Fuzzy smooth support vector machine with different smooth functions

Smooth support vector machine （SSVM） changs the normal support vector machine （SVM） into the unconstrained op- timization by using the smooth sigmoid function. The method can be solved under the Broyden-Fletcher-Goldfarb-Shanno （BFGS） algorithm and the Newdon-Armijio （NA） algorithm easily, however the accuracy of sigmoid function is not as good as that of polyno- mial smooth function. Furthermore, the method cannot reduce the influence of outliers or noise in dataset. A fuzzy smooth support vector machine （FSSVM） with fuzzy membership and polynomial smooth functions is introduced into the SVM. The fuzzy member- ship considers the contribution rate of each sample to the optimal separating hyperplane and makes the optimization problem more accurate at the inflection point. Those changes play a positive role on trials. The results of the experiments show that those FSSVMs can obtain a better accuracy and consume the shorter time than SSVM and lagrange support vector machine （LSVM）.

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.

A New Definition for Generalized First Derivative of Nonsmooth Functions

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

On Generalized High Order Derivatives of Nonsmooth Functions

In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.

Smoothing Approximations for Some Piecewise Smooth Functions

In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.

EXTENSION OF SMOOTHING FUNCTIONS TO SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.

Template-based smoothing functions for data smoothing in Geodesy

This paper is aimed at the derivation of a discrete data smoothing function for the discrete Dirichlet condition in a regular grid on the surface of a spheroid.The method employed here is the local L^2-seminorm minimization,through Euler-Lagrange method,for the Beltrami operator.The method results in a weighted average of the surrounding points in a te mplate based on the first order Taylor expansion of the unknown function under consideration.The coefficients of the weighted average are calculated and used to smooth the Geoid height data in Iran,derived from the EGM2008 geopotential model.

EFFICIENT ESTIMATION OF FUNCTIONAL-COEFFICIENT REGRESSION MODELS WITH DIFFERENT SMOOTHING VARIABLES

In this article,a procedure for estimating the coefficient functions on the functional-coefficient regression models with different smoothing variables in different coefficient functions is defined.First step,by the local linear technique and the averaged method,the initial estimates of the coefficient functions are given.Second step,based on the initial estimates,the efficient estimates of the coefficient functions are proposed by a one-step back-fitting procedure.The efficient estimators share the same asymptotic normalities as the local linear estimators for the functional-coefficient models with a single smoothing variable in different functions.Two simulated examples show that the procedure is effective.

Multilinear commutators of BMO functions and multilinear singular integrals with non-smooth kernels

This paper is concerned with certain multilinear commutators of BMO functions and multilinear singular integral operators with non-smooth kernels. By the sharp maximal functions estimates, the weighted norm inequalities for this kind of commutators are established.

Commutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels

This paper is devoted to studying the commutators of the multilinear singular integral operators with the non-smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained.

A new Leray formula for smooth functions on bounded domains in C^n

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.

New smooth gap function for box constrained variational inequalities

A new smooth gap function for the box constrained variational inequality problem （VIP） is proposed based on an integral global optimality condition. The smooth gap function is simple and has some good differentiable properties. The box constrained VIP can be reformulated as a differentiable optimization problem by the proposed smooth gap function. The conditions, under which any stationary point of the optimization problem is the solution to the box constrained VIP, are discussed. A simple frictional contact problem is analyzed to show the applications of the smooth gap function. Finally, the numerical experiments confirm the good theoretical properties of the method.

Effect of Chaihu Shugan decoction on gastric smooth muscle cell apoptosis in rats with functional dyspepsia

Objective:To investigate the effect of Chaihu Shugan decoction(CSD)on gastric smooth muscle cells(GSMCs)apoptosis in rats with functional dyspepsia(FD).Methods:48Sprague-Dawley(SD)rats were randomly assigned into six groups:a normal control group,a model group,apositive control(domperidone)group and low-,middle-and high-dose CSD groups.A rat model of FD was established by constantly squeezing their tails.The rats were administered CSD(0.16g/mL,0.32g/mL,0.64g/mL)or domperidone(0.3 g/L)via intragastric gavage for four weeks.The gastric emptying rate was detected at 4 weeks post-administration.Apoptosis of GSMCs was determined by terminal deoxynucleotidyl transferase dUTP nick end labeling(TUNEL)staining and the mitochondrial morphology was observed by transmission electron microscopy.The expression of Bcl-2and Bax was measured by immunohistochemistry.Results:FD resulted in marked reduction of gastric emptying rate,severe gastric tissue damage and mitochondria injury,but were reversed by CSD treatment(P<0.05).The apoptosis-induced protein Bax was markedly down-regulated by CSD,whereas the expression of the anti-apoptotic Bcl-2 protein was notably increased(P<0.05).Furthermore,CSD could protect the FD rats against GSMCs apoptosis manifested by a decreased in TUNEL-positive cells(P<0.05).Conclusion:CSD could alleviate GSMCs apoptosis in FD rats,possibly by the modulation of Bcl-2 and Bax expression,and the suppression of mitochondria injury.

Automatically Smoothing the Spectroscopic Data by Cubic B-Spline Basis Functions

In the present paper,a new criterion is derived to obtain the optimum fitting curve while using Cubic B-spline basis functions to remove the statistical noise in the spectroscopic data.In this criterion,firstly,smoothed fitting curves using Cubic B-spline basis functions are selected with the increasing knot number.Then,the best fitting curves are selected according to the value of the minimum residual sum of squares(RSS)of two adjacent fitting curves.In the case of more than one best fitting curves,the authors use Reinsch's first condition to find a better one.The minimum residual sum of squares(RSS)of fitting curve with noisy data is not recommended as the criterion to determine the best fitting curve,because this value decreases to zero as the number of selected channels increases and the minimum value gives no smoothing effect.Compared with Reinsch's method,the derived criterion is simple and enables the smoothing conditions to be determined automatically without any initial input parameter.With the derived criterion,the satisfactory result was obtained for the experimental spectroscopic data to remove the statistical noise using Cubic B-spline basis functions.

Equivalence between Modulus of Smoothness and K-Functional on Rotation Group SO(3)

In this paper we obtain the equivalence between modulus of smoothness and K-functional on rotation group SO(3).

A NEW FORMULATION OF BERNSTEIN-BEZIER BASED SMOOTHNESS CONDITIONS FOR pp FUNCTIONS

In this note, we establish a new formulation of smoothness conditions for piecewise polynomial (: =pp) functions in terms of the B-net representation in the general n-dimensional setting. It plays an important role for 2-dimensional setting in the constructive proof of the fact that the spaces of polynomial splines with smoothness rand total degree k≥3r+2 over arbitrary triangulations achieve the optimal approximation order with the approximation constant depending only on k and the smallest angle of the partition in [5].

A Smoothing Penalty Function Method for the Constrained Optimization Problem

In this paper, an approximate smoothing approach to the non-differentiable exact penalty function is proposed for the constrained optimization problem. A simple smoothed penalty algorithm is given, and its convergence is discussed. A practical algorithm to compute approximate optimal solution is given as well as computational experiments to demonstrate its efficiency.

Antioxidant Activities and Inhibitory Effects of <i>Auricularia Auricular</i>and Its Functional Formula Diet Against Vascular Smooth Muscle Cell <i>in Vitro</i>

The functional formula diet AHP, containing polysaccharides from Auricularia auricular, polyphenolic compounds from Hawthorn (Crataegus) and Pueraria radix, has been recently developed as a dietary intervention against dyslipidemia in our previous study. In the present study, its antioxidant activities and protective effects against proliferation of vascular smooth muscle cells (VSMCs) were investigated. AHP possessed the potent radical-scavenging effects against hydroxyl and superoxide radicals, and also the inhibitory effects against peroxidation of low density lipoprotein induced by Cu2+ in vitro. The protective effects of AHP against proliferation of VSMCs were evaluated through the methodology of serum pharmacology. The serum containing AHP significantly inhibited the proliferation of VSMCs induced by oxidized low density lipoprotein in a time- and dose-dependent manner, and also promoted the nitric oxide production of VSMCs. Our study indicated that this functional formula diet would be a potent alternative as a functional diet to prevent atherosclerosis at early stage.

A New Unified Path to Smoothing Nonsmooth Exact Penalty Function for the Constrained Optimization

We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.

Equivalence of <i>K</i>-Functionals and Modulus of Smoothness Generated by a Generalized Dunkl Operator on the Real Line

This paper is intended to establish the equivalence between K-functionals and modulus of smoothness tied to a Dunkl type operator on the real line.