Abstract. We extend Feynman's analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron “giant component” of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been “rewired” to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs.

We prove Barlow-Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant ρ which is theoretically estimated to be in the interval 5/4 ρ 3/2, with a numerical estimate ρ ≈ 1.306. This corresponds to the theoretical estimate of spectral dimension d S between 1.63 and 1.77, with a numerical estimate d S ≈ 1.74. On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant ρ T = 1/ρ, which is theoretically estimated to be in the interval 2/3 ρ T 4/5. This corresponds to the spectral dimension between 2.28 and 2.38. The difference between ρ and ρ T implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.

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Prognostic models, when feasible, are favored for avoiding unexpected maintenance. There is a need for a common language when discussing prognostic performance and behavior. The approach presented here considers model behavior in terms of two optimizable sub-problems for better performance assessment. The first evaluation construct considers how well the model tracks degradation over time and a second construct considers how effectively it improves operations. The right set of cost functions can determine the suitability to both objectives. The combined construct enables evaluation of a class of models which augment degradation physics with data-driven heuristics, supporting a more explainable recommendation.