Sum Of Gaussians Research Articles

Scaling laws and approximate expressions for the dynamic magnetic susceptibility of Brownian nanoparticles

We present simplified expressions for the out-of-phase component of the dynamic susceptibility χ″ of lognormal-sized magnetic nanoparticles under Brownian rotation. These expressions are based on transforming the general integral functions used for χ″ in the convolution of gaussian functions. χ″ can thus be expressed as a sum of gaussians with parameters directly related to those of the size distribution and to the saturation magnetization. The gaussian fit of χ″(ω) (where ω is the ac field frequency) is a simpler way to determine these structural and magnetic parameters as it avoids fitting χ″(ω) to an integral function. The expressions derived for χ″ suggest that χ″T data collapses in a ωη(T)/T scale (where T is the temperature and η the fluids viscosity), which is confirmed by numerical calculations. We also discuss the limits of validity of these approximations in real systems where both Néel and Brownian relaxation mechanisms coexist and we present further approximations for the relation of ωχ with the average volume (being ωχ the frequency at which χ″ is maximum). The ωη(T)/T scale can be used to qualitatively evaluate the dominance of the Brownian relaxation mechanism.

Output power and intracavity intensity profiles of a quasi-continuous end-pumped Nd:YVO4 self-Raman mini laser

The paper presents experimental investigation and modeling of an end-pumped quasi-continuous-wave YVO4/Nd:YVO4 mini self-Raman laser. The dependence of the Stokes output power on the pump power in the range from 3 to 17.5 W has been measured. As much as 1.76 W of an average Stokes power, corresponding to a total optical-to-optical conversion efficiency of about 10%, has been obtained. The transverse profiles of the laser (at the fundamental wavelength) and the Stokes beam intensity have been recorded at the output mirror and in the vicinity of the boundary between the pure and Nd-doped parts of the Raman crystal. These distributions have been approximated by the sum of Gaussian and super-Gaussian distributions with corresponding weights. We propose a model of such lasers that takes into account the features of intracavity self-frequency Raman conversion in lasers with highly inhomogeneous non-Gaussian spatial distributions of the pump, laser, and Stokes beam intensity in the cavity. The results of modeling are in good agreement with the experimental data.

Oligomeric Structure of Muscarinic and Adrenergic Receptors in Live Cells

We have determined the oligomeric size and configuration of fluorophore-tagged M1, M2, β1, and β2 receptors in the plasma membrane of Chinese hamster ovary (CHO) cells by examining the distribution of FRET efficiencies measured at the level of single pixels. The receptors were fused at the N-terminus to enhanced green or yellow fluorescent protein, and complementary pairs were co-expressed at different ratios of donor to acceptor (i.e., eGFP2-M1 and eYFP-M1, eGFP2-M2 and eYFP-M2, eGFP2-β1 and eYFP-β1, or eGFP2-β2 and eYFP-β2). Pixel-level emission spectra were recorded from images captured in a single plane; the relative contribution of each fluorophore then was determined by spectral deconvolution and used to calculate the corresponding apparent FRET efficiency. The distributions of efficiencies from 20 cells were analyzed in concert as a sum of Gaussians, with the number of components (n) and the relationships among the means taken as predicted for a dimer (n = 1), a triangular trimer (n = 2), and a tetramer configured as a square (n = 3) and a rhombus (n = 5). Distributions from each of the four receptors required 5 Gaussians, the means of which were related in the manner predicted for a rhombus. The inverse agonists atropine and timolol were without effect on the number of components or the relationship among the means detected for the M2 muscarinic receptor and the β2 adrenergic receptor, respectively. Homo-oligomers of the M1, M2, β1, and β2 receptors therefore appear to be rhombic tetramers in CHO cells. The configuration is unaffected by inverse agonists, at least in the case of the M2 and β2 receptors.

FRET Efficiency Distributions of Multistate Single Molecules

A simple analytic theory is developed to describe FRET efficiency histograms constructed from a photon trajectory generated by a molecule with multiple conformational states. The histograms are approximated by a sum of Gaussians with the parameters explicitly determined by the FRET efficiencies of the states and the rates of the transitions between the states. The theory, which has been tested against exact histograms for two conformational states and simulated data for three and four conformational states, accurately describes how the peaks in the histograms collapse as the bin time or the transition rates increase.

SU-GG-T-463: Study of the Magnitude of Detector Size Effect in the Measured Lateral Profiles of Proton Pencil Beam Spots

Purpose: Lateral profiles of therapeutic radiation beams measured by ionization chambers are known to be affected by their sizes. The purpose of this study is to quantify the detector size effect on the measurements of lateral profiles of the proton pencil beam spots (PPBS) of the scanning beam nozzle of the Proton Therapy Center at Houston. Methods and Materials: In-air and in-water lateral profiles of 72.5, 163.9 and 221.8 MeV energy PPBS were measured by a PTW 31014 Pinpoint ionization chamber (IC) (radius of 0.1 cm). The measured profiles were fitted to a sum of multiple Gaussians and were deconvolved with a Gaussian detector response kernel, K(x) = A. exp (- x2/2σκ2) with σκ taken as the radius of the IC. The deconvolved profiles are also given by the sum of Gaussians with modified σ2 = σm 2 - σκ2, where σm is the sigma of a Gaussian in the fitted measured profiles. The full width at half maxima (FWHM) and full width at 0.9 and 0.01 of the maxima, (FW0.9M, FW0.01M) of the profiles with and without the detector size corrections were compared to quantify the effect of the IC size on the in-air and inwater lateral profiles of PPBS. Results: The maximum differences between the uncorrected and corrected FWHM, FW0.9M and FW0.01M for the PPBS of all the three energies in this study, were found to be less than 0.3, 0.1, 0.5 mm for in-air and 0.2, 0.1, 0.3 mm for in-water lateral profiles respectively. The detector size corrected and the measured lateral profiles were found to be very close to each other. Conclusion: The analytical deconvolution method with a Gaussian detector response kernel indicates that the detector size has a rather small effect on the Gaussian like lateral profiles of PPBS measured with small ionization chambers.

SU-GG-T-597: General Analytical Solution for Photon Beam Profile Deconvolution

Purpose: To develop a general analytical solution for extracting “true” photon beam profiles with minimal volume averaging effect Method and Materials: We have derived several analytical functions suitable for photon beam profile fitting by examining the physics process of photon dose deposition. Beam profile measured with an ionization chamber can be described by a convolution equation with a detector response function of that chamber. A general analytical solution that minimizes the volume averaging effect of large volume chambers can be developed by linking the photon dose deposition process and beam profile measurement This work demonstrates that when dose deposition kernel and detector response function are carefully selected, photon beam profiles with minimal volume averaging effect can be easily recovered by a simple change of parameters in the analytical fitting function. Specific analytical fitting functions such as Gaussian, sum of multiple Gaussian and Lorentz functions are identified in this work along with solutions to extract the “true” beam profile. Results: Our analysis shows that detector response can be accurately modeled by a Gaussian or Lorentz function, any of three afore‐mentioned analytical fitting functions can accurately extract a “true” photon beam profile from profiles measured with a 3 mm radius ion chamber. The extracted profile perfectly matches profiles measured with a 0.8×0.8 mm2 diode detector.Conclusion: A general solution for minimizing volume averaging effect in a photon beam profile measurement with large size detectors is developed. It is found that three specific fitting functions and corresponding solutions presented in this work are ideal for recovering “true” photon beam profiles.

Oligomeric Size and Configuration of the M2 Muscarinic and β2 Adrenergic Receptors in Live Cells as Determined by Pixel-Level FRET

G protein-coupled receptors are known to form oligomers, but estimates of their size range from dimers to large arrays. We therefore have determined the oligomeric size of fluorophore-tagged M2 and β2 receptors in the plasma membrane of Chinese hamster ovary cells by examining the distribution of FRET efficiencies measured at the level of single pixels. Each receptor was fused at its N-terminus to enhanced green or yellow fluorescent protein and co-expressed as the complementary pair (EGFP2-M2 and EYFP-M2, or EGFP2-β2 and EYFP-β2). Pixel-level emission spectra were recorded from images captured in a single plane; the relative contribution of each fluorophore then was determined by spectral deconvolution and used to calculate the corresponding FRET efficiency. The distribution of efficiencies from each cell was analyzed as a sum of Gaussians. The number of Gaussians and the numeric relationship among the corresponding means (E) is determined by the number of combinatorial arrangements of FRET-productive pairs within a two-dimensional oligomer as predicted by the binomial theorem. A dimer will reveal a single efficiency or Gaussian, and a triangular trimer will reveal two; a square tetramer will reveal three efficiencies, and a rhombus will reveal five. Both the M2 receptor and the β2 receptor required five Gaussians to describe the distributions of efficiencies from several cells taken together. In each case, an efficiency (E) identified as the pair-wise efficiency (Ep) was related to the other four efficiencies in the manner predicted for a rhombus.

All-frequency rendering of dynamic, spatially-varying reflectance

We describe a technique for real-time rendering of dynamic, spatially-varying BRDFs in static scenes with all-frequency shadows from environmental and point lights. The 6D SVBRDF is represented with a general microfacet model and spherical lobes fit to its 4D spatially-varying normal distribution function (SVNDF). A sum of spherical Gaussians (SGs) provides an accurate approximation with a small number of lobes. Parametric BRDFs are fit on-the-fly using simple analytic expressions; measured BRDFs are fit as a preprocess using nonlinear optimization. Our BRDF representation is compact, allows detailed textures, is closed under products and rotations, and supports reflectance of arbitrarily high specularity. At run-time, SGs representing the NDF are warped to align the half-angle vector to the lighting direction and multiplied by the microfacet shadowing and Fresnel factors. This yields the relevant 2D view slice on-the-fly at each pixel, still represented in the SG basis. We account for macro-scale shadowing using a new, nonlinear visibility representation based on spherical signed distance functions (SSDFs). SSDFs allow per-pixel interpolation of high-frequency visibility without ghosting and can be multiplied by the BRDF and lighting efficiently on the GPU.

Computation of doses for large-angle Coulomb scattering of proton pencil beams

In this work we present a study of the impact of considering higher order terms in Molière's multiple Coulomb scattering (MCS) theory for the purpose of calculating scanning proton pencil beam lateral dose profiles in water. The proton beam profile in air, just before entering the target medium, was modeled with a sum of Gaussians fitted with measured data. The subsequent proton scattering in water was described using the three-term Molière distribution, which covers both small- and large-angle scatterings. We compared measured and computed lateral dose profiles at the 2 cm and at the near-Bragg peak depths for proton pencil beams with energies of 72.5 MeV, 121.2 MeV, 163.9 MeV and 221.8 MeV. At shallow depths, the Coulomb interaction model provided a good description of the profiles for all energies, except for 221.8 MeV. At the near-Bragg peak depths, the Coulomb interaction model provided a good description of the profiles only for the 72.5 MeV. The observed discrepancies may be attributed to the additional contributions from nuclear interactions, which may be quantified only after an accurate description of the MCS. The analysis presented in this work did not require user-adjustable parameters and may be carried out in a similar way for any other media, depths and proton energies.

Third-order dispersion in a pair of prisms

The general equation for third-order group velocity dispersion is derived in this paper, which contains all the terms of equation (11) in Arissian and Diels [1], as well as one extra term, which we have verified is not negligible. To verify our equation we have modeled the sum of Gaussian modulated frequencies. We show the electric field of bandwidth limited incident ultrashort pulses with a time duration of: 5 fs, 10 fs, 15 fs, 20 fs, 30 fs and 50 fs after propagating through a pair of identical isosceles prisms for a central frequency of 0.8 µm and 0.620 µm. The model is applied by using (a) the finite phase, (b) the phase approximated up to the second-order dispersion and (c) up to the third-order dispersion for each frequency component of the pulse. Graphs are presented for prisms made with fused silica and SF14.

SoftWAXS: a computational tool for modeling wide-angle X-ray solution scattering from biomolecules.

This paper describes a computational approach to estimating wide-angle X-ray solution scattering (WAXS) from proteins, which has been implemented in a computer program called SoftWAXS. The accuracy and efficiency of SoftWAXS are analyzed for analytically solvable model problems as well as for proteins. Key features of the approach include a numerical procedure for performing the required spherical averaging and explicit representation of the solute-solvent boundary and the surface of the hydration layer. These features allow the Fourier transform of the excluded volume and hydration layer to be computed directly and with high accuracy. This approach will allow future investigation of different treatments of the electron density in the hydration shell. Numerical results illustrate the differences between this approach to modeling the excluded volume and a widely used model that treats the excluded-volume function as a sum of Gaussians representing the individual atomic excluded volumes. Comparison of the results obtained here with those from explicit-solvent molecular dynamics clarifies shortcomings inherent to the representation of solvent as a time-averaged electron-density profile. In addition, an assessment is made of how the calculated scattering patterns depend on input parameters such as the solute-atom radii, the width of the hydration shell and the hydration-layer contrast. These results suggest that obtaining predictive calculations of high-resolution WAXS patterns may require sophisticated treatments of solvent.

Fast radial basis function interpolation with Gaussians by localization and iteration

Radial basis function (RBF) interpolation is a technique for representing a function starting with data on scattered points. In the vortex particle method for solving the Navier–Stokes equations, the representation of the field of interest (the fluid vorticity) as a sum of Gaussians is like an RBF approximation. In this application, there are two instances when one may need to solve an RBF interpolation problem: upon initialization of the vortex particles, and after spatial adaptation to replace a set of disordered particles by another set with a more regular distribution. Solving large RBF interpolation problems is notoriously difficult with basis functions of global support, due to the need to solve a linear system with a fully populated and badly conditioned matrix. In the vortex particle method, in particular, one uses Gaussians of very small spread, which lead us to formulate a method of solution consisting of localization of the global problem, and improvement of solutions over iterations. It can be thought of as an opposite approach from the use of compact support basis functions to ease the solution procedure. Compact support bases result in sparse matrices, but at the cost of reduced approximation qualities. Instead, we keep the global basis functions, but localize their effect during the solution procedure, then add back the global effect of the bases via the iterations. Numerical experiments show that convergence is always obtained when the localized domains overlap via a buffer layer, and the particles overlap moderately. Algorithmic efficiency is excellent, achieving convergence to almost machine precision very rapidly with well-chosen parameters. We also discuss the complexity of the method and show that it can scale as O(N) for N particles.

Anisotropic magnetic diffuse scattering in an easy-plane type antiferromagnet ErNi2Ge2

We report on neutron scattering studies of a rare earth intermetallic compound ErNi2Ge2. Polarized neutron scattering experiments revealed that the magnetic ordered moment m lies in ab-plane. Taking account of a lack of the third higher harmonic reflection, ErNi2Ge2 is considered to have a helical magnetic structure. The magnetic scattering profiles along the [100]*- and the [110]*-directions are well described by the sum of Gaussian and modified-Lorentzian terms, even far below TN, indicating that short-range orders coexist with a long-range order. Interestingly, the modified-Lorentzian-type diffuse scattering is not present in the profiles along the [001]*-direction. The anisotropy of the diffuse scattering suggests that the short-range-order consists of one dimensional long-range helices along the c-axis.

Some peculiarities of resist-profile simulation for positive-tone chemically amplified resists in electron-beam lithography

In the present work, we numerically modeled the processes of exposure and development of the CAMP6 chemically amplified resist during electron-beam lithography. The radial distributions of the absorbed electron energy in the resist for a zero-width δ-function 30keV electron beam are obtained by Monte Carlo simulation. These distributions (discrete data) are approximated by an analytical function (sum of double Gaussian and an exponential function). The values of the parameters of the function are calculated using an original Monte Carlo technique, and their dependencies on the resist thickness (d=100, 200, 600, 1000, and 1500nm) at two resist depths are presented. Using these parameters’ values, we performed a computer simulation of the process of developing the resist taking into account its peculiarities due to the complicated mechanism of resist removal from soluble resist areas. We obtained profiles at various development times of a single 100nm line.

A high resolution XPS study of sidewall functionalized MWCNTs by fluorination

Carbon nanotubes must be modified to magnify their application. Understanding the chemistry of carbon nanotubes is a crucial step towards their ultimate practical use. In this study, multi-walled carbon nanotubes (MWCNTs) were fluorinated at several different conditions. The change in the atomic structures of fluorinated MWCNTs was investigated using X-ray photoelectron spectroscopy (XPS). All core level spectra of MWCNTs were deconvoluted to several Pseudo-Voigt functions (sum of Gaussian–Lorentzian functions). The amount of doped fluorine increased with increasing doping fluorine partial pressure, and the fluorine atoms were covalently attached to the side-wall of the MWCNTs. However, the increasing rate of F1s component ratio became dull at 70%, as compared with ratios below 70%. This suggests that fluorine contents during the fluorination of carbon materials can be managed by controlling the fluorine mixing ratio.

Two Dimensional Wavelet Coefficient Statistics for Sea Bottom Classification

In this paper we examine the classification of different seafloors based on the analysis of images obtained by side-scan sonar. For this purpose, we apply various two dimensional multilevel wavelet decomposition schemes on images obtained from three different seafloor types, i.e., sand ripples, rocks and sands, and then we examine the statistics of the corresponding wavelet coefficients. The observed Probability Density Functions (pdf) are modeled using various theoretical distributions such as the Alpha Stable, Sum of Gaussians, Log-normal. The parameters of the fit are subsequently used to classify the side scan sonar images according to well known cluster analysis techniques. The use of the energy of the wavelet coefficients as a tool for side scan sonar image classification is also evaluated. A new unsupervised classification scheme based on the pdf fitting parameters is proposed.

Communication Limits Due to Photon Detector Jitter

Detector jitter, the random delay from the time a photon is incident on a single-photon-counting detector (SPD) to the time an electrical pulse is produced in response to that photon, is characterized for a number of SPDs. The jitter is modeled as a weighted sum of Gaussians. The performance in detector jitter is measured by determining the capacity of a communications channel utilizing a given detector. It is observed that the loss, measured as the ratio of the signal power required to achieve a specified capacity in the presence of jitter to that in the absence of jitter, goes as the square of the normalized jitter standard deviation (the standard deviation of the jitter in slotwidths). The loss is small when the normalized jitter is less than one, and becomes significant beyond that point. This loss must be taken into account when evaluating detectors for very high throughput channels.

The near-shore behaviour of shallow-water waves with localized initial conditions

We consider the behaviour of solutions to the nonlinear shallow-water equations which describe wave runup on a plane beach, concentrating on the behaviour at and just behind the moving shoreline. We develop regular series expansions for the hydrodynamic variables behind the shoreline, which are valid for any smooth initial condition for the waveform. We then develop asymptotic descriptions of the shoreline motion under localized initial conditions, in particular a localized Gaussian waveform: we obtain estimates for the maximum runup and drawdown of the wave, for its maximum velocities and the forces it is able to exert on objects in its path, and for the conditions under which such a wave breaks down. We show how these results may be extended to include initial velocity conditions and initial waveforms which may be approximated as the sum of several Gaussians. Finally, we relate these results tentatively to the observed behaviour of a tsunami.

From cloud overlap to PDF overlap

AbstractIn many applications, large‐scale numerical models of the atmosphere must represent small‐scale variability in clouds and water content. In particular, radiative‐transfer calculations need to know ‘cloud overlap’: that is, whether or not cloud patches at different altitudes overlap each other in the horizontal. Furthermore, it is important to know the overlap of horizontal variability in water content, or ‘PDF overlap’ (where PDF denotes ‘probability density function’). One approach to modelling PDF overlap is to stochastically generate profiles of water content that collectively have the desired statistical properties of the cloud field. Such approaches usually draw sample profiles using the Monte Carlo method and the independent‐column approximation.This paper proposes a new stochastic generator of cloud profiles. The primary novel advantage of our algorithm is that it can model correlations among multiple fields, such as water content and temperature. The algorithm is based on a simplified mixture of multiple normals (i.e. a sum of several Gaussians), which we call a ‘peg‐hat’ PDF because its normals are arranged like hats on pegs. The algorithm represents an entire vertical grid column by a single high‐dimensional sample point.The peg‐hat PDF is tested using large‐eddy simulations of boundary‐layer clouds: namely, a cumulus case, a case involving cumulus clouds rising into broken stratocumulus, and a marine stratocumulus case. For calculations of the histogram of liquid‐water path, the peg‐hat PDF outperforms a single‐normal PDF for cumulus clouds, which are often skewed. The peg‐hat and single‐normal PDFs perform comparably for stratocumulus. Copyright © 2007 Royal Meteorological Society

Level statistics in disordered linear networks

We consider a linear electrical impedance network, and study its frequency of nearest level statistics, P(s). We find that this probability density evolves from a sum of gaussians at small disorder, to a linear combination of Poisson and gaussian orthogonal ensemble for large values of the disorder.