直线上的布朗运动(二)

§4.Drift The methods above can be used to obtain analogous results for a Brownian motion with a constant drift,namely for the process: where X(t) is the standard Brownian motion and c is a nonzero constant.We may suppose c>0 for definiteness. The strong law of large numbers implies that almost surely The argument in §1 is still valid to show that exit from any given interval(a,b)is almost sure,but the analogue to Proposition 1 must be false.The reader should     

Rubber networks containing unattached macromolecules. III. Nonlinear stress relaxation in the system ethylene–propylene terpolymer with ethylene–propylene copolymer and behavior of extracted networks

Further stress relaxation experiments, mostly at 50°C, are reported on mixtures of crosslinkable ethylene–propylene terpolymer with saturated ethylene–propylene copolymer (molecular weights 3.6 and 45 × 10⁴) containing up to 50% by weight of copolymer, crosslinked by sulfur to leave the saturated copolymer unattached and free to reptate in the copolymer network. Stress relaxation was measured in small simple elongations (stretch ratio about 1.15) on samples which had been extracted to remove a large part of the unattached copolymer and dried. The relative increase in modulus at long times (10⁴ sec) increased with the proportion extracted; at short times (1 sec), extraction of the lower molecular weight copolymer increased the modulus to about the same extent but extraction of the higher molecular weight copolymer affected it very little. The relaxation modulus of the copolymer extracted from sample 50H (50% copolymer of high molecular weight), obtained by difference, agreed with that for the total copolymer except for a small difference probably attributable to molecular weight selectivity in the extraction. Stress relaxation was measured on sample 50H at six higher elongations up to a stretch ratio of 3. The dependence of stress on time and strain was consistent with an analysis based on the following assumptions: (a) linear additivity of the network and unattached copolymer contributions, (b) strain–time factorization of the stress contributions from the individual components, (c) a strain dependence for the unattached component corresponding to the presence of a Mooney–Rivlin C₂ term only, (d) a strain dependence for the network component which does not follow the Mooney–Rivlin equation but is dominated by a simple neo-Hookean term.     

A theory of the bion in the one-dimensional sine-Gordon system: I. Quantum statistical mechanics

Introducing an appropriate renormalization to the phase space of the bion, we study the quantum statistical mechanics of the bion in the one-dimensional sine-Gordon system. A bion representation is found for the partition function of the system in the limit of weak coupling, i.e., the free field. From the partition function, the bion distribution is calculated and an approximate analytic formula is obtained both at low and intermediate temperatures. The obtained bion distribution is neither the Bose distribution nor the Fermi distribution, nor the Boltzmann distribution at low temperatures. The self-energy of the bion at finite temperature is also discussed.     

System Dynamics in Shipping

A system which exhibits time-varying fluctuations in measures which describe its state and which comprises inter-dependent feedback loops is likely to be amenable to the modelling methods of system dynamics. The shipping industry is analysed from this point of view with the aim of identifying its dynamic characteristics and their causal mechanisms. Examples are given of observed dynamic behaviour within and between trade sectors.