Abstract:
Weighted and unweighted interpolations of general order are given for Jensen's integral inequality. Various upper-bound estimates are made for the differences between the interpolates and some convergence results erived. The results generalise and subsume a body of earlier work and employ streamlined proofs.

Abstract:
We present transparent proofs for some multivariate versions of both continuous and discrete Hardy-type inequalities. Our theorems subsume several known results and simplify the proofs of existing results.

Abstract:
We develop a method for systematically constructing Lagrangian functions for dissipative mechanical, electrical and, mechatronic systems. We derive the equations of motion for some typical mechatronic systems using deterministic principles that are strictly variational. We do not use any ad hoc features that are added on after the analysis has been completed, such as the Rayleigh dissipation function. We generalise the concept of potential, and define generalised potentials for dissipative lumped system elements. Our innovation offers a unified approach to the analysis of mechatronic systems where there are energy and power terms in both the mechanical and electrical parts of the system. Using our novel technique, we can take advantage of the analytic approach from mechanics, and we can apply these pow- erful analytical methods to electrical and to mechatronic systems. We can analyse systems that include non-conservative forces. Our methodology is deterministic and does does require any special intuition, and is thus suitable for automation via a computer-based algebra package.

Abstract:
In Parrondo's games, the apparently paradoxical situation occurs where individually losing games combine to win. The basic formulation and definitions of Parrondo's games are described in Harmer et al.. These games have recently gained considerable attention as they are physically motivated and have been related to physical systems such as the Brownian ratchet, lattice gas automata and spin systems. Various authors have pointed outinterest in these games for areas as diverse as biogenesis, political models, small-world networks, economics and population genetics. In this chapter, we will first introduce the relevant properties of Markov transition operators and then introduce some terminology and visualisation techniques from the theory of dynamical systems. We will then use these tools, later in the chapter, to define and investigate some interesting properties of Parrondo's games.

Abstract:
We transform the two-dimensional Dirac-Weyl equation, which governs the charge carriers in graphene, into a non-linear first-order differential equation for scattering phase shift, using the so-called variable phase method. This allows us to utilize the Levinson Theorem to find zero-energy bound states created electrostatically in realistic structures. These confined states are formed at critical potential strengths, which leads to us posit the use of `optimal traps' to combat the chiral tunneling found in graphene, which could be explored experimentally with an artificial network of point charges held above the graphene layer. We also discuss scattering on these states and find the zero angular momentum states create a dominant peak in scattering cross-section as energy tends towards the Dirac point energy, suggesting a dominant contribution to resistivity.

Abstract:
We propose an innovative fast-cycling accelerator complex conceived and designed to exploit at best the properties of accelerated ion beams for hadrontherapy. A cyclinac is composed by a cyclotron, which can be used also for other valuable medical and research purposes, followed by a high gradient linear accelerator capable to produce ion beams optimized for the irradiation of solid tumours with the most modern techniques. The properties of cyclinacs together with design studies for protons and carbon ions are presented and the advantages in facing the challenges of hadrontherapy are discussed.

Abstract:
Background Increasing evidence suggests that individual isoforms of protein kinase C (PKC) play distinct roles in regulating platelet activation. Methodology/Principal Findings In this study, we focus on the role of two novel PKC isoforms, PKCδ and PKCε, in both mouse and human platelets. PKCδ is robustly expressed in human platelets and undergoes transient tyrosine phosphorylation upon stimulation by thrombin or the collagen receptor, GPVI, which becomes sustained in the presence of the pan-PKC inhibitor, Ro 31-8220. In mouse platelets, however, PKCδ undergoes sustained tyrosine phosphorylation upon activation. In contrast the related isoform, PKCε, is expressed at high levels in mouse but not human platelets. There is a marked inhibition in aggregation and dense granule secretion to low concentrations of GPVI agonists in mouse platelets lacking PKCε in contrast to a minor inhibition in response to G protein-coupled receptor agonists. This reduction is mediated by inhibition of tyrosine phosphorylation of the FcRγ-chain and downstream proteins, an effect also observed in wild-type mouse platelets in the presence of a PKC inhibitor. Conclusions These results demonstrate a reciprocal relationship in levels of the novel PKC isoforms δ and ε in human and mouse platelets and a selective role for PKCε in signalling through GPVI.

Abstract:
We use very large cosmological N--body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model ($\tau$CDM) and a flat low density model with a cosmological constant ($\Lambda$CDM). Our simulations each use $10^9$ particles to follow the mass distribution within cubes of side $2h^{-1}$Gpc ($\tau$CDM) and $3h^{-1}$Gpc ($\Lambda$CDM) with a force resolution better than $10^{-4}$ of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from $r_0=12$ -- 13$h^{-1}$Mpc for samples with mean separation $d_{\rm c}=30h^{-1}$Mpc to $r_0=22$-- 27$h^{-1}$Mpc for samples with $d_{\rm c}=100h^{-1}$Mpc. The lower value here corresponds to $\tau$CDM and the upper to $\Lambda$CDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10% using the simple model of Sheth, Mo & Tormen (2000). This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the APM cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for $\Lambda$CDM. The observed power spectrum (for a cluster sample of mean separation $d_{\rm c}=31h^{-1}$Mpc) lies significantly above the predictions of both models.