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https://mathoverflow.net/questions/439235 | 3 | I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod\_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with the basic. Can you confirm that (1) we have absolute convergence, (2) the limit exists if $x$ is an integer, (3) if $x$... | https://mathoverflow.net/users/140356 | Curious infinite product, convergence, connection to prime numbers | I believe that the doubly infinite product $\prod \_{k=1} ^\infty \prod\_{m=1} ^\infty \left( 1 - \frac {x^2} {k^2 m^2} \right)$ is absolutely convergent and equals $f(x) \frac {\sin^2 (\pi x)} {(\pi x)^2 (1 - x^2)}$. Is that wrong? I am just using $\frac {\sin(\pi x)} {\pi x} = \prod\_{m=1} ^\infty \left(1 - \frac {x^... | 5 | https://mathoverflow.net/users/494014 | 439255 | 177,405 |
https://mathoverflow.net/questions/439249 | 1 | Let $f : \mathbb R\_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R\_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int\_0^\infty f(x)e^{-sx} \, dx.
$$
A classical Theorem due to Lerch [M. Lerch, Sur un point de la théorie des fonctions génératrices d’Abel, Acta Math. 27... | https://mathoverflow.net/users/223636 | Discrete uniqueness sets for the two-sided Laplace transform? | Of course, if $A$ has accumulation points then
your statement is correct, since you assume the function $Lf$ to be entire.
Otherwise, there are no restrictions on zeros of such functions $Lf$ (unless you somehow restrict your class $C$ of functions $f$). For example, let $f$ be an infinite sum of $\delta$-functions s... | 2 | https://mathoverflow.net/users/25510 | 439263 | 177,408 |
https://mathoverflow.net/questions/439266 | 5 | Traditionally in dependent type theory with axiom K or uniqueness of identity proofs, every type $A$ is 0-truncated, and thus the type of lists on $A$, $\mathrm{List}(A)$, is 0-truncated and the free monoid on $A$. However, in homotopy type theory, not every type is 0-truncated; for example, the circle type $\mathbb{S}... | https://mathoverflow.net/users/483446 | Are lists in homotopy type theory free $A_\infty$-spaces? | In an informal sense, the answer "should be yes", in the sense that if one ignore type theory and work with an $\infty$-topos one can make sense of the construction $List(A)$ either by the usual universal property of list objects or as $List(A) = \coprod\_\mathbb{N} A^n$ (both definitions can be shown to be equivalent)... | 7 | https://mathoverflow.net/users/22131 | 439267 | 177,409 |
https://mathoverflow.net/questions/439269 | 6 | The word "local" in category theory does not seem to have a precise definition in itself but it often appears as part of other terminology. To my understanding, it is then used in the following ways, which AFAIU are quite unrelated:
* A category is locally $P$ if all of its slice categories are $P$.
Example: [local... | https://mathoverflow.net/users/66017 | Overloading of the word "local" in category theory | The terminology situation is certainly unfortunate. There is an nLab page for [locally](https://ncatlab.org/nlab/show/locally), which lists the different usages of the term (it doesn't currently include those for "local" rather than "locally"). The page suggests "slice-wise" to disambiguate the first case, and "hom-wis... | 12 | https://mathoverflow.net/users/152679 | 439270 | 177,410 |
https://mathoverflow.net/questions/439264 | 1 | Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability distribution. Does the following inequality hold?
\begin{align}
\mathbb{E}\_\mathcal{F}\left[\sqrt{\sum\_{x\in\mathcal{X}}p^... | https://mathoverflow.net/users/68835 | Does the following expectation-based inequality hold? | No, the factor $\frac1{\sqrt A}$ is the best you can get.
Indeed, letting $\mathcal X=[n]:=\{1,\dots,n\}$ and $a\_x:=p^2(x|y)$, we have
$$R\_n:=\frac{E\sqrt{\sum\_{x\in\mathcal{X}}p^2(x|y)1[\mathcal{F}(x)=1]}}{\sqrt{\sum\_{x\in\mathcal{X}}p^2(x|y)}}
=E\sqrt{\frac
{\sum\_{i\in[n]} a\_i Y\_i}
{\sum\_{i\in[n]} a\_i} }... | 1 | https://mathoverflow.net/users/36721 | 439273 | 177,412 |
https://mathoverflow.net/questions/439238 | 1 | Let $V$ be a real vector space. Given a subset $A \subseteq V$, say that a point $x \in A$ lies in the *algebraic interior* of $A$ if every affine line $\ell$ that passes through $x$ has the property that $x \in (\ell \cap B)^\circ$. Here $\ell \cap B$ is a subinterval of $\ell \cong \mathbb{R}$, so we defined its inte... | https://mathoverflow.net/users/136356 | abstract description of the topology on a real vector space defined by the algebraically open sets | This is not true, already in $\mathbf R^2$. Indeed, if $V$ is finite-dimensional, then the Euclidean topology is the coarsest topology for which all linear¹ maps $V \to \mathbf R$ are continuous: choosing a basis $e\_1,\ldots,e\_n$ and taking the corresponding coordinate projections $\pi\_i \colon V \to \mathbf R$ show... | 2 | https://mathoverflow.net/users/82179 | 439274 | 177,413 |
https://mathoverflow.net/questions/439192 | 4 | I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds:
Let $c: [\lambda]^2 \to \kappa$ be such that for $\alpha < \beta < \gamma$, we have $c(\alpha, \gamma) \leq \max \lbrace c(\alpha, \beta) , c(\beta, \gamma) \rbra... | https://mathoverflow.net/users/495743 | Ramsey-like property with order condition | A coloring $c:[\kappa]^2\rightarrow\theta$ is *subadditive of the first kind* if for all $\alpha<\beta<\gamma<\kappa$, $c(\alpha,\gamma)\le\max\{c(\alpha,\beta),c(\beta,\gamma)\}$. It is *subadditive of the second kind* if for all $\alpha<\beta<\gamma<\kappa$, $c(\alpha,\beta)\le\max\{c(\alpha,\gamma),c(\beta,\gamma)\}... | 5 | https://mathoverflow.net/users/20033 | 439275 | 177,414 |
https://mathoverflow.net/questions/439277 | 1 | I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of scheme is of utmost importance in number theory but what about the cohomology of sheaves ? Why is étale cohomology useful in ... | https://mathoverflow.net/users/170999 | Sheaf cohomology in number theory | This is slightly too log to be a comment, but certainly too short to be a defintive answer. Apologees in advance.
I would suggest reading the volume 2 book ("*Cohomology of algebraic varieties*", by Danilov) of the EMS series "*Algebraic Geometry*". You will learn a lot about étale cohomology, in a non-technical way ... | 3 | https://mathoverflow.net/users/37214 | 439292 | 177,419 |
https://mathoverflow.net/questions/438021 | 1 | It is [well known](https://math.stackexchange.com/a/509018/656606) that an open subscheme of an affine scheme is not necessarily an affine one. But what are (if possible the most general) sufficient conditions for its affinity? And is it known how, in such cases, to directly describe its ring in terms of its correspond... | https://mathoverflow.net/users/148161 | Under what conditions is an open subscheme of an affine scheme affine and what ring corresponds to it? | Here is the best general result I'm aware of:
**Lemma.** *Let $A$ be a commutative ring with an ideal $I$, let $X = \operatorname{Spec} A$ with closed subset $Z = V(I)$ and open complement $U = D(I)$.*
1. *If $Z$ is a Cartier divisor, then $U$ is affine and dense in $X$.*
2. *If $A$ is Noetherian and $U$ is affine ... | 3 | https://mathoverflow.net/users/82179 | 439295 | 177,421 |
https://mathoverflow.net/questions/438983 | 1 | Let $\mathscr{C}:=\{\gamma : \mathbb{R}\_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\sigma$-algebra $\mathfrak{C}$ generated by all projections $\gamma \mapsto \gamma\_t$ (for $t\geq 0$ fixed). Let further
... | https://mathoverflow.net/users/160714 | Shift-ergodic stochastic processes in continuous time | Do you not also want that $\mathbb{P}\_Y$ is $\phi$-invariant?
In any case, yes there are extremely many continuous-time continuous-path real-valued stochastic processes whose law is ergodic under the time-$1$-shift map. Of course, the most trivial example would be where
$$ \mathbb{P}(Y\_t=c \ \ \forall t \geq 0) = 1... | 1 | https://mathoverflow.net/users/15570 | 439298 | 177,423 |
https://mathoverflow.net/questions/439278 | 8 | In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be *flat* if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this definition used in talks and the like as well, but I'm confused about where it comes from. I know two meanings of the word "f... | https://mathoverflow.net/users/158123 | Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself? | Here is a little context to maybe complement Tom's and Nick's answers.
---
The definition (2) in terms of being flat over $\pi\_\* \Bbb S$ is new - it's a specialization of a definition of flatness over $R$ that comes from derived / spectral algebraic geometry. Over $\Bbb S$ this is so rare that most examples are... | 8 | https://mathoverflow.net/users/360 | 439299 | 177,424 |
https://mathoverflow.net/questions/438839 | 4 | Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}\_{p}$. Let $\{ A\_{n}\}\_{n\in\mathbb{N}}$ be a projective family of $\mathcal{R}$ algebras equipped with maps $A\_{n+1}\rightarrow A\_{n}$, considerin... | https://mathoverflow.net/users/476832 | On inverse limits of $\pi$-adically complete algebras | Not sure about the main question, but regarding the "context" question, one can show the following:
If $A$ is any affinoid, and $f \in A$ any element with nowhere-dense zero locus, then $A^\circ = A\times\_{A\langle \pi^n/f \rangle} {A\langle \pi^n/f \rangle}^\circ$ for all sufficienly large $n$.
The point is that ... | 1 | https://mathoverflow.net/users/496798 | 439311 | 177,428 |
https://mathoverflow.net/questions/439310 | 2 | Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in practical computation utilizing barycentric subdivisions is the size of $Bary(S)$ is *much* larger than $S$, and so whe... | https://mathoverflow.net/users/408316 | Do there exist smaller simplicial models of barycentric subdivisions? | Consider an arbitrary finite simplicial complex $\ S.\ $ First, look at it purely combinatorially. Thus, let $\ \{a\ b\}\ $ be a 1-simplex of $\ S.\ $ Then define a subdivided simplicial complex $\ S(a\ b\ c),\ $ where $\ c\ $ is a new fixed vertex that didn't belong to $\ S.\ $ The simplexes that don't contain $\ \{a\... | 3 | https://mathoverflow.net/users/110389 | 439312 | 177,429 |
https://mathoverflow.net/questions/439320 | 3 | I am trying to observe the behavior of $x\_n \in (0,1)$ defined such that the function
\begin{equation}
f\_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)
\end{equation}
attains its maximum inside the interval $(0,1)$ at $x=x\_n$.
Upon using the Wolfram alpha, I have found out that as $n \to \infty$ it seems that $x\_... | https://mathoverflow.net/users/56524 | Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$ | The maximum $x\_n$ of
$$f\_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$$
is the smallest solution in $(0,1)$ of the equation
$$x=n x^n+\frac{1}{n}.$$
For $n\gg 1$ this gives $x\_n\rightarrow 1/n$.
The integral is given by
$$\int\_0^1 f\_n(x)dx=\text{Ei}(-1)+n^{-2}\,\Gamma (n-1,1)+1/e$$
$$\qquad\rightarrow \sqrt{2 ... | 6 | https://mathoverflow.net/users/11260 | 439321 | 177,430 |
https://mathoverflow.net/questions/439316 | 22 | It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin algebra, the existence of a Suslin line etc. This is not too surprising, since many results in this fields require a nont... | https://mathoverflow.net/users/493244 | Statements in differential geometry independent from ZFC | [Using the comments for context on undecidability/independence of ZFC]
A computably undecidable problem is whether or not a homology sphere has a metric of positive scalar curvature [Page 79 of *[Computers, Rigidity, and Moduli](https://www.google.com/books/edition/Computers_Rigidity_and_Moduli/Q5sAEAAAQBAJ)*].
| 19 | https://mathoverflow.net/users/1847 | 439326 | 177,432 |
https://mathoverflow.net/questions/437521 | 5 | Let $F\_n^{(k)}(x)= \sum\_j {\binom{n+(k-1)j}{kj} x^j}$ and $G\_n^{(k)}(x)= \sum\_j {\binom{n+j}{kj} x^j}.$
I am interested in the coefficients ${a\_{n,k,j}}$ such that
$$G\_n^{(k)}(x)=\sum\_{j\geq0 }{a\_{n,k,j}} F\_j^{(k)}((-1)^ { k}x).$$
Computations suggest the following:
Let $z\sum\_{j\geq 0}C\_{1,j}^{(k)}z^j$ ... | https://mathoverflow.net/users/5585 | A polynomial identity related to Catalan numbers | These assertions can be proved using (formal) generating functions.
Using that for $j\geq 0, k\geq 1$
\begin{align\*}
\sum\_{n\geq 0} {n-j+kj \choose kj} t^n &=\frac{t^j}{(1-t)^{kj+1} }\;\;\mbox{ and }\\
\sum\_{n\geq 0}{n+j \choose kj} t^n&=\frac{t^{kj-j}}{(1-t)^{kj+1}}\;\;\;,
\end{align\*}
gives that
\begin{align\*} ... | 3 | https://mathoverflow.net/users/48831 | 439349 | 177,438 |
https://mathoverflow.net/questions/416724 | 2 | Let $M$ be a 1-skeleton of a triangulation of a sphere with $V$ vertices and $E$ edges.
**Definition 1** A polyhedron is a map $M\to \mathbb R^3$ that is affine on edges (and non-degenerate on faces). The space of polyhedra is identified with (apparantly an open subset of) $\mathbb R^{3V}$.
**Definition 2** Two pol... | https://mathoverflow.net/users/13842 | Generic infinitesimal rigidity of polyhedra | In short, no, this cannot happen.
To be more specific. A standard definition for ``generic'' is that the coordinates of $P \in \mathbb{R}^{3V}$ form an algebraically independent set. It is a nice exercise proving that: (i) almost all (in the measure theory sense) points in $\mathbb{R}^{3V}$ are generic, and (ii) ever... | 3 | https://mathoverflow.net/users/122002 | 439361 | 177,443 |
https://mathoverflow.net/questions/439116 | 3 | (This is a natural continuation of a [previous post](https://mathoverflow.net/questions/438594/).)
**I. Quintic method**
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + 10)x +1 = 0$$
From [this post](https://mathoverflow.net/questi... | https://mathoverflow.net/users/12905 | Using the Lehmer quintic to solve $11$-degree equations and higher? | **Question 1:**
An irreducible Lehmer quintic $L(n)$ has Galois group ${\bf Z} / 5 {\bf Z}$.
Let $\sigma$ be a generator, and order the roots $x\_i$ ($i \bmod 5$) so that
$x\_{i+1} = \sigma(x\_i)$ for each $i$.
Note that $\prod\_i x\_i = -1$ because $L(n)$ has constant coefficient $1$.
Thus $\prod\_i x\_i^{a\_i} = (-... | 3 | https://mathoverflow.net/users/14830 | 439364 | 177,446 |
https://mathoverflow.net/questions/439324 | 2 | I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is ***not rigorous / wrong*** (as the power series for the Bernoulli numbers is not convergent), I wonder, why the final result - at least in the area $0<Im(z)\leq \frac{1}{2}... | https://mathoverflow.net/users/108459 | Fourier series of Eisenstein series — elegant and very good approximation | As Paul Garrett says, this reflects (for $k \geq 4$ even) the modularity of the Eisenstein series $E\_k(z) = -\frac{B\_k}{2k} + \sum\_{n=1}^{\infty} \sigma\_{k-1}(n) e(nz)$ with respect to the matrix $S = (\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix})$, which sends $z$ to $-1/z$ (so it preserves the imaginary ... | 2 | https://mathoverflow.net/users/6506 | 439368 | 177,447 |
https://mathoverflow.net/questions/439352 | 5 | I read that, with the Scott topology, suprema of sequences are topological limits (See page 1 of [this article](https://akjournals.com/view/journals/10473/88/1-2/article-p35.xml)).
Let $(X, \le)$ be a DCPO, and $D$ be a directed subset of $X$.
I can easily see that the identity net on $D$ converges to $\bigvee D$.
... | https://mathoverflow.net/users/55915 | Scott topology: Suprema of sequences are topological limits | $\newcommand\LL{\mathcal L}$**Brief answer:** The Scott topology is not Hausdorff, and therefore we have to deal here with the set of limits, rather than with the (unique) limit. Here, for the set $\LL\_D$ of limits of the identity net on $D$ we have
\begin{equation\*}
\LL\_D=L\_s:=\{y\in X\colon x\le s\},
\end{equat... | 7 | https://mathoverflow.net/users/36721 | 439376 | 177,450 |
https://mathoverflow.net/questions/439372 | 11 | Let $A$ be a set of cardinality $4n$. We define a pairing in $A$ to be a partition of $A$ into sets of cardinality $2$. What is the minimum number of pairings in $A$ such that every subset of $A$ of cardinality $4$ is the union of two pairs from at least one pairing?
This question is motivated by the computational pr... | https://mathoverflow.net/users/66833 | Minimum number of pairings that make all quadruples | It is a nice exercise in number theory.
Let $p$ be a prime slightly above $4n$. Let $1\le x<y<z<t\le 4n$.
Note that $0<(y+t)-(x+z)<4n<p$.
Now, there exists $a\in \mathbb Z\_p$ such that $(x+a)(z+a)=(y+a)(t+a)$ in $\mathbb Z\_p$ (this equation only pretends to be quadratic; in fact it is linear in $a$ with non-zer... | 18 | https://mathoverflow.net/users/1131 | 439377 | 177,451 |
https://mathoverflow.net/questions/438711 | 2 | I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
| https://mathoverflow.net/users/497573 | References for group of invariance of the Painlevé property | I assume "the book of Robert Conte" refers to *The Painlevé Handbook* by Robert Conte and Micheline Musette (Springer, 2008); the "groups of invariance of the Painlevé property" are briefly described in Appendix A of that book.
The book by Harold T. Davis *Introduction to Nonlinear Differential and Integral Equations... | 0 | https://mathoverflow.net/users/106467 | 439382 | 177,454 |
https://mathoverflow.net/questions/439220 | 1 | Let $A$ be a unital $C^{\*}$-algebra and $a \in A$ be normal, with spectrum $\sigma(a)$. Let $B = C^{\*}(a)$ be the $C^{\*}$-algebra generated by $1$ and $a$, which is abelian. Let $\hat{B}$ be the space of all homomorphisms $\chi: B \to \mathbb{C}$ with the weak\* topology, which makes it a compact Hausdorff topologic... | https://mathoverflow.net/users/152094 | Spectral theorem for unital $C^{*}$-algebras | To see that (1) continues to hold when B is replaced by A,
it suffices (by linearity and density of step functions) to consider only the case when g is the characteristic function of some measurable subset S of σ(a).
Indeed, in this case the right side equals the ν-measure of S, whereas the left side equals the μ-mea... | 1 | https://mathoverflow.net/users/402 | 439388 | 177,456 |
https://mathoverflow.net/questions/424813 | 0 | Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}\_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}\_t\}$-adapted quadratic variation process such that the Ito integral
$$
Z(t, \omega) := \int\_0^t \phi(s, \omega) d B(s, \... | https://mathoverflow.net/users/170508 | Distribution of local martingale is absolutly continuous to that of the Brownian motion? | Let $ B $ be a Brownian Motion, $\mu\_{B}=\mathsf{P}\circ B^{-1} $ and $ Z $ be a continuous local martingale, $ \mu\_{Z}=\mathsf{P}\circ Z^{-1}$.
Then $ \mu\_{Z}\ll \mu\_{B} $ if and only if $ \mu\_{Z}=\mu\_{B} $.
In this case, if $ Z $ may be expressed as Ito stochastic integral of a predicatble $ \phi $ with respect... | 3 | https://mathoverflow.net/users/103256 | 439389 | 177,457 |
https://mathoverflow.net/questions/439366 | 2 | Does there exist a sequence of strongly regular graphs with parameters $(n,d,\lambda,\mu)$ (so every pair of adjacent vertices have $\lambda$ common neighbours, and every pair of non-adjacent ones have $\mu$ common neighbours), with $d - \mu = \Omega(n)$ and $\mu - \lambda = \Omega(n)$?
| https://mathoverflow.net/users/141963 | Strongly regular graphs with certain parameters | I think, the answer is negative (I expect that by $\Omega(n)$ you mean something positive of order $n$). Denote $\mu-\lambda=k,d-\mu=\ell$.
The eigenvalues of the adjacency matrix of such a graph $G$ are $d$ and $t\_{1,2}$ which are the roots of a quadratic equation $t^2+kt-\ell=0$. The multiplicities $n\_{1},n\_2$ e... | 4 | https://mathoverflow.net/users/4312 | 439390 | 177,458 |
https://mathoverflow.net/questions/439359 | 8 | Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all geodesically-convex sets containing $K$.
Is $\text{conv}(K)$ bounded? Compact?
If it helps, I am interested in cases where the... | https://mathoverflow.net/users/97528 | Convex hulls of compact sets in a 2-manifold | *The convex hull of a compact set $K \subset M^2$ in a complete manifold need neither be bounded, nor closed.*
Both counterexamples are rotationally symmetric, and the second has a Euclidean metric outside of a compact region.
To prove that a convex hull need not be bounded, consider a sphere with an infinitely lon... | 5 | https://mathoverflow.net/users/103792 | 439391 | 177,459 |
https://mathoverflow.net/questions/439283 | 1 | Let $M$ be a surface with boundary and let $f\_t: M \to M, t \in [0,1]$ be a differentiable family of diffeomorphisms (I think this is usually called a diffeotopy). Suppose I have a Liouville form $\lambda\_0 \in \Omega\_1(M)$ such that $f\_0$ is exact, i.e. $f\_0^\*\lambda\_0 - \lambda\_0 = dF$ for some function $F: M... | https://mathoverflow.net/users/173545 | Deformation of a Liouville form with a diffeotopy | The statement is not true. For a function to be exact, it needs to preserve the symplectic form $d\lambda$. This puts strong conditions on the function, for instance it's differential should have determinant $1$ at fixed points. This is clearly not preserved in a diffeotopy.
| 0 | https://mathoverflow.net/users/173545 | 439401 | 177,462 |
https://mathoverflow.net/questions/439397 | 2 | Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]:
$$||\mu||\_0:= \sup\Bigg\{\int\_X fd \mu: f \in \mathrm{Lip}\_1(X), \sup\_{x \in X}|f(x)|\leq 1 \Bigg\}$$
where $\mathrm{Lip}\_1(X)$ deo... | https://mathoverflow.net/users/498220 | Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm | No, those sets are not open. Indeed, take any non-isolated point $x$ in $X$ and a sequence $(x\_n)\_{n\in\omega}$ that converges to $x$. For every $n$, consider the sign measure $\mu\_n=\delta\_{x}-\delta\_{x\_n}$, where $\delta\_p$ is the Dirac measures at a point $p\in X$. The definition of the Kantorovich-Rubinshtei... | 6 | https://mathoverflow.net/users/61536 | 439404 | 177,463 |
https://mathoverflow.net/questions/439350 | 0 | Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W\_R$. Each element of $W\_R$ is a vector of length $N$ with integer entries.
Firstly, I would like to know if it is true that $$\sum\_{\mu\in W\_R}m\_\mu\mu=A(R)(1,\ldots,1),$$ where $m\_\mu$ is the multiplicity of the wei... | https://mathoverflow.net/users/122036 | Sum of weights of an irreducible representation of $U(N)$ | As discussed in the [comments](https://mathoverflow.net/questions/439350/sum-of-weights-of-an-irreducible-representation-of-un#comment1133244_439350), your sum is a Weyl-fixed character, so trivial for $G = \operatorname{SU}(N)$ and a multiple of $\det = (1, \dotsc, 1)$ for $\operatorname U(N)$.
To be concrete, as I ... | 2 | https://mathoverflow.net/users/2383 | 439408 | 177,465 |
https://mathoverflow.net/questions/439400 | 0 |
>
> Let $f \in H\_{0}^{1}(0,1)$ and $\lambda >0$ big enough. Consider $0 <\alpha < 1$ and some $k > 0$. I would like to show the following inequality
> $$
> \int\_{\lambda^{-k}}^{1}|f(x)|^{2}dx \leq C\lambda^{-p}\int\_{\lambda^{-k}}^{1}x^{\alpha}|f^{\prime}(x)|^{2}dx
> $$
> for some $p> 0$. Here, $C> 0$ is constant.
... | https://mathoverflow.net/users/481556 | Perhaps an application of Hardy's inequality | $\newcommand\la\lambda\newcommand\al\alpha$If $C$ is allowed to depend on $\lambda$, just take $C=2\lambda^p$.
If $C$ is not allowed to depend on $\lambda$, take any nonzero $f\in H\_{0}^{1}(0,1)$ and let $\lambda\to\infty$. Then the left-hand side of your desired inequality will go to $\int\_0^1|f(x)|^2\,dx>0$ where... | 2 | https://mathoverflow.net/users/36721 | 439409 | 177,466 |
https://mathoverflow.net/questions/439308 | 6 | I am trying to understand a proof in [this paper](https://arxiv.org/abs/1512.01669) (specifically theorem 5.4). In it, a fact is used that every element of the $W^\*$-algebra $A$ is a linear combination of exponential unitaries.
I've tried to reason this fact out, but I don't follow where this fact comes from or why ... | https://mathoverflow.net/users/498367 | Every element of a $W^*$-algebra is a linear combination of exponential unitaries? | It is true in any unital $C^\ast$-algebra. Any element $x$ can be written as a linear combination of self-adjoints, by $x=\tfrac{1}{2}(x+x^\ast)+\tfrac{1}{2i}i(x−x^\ast)$. Moreover, if $y$ is a self-adjoint element with $\| y\| \leq 1$, and $\arccos\colon [−1,1]\to [0,π]$, then $u=\exp(i \arccos(y))$ is an exponential ... | 7 | https://mathoverflow.net/users/126109 | 439420 | 177,474 |
https://mathoverflow.net/questions/439313 | 1 | In the Lucas–Lehmer test with $ \quad p \quad $ an odd prime.
we know that $ \quad S\_0=4 \quad $ and $ \quad S\_i=S\_{i-1}^2-2 \quad $ for $\quad i>0 \quad$
$M\_p=2^p-1 \quad$ is prime if $ \quad S\_{p-2} \equiv 0 \bmod {(2^p-1)}$
after some observations i found a link with Triangle of coefficients of Chebyshev'... | https://mathoverflow.net/users/140242 | Lucas–Lehmer test and triangle of coefficients of Chebyshev's | By the [composition property](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Composition_and_divisibility_properties) of Chebyshev polynomials $T\_m(T\_n(x))=T\_{mn}(x)$. Since $x^2-2 = 2T\_2(\tfrac{x}2)$, we have $S\_i = 2T\_{2^i}(2)$ for all $i\geq 0$.
Furthermore, since $T\_{2^k}(x) = \frac{U\_{2^{k+1}-1}(x)}... | 3 | https://mathoverflow.net/users/7076 | 439421 | 177,475 |
https://mathoverflow.net/questions/439431 | 0 | Let $(X\_t)\_{t \ge 0}$ be a Lévy process on $\mathbb R$ with Lévy measure $\nu$.
Define the jump counting measure $N(t, A) = \lvert\{s \in [0, t] \mathrel: \Delta X\_s \in A\}\rvert$
where $\Delta X\_s$ denotes the jump height at time $s$.
For a fixed Borel set $A$ such that $0 \notin A$, let
$N\_t = N(t,A)$.
Ho... | https://mathoverflow.net/users/495513 | Lévy measure and jump behaviour of the corresponding Lévy process | This follows immediately from (say) the following statements in [Schilling - An Introduction to Lévy and Feller Processes](https://arxiv.org/abs/1603.00251):
* parts b) and c) of Lemma 9.4, stating that $N(\cdot,A)$ is a Poisson process of intensity $\nu(A):=EN(1,A)$, where $A$ is any Borel subset of $\mathbb R^d\set... | 2 | https://mathoverflow.net/users/36721 | 439438 | 177,479 |
https://mathoverflow.net/questions/439286 | 8 | Here's an idea for a knot-based Diffie–Hellman exchange:
* Public: random (oriented) knot $P$.
* Private: random (oriented) knots $A$ and $B$.
* Exchange: Alice sends (randomized or canonical representation of) $A'=A\oplus P$, Bob sends (randomized or canonical representation of) $B'=P\oplus B$. Here $\oplus$ is knot... | https://mathoverflow.net/users/129839 | Knot Diffie–Hellman | Here I assume that by “addition” of knots you mean the usual connect sum, as defined [here](https://en.wikipedia.org/wiki/Connected_sum#Connected_sum_of_knots). With that said, I think you correctly ask the relevant question: “Is factoring knots difficult?”
In favour of your idea is the fact that we do not (yet?) hav... | 8 | https://mathoverflow.net/users/1650 | 439442 | 177,480 |
https://mathoverflow.net/questions/439402 | 14 | This is in some sense a follow up to the question asked here [Properties of the category of compact Hausdorff spaces](https://mathoverflow.net/questions/382348/properties-of-the-category-of-compact-hausdorff-spaces)
To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\text{CHaus}$ of co... | https://mathoverflow.net/users/76299 | Is there a universal property characterizing the category of compact Hausdorff spaces? | I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts:
1. The category of compact Hausdorf topological space is the category of algebras for the ultrafilter monad.
2. the Ultrafilter monad is the codensity monad for... | 19 | https://mathoverflow.net/users/22131 | 439451 | 177,486 |
https://mathoverflow.net/questions/439195 | 9 | I am currently looking into how to construct a **skew-Hadamard matrix** of order 292. Where can I find such construction?
According to multiple papers (e.g. [Koukouvinos and Stylianou - On skew-Hadamard matrices](https://doi.org/10.1016/j.disc.2006.06.037) and [Seberry and Yamada - Amicable Hadamard matrices and amic... | https://mathoverflow.net/users/498306 | Construction of skew-Hadamard matrix of order 292 | The skew Hadamard matrix of order 292 can be constructed from skew supplementary difference sets (kindly supplied to us by Prof. Djokovic). This same construction is used, for example, in [Djokovic - Skew-Hadamard Matrices of Orders 436,580, and 988 Exist](https://arxiv.org/abs/0706.1973).
An implementation of this c... | 5 | https://mathoverflow.net/users/498306 | 439452 | 177,487 |
https://mathoverflow.net/questions/439437 | 0 | Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C\_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|\_\infty \leq 1\}.$
Prove or disprove that for all $U>0,\beta>0,$ there exist $\epsilon>0,C>0$ such that for all $\lambda \in \left]0,1\right... | https://mathoverflow.net/users/138491 | Integral with inequality | $\newcommand\EE{\mathcal E}\newcommand\la\lambda\newcommand\R{\mathbb R}\newcommand\ep\varepsilon$What you wanted us to prove is not true.
Indeed, take any $\phi\in\EE$ such that $\phi\ge1\_{[-1/2,1/2]}$. Write $A\gg B$ for $A\ge cB$, where $c$ is a universal positive real constant.
Then, for $w:=x-y\_2$,
\begin... | 3 | https://mathoverflow.net/users/36721 | 439456 | 177,489 |
https://mathoverflow.net/questions/439259 | 4 | Sarnak and Strömbergsson studied [Epstein zeta function](https://encyclopediaofmath.org/wiki/Epstein_zeta-function) $\zeta(L,s)=\sum\limits\_{0\neq v\in L}\langle v,v\rangle^{-s}$ of a number of highly symmetric lattices in their [Inventiones Math. paper](http://www2.math.uu.se/%7Eastrombe/papers/minima.pdf).
In partic... | https://mathoverflow.net/users/11100 | Epstein zeta function of Barnes-Wall and related lattices | It turns out this question, and more, is basically answered, for all Barnes-Wall lattices $\Lambda\_{n}$, in affirmative in [Spherical designs and zeta functions of lattices](https://arxiv.org/abs/math/0611735) - the published version [here](https://doi.org/10.1155/IMRN/2006/49620) with the extra assumption $s>n/2$.
... | 2 | https://mathoverflow.net/users/11100 | 439459 | 177,490 |
https://mathoverflow.net/questions/439458 | 1 | Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $G$-stable.
>
> Is there any "nice" description of the invariant ring $\mathbb{K}[Y]^G$ in terms of the invariant ri... | https://mathoverflow.net/users/338456 | Invariant ring of the subvariety | As we discussed in the comments, by linear reductivity of $G$, there is a $G$-module splitting of the surjection $\mathbb K[X] \to \mathbb K[Y]$; so the restriction map $\mathbb K[X]^G \to \mathbb K[Y]^G$ is a surjection. Thus $\mathbb K[Y]^G$ is a quotient of $\mathbb K[X]^G$, as both a $G$-module and an algebra; and ... | 5 | https://mathoverflow.net/users/2383 | 439464 | 177,492 |
https://mathoverflow.net/questions/439467 | 0 | Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C\_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|\_\infty \leq 1\}.$
Prove that for all $U>0,\beta>0,$ there exist $\epsilon>0,C>0$ such that for all $\lambda \in \left]0,1\right],u,v \in [0... | https://mathoverflow.net/users/138491 | Integral and inequality | At least when (say) $U=1$, $v=1$, and $u=0$, the integral
$$\int\_u^{v} \int\_{\mathbb{R}} \left(\int\_{\mathbb{R}} \phi\_x^\lambda(y\_1)p(v-r,y\_1-y\_2) \, dy\_1 \right)^2 \,dy\_2 \, dr$$
coincides (in view of the variable change $r\leftrightarrow v-r$) with the integral
$$\int\_0^{|u-v|} \int\_{\mathbb{R}} \left(\int... | 1 | https://mathoverflow.net/users/36721 | 439469 | 177,495 |
https://mathoverflow.net/questions/439463 | 0 | An issue from 3D tessellated geometry: Find the direction vector of a plane that minimizes the silhouette of a set of triangles. To say it another way, find the direction vector that is most perpendicular to a set of triangles. Each triangle area is half of the length of the normal vector of any two sides—call it $n\_i... | https://mathoverflow.net/users/498458 | Trigonometry/spherical angles/minimum-least-squares | If you use constrained optimization, you do get linear equations.
I set $g(x)=||x||^2$. For every $x \in \mathbb{R}^3$,
$$\nabla f(x) = \sum\_i 2(x \cdot n\_i)n\_i \text{ and } \nabla g(x) = 2x.$$
You look at (unit) vectors $x$ such that $\nabla f(x) = \lambda \nabla g(x)$ for some real number $\lambda$, namely you loo... | 0 | https://mathoverflow.net/users/169474 | 439476 | 177,499 |
https://mathoverflow.net/questions/439403 | 4 | $\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
* **Fact 1**: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)=\lim\limits\_{r\to 0}\frac{1}{|B\_r(x)|}\int\_{B\_r(x)}u(y)dy
$$
for Lebesgue-a.e. $x\in \Omega$.
* **Fact 2**:... | https://mathoverflow.net/users/33741 | Lebesgue differentiation theorem at boundary points for Sobolev traces | It is true - this is Theorem 5.7 in Evans and Gariepy’s *Measure Theory and Fine Properties of Functions* (2015 version).
Note that the theorem is stated for BV functions, but Sobolev functions are BV, so it holds also for your case.
| 6 | https://mathoverflow.net/users/173490 | 439479 | 177,500 |
https://mathoverflow.net/questions/439483 | 0 | (Asked previously in [MSE](https://math.stackexchange.com/questions/4625673/find-a-probability-density-from-the-moments))
Suppose a probability distribution $p(x)$ has moments $m\_n=\int p(x)x^ndx$ given by $m\_0=1$, $m\_1=1$, $m\_2=2$ and, for $n>0$,
$$m\_{n+1}={2n \choose n}.$$
The moment generating function exis... | https://mathoverflow.net/users/83671 | Probability distribution with shifted central binomial moments | The function $\mathbb R\ni t\mapsto g(t):=f(it)$ is not the characteristic function of any probability distribution.
Indeed, if it were, then we would have $|g(t)|\le1$ for all real $t$. However, in fact, $|g(10)|=1.316\ldots>1$.
So, $f$ is not the moment generating function of any probability distribution, and hen... | 2 | https://mathoverflow.net/users/36721 | 439489 | 177,503 |
https://mathoverflow.net/questions/439490 | 4 | I want to show that a certain Lie subgroup (i.e. generated by the exponential of elements in some Lie subalgebra) of a Lie group is closed. My knowledge of the subject of Lie groups is rudimentary, and I would really appreciate any pointers towards relevant results.
My question
-----------
**Is the following statem... | https://mathoverflow.net/users/7631 | Is a Lie subgroup whose center is closed, a closed subgroup itself? | No. Consider a group of the form $G=V\rtimes K$ where $V$ is a Euclidean group and $K$ is a compact 2-torus and $K$ acting faithfully on $D$, with no nonzero invariant vector. (For instance $G=(\mathbf{R}^2\rtimes\operatorname{SO}(2))^2$.) Let $D$ be a dense line in $K$: then $V\rtimes D$ is dense and non-closed in $G$... | 7 | https://mathoverflow.net/users/14094 | 439497 | 177,505 |
https://mathoverflow.net/questions/439502 | 1 | I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$$Tf(x)=\int \frac{e^{i|x-y|^{a}}}{|x-y|^{\alpha}}
f(y)dy$$
is studied. Here $0<\alpha<n$ and $a>0$, $a\neq 1$.
The author repea... | https://mathoverflow.net/users/116555 | Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$? | This is a standard property of self-adjoint (including convolution) operators, following from duality.
Let $Tf = k\*f$ and assume that $||Tf||\_{p} \leq C ||f||\_{p}$.
Then $$||Tf||\_{p'} = \sup\_{\substack{g \in L^p \\ ||g||\_{p}} =1} \left<Tf,g\right> =\sup\_{\substack{g \in L^p \\ ||g||\_{p}} =1} \left<f,Tg\righ... | 2 | https://mathoverflow.net/users/630 | 439506 | 177,508 |
https://mathoverflow.net/questions/439265 | 1 | The principle of induction over identity families, do not prohibit instances different from `refl: x == x` but its computation rule is only defined for this instance, i.e. `ind(C,c,x,x,refl) :≡ c(x)`.
If a function defined by path induction receives arguments different from the expected ones for the computation rule ... | https://mathoverflow.net/users/495133 | Computation over non-reflexivity | After analyzing, I have come with a conclusion: the path induction principle is not different from other axioms/rules that assert the existence of a function without providing an explicit definition, e.g. $succ: N \rightarrow N, inl: A \rightarrow A + B$.
As axioms, these functions do not need a definition, since a d... | 1 | https://mathoverflow.net/users/495133 | 439507 | 177,509 |
https://mathoverflow.net/questions/439293 | 6 | Let $X$ be a complex compact manifold, and write $\mathcal{O}\_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}\_X^{\times}$ be the subsheaf consisting of holomorphic functions. These are both sheaves of abelian groups. If we identify $H^1(X, \mathcal{O}\_X^{\times})$ with the Picard group $\text{Pic}... | https://mathoverflow.net/users/108274 | Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor | Let $\mathcal M$ be the sub-sheaf of $\mathcal I^0$ of sections mapping to (the restrictions of) $\alpha$. (I suppose that's what the OP has intended anyways.) That's easily seen to be an $\mathcal{O}\_X^\times$-torsor. By [Stacks Project Tag 0A6G](https://stacks.math.columbia.edu/tag/0A6G), if $X=\bigcup\_i U\_i$ is a... | 2 | https://mathoverflow.net/users/15782 | 439527 | 177,516 |
https://mathoverflow.net/questions/439518 | 2 | Consider $X$ a smooth cubic surface in $\mathbb{P}^3$, and let $l\_1,...,l\_6$ be six disjoint lines contained in $X$.
What is the linear system giving the blow-down map $X \to \mathbb{P}^2$, so that the lines $l\_k$ are contracted to points ?
The other way round is well-known : if $p\_1,\dots,p\_6$ are six points ... | https://mathoverflow.net/users/21030 | what is the linear system on a cubic surface giving the blow-down map to the plane | As you probably know, if a representation of $X$ as blowup is given,
$$
K\_X = -3h + \sum l\_i,
$$
where $h$ is the line class. Consequently, the linear system
$$
|-K\_X + \sum l\_i|
$$
gives the required contraction.
| 3 | https://mathoverflow.net/users/4428 | 439533 | 177,519 |
https://mathoverflow.net/questions/439474 | 1 | We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ has *property ${\bf B}$* if there is $S\subseteq V$ such that for all $e\in E$ with $|e|\geq 2$ we have $$(e\cap S) \neq \emptyset \neq (e\cap (V\setminus S)).$$
If $k\in \omega$, a hypergraph $H=(V,E)$ is $k$-*uniform* if all elements of ... | https://mathoverflow.net/users/8628 | Uniform hypergraphs with small edge intersections and propery ${\bf B}$ | There are counterexamples for every integer $k\ge3$.
In fact, if $2\le k\lt\omega$, there is a $k$-uniform hypergraph $H=(V,E)$ such that $|V|=\aleph\_0$, $\{e\_1,e\_2\}\in\binom E2\implies|e\_1\cap e\_2|\le1$, and $H$ has chromatic number $\chi(H)=\aleph\_0$.
Namely, let $V=\binom{\mathbb N}{k-1}$ and $E=\{\binom ... | 2 | https://mathoverflow.net/users/43266 | 439540 | 177,521 |
https://mathoverflow.net/questions/439526 | 5 | Let $\mathbb{G}$ be a compact quantum group (in the sense of Woronowicz) with discrete dual $\widehat{\mathbb{G}}$ which we view as a von Neumann algebraic locally compact quantum group (in the sense of Vaes-Kustermans). Let us denote its function algebra by $(\ell^\infty(\widehat{\mathbb{G}}), \hat{\Delta})$.
Consid... | https://mathoverflow.net/users/470427 | Completely isometric coaction of discrete quantum group is multiplicative? | Yes, such a map $\alpha$ is automatically multiplicative and thus defines an action of $\widehat{\mathbb{G}}$ on $M$.
As in the question, denote by $\alpha\_\gamma : M \to M \otimes B(H\_\gamma)$ the components of $\alpha$, for any irreducible unitary representation $\gamma$ of $\mathbb{G}$. Fix an irreducible repres... | 5 | https://mathoverflow.net/users/159170 | 439543 | 177,522 |
https://mathoverflow.net/questions/439551 | 1 | Let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on a complex Hilbert space $E$.
On $\mathcal{L}(E)^2$, we have two equivalent norms:
\begin{eqnarray\*}
N\_1(A\_1,A\_2)
&=&\sup\left\{\|A\_1x\|^2+\|A\_2x\|^2,\;x\in E,\;\|x\|=1\;\right\},
\end{eqnarray\*}
and
$$N\_2(A\_1,A\_2)=\sup\left\{|\langle A\_1x... | https://mathoverflow.net/users/113054 | Prove that $N_1(A_1,A_2)= N_2(A_1,A_2)$ when $A_1A_2=A_2A_1$ and $A_1$ and $A_2$ are normal operators | I will follow the OP's initial observation and notation. From the formula
$$\|A\_1 f\|^2+\|A\_2 f\|^2=\int\_X\left(|\varphi\_1|^2+|\varphi\_2|^2\right)|f|^2$$
it is clear that
$$N\_2(A\_1,A\_2)\leq N\_1(A\_1,A\_2)\leq\mathrm{ess}\sup\left(|\varphi\_1|^2+|\varphi\_2|^2\right).$$
Hence it suffices to show that, for any $... | 2 | https://mathoverflow.net/users/11919 | 439559 | 177,530 |
https://mathoverflow.net/questions/439344 | 3 | Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the sense of [Al Kelley - The stable, center-stable, center, center-unstable, unstable manifolds](https://doi.org/10.1016/0022... | https://mathoverflow.net/users/43097 | Analyticity of central stable manifolds | Quick answer to the first question: no, there is no reason why it should be analytic. Take *e.g.* the parametric vector field (written as a Lie derivative)$$X(x,y):=-x^3\partial\_x+(y+\alpha x)\partial\_y~~~,~\alpha\in\mathbb{R}.$$ Here the center manifold is a $C^\infty$ regular curve through the origin, tangent there... | 2 | https://mathoverflow.net/users/24309 | 439561 | 177,531 |
https://mathoverflow.net/questions/439187 | 4 | I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create branched covers of a given topological space or more precisely branched covers of surfaces.
If anyone shears related refere... | https://mathoverflow.net/users/490039 | Books for learning branched coverings | Montesinos wrote several papers defining the meaning of *branched coverings* and proving basic properties(not just between manifolds, but for general topological spaces):
*Montesinos-Amilibia, José María*, Branched folded coverings and 3-manifolds, Castrillón López, Marco (ed.) et al., Contribuciones matemáticas en h... | 4 | https://mathoverflow.net/users/39654 | 439567 | 177,532 |
https://mathoverflow.net/questions/439521 | 2 |
>
> Is there a general $\alpha$-tuple implementation that is of height $2$, that both doesn't require infinity of the naturals, and is at the same time stable under lack of Extensionality?
>
>
>
My own try to solve this question depends on a modified Holmes ordered pairs.
*Define:* $\langle x,y,z,..,s \rangle ... | https://mathoverflow.net/users/95347 | Are there known general tuple implementations that are 2 types high and withstand absence of extensionality and infinity? | This is a partial answer to this question:
I was always under the impression that a type-level pair must depend on having some kind of well ordered infinite class of objects, well at least this was the experience I found with the Quine-Rosser pair, and in addition it should presuppose Extensionality (like Quine-Rosse... | 1 | https://mathoverflow.net/users/95347 | 439569 | 177,534 |
https://mathoverflow.net/questions/439545 | 6 | Let $F$ be a number field and $\mathbb{A}$ its adele ring. $G$ be a classical group and $
\pi$ be a unitary cuspidal automorphic representation of $G(\mathbb{A})$.
Then I am wondering whether there is a known theorem on the local component of $\pi$. For example, the statement I am expecting is $\pi\_v$ is supercuspid... | https://mathoverflow.net/users/29422 | Local component of cuspidal automorphic representation | Let me work in the category of $L^2$-automorphic representations. Assuming your global representation $\pi$ is irreducible, about the only thing you can say about an arbitrary local component $\pi\_v$ is that it is an irreducible smooth representation of $G\_v$. For almost all $v$, you can also say $\pi\_v$ will be sph... | 8 | https://mathoverflow.net/users/6518 | 439574 | 177,536 |
https://mathoverflow.net/questions/374863 | 0 | If $(M,g)$ is a smooth Riemannian manifold and $c : [a,b] \to M$ is a smooth embedded simple curve on $M$, it is always possible to choose locally a Riemannian metric $g\_0$ on $M$ for which $c$ is a geodesic for $g\_0$. As I understand, this can be done by pulling back a tubular neighborhood of $c$ to a disc bundle al... | https://mathoverflow.net/users/94097 | Discs bundles along a curve and positive curvature | It appears to me that your question does not involve the metric $g$ at all. Any connected embedded curve in a smooth $n$-manifold has a tubular neighborhood diffeomorphic to $T = I\times D$,
where $I$ is an interval or a circle and $D$ is the unit $(n-1)$-dimensional disk. You can map $T$ into the unit sphere such that... | 3 | https://mathoverflow.net/users/613 | 439578 | 177,538 |
https://mathoverflow.net/questions/437768 | 1 | Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T\_1=\nabla^3f(x),$$ and the $n^2 \times n$ matrix $$T\_2=\frac{\partial \operatorname{vec}(\nabla^2f(x))}{\partial x^T}.$$ That relationship bein... | https://mathoverflow.net/users/484618 | Third order matrix differential norm | First, we will denote the $n^2\times n$ matrix of third-order derivatives as $\mathbf{K}$, which has the following structure:
$$
\mathbf{K}=\frac{\partial\operatorname{vec}(\mathbf{H})}{\partial\mathbf{x}^T}=\begin{bmatrix}
\frac{\partial f}{\partial \mathbf{x}\_1\mathbf{x}\_1\mathbf{x}\_1} & \cdots & \frac{\partial f... | 0 | https://mathoverflow.net/users/484618 | 439585 | 177,540 |
https://mathoverflow.net/questions/439494 | 0 | Let $P$ be a closed polygon defined by the sequence
$p\_0,\,\dots,\,p\_{n-1},p\_0$ of points.
>
> **Question:**
>
>
> how can one construct, with straightedge and compass alone, another sequence of points $q\_0,\,\dots,q\_{n-1}$ such that:
>
>
> * $q\_i$ lies on the bisector of $p\_i$ and $p\_{i+1}$
> * $q\_... | https://mathoverflow.net/users/31310 | Constructing a polygon from another with collinearity constraints | Let $L\_i$ be the bisector of $p\_i$ and $p\_{i+1}$, and let $f\_i \colon L\_i \to L\_{i+1}$ be the central projection through $p\_{i+1}$. This is a projective transformation with constructible coefficients. Your question is about the fixed points of the projective transformation $f\_n \circ \cdots \circ f\_1$. The coe... | 2 | https://mathoverflow.net/users/98590 | 439611 | 177,547 |
https://mathoverflow.net/questions/312876 | 6 | I, ask my question as a comment [in this post](https://mathoverflow.net/questions/311630/nash-isometric-embedding-for-noncompact-manifolds). Without answer I post a more detailed version.
I am looking for a reference about $C^\infty$ Nash isometric embedding for non compact manifold.
My question is what are exactly... | https://mathoverflow.net/users/9253 | Nash embedding for complete manifold | Your nice projection is usually called *positive reach*.
In order to have it, one has to have both curvature bounds and a lower bound on injectivity radius.
Plus the volume growth must be at most polynomial.
Say the Lobachevsky plane does not admit a smooth embedding positive reach; see this question: [Does a Riemannia... | 6 | https://mathoverflow.net/users/1441 | 439613 | 177,549 |
https://mathoverflow.net/questions/439576 | 0 | Let $X\sim \mathcal{N}(\mu,\sigma^2)$ be a Gaussian random variable with random mean $\mu\sim {\sf Bernoulli}(p)$, i.e., $\mu=1$ with probability $p$ and $\mu=0$ with probability $1-p$. In other words, to draw a realization of $X$, first one flips a coin to decide whether $\mu=0$ or $\mu=1$ and then, one samples from t... | https://mathoverflow.net/users/138242 | Maximal mutual information between a continuous and a discrete random variables | $\newcommand{\de}{\delta}\newcommand{\M}{\mathcal M}
\newcommand{\ep}{\varepsilon}
\newcommand{\thh}{\theta}\newcommand\I{\mathcal I}\newcommand{\Si}{\Sigma}$Here is the (slightly edited) definition of the mutual information given in the [answer](https://mathoverflow.net/questions/321364/mutual-information-between-cont... | 1 | https://mathoverflow.net/users/36721 | 439618 | 177,550 |
https://mathoverflow.net/questions/439624 | 2 | Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi\_t = \{ z = t\}$ at most twice:
\begin{equation}
\# \gamma(S^1) \cap \Pi\_t \leq 2 \quad \text{for all $t$.}
\end{equation}
>
> Does $... | https://mathoverflow.net/users/103792 | A geometric criterion for uniqueness in the Plateau problem? | The answer is no.
Take two parallel circles of unit radius in $z=\pm \epsilon$ with $\epsilon$ small. Tilt the two circles very slightly toward one another. This satisfies your hypotheses. There are clearly at least three minimal surfaces with this boundary the tilted flat disks and a stable and unstable annulus (obt... | 5 | https://mathoverflow.net/users/127803 | 439628 | 177,553 |
https://mathoverflow.net/questions/216338 | 7 | Let $f,g:(D^2,j\_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman similarity principle as in Floer--Hofer--Salamon):
**Lemma:** Suppose $f=g$ over a neighborhood of $0\in D^2$. Then $f... | https://mathoverflow.net/users/35353 | Does pseudo-holomorphic *submanifolds* satisfy unique continuation? | The desired statement is in fact a well known property of pseudo-holomorphic maps. Lemma 1.3.1 in [Singularities and positivity of
intersections of J-holomorphic curves](https://mathscinet.ams.org/mathscinet-getitem?mr=1274930) by Dusa McDuff says that for a pair of pseudo-holomorphic maps $u,v:(D^2,0)\to(M,p)$ with $d... | 2 | https://mathoverflow.net/users/35353 | 439651 | 177,559 |
https://mathoverflow.net/questions/439649 | 1 | Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space. Let $H=(H\_t, t\ge 0)$ be a stochastic process with continuous trajectories. Fix $T>0$. For $n \ge 1$, we define
$$
H\_{s,n} := \sum\_{i=1}^{2^n} H\left(\frac{(i-1) T}{2^n}\right) 1\_{ \left (\frac{(i-1) T}{2^n}, \frac{i T}{2^n} \right]}(s) \quad \forall s \... | https://mathoverflow.net/users/477203 | How is $\mathbb E[ \int_0^T H_s^2 \mathrm d s] < \infty$ important for this claim? | This statement is false in general.
E.g., let $T=1$ and $p\_k:=2^{-k}$ for integers $k=1,2,\dots$, so that $\sum\_{k=1}^\infty p\_k=1$. Let $A\_1,A\_2,\dots$ be pairwise disjoint events with respective probabilities $p\_1,p\_2,\dots$.
Let
\begin{equation}
H\_t:=\sum\_{k=1}^\infty 1\_{A\_k}\,2^k\,\sum\_{j=0}^{2^k}\Bi... | 2 | https://mathoverflow.net/users/36721 | 439654 | 177,560 |
https://mathoverflow.net/questions/439572 | 1 | I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^2$ for some given positive constant $m$.
Here $x^2$ is of course with respect to the above metric, and $x=(x\_0,x\_1,x\_2,... | https://mathoverflow.net/users/56524 | Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform? | If Lorentz invariance is not required:
Let $\phi$ be any smooth bump function $\phi:\mathbb{R}\to\mathbb{R}$ that is non-zero precisely on $(0,4m^2)$ (including the one you used in the question statement).
Let $f:\mathbb{R}^4\to \mathbb{R}$ be given by
$$ f(x\_0, x\_1, x\_2, x\_3) = \phi(x\_0^2 - x\_1^2 - x\_2^2 ... | 3 | https://mathoverflow.net/users/3948 | 439655 | 177,561 |
https://mathoverflow.net/questions/439635 | 14 | Suppose we have a diffeomorphism $f:{\mathbb{S}}^n\_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
>
> **Question.** Is it possible to extend it to a diffeomorphism $F:\mathbb{S}^n\to\mathbb{S}^n$?
>
>
>
I believe that when $n=2$ this shoul... | https://mathoverflow.net/users/121665 | Extending diffeomorphisms | The answer is positive and follows from Corollary 2 in
*Palais, Richard S.*, [**Extending diffeomorphisms**](http://dx.doi.org/10.2307/2032968), Proc. Am. Math. Soc. 11, 274-277 (1960). [ZBL0095.16502](https://zbmath.org/?q=an:0095.16502).
(A caveat: Palais is not entirely clear about the degree of smoothness he al... | 11 | https://mathoverflow.net/users/39654 | 439658 | 177,563 |
https://mathoverflow.net/questions/439071 | 5 | I am studying the enlightening article "The Picard Group of $\mathcal{M}\_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
**Setting**
Let $\mathcal{M}\_{1, 1, k}$ denote the stack of elliptic curves over an algebraically closed field $k$ with $char(k)=3$. We have
\begin{equation\*}
V... | https://mathoverflow.net/users/498211 | How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point | From Section 3 of the paper, we see that the moduli stack of elliptic curves (over a ring $A$) has a quotient stack presentation $$ \mathcal{M}\_{1,1,A} \simeq [U/G] $$ where $U = \operatorname{Spec} A[a\_{1},a\_{3},a\_{2},a\_{4},a\_{6},\Delta^{-1}]$ and $G = \operatorname{Spec} A[u^{\pm},r,s,t]$. The group law on $G$ ... | 2 | https://mathoverflow.net/users/15505 | 439663 | 177,564 |
https://mathoverflow.net/questions/438926 | 2 | To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the new primitives allowed to be used in instances of Separation and Replacement, I'll refer to this simply as $\sf ZFj$.
W... | https://mathoverflow.net/users/95347 | Can we interpret Reinhardt cardinals this way? | It is inconsistent. Call an ordinal b an independent critical point if for every for every Universe X,
if c∈X then there is a function f with domain X and an ordinal α, such that "f is an elementary
embedding from X to Vα" holds, for all x∈X f(x)∈X, and b is the least ordinal with f(b)≠b.
Note that the critical p... | 3 | https://mathoverflow.net/users/133981 | 439682 | 177,569 |
https://mathoverflow.net/questions/439206 | 5 | By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$J\_0(u)J\_0(v)+2\sum\_{n=1}^\infty J\_n(u)J\_n(v) \cos(n\alpha) = J\_0 \left( \sqrt{u^2+v^2-2uv \cos\alpha} \right)$$
What is the corresponding identity for
$$\sum\_{n=1}^\infty J\_n(u)J\_n(v)\sin(n\alpha)=\te... | https://mathoverflow.net/users/7154 | Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$? | I don't think a "closed form" expression exists, I tried several approaches. I guess the best one can do is to use a modified version of the OP's integral representation.
Denote
$$
w(\psi) = \sqrt{u^2+v^2-2uv\cos\psi},\tag{1}
$$
then
\begin{align}
S &= 2\sum\_{n=1}^\infty J\_n(u)J\_n(v)\sin(n\alpha) \tag{2a}\\
&= \fr... | 2 | https://mathoverflow.net/users/90413 | 439685 | 177,570 |
https://mathoverflow.net/questions/439665 | 6 | Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
**Definition:** let the group $G$ act by isometries on a metric space $(X,d)$. The *stable translation length* of $g\in G$ is $\tau(g) = \lim\_{n\rightarrow\infty} d(x,g^nx)/n$, (for any $x\in X$)... | https://mathoverflow.net/users/76590 | Translation length on annular curve graphs | I think that the answer is “yes, such a constant $K$ exists.”
Suppose that $g$ is the given mapping class in the stabiliser of $\gamma$. Replace $g$ by a large power (bounded by the topology of $S$) so that it fixes the curves of its canonical reducing system (and also their orientations and their sides). In each com... | 2 | https://mathoverflow.net/users/1650 | 439686 | 177,571 |
https://mathoverflow.net/questions/439681 | 3 | I see many mathematicians conflating the definitions of traveling waves and solitons, and I am unable to understand, from a mathematical point of view, the differences between these two types of solutions for a nonlinear dispersive PDE. All I know is the following:
Consider for example a nonlinear dispersive PDE whic... | https://mathoverflow.net/users/498602 | Mathematical difference between solitons and traveling waves for a non-linear dispersive PDE | A necessary requirement for a traveling wave $u(x,t)=f(x-ct)$ to be a "solitary wave" or "soliton" is that the two limits $\lim\_{s\rightarrow\pm\infty}f(s)=\alpha\_\pm$ exist. This is the condition of *shape invariance* and *localisation*. The stability under collision may or may not be added as extra condition, but i... | 6 | https://mathoverflow.net/users/11260 | 439694 | 177,572 |
https://mathoverflow.net/questions/439696 | 4 | A well known off-diagonal Ramsey result says that every $C\_3$-free graph $G$ on $N$ vertices has an independent set of size $\Omega(\sqrt{N\log N})$.
It is a conjecture of Erdos that every $C\_4$-free graph $G$ on $N$ vertices has an independent set of size $\Omega(N^{1/2+c})$ for some absolute constant $c>0$.
My ... | https://mathoverflow.net/users/130484 | Independent sets in graphs with girth $\ge g$ | Yes, and for any $c<1/2$. Namely, for $\delta>0$ assume that our graph does not have an independent set of size $N^{1-\delta}$. In particular this yields that the chromatic number is at least $N^{\delta}$, thus there is an induced subgraph $H$ with all degrees at least $N^{\delta}$ (proof: remove vertices of smaller de... | 4 | https://mathoverflow.net/users/4312 | 439698 | 177,573 |
https://mathoverflow.net/questions/439700 | 1 | Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X\_1,\ldots,X\_N)$ from $\mu$ as $\mu\_N$. Do we have the following for $p\geq 1$:
\begin{align}
\inf\_{\nu \in \mathcal{P}(S)} \mathbb{E}\_{X\si... | https://mathoverflow.net/users/176220 | Is the Wasserstein distance to the empirical measure minimized by the underlying distribution? | No. E.g., let $N=1$ and suppose that $X:=X\_1$ has a nondegenerate zero-mean distribution $\mu$ such that $E|X|^p<\infty$. Let $Y$ be an independent copy of $X$.
Then the expected $\mathcal W\_p$-distance from the empirical distribution to $\mu$ is
$$E\mathcal W\_p(\delta\_X,\mu)^p=E|X-Y|^p>E|X|^p=E\mathcal W\_p(\... | 1 | https://mathoverflow.net/users/36721 | 439704 | 177,574 |
https://mathoverflow.net/questions/439711 | 1 |
>
> Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is it then always true that $E|X-Y|>E|X|$?
>
>
>
To get the non-strict version of this inequality, condition on $X$ and then apply Jensen's inequality to the zero-mean random variable $Y$.
I... | https://mathoverflow.net/users/36721 | A strict inequality for the $L^1$-norm of a symmetrized zero-mean random variable | No, a symmetric random sign gives equality ($X$ is $\pm 1$ with probability $1/2$).
| 2 | https://mathoverflow.net/users/120845 | 439713 | 177,576 |
https://mathoverflow.net/questions/439710 | 3 | The [original post](https://math.stackexchange.com/questions/4158432/reference-request-for-a-riemannian-fokker-planck-equation) is in StackExchange but no one has answered it yet. I personally think it is more related to the research area so I put it in MathOverflow. Below is the question in the original post:
I am l... | https://mathoverflow.net/users/480283 | Reference request for a Riemannian Fokker-Planck equation | An early reference is [Coordinate-independent formulation of the Langevin equation](https://doi.org/10.1063/1.527396) (1986).
>
> A diffusion process on a compact Riemannian manifold is considered,
> and a coordinate-invariant Fokker-Planck equation is formulated. A
> covariant form of the Langevin equation is also... | 2 | https://mathoverflow.net/users/11260 | 439718 | 177,577 |
https://mathoverflow.net/questions/439716 | 3 | Consider the (non-unital) $\mathbb{C}$-algebra (point-wise multiplication) of $\mathcal{S}$ of Schwartz functions on $\mathbb{R}$.
**Question:** Does there exist some character (non-zero multiplicative functional to $\mathbb{C}$) $\omega$ of $\mathcal{S}$ that is *not* the evaluation map at any point in $\mathbb{R}$,... | https://mathoverflow.net/users/40789 | Characters of algebra of Schwartz functions | Let $m$ be a multiplicative functional. Let $A={\mathbb C}\oplus\mathcal S$ be the algebra $\mathcal S$ extended by the constant functions. This algebra is unital. Setting $m(f+\lambda)=m(f)+\lambda$ extends $m$ to a multiplicative functional on $A$.
For $f\in A$ let $s(f)=\sup\_{x\in\mathbb R}|f(x)|$. Assume $s(f)<1$.... | 4 | https://mathoverflow.net/users/473423 | 439722 | 177,578 |
https://mathoverflow.net/questions/439717 | 0 |
>
> Suppose that $Y$ is an independent copy of a random variable (r.v.) $X$ with a zero-mean nondegenerate distribution. Is there a non-tautological, preferably simple characterization of the cases when
> $$E|X-Y|=E|X|\,\text{?} \tag{1}\label{1}$$
>
>
>
This question is a modification of this [previous one](http... | https://mathoverflow.net/users/36721 | Equality cases in a certain case of Jensen's inequality | Since the distribution of the r.v. $X$ is zero-mean, $X$ integrable (hence also the copy $Y$).
The equality is attained iff the distribution of $X$ is carried by at most two points.
Indeed, by independence of $X$ and $Y$,
$$E\big(|X-Y|\big|X\big) = f(X) \text{ where } f(x) := E|x-Y|.$$
Hence
$$f(X) =E\big(|X-Y|\big... | 1 | https://mathoverflow.net/users/169474 | 439727 | 177,580 |
https://mathoverflow.net/questions/439728 | 1 | $\newcommand\Fl{\mathrm{Fl}}$Let $G$ be a connected, reductive algebraic group over $\mathbb{C}$. Fix a maximal torus $T$ and Borel subgroup $B$. Let $L$ be a generalized Levi ($L = Z\_G(s)^\circ$, for some semisimple element $s \in T$). Then $L$ is also a reductive group, and $B \cap L$ is a Borel subgroup in $L$. The... | https://mathoverflow.net/users/492133 | Inclusion of flag varieties and Schubert decomposition | As you suspect, the equality of inflated Schubert cells does hold for actual Levis $L$. Let $N$ be the unipotent radical of the parabolic subgroup of $G$ containing $B$ that has $L$ as a Levi componnet. Then $w^{-1}N w$ is contained in $B$ for all $w \in W\_L$, and $(B \cap L)w(w^{-1}N w)$ equals $B w$.
However, the ... | 2 | https://mathoverflow.net/users/2383 | 439730 | 177,581 |
https://mathoverflow.net/questions/400891 | 2 | Let $M$ be a $n$-dimensional compact Riemannian manifold, and $N$ a smooth submanifold of $M$ of dimension strictly less than $n$.
Denote by $N\_{\varepsilon}$ the $\varepsilon$-neighbourhood of $N$ - that is, the set $\{p \in M \ | \ d(x, N) < \varepsilon\}$.
Here $d$ denotes the Riemannian distance.
Let $\text{... | https://mathoverflow.net/users/173490 | Second derivative of the volume of the $\varepsilon$-neighbourhood of a submanifold | I guess we are interested in closed submanifolds; otherwise the question does not have much sense.
Suppose $k=\mathrm{codim}\, N$.
Note that $f(\varepsilon)=O(\varepsilon^k)$, so $f''(0)=0$ if $k\ge 3$.
It remains to consdier two cases $k=2$ and $k=1$.
Given a normal vector $v$ to $N$ denote by $\rho(v)$ the jacobi... | 3 | https://mathoverflow.net/users/1441 | 439732 | 177,582 |
https://mathoverflow.net/questions/439729 | 1 | Is it possible to have a properly embedded annulus $A$ in a genus two handlebody $V$ such that the two boundary curves of $A$ in $\partial V$ represent different isotopy classes of curves on $\partial V$? (I should also mention that in the case I care about, the curves bounding $A$ cannot be meridians, i.e. cannot boun... | https://mathoverflow.net/users/156387 | Properly embedded annuli in genus two handlebody? | Choose a non-trivial loop $a\subset \partial V$ and a disjoint meridian $b\subset \partial V$. Let $D\subset V$ be a disk with $\partial V = b$ and let $A'$ be an annulus in $V$ with $\partial A'$ equal to two parallel copies of $a$. Define $A$ to be the result of taking a boundary connected sum of $A'$ and $D$ paralle... | 2 | https://mathoverflow.net/users/284 | 439733 | 177,583 |
https://mathoverflow.net/questions/430449 | 0 | I've edited (ten days ago) a question on Physics Stack Exchange, this [*Mathematical characterization of gravitational geons*](https://physics.stackexchange.com/questions/726281/mathematical-characterization-of-gravitational-geons), post with identifier **726281** the users of the site were kind adding in the comment t... | https://mathoverflow.net/users/142929 | Mathematical characterization of gravitational geons as reference request, and their properties as main question | A gravitational geon is a space-time configuration that is bounded (asymptotically flat at spatial infinity) and stable (held together for all times by its own gravitational attraction). No such object is believed to exist, a geon should be radiating gravitational waves, lose energy, and decay for long times. No-go the... | 2 | https://mathoverflow.net/users/11260 | 439734 | 177,584 |
https://mathoverflow.net/questions/439632 | 2 | I have come across the following problem. Let $d\in\mathbb{N}$. Let $G$ be any $k$-regular connected directed graph with $n$ vertices, no parallel edges and no 2-cycles. For a vertex $v\in G$, let $e\_v$ denote the union of $\{v\}$ and the end vertices of edges starting at $v$. I would like to assure that there are seq... | https://mathoverflow.net/users/494777 | Bound for a sequence of vertices in a graph | Let $q$ be a prime power and let $P$ be a projective plane of order $q$. It has $q^2+q+1$ points and $q^2+q+1$ lines. Each point lies on $q+1$ lines, and each line has $q+1$ points. Each pair of lines has exactly 1 common point. Since the point-line incidence graph is regular and bipartite, there is a bijection $L$ fro... | 1 | https://mathoverflow.net/users/9025 | 439751 | 177,589 |
https://mathoverflow.net/questions/439329 | 2 | Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\\ \dfrac{\partial u}{\partial\nu} (x)=0, & x\in\partial\Omega\end{cases}$$
We assume $\Omega\subseteq\mathbb{R}^2$ to be a... | https://mathoverflow.net/users/61629 | Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$? | For (3), this follows from De-Giorgi-Nash-Moser so long as you are OK with dependence of the constant on the $C^{0, 1}$ character of the domain, not just e.g. the size.
This can be checked by running through the De Giorgi proof (all the energy inequalities are valid for the Neumann problem), or, with some extra care,... | 1 | https://mathoverflow.net/users/378654 | 439755 | 177,590 |
https://mathoverflow.net/questions/439719 | 0 | Let $X$ and $Y$ be two integral separated Noetherian Gorenstein schemes over a base field $k$ of arbitrary characteristic whose local rings are unique factorization domains and $f: X\to Y$ an étale map between them. Since we assumed these to be Gorenstein, their canonical bundles $\omega\_X$, resp. $\omega\_Y$ exist an... | https://mathoverflow.net/users/108274 | Relation between canonical bundles under étale maps | The usual definition of $\omega\_X^\bullet$ is $\pi\_X^!(k[0])$, where $\pi\_X \colon X \to \operatorname{Spec} k$ is the structure morphism. If $f \colon X \to Y$ is a map of $k$-schemes, we therefore get an isomorphism $f^! \omega\_Y^\bullet \stackrel\sim\to \omega\_X^\bullet$; see for instance [Tag [0ATX](https://st... | 5 | https://mathoverflow.net/users/82179 | 439756 | 177,591 |
https://mathoverflow.net/questions/439753 | 1 | The [hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula) give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ways to build the Ferrers diagram of $\lambda$ from the empty partit... | https://mathoverflow.net/users/76764 | hook length formula for plane partitions | I'll convert my comments to an answer.
You are asking about the number of linear extensions of a poset $P$ which is a finite order ideal (downwards closed set) in $\mathbb{N}^3$. With [SageMath](https://www.sagemath.org/) I was easily able to compute that the number of linear extensions of $P=[3]\times[3]\times[3]$ i... | 6 | https://mathoverflow.net/users/25028 | 439757 | 177,592 |
https://mathoverflow.net/questions/439687 | 1 | There is
>
> a unique simplicial set with a unique non-degenerate simplex in each dimension, (updated) and such that all faces of the non-degenerate simplex are non-degenerate.
>
>
>
Does it have a name, and what can be said about it ? What is it geometrically ? Can one think of it as a "fat" point ?
I have ... | https://mathoverflow.net/users/494312 | The simplicial set with a unique non-degenerate simplex in each dimension | As already pointed out in the comments, such a simplicial set is highly nonunique. For example, in addition to the simplicial set $S$ described in the second paragraph one could take the wedge of simplicial spheres, with one sphere for every dimension.
If we concentrate our attention on the simplicial set $S$ describ... | 1 | https://mathoverflow.net/users/402 | 439767 | 177,597 |
https://mathoverflow.net/questions/438367 | 8 | Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S\_n$.
Equivalently this is the number of ways that $G$ can act on the set $\{1,2,\ldots,n\}$.
Let $a\_{G,n}$ be the number of homomorphisms from $G$... | https://mathoverflow.net/users/470870 | Asymptotic number of permutation representations of a given group | In general it is not known whether $\alpha(G)$ exists as a limit (as opposed to limsup). For virtually solvable groups of finite rank (whose subgroup growth is at most polynomial) the relation between subgroup growth and representation growth given in the comments shows that $\alpha(G) \le 1$ (and equals it if the grou... | 5 | https://mathoverflow.net/users/32210 | 439768 | 177,598 |
https://mathoverflow.net/questions/438671 | 4 | The holomorphic function
$$F(\tau)=-\frac{1}{\vartheta\_4(\tau)}\sum\_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$
is a mock modular form for the congruence subgroup $\Gamma^0(4)$ of $\text{SL}(2,\mathbb Z)$ with shadow $\eta^3$. Here, $q... | https://mathoverflow.net/users/321953 | Evaluation of mock modular forms at elliptic points | The holomorphic function $F$ is related to the $q$-series $H^{(2)}$ of Mathieu moonshine in a simple way. We have
\begin{equation}
\begin{aligned}
H^{(2)}(\tau)&=2\frac{\vartheta\_2(\tau)^4-\vartheta\_4(\tau)^4}{\eta(\tau)^3}-\frac{24}{\vartheta\_3(\tau)}\sum\_{n\in\mathbb Z}\frac{q^{\frac{n^2}{2}-\frac18}}{1+q^{n-\fr... | 1 | https://mathoverflow.net/users/321953 | 439774 | 177,599 |
https://mathoverflow.net/questions/439794 | 4 | In my course on probabilistic graphical models, my professor made a claim which I find a little sus. In discussing the equivalence between Markov Random Fields and Factor Graphs, the following example was given:
An MRF over 3 variables was drawn as a triangle graph (3-cycle). This admits density function like $\psi(x... | https://mathoverflow.net/users/123034 | Can nonnegative functions $f(x,y,z)$ be written as a product of pairwise functions $u(x,y) v(y,z) w(x, z)$? | Let $f(x,y,z)=x^2+y^2+z^2$ for $(x,y,z) \in \mathbb{R}^3$. The only zero of $f$ is $(0,0,0)$.
If we had three functions $u,v,w$ such that $f(x,y,z)=u(x,y)v(y,z)w(z,x)$ for every $(x,y,z) \in \mathbb{R}^3$, at least one of the functions $u,v,w$ would vanish at $(0,0)$, so $f$ would vanish on a whole line.
This argum... | 17 | https://mathoverflow.net/users/169474 | 439799 | 177,603 |
https://mathoverflow.net/questions/439783 | 2 |
>
> **Question:** Does there exist an isomorphic predual of $\ell^1$, which does not have a quotient isomorphic to $c\_0$?
>
>
>
Thanks in advance.
---
Edit: The answer is no. Let $X$ be a Banach space such that $X^\*$ is isomorphic to $\ell^1$. If there didn't exist any surjective bounded $T:X\to c\_0$, t... | https://mathoverflow.net/users/164350 | $\ell^1$ predual with no $c_0$ quotient? | In my weak$^\*$ basic sequences paper with Rosenthal we proved that if $\ell\_1$ embeds into $X^\*$ and $X$ is separable, then $c\_0$ is isomorphic to a quotient of $X$.
| 2 | https://mathoverflow.net/users/2554 | 439801 | 177,605 |
https://mathoverflow.net/questions/439792 | 17 | Let $T\subset \mathbb{R}^n$ be a fixed simplex, $H\subset \mathbb{R}^n$ be a variable affine hyperplane. Is it true that the maximal area (i.e. the $(n-1)$-dimensional volume) of $T\cap H$ is attained when $H$ contains a facet of $T$?
| https://mathoverflow.net/users/4312 | Is a facet always a maximal area section of a simplex? | No, there is a $5$-dimensional simplex with a hyperplane section which is larger than any of its facets, see [Walkup, A simplex with a large cross-section, Am. Math. Monthly, January 1968](https://www.jstor.org/stable/pdf/2315102.pdf).
The idea is to squeeze a regular 5-simplex along the common perpendicular of two opp... | 23 | https://mathoverflow.net/users/98590 | 439818 | 177,610 |
https://mathoverflow.net/questions/439803 | 1 | I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded linear operators also enherit the property. We recall the definition here
>
> **Definition.** Let $X$ be a Banach sp... | https://mathoverflow.net/users/425509 | The space of linear operators between Hilbert spaces has martingale type 2 | As noted by Mikael de la Salle, $L(H,K)$ contains an isometric copy of $\ell^\infty$, which is not of martingale type $p$ for any $p\in(1,2]$, and hence $L(H,K)$ is not of martingale type $p$ for any $p\in(1,2]$.
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The answer to your highlighted **Problem** is also negative. Indeed, suppose that $H=K=\ell^2$ an... | 1 | https://mathoverflow.net/users/36721 | 439819 | 177,611 |
https://mathoverflow.net/questions/439816 | 0 | Recall that a probability $\mu$ measure on a metric space $(X,d)$ is called *Ahlfors $q$-regular* if there are $0<c\le C$ such that: for $\mu$-a.e.\ $x\in X$ one has
$$
cr^q \le \mu(B(x,r)) \le Cr^q,
$$
for any $0\le r\le \operatorname{diam}(X,d)$.
If $(X,d)$ is a finite metric space, then can it support an Ahlfors r... | https://mathoverflow.net/users/491352 | Can a measure on a finite metric space be Alhfors regular? | Take any $x\in X$ with $m:=\mu(\{x\})\in(0,\infty)$ (since $X$ is finite and $\mu$ is a probability measure, such a point $x$ exists). Let $R:=\min\{d(y,x)\colon y\in X\setminus\{x\}\}$. Then $R\in(0,\infty)$ and $B(x,r)=\{x\}$ for $r\in(0,R)$.
Letting now $r\downarrow0$, from $cr^q \le \mu(B(x,r)) \le Cr^q$ we get $... | 2 | https://mathoverflow.net/users/36721 | 439824 | 177,613 |
https://mathoverflow.net/questions/439347 | 7 | A common technique for constructing objects (sheaves) and morphisms in algebraic geometry is faithfully flat descent. Roughly speaking this consists on constructing an object or a morphism "locally" in a fpqc covering of a scheme, verifying that the local constructions form a descent datum, and using the theorem that s... | https://mathoverflow.net/users/143492 | Faithfully flat descent in complex analytic geometry | There will not be a way to directly translate faithfully flat descent into gluing over analytic opens; the former is genuinely stronger than the latter in the analytic category. However basically any faithfully flat descent statement that holds for schemes and makes sense for complex analytic spaces (including those in... | 3 | https://mathoverflow.net/users/109356 | 439830 | 177,615 |
https://mathoverflow.net/questions/303687 | 3 | Hairer in his [spdes notes](http://www.hairer.org/notes/SPDEs.pdf) on pg.6, says that GFF is the stationary solution of $u\_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda\_{k}} B\_{k}(t)e\_{k}(x)$$
for iid Brownian motions $B\_{k}$ and L2 basis $e\_{k}$. So that m... | https://mathoverflow.net/users/99863 | Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain | Yes, indeed, one nice reference is ["An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures"](https://arxiv.org/abs/1901.07420) section 2.5.1 Gaussian free field.
>
> The stochastic heat equation (2.3.1) does not admit an invariant measure, since we have seen that... | 1 | https://mathoverflow.net/users/99863 | 439839 | 177,617 |
https://mathoverflow.net/questions/439809 | 5 | Given a complete graph with $n$ nodes, if we want to use $n$ complete subgraphs to cover the graph and ask what is the minimum possible size of each complete subgraph, the answer is $\Theta(\sqrt{n})$: there are $\Theta(n^2)$ edges and each clique can cover at most $O(n)$ edges, so the lower bound for the complete subg... | https://mathoverflow.net/users/498695 | Cover a graph with small size complete graphs | Here is an example showing that the clique size $\sqrt{m/n}$ does not suffice.
Graphs on $n$ vertices without 4-cycles have less than $O(n^{3/2})$ edges, and there is a series of examples where this asymptotic bound is attained. If one wishes to cover all edges of such a dense graph with $n$ cliques of equal size, th... | 5 | https://mathoverflow.net/users/98590 | 439844 | 177,619 |
https://mathoverflow.net/questions/439843 | 0 | Let $R$ be an integral domain, $S$ be a finitely presented $R$ algebra. Then for a flat $R$ module $M$ which is also a finitely generated $S$ module I need to show that $M \otimes\_{R}T$ is a fintely presented $S\otimes\_{R}T$ module where $T$ is the quotient field of $R$.
My attempt :
Since $M$ is finitely generated... | https://mathoverflow.net/users/443060 | Proving finite presentation | I am following the notation in the comments. I do not have privilege to make a comment, do I should write and answer. Also from the comments I understand that you are working just with commutative rings.
Notice that $R[x\_1,\dots, x\_n]/(f\_1,\dots,f\_m)\otimes T$ is isomorphic to a quotient of $T[x\_1,\dots,x\_n]$ s... | 1 | https://mathoverflow.net/users/490959 | 439851 | 177,621 |
https://mathoverflow.net/questions/424313 | 3 | I am trying to derive the critical coupling strength for synchronisation in a network of phase oscillators with noisy input.
I am following the steps outlined in
Sakaguchi, Hidetsugu. ["Cooperative phenomena in coupled oscillator systems under external fields."](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.... | https://mathoverflow.net/users/483817 | Derivation of a series expansion | To simplify the notation I denote $\tilde{K}=K\sigma/D$ and $\tilde{\omega}=\omega/D$.
First expand
\begin{align}
&n(\psi , \omega)= \exp \bigl(-\tilde{K} +\tilde{\omega} \psi+\tilde{K} \cos \psi\bigr) n(0 , \omega)\nonumber\\
&\times\biggl[1+\frac{\bigl(e^{-2 \pi \tilde{\omega}}-1\bigr) \int\_{0}^{\phi} e^{(-\tilde{... | 3 | https://mathoverflow.net/users/11260 | 439860 | 177,622 |
https://mathoverflow.net/questions/439862 | 6 | So, I know that the existence of a scale (that is, a linear cofinal set in $({}^\omega \omega , \leq^\ast )$, where $\leq^\ast $ is eventual domination, is equivalent to $\mathfrak{b} = \mathfrak{d}$, and that we can obtain the latter through Martin's Axiom (see Hechler forcing).
My question is, what about cofinal tr... | https://mathoverflow.net/users/495743 | Cofinal trees in $({}^\omega \omega , \leq^\ast )$ | Hechler proved that you can force the existence of a cofinal subset of $\omega^\omega$ having any shape you like (subject to one or two necessary restrictions):
**Theorem:** (Hechler) Suppose that $(P,\leq)$ is a partially ordered set with the property that every countable subset of $P$ has a strict upper bound in $P... | 6 | https://mathoverflow.net/users/70618 | 439865 | 177,623 |
https://mathoverflow.net/questions/439744 | 2 | Let $X,Y$ be algebraic subsets of $\mathbb A^n.$ I would like to show that if $X$ and $Y$ intersect "transversely" then $I(X)+I(Y)$ is radical (so $I(X\cap Y)=I(X)+I(Y)$).
How to prove it?
"transversely" can mean that $T\_p\, X+ T\_p\, Y=T\_p\, \mathbb A^n$ for all $p\in X\cap Y$ and $X$ and $Y$ are smooth at all int... | https://mathoverflow.net/users/23935 | When a sum of the ideals is radical | (Adapting earlier comment into an answer.) Assume for simplicity that $X$ and $Y$ are irreducible, or in general take an irreducible component of each. Let $Z$ be an irreducible component of the intersection $X \cap Y$ and let $p \in Z$. Let $a = \dim X$, $b = \dim Y$.
Since $X$ and $Y$ are smooth at $p$, $\dim T\_p ... | 2 | https://mathoverflow.net/users/88133 | 439873 | 177,625 |
https://mathoverflow.net/questions/439869 | 3 | I have heard the expression recently that one should be careful when constructing cones in the homotopy category - namely, that this is not functorial. However, when working through some examples in cochain complexes, I was playing around with the following construction which I haven't been able to understand in relati... | https://mathoverflow.net/users/171987 | Why does this construction not give a functorial cone in the homotopy category of cochain complexes? | The main issue you run into is that the projection
$\Pi: Ch^\*\_h(k)^{\to} \to K(Ch^\*(k))^{\to}$
is not fully faithful, and so the Kan extension $L$ is not automatically an "extension": it does not satisfy $L \Pi(f) \simeq \mathrm{Cone}(f)$.
(This occurs prominently when the map $f$ is the map $0 \to k$. If $g = k[-... | 2 | https://mathoverflow.net/users/360 | 439874 | 177,626 |
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