Recently we discussed the Renner-Teller effect in triatomic molecules [J. Chem. Phys. 125, 094102 (2006)]. In that article the main message is that the Renner-Teller phenomenon, just like the Jahn-Teller phenomenon, is a topological effect. Now we extend this study to a tetra-atomic system, namely, the C(2)H(2) (+) ion, for which topological effects are revealed when one atom surrounds the triatom axis or when two atoms surround (at a time) the two-atom axis. The present study not only supports the findings of the previous study, in particular, the crucial role played by the topological D matrix for diabatization, but it also reveals new features which are expected to be more and more pronounced the larger the original collinear molecule. As already implied, shifting away two atoms from the collinear molecular axis does not necessarily abolish the ability of the remaining two atoms to form topological effects. Moreover, the study indicates that when the two hydrogens are shifted away, the CC axis produces two kinds of topological effects: (1) a Renner-Teller effect (characterized by a topological phase of 2pi) which is revealed when the two hydrogens surround, rigidly, this axis (as mentioned above), and (2) a Jahn-Teller effect (characterized by a topological phase of pi) which is revealed when one of the hydrogens surrounds this axis while the other hydrogen is clamped to its position.

In this article we present the first ab initio study of the conical intersections (cis) and their electronic nonadiabatic coupling terms (NACTs) for the {N,H2} system. Efforts were made to reveal the location of cis between the two lower, 1A′2 and 2A′2 states—to be designated as (1,2) cis—and the cis between the two upper, 2A′2 and 3A′2 states—to be designated as the (2,3) cis—of this system. We found that these cis are located along the collinear {NHH) arrangement. The study is carried out by analyzing two-state magnitudes such as the (1,2) and (2,3) adiabatic-to-diabatic transformation angles (known also as the mixing angles) and the corresponding topological phases (known also as the Berry phases or the Longuet-Higgins phases). In addition, a detailed three-state study is carried out. Here the emphasis is on driving the diagonal elements of the topological D matrix and analyzing situations for which the corresponding nonadiabatic coupling matrix is quantized. The reliability of two-state results is carefully examined by comparing them with corresponding outcomes derived for the three-state study. In addition we also calculated the potential-energy surfaces related to the two lower states and studied to what extent they are affected by the (1,2) ci. The results obtained in this treatment were found to be in full agreement with the NACT’s calculations.

A system of non-Hermite, non-orthogonal functions that are new for concrete applications is introduced to generate explicit and previously unknown, physically meaningful solutions for bar and plate equations. The approach allows in some cases to establish completeness of these sets of solutions. Various other separate results technically related to these basic development are presented. In the available literature there are no registered efforts to extract the two velocities from exact solutions of the equations of the processes in which the two parameters are explicitly present.

Two families of statistical measures are used for quantitative characterization of nonequilibrium patterns and their evolution. The first quantifies the disorder in labyrinthine patterns, and captures features like the domain size, defect density, variations in wave number, etc. The second class of characteristics can be used to quantify the disorder in more general nonequilibrium structures, including those observed during domain growth. The presence of distinct stages of relaxation in spatiotemporal dynamics under the Swift-Hohenberg equation is analyzed using both classes of measures.

We consider the necessity of half-off-shell transition-matrix elements (near-field) information on iterative approaches to the inverse acoustic scattering problem. We then show that, because of the manner in which half-off-shell effects enter, the recently introduced Volterra inverse series correctly predicts the first two moments of the interaction, while the Fredholm inverse series is correct only for the first moment. Finally, we demonstrate that the Volterra approach provides a method for exactly obtaining interactions that can be written as a sum of delta functions.

The disorder function formalism [Gunaratne et.al., Phys. Rev. E, {\bf 57}, 5146 (1998)]^M is used to show that pattern relaxation in an experiment on a vibrated layer of brass beads^M occurs in three distinct stages. During stage I, all lengthscales associated with ^M moments of the disorder grow at a single universal rate, given by $L(t) \sim t^{0.5}$. In stage II, pattern evolution is non-universal and includes a range of growth indices. Relaxation in the final stage is characterized by a single, non-universal index. We use analysis of patterns from the Swift-Hohenberg equation to argue that mechanisms that underlie the observed pattern evolution are linear spatio-temporal dynamics (stage I), non-linear saturation (stage II), and stochasticity (stage III)

We reconsider the Born–Oppenheimer–Huang treatment of systems of electrons and nuclei for the case of their interaction with time-dependent fields. Initially, we present a framework in which all expressions derived are formally exact since no truncations are introduced. The objective is to explore the general structure of the equations under the most unrestricted conditions, including the possibility that the electronic basis is dependent both on the nuclear coordinates and on time. We then derive an application of the theory applicable to cases of interaction with strong time-dependent fields. The method truncates the electronic basis only after the time-dependent interaction is taken into account in the electronic wave functions. This leads to theory which is similar to a Born–Oppenheimer-type truncation within the interaction picture.

We consider additional aspects of the recently derived “minimum uncertainty” (μ) wavelets. In particular, we show that they are fundamentally related to both the harmonic oscillator eigenstates and the canonical coherent states that play a fundamental role in quantum dynamics. In addition, we derive new raising and lowering operators that apply to the μ-wavelets. Finally, we explore in some detail the senses in which the μ-wavelets form complete sets that can be used in a variety of applications in quantum dynamics.

Periodic distributed approximating functionals are proposed and used to obtain a coordinate representation for the adiabatically adjusting principal axis hyperspherical coordinate kinetic energy operator. The approach is tested and accurate results for adiabatic surface functions for the reaction F+H2→HF+H are calculated and compared to those of some existing methods.

In this paper, we give an estimation for the smoothness of a new family of symmetric orthonormal wavelets arising from Hermite Distributed Approximating Functionals (HDAFs). Their corresponding associated low pass filters are symmetric with respect to the origin.

In this paper we present a non-separable multiresolution structure based on frames which is defined by radial scaling functions of the form of the Shannon scaling function. We also construct the resulting frame multiwavelets, which can be isotropic as well. Our construction can be carried out in any number of dimensions and for a great variety of dilation matrices.

in this article we present a nonseparable multiresolution structure based on frames which is defined by radial frame scaling functions. The Fourier transform of these functions is the indicator (characteristic) function of a measurable set. We also construct the resulting frame multiwavelets, which can be isotropic as well. Our construction can be carried out in any number of dimensions and for a big variety of dilation matrices.

We present a new inverse scattering series for quantum elastic scattering in three spherical dimensions. The new series, which converges absolutely, independent of the strength of the scattering interaction, results from a renormalization transformation of the Lippmann-Schwinger Fredholm integral equation to a Volterra form. A new feature of the formulation is that it does not require the determination of phase shifts and it can be applied even to integral cross-section measurements. The approach is illustrated by application to a simple example problem.

A robust and reliable implicit method is proposed for application in data interpolation. The algorithm is based on a recently developed analytic approximation method, namely the distributed approximating functionals (DAFs), which is known to have the "well-tempered" property of UNIFORMLY approximating a function and its derivatives. In comparison with the conventionally used local explicit interpolation algorithms, the implicit method achieves much more accurate interpolation results because it couples all sample values (both known and unknown) in the domain of interest using a set of simultaneous linear algebraic equations. Due to the fact that the well-tempered DAFs also are very good low-pass filters, the performance of the DAF-based implicit method is not affected very much by the high frequency noise in the input signal. As an application, the proposed algorithm is applied to signals that are corrupted with impulse noise.

The denoising characteristics for the representation of experimental data in terms of the Hermite Distributed Approximating Functionals (HDAF's) are analyzed with respect to signals corrupted with Gaussian noise. The HDAF performance is compared to both the ideal window and running averages representations of the same data. We find that the HDAF filter combines the best features of both. That is, the HDAF filter provides approximately the same noise reduction and bandwidth as the ideal filter while at the same time remaining limited in range in both the physical and Fourier spaces.

This paper deals with the design of interpolating wavelets based on a variety of Lagrange functions, combined with novel signal processing techniques for digital imaging. Halfband Lagrange wavelets, B-spline Lagrange wavelets and Gaussian Lagrange (Lagrange distributed approximating functional (DAF)) wavelets are presented as specific examples of the generalized Lagrange wavelets. Our approach combines the perceptually dependent visual group normalization (VGN) technique and a softer logic masking (SLM) method. These are utilized to rescale the wavelet coefficients, remove perceptual redundancy and obtain good visual performance for digital image processing.

An algorithm is introduced to extract an underlying image from a class of textures. It is assumed that the image is bandwidth limited and the noise is broad-band. The initial step of the algorithm extends the signal to a larger periodic image using "Distributed Approximating Functionals." The second step introduces a low-pass filter which allows the identification and elimination of the high-frequency components of the noise. The periodicity of the resulting image allows it to be Fourier filtered without aliasing. The feasibility of the algorithm is demonstrated on several noisy patterns generated in experiments and model systems. (c) 2000 American Institute of Physics.

New wavelet techniques are employed to improve the perceptual quality of images, and to enhance and detect the detail features in the region of interest (ROI). Distributed approximating functionals (DAFs) are used to construct a new class of interpolating wavelets, which enable better image processing performance. This paper is focused on previous improvements in DAF wavelet image processing. The combined perceptual techniques (such as visual group normalization and contrast nonlinear enhancement) produce natural high-quality images adapted to the human vision system. The underlying technologies significantly facilitate the creation of generic image processing and computer-aided diagnostic (CAD) systems

We design a new compactly-supported interpolating wavelet-distributed approximating functional (DAF) wavelet for biomedical signal/image processing. DAF class is a smooth, continuous interpolating function system which is symmetric and fast-decaying. DAF neural networks are designed for time varying electrocardiogram (EKG) signal filtering. The neural nets use the Hermite-DAF as the basis function and implement a 3-layer structure. DAF wavelets and the corresponding subband filters are constructed for image processing. Edge-enhancement normalization and device-adapted visual group normalization algorithms are presented which sharpen the desired image features (especially for digital mammography) without prior knowledge of the spatial characteristics of the images. We design a nonlinear multiscale gradient-stretch method for feature extraction of mammograms (such as the detection of ill-defined borders and spiculated lesions, etc). A fractal technique is introduced to characterize microcalcifications in localized regions of breast tissue. We employ a DAF wavelet-based multiscale edge detection and Dijkstra fractal technique to identify microcalcification regions, and use a stochastic thresholding method to detect the calcified spots. The combined perceptual techniques (such as regularization, visual group normalization and contrast nonlinear enhancement) produce natural high-quality images based on the human vision system. The underlying technologies significantly facilitate the creation of generic signal processing and computer-aided diagnostic (CAD) systems. The system is implemented in the JAVA language, which is cross-platform friendly and is facilitated for telemedicine application.

The quantum trajectory method (QTM) was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements (particles) moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares approach to extrapolate from the known function region into the neighboring regions. The QTM–DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region.