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Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form d x i d t = x i ϕ i (x 1 , ... , x N) , where N represents the number of species and xi , the abundance of species i. Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to ϕ i (x 1 , ... , x N) = r i − x i + (Γ x) i , play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, ri is the intrinsic growth of species i and Γ stands for the interaction matrix: Γij represents the effect of species j over species i. For large N , estimating matrix Γ is often an overwhelming task and an alternative is to draw Γ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology. [ABSTRACT FROM AUTHOR]

This exposition consists of three parts. Part I is an introduction to classical texture analysis. The harmonic method and the approach initiated by Roe, where the orientation distribution function (ODF) is always defined on the rotation group SO(3), is emphasized and given a systematic treatment. Basic concepts (e.g., the orientation density function) are made precise through their mathematical definition. The active view of rotations is implemented throughout. A conscientious effort is made to use machinery already available in mathematics and physics. The Wigner D -functions, whose properties are familiar in physics, are used instead of Bunge's and Roe's versions of generalized spherical harmonics. By including three mathematical appendices, it is hoped that engineering students would find Part I readable. The objectives of Parts II and III are threefold, namely: (i) To delve deeper into the mathematical foundations of the harmonic method. The Weyl method is used to prove that the Wigner D -functions are the matrix elements of a complete set of pairwise-inequivalent, continuous, irreducible unitary representation of SO(3). General formulas of the Wigner D -functions, valid for any parametrization of SO(3), are derived. An elementary proof (attributed to Wigner) of the Peter-Weyl theorem is presented. (ii) To provide mathematical prerequisites in group representations for research on representation theorems that delineate the effects of crystallographic texture on material properties defined by tensors or pseudotensors. (iii) To introduce tensorial Fourier expansion of the ODF and the tensorial texture coefficients. The classical ODF expansion in Wigner D -functions is recast as a special tensorial Fourier series. The relation between the tensorial and classical texture coefficients in this context is derived. [ABSTRACT FROM AUTHOR]

This is a survey of Harold Widom's work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy's groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy–Widom distributions in mathematics and physics, and close with Tracy and Widom's remarkable work on the asymmetric simple exclusion process. [ABSTRACT FROM AUTHOR]

A collection of scholarly scripts dealing with the mathematics and physics of peak profile functions in X‐ray powder diffraction has been written using the Wolfram language in Mathematica. Common distribution functions, the concept of convolution in real and Fourier space, instrumental aberrations, and microstructural effects are visualized in an interactive manner and explained in detail. This paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes. [ABSTRACT FROM AUTHOR]

If the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the Gaussian Orthogonal Ensemble (GOE) of random matrices. Previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation. The derivation of these expressions employed random point process theory, and in the theoretical development it was assumed that the modal generalised forces were uncorrelated. Although this assumption is often valid, there are cases in which correlations between the generalised forces can significantly affect the response variance, and in the present work the existing theory is extended to include correlations of this type. The extended theory is applicable to both single frequency responses and to band average responses, and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure. [ABSTRACT FROM AUTHOR]

A discussion of Nieuwenhuizen's description for the hidden variables of the detectors in the derivation of Bell's theorem is presented. This description prevents Bell's inequalities from being effected. However it will be argued, on mathematical and physical bases, that the flaws attributed by Nieuwenhuizen to Bell's probability distribution function are unjustified. [ABSTRACT FROM AUTHOR]

The nonlinear science community experienced a painful loss with the sudden death of our colleague and friend, Professor Tito Arecchi. Tito Arecchi was a pioneer of nonlinear optics and laser physics, as well as nonlinear dynamics. The Arecchi-Bonifacio equations are universally used to describe the dynamics of a single mode laser. [Extracted from the article]

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of $${C^*}$$ -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing $${C^*}$$ -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative $${C^*}$$ -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product $${C^{*}}$$ -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological $${K}$$ -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map). [ABSTRACT FROM AUTHOR]

We define a notion of semi-stability in the conditionally free probability and explain that the semi-stable measures are infinitely divisible. We also show that in the conditionally free probability stable measures are semi-stable, and that semi-stability for all r implies stability. [ABSTRACT FROM AUTHOR]

The concepts of Hamiltonian Feynman integrals in white noise analysis are used to realize as the first velocity-dependent potential of the Hamiltonian Feynman integrand for a charged particle in a constant magnetic field in coordinate space as a Hida distribution. For this purpose the velocity-dependent potential gives rise to a generalized Gauss kernel. Besides the propagators, the generating functionals are obtained. [ABSTRACT FROM AUTHOR]

Modelled as finite homogeneous Markov chains, probabilistic cellular automata with local transition probabilities in (0, 1) always posses a stationary distribution. This result alone is not very helpful when it comes to predicting the final configuration; one needs also a formula connecting the probabilities in the stationary distribution to some intrinsic feature of the lattice configuration. Previous results on the asynchronous cellular automata have showed that such feature really exists. It is the number of zero-one borders within the automaton's binary configuration. An exponential formula in the number of zero-one borders has been proved for the 1-D, 2-D and 3-D asynchronous automata with neighborhood three, five and seven, respectively. We perform computer experiments on a synchronous cellular automaton to check whether the empirical distribution obeys also that theoretical formula. The numerical results indicate a perfect fit for neighbourhood three and five, which opens the way for a rigorous proof of the formula in this new, synchronous case. [ABSTRACT FROM AUTHOR]

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial, and power-law degree distributions. The mean, mode, and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions. [ABSTRACT FROM AUTHOR]

We study the importance of local structural properties in networks which have been evolved for a power-law scaling in their Laplacian spectrum. To this end, the degree distribution, two-point degree correlations, and degree-dependent clustering are extracted from the evolved networks and used to construct random networks with the prescribed distributions. In the analysis of these reconstructed networks it turns out that the degree distribution alone is not sufficient to generate the spectral scaling and the degree-dependent clustering has only an indirect influence. The two-point correlations are found to be the dominant characteristic for the power-law scaling over a broader eigenvalue range. [ABSTRACT FROM AUTHOR]

While originally discovered in the context of the Gaussian unitary ensemble, the Tracy-Widom distribution also rules the height fluctuations of growth processes. This suggests that there might be other nonequilibrium processes in which the Tracy-Widom distribution plays an important role. In our contribution we study one-dimensional systems with domain wall initial conditions. For an appropriate choice of parameters, the profile develops a rarefaction wave while maintaining the initial equilibrium states far to the left and right, which thus serve as infinitely extended thermal reservoirs. For a Fermi-Pasta-Ulam type anharmonic chain, we will demonstrate that the time-integrated current has a deterministic contribution, linear in time t, and fluctuations of size t1/3 with a Tracy-Widom distributed random amplitude. [ABSTRACT FROM AUTHOR]