Search NASA⌕ Search

SEARCH · Search NASA

Results for “Basis function”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 577 records · Page 32
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Fourier Analysis and Design of a Shielded 120kW Inductive Wireless System

High-power inductive wireless power transfer (WPT) systems for EVs are designed to meet specifications such as stray field, power level, efficiency, misalignment tolerance, and ground clearance. These metrics are all heavily influenced by the coil geometry. Herein this paper proposes a coil design method based on the Fourier Analysis Method (FAM) which is an analytical method for directly designing coil geometries to meet stray field and power level requirements through an optimization of Fourier basis function coefficients. In this work, two 120 kW WPT proof-of-concept demonstrators with low stray field and high efficiency are built from FAM optimization results to validate the models and show the impact of the FAM design process. Experimental validation of the Gen. 2 demonstrator at 120 kW output power resulted in a measured DC/DC efficiency of 97.2% at alignment with a 125 mm airgap. At the 120 kW test point, the stray fields 80 cm away from the center of the airgap between the coil assemblies were 3.4 µT(rms) on the X-axis and 3.5 µT(rms) on the Y-axis, much lower than the 27 µT(rms) ICNIRP limit.

42 ENGINEERING↗
DOE OSTI2024

Fourier Analysis and Loss Modeling for Inductive Wireless Electric Vehicle Charging with Reduced Stray Field

With the growth of electric vehicle (EV) popularity, different charging options to increase user convenience and reduce charging times are being considered and researched. Among these, inductive wireless power transfer (WPT) systems for EVs are being designed to meet specifications such as stray field, power level, efficiency, misalignment tolerance, and ground clearance, which are all heavily influenced by the coil geometry. The proposed Fourier Analysis Method (FAM) is an analytical method to directly design coil geometries to meet stray field and power level requirements through an optimization of Fourier basis function coefficients. The outputs of the optimization are complex, planar coil geometries that meet the power level and stray field constraints with minimized loss factors. Contours of these potentials determine the coil conductor paths and loss models predict the system efficiency and performance over misalignment. The Fourier representation of the geometry is used to conveniently calculate the coupling over misalignment, external proximity effect loss, and ferrite loss. A 6.6 kW prototype WPT system with low stray field and high efficiency is built from the optimization results to validate the models and showcase the usefulness of the FAM design approach.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2018

Control of Three Degrees-of-Freedom Wave Energy Converters Using Pseudo-Spectral Methods

This study presents a solution to the optimal control problem of a three degrees-of-freedom (3DOF) wave energy converter (WEC). The three modes are the heave, pitch, and surge. The dynamic model is characterized by a coupling between the pitch and surge modes, while the heave is decoupled. The heave, however, excites the pitch motion through nonlinear parametric excitation in the pitch mode. This paper uses Fourier series (FS) as basis functions to approximate the states and the control. A simplified model is first used where the parametric excitation term is neglected and a closed-form solution for the optimal control is developed. For the parametrically excited case, a sequential quadratic programming approach is implemented to solve for the optimal control numerically. Numerical results show that the harvested energy from three modes is greater than three times the harvested energy from the heave mode alone. Moreover, the harvested energy using a control that accounts for the parametric excitation is significantly higher than the energy harvested when neglecting this nonlinear parametric excitation term.

16 TIDAL AND WAVE POWER↗
DOE OSTI2018

Parameterization of Large Variability Using the Hyper-Dual Meta-Model

One major problem in the design of aerospace components is the nonlinear changes in the response due to a change in the geometry and material properties. Many of these components have small nominal values and any change can lead to a large variability. In order to characterize this large variability, traditional methods require either many simulation runs or the calculations of many higher order derivatives. Each of these paths requires a large amount of computational power to evaluate the response curve. In order to perform uncertainty quantification analysis, even more simulation runs are required. The hyper-dual meta-model is introduced and used to characterize the response curve with the use of basis functions. The information of the response is generated with the utilization of the hyper-dual step to determine the sensitivities at a few number of simulation runs to greatly enrich the response space. This study shows the accuracy of this method for two different systems with parameterizations at different stages in the design analysis.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

General-Surrogate Adaptive Sampling Using Interquartile Range for Design Space Exploration

Abstract A surrogate model is a common tool to approximate system response at untested points for design space exploration. Adaptive sampling has been studied for improving the accuracy of surrogates iteratively by introducing additional samples (simulations and experiments). New samples are often selected based on the estimated uncertainty in the design space. While some surrogates such as kriging have readily available uncertainty models for their predictions, other surrogates do not. Consequently, there have been studies of using the process of leaving-samples-out (LSO) used in cross-validation tools to estimate prediction uncertainty, such as the universal prediction distribution (UPD). In this paper, an adaptive sampling scheme for general surrogates is proposed based on LSO, similar to cross-validation and interquartile range (IQR). Multiple submodels are first developed from LSO. The uncertainty is then estimated from the IQR of these surrogates at a given point. New samples are added iteratively at the point with maximum IQR for design space exploration. The proposed scheme is illustrated using kriging, radial basis function, and neural network surrogates. The proposed scheme is evaluated using four algebraic test functions. Multiple sets of initial samples were produced to account for randomness. For these test functions, the proposed scheme was found to be more accurate and robust than kriging with its own uncertainty model. The proposed scheme was more accurate than the UPD for three out of the four test functions. For a fixed number of samples, the IQR-based adaptive sampling also proved to be more accurate than all-at-once sampling in most cases even when the estimated uncertainty was only mildly correlated with prediction errors.

Engineering↗
DOE OSTI2020

A Novel Active Optimization Approach for Rapid and Efficient Design Space Exploration Using Ensemble Machine Learning

In this work, a novel design optimization technique based on active learning, which involves dynamic exploration and exploitation of the design space of interest using an ensemble of machine learning algorithms, is presented. In this approach, a hybrid methodology incorporating an explorative weak learner (regularized basis function model) that fits high-level information about the response surface and an exploitative strong learner (based on committee machine) that fits finer details around promising regions identified by the weak learner is employed. For each design iteration, an aristocratic approach is used to select a set of nominees, where points that meet a threshold merit value as predicted by the weak learner are selected for evaluation. In addition to these points, the global optimum as predicted by the strong learner is also evaluated to enable rapid convergence to the actual global optimum once the most promising region has been identified by the optimizer. Additionally, this methodology is first tested by applying it to the optimization of a two-dimensional multi-modal surface and, subsequently, to a complex internal combustion (IC) engine combustion optimization case with nine control parameters related to fuel injection, initial thermodynamic conditions, and in-cylinder flow. It is found that the new approach significantly lowers the number of function evaluations that are needed to reach the optimum design configuration (by up to 80%) when compared to conventional optimization techniques, such as particle swarm and genetic algorithm-based optimization techniques.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

A novel active optimization approach for rapid and efficient design space exploration using ensemble machine learning

In this work, a novel design optimization technique based on active learning, which involves dynamic exploration and exploitation of the design space of interest using an ensemble of machine learning algorithms, is presented. In this approach, a hybrid methodology incorporating an explorative weak learner (regularized basis function model) which fits high-level information about the response surface, and an exploitative strong learner (based on committee machine) that fits finer details around promising regions identified by the weak learner, is employed. For each design iteration, an aristocratic approach is used to select a set of nominees, where points that meet a threshold merit value as predicted by the weak learner are selected to be evaluated using expensive function evaluation. In addition to these points, the global optimum as predicted by the strong learner is also evaluated to enable rapid convergence to the actual global optimum once the most promising region has been identified by the optimizer. This methodology is first tested by applying it to the optimization of a two-dimensional multi-modal surface. The performance of the new active learning approach is compared with traditional global optimization methods, namely micro-genetic algorithm (pGA) and particle swarm optimization (PSO). It is demonstrated that the new optimizer is able to reach the global optimum much faster with a significantly fewer number of function evaluations. Subsequently, the new optimizer is also applied to a complex internal combustion (IC) engine combustion optimization case with nine control parameters related to fuel injection, initial thermodynamic conditions, and in-cylinder flow. It is again found that the new approach significantly lowers the number of function evaluations that are needed to reach the optimum design configuration (by up to 80%) when compared to particle swarm and genetic algorithm-based optimization techniques.

Machine learning, active learning, design optimiza↗
DOE OSTI2021

Hybrid Improved Empirical Mode Decomposition and Artificial Neural Network Model for the Prediction of Critical Heat Flux (CHF)

Three Hybrid artificial neural network (ANN) models namely radial basis function (RBF), generalized regression neural networks (GRNN), and multi-layer perceptron (MLP) combined with empirical mode decomposition (EMD) are developed for CHF predictive modelling using CHF experimental databases.First, the original experimental inputs data series are decomposed into several intrinsic mode functions (IMFs) and one residual by EMD, whose components are divided into high, medium and low components. The performance parameters of the hybrid models indicates that the root mean square error (RMSE) are 0.8831, 0.6522, and 0.4149; the mean absolute error (MAE) are 0.6697, 0.4636, and 0.1935. The values of the R-square of the developed prediction approach utilizing EMD-RBF, EMD-GRNN, and EMD-MLP models are 0.8553, 0.9302, and 0.9818, and the index of agreement are 0.9464, 0.9700, and 0.9894., The value of the R-square and the index of agreement of the proposed models are much higher than those of the simple models .The Pearson's test results show that the association strength between the measured and the predicted values of the proposed model EMD-MLP is the strongest. These results show the following: (a) compared with other related, recent studies, the prediction accuracy of the hybrid model EMD- MLP proposed in this research is the best hybrid model; (b) the proposed hybrid model (EMD-MLP) attains superior performance compared with simple models.

Djeddou, Messaoud↗
DOE OSTI2012

Anatomy structure creation and editing using 3D implicit surfaces

Purpose: To accurately reconstruct, and interactively reshape 3D anatomy structures' surfaces using small numbers of 2D contours drawn in the most visually informative views of 3D imagery. The innovation of this program is that the number of 2D contours can be very much smaller than the number of transverse sections, even for anatomy structures spanning many sections. This program can edit 3D structures from prior segmentations, including those from autosegmentation programs. The reconstruction and surface editing works with any image modality. Methods: Structures are represented by variational implicit surfaces defined by weighted sums of radial basis functions (RBFs). Such surfaces are smooth, continuous, and closed and can be reconstructed with RBFs optimally located to efficiently capture shape in any combination of transverse (T), sagittal (S), and coronal (C) views. The accuracy of implicit surface reconstructions was measured by comparisons with the corresponding expert-contoured surfaces in 103 prostate cancer radiotherapy plans. Editing a pre-existing surface is done by overdrawing its profiles in image views spanning the affected part of the structure, deleting an appropriate set of prior RBFs, and merging the remainder with the new edit contour RBFs. Two methods were devised to identify RBFs to be deleted based only on the geometry of the initial surface and the locations of the new RBFs. Results: Expert-contoured surfaces were compared with implicit surfaces reconstructed from them over varying numbers and combinations of T/S/C planes. Studies revealed that surface-surface agreement increases monotonically with increasing RBF-sample density, and that the rate of increase declines over the same range. These trends were observed for all surface agreement metrics and for all the organs studied--prostate, bladder, and rectum. In addition, S and C contours may convey more shape information than T views for CT studies in which the axial slice thickness is greater than the pixel size. Surface editing accuracy likewise improves with larger sampling densities, and the rate of improvement similarly declines over the same conditions. Conclusions: Implicit surfaces based on RBFs are accurate representations of anatomic structures and can be interactively generated or modified to correct segmentation errors. The number of input contours is typically smaller than the number of T contours spanned by the structure.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2012

Temporal and spectral imaging with micro-CT

Purpose: Micro-CT is widely used for small animal imaging in preclinical studies of cardiopulmonary disease, but further development is needed to improve spatial resolution, temporal resolution, and material contrast. We present a technique for visualizing the changing distribution of iodine in the cardiac cycle with dual source micro-CT. Methods: The approach entails a retrospectively gated dual energy scan with optimized filters and voltages, and a series of computational operations to reconstruct the data. Projection interpolation and five-dimensional bilateral filtration (three spatial dimensions + time + energy) are used to reduce noise and artifacts associated with retrospective gating. We reconstruct separate volumes corresponding to different cardiac phases and apply a linear transformation to decompose these volumes into components representing concentrations of water and iodine. Since the resulting material images are still compromised by noise, we improve their quality in an iterative process that minimizes the discrepancy between the original acquired projections and the projections predicted by the reconstructed volumes. The values in the voxels of each of the reconstructed volumes represent the coefficients of linear combinations of basis functions over time and energy. We have implemented the reconstruction algorithm on a graphics processing unit (GPU) with CUDA. We tested the utility of the technique in simulations and applied the technique in an in vivo scan of a C57BL/6 mouse injected with blood pool contrast agent at a dose of 0.01 ml/g body weight. Postreconstruction, at each cardiac phase in the iodine images, we segmented the left ventricle and computed its volume. Using the maximum and minimum volumes in the left ventricle, we calculated the stroke volume, the ejection fraction, and the cardiac output. Results: Our proposed method produces five-dimensional volumetric images that distinguish different materials at different points in time, and can be used to segment regions containing iodinated blood and compute measures of cardiac function. Conclusions: We believe this combined spectral and temporal imaging technique will be useful for future studies of cardiopulmonary disease in small animals.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2014

WE-E-18C-01: Multi-Energy CT: Current Status and Recent Innovations

Conventional computed tomography (CT) uses a single polychromatic x-ray spectrum and energy integrating detectors, and produces images whose contrast depends on the effective attenuation coefficient of the broad spectrum beam. This can introduce errors from beam hardening and does not produce the optimal contrast-to-noise ratio. In addition, multiple materials can have the same effective attenuation coefficient, causing different materials to be indistinguishable in conventional CT images. If transmission measurements at two or more energies are obtained, even with polychromatic beams, more specific information about the object can be obtained. If the object does not contain materials with k-edges in the spectrum, the x-ray attenuation can be well-approximated by a linear combination of two processes (photoelectric absorption and Compton scattering) or, equivalently, two basis materials. For such cases, two spectral measurements suffice, although additional measurements can provide higher precision. If K-edge materials are present, additional spectral measurements can allow these materials to be isolated. Current commercial implementations use varied approaches, including two sources operating a different kVp, one source whose kVp is rapidly switched in a single scan, and a dual layer detector that can provide spectral information in every reading. Processing of the spectral information can be performed in the raw data domain or in the image domain. The process of calculating the amount of the two basis functions implicitly corrects for beam hardening and therefore can lead to improvements in quantitative accuracy. Information can be extracted to provide material specific information beyond that of conventional CT. This additional information has been shown to be important in several clinical applications, and can also lead to more efficient clinical protocols. Recent innovations in x-ray sources, detectors, and systems have made multi-energy CT much more practical and improved its performance. In addition, this is a very active area of research and further improvements are expected through further technological improvements. Learning Objectives: Basic principles of multi-energy CT Current implementations of mutli-energy CT Data and image analysis methods in multi-energy CT Current clinical applications of dual energy CT5. recent innovations and anticipated advances in multi-energy CT.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2015

MO-F-CAMPUS-J-02: Automatic Recognition of Patient Treatment Site in Portal Images Using Machine Learning

Purpose: To investigate the method to automatically recognize the treatment site in the X-Ray portal images. It could be useful to detect potential treatment errors, and to provide guidance to sequential tasks, e.g. automatically verify the patient daily setup. Methods: The portal images were exported from MOSAIQ as DICOM files, and were 1) processed with a threshold based intensity transformation algorithm to enhance contrast, and 2) where then down-sampled (from 1024×768 to 128×96) by using bi-cubic interpolation algorithm. An appearance-based vector space model (VSM) was used to rearrange the images into vectors. A principal component analysis (PCA) method was used to reduce the vector dimensions. A multi-class support vector machine (SVM), with radial basis function kernel, was used to build the treatment site recognition models. These models were then used to recognize the treatment sites in the portal image. Portal images of 120 patients were included in the study. The images were selected to cover six treatment sites: brain, head and neck, breast, lung, abdomen and pelvis. Each site had images of the twenty patients. Cross-validation experiments were performed to evaluate the performance. Results: MATLAB image processing Toolbox and scikit-learn (a machine learning library in python) were used to implement the proposed method. The average accuracies using the AP and RT images separately were 95% and 94% respectively. The average accuracy using AP and RT images together was 98%. Computation time was ∼0.16 seconds per patient with AP or RT image, ∼0.33 seconds per patient with both of AP and RT images. Conclusion: The proposed method of treatment site recognition is efficient and accurate. It is not sensitive to the differences of image intensity, size and positions of patients in the portal images. It could be useful for the patient safety assurance. The work was partially supported by a research grant from Varian Medical System.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Improving lung cancer prognosis assessment by incorporating synthetic minority oversampling technique and score fusion method

Purpose: This study aims to investigate the potential to improve lung cancer recurrence risk prediction performance for stage I NSCLS patients by integrating oversampling, feature selection, and score fusion techniques and develop an optimal prediction model. Methods: A dataset involving 94 early stage lung cancer patients was retrospectively assembled, which includes CT images, nine clinical and biological (CB) markers, and outcome of 3-yr disease-free survival (DFS) after surgery. Among the 94 patients, 74 remained DFS and 20 had cancer recurrence. Applying a computer-aided detection scheme, tumors were segmented from the CT images and 35 quantitative image (QI) features were initially computed. Two normalized Gaussian radial basis function network (RBFN) based classifiers were built based on QI features and CB markers separately. To improve prediction performance, the authors applied a synthetic minority oversampling technique (SMOTE) and a BestFirst based feature selection method to optimize the classifiers and also tested fusion methods to combine QI and CB based prediction results. Results: Using a leave-one-case-out cross-validation (K-fold cross-validation) method, the computed areas under a receiver operating characteristic curve (AUCs) were 0.716 ± 0.071 and 0.642 ± 0.061, when using the QI and CB based classifiers, respectively. By fusion of the scores generated by the two classifiers, AUC significantly increased to 0.859 ± 0.052 (p < 0.05) with an overall prediction accuracy of 89.4%. Conclusions: This study demonstrated the feasibility of improving prediction performance by integrating SMOTE, feature selection, and score fusion techniques. Combining QI features and CB markers and performing SMOTE prior to feature selection in classifier training enabled RBFN based classifier to yield improved prediction accuracy.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-F-R-17: Advancing Glioblastoma Multiforme (GBM) Recurrence Detection with MRI Image Texture Feature Extraction and Machine Learning

Purpose: To test the potential of early Glioblastoma Multiforme (GBM) recurrence detection utilizing image texture pattern analysis in serial MR images post primary treatment intervention. Methods: MR image-sets of six time points prior to the confirmed recurrence diagnosis of a GBM patient were included in this study, with each time point containing T1 pre-contrast, T1 post-contrast, T2-Flair, and T2-TSE images. Eight Gray-level co-occurrence matrix (GLCM) texture features including Contrast, Correlation, Dissimilarity, Energy, Entropy, Homogeneity, Sum-Average, and Variance were calculated from all images, resulting in a total of 32 features at each time point. A confirmed recurrent volume was contoured, along with an adjacent non-recurrent region-of-interest (ROI) and both volumes were propagated to all prior time points via deformable image registration. A support vector machine (SVM) with radial-basis-function kernels was trained on the latest time point prior to the confirmed recurrence to construct a model for recurrence classification. The SVM model was then applied to all prior time points and the volumes classified as recurrence were obtained. Results: An increase in classified volume was observed over time as expected. The size of classified recurrence maintained at a stable level of approximately 0.1 cm{sup 3} up to 272 days prior to confirmation. Noticeable volume increase to 0.44 cm{sup 3} was demonstrated at 96 days prior, followed by significant increase to 1.57 cm{sup 3} at 42 days prior. Visualization of the classified volume shows the merging of recurrence-susceptible region as the volume change became noticeable. Conclusion: Image texture pattern analysis in serial MR images appears to be sensitive to detecting the recurrent GBM a long time before the recurrence is confirmed by a radiologist. The early detection may improve the efficacy of targeted intervention including radiosurgery. More patient cases will be included to create a generalizable classification model applicable to a larger patient cohort. NIH R43CA183390 and R01CA188300.NSF Graduate Research Fellowship DGE-1144087.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-F-I-41: Calibration-Free Material Decomposition for Dual-Energy CT

Purpose: To eliminate tedious phantom calibration or manually region of interest (ROI) selection as required in dual-energy CT material decomposition, we establish a new projection-domain material decomposition framework with incorporation of energy spectrum. Methods: Similar to the case of dual-energy CT, the integral of the basis material image in our model is expressed as a linear combination of basis functions, which are the polynomials of high- and low-energy raw projection data. To yield the unknown coefficients of the linear combination, the proposed algorithm minimizes the quadratic error between the high- and low-energy raw projection data and the projection calculated using material images. We evaluate the algorithm with an iodine concentration numerical phantom at different dose and iodine concentration levels. The x-ray energy spectra of the high and low energy are estimated using an indirect transmission method. The derived monochromatic images are compared with the high- and low-energy CT images to demonstrate beam hardening artifacts reduction. Quantitative results were measured and compared to the true values. Results: The differences between the true density value used for simulation and that were obtained from the monochromatic images, are 1.8%, 1.3%, 2.3%, and 2.9% for the dose levels from standard dose to 1/8 dose, and are 0.4%, 0.7%, 1.5%, and 1.8% for the four iodine concentration levels from 6 mg/mL to 24 mg/mL. For all of the cases, beam hardening artifacts, especially streaks shown between dense inserts, are almost completely removed in the monochromatic images. Conclusion: The proposed algorithm provides an effective way to yield material images and artifacts-free monochromatic images at different dose levels without the need for phantom calibration or ROI selection. Furthermore, the approach also yields accurate results when the concentration of the iodine concentrate insert is very low, suggesting the algorithm is robust with respect to the low-contrast scenario.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2015

Effect of the shape on the spin state and exchange in quantum dots. Feynman path integral analysis

The ab initio computer simulation of the mixed quantum states of 1–5-nm model ellipsoid quantum dots with “soft” walls containing two and three quantum-indistinguishable nonrelativistic electrons has been performed by the path integral method. The calculation has been carried out beyond the single-electron and mean-field approximations with the fundamentally exact inclusion of Coulomb and exchange correlations of all orders and the spin variable. Distributions over the eigenfunctions of the spin-squared operator, as well as the equilibrium spin numbers, have been obtained depending on the shape of a quantum dot and the temperature. The complete set of basis functions symmetrized in permutations according to the spin of the system has been obtained by application of the Young symmetry operators. The dependence of the energy on the shape of the quantum dot corresponds to the negative sign of the surface tension at its boundary. The calculation indicates that the spin magnetic susceptibility in the system of two electrons decreases strongly for spherical quantum dots (“pairing” of spins) and the temperature dependences have a pronounced maximum whose position depends on the shape of the quantum dot. For three electrons in an oblate quantum dot, the inversion of the energy levels of spin states is observed and affects the spin magnetic susceptibility. The results indicate a strong dependence of the energy of collective spin states of electrons on the detailed inclusion of exchange and Coulomb spatial correlations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Magnetic and ferroelectric ordering in BiMn{sub 2}O{sub 5} oxide

A group-theoretical analysis of the magnetic phase of BiMn{sub 2}O{sub 5} oxide is performed using the space symmetry group of the compound. Using the projection operator method, we determine the basis functions of the irreducible representation of the space group, which are expressed in terms of the magnetic vector components. This representation can govern two phase transitions from the paramagnetic state to the antiferromagnetic phase with close temperatures and ordering of the spins of manganese ions in two crystallographic positions. It is found from renorm group analysis of these transitions that when these transitions occur as second- order transitions, the electric polarization does not appear in the system because spin fluctuations in this case elevate the symmetry of the system. Polarization appears when at least one of these transitions becomes a first-order transition as a result of spin fluctuations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2011

Two-dimensional imaginary lobachevsky space. Separation of variables and contractions

The Inoenue-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗