Related papers: Theory of resistor networks: The two-point resista…

We propose two new kinds of infinite resistor networks based on the Fibonacci sequence: a serial association of resistor sets connected in parallel (type 1) or a parallel association of resistor sets connected in series (type 2). We show…

The phase diagram of binary mixtures of particles interacting via a pair potential of parallel dipoles is computed at zero temperature as a function of composition and the ratio of their magnetic susceptibilities. Using lattice sums, a rich…

Resilience of the most important properties of stochastic and regular (deterministic) small-world interconnection networks is studied. It is shown that in the broad range of values of the fraction of faulty nodes the networks under…

We consider random Gaussian eigenfunctions of the Laplacian on the standard torus, and investigate the number of nodal intersections against a line segment. The expected intersection number, against any smooth curve, is universally…

The notion of network connectivity is used to characterize the robustness and failure tolerance of networks, with high connectivity being a desirable feature. In this paper, we develop a novel approach to the problem of identifying critical…

The capacitance between two adjacent nodes on an infinite square grid of identical capacitors can easily be found by superposition, and the solution is found by explotting the symmetry of the grid. The mathematical problem presented in this…

We give explicit formulas for resistance distance matrices and Moore-Penrose inverses of incidence and Laplacian matrices of ladder, circular ladder, and M\"{o}bius ladder graphs. As a result, we compute the Kirchhoff index of these graphs…

The state of a 2-D random resistor network, resulting from the simultaneous evolutions of two competing biased percolations, is studied in a wide range of bias values. Monte Carlo simulations show that when the external current $I$ is below…

Random assemblies of magnetic nanowires represent a unique class of materials with promising applications in spintronics and information storage. These assemblies exhibit complex behavior due to the combination of magnetic dipolar…

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…

We present an extension of the framework introduced in [1] to treat multicomponent systems, showing that new degrees of freedom are necessary in order to obtain the desired boundary conditions. We then apply this extended framework to the…

We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.

The need of mathematically formulate relations between composite materials' properties and its resonance response is growing. This is due the fast technological advancement in micro-material manufacturing, present in chips for instance. In…

Existence, stability and dynamics of soliton complexes, centered at the site of a single transverse link connecting two parallel 2D (two-dimensional) lattices, are investigated. The system with the on-site cubic self-focusing nonlinearity…

A nonnegative function on the vertices of an infinite graph G which vanishes at a distinguished vertex o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We survey basic properties of potentials in…

The graphical notion of effective resistance has found wide-ranging applications in many areas of pure mathematics, applied mathematics and control theory. By the nature of its construction, effective resistance can only be computed in…

We consider the number of configurations of a surface in two dimensions that has a prescribed length and encloses a prescribed perimeter with respect to a baseline. An approximate analytical treatment in a semi--continuum compares…

This paper studies network resilience against structured additive perturbations to its topology. We consider dynamic networks modeled as linear time-invariant systems subject to perturbations of bounded energy satisfying specific sparsity…

A neutral network is a subgraph of a Hamming graph, and its principal eigenvalue determines its robustness: the ability of a population evolving on it to withstand errors. Here we consider the most robust small neutral networks: the graphs…

Despite a great deal of research, it is still not well-understood why trained neural networks are highly vulnerable to adversarial examples. In this work we focus on two-layer neural networks trained using data which lie on a low…