MULTIPLE INTERSECTIONS OF SPACE-TIME ANISOTROPIC GAUSSIAN FIELDS

Let X={X(t)∈R^(d),t∈R^(N)}be a centered space-time anisotropic Gaussian field with indices H=(H_(1),…,H_(N))∈(0,1)~N,where the components X_(i)(i=1,…,d)of X are independent,and the canonical metric√(E(X_(i)(t)-X_(i)(s))^(2))^(1/2)(i=1,…,d)is commensurate with■for s=(s_(1),…,s_(N)),t=(t_(1),…,t_(N))∈R~N,α_(i)∈(0,1],and with the continuous functionγ(·)satisfying certain conditions.First,the upper and lower bounds of the hitting probabilities of X can be derived from the corresponding generalized Hausdorff measure and capacity,which are based on the kernel functions depending explicitly onγ(·).Furthermore,the multiple intersections of the sample paths of two independent centered space-time anisotropic Gaussian fields with different distributions are considered.Our results extend the corresponding results for anisotropic Gaussian fields to a large class of space-time anisotropic Gaussian fields.

HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS

Let X^(H)={X^(H)(s),s∈R^(N_(1))}and X^(K)={X^(K)(t),t∈R^(N_(2))}be two independent time-space anisotropic random fields with indices H∈(0,1)^(N_(1)) and K∈(0,1)^(N_(2)),which may not possess Gaussianity,and which take values in R^(d) with a space metric τ.Under certain general conditions with density functions defined on a bounded interval,we study problems regarding the hitting probabilities of time-space anisotropic random fields and the existence of intersections of the sample paths of random fields X^(H) and X^(K).More generally,for any Borel set F⊂R^(d),the conditions required for F to contain intersection points of X^(H) and X^(K) are established.As an application,we give an example of an anisotropic non-Gaussian random field to show that these results are applicable to the solutions of non-linear systems of stochastic fractional heat equations.

ANISOTROPIC PLASTIC FIELDS AT A RAPIDLY PROPAGATING PLANE-STRESS CRACK-TIP

Under the condition that all the stress components at a crack-tip are the functions of 0 only, making use of the equations of steady-state motion. Hill anisotropic yield condition and stress-strain relations, we obtain the general solution of anisotropic plastic field at a rapidly propagating plane-stress crack-tip. Applying this general solution to four particular cases of anisotropy, the general solutions of these four particular cases are derived. Finally, we give the anisotropic plastic field at the rapidly propagating plane-stress mode I crack-tip in the case of X=Y=Z

SUPPLEMENTARY STUDY ON ANISOTROPIC PLASTIC STRESS FIELDS AT A STATIONARY PLANE-STRESS CRACK-TIP

Under the hypothesis that all the perfectly plastic stress components at a orach tip are the functions of θ only, making use of yield conditions and equilibrium equations. we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode Ⅰ Mode Ⅱ, Mode Ⅲ and Mixed Mode Ⅰ-Ⅱ cracks are obtained.

ANISOTROPIC PLASTIC FIELDS AT A RAPIDLY PROPAGATING CRACK-TIP

Under the condition that all the stress components at a crack-tip are the functions of only, making use of the equations of steady-slate motion, stress-strain relations and Hill anisotropic yield conditions, we obtain the general solutions at a crack-tip in both the cases of anti-plane and in-plane strains. Applying these general solutions to the concrete cracks, the anisotropic plastic fields at the rapidly propagating tips of mode III and mode I cracks are derived.

ANISOTROPIC PLASTIC STRESS FIELDS AT A SLOWLY PROPAGATING CRACK TIP

Under the condition that any perfeetly plastic stress components at a crack tip are nothing but the Junctions of 0 only, making use of equilibriumequations,Hill ani.sutropic yield condition and unloading stress-strain relations, in this paper, we derive the general analytical expressions of anisotropic plastiestress Jields at the slowly steadyhe slowly steady propagatin tips of plane and anti-phane strain,Applying these general analytical expressions to the concrete cracks the attchvtical expressions of anisotropie plastic stress fields at the slowly steady propagating tips of Motle I and Motle III cracks are obtained. For the isolropic plastic material, the anisotropic plastic stress fields at a slowly propagating crack tip become the perfeeby plastic mress fields

Magnetpolaron effect in two-dimensional anisotropic parabolic quantum dot in a perpendicular magnetic field

We study the two-dimensional weak-coupling Frohlich polaron in a completely anisotropic quantum dot in a perpendicular magnetic field. By performing a unitary transformation, we first transform the Hamiltonian into a new one which describes an anisotropic harmonic oscillator with new mass and trapping frequencies interacting with the same phonon bath but with different interaction form and strength. Then employing the second-order Rayleigh–Schrodinger perturbation theory, we obtain the polaron correction to the ground-state energy. The magnetic field and anisotropic effects on the polaron correction to the ground-state energy are discussed.

Three-dimensional stochastic seepage field for embankment engineering

Owing to the complexity of geo-engineering seepage problems influenced by different random factors, three-dimensional simulation and analysis of the stochastic seepage field plays an important role in engineering applications. A three-dimensional anisotropic heterogeneous steady random seepage model was developed on the basis of the finite element method. A statistical analysis of the distribution characteristics of soil parameters sampled from the main embankment of the Yangtze River in the Southern Jingzhou zone of China was conducted. The Kolomogorov-Smirnov test verified the statistical hypothesis that the permeability coefficient tensor has a Gaussian distribution. With the help of numerical analysis of the stochastic seepage field using the developed model, various statistical and random characteristics of the stochastic seepage field of the main embankment of the Yangtze River in the Southern Jingzhou zone of China were investigated. The model was also examined with statistical testing. Through the introduction of random variation of the upstream and downstream water levels into the model, the effects of the boundary randomness due to variation of the downstream and upstream water levels on the variation of simulated results presented with a vector series of the random seepage field were analyzed. Furthermore, the combined influence of the variation of the soil permeability coefficient and such seepage resistance measures as the cut-off wall and relief ditch on the hydraulic head distribution was analyzed and compared with the results obtained by determinate analysis. Meanwhile, sensitivities of the hydraulic gradient and downstream exit height to the variation of boundary water level were studied. The validity of the simulated results was verified by stochastic testing and measured data. The developed model provides more detail and a full stochastic algorithm to characterize and analyze three-dimensional stochastic seepage field problems.

Anisotropic Fractional Brownian Random Fields as White Noise Functionals

This investigation aims at a new construction of anisotropic fractional Brownian random fields by the white noise approach. Moreover, we investigate its distribution and sample properties （stationariness of increments, self-similarity, sample continuity） which will furnish some useful views to future applications.

On intersections of independent anisotropic Gaussian random fields

Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} be two independent anisotropic Gaussian random fields with values in R^d with indices H = （H1,... ,HN1） ∈ （0, 1）^N1, K = （K1,..., KN2）∈ （0, 1）^N2, respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let E1 R^N1, E2 R^N2 and F R^d be Borel sets. A necessary condition and a sufficient condition for P{（X^H（E1） ∩ X^K（E2）） ∩ F ≠ Ф} 〉 0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 x E2 x F in the metric space （R^N1＋N2＋d, ρ） are proved, whereρ is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.

Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of a Class of Anisotropic Random Fields

Let X = {X（t）：t ∈ R^N} be an anisotropic random field with values in R^d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

Stress fields caused by a dislocation in an anisotropic 3-layer system

We investigated the stress fields caused by a dislocation in an anisotropic 3-layer system. Based on the image method, the original 3-layer system is firstly decomposed into three infinite homogenous systems. The image dislocation densities used as unknowns are then strategically distributed in order to satisfy the boundary conditions. The resulting governing equations are singular Cauchy integral ones. Removing the singular terms yields non-linear Fredhom integral equations of the second kind. The obtained stress fields satisfy the boundary conditions, i.e., the traction free condition on the free surface and continuous conditions across the interfaces. Also, a comparison with previous results is made and good agreement is achieved. Numerical investigations show that under the plain strain condition, layer thickness and dislocation position play stronger roles in the stress fields than crystallographic orientation, and these effects more significantly affect the stress fields caused by an edge dislocation than by a screw dislocation.

Multi-impurity effects on the entanglement of anisotropic Heisenberg ring XXZ under a homogeneous magnetic field

The effects of multi-impurity on the entanglement of anisotropic Heisenberg ring XXZ under a homogeneous magnetic field are studied. The impurities make the equal pairwise entanglement in a ring compete with each other so that the pairwise entanglement exhibits oscillation. If the impurities are of larger couplings, both the critical temperature and pairwise entanglement can be improved.

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ? N } be a Gaussian random field with values in ? d defined by (1) $$ X(t) = (X_1 (t),...,X_d (t)),\forall t \in \mathbb{R}^N . $$ . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X.When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness （in the sense of Meyer- Watanabe） of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Magnetic coupling in ferromagnetic semiconductor GaMnAs/AlGaMnAs bilayer devices

We carefully investigated the ferromagnetic coupling in the as-grown and annealed ferromagnetic semiconductor GaMnAs/A1GaMnAs bilayer devices. We observed that the magnetic interaction between the two layers strongly affects the magnetoresistance of the GaMnAs layer with applying the out of plane magnetic field. After low temperature annealing, the magnetic easy axis of the A1GaMnAs layer switches from out of plane into in-plane and the interlayer coupling efficiency is reduced from up to 0.6 to less than 0.4. However, the magnetic coupling penetration depth for the annealed device is twice that of the as-grown bilayer device.