Calculation of Volterra kernels for solutions of nonlinear differential equations

We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of the solution operator A are continuous functions, and establish recurrence relations between the kernels allowing their explicit calculation. A practical tensor calculus is provided for the finite-dimensional case. In addition to analytically calculating the kernels, we present an algorithm to numerically obtain them from the output x(t) through sampling the input space by linear combinations of delta functions. We call this "differential sampling". It is a nonlinear analogue of the classical method of impulse response. We prove a continuity theorem stating that, in the finite-dimensional case, approximate delta functions give rise to approximate Volterra kernels and that continuity holds in the sense of weak convergence. Finally, we discuss a practical implementation of differential sampling and relate it to the Wiener method

Phonon distributions of a single bath mode coupled to a quantum dot

The properties of an unconventional, single mode phonon bath coupled to a quantum dot, are investigated within the rotating wave approximation. The electron current through the dot induces an out of equilibrium bath, with a phonon distribution qualitatively different from the thermal one. In selected transport regimes, such a distribution is characterized by a peculiar selective population of few phonon modes and can exhibit a sub-Poissonian behavior. It is shown that such a sub-Poissonian behavior is favored by a double occupancy of the dot. The crossover from a unequilibrated to a conventional thermal bath is explored, and the limitations of the rotating wave approximation are discussed.Comment: 21 Pages, 7 figures, to appear in New Journal of Physics - Focus on Quantum Dissipation in Unconventional Environment

Higher Powers in Gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalise the Einstein-Hilbert action by replacing the Ricci scalar, R, with some function, f(R). The general asymptotic behaviour of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption that the highest powers of R should dominate the Universe's early history, and that the lowest powers should dominate its future is found to be inaccurate. The behaviour of the general solution is complicated, and while it can be the case that single powers of R dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of R that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions investigated.Comment: 30 pages, 9 figure

Testing devices for the prevention and treatment of stroke and its complications

We are entering a challenging but exciting period when many new interventions may appear for stroke based on the use of devices. Hopefully these will lead to improved outcomes at a cost that can be afforded in most parts of the world. Nevertheless, it is vital that lessons are learnt from failures in the development of pharmacological interventions (and from some early device studies), including inadequate preclinical testing, suboptimal trial design and analysis, and underpowered studies. The device industry is far more disparate than that seen for pharmaceuticals; companies are very variable in size and experience in stroke, and are developing interventions across a wide range of stroke treatment and prevention. It is vital that companies work together where sales and marketing are not involved, including in understanding basic stroke mechanisms, prospective systematic reviews, and education of physicians. Where possible, industry and academics should also work closely together to ensure trials are designed to be relevant to patient care and outcomes. Additionally, regulation of the device industry lags behind that for pharmaceuticals, and it is critical that new interventions are shown to be safe and effective rather than just feasible. Phase IV postmarketing surveillance studies will also be needed to ensure that devices are safe when used in the ‘real-world’ and to pick up uncommon adverse events

Absence of Phase Transitions in Certain One-Dimensional Long-Range Random Systems

An Ising chain is considered with a potential of the form J(i,j)/|i-j|^α where the J(i,j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if α>1. This is to be contrasted with a ferromagnetic interaction which requires α>2. The basic idea is to prove that the "surface" free energy between two half-lines is finite, although the "surface" energy may be unbounded. For d-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if α>(1/2)d

Absence of Phase Transitions in Certain One-Dimensional Long-Range Random Systems

An Ising chain is considered with a potential of the form J(i,j)/|i-j|^α where the J(i,j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if α>1. This is to be contrasted with a ferromagnetic interaction which requires α>2. The basic idea is to prove that the "surface" free energy between two half-lines is finite, although the "surface" energy may be unbounded. For d-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if α>(1/2)d