Conformal invariants for determinants of laplacians on Riemann surfaces

For a Riemann surface with smooth boundaries, conformal (Weyl) invariant quantities proportional to the determinant of the scalar Laplacian operator are constructed both for Dirichlet and Neumann boundary conditions. The determinants are defined by zeta function regularization. The other quantities in the invariants are determined from metric properties of the surface. As applications explicit representations for the determinants on the flat disk and the flat annulus are derived.     

The interaction of water with solid surfaces: Fundamental aspects

The purpose of this review is to compare and discuss recent experimental and theoretical results in the field of H₂O-solid interactions. We emphasize studies of low (submonolayer) coverages of water on well-characterized, single-crystal surfaces of metals, semiconductors and oxides. We discuss the factors which influence dissociative versus associative adsorption pathways. When H₂O adsorbs molecularly, it tends to form three-dimensional hydrogen-bonded clusters, even at fractional monolayer coverages, because the strength of the attractive interaction between two molecules is comparable to that of the substrate-H₂O bond. The template effect of the substrate is important in determining both the local orientation and long-range order of H₂O molecules in these clusters. The influence of surface additive atoms (e.g., O, Br, Na, K) and also surface imperfections (e.g. steps and defects) on the surface structure and chemistry of H₂O is examined in detail. Some results on single-crystal substrates are compared with earlier measurements of H₂O adsorption on high-area materials.     

Local Gromov–Witten invariants of blowups of Fano surfaces

In this paper, using the degeneration formula we obtain a blowup formula of local Gromov–Witten invariants of Fano surfaces. This formula makes it possible to compute the local Gromov–Witten invariants of non-toric Fano surfaces from toric Fano surfaces, such as del Pezzo surfaces. This formula also verified an expectation of Chiang–Klemm–Yau–Zaslow in Section 8.3 of Chiang et al. (1999) [7]     

Differential invariants for cubic integrals of geodesic flows on surfaces

We construct differential invariants that vanish if and only if the geodesic flow of a two-dimensional metric admits an integral of third degree in momenta with a given Birkhoff–Kolokoltsov 3-codifferential.     

Two-scale modeling of adsorption processes at structured surfaces

We present an algorithm for the simulation of vicinal surface growth. It combines a lattice gas anisotropic Ising model with a phase-field model. The molecular behavior of individual adatoms is described by the lattice gas model. The microstructure dynamics on the vicinal surface are calculated using the phase-field method. In this way, adsorption processes on two different length scales can be described: nucleation processes on the terraces (lattice gas model) and step-flow growth (phase field model). The hybrid algorithm that is proposed here, is therefore able to describe an epitaxial layer-by-layer growth controlled by temperature and by deposition rate. This method is faster than kinetic Monte Carlo simulations and can take into account the stochastic processes in a comparable way.     

Dijkgraaf–Witten invariants of surfaces and projective representations of groups

We compute the Dijkgraaf–Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf–Witten invariants of surfaces of positive genus are positive integers.     

The relativistic Riemann invariants and the propagation of initial data surfaces

A method is described by which the relativistic Riemann invariants can be found for a fluid with an arbitrary equation of state, undergoing dissipation and moving in a general metric. Specific formulae are derived for a spherically symmetric system. Limiting cases defined by relativistic and non-relativistic gases, both warm, cold, fast and slow are examined. We prove that the invariants do exist, and a necessary and sufficient condition for their determination is the solution of a differential equation with the structure of an exterior one form of two components. The common parameter of these components is the characteristic space-time direction which is also derived in the process of determining the invariants. The characteristic surfaces, being the surfaces over which initial data is carried, all coalesce to the forward light cone in the extreme relativistic limit. Relativistic fluids emanating from receding sources appear to increase their internal kinetic energy as they decelerate.A non-linear distance-velocity relation for these waves is evident in the differential equations which are found. Their full meaning remains to be explored.     

The relativistic Riemann invariants and the propagation of initial data surfaces

A method is described by which the relativistic Riemann invariants can be found for a fluid with an arbitrary equation of state, undergoing dissipation and moving in a general metric. Specific formulae are derived for a spherically symmetric system. Limiting cases defined by relativistic and non-relativistic gases, both warm, cold, fast and slow are examined. We prove that the invariants do exist, and a necessary and sufficient condition for their determination is the solution of a differential equation with the structure of an exterior one form of two components. The common parameter of these components is the characteristic space-time direction which is also derived in the process of determining the invariants. The characteristic surfaces, being the surfaces over which initial data is carried, all coalesce to the forward light cone in the extreme relativistic limit. Relativistic fluids emanating from receding sources appear to increase their internal kinetic energy as they decelerate. A non-linear distance-velocity relation for these waves is evident in the differential equations which are found. Their full meaning remains to be explored.     

Quantum Riemann surfaces: II. The discrete series

We continue our study of noncommutative deformations of two-dimensional hyperbolic manifolds which we initiated in Part I. We construct a sequence of ^*-algebras which are quantizations of a compact Riemann surface of genus g corresponding to special values of the Planck constant. These algebras are direct integrals of finite-dimensional ^*-algebras.Supported by DOE under Grant DE-FG02-88ER25065.     

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.     

Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. We show that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner. The change in sample spacing with distance reflects the structure of Fresnel diffraction. This sampling grid also provides a simple and robust basis for accurate and efficient computation, which naturally handles the challenges of sampling chirplike kernels.     

Processing optical measurements using a regressive Fourier series: A review

In this article a generalized regressive Fourier series approach is shown which does not suffer from the so-called leakage effects prone to frequency domain approaches. This parametric approach will be used to tackle a number of optical techniques. Due to the fact that the technique offers an approximation and not an estimate of the data, it is possible to reduce the spatial data of operational mobility shapes measured by e.g. a laser Doppler vibrometer (LDV). It also allows a reduction of measurement time of this former technique. The LDV will also be used to visualize and localize acoustic sources.     

Power series expansions of quantum-mechanical potential surfaces

Power series expansions of quantum-mechanical potential surfaces for H₂O and LiH₂ are generated using conventional valence displacement coordinates and alternative distance [(r-r _e)/r] and angle [(sin (θ/2) — sin (θ_e/2))/sin (θ_e/2)] variables. Power series in the new variables are demonstrated to be superior in terms of the quality of the fit to the energygeometry data, predicted equilibrium geometry and energy, and stability of expansion coefficients.     

Stability of droplets and channels on homogeneous and structured surfaces

Wetting of structured or imprinted surfaces which leads to a variety of different morphologies such as droplets, channels or thin films is studied theoretically using the general framework of surface or interface thermodynamics. The first variation of the interfacial free energy leads to the well-known Laplace equation and a generalized Young equation which involves spatially dependent interfacial tensions. Furthermore, we perform the second variation of the free energy for arbitrary surface patterns and arbitrary shape of the wetting morphology in order to derive a new and general stability criterion. The latter criterion is then applied to cylindrical segments or channels on homogeneous and structured surfaces. Received 4 August 1999     

Ince-Gaussian series representation of the two-dimensional fractional Fourier transform

We introduce the Ince-Gaussian series representation of the two-dimensional fractional Fourier transform in elliptical coordinates. A physical interpretation is provided in terms of field propagation in quadratic graded-index media whose eigenmodes in elliptical coordinates are derived for the first time to our knowledge. The kernel of the new series representation is expressed in terms of Ince-Gaussian functions. The equivalence among the Hermite-Gaussian, Laguerre-Gaussian, and Ince-Gaussian series representations is verified by establishing the relation among the three definitions.     

The string prediction models as invariants of time series in the forex market

In this paper we apply a new approach of string theory to the real financial market. The models are constructed with an idea of prediction models based on the string invariants (PMBSI). The performance of PMBSI is compared to support vector machines (SVM) and artificial neural networks (ANN) on an artificial and a financial time series. A brief overview of the results and analysis is given. The first model is based on the correlation function as invariant and the second one is an application based on the deviations from the closed string/pattern form (PMBCS). We found the difference between these two approaches. The first model cannot predict the behavior of the forex market with good efficiency in comparison with the second one which is, in addition, able to make relevant profit per year. The presented string models could be useful for portfolio creation and financial risk management in the banking sector as well as for a nonlinear statistical approach to data optimization.     

Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces

This paper describes two techniques to accelerate the Fourier fringe analysis technique. The two 2-D Fourier transforms required for the technique are usually a bottleneck for good-quality images in which unwrapping is straight forward. Automation of the filtering operation is an operation that very often requires user interaction when two peaks at similar frequencies are present in the Fourier transform. The techniques presented in this paper attempt to solve these issues. The first technique proposes a reduction in the number of pixels calculated by the Fourier transform. The second, automates the filtering operation to avoid user interaction.