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https://mathoverflow.net/questions/326384 | 3 | Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta\_D$ on a set of cardinaltity 4 is the matrix
$$\Delta\_D := \left( \begin{matrix} 2 & -1 & \\
-1 & 2 & -1 \\
& -1 & 2 & -1 \\
& & -1 &2
\end{matrix}\right)\in \mathbb C^{4 \times 4}.$$
This matrix is s... | https://mathoverflow.net/users/137516 | Rate of convergence for eigendecomposition | Even though this is not a circulant matrix, its eigenvalues and eigenvectors are known in closed form; see for instance [this Wikipedia article](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors_of_the_second_derivative#Pure_Dirichlet_boundary_conditions_2), which gives an expression for the eigenvectors and e... | 2 | https://mathoverflow.net/users/1898 | 326395 | 140,498 |
https://mathoverflow.net/questions/325798 | 7 | Let us consider the Schrödinger operator
$$
H\_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x)
$$
on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a parameter.
It is easy to see that (up to normalization)
$$
\psi\_0^h(x)=e^{h\cos(x)}
$$
is an eigenfunction for the e... | https://mathoverflow.net/users/124450 | Monotonicity of Schrödinger Eigenvalues | As requested in the comment, let me explain what I could extract from the literature on this problem. (This is not the solution asked for in the OP, but what I have would be too long for a comment.)
The potential in the OP is known as the "Razavy potential" or "double cosine potential". A recent study is [Exact Solut... | 2 | https://mathoverflow.net/users/11260 | 326396 | 140,499 |
https://mathoverflow.net/questions/326375 | 5 | Let $X$ be an Artin stack over $k=\mathbf{C}$. I've heard that $X$ has a topological realisation, but I've not been able to find an explicit decription.
My first guess would be: take a smooth cover $U\to X$ ($U$ is a scheme), then consider the simplicial space
$$\cdots \substack{\rightarrow\\[-1em] \rightarrow \\[-1e... | https://mathoverflow.net/users/119012 | Topological realisation of a stack (explicit description) | Let $\mathrm{Ét}\_\mathbb{C}$ be the étale $\infty$-topos of schemes over $\mathbb{C}$, that is the $\infty$-categories of étale sheaves of $\infty$-groupoids over $\mathbb{C}$. This contains necessarily as a subcategory the sheaves of 1-groupoids, that is the étale stacks over $\mathbb{C}$ and so in particular the Art... | 6 | https://mathoverflow.net/users/43054 | 326398 | 140,500 |
https://mathoverflow.net/questions/324812 | 10 | Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.
I know that to construct the Jacobian variety associated to $C$, one integrates a basis of global holomorphic differential forms over the contours of the curve's homology group. I'm looking for... | https://mathoverflow.net/users/120369 | The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve | I'm getting back to the question of describing holomorphic $1$-forms on a plane curve.
**Affine curves:** Let $X$ be a smooth curve in $\mathbb{A}^2$, given by the equation $F(x,y)$. Then $F\_x dx + F\_y dy=0$. Since $F=0$ is smooth, the functions $F\_x$ and $F\_y$ have no common zero on $X$ and we can define a $1$-... | 8 | https://mathoverflow.net/users/297 | 326416 | 140,506 |
https://mathoverflow.net/questions/304722 | 7 | To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l\_n^\alpha$ for all intervals $I$ with $|I| = l\_n$, for some $0 \leq \alpha \leq 1$ and a certain sequence $l\_1 \geq l\_2 \geq \dots \to 0$, but for which... | https://mathoverflow.net/users/45220 | Examples of probability measures with `fake' decay | Later on in my research I answered this question in the negative:
Consider a sequence of dyadic scales $\{ r\_k \}$, such that $r\_k/r\_{k+1} \geq 4$. Consider a sequence of dyadic lengths $\{ r\_k \}$, and construct a set $X$ by a Cantor-type decomposition, defined as $\lim\_{k \to \infty} X\_k$ where $X\_k$ is a un... | 2 | https://mathoverflow.net/users/45220 | 326419 | 140,509 |
https://mathoverflow.net/questions/326391 | -1 | Consider the set of all discrete functions from $\mathbb F\_q^2$ to $\mathbb F\_q$ (denote it by $\mathcal F(\mathbb F\_q^2, \mathbb F\_q)$, the cardinality $|\mathcal F(\mathbb F\_q^2, \mathbb F\_q)| = q^{q^2}$). Some of the functions can be computed just from the *value* of others, i.e.
$$
f(x,y) = \phi\_{f,g} (g(x,y... | https://mathoverflow.net/users/101533 | Discrete functions as functions of value of another function | This is essentially counting the number of partitions of a set of $q^2$ labelled elements into $q$ unlabelled groups. (This generalizes to counting surjective maps from one labelled set onto a smaller set.) This is because you only ask for a "rewriting of the output" by a unary function on the range set. I believe the ... | 1 | https://mathoverflow.net/users/3402 | 326420 | 140,510 |
https://mathoverflow.net/questions/326387 | 8 | On triangulated categories we have a notion of [Bridgeland stability conditions](http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf).
Is there any known notion of "derived stability conditions" on a stable $\infty$-category $C$ such that they become Bridgeland stability conditions after passin... | https://mathoverflow.net/users/137379 | $\infty$-categorical understanding of Bridgeland stability? | Yes, this is chapter 7 of [Fosco Loregian's thesis](http://www.math.muni.cz/~loregianf/stuff/main.pdf), linked from his webpage. The paper Simone Virili linked to is one of 3 papers making up the thesis. Specifically, Section 7.2.1 discusses the topology, after Bridgeland, and proves the comparison you asked for. The s... | 6 | https://mathoverflow.net/users/11540 | 326422 | 140,511 |
https://mathoverflow.net/questions/326411 | 1 | Let $X\_1, X\_2, \ldots$ be a sequence of positive iid random variables with mean $\mu$ whose distribution admits a moment generating function in a neighborhood of zero.
Let $N\_t$ be the associated renewal process given by
$$N\_t = \sup \left\{ m \geq 0: \sum\_{i=1}^m X\_i \leq t \right\}.$$
I am looking for DKW ... | https://mathoverflow.net/users/38424 | DKW type inequality for renewal processes | $\newcommand{\si}{\sigma}
\newcommand{\ep}{\varepsilon}
$
Without loss of generality, $\ep\ge4/n$, because otherwise the bound $Ke^{-n\ep^2}$ on a probability (with an unspecified constant $K$) is trivial.
Let $S\_m:=\sum\_1^m X\_i$. Then for $t\ge0$
\begin{equation}
S\_{N\_{nt}}\le nt<S\_{N\_{nt}+1}.
\end{equation}... | 3 | https://mathoverflow.net/users/36721 | 326434 | 140,516 |
https://mathoverflow.net/questions/259563 | 4 | In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available [here](http://www.dblab.ntua.gr/~gtsat/collection/scheduling/Balls%20into%20Bins%20A%20Simple%20and%20Tight%20Anal.pdf) strong upper and lower bounds are proved about the number $M$ of balls in a maximally loaded bin when $m$ balls ... | https://mathoverflow.net/users/17773 | Variance of load in maximally loaded bin, if $m$ balls are thrown into $n$ bins | Theorem 4.4 in Devroye's book *On Bucket Algorithms* answers this question. His $n,m$ are opposite to those in the question and uniformity is not assumed. Subject to some technical conditions on a nonlinear, nonnegative, convex fumction $g(x)$ on $[0,\infty)$ which are satisfied here, with $g(x)=x^2,$
his result can b... | 0 | https://mathoverflow.net/users/17773 | 326439 | 140,519 |
https://mathoverflow.net/questions/326414 | 6 | Consider the familiar sequence of [Fibonacci numbers](https://en.wikipedia.org/wiki/Fibonacci_number): $F\_0=0, F\_1=1, F\_n=F\_{n-1}+F\_{n-2}$.
Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a different approach. Hence,
>
> **QUESTION.** Is there a combinator... | https://mathoverflow.net/users/66131 | Products and sum of cubes in Fibonacci | $F\_n$ is the number of compositions (ordered partitions) of $n-1$ into
parts equal to 1 or 2. The number of triples $(a,b,c)$ of such
compositions is $F\_n^3$. The number such that $a,b,c$ all begin with 1
is $F\_{n-1}^3$. The number such that $a,b,c$ all begin with 2 is
$F\_{n-2}^3$. Otherwise either one of $a,b,c$ b... | 15 | https://mathoverflow.net/users/2807 | 326440 | 140,520 |
https://mathoverflow.net/questions/326436 | 5 | As a follow up to my previous two MO questions, [here](https://mathoverflow.net/questions/324816/oddity-of-generalized-catalan-numbers-part-i) and [here](https://mathoverflow.net/questions/325057/oddity-of-q-catalan-polynomials-part-ii), let's consider the below inquiry.
Define the **Fibonacci-Catalan** numbers by $F... | https://mathoverflow.net/users/66131 | "Oddity" of Fibonacci-Catalan numbers | Let $\alpha=(1+\sqrt{5})/2,\beta=(1-\sqrt{5})/2$, then by Binet formula for Fibonacci numbers we have $F\_n=(\alpha^n-\beta^n)/(\alpha-\beta)=:P\_n(\alpha,\beta)$. Factorize our Catalan-like expression onto cyclotomics:
$$
\frac{\prod\_{j=1}^{2n} P\_j(x,y)}{\prod\_{i=1}^{n+1}P\_i(x,y)\cdot \prod\_{i=1}^{n}P\_i(x,y)}=\p... | 11 | https://mathoverflow.net/users/4312 | 326449 | 140,522 |
https://mathoverflow.net/questions/326400 | 14 | By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible [in this sense](https://ncatlab.org/nlab/show/accessible+model+category) (and just in case: even by using Vopěnka's principle).
| https://mathoverflow.net/users/24563 | Example of non accessible model categories | I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky *Locally presentable and accessible categories* the locally presentable category $\bf Gra$ of graphs has a reflective subcatego... | 8 | https://mathoverflow.net/users/49 | 326451 | 140,523 |
https://mathoverflow.net/questions/326425 | 1 | Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\pi^{-1}(X\_{\mathrm{sm}}) \cong X\_{\mathrm{sm}}$, where $X\_{\mathrm{sm}}$ is the regular locus of $X$ and the isomorphi... | https://mathoverflow.net/users/58203 | Infinitesimal deformation of strict transform | This is an answer that uses the language of Artin rings as in Michael Schlessinger's papers, Michael Artin's papers, etc. The statements are relative to a fixed square-zero extension of Artinian local rings, $$0 \to M \hookrightarrow A'\twoheadrightarrow A \to 0.$$ Denote the residue field $A/\mathfrak{m} = A'/\mathfra... | 3 | https://mathoverflow.net/users/13265 | 326472 | 140,528 |
https://mathoverflow.net/questions/326038 | 1 | We say that a finite, simple, undirected graph $G=(V,E)$ is *contraction-sensitive* if collapsing any $2$ non-adjacent points increases the [Hadwiger number](https://en.wikipedia.org/wiki/Hadwiger_number). An example of such a graph [is the icosahedron](https://mathoverflow.net/questions/325655/contracting-non-adjacent... | https://mathoverflow.net/users/8628 | Are contraction-sensitive graphs necessarily vertex-transitive? | If you pick one triangle of an icosahedron and link all its 3 vertices to an additional vertex, the resulting graph is still contraction-sensitive, as @David Roberson has kindly checked in Sage. So this is a counterexample.
| 2 | https://mathoverflow.net/users/29783 | 326485 | 140,536 |
https://mathoverflow.net/questions/326470 | 32 | I originally asked this question on math.stackexchange.com [here](https://math.stackexchange.com/questions/3155201/does-every-first-order-theory-have-a-finitely-axiomatizable-conservative-exten).
---
There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely... | https://mathoverflow.net/users/4613 | Does "every" first-order theory have a finitely axiomatizable conservative extension? | Essentially, yes. An old result of Kleene [1], later strengthened by Craig and Vaught [2], shows that every recursively axiomatizable theory in first-order logic without identity, and every recursively axiomatizable theory in first-order logic with identity that has only infinite models, has a finitely axiomatized cons... | 27 | https://mathoverflow.net/users/12705 | 326486 | 140,537 |
https://mathoverflow.net/questions/326366 | 0 | Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A\_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.
Let $X(G)\_{\mathbb{Q}}$ be the addtive group of homomorphisms $\chi: G \to GL(1)$ that are defiend over $\mathbb{Q}$.
Then I am wondering why $X(G)\_{\mathbb{Q}}$ is a free ... | https://mathoverflow.net/users/29422 | Does every character from group factor through largest central subgroup? | Every connected reductive group $G$ is a product $Z\cdot G'$ where $Z$ is the connected center of $G$ and $G'$ is the derived group of $G$; moreover $Z\cap G'$ is finite. This is a standard result. See for example 19.25 or 21.61 of Milne, Algebraic Groups, CUP 2017.
| 1 | https://mathoverflow.net/users/137572 | 326491 | 140,540 |
https://mathoverflow.net/questions/326504 | 7 | Very often, in topology, one restricts to a [coreflective](https://ncatlab.org/nlab/show/coreflective+subcategory) [Cartesian closed](https://ncatlab.org/nlab/show/cartesian+closed+category) subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the internal c... | https://mathoverflow.net/users/5801 | The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$ | The relevant paper is [Cartesian closed coreflective subcategories of the category of topological spaces](https://www.sciencedirect.com/science/article/pii/0166864191900046) by Juraj Cincura. The first line of the abstract says "Answering the first part of Problem 7 in [10] we prove that there is no largest Cartesian c... | 7 | https://mathoverflow.net/users/11540 | 326511 | 140,547 |
https://mathoverflow.net/questions/326463 | 3 | Let $E\_{ij}(x)\in \mathrm{Mat}\_{7\times7}(\overline{\mathbb{F}}\_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E\_{12}(x)+E\_{34}(x)+E\_{56}(x),\quad b(y)=1+E\_{23}(y)+E\_{45}(y)+E\_{67}(y),$$ for two indeterminates $x,y$. Set $$G\_8=\langle a(x), b(y) \mid x, y\in\overline{\... | https://mathoverflow.net/users/2024 | Is this unipotent group, over characteristic 2, connected? | Good news and bad news here: This group is connected and Zariski closed for nice general reasons. However, I assume that Dror considered this group because $a(1)$ and $b(1)$ generate a subgroup isomorphic to the dihedral group of order $16$, and are thus relevant to [my question](https://mathoverflow.net/questions/2627... | 5 | https://mathoverflow.net/users/297 | 326522 | 140,550 |
https://mathoverflow.net/questions/326520 | 6 | Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\mathbf{C}\to \mathbf{C}$. Is there a canonical morphism
$$(A\odot X)\oplus (B\odot Y)\to (A\otimes B)\odot (X\oplus Y)$... | https://mathoverflow.net/users/124042 | Two monoidal structures and copowering | **No.** Consider the case where $(M,\otimes,1)$ is $(\mathbf{Set},\times,1)$, so the enrichment is vacuous, and $(C,\oplus,0)$ is $(\mathbf{Set},+,0)$, with copowering $\odot$ given by $\times$.
Then the morphism you ask for would give a map
$$(A \times X) + (B \times Y) \longrightarrow (A \times B) \times (X + Y) $$... | 9 | https://mathoverflow.net/users/2273 | 326524 | 140,551 |
https://mathoverflow.net/questions/326514 | 0 | What I have in mind is a mechanical system that is described by an implicit system of ODEs or a system of DAEs (*differential algebraic equations*). The system is asymptotically stable, meaning that there exists an attracting orbit that the system converges to for all initial values (except possibly for the trivial ini... | https://mathoverflow.net/users/121477 | Nonsmooth dynamical system (DAE) - systematic way to calculate period numerically? | I would program this using an adaptive step size approach where the step size decreases as some power (square root works well, empirically) of the error associated with an assumed period. Specifically:
Start with a point $y\_0$ in the dynamics space that is "near" the asymptotic periodic orbit, some time step size $\... | 0 | https://mathoverflow.net/users/82067 | 326527 | 140,553 |
https://mathoverflow.net/questions/326534 | 6 | In Krause and Nikolaus' "Lectures on topological Hochschild homology and cyclotomic spectra" (which can be found [here](https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/papers.html)), there is a lemma I'm trying to understand, but one line in the proof eludes me. The lemma and proof are as follows:
>
> **Lem... | https://mathoverflow.net/users/29322 | $p$-adic equivalence of spectra with $G$-action | I don't know if "$2$-stage nilpotent" is a standard term for this, but I'm sure that what the author means is this: a group $A$ that $G$ is acting on has a subgroup $B$ such that the action of $G$ fixes every element of $B$ and such that also the resulting action of $G$ on $A/B$ is trivial. This is true for $A=H\_n(Z;F... | 9 | https://mathoverflow.net/users/6666 | 326536 | 140,554 |
https://mathoverflow.net/questions/326513 | 2 | Consider a perceptron $F(x) = \phi(x \* w - b), \ x \in \mathbb{R}^n,$ (with Heaviside activation function $\phi$) and a dataset consisting of a finite subset $\Omega \subseteq \mathbb{R}^n$ with labels $y: \Omega \rightarrow \{0, 1\}$. Clearly, $F$ can fit the dataset exactly (in the sense that there are $w$ and $b$ s... | https://mathoverflow.net/users/137583 | Can a Multilayer Perceptron fit any binary function? | Denote the points in $\Omega$ as $\omega\_i$, $1\leq i \leq m$.
As every finite point set has an extreme point, it is possible to arrange the points in an order such that $\omega\_i$ is separable from $\omega\_{i+1},\omega\_{i+2},...,\omega\_{m}$ by a hyperplane.
In other words, there exists $a\_i=ϕ(w\_i∗x−b\_i)$... | 1 | https://mathoverflow.net/users/125498 | 326557 | 140,557 |
https://mathoverflow.net/questions/325674 | 1 | What happens if the usual geodesic equation on an n-manifold is directly modified from a source dimension 1 space (giving a path) to a dimension 2 space (giving a surface). I suspect that this gives a totally geodesic surface, but I was hoping for somewhere to discuss modifications (adding extra terms like a connection... | https://mathoverflow.net/users/29625 | Surfaces extending modified geodesic paths | Assume that $\gamma\colon\Sigma\to M$ is a map from a surface to an $n$-manifold, both equipped with Riemannian metrics and theire Levi-Civita connections. Then there exists a connection $\nabla^\gamma$ along $\gamma$ induced by $\nabla^M$. Let me rewrite your equation as $\nabla^\gamma\_X(d\gamma\circ Y)=d\gamma(\nabl... | 1 | https://mathoverflow.net/users/70808 | 326559 | 140,559 |
https://mathoverflow.net/questions/326555 | 2 | Let $u$ be a solution of $-\Delta u= f \text{ in } \Omega$ with $u=0 \text{ on } \partial \Omega.$ If $f\in L^2(\Omega)$, then $u\in H^2(\Omega)\cap H^{1}\_{0}(\Omega)$provided $\Omega$ is a smooth bounded domain.
Does this type of theorem holds true for the equation $(-\Delta)^s u= f \text{ in } \Omega$ with $u=0 \... | https://mathoverflow.net/users/127663 | Boundary regularity type results of fractional laplacian | In your case, $u$ is locally in the fractional Sobolev space $H^{2s}$ (inside $\Omega$), and globally in $H^s$. The latter is either an assumption (in the weak formulation of the problem) or a proposition (when a different notion of a solution is used). The former follows already from the result given in Stein's book:
... | 2 | https://mathoverflow.net/users/108637 | 326563 | 140,560 |
https://mathoverflow.net/questions/326541 | 5 | Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $\bigcup\_{A\in\mathfrak{A}}A$ is Borel measurable?
| https://mathoverflow.net/users/137602 | Ordered union of Borel sets | ### Construction
Consider some non-Borel set $Y \subset [0,1]$ (e.g. Vitali set).
Enumerate $Y$ using ordinals as $Y=\{x\_\alpha\}\_{\alpha < \beta}$.
Let $m$ denote the smallest ordinal such that $X=\{x\_\alpha\}\_{\alpha < m}$ is non-Borel. Note that $m\ge \omega\_1$, since otherwise $X$ would be at most countabl... | 7 | https://mathoverflow.net/users/44463 | 326566 | 140,561 |
https://mathoverflow.net/questions/326575 | 6 | Let $(M,\otimes,1)$ be closed monoidal category and $C$ an $M$-enriched category. Assume we have $C$-objects $X$ and $X'$ and a morphism $f:1\to C(X,X')$ in $M$. We call $f$ an *isomorphism* if there is a $g:1\to C(X',X)$ in $M$ such that
$$1\cong 1\otimes 1\stackrel{g\otimes f}{\to} C(X',X)\otimes C(X,X')\to C(X,X)$$
... | https://mathoverflow.net/users/124042 | Isomorphisms in enriched categories | Unless I've misunderstood something, I think this is all fine. You would need the enriched naturality of $f^\*$ in order to conclude that the collection of $f^\*\_Y$ was induced by some $f$, but in your setup you already assumed this. It's hard to gauge from your question how much you know about enriched categories, so... | 6 | https://mathoverflow.net/users/11540 | 326579 | 140,563 |
https://mathoverflow.net/questions/326413 | 4 | Consider the usual Monge-Kantorovich transportation problem where $X$ and $Y$ are Polish spaces, $\mu$ and $\nu$ are probability measures on $X$ and $Y$, and $c:X\times Y \to \mathbb{R}^+ \cup \{+\infty \}$ is a lower semi-continuous cost function. The Kantorovich duality theorem states that the transportation cost bet... | https://mathoverflow.net/users/70190 | Monge-Kantorovich duality with a $\{0,1\}$ cost function | The strong Kantorovich duality, i.e. the existence of dual solutions, holds whenever $c$ is lower semicontinuous and **real-valued**, and the optimal coupling has finite cost, see Theorem 5.10(ii) in Villani "Optimal Transport old and new". In contrast, for the weak duality ("$\inf = \sup$") it suffices to assume $c$ i... | 2 | https://mathoverflow.net/users/123897 | 326583 | 140,565 |
https://mathoverflow.net/questions/325108 | 3 | In "[Stochastic modified equations and adaptive stochastic gradient algorithms](https://arxiv.org/abs/1511.06251)" (Li et. al 2015) the authors approximate stochastic gradient descent, as in
$$x\_{k+1} = x\_k - \eta u\_k \nabla f\_{\gamma\_k}(x\_k),$$
by an SDE, a so called *stochastic modified equation* (SME)
$$... | https://mathoverflow.net/users/78650 | Why control a continuous approximation of stochastic gradient descent instead of just the SGD? | Introducing a SDE to approximate a random process is very natural as soon as the "jumps" (here the learning rate) are small and has been a standard procedure since Langevin (1908)
The first advantage is that we obtain a "universal model" that doesn't depend on the particularity of the discrete model. The only random o... | 1 | https://mathoverflow.net/users/99045 | 326597 | 140,572 |
https://mathoverflow.net/questions/325885 | 10 | Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian densities, i.e.
$$
f(x)
\approx
G\_n(x)
:=\sum\_{i=1}^n a\_{i,n} g(x;\mu\_{i,n},\sigma\_{i,n}^2),
\quad
a\_{i,n}\in\math... | https://mathoverflow.net/users/99132 | Approximation of a compactly supported function by Gaussians | **Case 1:** Let $f\geq 0$. We consider the following easier problem :$$ \|f-G\_n^\*\|\_{L^1}\leq \inf \{\|f-G\_n\|\_{L^1}:a\_{i,n}\geq 0, \mu\_{i,n}\in \mathbb{R}, \sigma\_{i,n}>0 \}. $$
Since $f\geq 0$, we have $G\_n^\*\rightarrow f $. In this case, for any $n$ we have $\mu\_{i,n}^\*\in [a,b]$. Intuitively, otherwise ... | 2 | https://mathoverflow.net/users/99045 | 326606 | 140,575 |
https://mathoverflow.net/questions/326477 | 0 | I have a question about the explanation of the data defining a so called equivariant sheaf $F$ on a scheme X from wiki: <https://en.wikipedia.org/wiki/Equivariant_sheaf>. Let denote by $\sigma: G \times\_S X \to X$ an action of a group scheme $G$ on $X$ . Then a $O\_X$-module $F$ is called equivariant if there exist in... | https://mathoverflow.net/users/108274 | Equivariant Sheaf: Explanation on Stalks | I am just posting my comment as an answer. For a scheme $S$, one definition of a group $S$-scheme is a datum of $S$-schemes, $$(\pi:G\to S, m:G\times\_S G\to G, i:G\to G, e:S\to G),$$ of an $S$-scheme $G$ and $S$-morphisms $m$, $i$, and $e$ such that for every $S$-scheme $T$, the induced datum of sets, $$(G(T),m(T):G(T... | 3 | https://mathoverflow.net/users/13265 | 326608 | 140,576 |
https://mathoverflow.net/questions/326601 | 6 | It is well-known that the following Hamiltonian system
\begin{eqnarray}
\left\{\begin{array}{rcl}
\frac{dx}{dt}&=&y,\\
\frac{dy}{dt}&=&x(-1+x^2),
\end{array}\right.
\end{eqnarray}
with
$$ H(x,y)=\frac{x^2}2+\frac{y^2}{2}-\frac{x^4}{4}$$
has the solution
$$ x=\tanh\left(\frac{t}{\sqrt2}\right),\qquad y=\frac1{\sqrt2}\... | https://mathoverflow.net/users/137624 | Explicit solution of a Hamiltonian system | Edited.
Surprisingly, there is an elementary solution, if I made no mistake in
the following computation.
Your equation is equivalent to
$$\left(\frac{dx}{dt}\right)^2=\frac{1}{3}(x^6-3x^2+2),$$
(I just plugged $y=dx/dt$ to your Hamiltonian, and used its value $H=1/3$.)
This equation is separable,
$$\frac{t}{\sqrt{... | 12 | https://mathoverflow.net/users/25510 | 326610 | 140,577 |
https://mathoverflow.net/questions/326588 | 2 | Suppose $(x\_t)\_t$ is a bounded $\mathbb F\_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E\_t f(s,x\_s) \le f(s,\mathbb E(x\_s)) = f(s,x).$$
Does an analogous property also hold for any bounded $\mathbb F\_t$ stopping time? More precisely, suppose $T \in (0,... | https://mathoverflow.net/users/78761 | Concavity, martingales and stopping time | I assume $\mathbb{F}\_t$ is the underlying filtration and $\mathbb{E}\_t(\cdot)$ stands for $\mathbb{E}(\cdot | \mathbb{F}\_t)$. Also, $f(s, x)$ in the right-hand side should read $f(s, x\_t)$.
I also leave aside technicalities, such as whether $x\_\tau$ is measurable (which, as far as I remember, may fail to be true... | 1 | https://mathoverflow.net/users/108637 | 326611 | 140,578 |
https://mathoverflow.net/questions/326613 | 12 | While thinking about constructive mathematics, I stumbled on the following notion, and I would like to know if it has a standard name, a simpler equivalent, or has appeared in the literature:
Say that a set $X$ is **mixable** (temporary term, for lack of a better one) when it satisfies the following property:
>
>... | https://mathoverflow.net/users/17064 | Does this "mixable" property have a standard name in constructive mathematics? | According to [this proposition](https://ncatlab.org/nlab/show/flabby+sheaf#characterization_using_the_internal_language), the property in question is (by "a typical argument with Zorn's lemma") equivalent to flabbiness for sheaves on topological spaces. Thus, the term "flabby" is sometimes used more generally inside co... | 8 | https://mathoverflow.net/users/49 | 326618 | 140,579 |
https://mathoverflow.net/questions/301280 | 8 | Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror Calabi-Yau.
In the paper 'Arithmetic mirror symmetry for the 2-torus', authors construct a $\mathbb{Z}$-linear equiv... | https://mathoverflow.net/users/nan | Arithmetic symplectic geometry via mirror symmetry? | Yes. Recently Auroux (jointly with Efimov and Katzarkov) has proposed a definition of the Fukaya category for trivalent configurations of rational curves. If $\Sigma\_g$ is a genus $g$ Riemann surface with $g\geq2$, then its mirror is a trivalent configuration of $3g-3$ rational curves meeting in $2g-2$ triple points. ... | 2 | https://mathoverflow.net/users/43423 | 326622 | 140,581 |
https://mathoverflow.net/questions/326602 | 30 | I am wondering if there is some example of a mathematician or physicist who published other papers at the same time as their PhD work and independently of it which actually eclipsed the content of the PhD thesis.
The only semi-example I can think of immediately is Einstein, whose other publications in 1905 (especiall... | https://mathoverflow.net/users/119114 | Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis | [Anatoly Karatsuba](https://en.wikipedia.org/wiki/Anatoly_Karatsuba) discovered the
[Karatsuba algorithm](https://en.wikipedia.org/wiki/Karatsuba_algorithm) in 1960, and reported it to Kolmogorov who published it under his (Karatsuba's) name without his knowledge. It seems fair to say that this first example of a "div... | 27 | https://mathoverflow.net/users/11260 | 326624 | 140,582 |
https://mathoverflow.net/questions/326577 | 1 | Is there any analog of studying spectral properties of automorphisms of von Neumann algebra? Does it make sense, if anybody knows please give a reference.
| https://mathoverflow.net/users/136400 | Analogue of spectral values of automorphisms in vN algebra | I'm not really sure what you are asking, but is the following of interest:
<https://mathscinet.ams.org/mathscinet-getitem?mr=348518>
<https://www.sciencedirect.com/science/article/pii/0022123674900342>
```
Arveson, William
On groups of automorphisms of operator algebras.
J. Functional Analysis 15 (1974), 217–243. ... | 2 | https://mathoverflow.net/users/406 | 326646 | 140,591 |
https://mathoverflow.net/questions/326374 | 11 | $\newcommand{\Sig}{\Sigma}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\distSO}[1]{\dist(#1,\SO)}$
$\newcommand{\distO}[1]{\text{dist}(#1,\On)}$
$\newcommand{\tildistSO}[1]{\operatorname{dist}\_{\til d}(#1,\SO)}$
$\newcommand{\SOn}{\operatorname{SO}\_n}$
$\newcommand{\On}{\operatorname{O}\_n}$
$\newcommand{... | https://mathoverflow.net/users/46290 | Is there a "formula" for the point in $\text{SO}(n)$ which is closest to a given matrix? | Not quite an answer, but we can compute $Q$ using a single suitably chosen real parameter $\lambda.$
For any $\lambda>0,$ for all matrices $A$ with singular values $\sigma\_1\leq\sigma\_2\leq\cdots\leq\sigma\_n$ satisfying $0<\sigma\_1<\lambda<\sigma\_2,$
$$Q(A)=O(A)O(A^TA-\lambda^2 I)$$
where $I$ is the identity mat... | 1 | https://mathoverflow.net/users/112284 | 326656 | 140,596 |
https://mathoverflow.net/questions/326452 | 1 | There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. How the norm on L^{P} space related to weight $\varphi$? I am reading this:<https://arxiv.org/pdf/0806.3635.pdf>, I hav... | https://mathoverflow.net/users/136400 | Regarding Haagerup $L^{P}$ spaces | For general background on Haagerup $L^p$ spaces, I rather like Terp's lic.scient. thesis. This is old (e.g. is typeset, not LaTeX), but very self-contained and easy to read. It is fortunately available [here](https://www.fuw.edu.pl/~kostecki/scans/terp1981.pdf).
I'm not aware of a more modern "introduction" in this s... | 3 | https://mathoverflow.net/users/406 | 326663 | 140,600 |
https://mathoverflow.net/questions/326678 | 9 | I am trying to understand [this survey article](https://arxiv.org/abs/1810.02664) by Le Gall on Brownian geometry, especially the statement of Theorem 1.
The basic statement of the theorem is
$$(m\_n,d\_n) \to (m\_{\infty}, d\_{\infty})$$
"in the Gromov–Hausdorff sense" as $n \to \infty$, where the convergence is in ... | https://mathoverflow.net/users/4558 | What does convergence in distribution "in the Gromov–Hausdorff" sense mean? | Following the notation of the paper, let $\mathbb{K}$ be the metric space of all compact metric spaces, equipped with the Gromov-Hausdorff metric $\mathrm{d\_{GH}}$. Then we can express convergence in distribution in the usual way: for every bounded continuous $F : \mathbb{K} \to \mathbb{R}$, we have $\mathbb{E}[F((m\_... | 11 | https://mathoverflow.net/users/4832 | 326679 | 140,605 |
https://mathoverflow.net/questions/326620 | 3 | Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?
| https://mathoverflow.net/users/117723 | Hyperbolic 3 manifold with trivial deformation of flat conformal structures | 1. Let $M$ be a compact hyperbolic 3-manifold and let $C(M)$ be the moduli space of flat conformal structures on $M$. The hyperbolic metric on $M$ defines a distinguished point $h\in C(M)$. One says that $h$ is **locally rigid** (in $C(M)$) if $h$ is an isolated point of $C(M)$. By Thurston's holonomy theorem, local ri... | 11 | https://mathoverflow.net/users/21684 | 326680 | 140,606 |
https://mathoverflow.net/questions/326698 | 0 | Can the Green function for the fractional Laplacian operator be estimated from above and below.
$$ \left\{\begin{aligned}
(-\Delta\_x)^{s} G(x, y)+ G(x, y)&= \delta\_{y}(x) &&\text{in } \Omega \\
G(x,y) & =0 &&\text{ in } \mathbb{R}^N\setminus \Omega
\end{aligned}
\right.$$ when $N\geq 2s$ with $s\in (0, 1).$
| https://mathoverflow.net/users/127663 | Estimates for Green function for fractional Laplacian | The usual Green function would rather satisfy $(-\Delta)^s G(\cdot, y) = \delta\_y(\cdot)$, and bounds for this one and bounded $C^{1,1}$ open sets have been obtained independently in:
>
> Z.-Q. Chen, R. Song, *Estimates on Green functions and Poisson kernels for symmetric stable process*, Math. Ann. 312(3) (1998),... | 1 | https://mathoverflow.net/users/108637 | 326701 | 140,614 |
https://mathoverflow.net/questions/326692 | 26 | At time of writing the first definition of a $ (p, q) $-tensor on [the Wikipedia page](https://en.wikipedia.org/wiki/Tensor) is as follows.
---
**Definition.** A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i\_1\dots i\_p}\_{j\_{1}\dots j\_{q}}[\mathbf{f}] $$
to each basis $\mathbf{f}$ of ... | https://mathoverflow.net/users/137577 | Why is the standard definition of a $(p, q)$-tensor so bizarre? | I think that the answer lies in the "educational culture" of physicists. Physicists are often used -well at least at the undergraduate level- to learn and perform complicated computations with abstract objects, without caring much about the structure and the abstract properties of the ambient spaces containing these ab... | 21 | https://mathoverflow.net/users/85967 | 326703 | 140,615 |
https://mathoverflow.net/questions/326650 | 1 | I have asked the following question on [Math.SE](https://math.stackexchange.com/questions/3150310/is-the-prouhet-thue-morse-constant-transcendental-in-any-integer-base-b2) some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.
---
The Prouhet-Thue-Morse constant, defined... | https://mathoverflow.net/users/103722 | Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$? | Michel Waldschmidt, [Words and transcendence](https://hal.archives-ouvertes.fr/hal-00407221/file/WordsTranscendence.pdf), writes in Section 3.1, on page 461, "Mahler also proved in 1929 that the so-called Prouhet–Thue–Morse–Mahler number in base $g\ge2$, given by $$\xi\_g=\sum\_{n\ge0}{a\_n\over g^n}$$
where $(a\_n)\_... | 3 | https://mathoverflow.net/users/3684 | 326710 | 140,618 |
https://mathoverflow.net/questions/326661 | 3 | During my research I came across this question :
**Question:** What's the value of $x\_p=(\dfrac{p-1}{2})! \mod p$ when $p>3$ is prime ?
*Remark:* It's easy to see $x\_p^2 \mod p=(-1)^{\dfrac{p+1}{2}} \mod p$ with $x\_p\in [1;p-1] \cap \mathbb N$
But it's not easy to find if $x\_p \leq\dfrac{p-1}{2}$ or $x\_p> \d... | https://mathoverflow.net/users/110301 | 1/2 Wilson's theorem | For the case $p\equiv 1\pmod{4}$, see the following paper of Chowla: On the class number of real quadratic fields. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 878. There the following is proved:
Let $h$ be the class number of $\mathbb{Q}(\sqrt{p})$, and let $\epsilon = (u+v\sqrt{p})/2$ be the fundamental unit of the corr... | 8 | https://mathoverflow.net/users/16510 | 326715 | 140,620 |
https://mathoverflow.net/questions/326080 | 4 | Yesterday I read the Quanta article [How a Strange Grid Reveals Hidden Connections Between Simple Numbers](https://www.quantamagazine.org/the-sum-product-problem-shows-how-addition-and-multiplication-constrain-each-other-20190206/) about the sum-product problem:
Let $A$ be a set of integers. Erdös and Szemerédi conj... | https://mathoverflow.net/users/7089 | Trick for the sum-product problem | Your question is a well-known and difficult open problem. See [Lower bounds for $|A+A|$ if $A$ contains only perfect squares](https://mathoverflow.net/questions/150142/lower-bounds-for-aa-if-a-contains-only-perfect-squares/150147#150147). To repeat my answer from that question, the best lower bound to date is:
$$|A^2+A... | 5 | https://mathoverflow.net/users/630 | 326724 | 140,622 |
https://mathoverflow.net/questions/326705 | 2 | Do there exist connected proper smooth $\mathbb{C}$-schemes $X\_i$ ($\forall i\in \mathbb{Z}\_{>0}$) with $\mathrm{dim}\_{\mathbb{C}}X\_i=i$ such that $X\_i$ admits an immersion into $X\_{i+1}$ and any connected proper smooth $\mathbb{C}$-scheme admits an immersion into $X\_j$ for some $j\in \mathbb{Z}\_{>0}$? If this ... | https://mathoverflow.net/users/nan | Embedding smooth proper schemes into smooth proper schemes | This is too long for a comment. You can arrange this by blowing up.
Let $(Y\_i,y\_i)\_{i\in \mathbb{Z}\_{>0}}$ be a countable collection of connected, proper, smooth $\mathbb{C}$-schemes with specified closed point $y\_i$ such that every connected, proper, smooth $\mathbb{C}$-scheme admits a closed immersion in the o... | 2 | https://mathoverflow.net/users/13265 | 326733 | 140,623 |
https://mathoverflow.net/questions/326468 | 4 | There are non-isomorphic finite groups with the same (complex) character table, as $D\_4$ and $Q\_8$.
$$\scriptsize\begin{array}{c|c}
\text{class}&1&2A&2B&2C&4 \newline
\text{size}&1&1&2&2&2 \newline \hline
\rho\_1 &1&1&1&1&1 \newline
\rho\_2 &1&1&-1&1&-1 \newline
\rho\_3 &1&1&1&-1&-1 \newline
\rho\_4 &1&1&-1&-1... | https://mathoverflow.net/users/34538 | Finite groups with the same character table *including* class types, and square-free order | (Turning my comments into an answer).
A finite group with all Sylow subgroups cyclic is called a *$Z$-group*.
According to review MR0470050 in MathSciNet, in [1] it is shown that a $Z$-group is determined by its character table. So the answer to question 2 would be no.
I do not have access to [1] and I have not... | 7 | https://mathoverflow.net/users/10146 | 326735 | 140,624 |
https://mathoverflow.net/questions/325950 | 6 | Let $T$ be a rooted tree. For any subtree $T' \subset T$ write $L(T')$ for the number of leaves of $T'$.
Further, for $T' \subset T$ define the *branch-depth* of a node $v \in T'$ as the number of nodes $w$ on the path from $v$ to $root(T')$ having more than a single child. The branch depth of $T'$ is then the maxima... | https://mathoverflow.net/users/25905 | Does a bounded branching/log depth dihotomy hold for rooted trees? | Call a tree on $x$ vertices *low* if its branch-depth is at most $\log x$.
We prove by induction that for every tree on $n$ vertices there are some numbers $a,b$ such that $n\le ab$ and the tree contains a binary subtree on $a$ vertices AND a low subtree on $b$ vertices. Suppose this is false for some $n$, and denote... | 3 | https://mathoverflow.net/users/955 | 326736 | 140,625 |
https://mathoverflow.net/questions/326716 | 3 | What can be said about the Hausdorff dimension of the image of a set by a $W^{1,1}$ map?
In other words,
what is the relationship between
$\mathrm{dim}\_H f(A)$ and $\mathrm{dim}\_H A$, where $f \in W^{1,1}$?
Does the result also hold if $f$ is a $BV$ function?
| https://mathoverflow.net/users/122620 | Hausdorff dimension and $W^{1,1}$ functions | If $f$ is Lipschitz then $\dim\_H f(A) \le \dim\_H A$, since $H^s\_\delta(f(A)) \le Lip(f) \cdot H^s\_\delta(A)$ for any $\delta>0$.
However there exists an absolutely continuous function $f\in W^{1,1}$ such that $\dim\_H(f(A)) > \dim\_H A$ for some set $A$. Let us construct such a function (by a modification of the ... | 2 | https://mathoverflow.net/users/44463 | 326741 | 140,627 |
https://mathoverflow.net/questions/326714 | 1 | Let $S\_n$ be the Selmer group of $E/K\_n$ where $K\_n/K$ is the $n$-th layer of the cyclotomic $\mathbb{Z}\_p$-extension of $K$ and $C\_n$ be the torsion part of $S\_n$.
By Cassels pairing, we know that there is a non-degenerate skew symmetric Galois equivariant pairing $C\_n \times C\_n \rightarrow \mathbb{Q}\_p/\ma... | https://mathoverflow.net/users/116598 | Cassels Pairing for Fine Selmer groups | Flach has defined in [1] a Cassels-Tate pairing on very general Selmer groups, and the kernels of the pairing on both sides are the maximal divisible subgroups. But you need to be careful when applying his results to the fine Selmer group of an elliptic curve. In general, the pairing pairs the Selmer group attached to ... | 2 | https://mathoverflow.net/users/35416 | 326743 | 140,628 |
https://mathoverflow.net/questions/326730 | 1 | Let $M$ be a type $II\_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II\_{1}$ ?
| https://mathoverflow.net/users/136400 | Subalgebras of $II_{1}$ factor | If $M$ is type $II\_1$ then it has a tracial state, and hence so does $B$ by restriction. So if $B$ is an infinite dimensional subfactor it must be type $II\_1$, by inspection of factor types. If $B$ is merely a subalgebra then it could have a type $I$ part, e.g. $\mathbb{C}p + qMq$ where $p$ and $q$ are nonzero projec... | 3 | https://mathoverflow.net/users/23141 | 326746 | 140,629 |
https://mathoverflow.net/questions/326745 | 20 | I stumbled across the following problem in high school:$$
x^2 + y^2 = n!
$$
I tested it within my laptop capabilities, watched a 3b1b video [Pi in prime regularities](https://youtu.be/NaL_Cb42WyY), where he explains how to find the number of integer solutions based on prime factors. There doesn't seem to be any above $... | https://mathoverflow.net/users/137694 | Circle $x^2 + y^2 = n!$ doesn't hit any lattice points for any $n$ except for $0$, $1$, $2$ and $6$ or does it? | For $n\geq 7$, [Erdős proved](https://users.renyi.hu/~p_erdos/1935-10.pdf) in 1932 that there is a prime $n/2<p\leq n$ of the form $p=4k+3$. From this he deduces (in the same paper) that $1!$, $2!$, $6!$ are the only factorials which can be written as a sum of two squares.
| 55 | https://mathoverflow.net/users/11919 | 326752 | 140,631 |
https://mathoverflow.net/questions/326767 | 8 | The set of size-$n$ unitary matrices span $\Bbb C^{n \times n}$ (this can be proven nicely using [polar decomposition](https://en.wikipedia.org/wiki/Polar_decomposition)). If we select a maximal linear subset of unitary matrices, then we have a basis of $\Bbb C^{n \times n}$ consisting of $n^2$ unitary matrices. My que... | https://mathoverflow.net/users/34894 | Is there always a complete, orthogonal set of unitary matrices? | Yes, consider the set of $n^2$ matrices generated by the shift $e\_i \mapsto e\_{i+1}$ and the diagonal matrix with entries $(1,\omega,\omega^2,\cdots,\omega^{n-1})$ where $\omega$ is a primitive $n$th root of unit.
| 14 | https://mathoverflow.net/users/908 | 326768 | 140,637 |
https://mathoverflow.net/questions/326806 | 7 | Given two numbers $p,q\in(0,1)$, we say that *$p$ can simulate $q$* if, given a biased coin with probability $p$, we can toss it a bounded number of times and use the results to simuate a biased coin with probability $q$.
In this [math.SE question](https://math.stackexchange.com/q/2901886/29780) the following results... | https://mathoverflow.net/users/34461 | What numbers can simulate 1/2? | Depends on what kind of "description" you want, but consider the question of the probabilities $p$ that can simulate $1/2$ with exactly $n$ flips. The events you have are:
* One event with probability $p^n$
* $n$ events with probability $p^{n-1}(1-p)$
* $\binom{n}{2}$ events with probability $p^{n-2}(1-p)^2$
* ...
* ... | 4 | https://mathoverflow.net/users/6427 | 326809 | 140,645 |
https://mathoverflow.net/questions/326052 | 7 | This is a continuation of [Classification of algebras of finite global dimension via determinants of certain 0-1-matrices](https://mathoverflow.net/questions/324534/classification-of-algebras-of-finite-global-dimension-via-determinants-of-certai/325637?noredirect=1#comment813583_325637) but this time with a concrete co... | https://mathoverflow.net/users/61949 | On a problem for determinants associated to Cartan matrices of certain algebras | I'll keep the version where we look at
$$D(w,v)=\det\left(I + Z+ \ldots + Z^{w-1}-Z^{w-1}\mathrm{diag}(\mathbf{v})\right)$$
and ask when is $D(w,v)=1$?
In order to give a complete answer to both conjectures we first introduce the quiver $Q\_{w,v}$ defined on the vertices $\{S\_1,S\_2,\dots,S\_n\}$ with edges $S\_i\... | 6 | https://mathoverflow.net/users/2384 | 326811 | 140,646 |
https://mathoverflow.net/questions/326822 | 2 | This is lemma 1.15. of Deligne, Mumford's paper.
Let $X$ be an irreducible stable curve over an algebraically closed field, $\phi$ an automorphism on $X$ which induces the identity on $\text{Pic}^0X$.
Then is $\phi$ the identity on $X$?
Let $X'$ be the normalization and $\phi'$ the action of $\phi$ on $X'$. (induced ... | https://mathoverflow.net/users/128235 | The injectivity of $\text{Aut}(X) \to \text{Aut} (\text{Pic}^0X)$ for a stable curve | Let $s$ be a double point of $X$, and let $p,q$ be the two points of $\pi ^{-1}(s)$ (ordering chosen). Let $\pi :X\_s\rightarrow X$ be the partial normalization of $X$ at $s$. There is an exact sequence of sheaves
$$1\rightarrow \mathcal{O}\_{X}^\*\rightarrow \pi \_\*\mathcal{O}\_{X\_s}^\*\xrightarrow{\ \varphi \ } \ka... | 6 | https://mathoverflow.net/users/40297 | 326825 | 140,649 |
https://mathoverflow.net/questions/326819 | 1 | This is a question posted in MSE before-<https://math.stackexchange.com/questions/3169269/references-on-equivalent-characterization-for-sobolev-spaces-of-functions-of-one>:
I cited a result which characterizes Sobolev spaces of functions of one variable as
>
> $ H^p(a,b):= \{ x \in C^{p-1} [a,b]:
> x^{(p-1)}(t) ... | https://mathoverflow.net/users/114334 | References on equivalent characterization for Sobolev spaces of functions of one variable | In your notation $H^k=W^{k,2}$, where $W^{k,p}$ represents the Sobolev space of functions whose derivatives of orders $\leq k$ are in $L^p$.
A function $f\in W^{1,p}(a,b)$ if and only if there is $g\in L^p(a,b)$ and a constant $\alpha$ such that $f(x)=\alpha+\int\_a^x g(t)\, dt$ for almost all $x\in (a,b)$. Since $f... | 3 | https://mathoverflow.net/users/121665 | 326828 | 140,651 |
https://mathoverflow.net/questions/326759 | 5 | Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there is some diffeomorphism $F:M\to M$ so that $F(p)=q$ and $F$ equals the identity outside a small neighborhood of $p$ and $q... | https://mathoverflow.net/users/69190 | Existence of an isotopy in Riemannian manifold | For the first question, which concerns just the smooth category without reference to metrics, the answer depends on the dimension of $M$. Before explaining this a small clarification is needed. By "a small neighborhood of $p$ and $q$" you probably mean a ball containing $p$ and $q$, otherwise one could take the neighbo... | 8 | https://mathoverflow.net/users/23571 | 326832 | 140,654 |
https://mathoverflow.net/questions/326785 | 2 | Let $K/k$ be a finite separable extension. If necessary, we can assume $[K : k] = 2$. Let $H$ be a $K$-closed subgroup of $\operatorname{GL}\_n$, and let $\tilde{H} = \operatorname{Res}\_{K/k}H$. Since $\tilde{H}$ is a linear algebraic group over $k$, it should embed into some $\operatorname{GL}\_m$. I was wondering if... | https://mathoverflow.net/users/38145 | If $H \subset \operatorname{GL}(n)$, can we realize $\operatorname{Res}_{K/k} H$ inside $\operatorname{GL}([K : k]n)$? | Thanks to LSpice for answering. In the quadratic case, we have $K = k(\sqrt{d})$ for $d \in k$. A basis $e\_1, ... , e\_n$ of an $n$-dimensional $K$-vector space has basis $e\_1, ... , e\_n, \sqrt{d} e\_1, ... , \sqrt{d}e\_n$ over $k$. For $g = (\alpha\_{ij}) \in \operatorname{GL}\_n(K)$ with $\alpha\_{ij} = a\_{ij} + ... | 1 | https://mathoverflow.net/users/38145 | 326840 | 140,656 |
https://mathoverflow.net/questions/325877 | 1 | In discussion of following questions [question1](https://mathoverflow.net/questions/324254/selective-ultrafilter-and-bijective-mapping), [question2](https://mathoverflow.net/questions/324466/on-infinite-combinatorics-of-ultrafilters), [question3](https://mathoverflow.net/questions/324835/the-example-of-the-idempotent-f... | https://mathoverflow.net/users/118366 | Some kind of idempotence of dense filter | The answer is negative. See proof given by Andreas Blass [here](https://mathoverflow.net/questions/325925/the-property-of-the-dense-subfilter-of-a-selective-ultrafilter). Main idea: filter $\cal{F}\otimes\cal{F}$ must contain $\cal{N}\otimes\cal{N}$ because any free filter contains $\cal{N}$ and thus $\cal{N}\subset\ca... | 1 | https://mathoverflow.net/users/118366 | 326844 | 140,658 |
https://mathoverflow.net/questions/326805 | 4 | The [Thomason model structure](https://ncatlab.org/nlab/show/Thomason+model+structure) on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau\_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \rightleftarrows \mathrm{Cat} \colon \mathrm{Ex}^2 \circ N,$$ where $\tau\... | https://mathoverflow.net/users/16109 | Thomason fibrant replacement and nerve of a localization | **Regarding Question 1**, the only time I can think of when a Thomason fibrant replacement can be taken to be a 1-categorical localization is when $C$ has the homotopy type of the classifying space of a discrete groupoid $G$, i.e. $|C|$ is *aspherical*.[1] That is:
* For any category $C$, we have $\Pi\_1(|C|) \simeq ... | 7 | https://mathoverflow.net/users/2362 | 326847 | 140,659 |
https://mathoverflow.net/questions/119570 | 8 | **Theorem.** If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry.
This is a classic result of Blaschke and Hessenberg (that I just learned thanks to Guillaume's comment.). A short simple proof of it can be... | https://mathoverflow.net/users/21123 | Convex bodies with symmetric shadows | The answers are **no** to Question 1, and **yes** to Question 2 (assuming $n\ge 2$).
Let $h$ be the support function of $K$. The projection of $K$ to a linear subspace $L$ is central symmetric iff the restriction of $h$ to $L$ is a sum of an even function and a linear function. Here and below "linear" means $\mathbb ... | 4 | https://mathoverflow.net/users/4354 | 326853 | 140,662 |
https://mathoverflow.net/questions/326851 | 3 | Is there a compact orientable Riemannian manifold which does not have a compact totally geodesic submanifold of codimension $1$?
| https://mathoverflow.net/users/36688 | Existence of compact totally geodesic submanifold of codimension $1$ | According to the answer of Petrunin <https://mathoverflow.net/a/309692/121665>, any metric on a 3-manifold admits arbitrary small $C^\infty$ deformation such that the obtained Riemannian manifold has no totally geodesic surfaces, even locally.
| 6 | https://mathoverflow.net/users/121665 | 326855 | 140,663 |
https://mathoverflow.net/questions/326779 | 2 | I'm curious about the problem of deciding if a given incomplete first-order theory has a stable completion from a descriptive set theory point of view. It seems likely that this problem is $\Pi\_1^1$-complete, but I can't quite prove it myself and I'm having difficulty finding a reference.
Given a partitioned formula... | https://mathoverflow.net/users/83901 | Complexity of deciding if an incomplete first-order theory has a stable completion | EDIT: Thanks to tomasz's comment I realized I was making this more complicated than it needed to be. Here is a simpler construction:
Let $\mathcal{L}=\{\leq\_i\}\_{i<\omega}$ be a countable sequence of binary relations. Given a tree $R\subseteq \omega^{<\omega}$, let $T\_R$ be the theory with the following axioms:
... | 1 | https://mathoverflow.net/users/83901 | 326858 | 140,664 |
https://mathoverflow.net/questions/326860 | 15 | Consider the Young diagram of a partition $\lambda = (\lambda\_1,\ldots,\lambda\_k)$. For a square $(i,j) \in \lambda$, define the *hook numbers* $h\_{(i,j)} = \lambda\_i + \lambda\_j' -i - j +1$ where $\lambda'$ is the *conjugate* of $\lambda$.
The [hook-length formula](https://en.wikipedia.org/wiki/Hook_length_form... | https://mathoverflow.net/users/66131 | hook-length formula: "Fibonaccized" Part I | Sam is correct of course about $q$-hook formula. Below is a short self-contained proof not relying on such advanced combinatorics.
Denote $h\_1>\ldots>h\_k$ the set of hook lengths of the first column of diagram $\lambda$. Then the multiset of hooks is $\cup\_{i=1}^k \{1,2,\ldots,h\_i\}\setminus \{h\_i-h\_j:i<j\}$ an... | 10 | https://mathoverflow.net/users/4312 | 326868 | 140,667 |
https://mathoverflow.net/questions/326803 | 2 | I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:
$r^{+}(\nabla^s) v = f$
where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ restricts a function to $[0,1]$, and $f:[0,1] \rightarrow \mathbb{R}$ with $f(t) = t^{-s}$ or more generally, $f(t) = t^... | https://mathoverflow.net/users/112531 | Solving Fractional Laplacian Equations with Boundary Condition | Here is a solution in an integral form. I suppose it can be written in terms of hypergeometric functions (or maybe Meijer G-functions), but I did not attempt to do that. Once this is done, extension to general $\Re k > -1$ should follow by analytic continuation.
For $k \in \mathbb{C}$ let
$$ f\_k(x) = \begin{cases} x... | 3 | https://mathoverflow.net/users/108637 | 326871 | 140,669 |
https://mathoverflow.net/questions/326867 | 6 | On page 199 of Dummit and Foote's Abstract Algebra (Here $\Phi(G)$ is the Frattini subgroup of a group $G$, not necessarily finite):
>
> If $N\unlhd G$, then $\Phi(N)\subseteq\Phi(G)$.
>
>
>
**First**, When every proper subgroup of $N$ is contained in a maximal subgroup of $N$, I know how to prove the statemen... | https://mathoverflow.net/users/131145 | Frattini subgroup is normal-monotone | Indeed the result is false.
Consider the affine group $G=\mathbf{Q}^\*\ltimes\mathbf{Q}$ and $N$ the normal subgroup $\mathbf{Q}$.
Since $N$ has no maximal proper subgroup $\Phi(N)=N$.
Since $\mathbf{Q}^\*$ is a maximal proper subgroup of $G$ and since the intersection of its conjugates is trivial, we have $\Phi... | 7 | https://mathoverflow.net/users/14094 | 326873 | 140,670 |
https://mathoverflow.net/questions/326885 | 5 | Given a ring $R$ with finite additive basis $\{e\_i\}\_{i=1}^{n}$, such that $e\_i e\_j=\sum c\_{ijk}e\_k$ with $c\_{ijk}\in \mathbb{N}$, we define the Perron-Frobenius dimension $FPDim(e\_i)$ of a basis element $e\_i$ to be the maximal positive real eigenvalue of matrix $M\_{e\_i}$, multiplication by $e\_i$. This exis... | https://mathoverflow.net/users/128502 | Is the Perron-Frobenius dimension of a G-Set given by its cardinality? | Lets fix an orbit $X = G/H$. It suffices to determine the asymptotic growth of the trace of multiplication by $X^n$, since the maximal positive eigenvalue(s) dominates the sum $\sum\_i \lambda\_i^n$.
This trace is the sum $$\sum\_{K \subset G} \langle G/K, X^n \times G/K \rangle,$$ where the sum is over conjugacy cl... | 6 | https://mathoverflow.net/users/52918 | 326892 | 140,679 |
https://mathoverflow.net/questions/326883 | 0 | Let $(T(t))\_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with dissipative self-adjoint generator $(\mathcal D(A),A)$. In particular, $T(t)$ is self-adjoint for all $t>0$. By the spectral theorem, $$T(t)=e^{tA}\;\;\;\text{for all }t\ge0.\tag1$$ Let $(H\_\lambda)\_{\lambda\g... | https://mathoverflow.net/users/91890 | If $A$ is a dissipative self-adjoint operator with spectral decomposition $(H_λ)$, then $e^{tA}x$ tends to the projection of $x$ onto $H_0$ as $t→∞$ | By definition, $$\langle T(t)x,E\_0x\rangle\_H=\left\|E\_0x\right\|\_H^2+\underbrace{\int\_0^\infty e^{-t\lambda}\:{\rm d}\underbrace{\langle E\_\lambda x,E\_0x\rangle\_H}\_{=\:\left\|E\_0x\right\|\_H^2}}\_{=\:0}\tag5$$ and hence $$\left\|T(t)x-E\_0x\right\|\_H^2=\left\|T(t)x\right\|\_H^2-\left\|E\_0x\right\|\_H^2\xrig... | 0 | https://mathoverflow.net/users/91890 | 326894 | 140,681 |
https://mathoverflow.net/questions/231791 | 14 | Briefly, my question is the following.
*does every countable ω-categorical, ω-stable structure with
disintegrated strongly minimal sets interpret in the countable pure set?*
By countable pure set I mean a structure with countable universe and equality relation only.
---
This is a repetition of this question ... | https://mathoverflow.net/users/87983 | ω-categorical, ω-stable structure with trivial geometry not definable in the pure set | The answer to my question was given to me by Ehud Hrushovski, and is as follows
(the following formulation is my own).
Let $D$ be the pure set.
For each pair $\{a,b\}$ of distinct elements of $D$ let $C\_{\{a,b\}}$ be a directed square, i.e., a directed cycle of length four, and let $f\_{\{a,b\}}$ map the two element... | 4 | https://mathoverflow.net/users/87983 | 326895 | 140,682 |
https://mathoverflow.net/questions/326835 | 3 | Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous self maps of some topological space?
Are there at least existing theorems or frameworks for questions related to thes... | https://mathoverflow.net/users/112954 | Category of continuous self maps | Following the comment of Tom Goodwillie, you can recognize the underlying set and the underlying maps from the monoid. This leads to <http://matwbn.icm.edu.pl/ksiazki/fm/fm66/fm6614.pdf>.
| 6 | https://mathoverflow.net/users/73388 | 326896 | 140,683 |
https://mathoverflow.net/questions/326897 | 2 | Given a Lie Algebra $\mathfrak{g}$, its Cartan Matrix $A$ and a finite representation $R$, is there a way of determining its highest weight $\Lambda$ in a simple way?
In my course, we consider $\mathfrak{g}=A\_2= \mathfrak{L}\_{\mathbb{C}}(SU(3))$. It is stated that the highest weight of the fundamental representatio... | https://mathoverflow.net/users/137774 | Highest weight of a representation of a Lie Algebra | To obtain the highest weight of a semisimple Lie algebra $\mathfrak{g}$ you first have to choose Cartan subalgebra $\mathfrak{h} \leq \mathfrak{g}$ and then set of positive roots (or alternatively choose a Borel subalgebra). Then the highest weight vectors of your representation $V$ are given by linear system
$$
\rho(... | 4 | https://mathoverflow.net/users/6818 | 326899 | 140,684 |
https://mathoverflow.net/questions/326898 | 1 | Let $\mathbb{Z}\_m = \mathbb{Z}/m\mathbb{Z}$. Let $A$ be an $k \times n$ matrix over $\mathbb{Z}\_m$. Let $f: \mathbb{Z}\_m^n \to \mathbb{Z}\_m^k$ be a linear map defined by $f(x) = Ax$, $x \in \mathbb{Z}\_m^n$. Are there some references about the condition such that $|\ker(f)|=m^p$ for some positive integer $p$?
For... | https://mathoverflow.net/users/11877 | Reference request: conditions for the cardinality of the kernel of a linear map from $\mathbb{Z}_m^n \to \mathbb{Z}_m^k$ is a power of $m$ | Let $\hat{f}: \mathbb{Z}^n \to \mathbb{Z}^k$ be a linear lift of $f$ and let $e\_1,\ldots, e\_k$ be the elementary divisors of $\text{im}(\hat{f})$ in $\mathbb{Z}^k$.
>
> $|\ker(f)|$ is a power of $m$ iff $\prod\_{i=1}^k \gcd(m,e\_i)$ is a power of $m$ (including $1 = m^0$).
>
>
>
Proof: By the isomorphism t... | 3 | https://mathoverflow.net/users/18571 | 326916 | 140,688 |
https://mathoverflow.net/questions/326907 | 15 | Let $G$ be a finitely generated subgroup of $GL(n,\mathbb{C})$. Assume that there exists a number field $k$ (i.e. a finite extension of $\mathbb{Q}$) such that for all $g \in G$, the eigenvalues of $g$ all lie in $k$. This implies that $g$ is conjugate to an element of $GL(n,k)$.
Question: must it be the case that so... | https://mathoverflow.net/users/137782 | Finitely generated matrix groups whose eigenvalues are all algebraic | At the positive side, if $G$ acts irreducibly on $\mathbf{C}^n$ and $k$ is an arbitrary subfield of $\mathbf{C}$, then the answer is yes (allowing some field extension $k'$ of degree dividing $n$). This even works assuming that $G$ is a multiplicative submonoid of $M\_n(\mathbf{C})$ (keeping the irreducibility assumpti... | 20 | https://mathoverflow.net/users/14094 | 326919 | 140,690 |
https://mathoverflow.net/questions/326922 | 9 | I have found the following Fibonacci Identity (and proved it).
If $F\_n$ denotes the nth Fibonacci Number, we have the following identity
\begin{equation}
F\_{n-r+h}F\_{n+k+g+1} - F\_{n-r+g}F\_{n+k+h+1} = (-1)^{n+r+h+1} F\_{g-h}F\_{k+r+1}
\end{equation}
where $F\_1 = F\_2 = 1$, $r \leq n$, $h \leq g$, and $n, g, k \... | https://mathoverflow.net/users/129192 | Is this a new Fibonacci Identity? | Here is an expanded comment of user44191. The basic observation is that one can extend $F\_n$ to all $n\in {\mathbb Z}$ by requiring $F\_{-n}=(-1)^{n+1}F\_n$. Then by [Vajda's formula](https://en.m.wikipedia.org/wiki/Cassini_and_Catalan_identities), one has $$F\_{n'+a'}F\_{n'+b'}-F\_{n'}F\_{n'+a'+b'}=(-1)^{n'}F\_{a'}F\... | 12 | https://mathoverflow.net/users/104791 | 326925 | 140,692 |
https://mathoverflow.net/questions/326889 | 2 | Consider the $[n]!\_q = \prod\limits\_{k = 1}^{n} \frac{q^k - 1}{q - 1} = \sum\limits\_{k = 0}^{\binom n 2} c\_k q^k$ and let $\{f\_n\}\_{n \in \mathbb{N}}$ be the sequence of the functions on $[0; 1]$ defined by the following $$f\_n(x) = \frac{c\_{\lfloor \binom n 2 x \rfloor}}{n!}$$
Is there a formula for $\lim\limit... | https://mathoverflow.net/users/118839 | $q$-factorial coefficient asymptotics | Let $Z\_n$ be the number of inversions of a random permutation in $S\_n$. Then for all $x\in\mathbb{R}$,
$$ \mathrm{Prob}\left(Z\_n<\frac 14 n^2+\frac 16xn^{3/2}\right)\to \mathcal{N}(x),
$$ the standard normal distribution. This goes back to Feller,
1945. See for instance Theorem 3.3.4 of
<https://www.routledgehandbo... | 4 | https://mathoverflow.net/users/2807 | 326930 | 140,695 |
https://mathoverflow.net/questions/326932 | 3 | We know the Poincare series are defined as the following:
The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is:
$$
P\_{m}^{k} (z) = \sum\_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}.
$$
The definition yields that, via the Petersson inner product, we obtain:
$$\langle f, P\_{m}^{k}\rangle = c\_{k,m}a\_{m},$$
whe... | https://mathoverflow.net/users/95608 | Why are Poincare series defined as they are? | Ultimately this is bound to be a matter of taste -- you can't "prove" that one normalisation is better than another -- but let me try to justify why it is conventional to normalise this way.
The point of Poincare series isn't just to prove there exists a modular form satisfying $\langle f, P^k\_m \rangle = (const) \t... | 5 | https://mathoverflow.net/users/2481 | 326943 | 140,700 |
https://mathoverflow.net/questions/326948 | 9 | Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^\*A)^{\frac{1}{2}}$ ?
PS. I think the answer is **No**. But I could not find any counterexample!
| https://mathoverflow.net/users/136860 | A Matrix Inequality for positive definite matrices | The answer is **No**. Here is a counter-example:
$$X=\begin{pmatrix} 9 & 3 \\ 3 & 1 \end{pmatrix},\qquad Y=\begin{pmatrix} 1 & 3 \\ 3 & 9 \end{pmatrix}.$$
| 14 | https://mathoverflow.net/users/8799 | 326961 | 140,703 |
https://mathoverflow.net/questions/326964 | 9 | Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum \_{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic formula?
| https://mathoverflow.net/users/137248 | Sums of two squares in arithmetic progressions | The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$\sum \_{n\leq x\atop {n\equiv a(q)}}r(n) =\pi x \cdot \frac{\eta\_{a}(q)}{q^2}+ R\_{q,a}(x)$$
where $\eta\_{a}(q) := \{ (x\_1,x\_2... | 17 | https://mathoverflow.net/users/31469 | 326968 | 140,704 |
https://mathoverflow.net/questions/326947 | 5 | Let $X=G/H$ be a homogeneous manifold, where $G$ and $H$ are connected Lie groups and assume there is given a $G$-invariant Riemannian metric on $X$.
Let $B(R)$ be the closed ball of radius $R>0$ around the base point $eH$ and let $b(R)$ denote its volume.
Is it rue that
$$
\lim\_{\varepsilon\to 0}\ \limsup\_{R\to\inft... | https://mathoverflow.net/users/nan | Volume of balls in homogeneous manifolds |
>
> The idea somehow being that volume growth is largest with constant negative curvature
>
>
>
is essentially the content of the Bishop–Cheeger–Gromov comparison theorem. See for instance Lemma 36 of Peter Petersen's *Riemannian Geometry*, 2ed. It states, in his notation, that in any complete Riemannian manifo... | 7 | https://mathoverflow.net/users/4832 | 326979 | 140,708 |
https://mathoverflow.net/questions/326596 | 7 | I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional [Noetherian scheme](https://en.wikipedia.org/wiki/Noetherian_scheme) and $\mathbf{Spt}(S)$ the category of spectra over $S$. After inver... | https://mathoverflow.net/users/12204 | Model category structure on spectra |
>
> Is there a model structure on Spt(S), having SH(S) as homotopy category, such that every object is fibrant? If so, could you provide a reference?
>
>
>
No, if the given model category of spectra Spt(S) (for which there are many
different, but Quillen equivalent, definitions) is not right proper,
because if a... | 5 | https://mathoverflow.net/users/402 | 326991 | 140,714 |
https://mathoverflow.net/questions/327008 | 4 | I am trying to understand the main result (Theorem 1.1) in [this](https://projecteuclid.org/download/pdf_1/euclid.aop/1024404289) paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables
$$
\frac{\sum X\_i}{\sqrt{n}\sqrt{\sum X\_i^2}}
$$
without any conditions on the moments of $X... | https://mathoverflow.net/users/137854 | Large Deviations for Self-Normalized Sums | The conditions $x>EX/\sqrt{EX^2}$ and $EX\ge0$ imply $x>0$. So,
$t[bX\_i -x(X\_i^2+b^2)/2]$ is a quadratic polynomial in $X\_i$ whose leading coefficient $-tx/2$ is less than $0$ (except for the trivial case $t=0$). Therefore this polynomial is bounded from above, and hence the random variable $e^{t[bX\_i -x(X\_i^2+b^... | 1 | https://mathoverflow.net/users/36721 | 327009 | 140,721 |
https://mathoverflow.net/questions/326934 | 4 | A. P. Robertson and W. Robertson in their ["Topological Vector Spaces"](https://rads.stackoverflow.com/amzn/click/com/0521298822) VII, 1.4, (and H.Jarchow in "Locally convex spaces", 4.6, Theorem 2) prove the following proposition:
>
> Let $E=\lim\_{n\to\infty}E\_n$ be an inductive limit of a **sequence** of locall... | https://mathoverflow.net/users/18943 | When is a totally bounded set of an inductive limit contained in a component of this limit? | No. There are only very few general results about uncountable inductive limits. The following example is stated (without proof) in an article of Komura [Some examples on linear topological spaces. Math. Ann. 153 (1964), 150–162]:
For an uncountable set $I$ and every countable $J\subseteq I$ let $E\_J=\{f:I\to \mathb... | 3 | https://mathoverflow.net/users/21051 | 327027 | 140,726 |
https://mathoverflow.net/questions/327011 | 5 | Whilst I am reading articles on unipotent completion to understand its basic construction, I found something confusing. Let $F$ be a free group of rank 2 whose generating letters are $x$ and $y$ and let $K$ be a field of characteristic 0. Then the unipotent completion of $F$ over $K$ would be the set of group-like elem... | https://mathoverflow.net/users/44005 | Unipotent completion of free group | You're absolutely right. The last map you mention is called the Magnus expansion. One of the reasons it's interesting is that this formula is valid over $\mathbb{Z}$ and induces an injective map $$F\rightarrow \mathbb{Z}\langle \langle X,Y\rangle\rangle$$
which you can use to prove that $F$ is residually torsion-free-n... | 3 | https://mathoverflow.net/users/13552 | 327030 | 140,729 |
https://mathoverflow.net/questions/327032 | 2 | I'm interested in solving a particular non-linear recurrence in two variables:
$$\lambda\_{j,k} = (j+k) \lambda\_{j,k-1} + \begin{pmatrix} j+k \\ 2 \end{pmatrix} \lambda\_{j-1, k-1}.$$
Here $j \geq -1$ and $k \geq 0$, and we have initial conditions:
$\lambda\_{-1,k} = 0$ for all $k$;
$\lambda\_{j,0} = 0$ for a... | https://mathoverflow.net/users/38359 | A non-linear recurrence relation in two variables | Denote $\lambda\_{j,k}=(j+k)! a\_{j,k}$ (unless $j=-1,k=0$). Then we get $a\_{j,k}=a\_{j,k-1}+a\_{j-1,k-1}/2$. Further denoting $a\_{j,k}=2^{-j}b\_{j,k}$ we get $b\_{j,k}=b\_{j,k-1}+b\_{j-1,k-1}$ that looks like a Pascal triangle recurrence. So $b\_{j,k}=\binom{k}j$ (check the initial conditions also) and $\lambda\_{j,... | 6 | https://mathoverflow.net/users/4312 | 327035 | 140,731 |
https://mathoverflow.net/questions/327037 | 15 | For smooth manifolds it is known that they can admit a unique, finitely many, or a continuum of distinct smooth structures (I don't know whether there are any examples admitting precisely a countably infinite number).
For complex manifolds there are examples of smooth manifolds admitting a unique complex structure ($... | https://mathoverflow.net/users/102428 | Examples of smooth manifolds admitting inbetween one and a continuum of complex structures | Examples of smooth manifolds admitting finitely many complex structures are provided by some holomorphically rigid complex varieties.
For instance, considering [fake projective planes](https://en.wikipedia.org/wiki/Fake_projective_plane) (i.e., smooth compact complex surfaces which are not the complex projective pla... | 20 | https://mathoverflow.net/users/7460 | 327038 | 140,732 |
https://mathoverflow.net/questions/327018 | 1 | Given two discrete distributions $P$ and $Q$ with the same support $x\_1,\cdots,x\_n$. Assume $K \in L^1(\mathbb{R})$ is a nonnegative function with $\int\_\mathbb{R} K(x)dx = 1$, and let $K\_h(x) = \frac{1}{h}K(\frac{x}{h})$.
I am wondering whether the following result holds:
$$\left| \int\_\mathbb{R} |P\* K\_h(x) ... | https://mathoverflow.net/users/137543 | L1 distance after Convolution | This is definitely true, and the point is that for any two distinct $x\neq y\in \mathbb R$ the measures $\delta\_x\*K\_h$ and $\delta\_y\* K\_h$ are asymptotically singular as $h\to 0$, i.e.,
$$
\| \delta\_x\*K\_h - \delta\_y\*K\_h \| \to 2 \qquad \text{as}\quad h\to 0 \;.
$$
| 1 | https://mathoverflow.net/users/8588 | 327041 | 140,733 |
https://mathoverflow.net/questions/327042 | 3 | Despite the apparent simplicity of the following question I couldn't find the answer so far.
I am looking to construct an elliptic fibration $X \to \mathbb{P}^1$ with $X$ smooth, and exactly two reduced singular fibres $X\_0, X\_1$ such that the monodromy action around each of the fibres on the first cohomology of th... | https://mathoverflow.net/users/2234 | elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1 | If you look at the global monodromy action $\pi\_1 ( \mathbb P^1 -\{0,1\}) \to SL\_2(\mathbb Z)$, you see that $\pi\_1 ( \mathbb P^1 -\{0,1\}) = \mathbb Z$ so the image is abelian and therefore has infinite index. On the other hand, if we consider the $j$ map $\mathbb P^1 \to X(1)$, the image of the fundamental group h... | 6 | https://mathoverflow.net/users/18060 | 327048 | 140,735 |
https://mathoverflow.net/questions/327017 | 7 | let $a,b,c,x,y,z>0$ such $x+y+z=a+b+c,abc=xyz$,and $a>\max\{x,y,z\}$,
I conjecture $$a^n+b^n+c^n\ge x^n+y^n+z^n,\forall n\in N^{+}$$
Maybe this kind of thing has been studied, like I found something relevant, but I didn't find the same one.[Schur convexity and Schur multiplicative convexity for a class of symmetric... | https://mathoverflow.net/users/38620 | Conjecture: $a^n+b^n+c^n\ge x^n+y^n+z^n$ | A triple $(a,b,c)$ with $a+b+c=s$, $abc=p$, and $ab+bc+ca=t$ is the triple of roots of $X^3-sX^2+tX-p=0$, i.e., of $-X^2+sX+p/X=t$. Fix $s$ and $p$; the left-hand part has just two local extrema on the right semiaxis. Let $a(t)>b(t)>c(t)$ be the three roots (defined for all values of $t$ when they exist and are sdistin... | 7 | https://mathoverflow.net/users/17581 | 327054 | 140,738 |
https://mathoverflow.net/questions/327050 | 7 | I have a sequence of groups $H\_n$ which I know to be extraspecial 2-groups of order $2^{2n+1}$. I also know the number of order 4 elements I have in $H\_n$ for every $n$. Precisely, the number of order 4 elements is given by $2\sum\_{i=0}^{4i \leq 2n-1} {2n+1 \choose 2+4i}$. Is there a slick way of determining which e... | https://mathoverflow.net/users/135861 | Classification of the Extraspecial 2-groups $H_n$ | This may be answered as follows: any extra-special $2$ group of order $2^{2n+1}$ is either the central product of $n$-copies of $D\_{8}$ or else is the central product of $n-1$ copies of $D\_{8}$ with one copy of $Q\_{8}.$ The first type has all its complex irreducible representations realizable over $\mathbb{R}$, whil... | 10 | https://mathoverflow.net/users/14450 | 327055 | 140,739 |
https://mathoverflow.net/questions/327015 | 6 | This is a natural follow-up to [my previous MO question](https://mathoverflow.net/questions/326860/hook-length-formula-fibonaccized), which I share with Brian Hopkins.
Consider the Young diagram of a partition $\lambda = (\lambda\_1,\ldots,\lambda\_k)$. For a square $(i,j) \in \lambda$, define the *hook numbers* $h\_... | https://mathoverflow.net/users/66131 | hook-length formula: "Fibonaccized": Part II | This is my answer to the original question (<https://mathoverflow.net/a/327022/50244>) whether these numbers are integers to begin with, it gives some combinatorial meaning as well:
Use the formulas
$F(n) = \frac{\varphi^n -\psi^n}{\sqrt{5}}$, $\varphi =\frac{1+\sqrt{5}}{2}, \psi = \frac{1-\sqrt{5}}{2}$. Let $q=\frac... | 10 | https://mathoverflow.net/users/50244 | 327069 | 140,742 |
https://mathoverflow.net/questions/327077 | 3 | I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}\_{n,\infty}$ denote the infinite $n$-Grassmanian and denote $\xi\_n $ its universal bundle. The cobordism spectrum $\mathrm{MGL}$ is made out of the universal Thom spaces.Th... | https://mathoverflow.net/users/12204 | Basic question on the cobordism spectrum | A simple way of seeing it is to explicitly spell out what we mean when we say that a spectrum is "presented" by a prespectrum. To say that that a spectrum $E$ is presented by
$$(E\_0,E\_1,...)$$
means that, in whatever model for motivic spectra you are using,
$$E\cong \mathrm{colim}\_k \Sigma^{-2k,-k}\Sigma^{\infty}E\_... | 3 | https://mathoverflow.net/users/43054 | 327080 | 140,743 |
https://mathoverflow.net/questions/327084 | 1 | Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal.
We know that for any smooth $R/I$-algebra $A\_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A\_0$.
We also know that for any map $A\_0\to B\_0$ of smooth $R/I$-algebras, there exist $R$-smooth algebras $A$ and $B$, and a map $A\... | https://mathoverflow.net/users/134710 | Lifts of smooth algebras | Yes. Let $i\colon Y=\mathbf{Spec}(B)\to \mathbf{Spec}(A)=X$ be the induced map of schemes, and let $K$ be the cokernel of
$$ i^\*\colon \mathcal{O}\_X \to i\_\* \mathcal{O}\_Y. $$
This is a coherent $\mathcal{O}\_X$-module whose support does not meet $X\_0$ by assumption. Thus after replacing $X$ with an affine open ne... | 2 | https://mathoverflow.net/users/3847 | 327085 | 140,745 |
https://mathoverflow.net/questions/327074 | 4 | Let $S$ be the set of complex $N\times N$ matrices that are traceless, unitary and hermitian.
A friend asked me the following question, motivated by a problem in condensed matter physics:
>
> Is it true that for every two matrices $A$, $B$ in $S$, the value of $\det(A+iB)$ is always imaginary?
>
>
>
Well, ... | https://mathoverflow.net/users/83671 | Determinant involving traceless unitary hermitian matrices | Yes. It's real when $N\equiv 0 \text{ mod } 4$ and imaginary when $N\equiv 2\text{ mod } 4$.
The square of the determinant is $\det(A+iB)^2=\det(1-1+i(AB+BA))=i^N\det(AB+BA)$, so for either parity of $N/2$ we need to show the Hermitian matrix $AB+BA$ has nonnegative determinant. Up to unitary transformation, $B=D$ an... | 6 | https://mathoverflow.net/users/112641 | 327097 | 140,751 |
https://mathoverflow.net/questions/327093 | 9 | Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $f\in Hom(A,B)$ and $g \in Hom(B,C)$ then $g\circ f$ may not exist in $Hom(A,C)$, but when the relevant compositions do exist, then composition is associative. '... | https://mathoverflow.net/users/137147 | A category-like structure without composition? | As Qiaochu says, one way to talk about categories with partially defined composition is to talk about categories enriched over the monoidal category $Par$ of sets and partial functions with the cartesian product (that is, the cartesian product in $Set$, which is not the cartesian product in $Par$). Since $Par$ is equiv... | 12 | https://mathoverflow.net/users/49 | 327099 | 140,753 |
https://mathoverflow.net/questions/327094 | 5 | This is [cross-posted from Math.SE](https://math.stackexchange.com/questions/3171813/what-does-rosenlicht-mean-by-a-point#comment6530445_3171813) at the recommendation of a commenter.
I'm reading M. Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" [[link]](https://www.jstor.org/stable/pdf/2372523.pd... | https://mathoverflow.net/users/12419 | What does Rosenlicht mean by a "point"? By $k(v_1,v_2)$? | Weil's foundations (which Rosenlicht uses) use a universal domain, which is an algebraically closed extension of $k$ of infinite transcendence degree. There is a discussion of universal domains [here](https://mathoverflow.net/questions/36979/some-arithmetic-terminology-universal-domain-specialization-chow-point). In th... | 11 | https://mathoverflow.net/users/137902 | 327104 | 140,756 |
https://mathoverflow.net/questions/327044 | 5 | Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}\_w$. Specifically, $\mathbb{R}\_w$ is the Radin forcing defined using the measure sequence $w=j\_{\alpha}(u)$, where $j\_\alpha$ is the ... | https://mathoverflow.net/users/102990 | Radin generics from iterated ultrapowers | See section 5 (called "Generating generic sequences by iterating j") of Radin's paper [Adding closed cofinal sequences to large cardinals](https://www.sciencedirect.com/science/article/pii/0003484382900237?via%3Dihub).
**More details:** The iteration is the usual one, using $j$ (see my comment below). Now the point i... | 4 | https://mathoverflow.net/users/11115 | 327112 | 140,760 |
https://mathoverflow.net/questions/327016 | 16 | I recently had a debate with my friend about how much of symplectic topology is about Fukaya category. I thought that for the most part, symplectic topology is not about Fukaya category. Now, to prove that I am right I need a mathematical example of Fukaya category failing to discern some symplectic topology.
We kno... | https://mathoverflow.net/users/nan | Does Fukaya see all symplectic topology? | This answer just provides some general comments pertatining to the question asked in the title of the post. (When I first started graduate school, I was somewhat skeptical of Fukaya categories, and in fact went around asking people when one could show that Fukaya catgories can actually distinguish Lagrangians *up to Ha... | 13 | https://mathoverflow.net/users/118831 | 327114 | 140,761 |
https://mathoverflow.net/questions/326381 | 6 | This is probably more a reference request than a real question. I was studying dg-categories in order to understand how one can derive a functorial [cone construction](https://en.wikipedia.org/wiki/Mapping_cone_(homological_algebra)) when a triangulated category (which for what concerns me is often $D^b(X)$, for some s... | https://mathoverflow.net/users/91572 | Functorial cones | The cone construction can be written down very explicitly, just following the definition of mapping cone of chain complexes. Good sources are in my opinion:
<https://arxiv.org/pdf/math/0401009.pdf> Definition 3.7
<https://arxiv.org/pdf/math/0210114.pdf> paragraph 2.9, where it is discussed how the cone is functoria... | 4 | https://mathoverflow.net/users/20883 | 327115 | 140,762 |
https://mathoverflow.net/questions/327111 | 0 | I'm wondering how to upper bound the following ratio of integrals:
$$\frac{\int\_{\Delta\_a}(\prod\_{i=1}^n\lambda\_i)^{p-1}\prod\_{i<j}|\lambda\_i-\lambda\_j|}{\int\_{\Delta\_b}(\prod\_{i=1}^n\lambda\_i)^{p-1}\prod\_{i<j}|\lambda\_i-\lambda\_j|}$$
in terms of $a,b,p,n$, where
$$\Delta\_a=\{(\lambda\_1,....,\lambda\_n)... | https://mathoverflow.net/users/123075 | Upper bound of a ratio of integrals | Let
\begin{equation}
J\_a:=\int\_{\Delta\_a}
\Big(\prod\_{i=1}^nx\_i\Big)^{p-1}\prod\_{i<j}|x\_i-x\_j|\,\prod\_{i=1}^n dx\_i.
\end{equation}
By the change of variables $x\_i=ay\_i$, we see that
\begin{equation}
J\_a=a^{np+n(n-1)/2}J\_1.
\end{equation}
Hence, your ratio of integrals is
\begin{equation}
J\_a/J\_b... | 5 | https://mathoverflow.net/users/36721 | 327130 | 140,765 |
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