The number of spanning trees of a graph G is called the complexity of G and denoted c(G). A classical result in algebraic graph theory explicitly diagonalizes the Laplacian of the n-cube C(n) and yields, using the Matrix-Tree theorem, an explicit formula for c(C(n)). In this paper we explicitly block diagonalize the Laplacian of the q-analog Cq(n) of C(n) and use this, along with the Matrix-Tre...

We introduce improved model for sparseness constrained nonnegative matrix factorization (sNMF) of amplitude mixtures nuclear magnetic resonance (NMR) spectra into greater number of component spectra. In proposed method selected sNMF algorithm is applied to the square of the amplitude of the mixtures NMR spectra instead to the amplitude spectra itself. Afterwards, the square roots of separated s...

Carbon black is a common polymer additive that is used for reinforcement and for its ability to enhance physical properties, such as conductivity. This article pertains to an X-ray scattering (SAXS) study of a conductive grade of carbon black and carbon black–polymer composites. The scattering pattern for such blacks displays a surface-fractal-like power-law decay over many decades in scatterin...

In the present work we generalize the dielectric breakdown model to describe dielectric breakdown patterns in both conductor-loaded and insulator-loaded composites. The present model is an extension of a previous one [F. Peruani et al., Phys. Rev. E 67, 066121 (2003)] presented by the authors to describe dielectric breakdown patterns in conductor-loaded composites. Particles are distributed at ...

Measuring the rate of return is an important issue for theory and practice of investments in the stock market. A common measure for rate of return is the logarithm of the ratio of successive prices (LogRatio). In this paper it is shown that LogRatio as well as arithmetic return rate (Ratio) have several disadvantages. As an alternative relative differences (RelDiff) are proposed to measure retu...

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The frac...

Given i.i.d. observations of a random vector X ∈ R, we study the problem of estimating both its covariance matrix Σ, and its inverse covariance or concentration matrix Θ = (Σ). When X is multivariate Gaussian, the non-zero structure of Θ is specified by the graph of an associated Gaussian Markov random field; and a popular estimator for such sparse Θ is the l1-regularized Gaussian MLE. This est...

Given i.i.d. observations of a random vector X ∈ R, we study the problem of estimating both its covariance matrix Σ∗, and its inverse covariance or concentration matrix Θ∗ = (Σ). We estimate Θ∗ by minimizing an l1-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to l1-penalized maximum likelihood, and the structure of Θ∗ is specified by ...

Let F be a strongly regular graph with adjacency matrix A. Let I be the identity matrix, and J the all-1 matrix. Let p be a prime. Our aim is to study the p-rank (that is, the rank over Fp, the finite field with p elements) of the matrices M = aA + bJ + cI for integral a, b, c. This note is based on van Eijl [8]. 1. The Smith normal form Let us write M ~ N for integral matrices M and N of order...