New answers tagged

1 vote

Has anyone attempted to generalize the notion of a higher differential of $ A $ and the sheaf of differentials $ \Omega_{A/k} $?

I have this strange feeling I have commented on this or an identical question recently, but can't place it. Here is one solution which may not be what you are looking for and you have mentioned it in ...
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0 votes

Reference request: “A random integral and Orlicz spaces”

I could not find it on the internet so I uploaded it here: https://www.transfernow.net/dl/20221126nnUfCto7
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1 vote
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First visit of intervals for an irrational rotation

No, it is not possible. In the following I will use $I_n=(a_n,b_n)$ instead of $[a_n,b_n)$ (this is not a problem, you can just increase $a_n$ a bit so that the statement with $I_n=(a_n,b_n)$ is ...
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10 votes
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Name for vector spaces with two algebra structures that satisfy the exchange law

If the operations have units ($a * 1 = a$ etc), then that is simply called a commutative algebra: https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton_argument Indeed in that case, $a*b = a \circ b = ...
2 votes

Domains that may require a good categorical background

Besides the CT-functional programming connection, in recent years a field of "applied category theory" (ACT) has emerged that seeks to apply category-theoretic ideas to fields beyond the ...
6 votes

Homology of spherical $3$-manifold group

The attaching map has to kill $\pi_3(S^3/G)$, and the map $\mathbb Z =\pi_3(S^3) \to \pi_3 (S^3/G) $ induced by the covering is an isomorphism, so $\pi_3$ is generated by the class of the covering 3-...
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1 vote

Cone unfolding of space curves

Pardon me for this bit of self-promotion, especially because this is only tangential to the OP's concerns. But cone unrolling and Anton Petrunin's mention of Alexandrov's developments, in conjunction ...
1 vote

Cone unfolding of space curves

Liberman used cylinder unfolding to study geodesics on convex surfaces. [Либерман, И. М. «Геодезические линии на выпуклых поверхностях». ДАН СССР. 32.2. (1941), 310—313.] Right now standard ...
19 votes
Accepted

The $9$th tetration of $-\sqrt2$

This is not a huge coincidence: the idea is that the sequence $a_n={}^{n}(-\sqrt{2})$ has small norm until $n=6$, then it gets out of hand for $n=7$ ($a_7\sim-33+29i$), so that $a_8=e^{a_7\ln(-\sqrt{2}...
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0 votes

Results on Boolean matrices

We can try to produce some of the basic theory of Boolean matrices here to see what makes sense and what does not. For this post, we do not lose much by generalizing to the Boolean algebras of the ...
7 votes

Mathematical analysis of Lewisian concepts, esp. natural properties

I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why ...
5 votes
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Reference for Calderon-Zygmund $L^p$ inequalities on the sphere

I don't know of an exact reference, and in general this sort of result (transfering a "classical" result from the analysis of PDE in $\mathbb{R}^n$ to Riemannian manifolds) is often quite ...
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0 votes

Sum of three squares as class numbers and Waldspurger's formula

This is not an answer, but I found a work that seems to be related to my question. There's a paper by Ting-Yi Pei on the Eisenstein series of weight 3/2. Author defines a weight 3/2 Eisenstein series (...
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11 votes
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Can we force $\mathfrak{r}<\mathfrak{s}$?

The inequality $\mathfrak{r} \leq \mathfrak{u}$ is provable in ZFC (because every base for an ultrafilter is a reaping family). Blass and Shelah proved the consistency of $\mathfrak{u} < \mathfrak{...
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3 votes

Domains that may require a good categorical background

The blog by Bartosz Milewski comes to my mind. It focusses on the interplay between Haskell and category theory.
2 votes
Accepted

On partial absolute continuity

The notation $L^2$ for Lebesgue measure is confusing. I denote $\lambda$ and $\lambda_2$ the Lebesgue measure on $\mathbb{R}$ and $\mathbb{R}^2$, respectively. The answer is no. Fix a discrete measure ...
4 votes

Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

This result is stated as Exercise 7 in Chapter 8 of Bröcker and Jänich's book Introduction to Differential Topology (p. 86 of the English translation). This may or may not count as a citeable ...
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1 vote

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

Vershynin, R. Spectral norm of products of random and deterministic matrices. Probab. Theory Relat. Fields 150, 471–509 (2011). This result is a sharp bound on the spectral norm of $W=BA$, where $A$ ...
0 votes

4D Duoprisms based on nonconvex polygons

The great duoantiprism can be constructed using a compound of two pentagonal-pentagrammic duoprisms by inserting additional edges, or by alternating a decagonal-decagrammic duoprism.
6 votes

Is there a citeable reference for star-shaped open subsets of R^n being diffeomorphic to R^n?

The result in the following paper implies that open star-shaped domainin $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. But, in your case, a diffeomorphism can be obtained along the same lines. K. ...
0 votes

Concentration inequality for the spectral norm of the product of normalized Gaussian (and subgaussian) matrices in high dimensions

The Marcenko-Pastur resut (see, e.g., https://www.sciencedirect.com/science/article/pii/S0047259X85710512) gives you the Stieljes equation of the limiting spectral distributions of matrices of the ...
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2 votes
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Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request

You can find quiver and relations (not sure if they are admissible always) here: http://www.math.uni-bonn.de/ag/stroppel/Quivers.pdf In particular the explicit algebra is only fully understoof for $...
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0 votes
Accepted

Differentiability of the fixed points of a family of contraction maps

I found the answer myself: One can simply apply the Banach space version of the implicit function theorem to the function $G(t,x) = x-F_t(x)$. The implicit function theorem shows that, given $G$ is in ...
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3 votes

A question on a result of Colin de Verdière

Vedrin Šahović in his unpublished thesis [Approximations of Riemannian manifolds with linear curvature constraints, 2009] proved that any compact metric space can be appoximated by hyperbolic ...
2 votes

Reference request for "Tangent relation" for functions between metric spaces

I don't know whether there are "good" references to what you are actually asking but at least in some kind of implicit sense this kind of "tangency" was already considered by ...
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2 votes

Mutual information in large deviation theory

There's a few results. First of all there is the classical Sanov's Theorem. One other result is about Gaussian measures. For a centered Gaussian measure $\mu_0$ on Banach space $\mathcal B$ we can ...
2 votes

Question about and good reference for Kahn and Markovic result

I think that a good reference could be the following: Proposition ([1, Proposition 4.1 (or 5.1 in the arXiv version)]): Let $M$ be a closed hyperbolic 3-manifold, and regard $\pi_1(M)$ as acting on $\...
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4 votes
Accepted

References for applications of Young diagrams/tableaux to Quantum Mechanics

Young diagrams or Young tableaux (the latter being diagrams with integers in each box) are used in particle physics to describe the states of indistinguishable fermions or bosons: $n$ ...
2 votes

Reference request: Kummer étale topology and tame topology

He did not say that the sites are equal as this claim is wrong in general, but that the categories of covers, i.e. finite morphisms which cover the log scheme (See definition 3.1. of the same paper), ...
3 votes
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Counting numerical semigroups by largest element of minimal generating set

A key observation is that two sets of generators $g, g'\subseteq [n]$ produce the same semigroup if and only if $\langle g\rangle \cap [n] = \langle g'\rangle \cap [n]$. Hence, the number of different ...
2 votes

Free idempotent monad associated to a monad

What follows is a long comment, and because the question is asking for a reference, it is also an irrelevant comment. Anyhow, the whole idea behind the Fakir construction is to build on a ...
4 votes
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Lower bounds for pattern complexity of aperiodic subshifts

The answer is no in a very strong sense: there does not exist such $C_d$ for $d \geq 3$ even for aperiodic minimal subshifts. As far as finding lower bounds goes, complexities of subshifts containing ...
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2 votes

Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

Yes, R&S proved to be influential in at least one sense, the mathematical work of Z. Mebkhout (part four is dedicated to him indeed: "À Zoghman Mebkhout l’ouvrier solitaire en témoignage de ...
5 votes
Accepted

Anosov flow on the 2-sphere

The usual definition of Anosov flow requires three invariant sub-bundles, so I guess you are actually asking about the 3-sphere? Plante and Thurston have proved in Plante, J. F.; Thurston, W. P., ...
0 votes
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Source on equality-free second-order logic (nontrivially construed)

It currently appears that this logic is not already treated in the literature. (I'm posting this answer to move this question off the unanswered queue, but if someone does find a source on it of ...
7 votes

Lower bounds for pattern complexity of aperiodic subshifts

There are a few things to clarify here. First of all, the two-dimensional version of Morse-Hedlund, i.e. that whenever $X$ contains a point with no period vector, $p_{m,n}(X) \geq mn+1$ holds for all $...
3 votes
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Catalogue of groups with short finite presentations

I would very much like to have such a database and would like to contribute to its development. Prompted by this question, we talked about what such a database could look like (e.g. in terms of groups ...
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2 votes
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Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941). (Alternative link at Project Euclid)
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1 vote

Attaching an ideal whose square is zero: does this operation have a name and a notation?

I have recently been enlightened by colleagues in the Mathematics Dept. here at HWU on this very question: this construction (in various guises, inessential variations for the purposes of the ...
12 votes
Accepted

Applications of equivariant homotopy theory to representation theory

There are decades and decades of algebraic results that use techniques from equivariant homotopy theory. Some examples ... (1) Quillen's work on ring theoretic aspects of the cohomology of finite ...
1 vote
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Explicit formula for Fibonacci numbers; compositions of $n$

Yes, this identity is well known. According to Singh's The so-called Fibonacci numbers in ancient and medieval India, the $s=1$ case has been known since at least the the 14th century. Since ...
1 vote

Axioms for the category of groups

The category of groups is the universal example of a cocomplete category equipped with a cogroup object. A similar statement holds for other types of algebraic structures. This is due to Freyd. See ...
9 votes
Accepted

Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?

As igorf pointed out in the comment, the answer to the first question is 'no'. A quick counterexample is by looking into the complement of Fox-Artin arc, which is not simply connected. See figure ...
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9 votes
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When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?

If both cofibrations and weak equivalences are stable under filtered colimits, then so are trivial cofibrations. This happens for instance if $\mathcal{M}$ is a presheaf category on an elegant Reedy ...
9 votes
Accepted

A simple proof of the fundamental theorem of Galois theory

At a first glance your approach reminds me of Meinolf Geck's American Mathematical Monthly article, see also the arxiv version of his article.
7 votes

Making a submanifold transverse to a vector field by an isotopy

The simplest case is when $M$ is a compact manifold with connected boundary $N$. If $N$ is nowhere tangent to $X$ then, by replacing $X$ by $-X$ if necessary, we can assume $X$ points outwards at ...
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0 votes

Coordinate principal bundle over a curve

The standard reference for where this comes from originally is I. Gelfand and D. Kazhdan, Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields, ...
4 votes

General form of bounded linear functionals on Banach spaces

For example: For the real Banach space $L^p(\mathbb R)$, with $1 < p < \infty$, the "conjugate space" is $L^q(\mathbb R)$ where $\frac{1}{p}+\frac{1}{q}=1$. For general linear ...
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2 votes

Discontinuous functions without removable discontinuities

I claim that one can always get rid of all removable discontinuities of a function to obtain a function without removable discontinuities. For generality, we shall work in the framework of general ...
2 votes
Accepted

Could certain closed covering determine a coherent sheaf?

It is not true even for $\mathbb{P}^n$. For instance, the tangent bundle $T_{\mathbb{P}^n}$ restricts to each line as $$ T_{\mathbb{P}^n}\vert_L \cong \mathcal{O}_L(2) \oplus \mathcal{O}_L(1)^{\oplus (...
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