[FRIAM] A poem: On Kronecker's Loom was: (Fwd: links for this morning's FRIAM: Special Unitary Groups and Quaternions)

Fri May 5 16:45:24 EDT 2023

In celebration of my new found understanding of the Kronecker operator
and FFTs with Laplacians, I asked GPT's help to pen some verse as a bard
might for a court :-)

For context, a couple weeks ago, Frank gave me a ride to Friam with my
volvo in the shop. this week I returned his coffee mug he left Here comes
the metaphor, consider our iterated events of pairwise actions in
friendship as a cyclic block matrix and our interactions with you adding
the diagonal and the full tapestry/matrix of our community the Kronecker
tiling ;-) Here's the result

On Kronecker's Loom: Tapestry

Weeks ago, Stephen's Volvo under repair,
Frank's BMW arrived, a a lift to share,
horizontal row as thread in friendship,
Begins the weave tales of minds entwined

Todays mug with logo returned, a vertical thread, a column sewn,
With knowledge and random chatter they create,
The tapestry of ideas, together grown,
In loom of thought, friendships elevate.

Ed joins the Diagonal, the knowledge grows,
With Frank, a formal language bond they know,
Symbolic language, to Stephen once opaque,
Now grasped, the power of the operator in hand.

In past, with clumsy nested loops, Stephen worked,
Yet, newfound insight in linear algebra gained,
C = np.kron(A, B) the operator, unshirked,
Through friendships forged, knowledge unrestrained.

Ideas exchanged, insights embraced,
Connections formed, in wisdom traced,
A realm of learning, friendships laced,
Block matrices, the foundation laid.

In Kronecker's touch, a pattern forms,
Integrating all the pairwise friendships made,
Their union, a dance that transforms,
The tapestry of knowledge, unswayed.

_______________________________________________________________________
Stephen.Guerin at Simtable.com <stephen.guerin at simtable.com>
CEO, https://www.simtable.com <http://www.simtable.com/>
1600 Lena St #D1, Santa Fe, NM 87505
office: (505)995-0206 mobile: (505)577-5828

---------- Forwarded message ---------
From: Stephen Guerin <stephen.guerin at simtable.com>
Date: Fri, May 5, 2023 at 2:31 PM
Subject: Re: [FRIAM] links for this morning's FRIAM: Special Unitary Groups
and Quaternions
To: The Friday Morning Applied Complexity Coffee Group <friam at redfish.com>

Thanks Roger and Ed!

I've spent some time with Ed and Frank discussing this and I've really
filled in some gaps in my knowledge of parallel algorithms. eg, I knew how
to solve n-body system with particle N^2/2 focus with some quadtree or
octree optimizations to get from n^2 to nlog(n). But the FFT transform on
laplacians solving Poisson equation was new to me and I can now see the
beauty. Today, Ed quickly threw out the Kronecker Operator/Product which
Frank knew but I didn't. Frank flashed me a wikipedia article
<https://en.wikipedia.org/wiki/Kronecker_product> on his phone with
symbolics that I couldn't immediately grok. But asking chatGPT to explain
the operator to a 3D graphics person I immediately got it and had the
benefit that I would usually implement this function with two inner loops
over rows and columnts instead of using Kronecker available in optimized
linear algebra/graphics libraries. Often this was happening under the hood
of my tools but didn't realize it.

As a 3D graphics developer, understanding the Kronecker matrix can be very
useful. The Kronecker product is often used in computer graphics and
computer vision applications, such as texture mapping, geometric
transformations, and image processing. Here are a few specific ways in
which Kronecker matrix can be useful to a 3D graphics developer:

   1. Texture mapping: The Kronecker product can be used to create
   repetitive patterns in textures, such as brick walls, tiles, or grass. By
   creating a base texture and applying a Kronecker product with a smaller
   texture, a developer can create a seamless and repeating texture that
   covers a larger surface.
   2. Geometric transformations: The Kronecker product can be used to
   perform geometric transformations, such as scaling, rotation, and
   translation, on 3D objects. By creating a Kronecker matrix with a
   transformation matrix, a developer can apply the transformation to every
   vertex of an object, resulting in a transformed object.
   3. Image processing: The Kronecker product can be used to perform image
   processing operations, such as blurring, sharpening, or edge detection, on
   3D images. By creating a Kronecker matrix with a filter matrix, a developer
   can apply the filter to every pixel of an image, resulting in a processed
   image.

In summary, the Kronecker matrix is a powerful tool that can be used in
various ways by 3D graphics developers. Whether it's creating textures,
transforming objects, or processing images, understanding the Kronecker
matrix can help a developer achieve their desired results more efficiently
and effectively.

_______________________________________________________________________
Stephen.Guerin at Simtable.com <stephen.guerin at simtable.com>
CEO, https://www.simtable.com <http://www.simtable.com/>
1600 Lena St #D1, Santa Fe, NM 87505
office: (505)995-0206 mobile: (505)577-5828

On Fri, Apr 28, 2023 at 7:50 PM Angel Edward <edward.angel at gmail.com> wrote:

> Most of my dissertation (1968) was on numerical solution of potential
> problems. One of the parts was a proof that some of the known iterative
> methods converged. The argument loosely went something like this. Consider
> the 2D Poisson equation on a square. If you use an N x N approximation with
> the usual discretization of the Laplacian
>
> u_ij = (u_i(j-1) + u_i(j+1) + u_(i_1)j + i_(j+1))/4
>
> i.e, the average of the surrounding points, the problem reduces to the
> solution of a set of N^2 linear equations
>
> Ax = b
>
> where x in a vector of the unknown {u_ij} arranged by rows or columns, b
> is determined by the boundary conditions and the right side of the Poisson
> equation. The interesting part is A which is block tridiagonal. With only a
> small error A can be made block cyclic. You can then diagonalize A with a
> sine transform and I was able to use that for proofs.
>
> A few years later when the FFT came about, we realized that we could use
> the FFT to do the sine transform and the resulting numerical method was as
> least as efficient as any other method people had come up with.
>
> Ed
>
> Here’s a reference from 1986 that I think was based on paper at a Bellman
> Continuum
>
> ``From Dynamic Programming to Fast Transforms,'' E. Angel, J. Math. Anal.
> Appl.,119,1986.
>
> Ed
> __________
>
> Ed Angel
>
> Founding Director, Art, Research, Technology and Science Laboratory (ARTS
> Lab)
> Professor Emeritus of Computer Science, University of New Mexico
>
> 1017 Sierra Pinon
> Santa Fe, NM 87501
> 505-984-0136 (home)   edward.angel at gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On Apr 28, 2023, at 8:18 AM, Stephen Guerin <stephen.guerin at simtable.com>
> wrote:
>
> Special Unitary Groups and Quaternions
>
> Mostly for Ed from the context of last week's Physical Friam if you're
> coming today.
>
> Discussion was around potential ways of visualizing the dynamics of SU(3),
> SU(2), (SU1) that highlights Special Unitary Groups. (wiki link from Frank
> <https://en.wikipedia.org/wiki/Special_unitary_group>), and can we
> foreground how quaternions are used in this process.
>
> and a related bit on forces, I'm searching for ways to
> visualize/understand how FFTs with Poisson equation
> <https://www.codeproject.com/Articles/5308623/Solving-Poisson-Equation>
> are used to compute the forces from scalar fields (eg gravitational force
> from mass density, electric force from charge, etc) and if there's any
> relation to Special Unitary Groups.
>
> -S
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