Single particle tracking in systems showing anomalous diffusion: the role of weak ergodicity breaking

Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.     

Relative weak mixing is generic

A classical result of Halmos asserts that among measure preserving transformations the weak mixing property is generic. We extend Halmos' result to the collection of ergolic extensions of a fixed, but arbitrary,aperiodic transformation T_0. We then use a result of Ornstein and Weiss to extend this relative theorem to the general(countable) amenable group.     

Intersection of Isomorphic Linear Codes

Given an (n, k) linear code over GF(q), the intersection of with a codeπ( ), whereπS_n, is an (n, k₁) code, where max{0, 2k−n}k₁k. The intersection problem is to determine which integers in this range are attainable for a given code . We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes.     

N.m.r. spectra of cyclic amines. II—Factors influencing the chemical shifts of α-protons in aziridines

Proton magnetic resonance spectra of trans and cis-2,3-diphenylaziridine (1 and 2) and their N-ethyl derivatives 3 and 4 were measured in carbon tetrachloride, chloroform, and benzene-d₆ at low temperatures (1 and 3) and in dry conditions (1 and 2). On the basis of these results it was concluded that an N-ethyl group exerts a shielding influence on a cis ring proton and a deshielding influence on a trans ring proton. From results obtained by measuring the ¹H n.m.r. spectra of 1–4 in deuterochlorofom-trifluoroacetic acid it was derived that the lone pair of the aziridine nitrogen exerts a shielding influence on cis related ring hydrogens. In most N-alkylaziridines the effect of the N-alkyl group predominates.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

Flame-retardant epoxy resins based on phthalide derivative

The phthalide-containing epoxy resins are synthesized and characterized in comparison with the bisphenol-A epoxy resins in terms of thermal properties. Although both resins contain comparable amounts of halogens, the resulting flame retardancy is higher in the phthalide-containing resin. The char formation upon pyrolysis is also enhanced by the phthalide functionality.     

A simple classical prediction of quantal resonances in collinear reactive scattering

Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!