A new property of Bernstein-Kantorovi? operators

In the present paper, we prove that the Bernstein-Kantorovič operators have the ability of preserving translation property in both C and L^p norms. Supported (in part) by the NSFC(10471130,10371024) of PRC and the Natural Science Foundation (Y604003) of Zhejiang Province.     

A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

We provide a local as well as a semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions. Our equation contains a Fréchet differentiable operator F and another operator G whose differentiability is not assumed. Using more precise majorizing sequences than before we provide sufficient convergence conditions for Newton-like methods to a locally unique solution of equation F(x)+G(x)=0. In the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results. In the local case a larger radius of convergence is obtained. Several numerical examples are provided to show that our results compare favorably with earlier ones. As a special case we show that the famous Newton-Kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the Newton-Kantorovich theorem.     

Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance

We give a verification theorem by employing Arrow's generalization of the Mangasarian sufficient condition to a general jump diffusion setting and show the connections of adjoint processes to dynamic programming. The result is applied to financial optimization problems.     

A new type SVM-projected SVM

Support vector machine (SVM), developed by Vapnik et al., is a new and promising technique for classification and regression and has been proved to be competitive with the best available learning machines in many applications. However, the classification speed of SVM is substantially slower than that of other techniques with similar generalization ability. A new type SVM named projected SVM (PSVM), which is a combination of feature vector selection (FVS) method and linear SVM (LSVM), is proposed in present paper. In PSVM, the FVS method is first used to select a relevant subset (feature vectors, FVs) from the training data, and then both the training data and the test data are projected into the subspace constructed by FVs, and finally linear SVM(LSVM) is applied to classify the projected data. The time required by PSVM to calculate the class of new samples is proportional to the count of FVs. In most cases, the count of FVs is smaller than that of support vectors (SVs), and therefore PSVM is faster than SVM in running. Compared with other speeding-up techniques of SVM, PSVM is proved to possess not only speeding-up ability but also de-noising ability for high-noised data, and is found to be of potential use in mechanical fault pattern recognition.     

Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula

We define a smooth functional calculus for a non-commuting tuple of (unbounded) operators A_j on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also extended to tuples of more general operators allowing smooth functional calculii. We also discuss the relation to the case with commuting operators.     

Probabilistic and Average Linear Widths of Sobolev Space with Gaussian Measure in L \infty-Norm

In this paper we investigate the probabilistic linear $(n,\delta)$-widths and $p$-average linear $n$-widths of the Sobolev space $W^r_2$ equipped with the Gaussian measure $\mu$ in the $L_{\infty}$-norm, and determine the asymptotic equalities \begin{eqnarray*} \lambda_{n,\delta}(W^r_2,\mu,L_{\infty}) &\asymp&\frac{\sqrt{\ln (n/\delta)}}{n^{r+(s-1)/2}},\\[3pt] \lambda^{(a)}_n(W^r_2,\mu,L_{\infty})_p &\asymp&\frac{\sqrt{\ln n}}{n^{r+(s-1)/2}}, \qquad 0 < p < \infty. \end{eqnarray*}     

Spectral Analysis of Powers of the Operator (Vf)(x)=q(x)∫0 xw(t)f(t),dt

Mathematical Notes -     

Approximation by a class of modified Szász-Durrmeyer operators

We modify Szász-Durrmeyer operators by means of three-diagonal generalized matrix which overcomes a difficulty in extending a Berens-Lorentz result to the Szász-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators inL _p are also presented by Ditzian-Totik modulus of smoothness.This project is supported by Zhejiang Provincial Foundation of China.     

B（X）上保点谱的可加映射和初等算子

记Ｘ为复数域上无限维Ｂａｎａｃｈ空间，Ｈ为无限维复可分Ｈｉｌｅｒｔ空间。本文给出Ｂ（Ｘ）上保持点谱的满射可加射具有的形式，以及Ｂ（Ｈ）上某些初等算子保点谱的充要条件。