Association of evaluation methods of the effective permittivity of heterogeneous media on the basis of a generalized singular approximation

Various methods for evaluation of the effective permittivity of heterogeneous media, namely, the effective medium approximation (Bruggeman’s approximation), the Maxwell-Garnett approximation, Wiener’s bounds, and the Hashin-Shtrikman variational bounds (for effective static characteristics) are combined on the basis of a generalized singular approximation.     

Two-dimensional scaling properties of experimental fracture surfaces

The self-affine properties of postmortem fracture surfaces in silica glass and aluminum alloy were investigated through the 2D height-height correlation function. They are observed to exhibit anisotropy. The roughness, dynamic, and growth exponents are determined and shown to be the same for the two materials, irrespective of the crack velocity. These exponents are conjectured to be universal.     

Bounds and exact theories for the transport properties of inhomogeneous media

We investigate the bounds of Milton on the transport coefficient of a two-component composite, in their application to the square and hexagonal arrays of cylinders, and the three cubic lattices of spheres. We show that, in all five cases, as more information is supplied about the geometry of the composite, the bounds converge to the precise point obtained from an exact theory specific to the geometry in question. We illustrate the use of the bounds in determining whether a set of known values of the transport coefficient adequately specifies the general behaviour of that quantity. We determine the values of two structure-dependent parameters for cell materials with spheroidal cells and the value of one parameter for hexagonal and square arrays of cylinders with missing array elements. These parameters determine bounds both on the transport and on the elastic properties of the respective materials.     

Magnetic Bag Like Solutions to the SU(2) Monopole Equations on {{\mathbb R}^{3}}

Solutions to the SU(2) monopole equations in the Bogolmony limit are constructed that look very much like Bolognesis conjectured magnetic bag solutions. Three theorems are also stated and proved that give bounds in terms of the topological charge for the radii of balls where the solutions Higgs field has very small norm.     

Quantum mechanics and mechanical properties: Towards twenty-first century materials

The use of materials with otherwise attractive properties is often limited by unacceptable mechanical performance. Fortunately, modern processing techniques are sometimes able to overcome such deficiencies, though a systematic and fundamental approach to materials development has yet to be devised. Recent advances in quantum-mechanical computational capabilities have fostered a growing number of applications that bear directly upon the mechanical properties of materials. After a brief discussion of the role of defect structures in mediating deformation behaviour, techniques for computing properties of solids within a quantum-mechanical framework are reviewed. Examples are cited where insight into macroscopic behaviour has been attained from the application of quantum-mechanical calculations to materials of technological importance.     

On the Bergman-Milton bounds for the homogenization of dielectric composite materials

The Bergman-Milton bounds provide limits on the effective permittivity of a composite material comprising two isotropic dielectric materials. These provide tight bounds for composites arising from many conventional materials. We reconsider the Bergman-Milton bounds in light of the recent emergence of metamaterials, in which unconventional parameter regimes for relative permittivities are encountered. Specifically, it is demonstrated that: (a) for nondissipative materials the bounds may be unlimited if the constituent materials have relative permittivities of opposite signs; (b) for weakly dissipative materials characterized by relative permittivities with real parts of opposite signs, the bounds may be exceedingly large.     

Upper Bounds to the Ground State Energies of the One- and Two-Component Charged Bose Gases

We prove upper bounds on the ground state energies of the one- and two-component charged Bose gases. The upper bound for the one-component gas agrees with the high density asymptotic formula proposed by L. Foldy in 1961. The upper bound for the two-component gas agrees in the large particle number limit with the asymptotic formula conjectured by F. Dyson in 1967. Matching asymptotic lower bounds for these systems were proved in references [10] and [11]. The formulas of Foldy and Dyson which are based on Bogolubov’s pairing theory have thus been validated.     

Extraction of structural information from measured transport properties of composites

We describe clearly Milton's bounds on the scalar transport properties of composite materials in the case when no geometrical information is known about the composite, but when a set of measured values of a transport property is available. We consider how the measured values may be used when they are subject to known errors. In this case, a statistical analysis is employed. We illustrate the various conclusions which may result from this analysis by considering six cases involving both two-and three-dimensional composites. In four of the six cases, estimates of the volume fraction of the composite in agreement with the known value are obtained. In the fifth case, the analysis pointed out a possible error in the experimental data, while in the sixth it revealed a critical dependence of transport properties on composite geometry due to filamentary linking between particles.     

Dark materials based on graphene sheet stacks

The effective medium properties of graphene sheet stacks are calculated, and it is shown that such stacks can have very low reflectivity and high absorbance. These properties make graphene-sheet-stack-based materials darker than recently studied carbon nanotube materials. Graphene stacks thus hold promise for realizing lower reflectivity coatings and enhanced photodetectors. The bounds of the effective medium approximation and the possible benefits of using graphene sheet stacks in a regime where this approximation does not hold are discussed.     

Bounds for the chaotic region in the Lorenz model

In a previous paper, the authors made an extensive numerical study of the Lorenz model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz model is chaotic is bounded for fixed r. In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional model of the chaotic regime depending on the three parameters.     

Indentation of transversely isotropic power-law hardening materials: computational modelling of the forward and reverse problems

Using a combination of dimensional analysis and large deformation finite element simulations of triple indentations of 120 materials, a framework for capturing the indentation response of transversely isotropic materials is developed. By considering 4800 combinations of material properties within the bounds of the original set of 120 materials, forward algorithms that predict the indentation response of materials and reverse algorithms that predict the materials’ elastic and plastic properties from experimentally measured indentation responses are formulated for both longitudinal and transverse indentations. Issues of accuracy, reversibility, uniqueness and sensitivity within the context of the indentation of transversely isotropic materials are addressed carefully. Using 1400 combinations of material properties, it is demonstrated that there is perfect reversibility between the material properties and their indentation responses as predicted by the forward and reverse algorithms. On average, the differences between the results of the finite element analysis and those predicted by the forward algorithms for longitudinal or transverse indentations are less than 1%, thus demonstrating the high accuracy and uniqueness of the forward analysis. For longitudinal and transverse indentations, the reverse algorithms provide accurate results in most cases with an average error of 3 and 6%, respectively. A sensitivity analysis with a ±2% variation in the material properties in the forward algorithm and ±2% variation in the indentation responses in the reverse algorithms demonstrated the robustness of the algorithms developed in the present study, with the longitudinal indentations providing relatively less sensitivity to variability in indentation responses as compared to the transverse indentations.     

Ultra-high pressure apparatus

This review applies to static pressure apparatus capable of developing pressures over about 25 kbar for purposes of scientific measurements of the physical and chemical behavior of matter, and in some cases for the high-pressure, high-temperature, synthesis of materials like diamond and cubic boron nitride. A brief history is presented, and major emphasis is given to geometry and stress/strain analysis and the properties of materials that are useful in ultra-high pressure (UHP) apparatuses. Examples are given, and analyzed, of various kinds of UHP apparatuses which have been used extensively in actual service. Finally there is an assessment of the future possibilities for realizing pressures greater than those that have been attained to date.     

Scaling exponents for fracture surfaces in homogeneous glass and glassy ceramics

We investigate the scaling properties of postmortem fracture surfaces in silica glass and glassy ceramics. In both cases, the 2D height-height correlation function is found to obey Family-Viseck scaling properties, but with two sets of critical exponents, in particular, a roughness exponent zeta approximately 0.75 in homogeneous glass and zeta approximately 0.4 in glassy ceramics. The ranges of length scales over which these two scalings are observed are shown to be below and above the size of the process zone, respectively. A model derived from linear elastic fracture mechanics in the quasistatic approximation succeeds to reproduce the scaling exponents observed in glassy ceramics. The critical exponents observed in homogeneous glass are conjectured to reflect the damage screening occurring for length scales below the size of the process zone.     

Variational treatment of the elastic constants of disordered materials

Using the method of Hill a variational principle is derived to obtain upper and lower bounds for the effective elastic constants of disordered materials, such as polycrystals or multiphase materials. All bounds previously known are rederived and especially new bounds are given being closer than the ones of Hashin and Shtrikman. In detail the elastic constants of polycrystals built of cubic single crystals and of multiphase materials are considered. The analogous bounds for the dielectric constant of polycrystals are also given.     

Bloch oscillations in one-dimensional photonic crystal coupled microcavity composed of single-negative materials

We have studied the propagation properties of electromagnetic waves in one-dimensional photonic crystal coupled microcavity structure composed of single-negative materials. Two Wannier-Stark ladders can be obtained by employing only one type of microcavity. The dynamics behavior of a Gaussian pulse transmitting through the structure is simulated theoretically. We demonstrated for the first time that even in this multilayered structure completely made of single-negative materials, the optical Bloch oscillations and Zener tunneling can also be attained.     

Some Propagation Properties of the Iwatsuka Model

In this paper we study a two dimensional magnetic field Schr?dinger Hamiltonian introduced in [7]. This model has some interesting propagation properties, as conjectured in [2] and at the same time is a special case of the class of analytically decomposable Hamiltonians [5]. Our aim is to start from a conjugate operator, intimately related to the band structure of the Hamiltonian and to prove existence of an asymptotic velocity in one spatial direction and a theorem giving minimal and maximal velocity bounds for the propagation associated to the Hamiltonian. A simple example of this model, with a very simple conjugate operator, has been given in [9]. At the same time, by using the Virial Theorem, we obtain a generalisation of the hypothesis in [7]. Received: 12 February 1997 / Accepted: 26 February 1997     

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps

Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure μ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold. In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.     

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λ_k}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η_j are valid. The constants η_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λ_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function. Mathematics Subject Classification (2000) 11M26.     

Painlevé property and integrability

For an n degree of freedom hyperelliptic separable hamiltonian, the pole series with n+1 free constants, through the Hamilton-Jacobi equation, bounds the degrees of the n-polynomials in involution. When all the pole series have no fewer than 2n constants, the phase space is conjectured to be just the direct product of 2n complex lines cut out by (2n−1) integrals.