On the Hausdorff dimension of fractal attractors

We consider such mappingsx _n+1=F(x_n) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...     

A sharp lower bound for the Hausdorff dimension of the global attractors of the 2D Navier-Stokes equations

For a special class of the Navier-Stokes equations on the two-dimensional torus, we give a lower bound in the formG ^2/3 (whereG is the Grashof number) for the Hausdorff dimension of its global attractor which is optimal up to a logarithmic term.Dedicated to the memory of Professor Nicholas D. Kazarinoff     

On the dimension of the attractors in two-dimensional turbulence

Using a new version of the Sobolev-Lieb-Thirring inequality, we derive an upper bound for the dimension of the universal attractor for two-dimensional space periodic Navier-Stokes equations. This estimate is optimal up to a logarithmic correction. The relevance of this estimate to turbulence and related results are also briefly discussed.     

The Hausdorff dimension of some fractals and attractors of overlapping construction

We introduce a method to show that the Hausdorff dimension of certain fractals of overlapping construction is “almost always” what would be observed if no overlapping occurred. The method is also used to examine the dimension of attractors in some noninjective, piecewise linear, “baker's”-type transformations.     

Explicit dimension estimates of attractors for the MHD equations in three-dimensional space

In this article we estimate the dimension of functional invariant sets that arise from the MHD equations, in terms of some physically relevant parameters. We follow the analysis of Constantin, Foias and Temam for 3D Navier-Stokes equations and derive comparable estimates.     

Hausdorff dimension and fluctuations for the largest cluster at the two-dimensional percolation threshold

Monte Carlo simulation shows the average mass of the largest cluster to increase asL ^1.9 at the percolation threshold inL × L square lattices,L290. This fractal dimension agrees with the finite-size scaling prediction/v for this exponent, in contrast to results of Halley and Thang Mai. The mean-square fluctuations in the mass of the largest cluster diverge with the same exponent/v1.8 as the susceptibility.     

Universal quadratures for boundary integral equations on two-dimensional domains with corners

We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high-accuracy solution of Dirichlet boundary value problems for Laplace’s equation and the Helmholtz equation on such domains under a mild assumption on the boundary data. Our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.     

Hausdorff dimension for the quantum harmonic oscillator

We show that the measured path of a nonrelativistic quantum particle in a harmonic oscillator potential is a fractal curve with Hausdorff dimension equal to two.     

Modification of the magnetoconductance of a two-dimensional electron gas by altering the boundary conditions in the third dimension

Substrate bias can be used as a powerful diagnostic tool for studying the magneto-transport in an inversion layer. One can accurately determine the mobile carrier threshold by such a study. Marked changes in the mobility occur with increasing confinement to the surface; a decrease for high electrons densities and an increase for low electron densities. The former was expected while the latter was not.     

Dissipative boundary conditions for ferromagnetic resonance equations

Surface scattering of electrons and magnons results in nonuniform dissipation of the energy of ferromagnetic resonance (FMR) modes. This surface damping may be included in proper boundary conditions for the magnetic differential equations of motion, which are derived phenomenologically in the same way as the usual spin-pinning conditions. An analysis of the local energy balance in non-uniform modes leads to the concept of spin-wave energy flow; the dissipative boundary conditions describe the flow of the coherent mode energy into the surface. The magnitude of the surface damping parameters is only crudely estimated. Numerical calculations of the FMR linewidth and the shapes of standing-spinwave spectra indicate that experimentally significant effects may appear with damping parameters near to the estimated upper limits.     

二维介质柱光子晶体波导吸收边界条件

设计优化了用于截断二维正方格子介质柱光子晶体波导的分布布拉格反射波导的结构.二维时域有限差分数值模拟结果显示，在上述两种波导联接处的反射系数可以在大部分光子晶体波导导波的频谱范围内降到1％以下.将这种分布布拉格反射波导和通常的吸收边界条件相结合可以构成用于光子晶体波导的吸收边界条件，其反射率可以降低到-40dB以下，吸收层的厚度可取为晶格长度的1.3倍. 关键词： 吸收边界条件 光子晶体波导 时域有限差分     

Euler equations with non-homogeneous Navier slip boundary conditions

We consider the flow of an ideal fluid in a 2D bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with non-homogeneous Navier slip boundary conditions. These conditions can be written in the form , , where the tensor is the rate of strain of the fluid’s velocity and is the pair formed by the normal and tangent vectors to the boundary. We establish the solvability of this problem for the class of solutions with L_p-bounded vorticity, p∈(2,∞]. To prove the solvability we realize the passage to the limit in Navier-Stokes equations with vanishing viscosity.     

Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

We study the bifurcation of radially symmetric solutions of +f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.Research supported in part by the NSF under Grant No. MCS-800 2337     

Implementation of nonreflecting boundary conditions for the nonlinear Euler equations

Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier–Bessel modes. This leads to an ‘exact’ nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier–Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.     

Calculating the Hausdorff dimension of tree structures

The dimension of certain tree structures is of importance in percolation theory, as well as in the theoretical treatment of many other branching processes. We present a method of determining the Hausdorff dimension of such structures by employing the technique of Mauldin and Williams. The dimension is calculated based on the probability of generation of each branch from its parent on the tree representing the process. We use this method to analyze the dimension of tree structures representing two-directional linear bonding between equally weighted monomers, and show how it can be used to model enzymatic reaction pathways.     

On multi-term fractional differential equations with multi-point boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic.     

On fractional Langevin differential equations with anti-periodic boundary conditions

In this paper, we provide existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with anti-periodic boundary conditions. The Caputo fractional as well as Caputo q-fractional operators are used to address the derivatives. The main results are verified by the help of Leray–Schaefer’s fixed point theorem. We construct an example to illustrate the feasibility of the main theorems. Our results are new and provide extensions to some known theorems in the literature.     

Hausdorff dimension of order preserving sets

Letg aC ² generic bimodal map of the circle. We prove that the closure of the union of the order preserving recurrent sets with irrational rotation number has Hausdorff dimension zero. This set contains order preserving periodic orbits with each rotation numberp/q in the rotation interval ofg.     

Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates

For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.