All Questions
23,894 questions
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Question about wall crossing for Hassett spaces
The context of this question is that one of Hassett's famous compactifications of $M_{0,n}$ by means of weighted stable marked curves. I imagine the answer to my question is well known, but I haven't ...
- 650
4
votes
1
answer
132
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Reference request for Bessel function of the second kind with matrix argument
As the title says, I would like to know if anyone could provide a reference which provides the definition and properties of the Bessel function of the second kind with matrix argument. If possible, I ...
- 188k
8
votes
3
answers
1k
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Ramanujan's Master Formula: A proof and relation to umbral calculus
The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...
- 10.5k
1
vote
1
answer
101
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Independence of sample mean and variance among samples with infinite second moment
It's a well-known fact in statistics that the independence of the sample mean and variance characterise the Gaussian distribution in the sense that, if $X_1, \dotsc, X_n$ are iid and $\sum_{j=1}^n X_j$...
- 91.8k
0
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2
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1k
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An observation on the Riemann $\xi$ function
Anyone seen these conclusions about the Riemann xi function or see any errors here?
With $\xi(s)$ the entire Landau Riemann xi function
defined by the Hadamard product representation
$$\xi(s) = (1/...
- 59
5
votes
1
answer
561
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interiors of positive cones in ordered Banach spaces
I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references.
I want to know several examples of ...
- 12.6k
6
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2
answers
1k
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Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?
For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, ...
8
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3
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779
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Computer program for counting graph homomorphisms
I would like to ask is there a computer program for counting graph homomorphisms?
- 75
2
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0
answers
179
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Isomorphic simplicial complexes are simple-homotopy-equivalent: reference?
Any two isomorphic simplicial complexes are simple-homotopy-equivalent.
This is a fairly simple result, but it is not obvious. Yet I have been surprisingly unable to find it in the literature on ...
- 33.8k
1
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0
answers
61
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Cardinal of finite orthogonal groups
Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.
By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...
- 81
4
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1
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123
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Borel measures on the Martin boundary and the Poisson-Martin representation theorem
I have been studying the construction of the Martin boundary on a discrete set $X$ admitting an irreducible transient random walk $(X,P)$ from Wolfgang Woess' book titled "Random Walks on Infinte ...
- 17k
0
votes
0
answers
120
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Equality of two measures on functional spaces
It is well known that if $\mu$ and $\nu$ are two measures on the space $C^0([0,1],\mathbb{R}^n)$ of continuous mappings from $[0,1]$ to $\mathbb{R^n}$ endowed with its Borel $\sigma$-algebra satisfy $$...
CommunityBot
- 1
4
votes
0
answers
110
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Bar construction of a commutative monoid
Let $M$ be a commutative monoid. Define the bar construction $BM$ as the thin geometric realization of $[p] \mapsto M^p$. I am looking for a reference for the fact that $BM$ is again a commutative ...
- 965
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5
answers
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Who says understanding physics helps mathematicians? (A reference request) [Take the word "who" literally.]
If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that ...
- 7,179
3
votes
1
answer
238
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Steenrod operations on classifying spaces
Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
- 35.9k
5
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1
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267
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Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions
Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
1
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2
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162
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Proof that the closed convex hull of a weakly convergent sequence has empty interior using properties of $c_0$?
Let $X$ be an infinite-dimensional Banach space, and $x_n\to 0$ weakly in $X.$ Let $K$ be the closed convex hull of $\{x_n\}.$
I remember a proof that $K$ has empty interior as follows: define a map $...
- 16.4k
31
votes
1
answer
2k
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Topology on space of hyperfunctions
This is a reference request, coming from someone with little knowledge of hyperfunctions:
Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
- 281
0
votes
0
answers
61
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Recognizability of a substitution implies aperiodicity
Is there a good reference, aside from the book of "Tilings and Patterns" by Grunbaum and Shephard, on the fact that recognizability\unique-composition of a tiling implies aperiodicity? I ...
- 633
7
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1
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296
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Is this known? As $p,q\to\infty$, most elements of the power set of $\{1,\dots,p\}\times\{1,\dots,q\}$ are in free $\Sigma_p\times\Sigma_q$-orbits
Let $p,q$ be nonnegative integers. The product of symmetric groups $\Sigma_p\times\Sigma_q$ acts on the power set $P(\{1, \dots ,p\}\times\{1, \dots ,q\})$ in the evident way. You can ask what ...
- 12.9k
6
votes
1
answer
227
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Forcing $\neg\square_{\omega_1}$ from a Mahlo cardinal, reference
In Jensen's The fine structure of the constructible hierarchy, it is stated that Solovay proved the consistency of $\neg\square_{\omega_1}$ by collapsing a Mahlo cardinal to $\omega_2$. I was ...
- 18.6k
1
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0
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86
views
Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as
$$(f*g)(R) = \...
- 2,306
17
votes
0
answers
677
views
Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
- 37.8k
9
votes
1
answer
337
views
Characterizing germs of smooth functions
There's a sheaf of smooth real-valued functions on $\mathbb{R}$, and its germ at $0$ is some vector space $V$. I would like to understand this space. There is a surjective linear map
$$ \phi \colon ...
- 281
1
vote
0
answers
154
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Kasparov's descent homomorphism for higher KK groups
I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism
$$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$
but what still ...
- 1
4
votes
1
answer
723
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Reference request: Schlessinger's Thesis
Does anyone have a copy of Schlessinger's Thesis (not his paper "Functors of Artin Rings")
As other posters have mentioned, this document is cited in Deligne-Rapoport's "Les schemas de ...
- 3,998
0
votes
0
answers
96
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Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
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1
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97
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Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms
The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...
- 7,817
4
votes
1
answer
311
views
If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$
Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...
- 285
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Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
- 12.9k
5
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1
answer
353
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Reference for "trick" on guessing solutions to quadratic recurrences with differential equations
Consider the recurrence
$$g(h) = g(h-1) - \frac{1}{4}g(h-1)^2,$$
for $h \geq 0$ and $g(0)=1$. This recurrence occurs in many applications (For example fast minimum cut algorithms, the Galton-Watson ...
- 869
3
votes
0
answers
141
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Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$
Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
- 435
4
votes
2
answers
354
views
Injectivity of a convolution operator
Let $p,\mu,\nu$ be probability density functions on
$\mathbb{R}$ such that
$$
\int_{\mathbb{R}}p(y-x) \nu(y) \, dy=\mu(x).
$$ Now, consider the operator $T:L^2(\mu)\to L^2(\nu)$ such that $$ Tf=f*p.$$ ...
- 133
6
votes
1
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330
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A generalization of derangement number
What is the number of $n \times n$ binary matrices with row and column sums 2 and with only zeros on the diagonal? This simple problem must have been treated somewhere, but I couldn't find any ...
- 13.4k
11
votes
1
answer
903
views
Abstract mathematical concepts/tools appeared in machine learning research
I am interested in knowing about abstract mathematical concepts, tools or methods that have come up in theoretical machine learning. By "abstract" I mean something that is not immediately related to ...
- 3,599
1
vote
1
answer
110
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Looking for definition of function spaces appearing in article of DiPerna & Lions
I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail:
Article: ...
- 12.9k
1
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0
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94
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Multivariate polynomial approximation
Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$.
Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
- 61
4
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1
answer
148
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Multivariate polynomial approximation of functions in Sobolev space
I found a result of the estimation error of polynomial approximation in page
6 of https://scg.ece.ucsb.edu/publications/theses/ARajagopal_2019_Thesis.pdf
The statement is for $f \in W^{k, p}\left([-1,...
- 360
3
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0
answers
151
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Reference for homotopy and homology theory of topological groups
I am looking for references which deal with the homotopy theory and homology theory of general topological groups, not necessarily compact, or anything. I am eyeing towards certain infinite-...
- 869
0
votes
0
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89
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Stein's Lemma for conditional expectation?
Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > ...
- 6,853
14
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1
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359
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The first two $k$-invariants of $\mathrm{pic}(KU)$ and $\mathrm{pic}(KO)$
$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\pic{pic}$Real and complex topological $K$-theories, $KO$ and $KU$, have Picard spectra $\pic(KO)$ and $\pic(KU)$ built from the $\mathbb{E}_\infty$-...
- 10.4k
2
votes
0
answers
116
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Reference for a coarse complexity notion
Throughout, I'm only interested in structures with domain $\mathbb{N}$, no primitive relations, and at least $0,\mathsf{Succ}$ as primitive functions. The length of $m\in\mathbb{N}$ is $\lfloor 1+\...
- 20.5k
9
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1
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1k
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Newman's Lemma or Diamond Lemma
I'd like to learn the Newman's Lemma or Diamond Lemma (the one used in abstract rewriting system), can someone recommend me some books where I can read it? I'd appreciate self-contained books with ...
- 33.8k
4
votes
0
answers
256
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Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation
In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
2
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0
answers
261
views
When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?
Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$:
$$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
- 20.2k
5
votes
2
answers
333
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Differing monoidal model structures on a fixed model category
Suppose that $\mathcal{C}$ is a model category (with a fixed model structure). Are there any known examples where $\mathcal{C}$ is a (symmetric) monoidal model category in two different ways? I.e., ...
- 64k
0
votes
0
answers
106
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How to prove that $f(x) := |x|^{\frac{\lambda - n}{p}}(1 - \psi(x))$ satisfies a specific property related to its limit at the origin
Disclaimer. I have asked this question a month ago on MSE (click here to access the original post) and even bountied it. I got an answer on MSE, but unfortunately I don't feel like it has enough ...
CommunityBot
- 1
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votes
5
answers
2k
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$A_\infty$-categories basic reference
Can anyone provide me with a basic reference on $A_\infty$ categories?
- 4,713
3
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0
answers
79
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Continuity of disintegrations in non locally compact spaces
Let $X$ and $Y$ be Radon spaces, $\mu$ a Borel probability measure on $X$, $F\colon X\to Y$ measurable. Then the disintegration theorem gives us a disintegration $\{\mu^y\}_{y\in Y}$ of $\mu$ with ...
- 63
15
votes
2
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889
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Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Fix a compact Riemannian manifold $M$ (leaving the metric implicit). What I'd like to know is if the corresponding Hodge decomposition of smooth $n$-forms
$$
\Omega^n(M) \simeq \mathcal{H}^n(M)\oplus ...
- 8,098