A combination of time-scale calculus and a cross-validation technique used in fitting and evaluating fractional models

In this work, we demonstrate how fractional calculus and time-scale calculus can be combined beautifully to solve and fit a modeling problem. In addition, a cross-validation technique is used to evaluate the fitted model. The specific application that we consider is the one-compartment model. The one-compartment model is a first-order differential equation that describes drug concentration over time. It turns out that approximating the solution by using a fractional model allows us to get more accurate results for model fitting. To quantitatively verify this insight, we compare between a first-order model and anα-order fractional model using real data for drug concentration. Then the mean squared error and a cross-validation method are used to determine the model that provides the best fit and predictions for unseen data.     

The Limit of Detection of a Weak Acid,in the Presence of Another,by Potentiometric Acid-Base Titrimetry and Deviation-Pattern Recognition

Abstract

When a solution containing a single monobasic weak acid is titrated potentiometrically with a strong base, non-linear regression analysis of the data yields a random scatter of the residuals around the best fit to the appropriate equation. If the same equation is used when a second weak acid is also present, systematic errors arise and yield a deviation plot having a characteristic shape. If the amplitude of that plot is substantially larger than the random error of measurement, the presence of the second acid can be detected, and its concentration can then be evaluated by non-linear regression onto the equation that takes its presence into account. The amplitude increases as the relative concentration of the second acid increases, and as the difference between the values of pK_a for the two acids increases. Examination of these dependences shows, for example, that it should be possible to detect 1 per cent of an impurity for which pK_a differs by 0.57 unit from that of the major component.     

Surface‐enhanced Raman spectra of individual multiwalled carbon nanotubes with small innermost diameters

The Raman spectra of individual multiwalled carbon nanotubes (MWCNTs) with the innermost diameters of 0.6–0.9 nm are studied by surface‐enhanced Raman scattering. The influences of small innermost diameters to Raman features are investigated. A clear and relatively sharp Raman peak appears at 1510 cm⁻¹ when the innermost diameter is close to 0.6 nm. Lorentzian fits of G band indicate that its splitting is affected by the small innermost diameter of MWCNT. Moreover, the splitting of 2iTO mode is also observed at 2800–3000 cm⁻¹. Copyright © 2012 John Wiley & Sons, Ltd.     

Polynomial fitting of DT-MRI fiber tracts allows accurate estimation of muscle architectural parameters

Fiber curvature is a functionally significant muscle structural property, but its estimation from diffusion-tensor magnetic resonance imaging fiber tracking data may be confounded by noise. The purpose of this study was to investigate the use of polynomial fitting of fiber tracts for improving the accuracy and precision of fiber curvature (κ) measurements. Simulated image data sets were created in order to provide data with known values for κ and pennation angle (θ). Simulations were designed to test the effects of increasing inherent fiber curvature (3.8, 7.9, 11.8 and 15.3 m(-1)), signal-to-noise ratio (50, 75, 100 and 150) and voxel geometry (13.8- and 27.0-mm(3) voxel volume with isotropic resolution; 13.5-mm(3) volume with an aspect ratio of 4.0) on κ and θ measurements. In the originally reconstructed tracts, θ was estimated accurately under most curvature and all imaging conditions studied; however, the estimates of κ were imprecise and inaccurate. Fitting the tracts to second-order polynomial functions provided accurate and precise estimates of κ for all conditions except very high curvature (κ=15.3 m(-1)), while preserving the accuracy of the θ estimates. Similarly, polynomial fitting of in vivo fiber tracking data reduced the κ values of fitted tracts from those of unfitted tracts and did not change the θ values. Polynomial fitting of fiber tracts allows accurate estimation of physiologically reasonable values of κ, while preserving the accuracy of θ estimation.     

A background elimination method based on linear programming for Raman spectra

In this paper, we consider a new background elimination method for Raman spectra. As a background is usually slowly varying with respect to wavelength, it could be approximated by a slowly varying curve. However, the usual curve‐fitting method cannot be applied because there is a constraint that the estimated background must be beneath a measured spectrum. To meet the requirement, we adopt a polynomial as an approximating function and show that background estimation could be converted to a linear programming problem which is a special case of constrained optimization. In addition, we present an order selection algorithm for automatic baseline elimination. According to the experimental results, it is shown that the proposed method could be successfully applied to experimental Raman spectra as well as synthetic spectra. Copyright © 2011 John Wiley & Sons, Ltd.     

A new approach to the level function

This article devotes to design a hybrid Level Set Method which compromises the advantages of the level functions designed in [2] and [3].     

提高CCD在激光三角测距中分辨率的方法

激光三角测距系统的精度主要取决于光斑象在探测器上的定位，用ＣＣＤ摄象机作为探测器时，光斑象的定位精度又取决于ＣＣＤ摄象机的分辨率。通常用光斑的采样灰度质心作为象点的准确位置可将ＣＣＤ的分辨率提高到亚象元级，但这种方法存在其固的的局限，边缘灰度跳变与高频干扰噪音影响。     

Least squares approximation by splines with free knots

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process.The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.Dedicated to Professor Dr.-Ing. habil. Dr. h.c. Helmut Heinrich on the occasion of his 90th birthdayResearch of the second author was partly supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1.     

两维位置灵敏探测器的能量和动量响应函数刻度

用弹性散射对多通道(e,2e)谱仪两维位置灵敏探测器的能量和动量响应函数进行了刻度,从位置灵敏探测器输出的两维位置信息中获得了弹性散射的能谱和角度谱,估计了在当前实验条件下的能量分辨和角度分辨.同时,我们用正交多项式的最小二乘法拟合得到了谱仪单路的能量和动量响应函数.     

用棱镜摄谱仪测波长实验数据处理方法

本文阐述了用棱镜摄谱仪测波长实验中心以拟合算法处理数据的方法。