Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

Fractional Lévy processes on Gel’fand triple and stochastic integration

In this paper, we investigate the long-range dependence of fractional Lévy processes on Gel’fand triple and construct stochastic integral with respect to fractional Lévy processes for a class of deterministic integrands.      

A parametric variational principle and residuality

We prove a parametric version of a smooth convex variational principle with constraints using a Baire category approach. We examine in depth the necessity of the assumptions of our variational principle by providing counterexamples.     

Tail Dependence Comparison of Survival Marshall–Olkin Copulas

The multivariate tail dependence describes the amount of dependence in the upper-orthant tail or lower-orthant tail of a multivariate distribution and can be used in the study of dependence among extreme values. We derive an explicit expression of tail dependence of multivariate survival Marshall–Olkin copulas, and obtain a sufficient condition under which tail dependencies of two survival Marshall–Olkin copulas can be compared. Some examples are also presented to illustrate our results.      

The Generalized Uncertainty Principle

The uncertainty principle lies at the heart of quantum physics, and is widely thought of as a fundamental limit of the measurement precision of incompatible observables. Here it is shown that the traditional uncertainty relation in fact belongs to the leading order approximation of a generalized uncertainty relation. That is, the leading order linear dependence of observables gives the Heisenberg type of uncertainty relations, while higher order nonlinear dependence may reveal more different and interesting correlation properties. Applications of the generalized uncertainty relation and the high order nonlinear dependence between observables in quantum information science are also discussed.     

Geometry dependence of the elongation behavior of a polypropylene melt through a micro die

A series of rheological experiments was performed for a polypropylene (PP) melt to explore its elongation behavior through a capillary die. Using an advanced twin-bore capillary rheometer with dies measuring 1.0, 0.5, and 0.25 mm in diameter, the experiments were performed at 210, 220, and 230 °C. The results indicated that the temperature of the PP melt had a significant effect on its extensional viscosity. The different decreases in the extensional viscosity values in the tested dies revealed the geometry dependence of the extensional viscosity. In the case of PP in the 0.25 mm die at 210 °C, the extensional viscosity values under different extensional strain rates were much higher than those in the other dies. Only in the 1.0 mm die did the relationship between the extensional viscosity of PP and its temperature obey the Arrhenius equation due to the larger die size which related to a slight size effect on its elongation behavior. The calculated deviations of the extensional viscosity in the tested dies demonstrated that the increasing pressure applied to the PP melt in the micro channel was related to the geometry dependence of the elongation behavior of the PP melt. The change in the extensional viscosity eventually relied on the interaction of the die geometry, the temperature, and the extensional stress of the PP melt.     

Determination of dynamic properties of flax fibres reinforced laminate using vibration measurements

Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced polymer (FFRP) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited in the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine. Overall, the Zener model fitted the experimental results well. The storage modulus was not frequency dependant, while the loss factor increased with frequency and reached a maximum value for a fibre orientation of 70°. The damping of FFRP was, respectively, 5 and 2 times higher than for equivalent carbon and glass fibres reinforced epoxy composites.     

Closed-packed lattice model for polymer solution systems considering the chain length dependence effect: Correlating LLE,VLE, and gel swelling behaviors using identical interaction energy parameters

We propose a new Helmholtz energy of mixing equation following the original Flory–Huggins (F–H) closed-packed lattice model. Also, to overcome F–H mean-field approximation, we introduce new universal constants to consider chain length dependence of polymer in solvent and consider specific interactions to describe strongly interacting polymer systems. Our proposed model successfully describes liquid–liquid equilibria (LLE) for binary polymer–solvent systems using identical interaction parameters which do not depend on the polymer molecular weight. We also describe vapor–liquid equilibria (VLE) for polymer/solvent systems and swelling equilibria of thermosensitive hydrogel systems using the same energy parameters obtained from LLE calculations.     

Temperature dependence of structure and dynamics of the hydrated Ca2+ ion according to ab initio quantum mechanical charge field and classical molecular dynamics

Simulations using ab initio quantum mechanical charge field molecular dynamics (QMCF MD) and classical molecular dynamics using two‐body and three‐body potentials were performed to investigate the hydration of the Ca²⁺ ion at different temperatures. Results from the simulations demonstrate significant effects of temperature on solution dynamics and the corresponding composition and structure of hydrated Ca²⁺. Substantial increase in ligand exchange events was observed in going from 273.15 K to 368.15 K, resulting in a redistribution of coordination numbers to lower values. The effect of temperature is also visible in a red‐shift of the ion‐oxygen stretching frequencies, reflecting weakened ligand binding. Even the moderate increase from ambient to body temperature leads to significant changes in the properties of Ca²⁺ in aqueous environment. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010