Variances of Surface Area Estimators Based on Pixel Configuration Counts

The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting—we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in $$O(t^2)$$ O ( t 2 ) as the lattice distance t tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically derived convergence order is optimal in general, but further improvements are possible in special cases.

Surface Wave Topography using The 4 Point FDM Simulator

The 2D topography proffers a new challenge of modeling surface waves with a 4-point finite difference (FDM) model. Topographic representation of wave propagation over a certain area will result in loss of accuracy of the numerical model. Then from this the need for appropriate modifications to reduce calculation errors. The existing approach requires value representation as an internal extrapolation solution for temporary exterior conditions. It is finally by providing boundary conditions and initial conditions in the system. However, the scheme sometimes becomes unstable for very irregular topography. 1D extrapolation along the parallel path is known to produce a simple and efficient scheme. During extrapolation, the stability of the 1D hyperbolic Schema improved by disregarding the nearest interior boundary point, which is less than half the lattice distance. Given the limited difference so that the stencils from both sides of the central evaluation point can be used as a 2D form modification if there are not enough inside points. So that in propagation space, waves that move and change according to changes in time. It will be following the wave nature of one source that travels in the x and y fields whose amplitude will change exponentially against propagation time. It is by the nature of surface wave motion.

STRUCTURE, ELECTRON AND OSCILLATORY PROPERTIES OF ZINC NITRATE AND ITS CRYSTAL HYDRATES

Within the generalized gradient aproximation of the Density Functional Theory (DFT) with the PBE exchange-correlation functional in the basis of localized atomic orbital of CRYSTAL14 program code, the study is conducted to evaluate the structural, electronic and oscillatory properties of zinc nitrate and its crystal hydrates Zn(NO3)2 • nH2O (n = 2,4,6), with its tested method using the zinc oxide. The first-principle structural study is performed at the full optimization of the lattice distance and atomic positions for the zinc nitrate in the cubic lattice and that of crystal hydrates - in monoclinic lattice. Elastic properties of the nitrate are studied and the mechanical stability is approved using the Born criteria. Electronic properties of rated structures are assessed by energetic (energy-band picture, full and partial density of states) and spatial electron distribution (electronic and deformation density, population density of atomic membranes and density of their overlapping). Crystal hydrates show the electrostatic pattern of nitrogroup interaction and water molecules, availability of localized valence bands and areas of vacant state of anion and cation origin. Oscillatory properties are studied by calculation of frequencies and intensity of IR-active normal long-wave oscillation. In crystal hydrates, the appearance of additional oscillation frequency O-H in terms of nitrate 3000 cm-1 above the IR-spectrum in water molecules and within the area 1200÷1600 cm-1 - of hybrid with nitrogroups.

Stick–slip behaviour on Au(111) with adsorption of copper and sulfate

Several transitions in the friction coefficient with increasing load are found on Au(111) in sulfuric acid electrolyte containing Cu ions when a monolayer (or submonolayer) of Cu is adsorbed. At the corresponding normal loads, a transition to double or multiple slips in stick–slip friction is observed. The stick length in this case corresponds to multiples of the lattice distance of the adsorbed sulfate, which is adsorbed in a √3 × √7 superstructure on the copper monolayer. Stick–slip behaviour for the copper monolayer as well as for 2/3 coverage can be observed at F N ≥ 15 nN. At this normal load, a change from a small to a large friction coefficient occurs. This leads to the interpretation that the tip penetrates the electrochemical double layer at this point. At the potential (or point) of zero charge (pzc), stick–slip resolution persists at all normal forces investigated.

A Monte Carlo Study of Copolymer Chain Conformations in Dilute Solutions in Good and Selective Solvents

Conformations of symmetric diblock copolymers AB in dilute solutions in good and selective solvents were studied by Monte Carlo simulations on a simple cubic lattice. Individual chain conformations were created by the self-avoiding walk algorithm. A modified thermal equilibration of the system based on the Metropolis acceptance criteria for energies of the system and the Rosenbluth weights of chain conformations was applied. Interactions of the nearest neighbours (r = l), where l is the lattice distance, and interactions for r = sqrt(2l) and r = sqrt(3l) were considered. Various structural characteristics of the whole copolymer chain and individual blocks A, B were obtained in the course of computer simulations. It was found that a moderate contraction of the worse soluble block B and a certain segregation of blocks occurs in dilute solutions in selective solvents for the block A, however neither that contraction, nor the segregation of blocks are extensive.

Conformations of Insoluble Blocks in Swollen Micellar Cores of Multimolecular Block Copolymer Micelles Studied by Monte Carlo Simulation Technique

Monte Carlo simulations of chain conformations in restricted spherical volumes with an increasing radius were performed on a tetrahedral lattice (ca 2 700 to 9 200 lattice sites) at relatively high densities of the occupied lattice sites. A simultaneous self-avoiding walk together with the equilibration algorithm similar to that of Siepmann and Frenkel were used to create the equilibrated multi-chain conformations. (a) A series of simulations was carried out for a constant average segment density, <gS> = 0.52, together with the three values of the radius of the sphere, R = 10 l, 12.5 l and 15 l (l is the lattice distance), and various numbers of chains, N ∈ <15, 86>, and chain lengths, L ∈ <31, 163>. The results give information on the system behavior and on the effects of: (i) multi-chain conformational correlations, which depend both on N and L, (ii) the L-dependent chain flexibility, and (iii) R-dependent external geometrical constraints. Another two series of data: (b) for a constant average segment density, <gS> = 0.36, a constant N = 21, and L proportional to R3, and (c) for <gS> = 0.36, L = 47 and N proportional to R3, are shown to give a supplementary detailed information on conformational behavior of individual chains. Various physical quantities (e.g. the densities of chain free ends, gF(r), or distributions of the tethered end-to-the free end distances, ρTF(rTF), etc.) were calculated in the course of computer simulations and their shapes and physical significance is discussed with respect to the changing values of N, L and R.

On anomalous vibrational spectra

In previous work on the vibrational spectrum of simple lattice models, it has been shown (Blackman 1937) that the spectrum becomes anomalous when special values are chosen for the force constants entering into the description of the model. As an example we may take the case of a square lattice of lattice distance d containing one particle per cell, in which the force constants α (for particles at a distance α) and γ (for particles at a distance d √2) were used. When γ / α tended to zero it could be shown that the spectrum changed from the two-dimensional to that of the linear chain. A similar result holds in the corresponding three-dimensional case. Although one would not expect any actual crystal to correspond to a limiting case (it would not be stable) it is conceivable that there should be crystals which approach the limiting case. In all cases which have hitherto been discussed, the anomaly is associated with the behaviour of the low-frequency end of the spectrum. This suggests that one could trace the effect to some property of the elastic continuum. An examination of the lattice mentioned above, for which the elastic continuum has two elastic constants c 11 and c 12 = c 44 , shows that the transition to the limiting case ( γ / α = 0) can be put in the form c 44 / c 11 → 0.