Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,544
questions
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divergence equation with prescribed normal trace
Let $\Omega \subset \mathbb{R}^n$ be a smooth domain and $\nu$ be the outer unit normal to $\partial \Omega$. Given $\phi \in L^{\infty}(\partial \Omega)$ such that $\int_{\partial \Omega} \phi d\...
0
votes
2
answers
182
views
Reference from the article "Random Ordinary Differential Equations", by J.L. Strand
In the article Random Ordinary Differential Equations, Journal of differential equations 7, 538-553 (1970), by J.L. Strand, reference number 6 refers to his PhD thesis: Stochastic Ordinary ...
0
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0
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132
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Reference request: mathematical expectation of a random object in a topological space
Recently I got interested in the following question: what does a mathematical expectation look like for a random object taking values in a topological space?
This turns out to be a difficult ...
9
votes
2
answers
1k
views
Notation for the set of all injections from $A$ into $B$
Is there a common notation for the set of all injections from $A$ into $B$?
Some set-theorists use $B^{(A)}$, e.g., A. Levy in his book Basic Set Theory.
But some combinatorists use $B^{\underline{A}...
2
votes
0
answers
141
views
Regular epi- and mono-morphisms for locally compact (Hausdorff) groups
I am interested in what the regular monomorphisms are in the category of locally compact (for me, always Hausdorff) groups (with continuous group homomorphisms).
It is easy to see that the equaliser (...
0
votes
1
answer
260
views
Alternative reference to Davenport's Analytic Methods for geometry of numbers?
I was wondering if someone would be willing to suggest an alternative reference to Davenport's book Analytic Methods for Diophantine Equations and Diophantine Equations. I like the book but I would ...
5
votes
2
answers
505
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
2
votes
1
answer
89
views
Stochastic ordering of empirical mean
Consider a Bernoulli distribution with mean $\mu \in (0,1)$ taking values in the set $\{0,1\}$. Suppose we draw $t \in \mathbb{N}$ independent and identically distributed (i.i.d.) samples from this ...
0
votes
1
answer
124
views
Lie algebra cohomology with values in injective module
I am looking for a proof of the following result: Let $\mathfrak{g}$ be a Lie algebra and $I$ an injective $\mathfrak{g}$-module. Then $\mathrm{H}^q(\mathfrak{g},I)=0$ $\forall q>0$. More precisely,...
5
votes
2
answers
513
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Reference book for understanding Hilbert Series/functions
For my bachelor thesis my goal is to understand the reasoning behind "Hilbert series" and how they connect to the idea of "dimension".
https://en.wikipedia.org/wiki/...
5
votes
0
answers
100
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wild julia sets
Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the ...
10
votes
1
answer
662
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Injectivity radius of manifolds with boundary
This question stems from the discussion in:
how to define the injectivity radius of manifolds with boundary?
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let ...
7
votes
2
answers
773
views
Is there a name for this equivalence relation?
Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition?
$\sim_{M,\mathscr{F}}\,=\...
5
votes
1
answer
348
views
Divisibility of certain polynomials
Consider the finite sums
$$F_n(q)=\sum_{k=1}^nq^{\binom{k}2}$$
with exponents the triangular numbers $\binom{k}2$. When $n$ is odd, it appears that $F_n(q)$ does not factorize over $\mathbb{Z}[q]$. On ...
11
votes
2
answers
602
views
'Continuity' of the étale topos
In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative ...
1
vote
1
answer
110
views
Bounds for Khukhro-Makarenko theorems
Let’s define the set of outer-commutator group words $OC \subset F_\infty = F[x_0, x_1, …, x_n, …]$ using the following recurrence:
$$\forall i \in \mathbb{N} \text{ } x_i \in OC$$
$$\forall u, v \...
3
votes
0
answers
84
views
Self-contained reference for projective embedding of moduli of polarized abelian varieties via modular forms
I've been working on reading and understanding Arakelov's '71 paper and he uses the fact that the moduli space of complex abelian varieties of dimension $g$ with polarization of degree $d$ admits an ...
3
votes
0
answers
82
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Cancellativity of a particular $2$-generated monoid presented by an infinite number of relations
Let $X = \{x, y\}$ be a two-element set, and let $H$ be the monoid defined by the presentation
$$
\langle x, y \mid x y^k x = y x y^{k+1} x y, \text{ for } k = 0, 1, 2, \ldots\rangle.
$$
That is, $H$ ...
0
votes
0
answers
62
views
Can we have relations of the same type of their composing ordered pairs?
Is it possible to define a binary function $P$ in the language of set theory that obeys the characteristic property of ordered pairs and such that for any two sets $A,B$, for any definable relation $R$...
8
votes
1
answer
641
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Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
5
votes
0
answers
270
views
When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?
I am looking for research or references on the following problem.
Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
0
votes
0
answers
122
views
Ask for some percolation reference textbook
I try to learn Bernoulli percolation recently. Could anyone provide some lecture notes or textbooks to enter this field? Thanks.
0
votes
0
answers
369
views
Is there a known shorter axiomatization of NF than this?
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...
3
votes
0
answers
133
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Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)
For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
25
votes
1
answer
3k
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Surreal exponentiation -- are the varying definitions equivalent? If not, is there agreement on which ones are better?
The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
4
votes
1
answer
230
views
Solving equations in the Brauer algebra
(First asked in MSE)
The Brauer algebra $B_n(x)$ is an algebra of matchings whose product is described here.
Given $A$ and $B$ two elements of $B_n(x)$, and given an integer $m$, there are in ...
1
vote
1
answer
994
views
A new generalisation of dimension? part 2
I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...
6
votes
1
answer
238
views
Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
3
votes
1
answer
121
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Image of a finite dimensional complex representations of $GL_n(\mathcal{O})$
EDIT Let $\mathcal{O}$ be the ring of integers in a non-Archimedean local field. Let $GL_n(\mathcal{O})$ be the (compact) group of $n\times n$ matrices with entries in $\mathcal{O}$ such that its ...
5
votes
1
answer
678
views
When are homomorphisms between Banach algebras contractions?
When are homomorphisms between Banach algebras contractions?
I recall from my student days that there are results which show that a positive answer to the above question holds under very general ...
1
vote
1
answer
161
views
mollifier satisfying moment conditions
I wish to find a mollifier $\psi\in C_0^{d+1}(-1,1)$ such that
$$
\int_{-1}^1 x^k \psi(x)dx = \begin{cases}
1, & k=0;\\
0, & k=1,\dots,d.
\end{cases}
$$
This paper (https://home.cscamm....
2
votes
1
answer
392
views
Explicit semi-stable theorem for elliptic curves over $p$-adic fields
In this paper of Maja Volkov, the authur metions a number called "défaut de semi-stabilité" on page 9. It is defined as $\text{dst}(E)=\frac{12}{\text{pgcd} (12,v_p(\Delta_E))}$ where $E$ is ...
3
votes
1
answer
365
views
Lower bound on Carmichael Function
What is the tightest lower bound currently known for the Carmichael function?
I imagine it must grow much more slowly than the Euler's totient function which according to here is bounded as
$$ \phi(...
8
votes
1
answer
547
views
History of the study of Verma modules in terms of Kazhdan Lusztig Theory
Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ ...
2
votes
1
answer
97
views
Concentration of measure on finite powers of $S^\infty$
I am wondering about a natural generalization of theorem 1.4 in the article Dvoretzky's theorem — Thirty years later by Milman. My first thought was to look at Milman's paper that he cites for the ...
1
vote
0
answers
109
views
The $p$-adic valuation of powers of consecutive integers
Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$.
Claim: ...
5
votes
2
answers
899
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$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$
Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho ...
1
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0
answers
89
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On Fourier expansions of Siegel automorphic forms
Please tell me references on Fourier expansions of (non-holomorphic) automorphic forms on the symplectic group of matrix size > 2. I can find formulas of "Fourier coefficients'' of non-holomorphic ...
3
votes
0
answers
131
views
What is a $C^\infty$ diffeomorphism from $\ell_2\setminus\{0\}$ to $\ell_2$ which is the identity outside a ball?
Let $\ell_2:=\{x=(x_n)_{n\in\mathbb N}:\ \|x\|^2:=\sum_n|x_n|^2<\infty\}$ with its natural norm. According to Wikipedia https://en.wikipedia.org/wiki/Kuiper%27s_theorem and to other sources, it is ...
4
votes
1
answer
382
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
1
vote
0
answers
104
views
A formula that proves that $G$ acts trivially on $H^*(G,M)$
If $G$ is a group and $M$ a $G$-module then for $n\geq 0$ we have an action of $G$ on the cochains from $C^n(G,M)$. If $s\in G$, $a\in C^n(G,M)$ then $(sa)_{s_1,\ldots,s_n}=sa_{s^{-1}s_1s,\ldots,s^{-1}...
15
votes
0
answers
714
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Is this "Homology" useful to study?
In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$.
Now we can ...
2
votes
0
answers
1k
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bounds on derivatives of mollifiers/mollified functions
Consider the standard mollifier
$$
\phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1.
$$
such that $\int\phi(x) = 1$.
Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am ...
19
votes
1
answer
2k
views
Is Van der Waerden's conjecture really due to Van der Waerden?
Van der Waerden's conjecture (now a theorem of Egorychev and Falikman) states that the permanent of a doubly stochastic matrix is at least $n!/n^n$.
The Wikipedia article, as well as many other ...
0
votes
1
answer
66
views
Empirical measurement of plant noise, for implementing Kalman Filter, using chirp data
I want to implement a Kalman Filter for the system:
$$ \dot x = Ax + Bu + w_p, \qquad y = Cx + w_m $$
where $w_p$ and $w_m$ are the plant noise and measurement noise respectively, which are both white ...
4
votes
0
answers
310
views
Steinberg relations for elementary subgroup of a Chevalley group over an arbitrary ring
Given a semisimple Lie algebra $\frak{g}$ of type $\Phi$ with a Lie algebra representation $\rho:\frak{g}\to \frak{gl}(v)$ and an arbitrary commutative ring one can associate the following gadgets:
...
7
votes
4
answers
2k
views
$E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
$E$ is a holomorphic vector bundle.
There is a Dolbeault operator $\bar{\partial}...
0
votes
0
answers
184
views
Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$
Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
6
votes
0
answers
174
views
Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary
I'm looking for a reference for a statement like:
Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form
$\mathcal L ...
2
votes
0
answers
53
views
References for linear relations on Hilbert spaces
I am trying to find a reference for linear relations (multivalued operators). I would like to have something which gives an introductory overview.
All I have found so far doesn't seem right for ...