Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
12,985
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Multipole expansion
In Simon's book Harmonic Analysis, example 3.5.12 shows:
Fix $x \in \mathbb{R}^{\nu}$ (with $\left.\nu \geq 3\right), x \neq 0$ and define $f$ on $\{y:| y|<| x |\}$ by
$$
f(y)=|x-y|^{-(\nu-2)}.
$$
...
- 11
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53
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Riemannian geometry of Grassmannian bundles
The Grassmannian bundle of a vector bundle $E$ is a smooth manifold where each fiber over the base space is replaced by the Grassmannian (of specified rank) of the fiber. I am interested in defining a ...
- 31
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29
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Reference request about the measure of nonconvexity
I'm actually working on the measure of nonconvexity and its application to obtain different extensions of the famous fixed-point theorem. I'm looking for some references in which the measure of ...
- 301
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56
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Positive values of Schur polynomials
Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials
$$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...
- 702
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Integral representation of $\frac{355}{113}-\pi$? [duplicate]
It is well known that
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
- 689
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Prove $u(t,\cdot) \in L^\infty(\mathbb R)$ for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$
Let us consider the PDE
$$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f\in C^2(\mathbb R$) and $f$ strictly convex.
Assume $u(0,\cdot) = u_0(\cdot) \...
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Another proof of Euler inequality via the half-angle formulas
The Euler's inequality is an immediate consequence of Euler's identity in a triangle,
$$OI^2=R^2−2Rr.$$
An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
2
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71
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Approximating a probability density with a point set
Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
&...
- 3,698
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144
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MacMahon Master Theorem for non-matching coefficients
Let $ A$ be a complex $ n$ by $ n$ matrix and $ x_1, \dots, x_n$ be a set of commuting variables. Let $ X_i = \sum_i a_{ij}x_j$. MacMahon's Master Theorem (MMT) states that
\begin{align}
[x_1^{...
- 689
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Index-decomposition identity for Chebyshev polynomials
Let $n$ be a positive integer. Denote $ \left[ n \right] \equiv \{1, \ldots , n \}$. Denote $ T_{n} \left( x \right) $ as the $n$-th Chebyshev polynomials of the first kind. Let $ m_{1}, \ldots , m_{n}...
- 129
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Sets measurable in every affine subspace
Take a non-measurable subset $S\subseteq [-1,1]$ and subtract $S\times \{0\}$ from the unit disk $B$ in $\mathbb{R}^2$. The set $X=B\setminus (S\times \{0\})$ is measurable by 2-D Lebesgue measure ...
- 34k
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1-d $L^1$ Maximal Function estimate
For $d \ge 1$, $1 < p \le \infty$, and $f ∈ L^p(\mathbb R^d)$, it is known that
there is a constant $C_{p,d} > 0$ such that
$$\Vert Mf\Vert _{L^{p}(\mathbb {R} ^{d})}\leq C_{p,d}\Vert f\Vert _{L^...
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+200
Regularity of the Robin function
I consider an analytic bounded domain $\Omega\subset \mathbb R^3$ and an the operator $L_a=-\Delta +a$ where $a$ is a function from $\Omega$ to $\mathbb R$. I assume the operator to be coercive, in ...
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projection formula for generalized multiplicative cohomology theory
Let $E^{\bullet}$ be a multiplicative generalized cohomology theory and $\cup$ be the induced cup product. It is known that we can define a Gysin map $\iota_*:E^{\bullet}(Z)\rightarrow E^{\bullet}(X)$ ...
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A constant ratio of integrals? Part II
This question is a follow up on my latest MO post which was addressed kindly by Iosif Pinelis. What is new here is that I need to correct the assumption by including a missing hypothesis. The context ...
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A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 38.3k
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Reference for universal elliptic curves
I've seen the following sentence come up in a few papers:
Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$.
This comes up in Deligne's construction of ...
2
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1
answer
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Weak form of $\text{CH}$ in $L(\mathbb{R})$
I was wandering whether this weak form of $\text{CH}$ holds in $L(\mathbb{R})$ provably in $\text{ZF}+\text{DC}$
$(\text{ZF}+\text{DC}) \ L(\mathbb{R})\vDash \forall X\subseteq\mathbb{R} ( X \text{ ...
- 917
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322
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On quasi-modular forms with integer Fourier coefficients
It is well-known that the ring $M$ of modular forms has the structure $M=\mathbb{C}[E_4,E_6]$, where $E_k$ are the Eisenstein series.
It is also known that one can define the concept of quasi-modular ...
- 5,731
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+50
Why and how is a representation "continuously decomposable"?
What I am asking may apply to a much more general setting and I am interested in the underlying level of generality of these statements, mostly with canonical references. However I state the question ...
- 5,115
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votes
1
answer
108
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Computing mth power residue symbols
Let's say I have a two odd primes, $p, q$ and $K$ is the field $\mathbb{Q}(\zeta_{pq})$. Let's say $\alpha \in \mathcal{O}$ is an arbitrary element in the ring of integers of $K$, $\frak{b} \subset \...
- 33
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142
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Can Set theory be interpreted in Relational Mereology?
In a posting to MathStackExchange, I've presented a theory of rudimentary relations having a rudimentary kind of membership together with a primitive ordered pairing with the aim for it to capture the ...
- 5,318
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66
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When do Borel propositional theories have topologically tame truth assignments?
Let $(P_r)_{r \in \mathbb{R}}$ be an $\mathbb{R}$-indexed family of propositional variables. Let $\mathcal{L}$ be the collection of all propositional sentences formed from the variables $(P_r)_{r \in \...
- 6,394
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votes
1
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211
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Explicit isomorphism between two realizations of $H^*_T(\operatorname{Gr}(k,n))$ (reference request)
Let $X$ be the Grassmannian variety $\operatorname{Gr}(k,n)$ of $k$-planes in $\mathbb{C}^n$. I'm aware of two ways to describe its $T$-equivariant cohomology:
(Quotient ring) $H_T^*(X)=\Lambda[e_1(x|...
- 248
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95
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A continuous map relating co-constructible reals
My question is the following:
Given $x,y \in \omega^\omega$ such that $x\equiv_c y$ is there an $L$-definable continuous map $\varphi: \omega^\omega\rightarrow \omega^\omega$ such that $\varphi(x) = ...
- 917
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39
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Finite dimensional subspaces of L^p, entropy estimates
The following follows from Proposition 9.6 in Approximation of zonoids by zonotopes, Acta Math. 162 (1989), no. 1-2, 73–141, by Bourgain, J., Lindenstrauss, J., and Milman, V.
Let $X\subset L^1(\mu)$ ...
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Understanding descending intersections of generic extensions
Let $B_{0}\supseteq B_{1}\supseteq\dots\supseteq B_{\alpha}\supseteq\dots\,\,\left(\alpha<\kappa\right)$ be a descending sequence of complete Boolean algebras, $B_{\kappa}:=\bigcap_{\alpha<\...
- 267
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Taming families of rate functions
$\newcommand\R{\mathbb R}$Let us say that a function $r\colon\R_+\to\R_+$ is a rate function if $r$ is nondecreasing and $r(x)\to\infty$ as $x\to\infty$.
Let us say that a family $(r_j)_{j\in J}$ of ...
- 78.3k
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votes
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views
"Gluing and copy" graphs
Consider the minimal class of (simple, undirected) connected graphs (strictly speaking, isomorphism classes of connected graphs) which contains a single vertex $K_1$, and is closed under following ...
- 88.5k
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Recommendations for distributed calculations of Groebner Bases
There are many computer algebra systems available which can compute a Groebner basis, including:
Mathematica
Singular
Macaulay2
Magma
Maple
CoCoA
However (please correct me if I've missed something) ...
- 51
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Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 269
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Where could I find the article about creating three-varible inequalities which holds for three sets of given numbers?
Where could I find the article about creating three-varible inequalities which holds for three sets of given numbers?
Sometimes I use scipy.optimize.lq (Least squares) on strong function like $z= \...
4
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1
answer
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Reference request: path integral approach to Gaussian processes
Are there any good, rigorous and preferably modern books or papers on path integral approach to Gaussian processes? I am interested in both introductory level and deeper monographs on the subject.
I ...
- 141
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0
answers
31
views
Return probabilities for heavy tailed random walks
I need a reference to an explicit asymptotic formula for the return probabilities for random walks on $\mathbb Z$ with heavy tailed symmetric step distributions. More specifically, for an explicit ...
- 15.5k
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votes
0
answers
67
views
Counting points on Elliptic Curves with CM by $\mathbb{Q}[\sqrt{-d}]$, $d=1,3$ (CM ring with non-trivial units)
Consider an elliptic curve $E/H$ with CM by the entire ring of integers $\mathcal{O}_K$ of $K=\mathbb{Q}[\sqrt{-d}]$ (and such that $j(E)=j(\mathcal{O}_K)$) such that $H$ is the Hilbert class field of ...
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Reference request for statement concerning free subgroups of $ \mathrm{SL}_2(\mathbb{Z}). $
I am interested in finding a reference for the following claim:
There exists a free subgroup $F_2$ of $\mathrm{SL}_2(\mathbb{Z})$ on two generators that does not contain any nontrivial unipotent ...
- 131
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138
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A question on terminology for sequences satisfying $\gcd(a_m,a_n)=a_{\gcd(m,n)}$
How do you refer to those sequences $\{a_{n}\}_{n \in \mathbb{Z}^{+}}$ of integers that satisfy the condition $\text{gcd}(a_{m}, a_{n}) = a_{\text{gcd}(m,n)}$ for every $(m,n) \in \mathbb{Z}^{+} \...
- 135
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votes
1
answer
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Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 269
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Paradoxical games with applications [closed]
Few weeks ago I read an article (in Spanish [1]) about an application of a paradox in game theory, it is the known as Parrondo's paradox, and Wikipedia has the corresponding article Parrondo's paradox....
- 55
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0
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23
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The Mellin transform as a mapping from a Hardy space to a weighted space
I posted this at MSE, but I have got no response.
As far as I know, the following fact must be published somewhere, and I would like to find a reference.
The Mellin transform is defined by $f\mapsto\...
- 1
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Motivic cohomology as $\mathit{Hom}$ in the category of geometric motives, with coefficient in a Chow motive
The main references for this question are
1 : V.Voevodsky's paper Triangulated categories of motives over a field
2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, ...
8
votes
1
answer
445
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Where was the Cantor normal form theorem first proved?
We all take for granted the theorem that every ordinal $\alpha > 0$ has a Cantor normal form, and there are plenty of proofs of it, some of which are on this site. However, where was it proved? Was ...
- 470
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2
answers
255
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Pullback of $w_1$ for 3-manifolds
Given closed $3$-manifolds $M$ and $N$
and an element $\alpha\in H^1(M;\mathbb{Z}_2)$,
when does there exist a map $f:M\to N$
such that $\alpha=f^*(w_1(N))$?
- 251
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76
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Explicit Abel addition formula?
There are many specific examples of specific cases of Abel's addition formula on Abelian integrals. Is there any reference that does a proof of the addition theorem in a way that allows one to ...
0
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1
answer
59
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Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?
This question is based on this Math.SE answer, so let's recall a few concepts dealt with there. If $\{a_n\}_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum_{n=0}^\infty a_nz^n$ ...
- 4,467
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votes
2
answers
272
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Are infinitary monads monadic?
As discussed here, Are monads monadic?, in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd_f(C) \rightarrow Endo_f(C)$ is ...
- 239
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56
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Reference request: Integrability condition on measures
Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$.
Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
- 269
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1
answer
223
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Under which conditions is the bar construction a conservative functor?
The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends ...
- 2,465
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0
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49
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Affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$
I saw the following results on affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$ without giving any references, where $\mathrm{SU}_3$ is the quasi-split inner form of special ...
- 53
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votes
0
answers
43
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Uniform integrability condition for measures on classical Wiener space
Consider the (non-locally compact) Banach space $(\mathcal{C}, \|\cdot\|)$ of continuous paths $x : [0,1]\rightarrow\mathbb{R}^d$ endowed with the $\mathrm{sup}$-norm $\|x\|=\sup_{t\in[0,1]}|x_t|$.
...
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