Davis? I thought I was at Riverside.

]]>Sigma is the notation for ‘sum’. So,

says that for each node that is not a terminal, the sum of the currents over all edges having as their target ( such that ) equals the sum of the currents over all edges having as their source ( such that ).

Or, in plain English, it says what I said:

the total current flowing into that node equals the total current flowing out.

Also:

if is the current of the edge but its also a function of time, why is it not labelled or something?

Do that if you like. Physicists often write simply to be the velocity of something even if the velocity depends on time. Mathematicians would say that stands for the function of time, while stands for the value of that function at a particular time. Either way, they’re happy leaving out the

]]>I think that the whole point of diakoptics/tearing is to solve the parts and then glue the solutions back together. So one tears systems, but glues solutions. This reminds me Joseph Goguen’s & Grant Malcolm’s work on modelling (de)composition of systems using sheaves in category theory. Malcolm also has interests in biological modelling and diagrams (the broad sense of the word, not just categories)

Another interesting aspect of Kron’s work is that when he treated moving systems like EM motors & generators using non linear coordinate transformations, making real use tensors and differential geometry.

]]>‘Diakoptics’ sounds a bit weird, but the Wikipedia article on it says:

Gabriel Kron’s Diakoptics (Greek

dia–through +kopto–cut,tear) or Method of Tearing involves breaking a (usually physical) problem down into subproblems which can be solved independently before being joined back together to obtain a solution to the whole problem.Gabriel Kron was an unconventional Engineer who worked for GE in the US until his death in 1968. He was responsible for the first load flow (electricity) distribution system in New York.

He was perhaps most famous for his Method of Tearing, a technique for splitting up physical problems into subproblems, solving each individual subproblem and then recombining to give an (unexpectedly) exact overall solution. The technique is efficient on sequential computers, but is particularly so on parallel architectures. Whether this holds for quantum parallelism is as yet unknown. It is peculiar as a decomposition method, in that it involves taking values on the “intersection layer” (the boundary between subsystems) into account. The method has been rediscovered by the parallel processing community recently under the name “Domain Decomposition”.

A multilevel hierarchical version of the Method, in which the subsystems are recursively torn into subsubsystems etc., was published by Keith Bowden in 1991.

I’ve heard of the ‘method of tearing’ before in Jan Willems’ expository articles on control theory. I don’t really know how this method works, but one of my main concerns these days is how to use category theory to describe the process of building a physical system out of interacting parts. ‘Tearing’ sounds like the reverse operation. I can’t help but think it’s logically secondary. But I need to get a sense of how people use it in practice.

]]>Yes, circuits are a bit more general than bond graphs, and that’s why I’m focusing on them here. Later I’ll describe a category where circuits are morphisms, and bond graphs will give an important subcategory.

]]>Nice to see somebody interested in Kron. I am particularly interested in possible connections between Kron’s work & Information Geometry. I am fairly confident that the connection exists. Amai’s supervisor, Kazuo Kondo, was deeply influenced by Kron. Kondo’s research group, the RAAG, was largely dedicated to extending Kron’s geometrical idea’s. Amari’s master’s and doctoral theses were titled *Topological and Information-Theoretical Foundation of Diakoptics and Codiakoptics*. and *Diakoptics of Information Spaces* respectively, “diakoptics” being the term Kron coined for his methods. Unfortunately, my understanding of both Information Geometry and Diakoptics is currently too limited to say anything useful about the connection. However, I think it should help tie together a number of threads in this forum, particularly given that Kron, Kondo & the RAAG formulated much of their ideas in terms of tensor geometry.

It seems to me that circuits are more general than bond graphs, since I do not know of a way to reperesent lattice shaped circuits into n-port networks and thus bond graphs

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