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https://mathoverflow.net/questions/303155 | 4 | In my paper I would like to include an example of easy concrete finitary statement which can be easily verified by probabilistic algorithm to any reasonable confidence level, but which looks completely hopeless to prove rigorously.
A candidate example is the statement "$10^{1500} + 2329$ is prime". This can be easily... | https://mathoverflow.net/users/31472 | Example of concrete statement which requires probabilistic algorithm | First of all, there is a conjecture in computer science that BPP = P, i.e., anything that can be done in randomized polynomial time can also be done in deterministic polynomial time. The hope is that you can replace true-random trials with repeated trials with a cryptographically secure pseudo-random number generator. ... | 3 | https://mathoverflow.net/users/1450 | 303223 | 132,755 |
https://mathoverflow.net/questions/303099 | 5 | Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or at least a category with weak equivalences, such that its $\infty$-categorical localization is the homotopy limit of th... | https://mathoverflow.net/users/7108 | Homotopy limit of model categories in the category of categories | Philippe Gaucher is right. This problem was solved by Julie Bergner, [here](https://arxiv.org/abs/1010.0717). I [recently asked a question](https://mathoverflow.net/questions/298833/homotopy-pullback-of-quillen-equivalence) that summarized some of her work on this problem. The point is that the homotopy limit of your d... | 2 | https://mathoverflow.net/users/11540 | 303225 | 132,756 |
https://mathoverflow.net/questions/303232 | 6 | I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions could be asked over other fields, and they may even be more interesting in that situation. Let $M\_g$ denote the moduli sp... | https://mathoverflow.net/users/102390 | Rationality of the moduli space of genus g curves | Your question is a plausible guess; indeed, Severi conjectured that $\mathcal{M}\_g$ is unirational for all $g$. He proved this conjecture for $g\leq 10$. Sernesi was first to prove unirationality of $\mathcal{M}\_g$ in genus 12, then Chang and Ran proved it in genera 11 and 13, Verra in genus 14. Bruno and Verra have ... | 7 | https://mathoverflow.net/users/nan | 303244 | 132,761 |
https://mathoverflow.net/questions/303227 | 8 | Let $G/H\cong PSL(2,11)$, and $\theta$ be an irreducible $\mathbb{C}$-character of $H$. Suppose $\theta$ is invariant in $G$ and $\theta(1)=9$.
>
> Question: Is $\theta$ extendible to $G$?
>
>
>
| https://mathoverflow.net/users/99750 | Is a $G$-invariant character $\theta$ of $H$ extendible to $G$? | The answer is yes (and some of the comments were moving in the right direction): Let $T$ be a transversal to $H$ in $G,$ and let $\sigma$ afford the representation of $H.$ For each $t \in T,$ there is a matrix $M\_{t} \in {\rm GL}(9,\mathbb{C})$ such that $\sigma(tht^{-1}) = M\_{t}\sigma(h)M\_{t}^{-1}$ for all $h \in H... | 12 | https://mathoverflow.net/users/14450 | 303257 | 132,764 |
https://mathoverflow.net/questions/303239 | 2 | Let $X$ be a smooth, projective curve of genus at least $2$, $x\_1, x\_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic bundle of rank $2$ on $X$, we mean a triple $(E,F\_1(E),(0,\alpha))$, where $E$ is a rank $2$ vector bundle on $E$ and $F... | https://mathoverflow.net/users/38832 | When is the moduli of generalized parabolic bundles with fixed determinant smooth? | It seems to me that the definition of generalized parabolic structure given in the question differs from the one given in the following paper (a GPB should fix a flag in the global sections of a twist of the vector bundle):
* U. Bhosle. Generalized parabolic bundles and applications. II. Proc. Indian Acad. Sci. Math... | 1 | https://mathoverflow.net/users/50846 | 303263 | 132,767 |
https://mathoverflow.net/questions/303259 | 4 | Given $\cal{A},\cal{B}$ two dense $\*$-algebras of two $C^\*$-algebras $A$ and $B$ respectively, together with a $\*$-algebra homomorphism $f:\cal{A} \to \cal{B}$, is it clear that $f$ extends to a bounded linear operator $f:A \to B$?
| https://mathoverflow.net/users/125790 | Extending maps from dense $*$-algebras of $C^*$-algebras | I believe this is a counter-example.
Let $\newcommand{\mc}{\mathcal}\mc A$ be the algebra of complex polynomials restricted to $[0,1]$, with closure $A=C[0,1]$. Let $X\in\mc A$ be the coordinate function, $X(t)=t$ for $t\in[0,1]$, so $X$ generates $\mc A$ as a $\*$-algebra.
Let $H$ be a Hilbert space and $x\in\mc B... | 8 | https://mathoverflow.net/users/406 | 303264 | 132,768 |
https://mathoverflow.net/questions/303156 | 1 | Let $(X\_{t})\_{t\geq 0}$ be a Bessel Process starting at $x>0$ of dimension $\delta>0$. Namely
\begin{align\*}
X\_{t}=x+W\_{t}+\frac{\delta-1}{2}\int\_{0}^{t}\frac{1}{X\_{s}}\, ds.
\end{align\*}
where $(W\_{t})\_{t\geq 0}$ is a Brownian Motion.
I am interested in how to find the distribution of the Hitting Time $\ta... | https://mathoverflow.net/users/115081 | Obtaining the distribution of the First Hitting time of the Bessel Process | Since the Bessel process has Brownian scaling, we have
$$
\mathbb{P}^x(\tau<t)=F(xt^{-\frac12})
$$
for some unknown function $F$, where $\mathbb{P}^x$ denotes the probability for the Bessel process started from $x$. Now, either from Fokker-Plank equation, or by Ito calculus, or conditioning on the exit time and positi... | 5 | https://mathoverflow.net/users/56624 | 303267 | 132,769 |
https://mathoverflow.net/questions/303245 | 9 | For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{\*})^\*$, for $\*$ denoting dual? If so, what is a good example.
| https://mathoverflow.net/users/125790 | The dual of a dual in a rigid tensor category | As Tobias said in his answer, a good place to look for examples is in endofunctor categories with composition as the monoidal product, where duals are adjoints. But another way to get a rigid monoidal category of such is to just restrict to the full subcategory of endofunctors $F = F\_0$ that fit into some infinite adj... | 9 | https://mathoverflow.net/users/49 | 303268 | 132,770 |
https://mathoverflow.net/questions/303199 | 6 | If I understand my PL topology correctly (and please correct me if I don't), if $K$ is a $k-$complex and $n\ge 2k+2$, then any two PL embeddings $a,b\colon K\to \mathbb{R}^n$ are isotopic. Therefore, the regular neighborhoods of $a(K)$ and $b(K)$ are homeomorphic, and we can think about them as "the" regular neighborho... | https://mathoverflow.net/users/25051 | Can one construct a regular neighborhood without an ambient space? | One might think of the abstract regular neighbourhood as a canonical way of thickening $K$ to a handle decomposition of sufficiently high dimension.
If $\dim K = 1$, then $K$ has a canonical $n$-dimensional thickening $M$ as soon as $n\ge 3$, where vertices and edges are replaced by 0-handles and 1-handles. Of cours... | 4 | https://mathoverflow.net/users/6205 | 303271 | 132,771 |
https://mathoverflow.net/questions/303224 | 5 | I would like to ask is there a computer program for counting graph homomorphisms?
| https://mathoverflow.net/users/17787 | Computer program for counting graph homomorphisms | **EDIT:** I have since found that the Digraphs package sometimes counts homomorphisms incorrectly. Perhaps this problem has been fixed in more recent versions though. I use Minion for counting homomorphisms instead.
The [Digraphs](https://gap-packages.github.io/Digraphs/) package for GAP has several functions for fin... | 9 | https://mathoverflow.net/users/18606 | 303284 | 132,775 |
https://mathoverflow.net/questions/303260 | 4 | Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure preserving (called automorphism).
Is it true that there exists a sequence of continuous measure-preserving transforma... | https://mathoverflow.net/users/49870 | automorphisms of a measurable space can be approximated by continuous measure preserving maps? | In fact, sequences of continuous piecewise linear transformations of $[0,1]$ suffice to approximate a.e. any measure preserving transformation of $[0,1]$ as proved [in thm 2.1 in this paper](https://www.emis.de/journals/HOA/IJMMS/Volume13_2/708257.pdf) (therefore in measure, by Severini-Egorov theorem; but in fact in t... | 5 | https://mathoverflow.net/users/6101 | 303288 | 132,776 |
https://mathoverflow.net/questions/303269 | 11 | Let $G$ a complex reductive algebraic group, $X$ be a smooth compact complex curve. The moduli stack $Bun\_G$ of principal $G$-bundles on $X$ is generally speaking not quasi-compact (so we do not have Verdier duality). However, Drinfeld has conjectured that a different functor, $Ps\text{-}Id\_{Bun\_G, !}$, gives us an ... | https://mathoverflow.net/users/nan | Decategorification of Gaitsgory's strange functional equation? | This is essentially the question studied in <https://arxiv.org/abs/1503.04705v4>.
A (possibly wrong) comment: Unlike $\operatorname{Eis}\_\*$, which decategorifes (under Drinfeld-Wang's conventions) to the function-theoretic Eisenstein series, there is no obvious decategorification of $
\operatorname{Eis}\_!.$ The st... | 1 | https://mathoverflow.net/users/51424 | 303289 | 132,777 |
https://mathoverflow.net/questions/291058 | 10 | Let us consider a bounded, Borel function $F\colon \mathbb R^d \to \mathbb R^d$. Assume it satisfies the following
[Osgood-like](https://www.encyclopediaofmath.org/index.php/Osgood_criterion) condition:
$$\tag{O}
\boxed{\vert \langle F(x) - F(y), x-y \rangle\vert \le \Vert x-y \Vert \rho( \Vert x-y \Vert)} \qquad \fo... | https://mathoverflow.net/users/119793 | Does this Osgood-like condition imply continuity? | You do not need such heavy high-tech as Korn's inequality or even Lebesgue measure theory for an elementary geometry homework. Let's say $F(0)=0$.
The first claim is that $F$ is bounded in some neighborhood of the origin. Indeed, just choose finitely many points $x\_j$ such that $x\_i-x\_j$ span the space. Then for ... | 11 | https://mathoverflow.net/users/1131 | 303300 | 132,780 |
https://mathoverflow.net/questions/302915 | 5 | Let $C\_n$ be the $n$-th Catalan Number and let $\mathcal{O}\_{s,j} = {{2s-j-1}\choose{j}} C\_{s-j}^2$. Then we want to consider $\mathcal{E}\_s = \sum\_{j=0}^{s-1} (-1)^j\mathcal{O}\_{s,j}$. We want to show that
$$\frac{\mathcal{E}\_s}{C\_s^2} \to 0 {\text { as }} s\to \infty.$$
Does anyone have any ideas of how... | https://mathoverflow.net/users/36934 | Show a sequence of sums involving Catalan Numbers converges | By 'magic' and a computer (see the book "A=B" by Petkovsek, Wilf and Zeilberger <https://www.math.upenn.edu/~wilf/AeqB.html>) the numbers $\mathcal{E}\_s$ satisfies the recurrence
$\sum\_{k=0}^3 P\_k(s) \mathcal{E}\_{s+k} = 0$, with polynomials $P\_k$ given by
$P\_0(s) = 2 s^3+s^2-8 s+5$, $P\_1(s) = -26 s^3-93 s^2-82 s... | 11 | https://mathoverflow.net/users/65801 | 303322 | 132,786 |
https://mathoverflow.net/questions/303331 | 1 | Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^\*$, we know that the decomposition of $V \otimes V^\*$ must contain a copy of the trivial module. Is it true that it will only ever contain a single copy of the trivial repr... | https://mathoverflow.net/users/125790 | Abstracting the properties of the category $\frak{g}$-modules | $\newcommand{\C}{\mathcal{C}}$
To expand on Victor's comment: you don't actually need $\C$ to be semi-simple (let's assume it's abelian to be safe). By definition the multiplicity of $1\_C$ in $V\otimes V^\*$ is the dimension of $$Hom\_\C(1\_\C,V\otimes V^\*)\cong Hom\_\C(V,V).$$
Now Schur Lemma is valid at this leve... | 2 | https://mathoverflow.net/users/13552 | 303336 | 132,791 |
https://mathoverflow.net/questions/303321 | 11 | Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{const}$ on each electrode). And the rest of the boundary is insulating material $du/d\vec n=0$ (Neumann BC). The electr... | https://mathoverflow.net/users/73701 | Which matrices can be realized as the Dirichlet-to-Neumann map for a given domain? | Let $A$ be this matrix. Because of the formula
$$\int\_D\sigma\nabla u\cdot\nabla v\, dx=\sum\_{i,j}a\_{ij}U\_iI\_j,$$
($U$ for voltages of $u$, $I$ for currents of $v$), we see three necessary conditions:
* the matrix must be symmetric,
* it must be positive semi-definite,
* and $A{\bf1}=0$.
Actually, if $U\ne \mu... | 9 | https://mathoverflow.net/users/8799 | 303337 | 132,792 |
https://mathoverflow.net/questions/303265 | 7 | Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$
for each object $X$ in $\cal{C}$.
Denoting this functor by
$$
hom\_X(-):\cal{C} \to \cal{C}
$$
it is tempting to ask if th... | https://mathoverflow.net/users/125790 | Enrichments vs Internal homs | As Mike says, it is indeed the case that a closed monoidal structure gives rise to a self-enrichment of a category. This is an often-used fact.
Here's an example of "wrong-way" self-enrichment: the category $\mathsf{Cat}$ of small categories is enriched in itself in the usual way since it is cartesian closed. But it ... | 4 | https://mathoverflow.net/users/2362 | 303338 | 132,793 |
https://mathoverflow.net/questions/281349 | 1 | Suppose that there exist continuous maps $f:Y\longrightarrow Z$ and $g:Z\longrightarrow Y$ so that $g\circ f\simeq 1\_Y$. Also, let $h :X\longrightarrow Y$ be a continuous map which induces an isomorphism of
fundamental groups. We know that $M\_{h}\simeq Y$ and $M\_{f\circ h}\simeq Z$, where $M\_{h}=\frac{X\times I \cu... | https://mathoverflow.net/users/114580 | Homotopy domination of mapping cylinders with a special condition | I think you can do the following : factor your map $F\colon M\_h\to M\_{f\circ h}$ as the inclusion $i\colon M\_h\hookrightarrow M\_h\cup\_Y M\_f$ followed by a map $q\colon M\_h\cup\_Y M\_f\to M\_{f\circ h}$. Then, construct a retract $r\colon M\_h\cup\_Y M\_f\to M\_h$. It is then enough to prove that the map $M\_h\cu... | 3 | https://mathoverflow.net/users/94123 | 303345 | 132,795 |
https://mathoverflow.net/questions/303273 | 2 | Let $S$ be a [numerical semigroup](https://en.m.wikipedia.org/wiki/Numerical_semigroup). Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the *Frobenius number* of $S$; let $g(S)=|\mathbb N \setminus S|$ be the *genus* of $S$; and let $m(S)=\min (S \s... | https://mathoverflow.net/users/nan | On a generating set of numerical semigroups of multiplicity three | Yes, this is **true** and it follows immediately from
>
>
> >
> > **[1, Corollary 4]** Two numerical semigroups with multiplicity three are equal if and only if they have the same Froebenius number and the same gender.
> >
> >
> >
>
>
>
together with
>
>
> >
> > **[1, Lemma 6]** Let $S$ be a numeric... | 0 | https://mathoverflow.net/users/84349 | 303350 | 132,796 |
https://mathoverflow.net/questions/303361 | 6 | Laver introduced the concept of termspace forcing. If $\mathbb P \* \dot{\mathbb Q}$ is a two-step iteration, then we can order the $\mathbb P$-names for elements of $\dot{\mathbb Q}$ by putting $\dot q\_1 \leq \dot q\_0$ when $1 \Vdash \dot q\_1 \leq \dot q\_0$. This defines the termspace partial order $T(\mathbb P,\d... | https://mathoverflow.net/users/11145 | distributivity of termspace forcing | Here is a ZFC counterexample.
Let $\mathbb{P}$ be the forcing to add a Cohen real $c$, and let $\omega\_1=\bigsqcup\_n S\_n$ be a partition of $\omega\_1$ into disjoint stationary sets $S\_n$. Let $\dot{\mathbb{Q}}$ be the forcing to kill the stationarity of all $S\_n$, for $n\in c$, the generic Cohen real added by ... | 11 | https://mathoverflow.net/users/1946 | 303374 | 132,804 |
https://mathoverflow.net/questions/303382 | 1 | I am working with a space with the following properties, and want to know if it is necessarily metrizable.
* countable union of compact nowhere dense sets
* T$\_4$$=$T$\_1$+normal
* separable
* Lindelöf
I ran this through pi-base and found no examples which were not metrizable. For what it's worth, if the first pro... | https://mathoverflow.net/users/95718 | Does this collection of properties imply metrizable? | The space $X=\mathbb Q\times\{0,1\}^\mathfrak c$ is a counterexample.
$X$ is the union of the compact nowhere dense sets $\{r\}\times\{0,1\}^\mathfrak c,\ r\in\mathbb Q.$
$X$ is Lindelöf because it's $\sigma$-compact.
$X$ is $\text T\_3$ because it's a product of $\text T\_3$ spaces.
$X$ is $\text T\_4$ (and pa... | 4 | https://mathoverflow.net/users/43266 | 303383 | 132,807 |
https://mathoverflow.net/questions/303394 | 1 | Here is a naïve question: Let $K$ an absorbing, symmetric and convex set of a vector space $X$ that contains 0 that is bounded in the sense that for any direction $x\not=0$, there exists some $n$ such that $nx \not \in K$. Then the Minkowski functional $p\_K(x) = \inf \{ \lambda>0: \lambda^{-1} x \in K \}$ defines a no... | https://mathoverflow.net/users/49265 | Criteria for Minkowski functionals to induce a complete space? | There is a simple sufficient condition---that there exists a suitable locally convex topology on the space which is weaker than your norm topology and for which $K$ is complete. This is the Grothendieck completeness theorem.
| 2 | https://mathoverflow.net/users/124227 | 303398 | 132,813 |
https://mathoverflow.net/questions/303391 | 4 | Based of the detailed attempt to solve the integral $\int e^{\sin(x)} dx$ I stumbled upon a connection between modified Struve and modified Bessel function of the first kind. But, I cannot find a confirmation in the literature, and the connection looks too nice to be easily missed.
The integral can be shifted to $e^{... | https://mathoverflow.net/users/nan | Is $\frac{\pi}{4}L_0(z) = \sum\limits_{n=1}^{+\infty} (-1)^{n+1} \frac{I_{2n-1}(z)}{2n-1}$ between Bessel and Struve known? | This relation is a special case of a more general one:
$$L\_\nu(z)=\frac{4}{\sqrt{\pi}\,\Gamma\left(\nu+\frac{1}{2}\right)}\sum\_{n=0}^\infty\frac{(-1)^n\,(2n+\nu+1)\,\Gamma(n+\nu+1)}{n!\,(2n+1)(2n+2\nu+1)}\,I\_{2n+\nu+1}(z),$$ which can be found in <https://arxiv.org/abs/1301.5432> (Integral representations and summat... | 5 | https://mathoverflow.net/users/32389 | 303407 | 132,814 |
https://mathoverflow.net/questions/303406 | 11 | Let $X$ be a complex algebraic variety. We can ask if $X$ is *normal* as an algebraic variety, but also, if its analytification is normal as a complex analytic space. Is there a relationship between the two?
Do we have
$$\text{algebraic normality} \implies\text{analytic normality}$$
or
$$\text{analytic normality} \i... | https://mathoverflow.net/users/125883 | Algebraic vs analytic normality | Over $\mathbf{C}$, algebraic normalization and analytic normalization are equivalent concepts. See
N. Kuhlmann: [Die Normalisierung komplexer Räume](http://dx.doi.org/10.1007/BF01451331), *Math. Ann.* **144** (1961), 110-125, [ZBL0096.27801](https://zbmath.org/?q=an:0096.27801).
Quoting directly from Satz 4, p. 12... | 15 | https://mathoverflow.net/users/7460 | 303409 | 132,815 |
https://mathoverflow.net/questions/303415 | 1 | It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum\_{n\geq 0} a\_nx^n$ where $(a\_n)\_n$ is solution of a linear recurrence with constant coefficients of length equal to the degree of the denumerator, and conversely.
My question regards similar properties o... | https://mathoverflow.net/users/24309 | Generating series of rational$\times \exp($rational$)$ | We have $$ \frac{d}{dx}\left ( f(x) e^{h(x)}\right) = \left( \frac{ f'(x)}{f(x)} + h'(x) \right) \left( f(x) e^{h(x)}\right).$$
Thus
$$ \sum\_n a\_n n x^{n-1} = \left( \frac{ f'(x)}{f(x)} + h'(x) \right) \sum\_n a\_n x^n .$$
If you express $\frac{ f'(x)}{f(x)} + h'(x)$ as a ratio of rational functions, put the de... | 9 | https://mathoverflow.net/users/18060 | 303416 | 132,817 |
https://mathoverflow.net/questions/303434 | -2 | While reading some (relatively old) papers on group-theory I encountered the following notations whose meanings I cannot understand:
If $W= G \wr H$ is the (unrestricted) wreath product of $G$ and $H$, $f$ an element of the basis $G^H$ and $h$ an element of $H$, what do
$f^H$ and $h^W$ mean?
Just for the sake of ... | https://mathoverflow.net/users/124793 | Notation on wreath product | If the action is written in exponential notation, then $f^H = \{f^h \mid h\in H\}$. Similarly $W$ acts on itself by conjugation which is also frequently written in exponential notation so that $h^W = \{w^{-1}hw \mid w\in W\}$. It is maybe better not to use $G^H$ for the set of all set maps $H\to G$ in this situation an... | 1 | https://mathoverflow.net/users/3041 | 303435 | 132,820 |
https://mathoverflow.net/questions/303445 | 2 | Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a subfield $K'\subset K$ which is complete with respect to some valuation $v'$, possibly different from the one induced from $K... | https://mathoverflow.net/users/88385 | Subfields of higher local fields | The answer to both questions is no, even in the case of higher local fields.
Q1. Let us take $K = \mathbb Q\_p((T))$ and $K' = \mathbb Q\_p$. Then $K'$ is a local subfield of $K$ but there is only one embedding of $K'$ in $K$, and as a subfield of $K$, $K'$ is not a nice local subfield.
Q2. Take again $K=\mathbb Q... | 1 | https://mathoverflow.net/users/88385 | 303446 | 132,824 |
https://mathoverflow.net/questions/303452 | 4 | Using: <https://arxiv.org/pdf/math/9910179.pdf> as a reference...
My question involves spelling out explicitly the comment in 4.2 -
"Equivalently, the datum of an $A\_\infty$-structure on a graded space $M$ is the datum of a differential $m\_1$ on $M$ and of a morphism of $A\_\infty$-algebras from $A$ to the opposi... | https://mathoverflow.net/users/124286 | A-infinity modules | Let $(A,(m\_i)\_i)$ be an $A\_\infty$-algebra and $(M,d\_M)$, a (co)chain complex.
$Q1$: We endow $L:=Hom\_k^\*(M,M)$ with an $A\_\infty$-structure where
$$
m\_1^L(f):=d\_M\circ f - (-1)^{\text{deg} f} f\circ d\_M
$$
$$
m\_2^L(f\_1\otimes f\_2):=f\_2\circ f\_1
$$
$$
m\_i^L=0 \text{ for all }i\geq3
$$
Now we s... | 3 | https://mathoverflow.net/users/124286 | 303468 | 132,836 |
https://mathoverflow.net/questions/303482 | 6 | Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}\_q$. Let $\pi\in V$ be a uniformizer.
For any integer $d,n\ge 0$, define:
$${\pi^d \choose n} := \frac{\pi^d\cdot(\pi^d -1)\cdot\ldots\cdot(\pi^d-n+1)}{n!}.$$
>
> * Is ${\pi^d\choose n}$ an element of $V$?
> * For ... | https://mathoverflow.net/users/nan | Binomial coefficients in discrete valuation rings | If $p=2=n$, $d=1$, $V=\mathbf{Z}\_2[\sqrt{2}]$ and $\pi=\sqrt{2}$, then
$$ \binom{\pi^d}{n} = \frac{\sqrt{2}(\sqrt{2} - 1)}{2} = 2^{-1/2}\cdot \mathrm{unit} \notin V. $$
| 6 | https://mathoverflow.net/users/3847 | 303483 | 132,840 |
https://mathoverflow.net/questions/303457 | 7 | Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.
>
> QUESTION: Prove $G\cong A\_4\times A\_4$.
>
>
>
By using Magma, we know there is only one group of order $144$ with an irreducible $\mathbb{C}$-character $\theta$ of degree $9$... | https://mathoverflow.net/users/99750 | On the structure of a finite group of order $144$ | Since $G$ has an irreducible character of degree $9 = |G|\_{3},$ we have $O\_{3}(G) = 1,$ so $G$ has more than one Sylow $3$-subgroup. If $G$ has only $4$ Sylow $3$-subgroups, then $G$ has a normal subgroup of order divisible by $6$ with a normal Sylow $3$-subgroup ( consider the permutation action of $G$ on $N\_{G}(S)... | 11 | https://mathoverflow.net/users/14450 | 303491 | 132,842 |
https://mathoverflow.net/questions/303425 | 10 | A friend of mine, obtained a lower bound for the trace norm of matrices described in [this question](https://mathoverflow.net/questions/302424/is-this-lower-bound-for-a-norm-of-some-complex-matrices-true) (for the special case $a\_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where
$$
f(n) := \int\_0^\infty ... | https://mathoverflow.net/users/53059 | Asymptotic behavior of an integral depending on an integer | This is an improvement of my previous post. I claim that
$$4\pi n-6\pi<f(n)<4\pi n-2\pi.$$
Starting from
$$f(n)-2\pi(n-1)=\int\_0^\infty \log\left(\frac{1+t}{2}+\frac{1+t}{2}\left(\frac{1-t}{1+t}\right)^n+n(n-1)\frac{t}{1+t}\right)\,t^{-3/2}\,dt,$$
we see that
$$f(n)-2\pi(n-1)<\int\_0^\infty \log\bigl(1+n^2 t\bigr)\,t^... | 7 | https://mathoverflow.net/users/11919 | 303494 | 132,843 |
https://mathoverflow.net/questions/303492 | 4 | Are metrizable subspaces of separable spaces separable?
Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.
| https://mathoverflow.net/users/125920 | Metrizable subspaces of separable spaces | The answer is no, if we interpret separable as "has a countable dense subset" (Fedor Petrov's answer appears to have interpreted it as possessing a countable base). Consider the [Moore plane](https://en.wikipedia.org/wiki/Moore_plane), or Niemytzki tangent disc topology, as it is called on page 100 of Steen & Seebach's... | 8 | https://mathoverflow.net/users/61785 | 303497 | 132,845 |
https://mathoverflow.net/questions/303379 | 11 | If $T$ is a countable complete first-order theory with infinite models, the number of countable models it has, $I(T,\omega)$, must be an element of $N=\{1,3,4,5,6,7,\dots,\omega,\omega\_1,2^\omega\}$ (although we don't know if $\omega\_1$ can happen). For which pairs $n,m\in N$ does there exist a countable complete the... | https://mathoverflow.net/users/83901 | Can the number of countable models of a complete first-order theory decrease after adding constants? | I believe that the number of models can indeed decrease. The following seems to be an example of $5 \to 3$:
Start with $T$ the model companion of the theory of "valued trees": the language has two sorts $M$ and $\Gamma$, $\Gamma$ is equipped with a linear order $\leq\_{\Gamma}$ and is a model of DLO. The sort $M$ has... | 9 | https://mathoverflow.net/users/19534 | 303502 | 132,847 |
https://mathoverflow.net/questions/303496 | 1 | Originally [asked](https://math.stackexchange.com/questions/2817700/compactness-of-operators-and-norming-sets) on MSE.
Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.
Let $D\subset \overline{B}\_{F^{\ast}}$ be norming, i.e., there is $r>0$ such that $\sup\limits\_{v\in D}|\left<f,v\right>|\... | https://mathoverflow.net/users/53155 | Compactness of operators and norming sets | If I understand you correctly, the following can be considered as a proof of the converse.
(1) We can assume that $D$ is closed, convex, and symmetric about zero (using Krein-Smulian theorem, see Dunford-Schwartz, volume 1, page 434).
(2) Under this assumption, by the result of Dixmier (Duke Math. J., 1948), the co... | 2 | https://mathoverflow.net/users/37822 | 303507 | 132,850 |
https://mathoverflow.net/questions/303343 | 0 | Let $X, Y$ be smooth, projective varieties, $L\_X$ and $L\_Y$ are very ample line bundles on $X$ and $Y$, respectively. If I understand correctly, $\mbox{pr}\_1^\*L\_X \otimes \mbox{pr}\_2^\*L\_Y$ is a very ample line bundle on $X \times Y$, where $\mbox{pr}\_1, \mbox{pr}\_2$ are the natural projections from $X \times ... | https://mathoverflow.net/users/38832 | Fiber product of projective varieties and ample line bundles | Let $D\_X$ and $D\_Y$ be the Chern classes of $L\_X$ and $L\_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr\_{1\*}pr\_1^\*D\_X=0$ and $pr\_{2\*}pr\_2^\*D\_Y=0$. Applying the projection formula we obtain:
$$pr\_1^\*D\_X^{\dim X+1}=D\_X^{\dim X}\cdot pr\_{1\*}pr\_1^\*D\_X=0$$
$$pr\_2^\*D\_Y^{\dim Y+1}=D\_X^... | 2 | https://mathoverflow.net/users/98256 | 303510 | 132,852 |
https://mathoverflow.net/questions/303477 | 1 | Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I\_U = \{A \mid A \cap U \neq \emptyset\}$ with $U$ ranging over open subsets of $\mathbf{X}$ together with sets of the form $D\_K = \{A \mid A \cap K = \emptyset... | https://mathoverflow.net/users/15002 | Compactness of the Fell topology and local compactness | Local compactness means that every point has a compact neighborhood. That is, every point is interior to a compact set. The Fell topology on the nonempty closed sets of a Hausdorff space is always compact, without any further assumption. However, local compactness is equivalent to the Fell topology being Hausdorff.
Y... | 3 | https://mathoverflow.net/users/35357 | 303513 | 132,854 |
https://mathoverflow.net/questions/303520 | -1 | Let two vectors, $\mathbf{x}, \mathbf{y}$ be related as: $0 \leq x\_i \leq \lambda y\_i$, for some $\lambda > 0$. That is, $\mathbf{x}$ is coodinate-wise dominated by a scaled version of $\mathbf{y}$.
Also let $\mathbf{M}$ be a positive semi-definite matrix. My question is, how are their matrix $\mathbf{M}$ induced w... | https://mathoverflow.net/users/125930 | Bound for psd-matrix weighted norm of two related vectors | This is trivially false even in one dimension. Let $M = [1], \lambda = 1$. Then $x = -2, y = 1$ provides a counterexample.
Now let's change the condition to $|x\_i| \le |\lambda y\_i|$ coordinate-wise. Then $x'\mathbf{M}x \leq \lambda^2 y'\mathbf{M}y$ is true if $M$ is diagonal positive semidefinite (I changed to usi... | 0 | https://mathoverflow.net/users/75420 | 303526 | 132,857 |
https://mathoverflow.net/questions/303471 | 2 | I'm a graduate student doing research on time-frequency analysis. I am considering the existence of a certain frame-like inequality. Let $H: L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be a compact self-adjoint operator, and for each $(x,\omega)\in \mathbb{R}^{d} \times \mathbb{R}^d,$ let $\pi(x,\omega): L^2(\math... | https://mathoverflow.net/users/119788 | Insights about a frame-like inequality | Yes, your starting inequality can only hold if $H$ has finite dimensional range. In fact, the dimension can be at most $q$.
To see this, let $Y$ denote the range of $H$. Your inequality implies (why?) that if $y \in Y$ with $\langle y, \pi(x\_n, \omega\_n) g\rangle =0$, then $y =0$. In other words, the linear map
$$
... | 1 | https://mathoverflow.net/users/59219 | 303527 | 132,858 |
https://mathoverflow.net/questions/288445 | 6 | A referee of a submitted paper requested details on the statement that $\int\_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, I think the simplest way to address the mentioned request would be just to refer to the following well-known, folklor-ish,... | https://mathoverflow.net/users/36721 | Mixtures of log-convex functions are log-convex: a reference | Late to the party, those are all good points made in the comments above. I just came across a reference, [Kingman1961](https://academic.oup.com/qjmath/article-abstract/12/1/283/1534893?redirectedFrom=PDF), "A CONVEXITY PROPERTY OF POSITIVE MATRICES", The Quarterly Journal of Mathematics, Volume 12, Issue 1, 1 January 1... | 4 | https://mathoverflow.net/users/1877 | 303538 | 132,862 |
https://mathoverflow.net/questions/261839 | 7 | In the paper ["Computing Persistent Homology"](https://doi.org/10.1007/s00454-004-1146-y) by Zomorodian and Carlsson, it is stated as Theorem 3.1 that:
>
> The correspondence $\alpha$ defines an equivalence of categories between the category of persistence modules of finite type over $R$ and the category of finitel... | https://mathoverflow.net/users/83274 | Correspondence between persistence module and graded module over $R[t]$ | Recently, René Corbet and Michael Kerber posted the preprint ["The Representation Theorem of Persistent Homology Revisited and Generalized"](https://arxiv.org/abs/1707.08864) which gives a proof of this statement with elementary methods. Among other things, the authors also give a short discussion on your question.
| 9 | https://mathoverflow.net/users/125939 | 303539 | 132,863 |
https://mathoverflow.net/questions/303537 | 2 | Let $B\_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B\_{\infty}(\Omega,X)$ by the set of all bounded measurable functions $f:\Omega\to X$ (meaning $f^{-1}(O)$ is measurable for every open $O$ in $X$).
>
> Is the space... | https://mathoverflow.net/users/84390 | A formula for vector valued measurable functions | For $\Omega=\mathbb N$ with the disrete $\sigma$-algebra measurability is no condition and $$\ell^\infty(\mathbb N)\hat\otimes\_\varepsilon X \cong C(\beta\mathbb N)\hat\otimes\_\varepsilon X\cong C(\beta\mathbb N,X)$$ is (via restriction to $\mathbb N$ isomorphic to) the space of *pre-compact* sequences in $X$.
| 3 | https://mathoverflow.net/users/21051 | 303544 | 132,865 |
https://mathoverflow.net/questions/303521 | 3 | Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\end{align}
[It is well known that](https://mathoverflow.net/questions/273970/a-cubic-system-with-two-nested-limit-cycles-with-oppos... | https://mathoverflow.net/users/36688 | Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$? | Your vector field can be written in polar coordinates as
$$X=X^r\partial\_r+X^{\varphi}\partial\_{\varphi}=r(4r^2-r^4-3)\partial\_r+(2-r^2)\partial\_{\varphi},$$
which exhibits the rotational symmetry.
The condition $\nabla\_X X=0$ implies that
\begin{align}
X^r\partial\_rX^r+\Gamma^r\_{rr}(X^r)^2+(\Gamma^r\_{r\varphi... | 3 | https://mathoverflow.net/users/69603 | 303545 | 132,866 |
https://mathoverflow.net/questions/258634 | 10 | $K$-theory is often billed as the "universal way to split exact sequences". But it seems we're too anxious to group-complete things to actually take the slogan at face value.
Consider the following $\infty$-categories:
* $\mathcal{W}$ - Waldhausen categories (or Waldhausen $\infty$-categories, if you prefer)
* $\ma... | https://mathoverflow.net/users/2362 | Waldhausen $K$-theory before group completion | I'm not sure about Waldhausen categories in general, but if you restrict attention to **stable** $\infty$-categories (with trivial Waldhausen structure in which all maps are cofibrations) then group completion is essentially forced on you by the splitting of exact sequences. In principle, this happens because for every... | 6 | https://mathoverflow.net/users/51164 | 303546 | 132,867 |
https://mathoverflow.net/questions/182622 | 6 | What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory?
(This is a severely reduced version of the original post, which can be found at my blog <https://tcjpn.wordpress.com/2016/11/... | https://mathoverflow.net/users/12178 | An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes | From 4gravitons' [Post on the Amplitudes 2017 Conference] [1](https://4gravitons.wordpress.com/2017/07/20/more-travel/): "Between the two of them, Nima (Arkani-Hamed) and Yuntao (Bai) covered an interesting development, tying the Amplituhedron together with the string theory-esque picture of scattering amplitudes pione... | 1 | https://mathoverflow.net/users/12178 | 303550 | 132,870 |
https://mathoverflow.net/questions/302972 | 6 | Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f\_n\colon S^{n-1}\to\mathbb R$ by
$$f\_n(x) := \sum\_{w^{n}=1}|x\_1w+ x\_2w^2\cdots+x\_{n-1}w^{n-1}|$$
Denote $z^\*\_n := \min\_{S^{n-1}} f\_n$.
Is it true that for sufficiently large $n$s (which depend on $S$),... | https://mathoverflow.net/users/53059 | An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices | The answer is no if there exists $z \in S$ with $z \neq 1$ and $\vert 1 - z\vert \leq \frac{1}{2}$ and $\bar{z} \in S$ :
Let $$p(x) = \sum\_{k=0}^{n-1} b\_k x^k$$ and $x\_k = p(w^k)$ , where $w=e^{\frac{2 \pi i}{n}}$ .
Then the $n b\_k$ are the eigenvalues of the circulant matrix.
Now choose
$$p(x) = 1 - \frac... | 3 | https://mathoverflow.net/users/17261 | 303555 | 132,873 |
https://mathoverflow.net/questions/303474 | 7 | I asked [this question](https://math.stackexchange.com/q/2811279/660) on Mathematics Stackexchange, but got no answer.
Let $K$ be a field and $n$ a positive integer. To a finite dimensional $K$-vector space $V$, equipped with a family $V\_1,\dots,V\_n$ of subspaces, we attach the map $d:\mathcal P(\{1,\dots,n\})\to\m... | https://mathoverflow.net/users/461 | Prescribing the dimension of intersections of sub-vector spaces | I find this easier to think about if we dualize everything: replace $d(S)$ with $\dim V - d(S)$ and each subspace with its annihilator in $V'$. Recall that the annihilator of a sum is the intersection of the annihilators. Thus your question is the following:
Assume $d:\mathcal{P}(\{1,\dots,n\}) \to \mathbb{N}$ is inc... | 9 | https://mathoverflow.net/users/20598 | 303558 | 132,875 |
https://mathoverflow.net/questions/303518 | 3 | Let $X$ be a smooth, projective variety, $V\_1, V\_2$ smooth, closed subvarieties of the same dimension and $E$ a locally free sheaf on $X$. There exist natural morphisms $$r\_1: H^i\_{V\_1}(E) \to H^i(E) \mbox{ and } r\_2: H^i\_{V\_2}(E) \to H^i(E). $$
Suppose that $V\_1 \cap V\_2=\emptyset$. The question is, for whic... | https://mathoverflow.net/users/38832 | Cohomology with compact support | For $i=0$ this holds essentially by the definition of $H^i\_V$.
For $i>0$ it generally fails. Let $X$ be a curve and $V\_i$ arbitrary, different points. Then the complements $X\setminus V\_i$ are affine and hence both $r\_1$ and $r\_2$ are surjective.
| 6 | https://mathoverflow.net/users/10076 | 303559 | 132,876 |
https://mathoverflow.net/questions/303561 | 5 | It's often said that homotopy type theory should be interpretable in any $\infty$-topos. But really it should be interpretable in any "predicative $\infty$-topos". I'm not quite sure what this means, but the prime example to me of an $\infty$-category which is "almost, but not quite, an $\infty$-topos" is the unstable ... | https://mathoverflow.net/users/2362 | How much homotopy type theory should be modeled by the unstable motivic category? | When you say "really it should be interpretable in any predicative $\infty$-topos", well it depends on what you mean by "homotopy type theory". Homotopy type theory is not a fixed system, but a subject that studies many different type theories. Some of those are conjectured to correspond to $(\infty,1)$-toposes, others... | 7 | https://mathoverflow.net/users/49 | 303563 | 132,877 |
https://mathoverflow.net/questions/301907 | 3 | I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction".
In that notes author defines a notion of generalized map between Lie groupoids.
>
> Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. By a generalized map from $\mathcal{G}$ to $\mathcal{H}$ we mean a homomorphism of Lie groupoi... | https://mathoverflow.net/users/118688 | Necessity/Motivation for generalised homomorpisms | Let me attempt a very simple-minded answer.
Say your objects of interest are orbifolds. You have an orbifold V and you want to describe it through a groupoid, usually an action groupoid $G\ltimes M \rightrightarrows M$. This means that you have an action of $G$ on $M$ and that $V\simeq G\backslash M$, i.e. $V$ is the... | 3 | https://mathoverflow.net/users/6032 | 303567 | 132,879 |
https://mathoverflow.net/questions/299801 | 11 | Consider the symmetric sequence $P\_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma\_n$-action. There is a natural topological operad structure on $P$ given by multiplication. That is,
* Let $p \in \Delta^{n-1}$ be given in coordinates by $p = (p^1,\dots,p^n)$.
* For each $i=1,\d... | https://mathoverflow.net/users/2362 | How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures? | Abusing notation, write m for the uniform distribution on each finite set and define p o p' = m for any p, p' not equal to the identity.
This is possibly a limit of operads of the formula you give where you choose homeomorphisms which tend (pointwise) to a constant function f(x) = c. But I think qualifies as an opera... | 2 | https://mathoverflow.net/users/110 | 303572 | 132,880 |
https://mathoverflow.net/questions/303071 | 3 | By a theorem of Goldman and Tucker it is known that if a linear program (LP) has a finite valued optimal solution, then there is an optimal primal/dual pair $(x,z)$ satisfying not only *complementary slackness* (i.e. $x\_iz\_i=0$), but also *strict complementary slackness* (i.e. *exactly one* of $x\_i$ and $z\_i$ vanis... | https://mathoverflow.net/users/108884 | Strict complementary slackness for semidefinite programs with strong duality | Studying certain semidefinite programs arising in spectral graph theory, I discovered a semi-definite primal/dual pair satisfying strong duality but not strict complementarity.
Let $E=\{12,23,34\}$ (the edge set of the path graph $P\_3$) and $E^{ij}:=(\mathbf e\_i-\mathbf e\_j)^\top(\mathbf e\_i-\mathbf e\_j)$. We us... | 5 | https://mathoverflow.net/users/108884 | 303585 | 132,883 |
https://mathoverflow.net/questions/303579 | 1 | Suppose $M$ is von Neumann algebra, $A$ is $\*$-algebra in $M$, further if $(A)\_1$, the unit ball of $A$ is strong operator closed, does it implies $A$ is von Neumann algebra? I started proving this using Kaplansky density theorem but stuck with the fact I cannot use uniform bounded principle for nets. Please give som... | https://mathoverflow.net/users/125816 | On topology in von Neumann algebras | Yes, on the unit ball the strong operator topology is stronger than the ultraweak topology, hence $(A)\_1$ is also closed in the ultraweak topology (which is the weak$^{\ast}$-topology) of $M$. It follows from Krein-Smulian theorem that $A$ is closed in the ultraweak topology, therefore it is a von Neumann algebra.
| 5 | https://mathoverflow.net/users/24953 | 303586 | 132,884 |
https://mathoverflow.net/questions/303584 | 0 | Take $\mathcal M\_D$ the space of measurable functions from a compact set $D\subseteq \mathbb R\_n$ to ℂ.
I'm wondering if a Stone-Weierstrass-like theorem holds in this space, with the convergence in measure.
In other words, suppose $\mathcal A \subseteq \mathcal M\_D$ is a closed proper subalgebra with identity(c... | https://mathoverflow.net/users/49870 | dense subalgebra in measurable functions set | Take $D = [0,1]$ and let $\mathcal{A}$ be the set of all measurable functions satisfying $f(x) = f(1-x)$ almost everywhere. It satisfies your conditions but contains the function $x(1-x)$ which isn't constant on any set with more than two points.
(This illustrates that "separates points", in the sense of the previou... | 2 | https://mathoverflow.net/users/4832 | 303604 | 132,891 |
https://mathoverflow.net/questions/303478 | 4 | This question has come out while reading J. Moser "*New Aspects in the Theory of Stability
of Hamiltonian Systems*". I'm particularly interested to the Appendix, where one investigates the stability of elliptic fixed points of Hamiltonian dynamical systems, in the time independent case. I start presenting the **framew... | https://mathoverflow.net/users/101308 | Symplectic forms and sign of eigenvalues | I have found the solution myself.
I write the system of linear ordinary differential equations in matrix form:
$$
\begin{pmatrix}
\dot{\vec{x}}(t)\\
\dot{\vec{y}}(t)
\end{pmatrix}
=\mathbb{J}
\begin{pmatrix}
\vec{x}(t)\\
\vec{y}(t)
\end{pmatrix}
$$
where $\mathbb{J}$ is the Jacobian matrix associated to the linear... | 1 | https://mathoverflow.net/users/101308 | 303605 | 132,892 |
https://mathoverflow.net/questions/303571 | 3 | Given some $X, Y\ge 1$ and some $d\le Y$ not a perfect square, is it possible to bound
$$\sum\_{p\le X}\left(\frac{d}{p}\right)?$$
As long as $Y$ is not too large compared to $X$, I would expect that there would be some cancellation as it shouldn't be possible to select some $d$ which is a quadratic residue modulo man... | https://mathoverflow.net/users/40983 | Sum of Legendre symbol over primes | My previous answer addressed character sums for all integers, not primes. Apologies for reading the question too quickly.
Concerning the real question over primes, as the commenters said, it is addressed by the prime number theorem for the given quadratic character. First, one can replace $d$ by its square-free part ... | 3 | https://mathoverflow.net/users/11919 | 303609 | 132,894 |
https://mathoverflow.net/questions/303607 | 4 | Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics?
In particular, under what circumstances can we construct a partition of unity subordinate to some open cover of a compact set in Euclidean space?
| https://mathoverflow.net/users/42302 | Partitions of unity in constructive mathematics | In my PhD thesis [modern intuitionistic topology](https://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf) in chapter 3 there is a comprehensive treatment of partitions of unity within BISH.
In essence the main theorem is that per-enumerable open covers of a metric space admit a subordinate partition of unity... | 5 | https://mathoverflow.net/users/101577 | 303610 | 132,895 |
https://mathoverflow.net/questions/303615 | 1 | I came across the statement ``Every closed and convex subset of a uniformly convex b-metric space is Chebyshev'' in [1]. Here, the term `convex' is in the sense of Takahashi. I tried looking up for the proof of the same without success. I would be grateful if someone can direct me to the proof. Thanks.
[1] H. Fukhar-... | https://mathoverflow.net/users/111987 | Every closed and convex subset of a uniformly convex metric space is Chebyshev? | It's not stated in the paper, but you need the metric space to be complete. This is implicit in all papers on fixed point theory since otherwise the fixed points only live in the completion. With this proviso the claim holds and the proof is the same as in Hilbert space:
Let $(X,d)$ be a complete metric space, and le... | 4 | https://mathoverflow.net/users/327 | 303617 | 132,896 |
https://mathoverflow.net/questions/303612 | 0 | Problem
=======
Given $X \in \mathbb{R}^{n \times n}$ where $X\_{ij} \sim \mathcal{N}(\mu\_{ij}, \sigma\_{ij}^2 I)$
Find the marginal distribution of each eigenvalue, using whatever you can.
Background
==========
In my field, I have a Bayesian inference framework that will obtain the $X$ distribution, but what ... | https://mathoverflow.net/users/120253 | How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution? | Here is a $2 \times 2$ example to consider. Since each entry has its own distribution, let three of them have standard deviation $0$ and the matrix be $\left[ \begin {array}{cc} 0&X\_{12}\\ 1&0\end {array} \right]$ with the mean and standard deviation of $X\_{12}=\mathcal{N}(\mu, \sigma^2)$ to be determined. The eigenv... | 0 | https://mathoverflow.net/users/8008 | 303621 | 132,898 |
https://mathoverflow.net/questions/303395 | 5 | I'm trying to understand the proof of Kantor's Singer cycle theorem, which asserts that if $G$ is a subgroup of $\operatorname{GL}(n,q)$ containing a Singer cycle then $\operatorname{GL}(n/s,q^s) \leq G \leq \operatorname{\Gamma L}(n/s,q^s)$ for some $s|n$ (see [1]). There is (at least) one part of the proof that I'm s... | https://mathoverflow.net/users/20598 | Kantor's Singer cycle theorem | Ah, point I was overlooking is that in this case, $|\Delta|$ must be *prime*: this comes out of the application of Burnside-Schur.
To spell out the rest of the proof (no doubt one way of many), we know that $B^\Delta$ permutes the eigenspaces of $K(\Delta)$ transitively. Since $|B^\Delta|$ is prime this means that th... | 1 | https://mathoverflow.net/users/20598 | 303630 | 132,902 |
https://mathoverflow.net/questions/300872 | 4 | Let $G$ be a Lie group and $C\_r^\*(G)$ and $C^\*(G)$ be its reduced and maximal group $C^\*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map
$$\lambda\_G:C^\*(G)\rightarrow C^\*\_r(G),$$
so $C\_r^\*(G)$ is a quotient of $C^\*(G)$. I am wondering whether $C\_r^\*(G)$ ... | https://mathoverflow.net/users/78729 | Relation between maximal and reduced group $C^*$-algebras | The answer is negative. For a counter-example, consider the free group on two generators $\mathbb F\_2$. It is well known that the full C\*-algebra $C^\*(\mathbb F\_2)$ is residually finite dimensional, RFD for short (meaning that its finite dimensional representations separate points). This property is evidently hered... | 7 | https://mathoverflow.net/users/97532 | 303631 | 132,903 |
https://mathoverflow.net/questions/303634 | 4 | We know from linear algebra that if an $n \times n$ matrix $A$ over a field $k$ is diagonalizable (that is, there exists $P \in GL\_n(k)$ such that $PAP^{-1}$ is a diagonal matrix), then this diagonal matrix is unique up to permutation of diagonal entries.
Are there any analogous results in the literature if we repla... | https://mathoverflow.net/users/94610 | Uniqueness of diagonalizing a matrix over $\mathbb{Z}_{p^k}$ | The answer is yes, that you can get an analogous theorem, because $\mathbb{Z}\_{p^k}$ is a local ring.
The equation $PAP^{-1} = B$ can be rewritten $PA = BP$. Write $A = \text{diag}(a), B = \text{diag}(b), a = \{a\_i\}, b = \{b\_j\}$. Then this implies that for any $i, j$, we have that $P\_{i, j} a\_i = P\_{i, j} b\_... | 5 | https://mathoverflow.net/users/44191 | 303636 | 132,904 |
https://mathoverflow.net/questions/303627 | 1 | I'm trying to characterize equivalence classes of matrices over a vector space.
Specifically, let $V$ be a vector space over a field $K$, let $M \in M\_n(V)$ be an $n \times n$ matrix with entries in $V$ and let $A,B \in M\_n(K)$ be $n \times n$ matrices with entries in $K$. I want to characterize the equivalence cla... | https://mathoverflow.net/users/78871 | Equivalence of matrices over a vector space | If $\dim V = d$, then this is equivalent to the problem of simultaneous similarity of $d$-tuples of $n \times n$ matrices over $K$. I'll prove the equivalence below, but let me first discuss the consequences for your question. The latter is a famous "wild" problem (in the formal sense of representation-wildness, see, e... | 2 | https://mathoverflow.net/users/38434 | 303638 | 132,905 |
https://mathoverflow.net/questions/303599 | 9 | Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does such a description exist for the exceptional Lie algebras
$$
\frak{g}\_2 \subseteq \frak{f\_4} \subseteq\frak{e}\_6 \sub... | https://mathoverflow.net/users/125941 | Young tableaux for exceptional Lie algebras | Young diagrams for the exceptional Lie algebras are considered in the book <https://press.princeton.edu/titles/8839.html> (Group Theory: Birdtracks, Lie's, and Exceptional Groups, by Predrag Cvitanovic).
| 4 | https://mathoverflow.net/users/32389 | 303642 | 132,907 |
https://mathoverflow.net/questions/302129 | 2 | I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book:
"Geometric Theory of Generalized Functions with Applications to General Relativity (Mathematics and Its Applications) (Volume 537)" by Michael Grosser, Michael Kunzinge... | https://mathoverflow.net/users/86432 | Schwartz distributions, Colombeau algebra and applications | For the basics (= Chapter 1) you only need (functional) analysis and distribution theory. Then you also need infinite dimensional analysis (convenient calculus) but that is also contained in Chapter 2. For the geometric setting you need differential geometry but nothing sophisticated. For the applications some knowledg... | 2 | https://mathoverflow.net/users/47189 | 303648 | 132,909 |
https://mathoverflow.net/questions/303644 | 1 | Assume that $M$ is a manifold and $X$ is a vector field on $M$.
Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L\_X \alpha =0$?
| https://mathoverflow.net/users/36688 | Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$ | You could try the vector field $X=x\partial\_x$ on the real projective line, so with affine coordinate $x$. Then in another affine chart $y=1/x$, $X=-y\partial\_y$, so $X$ is smooth everywhere. Every 1-form on the real projective line is closed. An invariant 1-form has to be $\alpha=C dx/x$. To be defined near $x=0$, i... | 7 | https://mathoverflow.net/users/13268 | 303649 | 132,910 |
https://mathoverflow.net/questions/303643 | 6 | Consider a projective representation of $\mathbb Z^2$ with $U(1)$ coefficients. I would like to find the covering group corresponding to this representation. For this, one needs to find the appropriate central extension by $\mathbb Z^2$ of the Schur multiplier $H^2(\mathbb Z^2, C^\times)$. I have some confusion in calc... | https://mathoverflow.net/users/125997 | Calculation of the Schur multiplier of $\mathbb Z^2$ | I think that we have
$$H\_2(\mathbf{Z} \times \mathbf{Z}, \, \mathbf{Z}) \simeq \mathbf{Z}, \quad H^2(\mathbf{Z} \times \mathbf{Z}, \, \mathbf{C}^{\times})\simeq \mathbf{C}^{\times}.$$
In general, by the Universal Coefficient Theorem there is an isomorphism $$H^2(G, \, \mathbf{C}^{\times})=H\_2(G, \, \mathbf{Z})^\*... | 7 | https://mathoverflow.net/users/7460 | 303651 | 132,911 |
https://mathoverflow.net/questions/303659 | 4 | Let $M$ be a smooth manifold, $f$ a smooth function on $M$, and $X$ a vector field on $M$ that ascends $f$, i.e. $df\_p(X\_p)>0$ for all $p\in M$. Are there choices of $(M,f,X)$ such that there is no Riemannian metric $g$ on $M$ in which $X=\nabla\_g f$, where $\nabla\_g$ is the gradient operator on $(M,g)$?
I'm look... | https://mathoverflow.net/users/69603 | Obstructions for vector fields to be gradient fields of a fixed function | Your condition implies that $f$ has no critical points and $X$ has no zeros. On the $(n-1)$-dimensional distribution $D$ defined by the kernel of $df$, choose any metric (in a way depending smoothly on the point). Then extend it to a metric on $M$ by declaring $X$ to be a unit vector orthogonal to $D$.
| 4 | https://mathoverflow.net/users/28128 | 303661 | 132,913 |
https://mathoverflow.net/questions/303505 | 5 | Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a factorization of $A(x)$ of the form
$$\tag{$\star$} \label{fact}
A(x) = C^\top(x^{-1})\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}... | https://mathoverflow.net/users/62673 | Finding a particular matrix factor | This is impossible. Let
\begin{equation\*}
C(x) = \left(
\begin{array}{cc}
a(x) & b(x) \\
c(x) & d(x)
\end{array} \right)
\end{equation\*}
and $\theta(x)=\det C(x)$. From
$$a(x^{-1})d(x)+b(x)c(x^{-1})=x,\,a(x)c(x^{-1})+c(x)a(x^{-1})=0$$
we have
$$a(x^{-1})=\frac{xa(x)}{\theta(x)}.$$
Assuming (1) and (2), the funct... | 1 | https://mathoverflow.net/users/9833 | 303666 | 132,916 |
https://mathoverflow.net/questions/303662 | 2 | Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. Topologically, this manifold is the cone of the Hopf fibration. Is there some other (topological) description of it?
| https://mathoverflow.net/users/114985 | What is symplectic cut of a 4-ball? | The answer is actually written in p.249 of Lerman paper Symplectic cuts. It says that, more generally, symplectic cut of 2n-ball along its boundary is symplectomorphic to $\mathbb{C}P^n.$ In particular, here we get $\mathbb{C}P^2.$
| 2 | https://mathoverflow.net/users/114985 | 303673 | 132,919 |
https://mathoverflow.net/questions/303679 | 5 | Inspired by [this question](https://mathoverflow.net/questions/150835/does-fermats-last-theorem-hold-in-the-ordinals) and its answer, I am curious whether or not Fermat's last theorem holds in the Grothendieck ring of the ordinals under Hessenberg (commutative) operations.
The excellent answer to the other question b... | https://mathoverflow.net/users/92164 | Does Fermat's last theorem hold in the Grothendieck ring of the ordinals? | It does hold. Assume for contradiction that $x^n+y^n=z^n$ is a solution with $x,y,z$ nonzero, and put $w=xyz$. It suffices to find a ring homomorphism $f$ to $\mathbb Z$ such that $f(w)$ is nonzero, as then $f(x)^n+f(y)^n=f(z)^n$ is a nontrivial solution in $\mathbb Z$ contradicting the Fermat–Wiles theorem.
Now, you... | 9 | https://mathoverflow.net/users/12705 | 303688 | 132,922 |
https://mathoverflow.net/questions/303684 | 7 | In one of the very first sentences in Hovey's "Model Categories", Ist chapter, we read that
>
> One can always invert these "weak equivalences"
> formally
> ,
> but
> there
> is a foundational
> problem
> with
> doing
> so,
> since
> the
> class
> of maps
> between
> two objects
> in the
> localiz... | https://mathoverflow.net/users/123432 | Class of maps in localized category may not be a set | When Hovey says that we "can always invert these 'weak equivalences' formally", the weak equivalences he's talking about aren't necessarily weak equivalences in a model category. He's introducing and motivating model categories by saying that they solve a problem: sometimes we'd like to formally invert some class of ar... | 6 | https://mathoverflow.net/users/33143 | 303689 | 132,923 |
https://mathoverflow.net/questions/303641 | 6 | The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\Omega(n^{k/2-1})$.
This approximation is very bad in limit with dimension. For example, if $n$ grows linearly with $k$ then ... | https://mathoverflow.net/users/10668 | The Gauss Circle Problem asymptotic in dimension | This problem was studied by J.E. Mazo and A.M. Odlyzko in [Lattice Points in High-Dimensional Spheres](http://www.dtc.umn.edu/~odlyzko/doc/arch/high.dim.spheres.pdf). If $n$ grows with $k$ faster than linearly then the volume asymptotic still applies. If it increases slower than linearly then for most locations of the ... | 6 | https://mathoverflow.net/users/11260 | 303699 | 132,927 |
https://mathoverflow.net/questions/303618 | 3 | When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about it is restricted to sets of a given sub-sigma-algebra, which is usually not finite. The article does not otherwise use ... | https://mathoverflow.net/users/1445 | Non-probabilist term for conditional expectation? | I quote "Functional Analysis for Probability and Stochastic Processes: An Introduction" by Adam Bobrowski, introduction to Chapter 3 "Conditional expectation":
$\newcommand{\cF}{\mathcal{F}}\newcommand{\cG}{\mathcal{G}}\newcommand{\PP}{\mathbb{P}}$
>
> The space $L^2(\Omega,\cF,\PP)$ of square integrable random var... | 3 | https://mathoverflow.net/users/9652 | 303706 | 132,930 |
https://mathoverflow.net/questions/303681 | 11 | Let $p>2$. I'd like to know the best possible lower and upper bound for $\|x\|\_p$ given that $x\in R^n$ and $\|x\|\_1$, $\|x\|\_2$, and $\|x\|\_\infty$ have fixed values.
It is well-known that
$$\|x\|\_\infty\le \|x\|\_p\le \|x\|\_2^{2/p}\|x\|\_\infty^{1-2/p}~~~
\Big[\le \|x\|\_2\le \|x\|\_1\Big].$$
But these bound... | https://mathoverflow.net/users/56920 | norm inequalities | By rescaling, without loss of generality (wlog) $\|x\|\_\infty=1$.
Let $X$ be a random variable such that $P(X=|x\_i|)=1/n$ for $x=(x\_1,\dots,x\_n)$ and all $i=1,\dots,n$. Then
\begin{equation\*}
\|x\|\_r^r=nm\_r,\quad m\_r:=EX^r,
\end{equation\*}
for $r\ge1$. Let us find the best possible bounds on $m\_p$ in term... | 10 | https://mathoverflow.net/users/36721 | 303713 | 132,933 |
https://mathoverflow.net/questions/303719 | 5 | A link to wikipedia for rough pat theory is: <https://en.wikipedia.org/wiki/Rough_path>
It appears path and signatures has one to one mapping in many cases. I understand that the signature is not dependent on initial condition and if there is any retraces or double back on itself in the path it cancels out in the signa... | https://mathoverflow.net/users/71105 | Under what condition we get back path from signatures in rough path theory? | Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside the circle, maybe up to a factor of $\frac{1}{2}$ or something.) What does get cancelled out is any segment where the pa... | 6 | https://mathoverflow.net/users/4832 | 303721 | 132,934 |
https://mathoverflow.net/questions/303723 | 6 | Let $n > 1$ and $m > 0$ be two integers and $P\_n$ be the $n^{th}$ prime.
>
> Prove: $$P\_{n+m} \ge P\_n + P\_m .$$
>
>
>
Can you give a hint, reference, comment, or proof?
| https://mathoverflow.net/users/122662 | Prove: If $P_n$ is $n$-$th$ prime number then $P_{n+m} \ge P_n+P_m$ | This is an expanded version of my previous answer. It shows, among other things, that the OP's conjecture contradicts the [$k$-tuple conjecture](http://mathworld.wolfram.com/k-TupleConjecture.html) for $k=459$.
**1.** Let $r\geq 0$ be a fixed integer. I claim that the following two statements are equivalent (for inte... | 13 | https://mathoverflow.net/users/11919 | 303726 | 132,935 |
https://mathoverflow.net/questions/301416 | 5 | Please accept apology if this question is vague. (Would you please comment rather then downvote, I may be stopped to ask more questions. I will delete my question if required.)
It is related to the link which describes white noise theory to deal with stochastic differential equation.
<https://www.duo.uio.no/bitstream... | https://mathoverflow.net/users/71105 | Is it possible to compare Rough path theory and White noise Theory? | I looked briefly at the paper you linked - it looked to be about Malliavin calculus.
Yes, both Malliavin calculus and rough path theory have these iterated integrals. However what those integrals mean are slightly different. In Malliavin calculus those iterated integrals are usually defined probabilistically - i.e. ... | 4 | https://mathoverflow.net/users/nan | 303734 | 132,938 |
https://mathoverflow.net/questions/303745 | 4 | Let $H\_n : (\langle 0,1 \rangle \cap \mathbb{Q})^n \to \langle 0,\log\_2 n \rangle, \; H\_n(P) = -\sum\_{i=1}^n P\_i \log\_2 P\_i, \; \sum\_i P\_i = 1$ be the information entropy on rationals. I am looking for an integer-valued analogue $H\_n' : \mathbb{N}^n\to\mathbb{Z}$ satisfying the following: Denote $ K\_x =\sum\... | https://mathoverflow.net/users/81305 | Is there an integer-valued analogue of information entropy? | Such a function cannot exist.
Let $I$ be the image of $H\_n$. Then $I$ is a countable dense subset of $[0, \log\_2 n]$. If such a function $H\_n'$ would exist, then $H\_n'(C\_a) = f\big(H\_n\big(\tfrac{C\_a}{K\_a}\big)\big)$, where $f$ is an injective monotonic function $f:I\to\mathbb{Z}$. Such a function $f$ cannot ... | 6 | https://mathoverflow.net/users/82587 | 303749 | 132,942 |
https://mathoverflow.net/questions/303739 | 0 | Let $V \in C^{1}(\mathbb{R}^n, \mathbb{R})$ consider the following PDE:
$$u\_t = grad[V(u)]$$
For $u \in C^{1}([0,1]^n\times [0,T),\mathbb{R}^n)$, with boundary conditions specified on the $n$-dimensional faces of the $n+1$-th cube $[0,1]^{n} \times [0,T)$. Where by "$grad$" I mean the gradient w.r.t. to the spatia... | https://mathoverflow.net/users/22810 | Well-posedness for equations of the form $u_t = grad[V(u)]$ and $u_{tt}=grad[V(u)]$? | The first equation (first order) is a quasilinear transport equation. Developping, it writes $u\_t+(Z\cdot\nabla\_x)u=0$ where $Z:=-(\nabla\_uV)\circ u$. The method of characteristics tells you that $u$ is constant along the integral curves of $Z$. Since $Z$ depends only on $u$, these curves are straight lines.
The b... | 3 | https://mathoverflow.net/users/8799 | 303750 | 132,943 |
https://mathoverflow.net/questions/303697 | 7 | The so-called Mean Ergodic Theorem goes back to von Neumann for Hilbert spaces. Later on, versions of this result in reflexive Banach spaces have also appeared (see, e.g., the book by Krengel, *Ergodic Theorems*, section 2.1. At the beginning of the same book (end of p.4), Krengel mentions that such a result is *not* t... | https://mathoverflow.net/users/126010 | A counterexample for the Mean Ergodic Theorem in $L_\infty$ | Let $I\_n$ be a collection of rapidly-shrinking intervals with $|I\_n|=1/n!$, say. Let $f\_0=0$.
Inductively define
$$
f\_n(x)=\begin{cases}
(-1)^n&\text{if $x\in\bigcup\_{i=0}^{n-1}\tau^i(I\_n)$;}\\
f\_{n-1}(x)&\text{otherwise.}
\end{cases}
$$
Let $f(x)=\lim\_{n\to\infty}f\_n(x)$. This limit exists by the first Borel... | 6 | https://mathoverflow.net/users/11054 | 303752 | 132,944 |
https://mathoverflow.net/questions/303683 | 6 | Let $L\neq 0$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. If $F$ is algebraically closed, then it is known that $L$ has no nontrivial restricted subalgebras if and only if $L$ is 1-dimensional.
Over arbitrary fields of positive characteristic, is there any description of restricted Lie alge... | https://mathoverflow.net/users/17582 | Restricted Lie algebras with no nonzero proper restricted subalgebras | For an element $a$ of $L$, denote by $\langle a \rangle\_p$ the restricted subalgebra generated by $a$.
Let $\mathbb{F}[t,\sigma]$ be the ring consisting of all polynomials $f=\sum\_{i\geq0}\alpha\_i t^{i}$ with respect to the usual sum and multiplication defined by the condition $t\cdot\alpha=\alpha^p t$ for every $\a... | 2 | https://mathoverflow.net/users/14653 | 303756 | 132,946 |
https://mathoverflow.net/questions/303758 | 16 | Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-module (Abelian group with compatible $G$-action) with the $G$-action inherited from the $G$-action on $X$.
Now for any ... | https://mathoverflow.net/users/81457 | "Rotated" version of the Atiyah-Hirzebruch spectral sequence | Good question. I think the answer is yes.
The unnamed spectral sequence is usually referred to as the *isotropy spectral sequence*. For a group $G$ acting on $X$ and an abelian group $A$ of coefficients as in your question, it has $E\_2$-page given by the sheaf cohomology $H^p(X/G; \mathcal{H}^q)$ of the orbit space,... | 14 | https://mathoverflow.net/users/8103 | 303769 | 132,948 |
https://mathoverflow.net/questions/303761 | 1 | Let $M$ be a complete Riemannian manifold. Now for a fixed $p\in M$, is there any non-constant smooth function $u:M\rightarrow\mathbb{R}$ such that
$$\int\_{B\_r}udV=0\ \forall 0\leq r<\infty,$$
where $B\_r$ is a ball with center at $p$ and radius $r$. I can not find such an example. I know that in $\mathbb{R}$, the ab... | https://mathoverflow.net/users/122445 | Example of a smooth function in a manifold whose integration vanishes | On the circle $x^2+y^2=1$, the function $x$ has this property for $p=(0,1)$ or $p=(0,-1)$.
On the sphere $|x|^2=1$ in $\mathbb{R}^{n+1}$, every nonzero linear function restricts to a function with this property, around any point $p$ at which it vanishes. On hyperbolic space, in the ball model, any linear function on th... | 7 | https://mathoverflow.net/users/13268 | 303778 | 132,949 |
https://mathoverflow.net/questions/247153 | 2 | Let $(X,d)$ be an uncountable infinite complete disconnected metric space (what I have in mind is something like $X=\{0,1,\ldots,n\}^{\mathbb{N}}$). I would like to know if the space $C^{\gamma}(X)$ of all continuous real functions satisfying $\mathrm{Hol}(f) = sup\_{x\neq y} (|f(x)-f(y)|)/d^{\gamma}(x,y)<+\infty$ endo... | https://mathoverflow.net/users/2386 | Is the Hölder Space with the Hölder Norm Reflexive? | I suppose you are assuming $X$ is compact? Because $\|f\|\_\infty$ need not be finite in general.
Anyway, for any infinite metric space $X$ the space ${\rm Lip}(X)$ is not isomorphic to a reflexive Banach space. This includes Holder spaces as a special case ($f$ is $\alpha$-Holder for the metric $d$ iff it is Lipschi... | 1 | https://mathoverflow.net/users/23141 | 303783 | 132,950 |
https://mathoverflow.net/questions/303791 | -6 | Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
where $a=3/2$ and $r$ is an arbitrary non-negative number. I want to prove the following statement for $f(x)$ but it see... | https://mathoverflow.net/users/123067 | Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ | If $r=0$, then $f(x)=0$ for all $x\ge0$. If $r>0$, then
$$f'(x)=-\frac{8 r (x+1) (8 r x+18 r+24 x+27)}{(2 x+3) (4 x+3) (2 r+2 x+3) (2 r+4 x+3)}<0$$
for $x\ge0$, $f(0)=2 \log \left(\frac{2 r}{3}+1\right)>0$, and $f(\infty-)=-r \log (4)<0$. So, your desired result follows.
| 1 | https://mathoverflow.net/users/36721 | 303795 | 132,954 |
https://mathoverflow.net/questions/303792 | 1 | Takahashi introduced the concept of convex structure in a metric space $(X,d)$ as a mapping $\mathcal{W}:X^2\times[0,1]\longrightarrow X$ satisfying
$$d\left(z,\mathcal{W}(x,y,\alpha)\right)\leq\alpha d(z,x)+(1-\alpha)d(z,y)$$
for all $x,y,z\in X$ and $\alpha\in[0,1]$. If the metric $d$ is induced by the norm $\|.\|\_X... | https://mathoverflow.net/users/111987 | convexity in linear metric spaces | Suppose $(X,d)$ is a linear metric space such that $B=\{x\in X: d(0,x)\leq 1\}$ is not convex (such as $L\_p$ or $\ell\_p^2$ when $0<p<1$). Since $B$ is not convex, there exist $x,y\in B$ and $0<\alpha<1$ such that $w=\alpha x+(1-\alpha)x\notin B$. Then with $z=0$, $$d(z,w)=d(0,w)>1,$$ since $w\notin B$. But $$d(z,x), ... | 1 | https://mathoverflow.net/users/nan | 303800 | 132,956 |
https://mathoverflow.net/questions/303759 | 4 | The [Fejer-Jackson inequality](https://link.springer.com/article/10.1023/A:1006560422934) as follows:
$$\sum\_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
>
> I conjecture that the inequality as follows holds:
>
>
> $$\sum\_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for al... | https://mathoverflow.net/users/122662 | $\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$ | Comment by Cherng-tiao Perng converted to an answer: It appears that Theorem A of [this paper](https://ac.els-cdn.com/S0898122111008054/1-s2.0-S0898122111008054-main.pdf?_tid=21771cc0-73cc-48ba-afa1-3ae25989a942&acdnat=1530100799_f17da2900208a8df38cfe852dc50b0c9) solves your problem.
| 7 | https://mathoverflow.net/users/2926 | 303801 | 132,957 |
https://mathoverflow.net/questions/303793 | 30 | Here is a naive outsiders perspective on set theory: A typical set-theoretical result involves constructing new models of set theory from given ones (typically with different theories for the original model and the resulting model). Typically, from a meta-perspective we are allowed (encouraged, or even required) to ass... | https://mathoverflow.net/users/15002 | How (non-)computable is set theory? | The question is extremely interesting, and I have looked into this kind of thing with various colleagues (including Russell Miller and Kameryn Williams), although our investigation has not yet resulted in any paper.
One important realization to which we had come was that the question is highly sensitive to whether th... | 25 | https://mathoverflow.net/users/1946 | 303810 | 132,962 |
https://mathoverflow.net/questions/303757 | 4 | Given that one can sample unitaries from the Haar measure over $U(n)$ (as in [F. Mezzadri, Notices of the AMS 54 (2007), 592-604](https://arxiv.org/abs/math-ph/0609050)), how can one sample from the uniform distribution over the following subgroup,
$$
\mathcal{B} = \{ A \in U(n) : [A, B] = 0 \},
$$
where $B$ is another... | https://mathoverflow.net/users/97715 | Haar unitaries with constraints | I assume all eigenvalues of $B$ are distinct, so that $B$ has a unique set of eigenvectors. Because of the constraint $[A,B]=0$, you wish to sample over unitary matrices $A$ that have the same set of eigenvectors as $B$, hence only the eigenvalues $e^{i\phi\_1},e^{i\phi\_2},\ldots e^{i\phi\_n}$ of $A$ matter. To achiev... | 2 | https://mathoverflow.net/users/11260 | 303822 | 132,967 |
https://mathoverflow.net/questions/303674 | 4 | Let $X$ be an $n$ dimensional smooth projective variety and $\mathcal{F}$ a globally generated rank $n$ locally free sheaf on $X$.
>
> Question: Is the locus of sections $s\in H^0(\mathcal{F})$ having a positive dimensional zero locus Zariski closed?
>
>
>
My idea: denote by $s\_1,\dots,s\_{m}$, $m=h^0(\mathc... | https://mathoverflow.net/users/64302 | Locus of global sections with positive dimensional zero locus is Zariski closed? | I put my comment as an answer. Yes, the result is true. Here is a standard argument: in $\mathbb{P}(H^0(\mathcal{F}))\times X$, consider the subvariety $Z$
of pairs $([s],x)$ such that $s(x)=0$.
This is clearly closed, hence projective, and you are looking at the sublocus of $\mathbb{P}(H^0(\mathcal{F}))$ where the fi... | 2 | https://mathoverflow.net/users/40297 | 303823 | 132,968 |
https://mathoverflow.net/questions/300376 | 2 | Let $V\subset H\subset V^\*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^\*),$$ where $f\in\ L^2(0,T;H)$.
**Can we let $f\in L^2(0,T;V^\*)$?**
This question is a special case ($A(t)=A$) of [Regularity of solution to a hyperbolic pde](htt... | https://mathoverflow.net/users/110050 | Solution of hyperbolic equations with $V^*$ data | Chapter 9 in Volume 1 of Lions/Magenes [1] treats this case, even for nonautonomous operators. One essentially gets (somewhat as expected?) a regularity shift just in the spatial components, so the solution $u$ will satisfy $(u,u') \in C([0,T];H \times V^\*)$. (This is provided the initial values for $(u,u')$ are in $H... | 1 | https://mathoverflow.net/users/85906 | 303834 | 132,972 |
https://mathoverflow.net/questions/303833 | 26 | We work in a separable metric space $(X,d)$. With $\overline{B}(x,r)$ I denote the closed ball around $x$ of radius $r$, and with $cl \ B(x,r)$ I denote the closure of the open ball. Clearly, we always have $cl \ B(x,r) \subseteq \overline{B}(x,r)$, and it is easy to construct examples where the inclusion is strict.
... | https://mathoverflow.net/users/15002 | Closed balls vs closure of open balls | The following theorem (or its corollary) implies negative answer to the original question.
>
> **Theorem.** For any point $x$ of a metric space $(X,d)$ the set $R\_x:=\{r>0:cl(B(x,r))\ne \bar B(x,r)\}$ has cardinality $|R\_x|\le w(X)$, where $w(X)$ stands for the weight of the topology of the metric space $(X,d)$.
... | 28 | https://mathoverflow.net/users/61536 | 303836 | 132,974 |
https://mathoverflow.net/questions/303711 | 6 | For each pair of functions $x,y\in L\_2[0,1]$ let us denote by $x\cdot y$ their pointwise product
$$
(x\cdot y)(t)=x(t)\cdot y(t),\quad t\in [0,1].
$$
It belongs to $L\_1[0,1]$ due to the Cauchy-Bunyakovsky inequality:
$$
x,y\in L\_2[0,1]\quad\Longrightarrow\quad x\cdot y\in L\_1[0,1].
$$
And for each pair of sets $... | https://mathoverflow.net/users/18943 | Is the unit ball in $L_p$, $1<p<2$, contained in a "compact perturbation" of the unit ball in $L_2$? | You are right, the answer to the question is no: if $1<p<2$, then there is no compact set $K \subset L\_2$ such that $$B\_p \subset \overline{\textrm{absconv}} (K \cdot B\_2).$$
Assume by contradiction that such $K$ exists, and define $C=\max\_K \|f\|\_2$. By duality (this is an equivalence) we have
$$ \|g\|\_q \leq... | 1 | https://mathoverflow.net/users/10265 | 303844 | 132,976 |
https://mathoverflow.net/questions/303735 | 5 | Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$.
I want to know when this matrix is also an involution, i.e., $P^2 = I$.
If there is any characterization of such $A$, please kindly share. Thank you.
| https://mathoverflow.net/users/33047 | Positive definite matrices diagonalised by orthogonal matrices that are also involutions | Let $n \times n$ matrix $\rm A$ be symmetric and positive definite. Since $\rm A$ is symmetric, it is diagonalizable. Hence, there exists a (non-singular) matrix $\rm P$ such that $\mathrm A = \mathrm P \,\mbox{diag} (\lambda\_1, \lambda\_2, \dots, \lambda\_n) \,\mathrm P^{-1}$, where $\lambda\_1, \lambda\_2, \dots, \l... | 5 | https://mathoverflow.net/users/91764 | 303845 | 132,977 |
https://mathoverflow.net/questions/303872 | 0 | Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number theory) numbers as even integers $ n $ such that $ n $ can be written in at least $ n $ ways as the difference between con... | https://mathoverflow.net/users/13625 | Sergei numbers : even integers n being a prime gap at least n times | In fact, even stronger fact is true: every Polignac number is a Sergei number and therefore there is a constant $C$ such that every interval of the form $[M,M+C]$ contains at least one Sergei number.
The corresponding result for Polignac numbers is proved in [this](https://arxiv.org/abs/1305.6289) article, Theorem 2.
... | 2 | https://mathoverflow.net/users/101078 | 303879 | 132,986 |
https://mathoverflow.net/questions/303861 | 31 | It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some number theoretic problems.
I was quite intrigued and ask for more explanation but the speaker himself said he did not u... | https://mathoverflow.net/users/25437 | What is the Katz-Sarnak philosophy? | The "Katz-Sarnak philosophy" is just the idea that statistics of various kinds for $L$-functions should, in the large scale limit, match statistics for large random matrices from some particular classical compact group.
First you need to decide what kinds of zeros to look at: the high zeros of an individual $L$-func... | 31 | https://mathoverflow.net/users/3272 | 303881 | 132,987 |
https://mathoverflow.net/questions/303876 | 7 | Recall that a category $\mathcal C$ has *amalgamation* if every span admits a cocone. If $\mathcal C$ has amalgamation, then does $Ind(\mathcal C)$ have amalgamation?
The "obvious way to show this" would be to induct on the presentability ranks of objects being amalgamated, presenting them as colimits of chains of ob... | https://mathoverflow.net/users/2362 | If $\mathcal C$ has amalgamation, does $Ind(\mathcal C)$ have amalgamation? | A counterexample is given in the paper *Disjoint Amalgamation in Locally Finite AEC* by Baldwin, Koerwien, and Laskowski [(link).](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/disjoint-amalgamation-in-locally-finite-aec/74050B19FF648BBC596C7A2C8423CB56) They're actually interested in findin... | 5 | https://mathoverflow.net/users/2126 | 303884 | 132,988 |
https://mathoverflow.net/questions/303865 | 2 | We have a set of elements $S=\{1,2,3, . . . , N\}$ and family $F$ of $N$ triples of elements in $S$ ($N$ is a multiple of 3). Each element of $S$ appears in exactly three triples. The elements in each triple are distinct. The intersection of two triples $T\_1, T\_2$ is either empty or has exactly two elements ( $|T\_1 ... | https://mathoverflow.net/users/8784 | Counting triples family with double shared elements | I'm guessing that what matters here is subsets of size 3 from S, and that F is a special collection of such sets. It turns out that F meeting the conditions is very special, and to see this, we relax the condition on N and assume it an arbitrary large enough positive integer, instead of having three divide N.
So pick... | 4 | https://mathoverflow.net/users/3402 | 303886 | 132,989 |
https://mathoverflow.net/questions/303809 | 3 | Let $$f(n) = 2 \sum\limits\_{a = 2}^{n - 1} \sum\limits\_{b = n + 1}^{n + a - 1}\frac{1}{ab} .$$
One can see that $$\lim\limits\_{n \to \infty}f(n) = 2\int\limits\_0^1 \frac{dx}{x} \int\limits\_1^{1 + x}\frac{dy}{y} = \frac{\pi^2}{6}.$$ This suggests that the double sum defining $f(n)$ can be transformed into a singl... | https://mathoverflow.net/users/106512 | Simplifying a double sum of inverses | We have
$$f(n+1)-f(n)=\frac1{n^2}+\frac2{n}\left(1+\frac 12+\ldots+\frac 1{n-1}\right)-\frac2{n+1}\left(1+\frac 12+\ldots+\frac 1{n}\right),$$ and $f(2)=0$. So
\begin{gather\*}
f(m)=\sum\_{n=2}^{m-1}(f(n+1)-f(n))=\sum\_{n=2}^{m-1}\frac1{n^2}-\frac2{m}\left(1+\frac 12+\ldots+\frac 1{m-1}\right)+1=\\=\sum\_{n=1}^{m-1}\fr... | 7 | https://mathoverflow.net/users/5712 | 303887 | 132,990 |
https://mathoverflow.net/questions/303898 | 1 | I posed a conjecture as follows that is a special case of [Second Hardy–Littlewood conjecture](https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture):
>
>
> >
> > For $n, x \ge 2$ be two integers then:
> > $$P\_{2n} \ge 2P\_n$$
> > and
> > $$\pi(2x) \le 2\pi(x)$$
> >
> >
> >
>
>
>
... | https://mathoverflow.net/users/122662 | Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture? | We have $\pi(2x)<2\pi(x)$ for any $x\geq 11$. For a proof see [here](https://math.stackexchange.com/questions/1738347/proving-that-pi2x-2-pix). Similarly, it follows from standard bounds that $P\_{2n}>2P\_n$ when $n$ is sufficiently large.
| 3 | https://mathoverflow.net/users/11919 | 303899 | 132,995 |
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