Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

Cosmology of the invisible axion

Single trapped ions represent elementary quantum systems that are well isolated from the environment. They can be brought nearly to rest by laser cooling, and both their internal electronic states and external motion can be coupled to and manipulated by light fields. This makes them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

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Quantum dynamics of single trapped ions

A cosmological bound on the invisible axion

Visual Reconstruction

Theory and Application of Mathieu Functions

Bessel Functions for Engineers

Ordinary Non-Linear Differential Equations in Engineering and Physical Sciences

Laplace transforms and their applications to differential equations

THIS book is intended as an introduction to operational calculus based upon the transform of f(x) defined by that is to say, the Laplace transform multiplied by p. The function p(p) so defined is identical with what Heaviside called the operational form. The advantage of the present treatment over Heaviside's is that the theorems of operation can be placed upon an impeccable basis without the necessity for acts of faith. The contents are : 1, the Laplace transform ; 2, rules of the operational calculus ; 3, solution of ordinary linear differential equations with constant coefficients ; 4, solution of partial linear differential equations with constant coefficients ; 5, evaluation of integrals ; 6, derivation of Laplace transforms ; 7, Laplace transform for a finite interval ; impulses.Modern Operational Calculus With Applications in Technical Mathematics. By Dr. N. W. McLachlan. Pp. xiv+218. (London: Macmillan and Co., Ltd., 1948.) 21s. net.

N. W. McLachlan

Papers

Theory and Application of Mathieu Functions

Bessel Functions for Engineers

Ordinary Non-Linear Differential Equations in Engineering and Physical Sciences

Laplace transforms and their applications to differential equations

Modern Operational Calculus - With Applications in Technical Mathematics