Generalised Error Functions from the Kerr Metric

Motivated by the effort to understand the mathematical structure underlying the Teukolsky equations in a Kerr metric background, a homogeneous integral equation related to the prolate spheroidal function is studied. From the consideration of the Fredholm determinant of the integral equation, a family of generalized error function is defined, with which the Fredholm determinant of the sinc kernel is also evaluated. An analytic solution of a special ease of the fifth Painlev~ transcendent is then worked out explicitly.

A New Approach for Solving Fuzzy Linear Multi-Criterion Problems: An Approach Based on Minimization of the Errors Functions

The main purpose of this paper is to build a new approach for solving a fuzzy linear multi-criterion problem by defining a function called “error function”. For this end, the concept of level set is used to construct the error function. In addition, we introduce the concept of deviation variable in the definition of the error function. The algorithm of the new approach is summarized in three main steps: first, we transform the original fuzzy problem into a deterministic one by choosing a specific level . second, we solve separately each uni-criteria problem and we compute the error function for each criteria. Finally, we minimize the sum of error functions in order to obtain the desired compromise solution. A numerical example is done for a comparative study with some existing approaches to show the effectiveness of the new approach.

Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.

Error Model of Rotary Ring Laser Gyro Inertial Navigation System

To improve the precision of inertial navigation system（INS） during long time operation,the rotation modulated technique（RMT） was employed to modulate the errorr of the inertial sensors into periodically varied signals,and,as a result,to suppress the divergence of INS errors.The principle of the RMT was introduced and the error propagating functions were derived from the rotary navigation equation.Effects of the measurement error for the rotation angle of the platform on the system precision were analyzed.The simulation and experimental results show that the precision of INS was ① dramatically improved with the use of the RMT,and ② hardly reduced when the measurement error for the rotation angle was in arc-second level.The study results offer a theoretical basis for engineering design of rotary INS.

SPARSE RECOVERY BASED ON THE GENERALIZED ERROR FUNCTION

In this paper,we offer a new sparse recovery strategy based on the generalized error function.The introduced penalty function involves both the shape and the scale parameters,making it extremely flexible.For both constrained and unconstrained models,the theoretical analysis results in terms of the null space property,the spherical section property and the restricted invertibility factor are established.The practical algorithms via both the iteratively reweighted■_(1)and the difference of convex functions algorithms are presented.Numerical experiments are carried out to demonstrate the benefits of the suggested approach in a variety of circumstances.Its practical application in magnetic resonance imaging(MRI)reconstruction is also investigated.

Adsorption Isotherm, Kinetic and Thermodynamic Modelling of <i>Bacillus subtilis</i>ATCC13952 Mediated Adsorption of Arsenic in Groundwaters of Selected Gold Mining Communities in the Wassa West Municipality of the Western Region of Ghana

This study investigated Bacillus subtilis ATCC13952 as an adsorbent for arsenic in groundwater. Batch experiments were used to determine the effect of contact time, adsorbent dose, arsenic (III) concentration, pH, and temperature on the process. The percentage of arsenic (III) removed was high at a contact time of four days, 3.0 mL of Bacillus subtilis ATCC13952, pH 8 and temperature of 35°C. The kinetics of the process showed the Elovich kinetics model as the best fit for the process. This indicates that arsenic removal was by chemisorption. The analysis of the nonlinear equilibrium isotherms and the error functions showed the Langmuir isotherm as best fit for the process. Mechanistic study of the process indicated bulk diffusion to be the rate-determining step. Thermodynamically, the process was favourable, spontaneous and feasible. When the community water samples were treated with the Bacillus subtilis ATCC13952 at the optimum contact time, adsorbent dose, pH and temperature, 99.96% - 99.97% of arsenic was removed across all sampling points within the studied communities. Hence, the results show that Bacillus subtilis ATCC13952 is an efficient adsorbent for arsenic in aqueous systems and the organism appears to hold the key to purging the environment of arsenic contamination.

Maxwell-Boltzmann Distribution Mathematica Based Practitioner’s Approach

Utilizing Mathematica this report shows how from a practitioner’s point of view useful quantities some known, and some unknown and fresh properties about the Maxwell-Boltzmann distribution are calculated. We shortcut circling the usage of antiquated incomplete tabulated error functions given in the textbooks and professional literature replacing them with efficient upgrades. And, utilizing the animation features of Mathematica displaying the temperature-dependence of the distribution function assists in visualizing the character of the distribution.

Additive predictions of aboveground stand biomass in commercial logs and harvest residues for rotation age Pinus radiata plantations in New South Wales,Australia

Two systems of additive equations were developed to predict aboveground stand level biomass in log products and harvest residue from routinely measured or predicted stand variables for Pinus radiata plantations in New South Wales,Australia.These plantations were managed under three thinning regimes or stand types before clear-felling at rotation age by cut-to-length harvesters to produce sawlogs and pulpwood.The residue material following a clear-fell operation mainly consisted of stumps,branches and treetops,short off-cut and waste sections due to stem deformity,defects,damage and breakage.One system of equations did not include dummy variables for stand types in the model specification and was intended for more general use in plantations where stand density management regimes were not the same as the stand types in our study.The other system that incorporated dummy variables was for stand type-specific applications.Both systems of equations were estimated using 61 plot-based estimates of biomass in commercial logs and residue components that were derived from systems of equations developed in situ for predicting the product and residue biomass of individual trees.To cater for all practical applications,two sets of parameters were estimated for each system of equations for predicting component and total aboveground stand biomass in fresh and dry weight respectively.The two sets of parameters for the system of equations without dummy variables were jointly estimated to improve statistical efficiency in parameter estimation.The predictive performances of the two systems of equations were benchmarked through a leave-one-plot-out cross validation procedure.They were generally superior to the performance of an alternative two-stage approach that combined an additive system for major components with an allocative system for sub-components.As using forest harvest residue biomass for bioenergy has increasingly become an integrated part of forestry,reliable estimates of product and residue biomass will assist harvest and management planning for clear-fell operations that integrate cut-to-length log production with residue harvesting.

Approximate Analytical Solutions to the Heat and Stokes Equations on the Half-Line Obtained by Fokas’ Transform

In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.

Effects of mask-alignment error on point spread function for multi-level Fresnel diffractive lenses

The full aperture complex amplitude transmittance function of a multi-level diffraction lens with mask- alignment errors was derived based on scalar diffraction theory. The point spread function （PSF） was calculated by the Kirchhoff diffraction integral. It is found that the radius of the Airy disk increases with the increase of the error in the direction of misalignment, and the image center shifts along the direction of misalignment. A fourlevel diffractive lens with a diameter of 80 mm was fabricated, and its PSF and diffraction efficiency of ＋1st order were calculated and measured. The distribution of PSF is consistent with the calculated results, and the tested diffraction efficiency is slightly smaller than the calculated value; the relative error is 5.71%.

Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev's Method

The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function.Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_(2)^((m,0)) is obtained,moreover the existence and uniqueness of the solution of this system are discussed.Next,the discrete analogue D_(m)(hβ)of the differential operatord^(2m)/dx^(2m)-1 is constructed.Further,Sobolev's method of construction of optimal quadrature formulas in the space W_(2)^((m,0)),which based on the discrete analogue D_(m)(hβ),is described.Next,for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Finally,at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_(2)^((3,0))for the cases m=1 and m=3 are presented.

A combination of damage locating vector method （DLV） and differential evolution algorithm （DE） for structural damage assessment

In this study, a two-stage method is presented for identifying multiple damage scenarios. In the first stage, the damage locating vector （DLV） method using normalized cumulative energy （nce） is employed for damage localization in structures. In the second stage, the differential evolution algorithm （DE） is used for damage severity of the structures. In addition, in the second stage, a modification of an available objective function is made for handing the issue of symmetric structures. To verify the effectiveness of the present technique, numerical examples of a 72-bar space truss and a one-span steel portal frame are considered. In addition, the effect of noise on the performance of the identification results is also investigated. The numerical results show that the proposed combination gives good assessment of damage location and extent for multiple structural damage cases.

Effect of refractive correction on ocular optical quality measurement using double-pass system

Background Optical Quality Analysis System II （OQAS, Visiometrics, Terrassa, Spain） that uses double-pass （DP） technique is the only commercially available device that allows objective measurement of ocular retinal image quality. This study aimed to evaluate the impact of spectacle lenses on the ocular optical quality parameters and the validity of the optometer within OQAS. Methods Seventy eyes of healthy volunteers were enrolled. Optical quality measurements were performed using OQAS with an artificial pupil diameter of 4.0 mm. Three consecutive measurements were obtained from spectacle correction corresponding to subjective refraction and from the OQAS built-in optometer separately. The modulation transfer function cutoff frequency, the Strehl ratio, the width of the point spread function （PSF） at 10% of its maximal height （PSF10）, and the width of the PSF at 50% of its maximal height （PSF50） were analyzed. Results There was no significant difference in any of the parameters between the spectacle correction and the optometer correction （all P 〉0.05, paired t-test）. A good agreement was found between both the methods and a good intraobserver repeatability in both the correction methods. Difference in best focus between two methods was the only parameter associated significantly with optical quality parameter differences. Best focus difference, built-in optometer correction with or without external cylindrical lens, and age were associated significantly with PSF10 difference. No linear correlation between refractive status and optical quality measurement difference was observed. A hyperopic bias （best focus difference of （0.50±0.44） D） and a relatively better optical quality using spectacle correction in high myopia group were found. Conclusions OQAS based on DP system is a clinically reliable instrument. In patients with high myopia, measurements using built-in optometer correction should be considered and interpreted with caution.