Random tilings of spherical 3-manifolds

A set of random tilings for the compact Euclidean 3-manifolds have been considered recently. In this paper, non-deterministic triangulations of spherical 3-manifolds based on recursive procedures are introduced. Simplicial structures for lens, prism and Poincaré spherical dodecahedron spaces are described. The configurational entropies for the random structures depend only on the information in 2D and are consistent with the Bekenstein–Hawking bound.     

Crystals and quasicrystals: A continuum model

We construct the first model of particles in the plane with completely symmetric, short range, two body interactions which has quasiperiodic, but no periodic, ground states.Supported in part by NSF Grant No. DMS-8501911     

A spiral growth model for quasicrystals in two dimensions

Spiral space filling geometrical constructions using rhombuses in two dimensions are considered as plausible mechanisms for quasicrystal growth. These models will show staircase-like features which may be observed experimentally.     

An exactly solvable model for two-dimensional non-Hermitian quasicrystals

Ultrasound focusing in three-dimensional(3 D)space is of crucial and enduring significance in a variety of biomedical and industrial applications.Conventional ultrasound focusing based on active phase array or passive geometry of bulky size is unable to realize the 3 D arbitrary focusing with subwavelength resolution.Acoustic metamaterial of complex deep-subwavelength microstructure has facilitated the advanced airborne-sound-focusing but is inevitably not applicable for underwater ultrasound,restricted by the law between the multi-modes coupling/thermal viscosity and the feature size of the structure.Here,we aim to circumvent the restriction by increasing the feature size of the metamaterial while keeping the compact overall geometry,and realize the robust subwavelength ultrasound focusing with the sparse metalens of the wavelength-scale meta-atom.We theoretically propose and demonstrate numerically and experimentally the broadband arbitrary ultrasound focusing in 3 D space.The axial and off-axis ultrasound focusing with the subwavelength resolution(FWHM<0.58λ)are achieved by the spatially sparse and compact metalens within one-octave bandwidth.With advantages of 3 D freewheeling focusing,subwavelength resolution,spatial sparsity,geometric simplicity,and broadband,the sparse metalens would offer more initiatives to advanced researches in ultrasound focusing and empower applications such as precise biomedical imaging and therapy,nondestructive evaluation,integrated and multiplexed ultrasound devices.     

An exactly solvable model for two-dimensional non-Hermitian quasicrystals

The phenomenon Anderson localization explains the metalinsulator transition in a material with the increase of disorder and its electrons’transport change from diffusive into localized.The study of the Anderson localization has been extended to many fields of physics,including the quasiperiodic or incommensurate systems.     

Piezoelectricity in quasicrystals: A group-theoretical study

Group-theoretical methods have been accepted as exact and reliable tools in studying the physical properties of crystals and quasicrystalline materials. By group representation theory, the maximum number of non-vanishing and independent second-order piezoelectric coefficients required by the seven pentagonal and two icosahedral point groups — that describe the quasicrystal symmetry groups in two and three dimensions — is determined. The schemes of non-vanishing and independent second-order piezoelectric tensor components needed by the nine point groups with five-fold rotations are identified and tabulated employing a compact notation. The results of this group-theoretical study are briefly discussed.      

A basis for the synthesis of quasicrystals

It has been established that quasicrystals with icosahedral point group symmetry occur in a rapidly solidified Mg₃₂ (Al, Zn)₄₉ alloy chosen on the basis of its equilibrium crystal structure. This alloy has a natural tendency to form icosahedral atomic clusters stabilised by size difference amongst constituent atoms. Results highlight the relationship between equilibrium crystal structure and the tendency to form quasicrystals.     

Hardness of Al-based quasicrystals evaluated via cluster-plus-glue-atom model

In this paper, the hardness of ternary Al-based quasicrystals was assessed through an application of the cluster-plus-glue-atom model. In this model, any structure is decomposed into a first-neighbour strongly bonded cluster part and second-neighbour weakly bonded glue-atom part so that the overall structural information is condensed into a local structural unit [cluster](glue atom)_x. For quasicrystals, the averaged local units are formulated as [icosahedron] TM_0,1(Transition Metal) and could be visualized as single icosahedron packing. Then, the hardness of quasicrystals was related to the rupture of weak inter-cluster bonds. Typically, theoretical hardness values of 8–9?GPa were obtained using 19 broken inter-cluster bonds, which accounts for about half of all the surface bonds of an icosahedron in the Mackay-type environment. The unit cluster formulas would act as rigid units during deformation and cracking.     

A Dugdale–Barenblatt model for a strip with a semi-infinite crack embedded in decagonal quasicrystals

The present study is to determine the solution of a strip with a semi-infinite crack embedded in decagonal quasicrystals, which transforms a physically and mathematically daunting problem. Then cohesive forces are incorporated into a plastic strip in the elastic body for nonlinear deformation. By superposing the two linear elastic fields, one is evaluated with internal loadings and the other with cohesive forces, the problem is treated in Dugdale-Barenblatt manner. A simple but yet rigorous version of the complex analysis theory is employed here, which involves conformal mapping technique. The analytical approach leads to the establishment of a few equations, which allows the exact calculation of the size of cohesive force zone and the most important physical quantity in crack theory: stress intensity factor. The analytical results of the present study may be used as the basis of fracture theory of decagonal quasicrystals.     

Local rules for quasicrystals

The relationship of local ordering and long-range order is studied for quasicrystalline tilings of plane and space. Two versions of the concept of local rules are introduced: strong and weak. Necessary conditions of the existence of strong local rules are found. They are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals. For planar quasicrystals the quadratic irrationalities play an important role. For three-dimensional quasicrystals not only quadratic but also cubic irrationalities are allowed. The existence of weak local rules is established for almost all two-dimensional quasicrystals based on quadratic irrationalities and for the three-dimensional quasicrystal having icosahedral symmetry.     

The isoperimetric problem for pinwheel tilings

In aperiodic pinwheel tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)² arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; ifP andQ are points separated by large Euclidean distanceR, then the shortest path along tile edges fromP toQ has lengthR+o(R).Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113.Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship and Texas ARP Grant 003658-037.     

Random matrix model for disordered conductors

We present a random matrix ensemble where real, positive semi-definite matrix elements, x, are log-normal distributed, exp[−log²(x)]. We show that the level density varies with energy, E, as 2/(1+E) for large E, in the unitary family, consistent with the expectation for disordered conductors. The two-level correlation function is studied for the unitary family and found to be largely of the universal form despite the fact that the level density has a non-compact support. The results are based on the method of orthogonal polynomials (the Stieltjes-Wigert polynomials here). An interesting random walk problem associated with the joint probability distribution of the ensuing ensemble is discussed and its connection with level dynamics is brought out. It is further proved that Dyson’s Coulomb gas analogy breaks down whenever the confining potential is given by a transcendental function for which there exist orthogonal polynomials.     

Weak matching rules for quasicrystals

Weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in perp-space are uniformly bounded. It is shown here that weak matching rules exist forN-fold symmetric tilings, whereN is any integer not divisible by four. The result suggests that, contrary to previous indications, quasicrystalline ground states are not confined to those symmetries for which the incommensurate ratios of wavevectors are quadratic irrationals. An explicit method of constructing weak matching rules forN-fold symmetric tilings in two dimensions is presented. It is shown that the generalization of the construction yields weak matching rules in the case of icosahedral symmetry as well.