Can one hear the dimension of a fractal?

We consider the spectrum of the Laplacian in a bounded open domain of ⁿ with a rough boundary (i.e. with possibly non-integer dimension) and we discuss a conjecture by M. V. Berry generalizing Weyl's conjecture. Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function. The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones.Dedicated to the memory of Mark Kac     

Eigenfrequency spacing analysis and eigenmode breakdown for semi-stadium-type 2-D fields

This paper describes eigenfrequency statistics including modal patterns and degrees of freedom for a semi-stadium-type 2-D field. The authors numerically investigated sound fields surrounded by 2-D semi-stadium-type boundaries as examples of boundaries where chaotic properties are hidden, in order to understand the characteristics of complex sound fields and to gain new insight into sound-field design. One limit of the semi-stadium boundaries is a rectangular boundary that gives a regular field, while another limit is a stadium boundary in which the chaotic property emerges. The numerical results show that eigenfrequency spacing in all the cases can be expressed as a family of Γ distributions extended to a non-integer degree of freedom. This fractal degree of freedom can be interpreted as the degree of freedom of the sound field. For the regular limit case, i.e., a rectangular case, the distribution is the exponential distribution with the freedom of unity, while in the chaotic case, i.e., the stadium case, it is the Wigner distribution with a degree of freedom of two. The authors, however, analyze the semi-stadium sound fields. The analysis of the fluctuations in the distribution function for the eigenfrequencies showed that the skewness decreases as the boundary approaches the stadium condition. Modal patterns also clearly showed breaks of the regular pattern of nodal lines in a rectangular case as the boundary was deformed from the rectangular to the stadium condition. The breaks of the modal pattern could be also confirmed by decreasing of the skewness for the sound pressure distribution.     

Direct experimental determination of grain boundary excess volume in metals

The grain boundary excess volume, i.e., the grain boundary expansion, e{GB}, was experimentally determined for high-angle grain boundaries in nickel using the direct technique of high-precision difference dilatometry. Values of e{GB}=(0.35±0.04)×10{-10} m and e{GB}=(0.32±0.04)×10{-10} m were obtained by measuring the removal of grain boundary volume upon grain growth for two different types of ultrafine-grained samples. The results are discussed in comparison to values obtained so far from indirect techniques and from computer simulations. It demonstrates the strength of the presented novel, direct approach for grain boundary expansion measurements.     

Staggered fermions and rotation-gauge invariance

We study the rotational invariance of the staggered fermions on the lattice by considering them as spin-zero fields in a quenched gauge field background. The non-integer spin fermionic fields are reconstructed. Bilinear operators (i.e. meson fields) in terms of the quark fields on a cube are classified according to the representations of the discrete rotational group.     

晶间界面附近区域中的弹性场——各向异性理论

根据现代关于晶间界面的研究可知,晶界和相界多具有二维周期结构。由此出发,采用Fourier级数展开的表示,在给定了界面结构,在数学上即是形式地给定了边界条件的情况下,求出了两个各向异性物相间相界附近的弹性场。并以Ni中扭转晶界为例,作了应力场的数值计算。本文的方法和方程对于研究各种类型的晶界和相界均普遍可用,为处理这类问题提供了一种有效的简捷手段。 关键词：     

Network analysis of the Íslendinga sögur – the Sagas of Icelanders

It is well-known that cooperative properties such as magnetic ordering can depend on the samples’ dimensions (Ds) in a qualitative way. However, there have been no samples with well-defined non-integer Ds. The dimension of a given sample has been always discussed on the anisotropy of the electronic/crystal/magnetic structures, which has no definition suitable for quantitative discussion on dimensions vs. properties. On the other hand a particular type of porous samples, i.e. fractal bodies, can have well-defined non-integer Ds dependent exclusively on the geometrical feature of structures, and physical properties of such materials remains unexplored. This paper reports on magnetic ordering in samples covering 2.5 ≤ D ≤ 3, in addition to a way of precise control of the fractal dimensions of given samples simply by wax (alkylketene dimer). The results show that the magnetic ordering temperatures, i.e. Néel temperatures (T _N s), of CoO depend on D, and rapidly enhance immediately below D = 3. This means that one can control or enhance the critical temperature simply by tuning D with keeping the remaining magnetic properties unchanged.     

Matrix continued fraction solutions of the Kramers equation and their inverse friction expansions

The distribution function in position and velocity space for the Brownian motion of particles in an external field is determined by the Kramers equation, i.e., by a two variable Fokker-Planck equation. By expanding the distribution function in Hermite functions (velocity part) and in another complete set satisfying boundary conditions (position part) the Laplace transform of the initial value problem is obtained in terms of matrix continued fractions. An inverse friction expansion of the matrix continued fractions is used to show that the first Hermite expansion coefficient may be determined by a generalized Smoluchowski equation. The first terms of the inverse friction expansion of this generalized Smoluchowski operator and of the memory kernel are given explicitly. The inverse friction expansion of the equation determining the eigenvalues and eigenfunctions is also given and the connection with the result of Titulaer is discussed.     

An exact solution to the problem of current transport through the grain boundary in a metal

The problem of electrical resistance of a planar boundary between two crystalline grains of a polycrystalline metal is solved analytically. The solution is derived using the Case method. The electric field and the electron distribution function are represented in the form of an eigenfunction expansion of the corresponding characteristic equation. It is demonstrated that the electric field at the interface increases drastically when the Debye frequency substantially exceeds the frequency of electron collisions in the bulk of the metal. It is found that the resistance of the grain boundary does not depend on the frequency of electron collisions in the bulk, i.e., on their mean free path.     

Quantum field-theory models on fractal spacetime

The present work explores the possibility of giving a non-perturbative definition of the quantum field-theory models in non-integer dimensions, which have been previously studied by Wilson and others using analytic continuation of dimension in perturbation integrals. The method employed here is to base the models on fractal point-sets of non-integer Hausdorff-Besicovitch dimension. Two types of scalar-field models are considered: the one has its propagator (=covariance operator kernel) given by a proper-time or heat-kernel representation and the other has a hierarchical propagator. The fractal lattice version of the proper-time propagator is shown to be reflection-positive. The hierarchical models are introduced and their properties discussed on an informal basis.This paper is based largely on the thesis presented by the author for the degree doctor of philosophy in physics at the Ohio State University in 1987     

An electrochemomechanical theory of defects in ionic solids. Part II. Examples

In this paper, a boundary value problem formulated previously by N. Swaminathan, J. Qu, Y. Sun, Phil. Mag. 2006 (this volume) is solved to obtain the defect, stress distributions in a planar electrolyte in solid oxide fuel cells. It is found that analytical solutions from the coupled theory reduce to the known results from electrochemistry literature when the coefficient of compositional expansion is set to zero, i.e. when there is no coupling between electrochemical and mechanical fields. For the case of non-zero coefficient of compositional expansion, extensive numerical results are presented which show that there is a significant interaction between mechanical stresses and defect distribution.     

Phase diagram of the helical structure of a two-subsystem frustrated antiferromagnet

The expansion of the thermodynamic potential for the two-subsystem antiferromagnet with frustrated intersubsystem isotropic exchange is obtained. It is demonstrated that this expansion contains the first derivatives with respect to the antiferromagnetic vectors of the subsystems, i.e., the Lifshitz invariant. The equation for the temperature-field boundary of the helical phase for the two-subsystem frustrated antiferromagnet is derived by linearizing the variational equations for the minimum free energy within the mean-field approximation. Relationships are obtained for the critical field at T = 0, the angle of canting of moments of the antiferromagnetic sublattices, and the temperature of spontaneous appearance of helical ordering in the absence of an external field. It is revealed that there is a second higher temperature of formation of the helical magnetic structure induced by the magnetic field with the wave vector of the helix nonmonotonically depending on the external field. The phase boundary of the helical phase and the temperature dependence of the orientation of moments of the magnetic subsystem with weak exchange interaction are determined using numerical minimization of the free energy. It is shown that the transition to the commensurate phase is a first-order transition with a small magnetization jump. A comparative analysis of models with different spatial displacements of ions in the subsystems along the direction of the vector of the helical structure is performed. A criterion is proposed for the choice of the direction of the vector of the incommensurate magnetic structure.     

Perfect fluid spheres in general relativity

Spherically symmetric perfect fluid distributions in general relativity have been investigated under the assumptions of (i) uniform expansion or contraction and (ii) the validity of an equation of state of the formp=p(ρ) with nonuniform density. An exact solution which is equivalent to a solution found earlier by Wyman is obtained and it is shown that the solution isunique. The boundary conditions at the interface of fluid distribution and the exterior vacuum are discussed and as a consequence the following theorem is established:Uniform expansion or contraction of a perfect fluid sphere obeying an equation of state with nonuniform density is not admitted by the field equations. It is further shown that the Wyman metric is not suitable on physical grounds to represent a cosmological solution.     

The relationship between chemical composition distributions and specific grain boundary conductivity in Y-doped BaZrO3 proton conductors

The relationship between chemical composition at a grain boundary and specific grain boundary conductivity is studied in Y-doped BaZrO₃ ceramics sintered at 1800 °C. Y enrichment at the grain boundary as yttrium concentration increased and Ba deficiency in ceramics as an increase of sintering time are observed. At the grain boundary, phase segregation, i.e. existence of secondary phase and amorphous phase which prevent proton migration, is not observed. The results indicate that at the grain boundary, concentration of Ba vacancy and Y substituted into Zr site is changed. Grain boundary conductivity shows significant dependences to Ba deficiency and Y concentration in ceramics. It is due to significant dependences of specific grain boundary conductivity to Ba deficiency and Y concentration in ceramics. The results indicate that proton migration across the grain boundary is sensitive to concentration of those effectively charged defect, i.e. Y substituted into Zr site and Ba vacancy at grain boundary.     

A generalization of Wigner's R-matrix method

A generalization of the Wigner's non-relativistic R-matrix theory of scattering by a central potential field is proposed. The idea is to use an R-matrix expansion basis generated by a Sturm-Liouville problem with an eigenparameter included both in a differential equation and in a boundary condition (in the standard theory an R-matrix basis is obtained by solving an eigenvalue problem with fixed boundary conditions). A partial fraction expansion of an R^(η)-matrix introduced is derived and shown to converge faster than a partial fraction expansion of Wigner's R-matrix used in the standard theory.     

Use of the integer and non-integer n-dimensional Debye functions in computing thermal expansivity,with applications to Si and Ge semiconductors

In the present study, a novel method is proposed to accurately calculate the thermal expansivity of semiconductors using the analytical expression obtained for the integer and non-integer n-dimensional Debye functions. The proposed approach is novel as it uses the non-integer n-dimensional Debye functions for the accurate calculations of the thermal expansivities of semiconductors. As an example of the application, the calculation is performed for the thermal expansivities of the Si and Ge semiconductors. Analysis using computer simulation proved that the formula was in excellent agreement with the results reported in the literature.     

高能碰撞实验数据非整数分割时的阶乘矩分析

在对相空间用非整数分割作阶乘矩分析时,由二粒子或多粒子关联所引起的非均匀因素的影响必须加以认真考虑.本文对高能碰撞末态粒子的阶乘矩在相空间进行非整数分割时的分析方法进行了仔细研究,建立了一套完整的对非整数分割下阶乘矩的分布进行修正的方法.用NA22实验组的实验数据计算,证明其方法是正确的.     

具有小Knudsen数的Boltzmann方程的奇异扰动解法

本文提出一种解小Knudsen数的Boltzmann方程的方法,它可以消去Hilbert展开和Enskog展开中的久期项改进正规解,消去Grad展开中的久期项改进初始层解,并把求边界层解归结为求解线性代数方程组。 关键词：     

半空间球面波函数叠加法重构声源直接辐射声场

提出了基于半空间球面波函数叠加的声场重构方法,以重构含有限声阻抗边界半空间中声源直接辐射的声场.在半空间中多极子声源声压场的解析解的基础上,构造出以边界声阻抗为参量的半空间球面波函数的正交基;通过求逆获得半空间总声压解的基函数系数,同时也获得声源直接辐射声场即自由空间中的基函数系数,进而重构出声源直接辐射的声场.在边界...     

Pressure limits for the vacuum arc deposition technique

Observations of the cathodic copper plasma expansion at low pressures of He, Ar, and SF₆ showed that, for background gas mass densities of ρ_g=1 to 4×10^-4 kg/m ³ and higher, the plasma and gas are separated into two volumes. A shock wave acts as a boundary between the two volumes. The boundary attains a stationary position once its expansion velocity decreases to the velocity of sound in the background gas. This position corresponds to a distance R_c to the cathode that agrees with a snowplow expansion model, giving R_c ^βf=E_r, where f is a function of the arc current and background gas characteristics, E _r is the erosion rate of the cathode, and β varies between 2.1 and 2.5. The interaction model is based on kinetic energy exchanges between two gas-like volumes without other energy losses. A maximum pressure limit for vacuum arc deposition is set for ρ_g /I=2 to 9×10^-6 kg/m³ A