Examples of multiplier Hopf algebras

A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $Delta: A to M(A otimes A)$ such that

(i) the following subsets of $M(A otimes A)$ are contained in $A otimes A$:

$$Delta(A)(1 otimes A), quad Delta(A)(A otimes 1), quad (A otimes 1) Delta(A), quad (1 otimes A)Delta(A).$$

(ii) $Delta$ is coassociative.

(iii) The following maps $A otimes A to A otimes A$ are bijective:
$$a otimes b mapsto(a otimes 1)Delta(b), quad a otimes b mapsto Delta(a)(1 otimes b)$$

Examples include:

  • Any Hopf-algebra.
  • The finitely supported functions on a group $X$. Here, the comultiplication is given by
    $$Delta(f)(x,y) = f(xy)$$

Are there any easy other examples?

compactness – Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds

A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $Gamma$ on $G$ and the solvmanifold is $G/Gamma$. Moreover, if there are left-invariant structures on $G$ (e.g. metrics, complex strucutres etc.), those structures define analogous structure on the compact quotient. Moreover, left-invariant structures are viewed as tensors on the Lie algebra $mathfrak{g}$ of $G$.

I would like to know if there are known examples of solvmanifold with pseudo-Riemannian (or semi-Riemannian or indefinite) Einstein non-Ricci-flat metrics.

By the previous discussion, this question can be rephrase as follows: Are there examples of solvable Lie algebras admitting indefinite metrics (i.e. indefinite non-degenerate bilinear form) which are Einstein non-ricci-flat? Do those examples admit compact quotients?

Finally, remind that a nilpotent Lie group is a solvable Lie group, and a result of Maltcev assure that nilpotent Lie group with rational constant structures admits a co-compact lactice (hence a compact quotient).

Building An Ehr System: Definition, Benefits, Problems, And Examples

Innovation and healthcare always go hand in hand. That’s why doctors heavily rely on Electronic Health Record systems to be able to concentrate on patients. Thinking about building an ehr system? Of course, with great technology comes a lot of effort to understand it and adopt. Here we clear this up

dg.differential geometry – Examples of group actions on statistical manifolds

A statistical manifold $(M,g,nabla)$ is a Riemannian manifold with a torsion-free affine connection $nabla$ such that $nabla g$ is symmetric in all entries. Equivalently, there is a dual affine connection $nabla^*$ such that $X(g(Y,Z)) = g(nabla_X Y,Z) + g(Y,nabla^*_X Z)$, with $X,Y,Z in mathcal{X}(M)$.
In the preprint https://arxiv.org/pdf/2005.13927.pdf there is a definition of homogenous statistical manifold (Definition 1.4.1). One of the conditions says that the action of $G$ (by isometries) is affine with respect to $nabla$. I am not familiar with actions on statistical manifolds. I am supposing that it means that, as a map $nabla : mathcal{X}(G) times mathcal{X}(G) to mathcal{X}(G)$, the connection verifies $nabla_X Y = (h^{-1})_* (nabla_{h_* (X)} h_*(Y))$ (?). Here $nabla_X Y := nabla(Y,X)$ and $h_*$ indicates de tangent action induced by the left translation by an element $h$ in $G$.

Thus there is an induced quotient homogeneous statistical structure $(G/H, bar g, bar nabla)$, right?. Besides the examples presented in that paper, are there others (more geometrically inspired) known examples of homogeneous non-trivial statistical manifolds?
The structure is said trivial if $nabla = nabla^*$ and in that case they coincide with the Levi-Civita connection, otherwise it is non-trivial.

reference request – Examples of matrices with all subdeterminants bounded away from $0$

For any distinct integers $x_1,ldots,x_m$ you can take the matrix whose
$i$-th row $(1 leq i leq m)$ is $(1,x_i,x_i^2,ldots,x_i^{n-1})$.
Each $n times n$ submatrix is
with distinct rows, so has nonzero determinant, which being an integer
must have absolute value at least $1$.

In fact the absolute value is
at least $1! 2! cdots (n-1)!$; replacing each entry $x_i^{j-1}$ by
$x_i choose j-1$ applies an invertible column transformation
and retains integrality, so we still have all $n times n$ submatrices
with nonzero integral determinant, and the entries are overall smaller.

Or, if the $x_i$ are all distinct modulo some prime $p$,
replace each $x_i^{j-1}$ by its remainder mod $p$ to get
a matrix with all entries in $(0,p)$ and all $n times n$ submatrices
irreducible mod $p$ and thus again with determinant equal to
some nonzero integer.

Any good VR concept document examples?

Just wondering if anybody could share some examples of VR concept documents, for either games or productivity apps. And would be nice to see some from successful products, especially from larger companies.

Is there much of a difference between VR concept documents and traditional software ones?

Examples of material design at real sites

For inspiration I’d like to have a look at enough examples of material design for web sited and of that framework by Google (https://github.com/material-components/material-components-web).

I have been trying. Trying hard. But I have found only one example. And it is https://design.google/.

One example is for me not enough.

I try to google something “material design example”. But if I inspect the code of those sites suggested by bloggers, I can’t see any material design there.

Wikipedia lists some sites using material design. But I can’t recognize it.

First of all, I try to find grid classes. That is I find for “mdc-layout-grid” in the code. Nothing. Then I try to reveal any CSS grid at the site. It may either be absent at all. Or the number of columns doesn not comply with material design guidline.

Then I go to GitHub and try to find something there. There are some code really found. But it also didn’t help me find any good working sites to scrutinize.

Could you show me some: 5, 10, 20 examples of web sites using material design and that frontend framework by Google?

hardware – Are there any examples of multi-screen or multi-device design patterns?

There are increasingly more offices and work environment setups that allow employees to use multiple screens to do their work, and there are also many people who use multiple devices at the same time.

I am wondering if there are any example designs or design patterns that have been used to cater to these types of user behaviour. I think only creating multiple instances of the application (i.e. different windows) or having different components of the design float around is probably not taking advantage of the screen space or the multi-modal way of interaction with the application, so I would like to know if anyone has seen good examples that are around.

UPDATE: The Aurora 7 Laptop from Expanscape seems to have taken multi-screen design to another level altogether

What are some examples of why Python is used with Microsoft Access databases?

I see lots about how to connect Python with Access database, but not anything on why one does it. (Clearly you can see that I am not familiar with Python.) I’ve created CRM applications using Access for both the front-end and back-end. The VBA in Access has allowed me to develop useful applications (for very small businesses!!), but I am always interested in learning what else I could do to make the applications more robust or expansive. I hope I’m not asking too broad of a question- I realize I’m very naive- I just need somewhere to start.
Thank you,

website design – File cart (document basket) – good UI examples

can anyone provide a website with a good example of a well designed file cart?
I’ve researched the topic but there’s plenty of articles about shopping carts and not file carts. The only examples of file carts I’ve found were the simple ones which allowed the users to share files via email or merge files.
However I struggle to find a case which would have more elaborate functionalities, like e.g. ability to create a submittal based on contents in file cart, which would require ability to sort and order assets and adding customizable cover page (e.g. adding a project title).
One more question:

  1. Would it be beneficial to ensure that merged .pdf file size gets optimized before a user downloads it (should the user have a choice to download the original or optimized size)?

Any feedback will be highly appreciated. Thanks!