The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

The blood-brain barrier (BBB) limits the delivery of systemically administered drugs to the brain. Methods to circumvent the BBB have been developed, but none are used in standard clinical practice. The lack of adoption of existing methods is due to procedural invasiveness, serious adverse effects, and the complications associated with performing such techniques coincident with repeated drug administration, which is customary in chemotherapeutic protocols. Pulsed ultrasound, a method for disrupting the BBB, was shown to effectively increase drug concentrations and to slow tumor growth in preclinical studies. We now report the interim results of an ultrasound dose-escalating phase 1/2a clinical trial using an implantable ultrasound device system, SonoCloud, before treatment with carboplatin in patients with recurrent glioblastoma (GBM). The BBB of each patient was disrupted monthly using pulsed ultrasound in combination with systemically injected microbubbles. Contrast-enhanced magnetic resonance imaging (MRI) indicated that the BBB was disrupted at acoustic pressure levels up to 1.1 megapascals without detectable adverse effects on radiologic (MRI) or clinical examination. Our preliminary findings indicate that repeated opening of the BBB using our pulsed ultrasound system, in combination with systemic microbubble injection, is safe and well tolerated in patients with recurrent GBM and has the potential to optimize chemotherapy delivery in the brain.

Clinical trial of blood-brain barrier disruption by pulsed ultrasound

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equation

We derive an implicit-explicit (IMEX) formalism for the three-dimensional (3D) Euler equations that allow a unified representation of various nonhydrostatic flow regimes, including cloud resolving and mesoscale (flow in a 3D Cartesian domain) as well as global regimes (flow in spherical geometries) This general IMEX formalism admits numerous types of methods including single-stage multistep methods (eg, Adams methods and backward difference formulas) and multistage single-step methods (eg, additive Runge--Kutta methods) The significance of this result is that it allows a numerical model to reuse the same machinery for all classes of time-integration methods described in this work We also derive two classes of IMEX methods, one-dimensional and 3D, and show that they achieve their expected theoretical rates of convergence regardless of the geometry (eg, 3D box or sphere) and introduce a new second-order IMEX Runge--Kutta method that performs better than the other second-order methods considered We

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Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)

https://faculty.nps.edu/fxgirald/Homepage/Publications_files/Kelly_Giraldo_JCP_2012_published.pdf

Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode

Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation [“Time domain wave-equations for lossy media obeying a frequency power-law,” J. Acoust. Soc. Am. 96, 491–500 (1994)] is an approximation to this equation. This paper derives analytical time-domain Green’s functions in power-law media for exponents in this range. To construct solutions, stable law probability distributions are utilized. For exponents equal to 0, 1∕3, 1∕2, 2∕3, 3∕2, and 2, the Green’s function is expressed in terms of Dirac delta, exponential, Airy, hypergeometric, and Gaussian functions. For exponents strictly less than 1, the Green’s functions are expressed as Fox functions and are causal. For exponents greater than or equal than 1, the Green’s functions are expressed as Fox and ...

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Analytical time-domain Green's functions for power-law media.

An analytical expression is derived for time-harmonic calculations of the near-field pressure produced by a circular piston. The near-field pressure is described by an efficient integral that eliminates redundant calculations and subtracts the singularity, which in turn reduces the computation time and the peak numerical error. The resulting single integral expression is then combined with an approach that divides the computational grid into sectors that are separated by straight lines. The integral is computed with Gauss quadrature in each sector, and the number of Gauss abscissas in each sector is determined by a linear mapping function that prevents large errors from occurring in the axial region. By dividing the near-field region into 10 sectors, the raw computation time is reduced by nearly a factor of 2 for each expression evaluated in this grid. The grid sectoring approach is most effective when the computation time is reduced without increasing the peak error, and this is consistently accomplished with the efficient integral formulation. Of the four single integral expressions evaluated with grid sectoring, the efficient formulation that eliminates redundant calculations and subtracts the singularity demonstrates the smallest computation time for a specified value of the maximum error.

An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston.

Extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power‐law dependence on frequency. To describe this power‐law behavior in tissue, a hierarchical fractal network model is proposed. The viscoelastic and self‐similar properties of tissue are captured by a constitutive equation based on a lumped‐parameter infinite‐ladder topology involving alternating springs and dashpots. In the low‐frequency limit, this ladder network yields a stress‐strain constitutive equation with a time‐fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, the fractional partial differential equations previously proposed by [M. Caputo “Linear models of dissipation whose Q is almost frequency independent‐‐II,” Geophys. J. R. Astron. Soc. 13, 529–539 (1967)] and [MG. Wismer “Finite element analysis of broadband acoustic pulses through inhomogeneous media with power‐law attenuation,” J. Acoust. Soc. Am. 120, 3493–3502 (2006)] are derived. The resulting attenuation coefficient is a power‐law with exponents ranging between 1 and 2, while the phase velocity is in agreement with the Kramers–Kronig relations. The power‐law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry.

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James F. Kelly

Papers

Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA)

Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode

Analytical time-domain Green's functions for power-law media.

An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston.

Fractal ladder models and power law wave equations.