This paper describes the design and operation of a 40 spatial channel Thomson scattering system that uses multiple 20-Hz Nd:YAG lasers to measure the electron temperature and density profiles periodically throughout an entire plasma discharge. As many as eight lasers may be fired alternately for an average measurement frequency of 160 Hz, or they may be fired in rapid succession (<10 kHz), producing a burst of pulses for measuring transient events. The high spatial resolution (1.3 cm) and wide dynamic range (10 eV–20 keV) enable this system to resolve large electron density and temperature gradients formed at the plasma edge and in the scrape-off layer during H-mode operation. These features provide a formidable tool for studying L–H transitions, edge localized modes (ELMs), beta limits, transport, and disruptions in an efficient manner suitable for large tokamak operation where shot-to-shot scans are impractical. The scattered light is dispersed by interference filter polychromators and detected by silicon avalanche photodiodes. Laser control and data acquisition are performed in real time by a VME-based microcomputer. Data analysis is performed by a MicroVAX 3400. Additional features of this system include real-time analysis capability, full statistical treatment of error bars based on the measured background light, and laser beam quality and alignment monitoring during plasma operation. Results of component testing, calibration, plasma operation, and error analysis are presented.

A generalization of the Cucker-Smale model for collective animal behaviour is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modelling: (i) stochasticity (imperfections) of individual behaviour; and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a byproduct, a new result regarding the asymptotic behaviour of delayed geometric Brownian motion is obtained. In the second part of the paper results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behaviour of the system -namely, that an introduction of intermediate time delay may facilitate flocking.

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity. For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.

Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai [10,11]. The model describes the pressure field thanks to Darcy's type equation and the dynamics of the conductance network under pressure force effects with a diffusion rate D representing randomness in the material structure.We prove the existence of global weak solutions and of local mild solutions and study their long term behavior. It turns out that, by energy dissipation, steady states play a central role to understand the pattern capacity of the system. We show that for a large diffusion coefficient D, the zero steady state is stable. Patterns occur for small values of D because the zero steady state is Turing unstable in this range; for D = 0 we can exhibit a large class of dynamically stable (in the linearized sense) steady states.

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai [8,10], which models the formation of biological transport networks. The model describes the pressure field using a Darcy's type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. The analytical part extends the results of [7] regarding the existence of weak and mild solutions to the whole range of meaningful relaxation exponents. Moreover, we prove finite time extinction or break-down of solutions in the spatially one-dimensional setting for certain ranges of the relaxation exponent. We also construct stationary solutions for the case of vanishing diffusion and critical value of the relaxation exponent, using a variational formulation and a penalty method.The analytical part is complemented by extensive numerical simulations. We propose a discretization based on mixed finite elements and study the qualitative properties of network structures for various parameters values. Furthermore, we indicate numerically that some analytical results proved for the spatially one-dimensional setting are likely to be valid also in several space dimensions.

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We introduce a Cucker-Smale-type model for flocking, where the strength of interaction between two agents depends on their relative separation (called "topological distance" in previous works), which is the number of intermediate individuals separating them. This makes the model scale-free and is motivated by recent extensive observations of starling flocks, suggesting that interaction ruling animal collective behavior depends on topological rather than metric distance. We study the conditions leading to asymptotic flocking in the topological model, defined as the convergence of the agents' velocities to a common vector. The shift from metric to topological interactions requires development of new analytical methods, taking into account the graph-theoretical nature of the problem. Moreover, we provide a rigorous derivation of the mean-field limit of large populations, recovering kinetic and hydrodynamic descriptions. In particular, we introduce the novel concept of relative separation in continuum descriptions, which is applicable to a broad variety of models of collective behavior. As an example, we shortly discuss a topological modification of the attraction-repulsion model and illustrate with numerical simulations that the modified model produces interesting new pattern dynamics.

We present an overview of recent analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological transportation networks. The model describes the pressure field using a Darcy type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro-and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.