does there exist a mathematically rigorous form of diakoptics that would be considerd mathematically rigorous by a pure mathematician?
That’s a really good refinement of the question — because what an engineer calls “rigorous” (derivable, works on real networks, convergence in practice) and what a *pure mathematician* calls “rigorous” (axiomatically precise, grounded in abstract structures, theorems and proofs about existence/uniqueness, convergence, categorical equivalence, etc.) can be very different things.
Here’s the situation with **diakoptics**:
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### 1. Kron’s original formulation
* Kron’s *diakoptics* was brilliant but informal. It used analogies with tensors and topology, but definitions were heuristic.
* For a pure mathematician, this work does **not** qualify as rigorous — it lacked axioms, precise theorems, and formal proofs.
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### 2. Rigorous re-interpretations (engineering mathematics)
* In the 1960s–70s, Branin, Seshu, and others rephrased diakoptics in graph theory and algebraic topology terms.
* That step made it rigorous in the sense of applied mathematics:
* Networks as graphs (finite sets, incidence matrices).
* Equations as linear systems.
* Tearing and reconnecting as block elimination/Schur complements.
* From the standpoint of applied mathematics, this *is* rigorous.
But to a pure mathematician, this is just “linear algebra and graph theory applied to networks.” It doesn’t elevate *diakoptics* to a new branch of mathematics — it reduces it to already-rigorous, well-developed tools.
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### 3. Modern formal equivalence
* **Schur complements** and **domain decomposition methods** (in PDE theory) are *completely rigorous*. They live inside functional analysis, operator theory, and numerical analysis with convergence theorems.
* In fact, one can prove theorems like: *“The Kron reduction of a symmetric positive semidefinite Laplacian matrix is itself a symmetric positive semidefinite Laplacian matrix.”* These theorems appear in spectral graph theory, circuit theory, and PDE decomposition.
* This is mathematically rigorous by pure math standards: it is well-defined, theorem-proof, and ties into established structures like spectral graph theory and category theory.
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### 4. Philosophical point
So the answer is:
* There is **no separate “pure mathematical theory of diakoptics”** in the sense of a standalone, axiomatically developed discipline (like topology or measure theory).
* But diakoptics *can be and has been rigorized* in existing pure mathematical frameworks:
* **Graph theory** (cut-sets, Laplacians, Kron reduction).
* **Linear algebra** (block matrices, Schur complements).
* **Operator theory / PDE theory** (domain decomposition, boundary traces, variational formulations).
* Within those frameworks, it *does* meet the standards of rigor a pure mathematician would accept. But a pure mathematician would likely just say: “This is Schur complement theory and domain decomposition; ‘diakoptics’ is the engineering name.”
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**Conclusion:**
Yes, there is a mathematically rigorous form of diakoptics acceptable to pure mathematicians, but only by embedding it into existing rigorous theories (spectral graph theory, operator theory, domain decomposition). There is no independent “pure math theory of diakoptics” in the way Kron imagined.
Puzzels
**Conclusion:**
