Substitution Delone Sets

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems reformulate

Complexity Results for MCMC derived from Quantitative Bounds

This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization approach, which establishes a generalized drift condition defined in a subset of the state space. The subset is called the ``large set'' and is chosen to rule out some ``bad'' states which have poor drift property when the dimension gets large. Using the ``large set'' together with a ``centered'' drift function, a quantitative bound can be obtained which can be translated into a tight complexity bound. As a demonstration, we analyze a certain realistic Gibbs sampler algorithm and obtain a complexity upper bound for the mixing time, which shows that the number of iterations required for the Gibbs sampler to converge is constant under certain conditions on the observed data and the initial state. It is our hope that this modified drift-and-minorization approach can be employed in many other specific examples to obtain complexity bounds for high-dimensional Markov chains.Comment: 42 page

The CR Paneitz Operator and the Stability of CR Pluriharmonic Functions

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in $\mathbb{C}^2$ are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.Comment: 11 pages; final versio

Gradual Spread of Market-Led Industrialization

The paper introduces asymmetric production conditions between firms and asymmetric transaction conditions between countries into the Murphy-Shleifer-Vishny model of industrialization. It explores a general equilibrium mechanism that generates circular causation loop that each firm's profitability and its decision of involvement in a network of industrial linkages and trade flows is determined by the size of the network, while the network size is in turn determined by all firms' decisions of participation. It shows that the very function of the market is networking relevant self-interested decision makers and utilize the network effects of industrialization, though this function is not perfect. Hence, market led industrialization will gradually spread until the whole world economy is integrated in a single network of trade and industrial linkages as transaction conditions are improved. Also, this general equilibrium mechanism predicts empirical observation that temperate zone is involved in this industrialization process more early than the tropic zone because of its better climate and public health conditions. This paper devises a new approach to specifying zero profit condition for a marginal modern firm, while keeping original feedback loop between positive profit and the extent of the market of the MSV model. Hence, this new method and the trade off between economies of scale and transaction costs can be used to endogenize the number of modern sectors and increases applicability of this type of models which is featured with compatibility between economies of scale and competitive market.globalization, industrialization, market-led development

Extremal metrics for the Q′{Q}^\prime-curvature in three dimensions

We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.Comment: Final version; Corrects minor typos; This is an announcement of the main results of arXiv:1511.05013; 5 page

Effective thermal conductivity of polycrystalline materials with randomly oriented superlattice grains

A model has been established for the effective thermal conductivity of a bulk polycrystal made of randomly oriented superlattice grains with anisotropic thermal conductivity. The in-plane and cross-plane thermal conductivities of each superlattice grain are combined using an analytical averaging rule that is verified using finite element methods. The superlattice conductivities are calculated using frequency dependent solutions of the Boltzmann transport equation, which capture greater thermal conductivity reductions as compared to the simpler gray medium approximation. The model is applied to a PbTe/Sb_2Te_3 nanobulk material to investigate the effects of period, specularity, and temperature. The calculations show that the effective thermal conductivity of the polycrystal is most sensitive to the in-plane conductivity of each superlattice grain, which is generally four to five times larger than the cross-plane conductivity of a grain. The model is compared to experimental measurements of the same system for periods ranging from 287 to 1590 nm and temperatures from 300 to 500 K. The comparison suggests that the effective specularity increases with increasing annealing temperature and shows that these samples are in a mixed regime where both Umklapp and boundary scattering are important