Questions tagged [reference-request]

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

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2 votes
1 answer
244 views

Application of Yamabe and Liouville type equation

Let $\Omega$ be a domain in $\mathbb{R}^n$. I am interested in the following critical elliptic partial differential equations (PDEs): The Yamabe Type Equation (for $n>2$): \begin{equation} -\...
3 votes
1 answer
276 views

Prove positivity of a binomial sum

Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
1 vote
1 answer
96 views

Devaney chaos and topological entropy

I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the ...
0 votes
0 answers
34 views

Enumeration of uniform polyhedra

[I already asked this question on MSE (here) but got no answer so I am trying here] It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with $75$ uniform ...
3 votes
1 answer
105 views

Generic absoluteness

In Theorem 14 of "On The Question Of Absolute Undecidability" Peter Koellner describes a generic absoluteness result which could be summed up as $\Sigma^2_1(\Gamma^\infty)$-generic ...
0 votes
0 answers
55 views

Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1. Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
4 votes
3 answers
4k views

Green's function on sphere

Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
2 votes
1 answer
174 views

References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment of tilted distributions using the following notion of "tilting" (or equivalent): ...
1 vote
0 answers
47 views

Reference on faithfully exact functors between abelian categories

I am looking for a reference of the following result (or any subresult) in any book or notes: Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...
0 votes
0 answers
49 views

Random walks on groups

I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
1 vote
0 answers
30 views

Optimal orbital tranfers: reference for statements & proofs

I know that Pontryagin's contribution to optimal control in the 1950'ies was allegedly inspired by the then-nascent rocket industry and the question of how to get a rocket into orbit with minimum fuel....
1 vote
1 answer
76 views

A Kolmogorov inequality for sums of contiguous subsequences

If $X_1, \ldots X_n$ are independent real-valued random variables such that $E[X_k] = 0$ and $E[X_k^2]$ is finite for each $k$, Kolmorogov's inequality gives an upper bound on $P[\max_{1\le k \le n}|...
2 votes
0 answers
42 views

Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
2 votes
0 answers
90 views

Is there a name for a normal, projective variety where every effective divisor is ample?

Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
4 votes
0 answers
50 views

Coloured Jones polynomial of the mirror image of a multicomponent link

This question has been reposted from MathStackExchange It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...
1 vote
2 answers
269 views

Closed formula for Hermite polynomials

Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula $$ H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) . $$ Each $H_k(x)$ is a polynomial of exact ...
3 votes
1 answer
337 views

Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?

(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.) Imagine an introductory probability course ...
1 vote
1 answer
138 views

Special Darboux chart for tranverse Lagrangians

In the notes of Fukaya-Oh-Ohta-Ono (Lagrangian intersection Floer theory), Chapter 10 §54.1, it is stated: Let $L_{1}$ and $L_{2}$ be a pair of oriented Lagrangian submanifolds in $(M, \omega)$ that ...
5 votes
0 answers
79 views
+50

Reference request for equivalences between different models of Lax limits

There are several models for Lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
32 votes
3 answers
1k views

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

Updated on Feb.16.2024 Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
2 votes
3 answers
848 views

The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$. I am interested in the following questions: How ...
1 vote
1 answer
227 views

Numerical estimation of partial derivatives of convolved functions when closed forms do not exist

Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
6 votes
2 answers
304 views

Are finite projective modules over $R[t]$ free when $R$ is DVR?

Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$? As far as I understand, this should be a ...
2 votes
0 answers
59 views

Different definitions of the thick affine flag variety

I have seen several different definitions of the so called "thick" affine flag variety associated to an affine Lie algebra, and I am having trouble seeing why they are the same. Some ...
2 votes
0 answers
77 views

A recursive description of the smallest divisor-closed subsemigroup containing a set

Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid). ...
2 votes
0 answers
190 views

Squares whose differences are squares

EDIT. I've just noticed a thread from 2011 in the "Related" column on the right (click me), where a closely related question is being discussed (the main difference seems to be that, in ...
4 votes
1 answer
445 views

Turing Machine which generates order on the set of its states

This question is related to this one Do Turing Machines generates any nontrivial lattice on the set o symbols or states? The Turing machine (TM) is an abstract model for effective implementation of (...
4 votes
1 answer
338 views

How much choice is needed to prove the completeness of equational logic?

All the proofs of the completeness of (Birkhoff's) equational logic I have read seem to pick representatives for equivalence classes of terms and hence require the axiom of choice. Is AC (or a weak ...
22 votes
5 answers
4k views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
0 votes
1 answer
192 views

Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia

I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
7 votes
1 answer
470 views

Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
14 votes
1 answer
686 views

What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions

Consider the following equivalence relation on topological spaces: $X\sim Y$ $:\Longleftrightarrow$ there are bijective continuous maps $\phi:X\to Y$ and $\psi:Y\to X$. Note that there are no ...
2 votes
1 answer
351 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
32 votes
23 answers
7k views

Early two-author math papers

The middle of the twentieth-century featured several famous papers with two authors. For example, Eilenberg and Mac Lane's papers introducing categories and Eilenberg-MacLane spaces appeared in 1945. ...
4 votes
0 answers
142 views

Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$

It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$ for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
7 votes
0 answers
106 views

What are these generalizations of the principles of omniscience called?

I will give some principles that are slightly stronger versions of the principles of omniscience. Despite being about the natural numbers, they imply their analytic versions! Under countable choice (...
1 vote
1 answer
116 views

Iwahori action on the $p$-ordinary line of a principal series representation

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\diag{diag}\DeclareMathOperator\Ind{Ind}\newcommand\Iw{\mathrm{Iw}}\DeclareMathOperator\ord{ord}$Let $F$ be a $p$-adic local field, i.e. a finite ...
2 votes
0 answers
81 views

Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it. We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
4 votes
0 answers
145 views

Every stable homotopical functor factors through $\mathbf{SH}$

In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...
4 votes
2 answers
399 views

Reference request - Pillai-Selberg Theorem

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
2 votes
1 answer
107 views

Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations

Does anyone know the parabolic regularity result that Ben-Artzi used in the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces used to prove that the solution to the ...
-3 votes
0 answers
52 views

Are there any references for finding pi using polygons inscribed in a circle? [closed]

Polygons inscribed in a circle of a diameter of 1 unit: Let there be an equilateral triangle inscribed in a circle and the measure of the sides of the triangle is a measure of an angle of sin of 60°, ...
3 votes
0 answers
140 views

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 ...
9 votes
1 answer
354 views

About certain infinite products with the property $f(a)=f(1/a)$

In the paper "Transformations of infinite series" Bryden Cais gives the following transformations of infinite products With some modification of Cais's method using contour integration one can obtain ...
13 votes
4 answers
966 views

Source for analysis of identification of structures in learner's mind and mathematical structures?

Concerning the structure of the learner's mind, psychologist Piaget claimed that There exists, as a function of the development of intelligence as a whole, a spontaneous and gradual construction of ...
10 votes
0 answers
363 views

Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
-3 votes
0 answers
70 views

What are the possible applications in maths and physics of vector fields along smooth maps? [migrated]

I am currently working on a problem related to singularities of mappings between manifolds with metrics and the interplay of metric singularities with mapping singularities. Given a smooth map $F:M\...
3 votes
1 answer
148 views

Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative rational

Let a (rational) Puiseux monoid be a non-trivial submonoid of the non-negative rational numbers under (the usual operation of) addition. It is not difficult to show that, if $f \colon H \to K$ is a (...
7 votes
1 answer
217 views

Computing $\pi_1$ of the complement of a non-singular plane curve

The following is a well-known fact: Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$. This was further ...
17 votes
11 answers
4k views

Applications of measure, integration and Banach spaces to combinatorics

I'm going to be teaching a Master's level analysis course (measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is ...

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