Variational Bayes and the reduced dependence approximation for the autologistic model on an irregular grid with applications

Discrete Markov random field models provide a natural framework for representing images or spatial datasets. They model the spatial association present while providing a convenient Markovian dependency structure and strong edge-preservation properties. However, parameter estimation for discrete Markov random field models is difficult due to the complex form of the associated normalizing constant for the likelihood function. For large lattices, the reduced dependence approximation to the normalizing constant is based on the concept of performing computationally efficient and feasible forward recursions on smaller sublattices which are then suitably combined to estimate the constant for the whole lattice. We present an efficient computational extension of the forward recursion approach for the autologistic model to lattices that have an irregularly shaped boundary and which may contain regions with no data; these lattices are typical in applications. Consequently, we also extend the reduced dependence approximation to these scenarios enabling us to implement a practical and efficient non-simulation based approach for spatial data analysis within the variational Bayesian framework. The methodology is illustrated through application to simulated data and example images. The supplemental materials include our C++ source code for computing the approximate normalizing constant and simulation studies.

The Galerkin finite element approximation of the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. Cable equations with a fractional order temporal derivative have been introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, the fractional cable equation involving two integro-differential operators is considered. The Galerkin finite element approximations of the fractional cable equation are proposed. The main contribution of this work is outlined as follow: • A semi-discrete finite difference approximation in time is proposed. We prove that the scheme is unconditionally stable, and the numerical solution converges to the exact solution with order O(Δt). • A semi-discrete difference scheme for improving the order of convergence for solving the fractional cable equation is proposed, and the numerical solution converges to the exact solution with order O((Δt)2). • Based on the above semi-discrete difference approximations, Galerkin finite element approximations in space for a full discretization are also investigated. • Finally, some numerical results are given to demonstrate the theoretical analysis.

Composition-dependent structural and electronic properties of α-(Si1−xCx)3N4

The highly unusual structural and electronic properties of the α-phase of (Si1-xCx)3N4 are determined by density functional theory (DFT) calculations using the Generalized Gradient Approximation (GGA). The electronic properties of α-(Si 1-xCx)3N4 are found to be very close to those of α-C3N4. The bandgap of α-(Si 1-xCx)3N4 significantly decreases as C atoms are substituted by Si atoms (in most cases, smaller than that of either α-Si3N4 or α-C3N4) and attains a minimum when the ratio of C to Si is close to 2. On the other hand, the bulk modulus of α-(Si1-xCx)3N 4 is found to be closer to that of α-Si3N 4 than of α-C3N4. Plasma-assisted synthesis experiments of CNx and SiCN films are performed to verify the accuracy of the DFT calculations. TEM measurements confirm the calculated lattice constants, and FT-IR/XPS analysis confirms the formation and lengths of C-N and Si-N bonds. The results of DFT calculations are also in a remarkable agreement with the experiments of other authors.

Effect of doping with Co and/or Cu on electronic structure and optical properties of ZnO

This paper reports on ab initio numerical simulations of the effect of Co and Cu dopings on the electronic structure and optical properties of ZnO, pursued to develop diluted magnetic semiconductors vitally needed for spintronic applications. The simulations are based upon the Perdew-Burke-Enzerh generalized gradient approximation on the density functional theory. It is revealed that the electrons with energies close to the Fermi level effectively transfer only between Cu and Co ions which substitute Zn atoms, and are located in the neighbor sites connected by an O ion. The simulation results are consistent with the experimental observations that addition of Cu helps achieve stable ferromagnetism of Co-doped ZnO. It is shown that simultaneous insertion of Co and Cu atoms leads to smaller energy band gap, redshift of the optical absorption edge, as well as significant changes in the reflectivity, dielectric function, refractive index, and electron energy loss function of ZnO as compared to the doping with either Co or Cu atoms. These highly unusual optical properties are explained in terms of the computed electronic structure and are promising for the development of the next-generation room-temperature ferromagnetic semiconductors for future spintronic devices on the existing semiconductor micromanufacturing platform.

Single layer lead iodide: Computational exploration of structural, electronic and optical properties, strain induced band modulation and the role of spin-orbital-coupling

Graphitic like layered materials exhibit intriguing electronic structures and thus the search for new types of two-dimensional (2D) monolayer materials is of great interest for developing novel nano-devices. By using density functional theory (DFT) method, here we for the first time investigate the structure, stability, electronic and optical properties of monolayer lead iodide (PbI2). The stability of PbI2 monolayer is first confirmed by phonon dispersion calculation. Compared to the calculation using generalized gradient approximation, screened hybrid functional and spin–orbit coupling effects can not only predicts an accurate bandgap (2.63 eV), but also the correct position of valence and conduction band edges. The biaxial strain can tune its bandgap size in a wide range from 1 eV to 3 eV, which can be understood by the strain induced uniformly change of electric field between Pb and I atomic layer. The calculated imaginary part of the dielectric function of 2D graphene/PbI2 van der Waals type hetero-structure shows significant red shift of absorption edge compared to that of a pure monolayer PbI2. Our findings highlight a new interesting 2D material with potential applications in nanoelectronics and optoelectronics.

Bias reduction using stochastic approximation

The paper studies stochastic approximation as a technique for bias reduction. The proposed method does not require approximating the bias explicitly, nor does it rely on having independent identically distributed (i.i.d.) data. The method always removes the leading bias term, under very mild conditions, as long as auxiliary samples from distributions with given parameters are available. Expectation and variance of the bias-corrected estimate are given. Examples in sequential clinical trials (non-i.i.d. case), curved exponential models (i.i.d. case) and length-biased sampling (where the estimates are inconsistent) are used to illustrate the applications of the proposed method and its small sample properties.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Study on microwave dielectric properties of epoxy resin mixtures used for rapid product development

This paper presents a preliminary study on the dielectric properties and curing of three different types of epoxy resins mixed at various stichiometric mixture of hardener, flydust and aluminium powder under microwave energy. In this work, the curing process of thin layers of epoxy resins using microwave radiation was investigated as an alternative technique that can be implemented to develop a new rapid product development technique. In this study it was observed that the curing time and temperature were a function of the percentage of hardener and fillers presence in the epoxy resins. Initially dielectric properties of epoxy resins with hardener were measured which was directly correlated to the curing process in order to understand the properties of cured specimen. Tensile tests were conducted on the three different types of epoxy resins with hardener and fillers. Modifying dielectric properties of the mixtures a significant decrease in curing time was observed. In order to study the microstructural changes of cured specimen the morphology of the fracture surface was carried out by using scanning electron microscopy.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.