Combinatorial laplacians and positivity under partial transpose

Density matrices of graphs are combinatorial laplacians normalized to have trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320 (2006)). If the vertices of a graph are arranged as an array, then its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for separability of density matrices of graphs, is equivalent to the PPT-criterion. As such it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (for an alternative proof see Wu, \emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22). Finally we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.Comment: 19 pages, 7 eps figures, final version accepted for publication in Math. Struct. in Comp. Sc

`The frozen accident' as an evolutionary adaptation: A rate distortion theory perspective on the dynamics and symmetries of genetic coding mechanisms

We survey some interpretations and related issues concerning the frozen hypothesis due to F. Crick and how it can be explained in terms of several natural mechanisms involving error correction codes, spin glasses, symmetry breaking and the characteristic robustness of genetic networks. The approach to most of these questions involves using elements of Shannon's rate distortion theory incorporating a semantic system which is meaningful for the relevant alphabets and vocabulary implemented in transmission of the genetic code. We apply the fundamental homology between information source uncertainty with the free energy density of a thermodynamical system with respect to transcriptional regulators and the communication channels of sequence/structure in proteins. This leads to the suggestion that the frozen accident may have been a type of evolutionary adaptation

Zeta functions of quantum graphs

In this article we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle. We start by considering the special case of the star graph with Neumann matching conditions at the center of the star. We then extend the technique to allow any matching conditions at the center for which the Laplace operator is self-adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions. In the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the Neumann star graph. In the second half of the article we apply the zeta function to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graphs. These have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature.Comment: 32 pages, typos corrected, references adde

Dynamical processes on metric networks

The structure of a network has a major effect on dynamical processes on that network. Many studies of the interplay between network structure and dynamics have focused on models of phenomena such as disease spread, opinion formation and changes, coupled oscillators, and random walks. In parallel to these developments, there have been many studies of wave propagation and other spatially extended processes on networks. These latter studies consider metric networks, in which the edges are associated with real intervals. Metric networks give a mathematical framework to describe dynamical processes that include both temporal and spatial evolution of some quantity of interest -- such as the concentration of a diffusing substance or the amplitude of a wave -- by using edge-specific intervals that quantify distance information between nodes. Dynamical processes on metric networks often take the form of partial differential equations (PDEs). In this paper, we present a collection of techniques and paradigmatic linear PDEs that are useful to investigate the interplay between structure and dynamics in metric networks. We start by considering a time-independent Schr\"odinger equation. We then use both finite-difference and spectral approaches to study the Poisson, heat, and wave equations as paradigmatic examples of elliptic, parabolic, and hyperbolic PDE problems on metric networks. Our spectral approach is able to account for degenerate eigenmodes. In our numerical experiments, we consider metric networks with up to about $10^4$ nodes and about $10^4$ edges. A key contribution of our paper is to increase the accessibility of studying PDEs on metric networks. Software that implements our numerical approaches is available at https://gitlab.com/ComputationalScience/metric-networks.Comment: 33 pages, 12 figure

Noncommutative field theories on Rλ3R^3_\lambda: Towards UV/IR mixing freedom

We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of the algebra of functions on $\mathbb{R}^3$ which yields a "foliation" of $\mathbb{R}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to $\mathbb{R}^3_\lambda$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.Comment: 31 pages, 4 figures. Improved version. Sections 5.1 and 5.2 have been clarified. A minor error corrected. References adde

The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problem associated with the second derivative acting on a space of vector-valued functions. We present two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well-known in some special cases) by an abstract rank condition. The latter seems to apply particularly well to a specific boundary condition, sometimes dubbed "anti-Kirchhoff" in the literature, that arise in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities. In either case our results yield, among other, some results on the symmetry of the spectrum

Numerical Methods for Parabolic Partial Differential Equations on Metric Graphs

The major motivation for this work arose from the problem of simulating diffusion type processes in the human brain network. This thesis addresses numerical methods for parabolic partial differential equations (PDEs) on network structures interpreted as metric spaces (metric graphs). Such domains frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices of the metric graph. Quantum graphs are popular models for thin, branched structures, and there is a great interest in their studies also from the theoretical point of view. The present work aims to bridge the gap between the theoretical work and the practical usage of quantum graph models by studying arising numerical problems. The main focus is on initial boundary value problems governed by (semilinear) parabolic partial differential equations that involve a second order spatial derivative posed on the edges of the graph. The particularity of these problems are the coupling conditions of the PDEs on their common vertices. The two central methods studied in this thesis are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. Both approaches follow the method of lines, i.e., Galerkin’s method is applied for the spatial discretization resulting in a system of ordinary differential equations. Spectral accuracy can be obtained with the spectral discretization in space for sufficiently smooth functions that fulfill certain coupling conditions at the vertices. In the finite element approach, the semidiscretization is solved with classical implicit-explicit time stepping methods combined with a graph specific multigrid solver for the arising systems of linear equations in each time step. In the spectral method, the stiffness matrix is diagonal such that exponential integrators can be applied efficiently to solve the semidiscretized system. The difficulty of the spectral method, by contrast, is the computation of an eigenfunction basis. The computation of quantum graph spectra thus is the last important aspect of this work. The problem of computing eigenfunctions can be reduced to a nonlinear eigenvalue problem (NEP). In the particular case of equilateral graphs, the NEP even simplifies to a linear eigenvalue problem in the size of the number of vertices of the underlying graph. The proposed NEP solver applies equilateral approximations combined with a nested iteration approach to obtain initial guesses for a Newton-trace iteration. Human connectomes interpreted as metric graphs are consulted to test the applicability of the methods to real world, large scale problems. Experiments on simulating distribution of tau proteins in the brain of Alzheimer’s disease patients complete this work