reference request – Relations of Lambert W function with Hypergeometric function


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magento2.4.1 – Magento 2 – How to solve report.INFO: Broken reference error

How to solve the following error,

[2021-06-17 03:28:43] report.INFO: Broken reference: the 'catalog.compare.sidebar' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'sale.reorder.sidebar' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'wishlist_sidebar' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'multiple-wishlist_sidebar' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'yotpo_bottomline' element cannot be added as child to 'product.info.main', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'paypal.partner.right.logo' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'bml.right.logo' element cannot be added as child to 'sidebar.additional', because the latter doesn't exist [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'yotpo_bottomline' tries to reorder itself towards 'product.info.addto', but their parents are different: 'product.info.main' and '' respectively. [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'wish-list-link-custom' tries to reorder itself towards 'minicart', but their parents are different: 'page.top' and 'header-wrapper' respectively. [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'navigation.sections' tries to reorder itself towards 'logo', but their parents are different: 'header-wrapper' and 'header.panel' respectively. [] []
[2021-06-17 03:28:43] report.INFO: Broken reference: the 'header.links' tries to reorder itself towards 'navigation.sections', but their parents are different: 'header.panel' and 'header-wrapper' respectively. [] []
[2021-06-17 03:28:48] report.INFO: Broken reference: the 'yotpo_bottomline' element cannot be added as child to 'product.info.main', because the latter doesn't exist [] []
[2021-06-17 03:28:48] report.INFO: Broken reference: the 'yotpo_bottomline' tries to reorder itself towards 'product.info.addto', but their parents are different: 'product.info.main' and '' respectively. [] []
[2021-06-17 03:28:48] report.INFO: Broken reference: the 'wish-list-link-custom' tries to reorder itself towards 'minicart', but their parents are different: 'page.top' and 'header-wrapper' respectively. [] []
[2021-06-17 03:28:48] report.INFO: Broken reference: the 'navigation.sections' tries to reorder itself towards 'logo', but their parents are different: 'header-wrapper' and 'header.panel' respectively. [] []
[2021-06-17 03:28:48] report.INFO: Broken reference: the 'header.links' tries to reorder itself towards 'navigation.sections', but their parents are different: 'header.panel' and 'header-wrapper' respectively. [] []

reference request – Foliations induced by Vector Fields without Singularities

A well-known type of foliations is the one that is induced by vector fields without singularities. However, I have already read this type of foliations from Geometric Theory of Foliations, Page 28. Unfortunately, I couldn’t understand it very well. Therefore, I am actually looking for highly recommended references for the foliations induced by vector fields without singularities. Thanks in advance.

reference request – Duality of maps on bounded vs trace-class operators (Schrödinger-Heisenberg dual)

$newcommandcalH{mathcal H}
newcommandcalK{mathcal K}
newcommandtr{operatorname{Tr}}$
I am looking for a (citable) reference for the following fact:

  • Bounded linear maps $g:T(calH)to T(calK)$ (bounded w.r.t. trace norm) stand in 1-1 correspondence with normal bounded linear maps $f:B(calK)to B(calH)$ via $tr g(rho)a=tr rho f(a)$ for all $rho,a$. (Here $T(calH)$ are trace-class operators and $B(calH)$ are bounded operators. And $calH,calK$ are Hilbert spaces.)
  • And then $f$ is completely positive iff $g$ is.

This is mentioned as a standard result e.g. in this comment. I have seen $f$ referred to as the Schrödinger-Heisenberg dual of $g$ but all references were for finite-dimensional Hilbert spaces $calH,calK$.

reference request – A group acts on a groupoid

Let $G$ be a group. Let $(Pi,circ)$ be a groupoid. Suppose I have a $G$-action on every morphism space $Pi(p,q)$, denoted by $Gtimes Pi(p,q)to Pi(p,q)$, $(g, sigma)mapsto gcdot sigma$. (For simplicity, we may assume $G$ is abelian.)
Suppse these $G$-actions further satisfy the natural properties (1) $gcdot (sigma_1circ sigma_2)=(gcdot sigma_1)circ sigma_2=sigma_1circ (gcdot sigma_2)$; (2) $g_1g_2cdot sigma=g_1cdot(g_2cdot sigma)$; (3) $(gcdot sigma)^{-1}=g^{-1}cdot sigma^{-1}$ (I might miss some.)

Is there a nice conceptual way to describe the above situation? I think what I ask is standard, but I fail to find a reference.

reference request – Automorphisms of the topological field $mathbb{C}_p$ of $p$-adic complex numbers?

I am interested to see what is currently known about the automorphisms of the topological field $mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-adic norm extended from $mathbb{Q}_p$). By automorphisms of topological fields I mean ones that are not only field isomorphisms but also are homeomorphisms.

Unless I’ve made some mistake it would appear that any automorphism of $mathbb{C}_p$ is fixed on $mathbb{Q}_p$ by continuity, so this would then boil down to finding the number of different ways to extend the identity map on $mathbb{Q}_p$ to its algebraic closure (since it would then extend uniquely to $mathbb{C}_p$ from $mathbb{Q}_p^text{alg}$).

I have not been able to find any literature on this, so maybe I am searching the wrong terms. I would very much appreciate being pointed in the direction of any relevant literature.

Many thanks

reference request – How does one know what is the right Riemannian metric matrix $G$ to equip for a given manifold $M$?

Let $(M, g)$ be a finite-dimensional Riemannian manifold. A Riemannian metric is expressed by a symmetric positive definite matrix $G(x)$ where $x$ lies in a tangent space of $M$.

My question is, given a space $M$, how do you systematically choose/construct/come up with an appropriate $G$?

One idea I’ve thought of is to come up with an arbitrary positive definite matrix. But then it leads to the following questions: 1. for example, let $M$ be the positive orthant: $R^n_{++}= {x in R^n: x_i > 0, i = 1,…,n}$, then I can take $G(x) =left(frac{1}{x_i}right)$ or $G(x) = left(frac{1}{x^2_i}right)$, but which one is more suitable and why? 2. how do I generalize this idea so to come up with $G$ for other types of $M$?

Is this a trivial or hard question?

reference request – Video lectures about Information Theory and Thermodynamics!

I’d like to know if there is any video courses that relate information theory and thermodynamics, e.g. courses cover laws of thermodynamics, Shannon’s entropy, Kolomogorov Complexity, Landauer’s Principle, and anything related to them in one course.

Thank you!

8 – Access a user entity reference in a commerce-product template file

I have a field_intervenantfield which contains a user entity reference.
In the commerce-product template file, I want to access the description field of each intervenant.

How can I achieve it?

I tried reading the following values, but none of them returns what I am looking for.

  • product.field_intervenants.entity.field_description
  • product.field_intervenants.0.entity.field_description
  • product_entity.field_intervenants.0.entity.field_description
  • product_entity.field_intervenants.field_description
  • product_entity.field_intervenants.field_description.value

Do I need to loop through it, or am I missing something else?

8 – Access user entity reference in a commerce-product twig

I have a field named field_intervenant who is a reference to my Users.
In my commerce-product twig I want to access to the description field of each of my ‘intervenant’, does anyone know the way to do it ?
Here is what I tried :

product.field_intervenants.entity.field_description
product.field_intervenants.0.entity.field_description
product_entity.field_intervenants.0.entity.field_description
product_entity.field_intervenants.field_description
product_entity.field_intervenants.field_description.value

As long as some others but never gave me back something, and I can’t find it in my kint aswell, do I need to loop through it or am I missing something else?
Thanks for your help!