y,u)) \land \forall y \in d \exists x \in a (\psi(x,... | https://mathoverflow.net/users/312621 | Subset Collection axiom | Subset collection is typically used as a stronger version of exponentiation, rather than a weaker version of power set, so that is why fullness is usually the best way of understanding it.
If you want to think of it as a special case of power set, then what it is saying is that given sets $a$ and $b$ it asserts the e... | 3 | https://mathoverflow.net/users/30790 | 405912 | 166,396 |
https://mathoverflow.net/questions/405921 | 4 | Start with [this triangle (OEIS A118981)](https://oeis.org/A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version):
$$
T(0,0) = 2;T(1,0) = 1;T(1,1) = 1;
$$
$$T(n,k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k-2), n>1 $$
The sequence in qu... | https://mathoverflow.net/users/174962 | Why do convoluted convolved Fibonacci numbers pop up from this triangle? | We have $$T(n,k) = [x^ny^{n-k}]\ \frac{2 - (1+y)x}{1-(1+y)x-x^2}=[y^{n-k}]\ L\_n(1+y),$$
where $L\_n$ is the $n$-th [Lucas polynomial](https://en.wikipedia.org/wiki/Fibonacci_polynomials).
For $k<n$, we have an explicit formula:
\begin{split}
T(n,k) &= \sum\_{i=0}^{\lfloor n/2\rfloor} \frac{n}{n-i}\binom{n-i}i \binom... | 4 | https://mathoverflow.net/users/7076 | 405928 | 166,401 |
https://mathoverflow.net/questions/405929 | 3 | Given an arbitrary set $X$, let $\beta X$ be the set of all ultrafilters over $X$. Consider endowing $\beta X$ with a topology consisting of the following open sets:
$$
\{\mathcal{U} \in \beta X : A \in \mathcal{U}\}
$$
where $A$ ranges over subsets of $X$. Now let $\kappa < |X|$ be an infinite cardinal, and assume the... | https://mathoverflow.net/users/146831 | Is the set of $\kappa$-complete ultrafilters closed in $\beta X$? | If $\kappa=\aleph\_0$ then yes: every ultrafilter is $\aleph\_0$-complete.
If $\kappa>\aleph\_0$ then no, if $\lambda X$ is nonempty. Split $X$ into countably many sets $\{X\_n:n\in\mathbb{N}\}$, of the same cardinality as $X$ itself.
Take a $\kappa$-complete $u\_n$ on $X\_n$ for each $n$.
For every free ultrafilter ... | 9 | https://mathoverflow.net/users/5903 | 405932 | 166,403 |
https://mathoverflow.net/questions/405943 | 5 | I am trying to get a hold on a copy of a 1988 paper by M. Naimi. It appears in MR as [MR0950949](https://mathscinet.ams.org/mathscinet-getitem?mr=0950949) and in zbMATH as [Zbl 0669.10066](https://www.zbmath.org/?q=an%3A0669.10066). Its title is "Les entiers sans facteurs carré $\le x$ dont leurs facteurs premiers $\le... | https://mathoverflow.net/users/31469 | Copy of paper by M. Naimi on squarefree friable integers | Simply trying to put [Naimi "Les entiers sans facteurs carré"](https://www.google.com/search?q=Naimi+%22Les%20entiers%20sans%20facteurs%20carr%C3%A9%22) into Google leads to a [PDF file](https://www.imo.universite-paris-saclay.fr/%7Ebiblio/numerisation/docs/39_SEMINAIRE/pdf/39_SEMINAIRE.pdf) which contains the correspo... | 6 | https://mathoverflow.net/users/8250 | 405956 | 166,409 |
https://mathoverflow.net/questions/405968 | 2 | Let ${\cal P}(\omega)/({\rm fin})$ be the quotient of the Boolean algebra ${\cal P}(\omega)$ where two sets are considered to be equivalent if they differ by a finite number of elements.
It turns out that ${\cal P}(\omega)/({\rm fin})$ is an atomless Boolean algebra. Is the Dedekind-MacNeille completion isomorphic to... | https://mathoverflow.net/users/8628 | Dedekind-MacNeille completion of ${\cal P}(\omega)/({\rm fin})$ | No. $\overline{P(\omega)/\text{fin}}$ is not isomorphic to $P(X)$ for any set $X$ because $P(\omega)/\text{fin}$ is atomless.
We shall write $\overline{B}$ for the Dedekind-MacNeille completion of a Boolean algebra $B$, and we shall write $B^{+}$ for $B\setminus\{0\}$. A minimal element in $B^{+}$ is said to be an at... | 6 | https://mathoverflow.net/users/22277 | 405971 | 166,411 |
https://mathoverflow.net/questions/405967 | 15 | Does there exist a concrete example of a finitely presented group that contains an isomorphic copy of $\operatorname{GL}\_n(\mathbb{Z})$ for every $n\in\mathbb{N}$? I think the Higman embedding theorem implies such a group must exist, but probably not in an especially constructive way, so I'm curious if there's a concr... | https://mathoverflow.net/users/164670 | Finitely presented group containing every $\mathrm{GL}_n(\mathbb{Z})$ | The idea suggested by Anthony Genevois works.
I'm using that for any set $S$, the finitary linear group $\mathbf{Z}$ has the presentation with generators $e\_{st}$ for distinct $s,t\in S$, and relators $[e\_{pq},e\_{qr}]=e\_{pr}$ for any distinct $p,q,r\in S$, and $[e\_{pq},e\_{st}]=1$ for any distinct $p,q,s,t\in S$... | 12 | https://mathoverflow.net/users/14094 | 405980 | 166,414 |
https://mathoverflow.net/questions/405866 | 5 | Original question:
For compact metric spaces, plenty of subtly different definitions converge to the same concept. Overtness can be viewed as a property dual to compactness. So is there a similar story for overt metric spaces?
Edit: Since overtness is trivially true assuming the Law of the Excluded Middle, clearly th... | https://mathoverflow.net/users/174368 | What does overtness mean for metric spaces? | David Roberts has rubbed the magic lamp and the genie appears!
Even though the notion of overtness does depend on the strength of the ambient logic,
I believe the question here is with the notion of metric space, rather than the choice of a model of mathematics.
The natural answer is that any metric space has enoug... | 6 | https://mathoverflow.net/users/2733 | 405981 | 166,415 |
https://mathoverflow.net/questions/304399 | 15 | Let $\mathrm{CombModCat}$ be the category of combinatorial model categories with left Quillen functors between them. By Dugger's theorem and the appendix of Lurie's "Higher Topos Theory" it ought to be true that its localization at the class of Quillen equivalences is equivalent to the homotopy category of presentable ... | https://mathoverflow.net/users/381 | Localizing $\mathrm{CombModCat}$ at the Quillen equivalences | The answer is affirmative and is provided by the paper
[Combinatorial model categories are equivalent to presentable quasicategories](https://arxiv.org/abs/2110.04679).
Among other things, it proves that the relative categories of combinatorial model categories, left Quillen functors, and left Quillen equivalences;
p... | 6 | https://mathoverflow.net/users/402 | 405993 | 166,420 |
https://mathoverflow.net/questions/405969 | 1 | Let $p$ and $q$ be integers.
Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a(0)=a(1)&=1\\
a(2n)& = pa(n)+qa(2n-2... | https://mathoverflow.net/users/231922 | Subsequences of odd powers | Quite similarly to [my answer](https://mathoverflow.net/q/405210) to the previous question, we have that for $n=2^tk$ with odd $k$,
$$
a(n)=\sum\_{i=0}^t \binom{t}{i}p^{t-i}q^i a(2^i(k-1)+1).
$$
It further follows that for $n=2^{t\_1}(1+2^{t\_2+1}(1+\dots(1+2^{t\_s+1}))\dots)$ with $t\_j\geq 0$, we have
\begin{split}... | 3 | https://mathoverflow.net/users/7076 | 405996 | 166,421 |
https://mathoverflow.net/questions/405992 | 2 | Let $J(x)$ be Riemann's prime counting function given by $\frac{1}{2}w(x) + \sum\_{n < x} w(n)$, where $w(p^k) = \frac{1}{k}$ when $p$ is a prime number and $k$ is a positive integer, and $w$ vanishes everywhere else.
Let the logarithmic integral $\newcommand{\li}{\mathrm{li}} \li(x)$ be some antiderivative of $\frac... | https://mathoverflow.net/users/3902 | Constant in logarithmic integral in prime counting | See Edwards' book *Riemann's Zeta Function*. He introduces $J(x)$ on p. 22 (in fact Edwards created the notation $J(x)$, which Riemann had written as $f(x)$). On p. 26 he defines ${\rm Li}(x)$ to be $\int\_0^x dt/\log t$, with the integral defined across the point 1 using a Cauchy principal value. On p. 33, equation (3... | 5 | https://mathoverflow.net/users/3272 | 406001 | 166,422 |
https://mathoverflow.net/questions/299365 | 16 | * Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences.
* Let $\mathbf Q$ be the corresponding $\infty$-category (in whatever foundations you prefer).
* Let $Pres^L$ be the $\infty$-category of presentable $\infty$-categori... | https://mathoverflow.net/users/2362 | Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories? | As pointed out in the answer to [Localizing $\mathrm{CombModCat}$ at the Quillen equivalences](https://mathoverflow.net/questions/304399/localizing-mathrmcombmodcat-at-the-quillen-equivalences/405993#405993),
the answer to Question 1
is affirmative and is provided by the paper
[Combinatorial model categories are equiva... | 5 | https://mathoverflow.net/users/402 | 406002 | 166,423 |
https://mathoverflow.net/questions/342840 | 16 | We know by Karol Szumiło's thesis (<https://arxiv.org/pdf/1411.0303.pdf>) that there is an equivalence between the two fibration categories of cofibration categories on one side and cocomplete $\infty$-categories on the other. By a dual argument, we obtain an equivalence between fibration categories and complete $\inft... | https://mathoverflow.net/users/134438 | Correspondence between classes of model categories and classes of $\infty$-categories | As pointed out in the answer to [Localizing $\mathrm{CombModCat}$ at the Quillen equivalences](https://mathoverflow.net/questions/304399/localizing-mathrmcombmodcat-at-the-quillen-equivalences/405993#405993),
the answer to Question 1 whether
>
> there is also an equivalence between some model-like structures of com... | 3 | https://mathoverflow.net/users/402 | 406004 | 166,424 |
https://mathoverflow.net/questions/406014 | 2 | Let $H$ be a separable Hilbert space of infinite dimension and let $(e\_n)\_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha\_n)\_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are interested in whether or not the unit ball $$B\_1 := \{h \in H \mid 1 \geq \|h\|\_H\}$$ is a subset of the closed sym... | https://mathoverflow.net/users/409412 | Geometry in Hilbert spaces / spheres in high dimensions | Set $c\_n:=\frac{1}{\alpha\_n}$. Clearly, for any $N>0$ one must have $\DeclareMathOperator{\spa}{span}$ $\DeclareMathOperator{\conv}{conv}$ $\newcommand{\bsB}{\boldsymbol{B}}$
$$
\bsB\_1\cap\spa\{e\_1,\dotsc, e\_N\} \subset\conv\{ \pm\alpha\_1 e\_1,\dotsc,\pm\alpha\_N e\_n\}.
$$
This happens if and only if
$$
f\_N(\al... | 3 | https://mathoverflow.net/users/20302 | 406029 | 166,430 |
https://mathoverflow.net/questions/405978 | 4 | Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K\_X$ is not necessarily invertible, it is only reflexive.
>
> **Question.** Is the pull-back $f^\*K\_X$ invertible? If not, can we say that $f^\*... | https://mathoverflow.net/users/45397 | Is the pull-back of canonical sheaf invertible (modulo torsion)? | I think that $f^\*K\_X$ is not invertible in general. For instance, take as $X$ a quotient surface singularity of type $\frac{1}{4}(1, \, 1)$. Then straightforward computations give $$K\_Y=f^\*K\_X - \frac{1}{2}E,$$
where $E$ is the exceptional divisor. We infer that $$f^\*K\_X = K\_Y + \frac{1}{2}E$$ is not an inverti... | 2 | https://mathoverflow.net/users/7460 | 406031 | 166,432 |
https://mathoverflow.net/questions/406025 | 1 | (I asked this [on MSE](https://math.stackexchange.com/questions/4268290/relation-between-two-notions-intermediate-between-pointwise-convergence-and-u) a week ago, but did not get any answers there, so I'm trying here.)
Let $X$ be a topological space. I will define four ways in which a sequence $(f\_n)$ of *continuous... | https://mathoverflow.net/users/17064 | Relation between two notions intermediate between “pointwise convergence” and “uniform convergence” | I think the following is a counter-example for the equivalence of (2) and (3):
Consider the [Baire space](https://en.wikipedia.org/wiki/Baire_space_(set_theory)) $\mathbf N^\mathbf N$ and let $f\_n$ be the following function: If your sequence $x$ starts with $k$ zeroes followed by the entry $n$ then $f\_n(x) = 1/(k+1... | 2 | https://mathoverflow.net/users/409730 | 406042 | 166,436 |
https://mathoverflow.net/questions/406045 | 1 | **Question.** What restrictions are known on closed manifolds $(M^3,g)$ that contain a sequence of embedded minimal surfaces $(\Sigma\_j \mid j \in \mathbf{N})$ with
\begin{equation}
\mathrm{area} \, \Sigma\_j \to \infty \, \, \text{ but } \, \, \mathrm{index} \, \Sigma\_j \leq C
\end{equation}
for some constant $C$ an... | https://mathoverflow.net/users/103792 | Minimal surfaces with increasing area but bounded Morse index | Positive scalar curvature implies that if $\textrm{index}(\Sigma\_j)\leq I$ then $\Sigma\_j$ have bounded area and genus. This is proven here <https://arxiv.org/pdf/1509.06724.pdf> (Theorem 1.3). That paper also contains some other examples related to the Colding--Minicozzi looping example.
A natural generalization i... | 2 | https://mathoverflow.net/users/1540 | 406049 | 166,438 |
https://mathoverflow.net/questions/406018 | 0 | Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p\_{0},p\_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p\_{0}\neq p\_{1}$ such that
\begin{eqnarray\*}
L(\widetilde{y})(p\_{0}) &=&0\Leftrightarrow y=0\text{ on }(0,\frac{1}{2}), \\
L(... | https://mathoverflow.net/users/106804 | Laplace transform injectivity for different values of $p$ |
>
> **Claim:** Assuming that $L$ is the Laplace transform, there is no complex $p\_0$ such that for all $y\in L^2(0,1)$ we have the implication $L(\tilde y)(p\_0)=0\implies y=0$ on $(0,1/2)$.
>
>
>
So, your conditions can never be fulfilled, and therefore they imply any statement, be it true or false.
*Proof o... | 1 | https://mathoverflow.net/users/36721 | 406054 | 166,441 |
https://mathoverflow.net/questions/406052 | 6 | If $X$ is a smooth, geometrically connected, projective curve of genus $g$
over $\mathbb{F}\_q$, then the [zeta function](https://en.wikipedia.org/wiki/Weil_conjectures) of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of degree $2g$. Denote the $P(s)$ of $X$ by $P\_X$.
By the Riemann hypoth... | https://mathoverflow.net/users/125498 | Could the Weil zeroes of curves be evenly distributed? | If $q$ is a prime for which $2$ is a primitive root, then I claim that the Frobenius eigenvalues of the curve $y^2-y = x^q$ over $\mathbb{F}\_2$ have spacing exactly $\tfrac{2 \pi}{q-1}$. This, if [Artin's conjecture](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) holds, such examples exist over... | 9 | https://mathoverflow.net/users/297 | 406059 | 166,443 |
https://mathoverflow.net/questions/406055 | 7 | In John Cremona's book, he defines the *modular symbol* of an elliptic curve in the following way.
Let $E/\mathbf{Q}$ be an elliptic curve and let $f\_E$ be the modular form associated to $E$. The *modular symbol* associated to $E$ is the map $\mathbf{Q} \to \mathbf{Q}$ given by sending any $r \in \mathbf{Q}$ to the ... | https://mathoverflow.net/users/394740 | How do you compute modular symbols? | There are two ways.
You can calculate a sufficiently good approximation by integrating numerically the modular form. Since the value of the modular symbol is known to be a rational number and we know a bound for the denominator (by the Manin-Drinfeld Theorem), we know to what precision we need to evaluate the integra... | 15 | https://mathoverflow.net/users/5015 | 406062 | 166,445 |
https://mathoverflow.net/questions/406060 | 6 | This is a [crosspost](https://math.stackexchange.com/questions/4272807/connections-between-0-toposes-and-1-toposes-grothendieck-and-elementary) from math.stackexchange.
A Grothendieck $0$-topos is the same as a frame (see [here](https://ncatlab.org/nlab/show/frame#As0Topos)). Another relationship between Grothendieck... | https://mathoverflow.net/users/409911 | Connections between $0$-toposes and $1$-toposes (Grothendieck and elementary) | 1. Yes and no. As Jonas Frey pointed out on MSE, you can take the category of finite sheaves on any Heyting algebra $H$ to get an elementary topos. However, unlike in the case of frames and Grothendieck toposes, the Heyting algebra $H$ will not in general coincide with the Heyting algebra of subterminal objects in this... | 7 | https://mathoverflow.net/users/49 | 406063 | 166,446 |
https://mathoverflow.net/questions/405510 | 1 | Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ respectively. We are interested in the quantity $p\_t(A,x,t\_A|A)$, which denotes the probability that, at time $t$, the Markov pr... | https://mathoverflow.net/users/168672 | Occupation times for two-state Markov processes | Without loss of generality, let the final time to be $t=1$ (if it is not, we can make it so by rescaling time as $\alpha'=t\alpha$ and $\beta'=t\alpha$).
Then, consider a single trajectory of the process that starts and ends on state $A$ and makes $x$ jumps ($x$ has to be even). We can represent any such trajectory i... | 1 | https://mathoverflow.net/users/76565 | 406082 | 166,448 |
https://mathoverflow.net/questions/405356 | 0 | Let $p$ and $q$ be integers.
Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a(0)=a(1)&=1\\
a(2n)& = pa(n)+qa(2n-2... | https://mathoverflow.net/users/231922 | Generating function for partial sums of the sequence | As proved in [this answer](https://mathoverflow.net/q/405996), if we represent $n$ as $n=2^{t\_1}(1+2^{t\_2+1}(1+\dots(1+2^{t\_m+1}))\dots)$ with $t\_j\geq 0$, then
\begin{split}
a(n) = P(\ell)^{t\_1}\prod\_{j=1}^{\ell-1} P(\ell-j)^{t\_{2j}+t\_{2j+1}+1},
\end{split}
where $\ell:=\left\lfloor\frac{m+1}2\right\rfloor$ an... | 0 | https://mathoverflow.net/users/7076 | 406085 | 166,449 |
https://mathoverflow.net/questions/406021 | 4 |
>
> Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology sheaves $\mathcal{H}^n(A)$ such that the following three conditions hold true:
>
>
> 1. $\mathcal{H}^n(A)=0$ unl... | https://mathoverflow.net/users/66686 | Perverse sheaves on the complex affine line | It sounds like you are stuck computing the stalks of $\mathcal D A$. To do this, you can use that the homology of the stalk of $\mathcal D A$ at $p$ is the dual of $$H^\*(X, X-p ; A) = H^\*(U,U-p;A)$$ for any neighborhood $U$ of $p$. This follows from $i^\*\mathcal D A = \mathcal D i^! A$ and the exact triangle $i\_\* ... | 3 | https://mathoverflow.net/users/382874 | 406087 | 166,450 |
https://mathoverflow.net/questions/406086 | 9 | I am considering a classical mechanics problem with a fairly complicated system where I think it might be possible to simplify the calculations using the formalism of screw theory and screw algebras, but I cannot find many resources where such a type of calculation is explained in a clear and general way (for example, ... | https://mathoverflow.net/users/119114 | Resources on screw theory in classical mechanics | This is my impression. Maybe I have this wrong.
In the following, $\mathbb D^3$ is the free 3D linear space (or *free module*) over the dual numbers. We may use the terms *infinitesimal part* and *standard part* to refer to the two parts of a dual number. An element of this space is referred to as a *screw*.
A scre... | 7 | https://mathoverflow.net/users/75761 | 406088 | 166,451 |
https://mathoverflow.net/questions/405216 | 0 | Given a productive set, there is a collection of c.e. sets union of which is the productive set, as we know that every c.e. set is with a c.e. function with a index.
My question: is the set of the indices of c.e.sets that cover a productive set also productive one?
| https://mathoverflow.net/users/14024 | Is set of the indices of c.e.sets that cover a productive set also productive one? | No, it isn't. This is a consequence of the fact that we have a choice of infinitely many equivalent indices for each c.e. set. Let $E$ be a set of indices such that
$$(e, j \in E \land e \neq j) \implies W\_e \neq W\_j $$
$$ P = \bigcup\_{e \in E} W\_e$$
For any $e$ let $e' \neq e$ with $W\_e = W\_{e'}$ (note that ... | 1 | https://mathoverflow.net/users/23648 | 406093 | 166,453 |
https://mathoverflow.net/questions/406073 | 1 | The number $p\_1(n)$ of [overpartitions of $n$](https://oeis.org/A015128) is generated by
$$\sum\_{n\geq0}p\_1(n)\,q^n=\prod\_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$
Let $t\in\mathbb{N}$. Now, extend this to construct a family of sequences $p\_t(n)$ as generated by the product
$$\sum\_{n\geq0}p\_t(n)\,q^n=\prod\_{k=1}^{\i... | https://mathoverflow.net/users/66131 | Log-concavity of sequence related to overpartitions | This follows from the fact that $p\_1(n)$ is a log-concave sequence, together with the fact that the convolution of log concave sequences is also a log concave sequence.
The first fact is proven in B. Engel "Log-concavity of the overpartition function", Ramanujan J 43, 229–241 (2017).
The second is proven in S. Hog... | 4 | https://mathoverflow.net/users/2384 | 406094 | 166,454 |
https://mathoverflow.net/questions/406095 | 5 | Let $\pi:\mathcal X\to B$ be a deformation of a compact complex manifold $X=\pi^{-1}(0)$, then for any $t\in B$, the first Chern class $c\_1(X)=c\_1(X\_t)$?
I know the Chern class of a manifold depends on the complex structure, for example, the same diffeomorphism type of $\mathbb C^1$ with a different complex struct... | https://mathoverflow.net/users/99826 | Deformation invariance of Chern classes | This is actually true for *all* Chern classes, but you must first say how you identify $H^\*(X,\mathbb{Z})$ and $H^\*(X\_t,\mathbb{Z})$. There is no problem for small deformations, that is if $B$ is a ball (say). Then the restriction maps $H^\*(\mathscr{X},\mathbb{Z})\rightarrow H^\*(X,\mathbb{Z})$ and $H^\*(\mathscr{X... | 8 | https://mathoverflow.net/users/40297 | 406099 | 166,455 |
https://mathoverflow.net/questions/405994 | 1 | It is clear that the Verma modules are indecomposable modules for the Virasoro algebra with the $L\_0$-weights bounded. I am wondering if the Verma modules exhaust all such indecomposable modules. Does there exist other types of indecomposable modules whose weights are bounded?
Any information and comments will be ap... | https://mathoverflow.net/users/164854 | Indecomposable modules for Virasoro algebra whose weights are bounded | Yes there are other indecomposable modules where $L\_0$ is bounded from below. Logarithmic theories provide many examples.
The simplest example I can think of: take a Verma module generated by a primary state of conformal dimension $\Delta$, and formally take the derivative of this state with respect to $\Delta$. The... | 1 | https://mathoverflow.net/users/12873 | 406109 | 166,458 |
https://mathoverflow.net/questions/406112 | 6 | I have encountered an algebraic group $G$ over $\mathbb C$ such that there is a Zariski open orbit for the adjoint action of $G$ on the nilpotent radical $\mathfrak n$ of its Lie algebra, but there is an ideal $\mathfrak n'\subset\mathfrak n$ such that the adjoint action of $G$ on $\mathfrak n'$ does not have open orbi... | https://mathoverflow.net/users/41291 | Non-hereditarily locally transitive linear algebraic groups | (1) Let $G$ be the pointwise stabilizer of the first coordinate line $L\subseteq V$ in $\mathrm{GL}\_2$ (so $G$ is 2-dimensional, connected, non-reductive). Then $G$ has an open orbit on the plane $V$ (the complement of $L$) but acts trivially on $L$, hence with no open orbit.
This answer the main question, although ... | 9 | https://mathoverflow.net/users/14094 | 406116 | 166,459 |
https://mathoverflow.net/questions/406122 | 3 | What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module.
The following question is my motivation:
[Faithful flatness for rings](https://mathoverflow.net/questions/403926/faithful-flatness-for-rings)
But please note that I am looking for ... | https://mathoverflow.net/users/351872 | RIng that is flat over a subring as a right module but not as a left module | Take the associative algebra over a field $k$, with generators $x$ and $y$ subject to the relation $xy=0$. This admits a basis consisting of monomials of the form $y^a x^b$. It thus contains a subring $k[x]$, and is flat (even free) over this subring as a right module, on the basis $y^a$, $a\geq 0$. As a left module, i... | 10 | https://mathoverflow.net/users/39747 | 406123 | 166,460 |
https://mathoverflow.net/questions/406115 | 5 | $\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^\*$ is a closed subspace of $\Lip\_0(X)$. ($\Lip\_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ satisfying $f(0)=0$ with Lipschitz constant as norm).
Is there anything known about the complementability of $X^\*$... | https://mathoverflow.net/users/12248 | Dual Banach space $X^*$ complemented in $\mathrm{Lip}_0(X)$? | Corollary 5.4 in my book (*Lipschitz Algebras*, 2nd edition): Let $V$ be a Banach space. Then there is a norm 1 linear projection from ${\rm Lip}\_0(V)$ onto $V^\*$. If $V$ is separable then there is a weak\* continuous norm 1 linear projection.
This follows from Theorem 2 of "On nonlinear projections in Banach space... | 10 | https://mathoverflow.net/users/23141 | 406124 | 166,461 |
https://mathoverflow.net/questions/406125 | 5 | Let $(X,d)$ be a metric space and let $H^\alpha$ denote the $\alpha$-dimensional Hausdorff measure on $X$, where $\alpha$ is the Hausdorff dimension of $X$. Is there any simple condition on $X$ that allow me to conclude that $H^\alpha$ is locally finite??
| https://mathoverflow.net/users/321882 | Hausdorff measure | Not true for all $X$.
Taking the gauge function $\phi(t) = t^{1/2}/\log|t|$, construct a Cantor set $X$ in $\mathbb R$ using Hausdorff's original method so that $H^\phi(X) = 1$. Then the Hausdorff dimension of $X$ is $1/2$, but the "natural measure" on it is $H^\phi$, not $H^{1/2}$. Now every open set $U \subseteq X... | 7 | https://mathoverflow.net/users/454 | 406130 | 166,462 |
https://mathoverflow.net/questions/406127 | 9 | Throughout this post $G$ denotes $GL\_{n}(\mathbb{F})$ where $\mathbb{F}$ denotes the finite field of $q$ elements.
I'm currently reading the aforementioned book to understand how the irreducible representations of $G$ are constructed. To get started, we need to understand the conjugacy class of $G$.
In the book he... | https://mathoverflow.net/users/156604 | Chapter 4 Section 2 of Macdonald's Symmetric Functions and Hall Polynomials | The point is that the same polynomial $f\_i$ may occur many times. So for each polynomial $f$, we have a partition whose parts are the exponents $e\_i$ in all the different cases where $f\_i=f$. So this is not a partition of $n$ or of $d\_i$.
Assume for a moment that $f(u)=u-a$ for some $a$; then the resulting partit... | 9 | https://mathoverflow.net/users/66 | 406133 | 166,465 |
https://mathoverflow.net/questions/406135 | 0 | We want to prove that
the unique solution to the following difference equation is the null one:
$$
au(x)+b\mathbf{1}\_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}\_{(\frac{1%
}{2},1)}(x)u(x-\frac{1}{2})=0,\text{ }x\in (0,1).
$$
Extending $u$ by zero outside $(0,1)$ and taking the Laplace transform yields
$$
a\int\_... | https://mathoverflow.net/users/106804 | Unique zero solution to a difference equation via Laplace transform | Your condition can be rewritten as the system of three equations:
$$au(x)+bu(x+1/2)=0\ \forall x\in(0,1/2), \tag{1}$$
$$au(x)+cu(x-1/2)=0\ \forall x\in(1/2,1), \tag{2} $$
$$au(1/2)=0. \tag{3} $$
In turn, (2) can be rewritten as
$$cu(x)+au(x+1/2)=0\ \forall x\in(0,1/2). \tag{2a} $$
So, if the determinant $a^2-bc$ of ... | 1 | https://mathoverflow.net/users/36721 | 406139 | 166,466 |
https://mathoverflow.net/questions/406129 | 2 | If $X\neq \emptyset$ is a set, we say ${\cal C}\subseteq {\cal P}(X)\setminus \{\emptyset\}$ is a *cover* if $\bigcup{\cal C} = X$, and ${\cal C}$ is said to be *minimal* if for all $D\in {\cal C}$ , the collection ${\cal C}\setminus\{D\}$ is no longer a cover. (Equivalently: for every member $D$ of a minimal cover, th... | https://mathoverflow.net/users/8628 | $\min$-compactness vs metacompactness | Using Zorn's lemma, one can show that every point-finite cover of a set has a minimal subcover. Therefore meta-compactness implies min-compactness. I claim that there are min-compact spaces that are not meta-compact (my example is not $T\_{1}$).
Suppose that $X$ is a topological space.
Then we say that a subset $A\... | 2 | https://mathoverflow.net/users/22277 | 406140 | 166,467 |
https://mathoverflow.net/questions/406090 | 2 | Suppose I have a (incomplete) theory $T$ (e.g. PA) which I skolemize to get a theory $T\_S$ in the expanded language. I now build $T'$ by adding to $T\_S$ any sentence $(\forall x)\phi(x)$ where I can prove (say using ZFC as my background set theory) that for all terms $t$ in my skolemized language $ T\_S \vdash\phi(t)... | https://mathoverflow.net/users/23648 | Is adding all sentences true of terms in skolemized theory conservative? | So I can mark this answered (when I can tmw) I'm posting [Emil Jeřábek's](https://mathoverflow.net/users/12705/emil-je%c5%99%c3%a1bek) comment as an answer but they deserve all the credit.
Suppose that $(\forall x)\phi(x)$ is in $T' - T\_S$. That means we've proved that $T\_S \vdash \phi(t)$ for each term $t$ in our ... | 2 | https://mathoverflow.net/users/23648 | 406145 | 166,468 |
https://mathoverflow.net/questions/406146 | 1 | Whenever we have a continuous map of topological spaces $f: X \to Y$ and a sheaf $\mathcal{F}$ on $Y$ (of abelian groups for example), we get an induced pullback map
$$ H^n(Y, \mathcal{F}) \to H^n(X, f^\* \mathcal{F}), $$
which is described for example in Iversen, *Cohomology of Sheaves*, II.5.1. I would like to have a... | https://mathoverflow.net/users/173545 | What is the pullback morphism on sheaf cohomology of local systems in terms of representations of the fundamental group? | The circle $S^1$ is a $K(\mathbb{Z},1)$ space, so the cohomology of a local system on it can be identified with group cohomology. Let $R=\mathbb{Z}[T,T^{-1}]$ be the group ring of $\mathbb{Z}$. Given a local system $\mathcal{L}$,
$$H^i(S^1,\mathcal{L})= Ext\_R^i(\mathbb{Z}, \mathcal{L})$$
This can be computed as using ... | 3 | https://mathoverflow.net/users/4144 | 406151 | 166,470 |
https://mathoverflow.net/questions/406149 | 6 | Consider a sufficiently nice topological space $X$ as well as topological groups $G$ and $H$. Consider the functor $F$ that associates to $X$ the set of all isomorphism classes of all principal $H$-bundles $\pi\_H\colon P\_H\twoheadrightarrow P\_G$ on all principal $G$-bundles $\pi\_G\colon P\_G\twoheadrightarrow X$ on... | https://mathoverflow.net/users/411120 | Classifying space of bundles over bundles | If I understand the question right, I think the classifying space can be described like this. Let $Map(G,BH)$ be the space of all continuous maps from $G$ to $BH$. Make the group $G$ act continuously on that function space (using the usual free transitive action of $G$ on the space $G$), and form the associated bundle ... | 8 | https://mathoverflow.net/users/6666 | 406155 | 166,471 |
https://mathoverflow.net/questions/406118 | 7 | Let $\mathcal{C}$ be a category and $S$ a collection of morphisms in $\mathcal{C}$. The localisation $\mathcal{C}[S^{-1}]$ is defined via a functor $F: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ and the following universal property. For all $\varphi\in S$ the morphism $F(\varphi)$ is an isomorphism. Now let $G: \... | https://mathoverflow.net/users/145920 | Localisation of categories, but instead of "isomorphisms" I want "morphisms with right inverse". Construction via calculus of fractions possible? | I think there is a relatively good reason why such a thing shouldn't exists.
In general when you freely add right inverse or inverse, the general arrows of the resulting category will be zig-zag in the original category where the arrow in the wrong directions are all in S, and are there to represent composition with ... | 6 | https://mathoverflow.net/users/22131 | 406157 | 166,473 |
https://mathoverflow.net/questions/406158 | -1 | Let $X$ be a Banach space. Let $\mathcal{F}$ be the family of all the bounded subsets of $X$. If $I$ is the identity map on $X$, we shall denote by $\operatorname{span}\{I\}$ the vector space generated by $I$.
In [Measures of Weak Compactness and Fixed Point Theory](https://arxiv.org/abs/math/0310422), Barroso and O'... | https://mathoverflow.net/users/102228 | Definition of a $\psi$-Banach space | The definition makes no sense due to the mixing up of "relatively" (weakly) compact and (not relatively) compact.
I guess that what you mean is:
$\psi$ is strongly-weakly proper on closed balls (that is, the intersection of preimages of weakly compact sets with closed balls are compact). For strongly-weakly continu... | 2 | https://mathoverflow.net/users/165275 | 406162 | 166,474 |
https://mathoverflow.net/questions/357785 | 5 | Is there a simple construction of a Sullivan minimal model $\Lambda U \rightarrow V$ in the case that $H^1(V)\neq 0$? Do you have a reference? I envisage a degree-wise construction as in the case of $H^1(V)=0$ explaining how to deal with the new phenomenon. The existence is proven very generally in *Felix, Halperin, Th... | https://mathoverflow.net/users/43645 | Sullivan minimal model in the case of $H^1(V)\neq 0$ | Gelfand and Manin explain it very nicely in their book "Methods of Homological Algebra", last chapter.
| 3 | https://mathoverflow.net/users/6635 | 406165 | 166,476 |
https://mathoverflow.net/questions/406164 | 4 | If the product of two functions is smooth, then how quickly must one function decay when the other is non-smooth? Suppose we have two functions $f,h$ on $\mathbb{R}$ such that:
* $h$ is Lipschitz continuous
* $f$ is smooth (i.e. $C^\infty$), and
* $fh$ is a smooth function.
What can we conclude about $f$ from this ... | https://mathoverflow.net/users/112378 | If $fh$ is smooth and $h$ Lipschitz, what can be said about $f$? | $\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$Here are answers to your three questions (the latter two of them partial).
**Answer 1:** Yes, for any real $a$ and any $k\in\{0,1,\dots\}$,
\begin{equation\*}
\text{if $h$ does not have all the derivatives at $a$, then $f^{(k)}(a)=0$.}\tag{1}
\end{equation\*}
Inde... | 4 | https://mathoverflow.net/users/36721 | 406168 | 166,478 |
https://mathoverflow.net/questions/406175 | 6 | Let $ G $ be a simple linear algebraic group. Let $ G\_\mathbb{R} $ be the real points of $ G $. Let $ G\_\mathbb{Z} $ be the integer points of $ G $. Is $ G\_\mathbb{Z} $ a maximal closed subgroup? In other words, are the only closed subgroups $ H $ of $ G\_\mathbb{R} $ such that
$$
G\_\mathbb{Z} \subset H \subset G\_... | https://mathoverflow.net/users/387190 | Is the group of integer points of a simple real linear algebraic group a maximal closed subgroup? | Your guess is correct: if $H$ is a closed subgroup containing $\operatorname{SL}\_n({\mathbb Z})$,then it is a Lie subgroup. If $\mathfrak h$ is its Lie algebra, then $\mathfrak h$ is stable under the adjoint action of $SL\_n(\mathbb Z)$ and hence under all of $\operatorname{SL}\_n(\mathbb R)$ (since $\operatorname{SL}... | 7 | https://mathoverflow.net/users/23291 | 406177 | 166,482 |
https://mathoverflow.net/questions/406176 | 9 | Let $F\_n$ be a free group of rank $n \geq 2$. The group $F\_n$ acts on its commutator subgroup $[F\_n,\, F\_n]$ by conjugation. Let $G = [F\_n,\, F\_n] \rtimes F\_n$. It's not hard to see that $G$ is finitely generated.
>
> **Question**: Is $G$ finitely presentable? Presumably not, but I can't seem to prove it.
> ... | https://mathoverflow.net/users/411409 | Finite presentability of semi-direct product of free group and its commutator subgroup | This group is not finitely presentable. Indeed, write $F$ for the given free group and $F'$ for its derived subgroup. The map
$$F\ltimes F'\to F\times F,\quad (f,g)\mapsto (f,fg)$$
is an injective group homomorphism, and its image is the fibre product of two copies of $F$ over the abelianization map $F\to F/F'$. (Alter... | 9 | https://mathoverflow.net/users/14094 | 406187 | 166,484 |
https://mathoverflow.net/questions/406171 | 21 | I've heard tell the following anecdote involving Pierre Gabriel and [Jacques Tit](https://euromathsoc.org/news/jacques-tits-1930---2021-46) at least twice in a lapse of four years or so:
>
> When P. Gabriel presented the theorem in a conference [sometime around 1970], he said something like this: "OK, this algebra ... | https://mathoverflow.net/users/1593 | Reference request: a tale of two mathematicians | Here is a recent talk by Ringel (in German):
[Algebra und Kombinatorik](https://www.math.uni-bielefeld.de/%7Eringel/lectures/unger-text/hagen.htm).
The related part is:
>
> Besonderes Aufsehen hat ein Ergebnis erregt, das meist Satz von Gabriel genannt wird: *Ein zusammenhängender Köcher ist genau dann darstellun... | 18 | https://mathoverflow.net/users/61949 | 406188 | 166,485 |
https://mathoverflow.net/questions/406043 | 4 | Let $\Gamma$ be a connected graph, let $\lambda \ge 1$ and $c \ge 0$ be some constants. Recall that a combinatorial path $p$ in $\Gamma$ is said to be *$(\lambda,c)$-quasigeodesic* if for every combinatorial subpath $q$ of $p$ one has $$\ell(q) \le \lambda d(q\_-,q\_+)+c,$$
where $\ell(q)$ is the length of $q$, $q\_-$ ... | https://mathoverflow.net/users/7644 | Shortcutting quasigeodesics | As suggested, I am turning my comments into an answer. The answer to both questions is negative for any $\lambda > 1$, and positive for $\lambda = 1$.
For $\lambda = 1+\epsilon$ note that in $\mathbb Z^2$, the concatenation of the (unique) geodesics from $(0,0)$ to $(0,k)$, from $(0,k)$ to $(n,k)$ and from $(n,k)$ to... | 4 | https://mathoverflow.net/users/97426 | 406192 | 166,487 |
https://mathoverflow.net/questions/406197 | 0 | Let us look at the subspace of smooth complex functions of $L^2(\mathbb{R}^n,\mathbb{C})$, call $H^s$ the Sobolev spaces. By the diamagnetic inequality $\lvert \nabla \lvert\psi\rvert\rvert (x) \le \lvert\nabla \psi\rvert(x)$ (a proof is [here](https://physics.stackexchange.com/questions/100948/is-there-a-physical-intu... | https://mathoverflow.net/users/nan | 'Diamagnetic' inequality for negative Sobolev spaces | This is not true in the case of $H^{-1}(\Omega) = H\_0^1(\Omega)^\*$ (real spaces), where $\Omega \subset \mathbb R^d$ is bounded:
* In <https://math.stackexchange.com/questions/336834/decomposition-of-functionals-on-sobolev-spaces>, we see that $|\psi|$ cannot be defined for all $\psi \in H^{-1}(\Omega)$.
* Even if ... | 0 | https://mathoverflow.net/users/32507 | 406201 | 166,490 |
https://mathoverflow.net/questions/406190 | 4 | I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this line for the Dirichlet problem at infinity for harmonic functions (see [M. T. Anderson's work on Laplace-Beltrami operato... | https://mathoverflow.net/users/411616 | Elliptic equations in asymptotically hyperbolic manifolds | The definite reference for this is the monograph by John Lee "Fredholm operators and Einstein metrics on conformally compact manifolds", Mem. Am. Math. Soc. Series Profile 864, 83 p. (2006), that you can also find on the arXiv: <https://arxiv.org/abs/math/0105046>
This is limited to "natural" differential operators b... | 2 | https://mathoverflow.net/users/24271 | 406203 | 166,491 |
https://mathoverflow.net/questions/406161 | 6 | I am very curious to study [arXiv:1310.7930](https://arxiv.org/abs/1310.7930) (henceforth:DCCT) but am not sure if I have the pre-requisites. I am familiar with basic algebraic topology (singular cohomology, classifying spaces, characteristic classes), differential geometry (bundles, connections, Chern--Weil theory) an... | https://mathoverflow.net/users/40386 | Learning roadmap to 'Differential cohomology in a cohesive $\infty$ topos' | I haven't read all of DCCT so take this with a grain of salt, but after having spent a lot of time with it, this is how I would recommend getting started on the abstract stuff.
First one must learn classical Grothendieck Topos Theory. Chapter 1.2 of DCCT gives a pretty good motivation and some nice examples of sheave... | 5 | https://mathoverflow.net/users/124010 | 406206 | 166,492 |
https://mathoverflow.net/questions/406204 | 1 | **Motivation.** At a recent gathering of co-workers, somebody said that he thinks that of all the people present that he knows, about half are happy with the current COVID19 measures, and the other half are not. Which made me put this into a graph question: Given a finite, simple, undirected graph $G = (V,E)$, we say t... | https://mathoverflow.net/users/8628 | Discrepancy number of a graph | Yes, there is such a graph. Let $A = \{1,\dots,2n\}$, let $B = \{b \subseteq A \colon |b|=n\}$, and let $G$ be a graph with vertex set $A \uplus B$ and an edge between $a\in A$ and $b\in B$ if $a \in b$.
For any set $C$ there are either $n$ vertices in $A \cap C$ or $n$ vertices in $A\setminus C$, and thus there is a... | 3 | https://mathoverflow.net/users/97426 | 406209 | 166,494 |
https://mathoverflow.net/questions/406200 | 1 | Hilbert introduced a construct $\epsilon x. P(x)$ for a predicate $P$ such that
$$\exists x. P(x) \implies P(\epsilon y.P(y))$$
Obviously, this is equivalent to the axiom of global choice. With this operator, we can form complex propositions conveniently:
$$Q(\epsilon x. P(x)) \text{ equiderivable to } \forall x. P... | https://mathoverflow.net/users/136535 | A dual to Hilbert's $\epsilon$ operator | Nothing like this can happen.
Take $P(x)\equiv\top$, and let $S$ be any formula such that both $\exists x S(x)$ and $\exists x\neg\ S(x)$ are derivable. Then both $$(\*)\_1:\quad\exists x(P(x)\wedge S(x))$$ and $$(\*)\_2:\quad \exists x(P(x)\wedge\neg S(x))$$ are themselves derivable. However, we can't possibly have ... | 4 | https://mathoverflow.net/users/8133 | 406210 | 166,495 |
https://mathoverflow.net/questions/406225 | 6 | I'm just studying Lie algebras. If $A$ is a $k$-algebra (not necessarily Lie or associative, just a bilinear law), it is straightforward to check that any derivation algebra of $A$ is a Lie algebra. I suppose that the converse is not true, but I can't find a counterexample.
1. Is there any example of a Lie algebra wh... | https://mathoverflow.net/users/173535 | Is there any example of a Lie algebra which is not a derivation algebra? | $\newcommand{\r}{\mathfrak{r}}\newcommand{\h}{\mathfrak{h}}\newcommand{\g}{\mathfrak{g}}\newcommand{\a}{\mathfrak{a}}$The 2-dimensional abelian Lie algebra $\a\_2$ is not isomorphic to the derivation Lie algebra of any Lie algebra.
Suppose otherwise. Let $\g$ be such a Lie algebra. We discuss according to whether the... | 8 | https://mathoverflow.net/users/14094 | 406227 | 166,499 |
https://mathoverflow.net/questions/406213 | 4 | **Context and mathematical maturity:** I have knowledge of the usual engineering math courses, meaning differential+integral+vector calculus, linear algebra, probability and statistics, etc. and some pure math courses like analysis. I am a pure mathematics enthusiast, so I have been working through some texts in pure m... | https://mathoverflow.net/users/324468 | Geometry book recommendation | Your requirements: "much preferably if it is not at a high school level" and
"My dream would be to read Hilbert's Geometry and Imagination" are contradictive. The book of Hilbert and Cohn-Vossen, Geometry and imagination IS on the high school level.
Other highly recommended geometry books on the same level are
Marcel... | 2 | https://mathoverflow.net/users/25510 | 406229 | 166,500 |
https://mathoverflow.net/questions/403966 | 3 | In a comment made by Gjergji Zaimi to [this](https://mathoverflow.net/q/61615/167834) older question, it is conjectured that $\frac 73$ is the threshold separating countability and uncountability of the sets of infinite binary words having a given [critical exponent](https://en.wikipedia.org/wiki/Critical_exponent_of_a... | https://mathoverflow.net/users/167834 | Words with critical exponent $< \frac 73$ | [From user Ben's comment]
In the paper "The ubiquitous Prouhet-Thue-Morse sequence" ([DOI link behind paywall](https://doi.org/10.1007/978-1-4471-0551-0_1), [preprint version](https://cs.uwaterloo.ca/%7Eshallit/Papers/ubiq15.pdf)), Allouche and Shallit claim that there are uncountably many overlap-free binary words, ... | 1 | https://mathoverflow.net/users/14094 | 406232 | 166,502 |
https://mathoverflow.net/questions/406224 | 0 | Let $s, \delta \in (0,1)$. Consider the outer measure on $\mathbb{R}$, $\mu^s\_{\delta}$, defined by
\begin{align\*}
\mu^s\_{\delta}(E):=\inf \left\{\sum\_{j}\lvert I\_{j}\rvert^s: E \subset \bigcup\_{j} I\_{j}: I\_{j} \text { closed intervals, } \lvert I\_j\rvert\leq\delta\right\}.
\end{align\*}
For an interval $I \s... | https://mathoverflow.net/users/197849 | An outer measure defined on $\mathbb {R}$ | $\newcommand\de\delta\newcommand\ol\overline$Your goal cannot be attained in general. Indeed, suppose that $\de\in(0,1/2)$. Take any interval $E$ of length $\de\_1:=|E|\in(\de,2\de)$. Then for the closure $\ol E$ of $E$ and some closed intervals $I\_1$ and $I\_2$ of lengths $|I\_1|=\de$ and $|I\_2|=\de\_1-\de\le\de$ we... | 3 | https://mathoverflow.net/users/36721 | 406236 | 166,504 |
https://mathoverflow.net/questions/406248 | 0 | Let $H$ and $K$ be infinite dimensional (separable) Hilbert spaces and $X=B(H,K)$ denote the space of bounded linear operators. For $T\_1, T\_2$ in $X$, we define $D\_{T\_1,T\_2}:X \to X$ as $D\_{T\_1,T\_2}(T)=TT\_1^\*T\_2$. Finally define $V^0=\operatorname{span}\{D\_{T\_1,T\_2}, T\_1, T\_2 \in X\}$. One can check tha... | https://mathoverflow.net/users/129638 | Trying to recognise a $C^*$-algebra | Assuming $H$ is separable? If $K$ is finite dimensional and $H$ is infinite dimensional then this gives you the compact operators on $H$. Otherwise you get $B(H)$.
If $H$ can have uncountable dimension then there are more possibilities because there are more closed ideals of $B(H)$.
Some details: $B(H)$ acts on $X ... | 4 | https://mathoverflow.net/users/23141 | 406253 | 166,513 |
https://mathoverflow.net/questions/406265 | 2 | Call a function$$\mathbb{N}\times \mathbb{N}\to \{0, 1\}, \quad (n, m)\to f(n, m)$$computable in polynomial time in $\log n+\log m$ a PTIME family.
Given a PTIME family $f$ call a computable function $g:\mathbb{N}\to \mathbb{N}$ such that $f(n, g(n))=1$ for all $n\in \mathbb{N}$ a solution of $f$.
Is there a PTIME ... | https://mathoverflow.net/users/nan | Family of PTIME sets where it is hard to name elements | This is an open problem.
If $\mathrm{TFNP\ne FP}$, then a TFNP problem outside FP directly gives what you call a PTIME family (with solutions polynomially bounded in terms of the input) whose solutions cannot be computed in polynomial time.
If P = NP, then no such problem exists: the assumption implies that given $... | 4 | https://mathoverflow.net/users/12705 | 406266 | 166,516 |
https://mathoverflow.net/questions/405670 | 1 | I've been trying to write a test function for Fibonacci pseudo-primes with large $n$. Fibonacci pseudoprimes are composite numbers such that $V\_n(P,Q) \equiv P \mod n$ for $P>0$ and $Q =\pm 1$, with $V\_n$ Lucas sequence.
As such I need to compute Lucas sequence for large $n$. They are defined by:
$U\_n(P,Q)=\frac... | https://mathoverflow.net/users/402326 | Computing Lucas sequence for large n | Following Emil remark, I used other relations to obtain a $O(\log n)$ algorithm:
$\begin{cases}U\_{2n}=U\_nV\_n\\
V\_{2n}=V\_n^2-2Q^n\\
U\_{2n+1} = U\_{n+1}V\_n - Q^n\\
V\_{2n+1} = V\_{n+1}V\_n -PQ^n\end{cases}$
This concluded in the following python code:
```
def lucas_sequence(p, q, n):
if n == 0:
... | 0 | https://mathoverflow.net/users/402326 | 406272 | 166,517 |
https://mathoverflow.net/questions/406270 | 4 | Context: I'm trying to deal with presentations in the framework of Gonthier et al. formalization of the group theory in the proof assistant Coq. It was used to machine check the Feit-Thompson odd order theorem. In this formalisation, all groups are assumed to be **finite** and it would be a lot of work to remove the as... | https://mathoverflow.net/users/37190 | Group presentation in the category of finite group | A well-known example, due to G. Higman, of an infinite finitely presented group with no nontrivial finite quotients is $$G = \langle a,b,c,d \mid a^{-1}ba=b^2,b^{-1}cb=c^2,c^{-1}dc=d^2,d^{-1}ad=a^2 \rangle.$$
Proving it has no nontrivial finite quotients is elementary and a nice exercise. The idea is to prove that $f(a... | 8 | https://mathoverflow.net/users/35840 | 406273 | 166,518 |
https://mathoverflow.net/questions/405784 | 14 | Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(T)^2}$ is not constant). Does there necessarily exist $t\in\mathbb{Q}$ such that the Mordell-Weil group of $\mathcal{E}\... | https://mathoverflow.net/users/404359 | Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization? | I suspect that this is unknown in general. I would guess that any method which produces at least one elliptic curve of positive rank should also produce infinitely many of positive rank, which as you already mention is an open problem.
Anyway, here is some discussion around this problem and a particular open case whi... | 9 | https://mathoverflow.net/users/5101 | 406276 | 166,519 |
https://mathoverflow.net/questions/406181 | 1 | Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v\_i=(x\_1,x\_2,\ldots,x\_t)$, where $x\_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors.
Then we construct the following matrix:
$A=(a\_{ij})\_{2^{ct},2^{ct}}$, where $a\_{ij}=1$ if $\bigvee\_{1\leq l\leq t}v\_{i}(l)\wedge v\_{... | https://mathoverflow.net/users/178444 | Upper bound of rank of a matrix | If I understood your question correctly, it can be reformulated as follows:
The vectors $v\_i$ are in bijective correspondence with subsets of $X=\{1,\cdots, c\} \times \{1,\cdots, t\}$ by sending a vector $v\_i=(x\_1,x\_2,\cdots, x\_t)$ to the subset $S=\{(k,l)\in X:k\in x\_l\}$.
You can then see the matrix $A$ as a $... | 2 | https://mathoverflow.net/users/160416 | 406283 | 166,523 |
https://mathoverflow.net/questions/406274 | 5 | **Motivation:**
This is a toy model of how a closed system will always evolve towards the distribution of maximal entropy, where no further transfer of heat/energy is possible.
**Problem set up:**
Fix a positive integer $N$, and denote by $[N]$ the set $\{1, \dots, N\}$.
Let $\mathcal L := [N] \times [N]$ be a ... | https://mathoverflow.net/users/173490 | A toy model of heat death | The Ehrenfest model (in discrete time, for simplicity) is just a Markov chain with the finite state space $\{0,1,\dots, N\}$ and the transition probabilities
$$
p(k,k-1)=k/N, \quad p(k,k+1)=1-k/N
$$
described by a single transition (averaging) operator $P$. Its stationary distribution $m$ is the binomial one with the p... | 2 | https://mathoverflow.net/users/8588 | 406286 | 166,525 |
https://mathoverflow.net/questions/406153 | 3 | The Dedekind–MacNeille Completion is the generalized way of completing an arbitrary lattice $L$. We will call $C$ the Dedekind–MacNeille completion of $L$ (I will not go into the details of the Completion but see the comment below.) The completion includes an embedding $i: L \to C$ which preserves existing arbitrary me... | https://mathoverflow.net/users/312621 | Constructive lattice completion | Joseph Van Name’s answer is correct for the general case. However, the OP indicated in comments that they are interested in the case where $L$ is a Heyting algebra, and then it turns out that the Dedekind–MacNeille completion *can* be constructed by taking the set of complete ideals of $L$ ordered by inclusion. (I’m fr... | 2 | https://mathoverflow.net/users/12705 | 406287 | 166,526 |
https://mathoverflow.net/questions/405926 | 5 | Cross-posted from MSE
<https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups>
$\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{SU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SO{SO}$Let $\G$ be a Lie group.
I am interested in finite m... | https://mathoverflow.net/users/387190 | Finite maximal closed subgroups of Lie groups | $\newcommand{\G}{\mathcal{G}}
\newcommand{\K}{\mathcal{K}} $Question: When does $ \G $ admit a finite maximal closed subgroup?
Answer : Must be one of the following two cases
1. $ \G $ is compact and simple
2. $ \G $ is not compact in which case $ \G $ cannot be connected and moreover the component group $ \G/\G^\c... | 1 | https://mathoverflow.net/users/387190 | 406300 | 166,533 |
https://mathoverflow.net/questions/406297 | 4 | Recall that a prime $p$ is a Wieferich prime if $p^2|2^{p-1}-1$. The only known Wieferich primes are $p=1093$ and $p=3511$. A prime $p$ is a generalized Wieferich prime to base $q$ if $p^2|q^{p-1}-1$.
It is strongly suspected that there are infinitely many Wieferich primes to base $q$ for any $q>1$. In the other dire... | https://mathoverflow.net/users/127690 | Ruling out an extremely specific class of Wieferich-like primes | This is impossible from standard results about cyclotomic polynomials and their factors. Note that $\frac{q^{p-1}-1}{p-1}$ is just
$$
\prod\_{d|(p-1), d>1}\Phi\_d(q),
$$
where $\Phi\_d(x)$ is the $d$th cyclotomic polynomial. Taking $p>3$, there are two such divisors $d$; namely $2$ and $p-1$. By an old result attribute... | 8 | https://mathoverflow.net/users/3199 | 406307 | 166,537 |
https://mathoverflow.net/questions/401499 | 1 | Let $\mathfrak{G}$ be the class of all finite directed and undirected graphs. Let $A,B\subseteq \mathfrak{G} $, $A$ and $B$ are closed under graph isomorphisms, and $A \cap B = \varnothing$. Consider the following two player game. On the graph $G \in \mathfrak{G} $ with the starting vertex $u$, play as follows: the fir... | https://mathoverflow.net/users/175589 | Complexity of games with graph classes | No, it is not possible to go above PSPACE, because all positional games with an exponentially large game tree are in PSPACE; you can just check all the options. And the game when $A=B=\emptyset$ is indeed PSPACE-hard, I've found a gadget to reduce the original Generalized Geography to it.
Update: Some of my students ... | 1 | https://mathoverflow.net/users/955 | 406317 | 166,540 |
https://mathoverflow.net/questions/406301 | 4 | $\def\f#1{\text{#1}}$Does $\f{ACA}$ prove that any two internally complete ordered fields are isomorphic?
Here internal completeness is expressed roughly as "every sequence of reals with an upper bound has a least upper bound", where the sequences and reals are coded appropriately.
This is an attempt to express the... | https://mathoverflow.net/users/50073 | Does ACA prove categoricity of the reals? | The current version of the question is flawed: the proposed categoricity principle is in fact classically **false**. Roughly speaking, any "interesting" theory of the appropriate type is going to have lots of non-isomorphic countable $\omega$-models. *(For massive overkill let $M,N$ be countable elementary submodels of... | 6 | https://mathoverflow.net/users/8133 | 406325 | 166,544 |
https://mathoverflow.net/questions/406275 | 3 | In my research in Quantum Field Theory, I have encountered two questions that involve partial Bell polynomials:
1. Let $u$ and $x\_i$ be indeterminates. I have checked that the following conjectured identity
$$
\sum \_{m=0}^\infty u^m \frac{(m+1)!}{(2m+2)!} B\_{2m+2,m+2}(x\_1,\dots,x\_{m+1})=
\frac{1}{2} \left( \su... | https://mathoverflow.net/users/98550 | Two questions about Bell polynomials | With respect to the first question, I'm not sure about whether the given identity is known, but here is a proof. It also expresses the involved series in terms of reversion of the series $\frac{t}{g(t)}$ as defined below.
It is convenient to express the given identity in terms of [ordinary Bell polynomials](https://e... | 7 | https://mathoverflow.net/users/7076 | 406326 | 166,545 |
https://mathoverflow.net/questions/406221 | 3 | I have asked [this question](https://math.stackexchange.com/questions/4276271/chebyshev-polynomials-and-ballot-number) a short time ago on mathstackexchange, but it has already fallen into the abyss of answered and uncommented questions. So I take the risk to ask it on mathoverflow.
Playing with the various expressio... | https://mathoverflow.net/users/37214 | Chebyshev polynomials and ballot numbers | Here is a combinatorial proof.
It is more convenient to prove the equivalent formula (obtained by setting $n=m+2j$)
$$\sum\_k \binom{k}{j}\binom{m+2j}{2k}=2^{m-1}\frac{m+2j}{m+j}\binom{m+j}{j}.\tag{$\*$}$$
To motivate the proof we first prove the closely related but slightly easier
formula
$$\sum\_k \binom{k}{j}\bi... | 7 | https://mathoverflow.net/users/10744 | 406329 | 166,546 |
https://mathoverflow.net/questions/406291 | 1 | My question is about part of the Borel Transgression Theorem on spectral sequences, translated from [1, Theorem 13.1] (see also [2] for a translated version of the whole paper):
>
> Let $B^\bullet := \bigoplus\_{p \in \mathbb{N}\_0} B^p$ and $P^\bullet := \bigoplus\_{q=0}^m P^q$ be finite-dimensional, graded vector... | https://mathoverflow.net/users/126256 | Borel's transgression theorem for spectral sequences | Mimura and Toda's statement in *The Topology of Lie Groups*, Theorem VII.2.9 (p. 378), requires less severe degree constraints. They start with a Serre fibration $F \to E \to B$ with $B$ simply-connected, and demand only that
* there be an oddly-graded space $P$ of "generators" of degrees no more than some fixed $N$ ... | 3 | https://mathoverflow.net/users/5792 | 406335 | 166,549 |
https://mathoverflow.net/questions/406083 | 3 | I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an answer to either.
First question
--------------
Let $M$ be a real analytic manifold, and let $P$ be a linear partial ... | https://mathoverflow.net/users/36720 | do hyperfunction solutions always exist? | The answer to both questions is Yes. It has been known (Grainger, Kohn, Stein, Proc. Nat. Academy USA, Vol. 72, No. 9, 3287-3289 (1975)) that when $f$ is Baire category 1, then the Lewy operator is locally solvable. This means that if the 'size'of the test space is reduced making its dual larger, then solvability occur... | 5 | https://mathoverflow.net/users/167228 | 406345 | 166,553 |
https://mathoverflow.net/questions/406314 | 2 | I am looking at the following convolution integral (with $\theta > 0$):
$$ f(t) = \int\_{-\infty}^t \exp(-(t-s)\theta) g(s) ds = (\textbf{1}\_{\{.\geq 0 \}} \exp(-.\theta) \* g)(t)$$
I want to find conditions on $g$ s.t.
$$ f \in C^{k, \alpha},$$ by which I denote the Hoelder space. That means, f should be $k$-times di... | https://mathoverflow.net/users/336584 | Assumptions under which convolution is in Hölder class | *Edit: In the first part, $C^{k,\alpha}$ is understood locally. For a global result, see the final part.*
The equation reads $$f(t) = \int\_{-\infty}^t e^{-(t - s)\theta} g(s) ds,$$ is equivalent to $$e^{t \theta} f(t) = \int\_{-\infty}^t e^{s \theta} g(s) ds,$$ so if $g$ is $C^{k-1,\alpha}$, then $s \mapsto e^{s \th... | 2 | https://mathoverflow.net/users/108637 | 406349 | 166,554 |
https://mathoverflow.net/questions/406315 | 21 | The typical application of Sylow's Theorem is to count subgroups. This makes it difficult to search the web for other applications, since most hits are in the context of qualifying exams.
>
> What are other uses of Sylow's Theorem? In particular, are there famous/common instances where the goal is to find two non-c... | https://mathoverflow.net/users/13923 | Atypical use of Sylow? | There are many examples of using Sylow $p$-subgroups to understand the structure of general finite groups. KConrad's answer indicates some of these, and others are mentioned in comments. A finite group $G$ with Sylow $p$-subgroup $P$ is said to have a normal $p$-complement if there is a normal subgroup $K$ (necessarily... | 23 | https://mathoverflow.net/users/14450 | 406350 | 166,555 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.