Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

Filter by
Sorted by
Tagged with
1 vote
0 answers
27 views

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 $$ f(y)=|x-y|^{-(\nu-2)}. $$ ...
user avatar
1 vote
0 answers
53 views

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 ...
user avatar
0 votes
0 answers
29 views

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 ...
user avatar
  • 301
2 votes
0 answers
56 views

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}(\...
user avatar
6 votes
1 answer
391 views

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 ...
user avatar
1 vote
0 answers
27 views

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) \...
user avatar
  • 135
1 vote
0 answers
54 views

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 ...
user avatar
2 votes
0 answers
71 views

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? &...
user avatar
  • 3,698
4 votes
1 answer
144 views

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^{...
user avatar
1 vote
0 answers
32 views

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}...
user avatar
0 votes
0 answers
33 views

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 ...
user avatar
-1 votes
0 answers
28 views

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^...
user avatar
1 vote
0 answers
55 views
+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 ...
user avatar
  • 812
3 votes
1 answer
96 views

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 $\iota_*:E^{\bullet}(Z)\rightarrow E^{\bullet}(X)$ ...
user avatar
  • 93
5 votes
1 answer
512 views

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 ...
user avatar
5 votes
1 answer
307 views

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_{...
user avatar
4 votes
2 answers
192 views

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 ...
user avatar
2 votes
1 answer
125 views

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{ ...
user avatar
  • 917
1 vote
0 answers
322 views

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 ...
user avatar
9 votes
0 answers
138 views
+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 ...
user avatar
3 votes
1 answer
108 views

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 \...
user avatar
1 vote
0 answers
142 views

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 ...
user avatar
3 votes
0 answers
66 views

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 \...
user avatar
  • 6,394
4 votes
1 answer
211 views

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|...
user avatar
2 votes
1 answer
95 views

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) = ...
user avatar
  • 917
1 vote
0 answers
39 views

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 $X\subset L^1(\mu)$ ...
user avatar
5 votes
0 answers
109 views

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<\...
user avatar
  • 267
1 vote
0 answers
35 views

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 ...
user avatar
10 votes
3 answers
791 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 ...
user avatar
  • 88.5k
3 votes
0 answers
53 views

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) ...
user avatar
1 vote
0 answers
81 views

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 ...
user avatar
  • 269
-1 votes
0 answers
70 views

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= \...
user avatar
4 votes
1 answer
95 views

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 ...
user avatar
0 votes
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 ...
user avatar
  • 15.5k
0 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 ...
user avatar
3 votes
1 answer
132 views

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 ...
user avatar
2 votes
0 answers
138 views

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}^{+} \...
user avatar
  • 135
2 votes
1 answer
104 views

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 ...
user avatar
  • 269
-1 votes
1 answer
134 views

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....
user avatar
0 votes
0 answers
23 views

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\...
user avatar
3 votes
1 answer
74 views

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, ...
user avatar
8 votes
1 answer
445 views

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 ...
user avatar
  • 470
2 votes
2 answers
255 views

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 $\alpha=f^*(w_1(N))$?
user avatar
0 votes
0 answers
76 views

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 ...
user avatar
0 votes
1 answer
59 views

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 $\sum_{n=0}^\infty a_nz^n$ ...
user avatar
6 votes
2 answers
272 views

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 ...
user avatar
  • 239
0 votes
0 answers
56 views

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 ...
user avatar
  • 269
7 votes
1 answer
223 views

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 ...
user avatar
1 vote
0 answers
49 views

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 ...
user avatar
0 votes
0 answers
43 views

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 $\|x\|=\sup_{t\in[0,1]}|x_t|$. ...
user avatar
  • 269

1
2 3 4 5
260