Gaussian Formula Research Articles

Some Gaussian binomial sum formulæ with applications

We introduce and compute some Gaussian q-binomial sums formulae. In order to prove these sums, our approach is to use q-analysis, in particular a formula of Rothe, and computer algebra. We present some applications of our results.

Corneal Power Measurement With the Dual Scheimpflug-Placido Topographer After Myopic Excimer Laser Surgery

To evaluate the ability of the Sirius Scheimpflug-Placido Topographer (Costruzione Strumenti Oftalmici, Florence, Italy) in predicting corneal power in eyes with previous myopic corneal refractive surgery. The following corneal refractive power measurements were obtained with Sirius in 38 eyes with prior myopic excimer laser surgery: simulated keratometry (SimK), true net power based on the Gaussian optical formula (TNP), and the mean pupil power (MPP) calculated by ray tracing within the diameter of the entrance pupil range from 2.5 to 6 mm. These values, along with the corresponding keratometric value, which was based on the equation that converted the MPP within 4.5-mm entrance pupil diameter to a keratometric reading in 75 myopic eyes, were compared using the clinical history method. SimK significantly overestimated corneal power by 0.67 ± 0.52 diopters (D), whereas TNP significantly underestimated corneal power by -1.05 ± 0.48 D. The only measurements that showed no statistically significant difference by the clinical history method were the 5 mm MPP, 5.5 mm MPP, and KMPP4.5POST, and the 95% limits of agreement (LOA) were -1.10 to 0.62, -0.91 to 0.89, and -0.67 to 1.05 D, respectively. The Sirius Scheimpflug-Placido Topographer is an alternative method of measuring corneal power in eyes with previous corneal refractive surgery for myopia when preoperative data are not available. The best options seem to be the 5 mm MPP, 5.5 mm MPP, and the keratometric reading calculated from the regression formula.

Long-term changes in intraocular lens position and corneal curvature after cataract surgery and their effect on refraction

To evaluate the role of intraocular lens (IOL) position shift and changes in corneal curvature on long-term refractive shift after cataract surgery. Rotterdam Ophthalmic Institute, Rotterdam, the Netherlands. Prospective cohort study. Patients who had routine cataract surgery with implantation of a hydrophobic acrylic 1-piece IOL (Acrysof SA60AT) in the capsular bag were enrolled. Measurements were performed preoperatively and 1 month, 3 months, and 1 year postoperatively. Refraction was measured with the ARK-530A autorefractor. The IOL position and corneal curvature were measured with the Lenstar LS-900 biometer. The refractive effect of changes in IOL position and corneal curvature was calculated with a Gaussian optics-based thin-lens formula and correlated with the measured refractive shift. The study group comprised 59 eyes of 59 patients. The median measured absolute refractive change was 0.25 diopter (D). The IOL position showed a statistically significant mean posterior shift of 0.033 mm ± 0.060 (SD) between 1 month and 1 year postoperatively (P < .01), of which the median calculated absolute refractive effect was 0.05 D. This did not correlate with the measured refractive shift (Pearson r = 0.10, P = .46). Natural fluctuations in corneal curvature caused a median calculated absolute refractive effect of 0.17 D, which correlated well with the measured refractive shift (Pearson r = .55, P < .001). Long-term changes in refraction after cataract surgery resulted from natural fluctuations in corneal curvature rather than from IOL position shift. These fluctuations limit the accuracy with which the refractive outcome can be planned. No author has a financial or proprietary interest in any material or method mentioned.

A new method for analyzing the collimation angle of a neutron Soller collimator**Supported by National Natural Science Foundation of China (11205248)

A new method for analyzing the collimation angle of a neutron Soller collimator is described. A Gaussian distribution formula is used to define the angular distribution function of the neutron source and the neutron transmission function of the Soller collimator. A relationship between the FWHM of the collimator rocking curve and the collimation angle is derived. Using this method, some rocking curve experiment results are analyzed. The results show that the new function can be a good theoretical model for fitting the experimental data, especially for the data of two collimators with different collimation angles.

Asymptotical formulas for Gaussian and generalized hypergeometric functions

In this paper, we present several generalizations and refinements for the asymptotic formulas of Gaussian and generalized hypergeometric functions.

Numerical simulation for flow‐induced stress in injection/compression molding

Flow‐induced stress in compression stage is complex because compression flow combines both shear flow and extensional flow characters and involves the boundary change in melt front and compressing directions. To describe this kind of flow effectively, the theoretical model was established in terms of three‐dimensional compressible and viscoelastic fluid, and the material derivative which includes mesh moving velocity was introduced to modify the convective terms in governing and constitutive equations based on Arbitrary Lagrangian Eulerian description. The corresponding variational equations were derived via Gauss formula and the finite element method was used to discretize the flow variables with mixed elements. In order to decouple the interdependence of velocity, stress, and temperature, a new approach involves twofold iterations was proposed to solve nonlinear flow problem. A new mesh updating scheme was presented to keep consistent with compressed regions. Numerical results indicate the compression can reduce the flow‐induced stress significantly compared to conventional injection molding, increase compressing velocity enlarges flow‐induced stress, and high temperature decreases flow‐induced stress. The proposed method was verified by photoelastic pictures and measured pressures and temperatures during molding. POLYM. ENG. SCI., 56:287–298, 2016. © 2015 Society of Plastics Engineers

Symbol error rate analysis for colour‐shift keying modulation in visible light communication system with RGB light‐emitting diodes

The symbol error rate (SER) is investigated for colour-shift keying (CSK) modulation in visible light communication system with red/green/blue LEDs. A new concept of pseudo signal-to-noise ratio is proposed to illustrate the relationship between the signal power and the noise power since the traditional electrical signal-to-noise ratio fails to capture the properties of CSK signals. The authors map the three-dimensional (3D) signal space of CSK signals to a 2D plane because the decision region of constellations in 2D space is easy to analyse and is proved to be equivalent to that of 3D signal space. The symbol error probability for a certain constellation is calculated as the summation of the probabilities of the received signals lying outside the decision region. As the SER is formulated as the weighted sum of error probabilities for all constellations, an analytical expression for SER is derived for certain constellation distribution. Moreover, a general formula for 22n -CSK modulation is derived by exploiting the distribution regularity of 4CSK and 16CSK. The SER expression for CSK modulation is evaluated approximately using the numerical Gaussian formula. Simulation results show that the analytical expression can predict SER performance accurately.

Theoretical study of photo-double ionization of CO2 in the equal energy sharing regime using Dyson orbitals and the parameterized three center continuum wave function

The theoretical procedure for the determination of the multiply differential cross sections (MDCS) applied recently [, ] to the vertical photo-double ionization of a diatomic N2 molecule is extended to the three center CO2 molecule producing dissociative dication. Dyson orbitals for the bound active electrons are introduced, and the parameterized three-center continuum wave function in its correlated form is used to describe the two ejected electrons. The variation of the MDCS for the coincidence detection of the two ejected electrons, for a randomly orientated internuclear axis is studied. The comparison of our results to those obtained by the Gaussian parametrization formula shows the similarities and the differences between the behavior of an atomic target and CO2.

Scene distance measurement method based on light field imaging.

Because a light field camera first takes an image and then refocuses it, we propose a scene distance measurement method based on light field imaging spatial refocusing principles, which was used to generate sequences of refocused images from raw spatial scene images taken by a Lytro1 light field camera. We selected the window of the object whose scene distance needed to be measured, used image-resolution evaluation functions to choose one of the clearest images among the refocused image sequences, and employed Gaussian formula to deduce the scene distance measurement formula. Finally, we used the scene distance measurement method to measure an object on the scene and theoretically analyzed its measurement range and accuracy. The experimental results showed that this scene distance measurement method provides the scene distance information.

Keratometric index obtained by Fourier-domain optical coherence tomography.

PurposeTo determine the keratometric indices calculated based on parameters obtained by Fourier-domain optical coherence tomography (FD-OCT).MethodsThe ratio of anterior corneal curvature to posterior corneal curvature (Ratio) and keratometric index (N) were calculated within central 3 mm zone with the RTVue FD-OCT (RTVue, Optovue, Inc.) in 186 untreated eyes, 60 post-LASIK/PRK eyes, and 39 keratoconus eyes. The total corneal powers were calculated using different keratometric indices: Kcal based on the mean calculated keratometric index, K1.3315 calculated by the keratometric index of 1.3315, and K1.3375 calculated by the keratometric index of 1.3375. In addition, the total corneal powers based on Gaussian optics formula (Kactual) were calculated.ResultsThe means for Ratio in untreated controls, post-LASIK/PRK group and keratoconus group were 1.176 ± 0.022 (95% confidence interval (CI), 1.172–1.179), 1.314 ± 0.042 (95%CI, 1.303–1.325) and 1.229 ± 0.118 (95%CI, 1.191–1.267), respectively. And the mean calculated keratometric index in untreated controls, post-LASIK/PRK group and keratoconus group were 1.3299 ± 0.00085 (95%CI, 1.3272–1.3308), 1.3242 ± 0.00171 (95%CI, 1.3238–1.3246) and 1.3277 ± 0.0046 (95%CI, 1.3263–1.3292), respectively. All the parameters were normally distributed. The differences between Kcal and Kactual, K1.3315 and Kactual, and K1.3375 and Kactual were 0.00 ± 0.11 D, 0.21 ± 0.11 D and 0.99 ± 0.12 D, respectively, in untreated controls; -0.01 ± 0.20 D, 0.85 ± 0.18 D and 1.56 ± 0.16 D, respectively, in post-LASIK/PRK group; and 0.03 ± 0.67 D, 0.56 ± 0.70 D and 1.40 ± 0.76 D, respectively, in keratoconus group.ConclusionThe calculated keratometric index is negatively related to the ratio of anterior corneal curvature to posterior corneal curvature in untreated, post-LASIK/PRK, and keratoconus eyes, respectively. Using the calculated keratometric index may improve the prediction accuracies of total corneal powers in untreated controls, but not in post-LASIK/PRK and keratoconus eyes.

A classification theorem for hypersurfaces of Minkowski space

Let Mn be a compact hypersurface of a Minkowski space (Vn+1, F). In this paper, using the Gauss formula of the Chern connection for Finsler submanifolds, we prove that if the second mean curvature H2 of M is constant and the norm square S of the second fundamental form of M satisfies S  n(n−1) n−2 H2, then M with the induced metric is isometric to the standard Euclidean sphere. This generalizes the result of [2] from the Euclidean to the Minkowski space.

Applications of the Gaussian kinematic formula to CMB data analysis

The Gaussian Kinematic Formula (GKF, see Adler and Taylor (2007,2011)) is an extremely powerful tool allowing for explicit analytic predictions of expected values of Minkowski functionals under realistic experimental conditions for cosmological data collections. In this paper, we implement Minkowski functionals on multipoles and needlet components of CMB fields, thus allowing a better control of cosmic variance and extraction of information on both harmonic and real domains; we then exploit the GKF to provide their expected values on spherical maps, in the presence of arbitrary sky masks, and under nonGaussian circumstances. All our results are validated by numerical experiments, which show a perfect agreement between theoretical predictions and Monte Carlo simulations.

Assessing the corneal power change after refractive surgery using Scheimpflug imaging.

To investigate whether the Pentacam HR could accurately predict the surgically induced refractive change in patients operated with small-incision lenticule extraction (SMILE) for myopia or femto-second laser in situ keratomileusis (FS-LASIK) for myopia or hyperopic astigmatism. Data from three groups consisting of (1) 410 myopic eyes of 410 patients operated with SMILE, (2) 111 myopic eyes of 111 patients operated with FS-LASIK, and (3) 40 eyes of 40 patients with hyperopic astigmatism operated with FS-LASIK were retrospectively analysed. The change in manifest refraction due to surgery was compared with the objectively measured change in corneal power by the Pentacam HR in three different ways: Sagittal Power (calculated as for placido topographers), True Net Power (calculated by a Gaussian optics formula), and Total Corneal Refractive Power (calculated by ray tracing). Multiple linear regression analysis was performed to investigate which parameters influenced the Pentacam HR's prediction of the change in subjective refraction due to surgery. The Total Corneal Refractive Power Apex, Zone calculation in a diameter of 4.0 mm effectively predicted the surgically induced refractive change for all three patient groups. The spherical equivalent was predicted with an error of 0.08 ± 0.41 D for the SMILE eyes, 0.05 ± 0.61 D for the myopic eyes operated with FS-LASIK, and -0.15 ± 0.49 D for the hyperopic astigmatic eyes treated with FS-LASIK. Regression showed that preoperative refractive error had a significant impact on the prediction error of the Pentacam HR. Ray tracing calculations based on Scheimpflug imaging accurately assessed the change in manifest refraction due to corneal laser surgery.

Visual Attention Distribution and its Application in the Gesture Interaction System

Aiming at making the simulation of human visual attention behavior more truly in computer, starting from analyzing operator’s cognitive model, a gesture tracking algorithm is put forward based on the distribution model of visual attention. To begin with, analyzing the change of the operator human eye sight, a visual attention model was built. Secondly, the basic characteristics of visual attention model were studied. Finally, the three Gauss formula is used to describe the model. Experimental results show that the algorithm can effectively improve the speed and tracking accuracy of gesture interaction.

Exact and Asymptotic Features of the Edge Density Profile for the One Component Plasma in Two Dimensions

There is a well known analogy between the Laughlin trial wave function for the fractional quantum Hall effect, and the Boltzmann factor for the two-dimensional one-component plasma. The latter requires analytic continuation beyond the finite geometry used in its derivation. We consider both disk and cylinder geometry, and focus attention on the exact and asymptotic features of the edge density. At the special coupling \Gamma := q^2/k_BT=2 the system is exactly solvable. In particular the k-point correlation can be written as a k \times k determinant, allowing the edge density to be computed to first order in \Gamma - 2. A double layer structure is found, which in turn implies an overshoot of the density as the edge of the leading support is approached from inside the plasma. Asymptotic analysis shows that the deviation from the leading order (step function) value is different for into the plasma than for outside. For general \Gamma, a Gaussian fluctuation formula is used to study the large deviation form of the density for N large but finite. This asymptotic form involves thermodynamic quantities which we independently study, and moreover an appropriate scaling gives the asymptotic decay of the limiting edge density outside of the plasma.

Research on Peak-detection Algorithm for High-precision Demodulation System of Fiber Bragg Grating

In order to improve the detection accuracy of wavelength, the filtering and curve fitting technologies were applied in the FBG wavelength demodulation system based on tunable F-P filter. These methods could realize the accurate peak-location of output signals of the photo detector. According to the characteristics of noise, the FIR low-pass filter was designed to filter the obtained light power signals so as to provide the input signals with high SNR for the peak-detection algorithms. By analyzing and comparing several typical peak-searching algorithms, the algorithm of Gauss formula nonlinear curve fitting (L-M) was chosen to fit the digitized light power signals. The experimental results show that L-M fitting algorithm reduces the mean square error by 7.5% compared with the Gauss fitting algorithm. For the Gauss signal in the wavelength demodulation system designed in the paper, the L-M algorithm has lower mean square error than other peak-searching algorithms. This algorithm is suitable for FBG wavelength demodulation system based on tunable F-P filter. It can efficiently raise the accuracy of wavelength demodulation system.

Atmospheric dispersion modeling near a roadway under calm meteorological conditions

Atmospheric pollutant dispersion near sources is typically simulated by Gaussian models because of their efficient compromise between reasonable accuracy and manageable computational time. However, the standard Gaussian dispersion formula applies downwind of a source under advective conditions with a well-defined wind direction and cannot calculate air pollutant concentrations under calm conditions with fluctuating wind direction and/or upwind of the emission source. Attempts have been made to address atmospheric dispersion under such conditions. This work evaluates the performance of standard and modified Gaussian plume models using measurements of NO2, PM10, PM2.5, five inorganic ions and seven metals conducted near a freeway in Grenoble, France, during 11–27 September 2011. The formulation for calm conditions significantly improves model performance. However, it appears that atmospheric dispersion due to vehicle-induced turbulence is still underestimated. Furthermore, model performance is poor for particulate species unless road dust resuspension by traffic is explicitly taken into account.

Gaussian Cubature Arising from Hybrid Characters of Simple Lie Groups

Lie groups with two different root lengths allow two mixed sign homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.

Validation of an operational advanced Gaussian model with the Copenhagen and Kincaid datasets

A practical model for evaluating ground–level concentrations from elevated sources that applies a new Gaussian formulation for transport and vertical diffusion is presented. In the Gaussian formula, the source height is replaced by a virtual source height, which is expressed by simple functions of the vertical profiles of wind and turbulent diffusivity. The performances of the model are evaluated by comparing different lateral diffusion parameterisations.

Fractional partial differential equation denoising models for texture image

In this paper, a set of fractional partial differential equations based on fractional total variation and fractional steepest descent approach are proposed to address the problem of traditional drawbacks of PM and ROF multi-scale denoising for texture image. By extending Green, Gauss, Stokes and Euler-Lagrange formulas to fractional field, we can find that the integer formulas are just their special case of fractional ones. In order to improve the denoising capability, we proposed 4 fractional partial differential equation based multiscale denoising models, and then discussed their stabilities and convergence rate. Theoretic deduction and experimental evaluation demonstrate the stability and astringency of fractional steepest descent approach, and fractional nonlinearly multi-scale denoising capability and best value of parameters are discussed also. The experiments results prove that the ability for preserving high-frequency edge and complex texture information of the proposed denoising models are obviously superior to traditional integral based algorithms, especially for texture detail rich images.