Boundary-layer of torsion in a piezoelectric material with a symmetry of order six

An analytical model for the boundary-layer of torsion in a piezoelectric thin plate with a symmetry of order six is proposed on the basis of the method of asymptotic expansions. The local behaviour of the three-dimensional solution of the problem of linear piezoelectricity is formulated in the vicinity of the lateral contour of the plate, and the solution of the problem for the boundary-layer of torsion is then obtained. However, we denote that full proof of the theorems is not given in this paper. Only hints of the proof are given.     

A time delay model for the diffusion of a new technology

In this paper, we propose a mathematical model with time delay to describe the process of diffusion of a new technology. This model is suitable for modeling diffusion processes of all those technologies that require great initial investments and public subsidies, such as technologies used for producing renewable energy. We consider external factors, such as the government policy and the production costs, that influence the decision of adoption of the new technology. We also consider the internal influence from adopters. The adoption process is described by a delay differential equation. The time delay represents the evaluation stage at which the potential consumers decide whether to adopt the new technology or not. A qualitative analysis is carried out in order to assess the stability of the equilibrium for certain parameters and to find the final level of adopters.     

ESTIMATING EXTREME-VALUE INDEX FROM RECORDS

1.IntroductionLet{X.,n21}beasequenceoflidry'swithanondegeneratedistributionfunctionF(x).Supposethereexistsomeconstantsan>0,b.6RandsomeacERsuchthatwhereG.standsforoneoftheextremevaluedistributions:Heretheindex7ERisarealparameter(interpret(1 box)--'/ryase--"for7~0).Theestimationoftheextreme-valueindex7isveryimportantbothintheextremevaluetheoryandinpractice.Manystatistics,suchasHillestimator(forcaseac>0),PickandsestimatorandDekkers-EinmahLdeHaan'smomentestimatorwhicharebasedonafinitesample,…     

ASYMPTOTIC ERROR EXPANSION AND DEFECT CORRECTION FOR SOBOLEV AND VISCOELASTICITY TYPE EQUATIONS

1.IntroductionLetObearectangulardomain.WeareconcernedwiththeRichardsonextrapolationanddefectcorrectionofthefiniteelementapproximationstothesolutionsofthefollowingsimpleSobolevtypeequationandviscoelasticitytypeequationTheextrapolationtechniqueusedforthefin…     

Stability of inclined planar flames as a local approximation of weakly curved flames

To evaluate the effect of vorticity usually generated by curved flames on the flame stability, laminar premixed planar flames inclined in the gravitational field is asymptotically examined. The flame structure is resolved by a large activation energy asymptotics and a long wave approximation. The coupling between hydrodynamics and diffusion processes is included and near-unity Lewis number is assumed. The results show that as the flame is more inclined from the horizontal plane it shows more unstable characteristics due to not only the decrease of the stabilizing effect of gravity but also the increase of the destabilizing effect of rotational flow. Unlike the planar flame propagating downward with the right angle to the upstream flow, the obtained dispersion relation involves the Prandtl number and shows the destabilizing effect of viscosity. The analysis predicts that the phase velocity of unstable wave depends on the Lewis number as well as the flame angle and, especially for unity Lewis number, it is the same with tangential velocity at the reaction zone. For relatively short wave disturbances, still much larger than flame thickness, the most unstable wavelength is nearly independent on the flame angle and the flame can be stabilized by gravity and diffusion mechanism.     

Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables.     

En route to Background Independence: Broken split-symmetry,and how to restore it with bi-metric average actions

The most momentous requirement a quantum theory of gravity must satisfy is Background Independence, necessitating in particular an ab initio derivation of the arena all non-gravitational physics takes place in, namely spacetime. Using the background field technique, this requirement translates into the condition of an unbroken split-symmetry connecting the (quantized) metric fluctuations to the (classical) background metric. If the regularization scheme used violates split-symmetry during the quantization process it is mandatory to restore it in the end at the level of observable physics. In this paper we present a detailed investigation of split-symmetry breaking and restoration within the Effective Average Action (EAA) approach to Quantum Einstein Gravity (QEG) with a special emphasis on the Asymptotic Safety conjecture. In particular we demonstrate for the first time in a non-trivial setting that the two key requirements of Background Independence and Asymptotic Safety can be satisfied simultaneously. Carefully disentangling fluctuation and background fields, we employ a ‘bi-metric’ ansatz for the EAA and project the flow generated by its functional renormalization group equation on a truncated theory space spanned by two separate Einstein–Hilbert actions for the dynamical and the background metric, respectively. A new powerful method is used to derive the corresponding renormalization group (RG) equations for the Newton- and cosmological constant, both in the dynamical and the background sector. We classify and analyze their solutions in detail, determine their fixed point structure, and identify an attractor mechanism which turns out instrumental in the split-symmetry restoration. We show that there exists a subset of RG trajectories which are both asymptotically safe and split-symmetry restoring: In the ultraviolet they emanate from a non-Gaussian fixed point, and in the infrared they loose all symmetry violating contributions inflicted on them by the non-invariant functional RG equation. As an application, we compute the scale dependent spectral dimension which governs the fractal properties of the effective QEG spacetimes at the bi-metric level. Earlier tests of the Asymptotic Safety conjecture almost exclusively employed ‘single-metric truncations’ which are blind towards the difference between quantum and background fields. We explore in detail under which conditions they can be reliable, and we discuss how the single-metric based picture of Asymptotic Safety needs to be revised in the light of the new results. We shall conclude that the next generation of truncations for quantitatively precise predictions (of critical exponents, for instance) is bound to be of the bi-metric type.     

Oscillation criteria for second order nonlinear neutral differential equations

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form

(r(t)(x(t)+p(t)x(t-τ))^′)^′+q(t)f(x(t),x(σ(t)))=0.