THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

https://www.chester.ac.uk/sites/files/chester/rep377.pdf

Analysis of Fractional Differential Equations

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Signal sequences: The limits of variation

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

The importance of the correct determination of the relaxation times, entering the electron–phonon coupling, is crucial for a proper evaluation of the rise of the crystal lattice temperature induced...

Assessment of the Constant Phonon Relaxation Time Approximation in Electron–Phonon Coupling in Graphene

Abstract Charge transport in graphene is crucial for the design of a new generation of nanoscale electron devices. A reasonable model is represented by the semiclassical Boltzmann equations for electrons in the valence and conduction bands. As shown by Romano et al. (J. Comput. Phys., 2015), the discontinuous Galerkin methods are a viable way to tackle the problem of the numerical integration of these equations, even if efficient DSMC with a proper inclusion of the Pauli principle have been also devised. One of the advantages of the solutions obtained with deterministic approach is of course the absence of statistical noise. This fact is crucial for an accurate estimation of the low field mobility as proved by Majorana et al. (J. Math. Industry, 2016) in the case of a unipolar charge transport in a suspended graphene sheet under a constant electric field. The mobility expressions are essential for the drift-diffusion equations which constitute the most adopted models for charge transport in CAD. Here the analysis by Majorana et al. (J. Math. Industry, 2016) is improved in two ways: by including the charge transport both in the valence and conduction bands; by taking into account the presence of an oxide as substrate for the graphene sheet. New models of mobility are obtained and, in particular, relevant improvements of the low field mobility are achieved.

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Improved mobility models for charge transport in graphene

The aim of this work is to simulate charge transport in a monolayer graphene on different substrates. This requires the inclusion of the scatterings of charge carriers with impurities and phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As mathematical model, the semiclassical Boltzmann equation is assumed and the results are based on Direct Simulation Monte Carlo (DSMC) method. A crucial point is the correct inclusion of the Pauli Exclusion Principle (PEP). Most simulations use the approach proposed by Jacoboni e Lugli which, however, allows an occupation number greater than one with an evident violation of PEP. Here the Monte Carlo scheme devised by Romano et al. (J. Comput. Phys. 302 , 267--284, 2015) is employed. It predicts occupation numbers consistent with PEP and therefore is physically more accurate. Two different substrates are investigated: SiO 2 and hexagonal boron nitride (h-BN). We adopt the model for charge-impurities scattering described by E. H. Hwang and S. Das Sarma (Phys. Rev. B 75 , 205418, 2007). In such a model a crucial parameter is the distance d between the graphene layer and the impurities of the substrate. Usually d is considered constant. Here we assume that d is a random variable in order to take into account the roughness of the substrate and the randomness of the location of the impurities. Our results confirm that h-BN is one of the most promising substrate also for the high-field mobility on account of the reduced degradation of the velocity due to the remote impurities. This is in agreement with results shown by Hirai et al. (J. Appl. Phys. 116 , 083703, 2014) where only the low-field mobility has been investigated.

https://cab.unime.it/journals/index.php/AAPP/article/download/AAPP.97S1A6/AAPP97S1A6

High-field mobility in graphene on substrate with a proper inclusion of the Pauli exclusion principle

In this work, we propose a hierarchy of macroscopic models for the description of phonon transport in graphene. They are devised starting from the phonon Boltzmann transport equation by using the moments method and the maximum entropy principle. The state variables are the phonon energy densities and the fluxes of integer powers of the energy. We underline that the full energy dispersion relations are used. We test the validity of the models by analysing the steady-state and dynamical thermal conductivity, and the heat transport in zigzag and armchair graphene nanoribbons under a fixed temperature gradient.

A hierarchy of macroscopic models for phonon transport in graphene

In this paper we present a hydrodynamical model for the charge and the heat transport in graphene. The macroscopic variables are moments of the electron, hole and phonon distribution functions, and their evolution equations are derived from the Boltzmann equations by integration. The system of equations is closed by means of the maximum entropy principle and all the main scattering mechanisms are taken into account. Numerical simulations are presented in the case of a suspended graphene monolayer.

Vittorio Romano

Papers

Assessment of the Constant Phonon Relaxation Time Approximation in Electron–Phonon Coupling in Graphene

Improved mobility models for charge transport in graphene

High-field mobility in graphene on substrate with a proper inclusion of the Pauli exclusion principle

A hierarchy of macroscopic models for phonon transport in graphene

A comprehensive hydrodynamical model for charge transport in graphene