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January 15,2011
Volume 4,Issue 1
Pages -
Subject Area:Differential Dynamical Systems,Functional Analysis,Applied Statistical Mathematics,Applied Mathematics

Title: A Hilbert-type integral inequality with some parameters on the finite interval

Authors: CHEN Guangsheng
PP: 82-86
Abstract: By introducing two independent parameters λ1, λ2 and two pairs of conjugate exponents, defines two weight functions, using Holder’s inequality and generalized arithmetic-geometric inequality, establishes a Hilbert-type integral inequality with some parameters on the finite interval (a, b)（0 Keywords: mathematical analysis；Hilbert-type integral inequality; weight function; Holder’s inequality; equivalent form

Title: Technology of adding rows and columns to augmented matrix for solving systems of linear equations

Authors: CHEN Bihong
PP: 77-81
Abstract: In this paper a technology is proposed that adds rows and columns to the augmented matrix. Adding rows is to increase the rank of the augmented matrix, and adding columns is to increase the column number of augmented matrix to keep entire information of the linear system. By constantly adding rows and columns and deleting zero rows, and finally generating the general solution in the matrix, such method has universal property and is easy to understand and get general solution by programming. It has been used in online calculation of systems of linear equations on the internet.
Keywords: linear algebra; matrix; linear space

Title: Numerical solution of European call option pricing model with semidiscretization technique

Authors: NIU Chenghu
PP: 72-76
Abstract: For the complicated formula is hard to be operated, a numerical method of European call option pricing model is provided in this paper. Partial differential equation, which adopts fourth-order Lagrange interpolating polynomial to expand boundary values making all the neighbors mesh internal nodes in domain, is discretized by the semidiscretization technique. The examples show that numerical results coincide with the theoretical results, and the algorithm can provide accurate results with little calculating works.
Keywords: financial mathematics; option pricing; semidiscretization technique; European call option; numerical solution; Black-Scholes model

Title: Independent identical distribution testing of pseudo-random numbers

Authors: ZOU Yongjie
PP: 67-71
Abstract: In this paper, BDS test can effectively detect whether pseudo-random numbers are independently and identically distributed.First,100 000 pseudo-random numbers were generated by linear congruence generator in Matlab, and divided it into 100 groups with each one have 1 000, then tested every groups by BDS method.The experiment showed that BDS test was more strictly than corration coefficient and chisquare test methods.
Keywords: statistical simulation; pseudo-random numbers; Monte Carlo simulation; BDS test

Title: Simulation and comparison of Lasso and other variable selection methods

Authors: HU Yirui, QU Ronghua, CHEN Siyang, XU Jiajing, TONG Xingwei
PP: 63-66
Abstract: Objective: To bring in a new variable selection method: Lasso (the least absolute shrinkage and selectionator) based on shrinkage estimate, and then compare it with other variable selection methods. Methods: First, analyzing several common variable selection methods such as stepwise regression, Akaike information criterion (AIC) and Bayesian information criterion (BIC). After stochastic simulation, comparing correct and incorrect rates of those methods. Results: The stochastic simulation showed that Lasso’s incorrect rate had decreased to 0 when n=200, with obvious advantages. As the simulation times increased, the average correct rate of Lasso (0.951) had reached a relatively higher level. Conclusion: Lasso method is easy to understand, and it has broad application prospects in variable selection.
Keywords: theoretical statistics; variable selection; Lasso; Akaike information criteria; Bayesian information criterion; stepwise selection

Title: SQP-Filter method for constrained Minimax problem

Authors: XIE Yajun, MA Changfeng
PP: 56-62
Abstract: In this paper, a SQP-Filter method is proposed for solving Minimax problems with equality and inequality constraints. In each step, a quadratic programming sub-problem is solved to get the search direction which is used as line search. This method avoids the difficulty of selecting the penalty factor and overcomes Maratos effects successfully. Its global convergence is obtained under some suitable conditions.
Keywords: operations research; Minimax problem; SQP-Filter method; global convergence

Title: Stability and Hopf bifurcation of differential equation model of price with time delay

Authors: WANG Yong, ZHAI Yanhui, TANG Zhanfeng
PP: 49-55
Abstract: By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the linear stability of the system is studied and Hopf bifurcations are established. In particular, the formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Finally, the numerical simulation verifies the theoretical results.
Keywords: differential equation and dynamical system; time delay; differential equation model of price; stability; Hopf bifurcation

Title: Toeplitz operators with unbounded symbols on weighted Dirichlet space

Authors: HE Zhonghua, CAO Guangfu
PP: 43-46
Abstract: This paper constructs a class of unbounded functions on D such that Toeplitz operator with these symbols is compact. Then it also constructs a function φ in L2(D, dμ) which is unbounded on any neighborhood of each boundary point of D, but Tφ is a trace class operator on weighted Dirichlet space.
Keywords: operator theory; Toeplitz operator; weighted Dirichlet space; unbounded symbol; trace class operator; compact operator

Title: Legendre spectral method for solving Abel integral equation

Authors: ZHAO Zhenyu, QI Jiejing
PP: 37-42
Abstract: In this paper, a problem of numerical inversion of Abel transform is considered. A new method for numerical differentiation is introduced and a reasonable method to deal with singular integral is given. The theoretical results and numerical examples show that the new method is effective.
Keywords: computational mathematics; ill posed problem; Legendre spectral method; numerical differentiation; Abel integral equation

Title: Numerical simulations of subsonic-transonic flows in cavities using Gao’s scheme

Authors: LI Mingjun, ZHU Ke
PP: 29-36
Abstract: Some subsonic-transonic flows in cavities with different ratio of length to depth are simulated numerically using Gao’s upwind finite-volume schemes (GUVS) coupling with compressible and incompressible SIMPLE (semi-implicit press linked equations) algorithm in this paper, GUVS is product of reconstructing first order upwind finite volume scheme by Gao’s algorithm. Some subsonic-transonic flows in cavities with different ratio of length to depth are simulated numerically using Gao’s sixth-order upwind finite-volume scheme on structured and unstructured collocated grids. Some excellent numerical results are obtained. For example, the numerical results for subsonic flows in rectangle cavity (i.e lid-driven cavity flows) are quite well in agreement with the Benchmark solutions given by Ghia et al. using multigrid technique, flow characteristics of subsonic-transonic flows in cavities with different ratio of length to depth are also presented. Excellent properties of GUVS are proved by analysis and numerical tests.
Keywords: computational fluid dynamics; flows in cavities; finite volume scheme; SIMPLE algorithm

Title: Order of polynomial xp－x－a over prime field Fp

Authors: WU Genqiang, LI Yanchao
PP: 26-28
Abstract: Let p be a prime and a an element in prime field Fp. The order of the polynomial xp－x－a over Fp was treated. By the analyses of many special cases, a conjecture that the order of xp－x－a over Fp is pp－1, which is then a primitive polynomial over Fp, was proposed. From several different directions, this problem was analyzed and discussed deeply. Several necessary and sufficient conditions and a sufficient condition which is an equation were got to prove this conjecture.
Keywords: algebra; finite fields; primitive polynomial; order of a polynomial

Title: Covariance analysis of teaching effect evaluation and its SPSS realization

Authors: YI Xiuying, WANG Sanbao
PP: 21-25
Abstract: The paper introduces the teaching evaluation method based on the analysis of covariance (ANCOVA), the data come from the student achievement. It uses the regression analysis method to eliminate the influence of foundational achievement, based on this, uses the variance analysis again. In the SPSS environment, statistics examination is implemented to the correlation data, and the statistical analysis result is obtained. The fact proves that combines regression analysis and variance analysis with achievement to evaluate the teaching, it can reflect the teaching effect fairly and objectively, and make the conclusion more realistic.
Keywords: experimental analysis and experimental design; statistics； evaluate； covariance； SPSS

Title: Standard credit default swap pricing model with incomplete information

Authors: XIONG Yuying, LUO Hua
PP: 17-20
Abstract: This paper attempts to consider when the market information is incomplete, investors can not know exactly a company’s true value, investors gather relevant information, and use the obtained information to estimate the right information through Bayesian learning. On this basis the paper studies the pricing of standard credit default swaps (CDS) and gets it’s pricing formula.
Keywords: finance; credit default swaps; Bayesian learning; incomplete information; pricing formula

Title: Approximate controllability of 1-d parabolic system with switched internal controllers

Authors: ZHANG Fengyan, JIA Chaohua
PP: 11-16
Abstract: This paper considers the 1-d parabolic system with two internal controls. At each moment one of the controls must be switched off, that is, its control value must be equal to zero. There must exist two switching controls making the system controllable approximately with the pointwise controls. The main research frame is calculus of variations. This paper first minimizes some quadratic functional, and then uses the point of minimum to construct the appropriate control functions, which make the system approximately controllable with just one effective at arbitrary instants of time.
Keywords: control theory; switching control; approximate controllability; parabolic system

Title: Distance estimation for zeros of solutions of first-order multi-delay differential equations

Authors: WANG Huimin, JIANG Wei
PP: 4-10
Abstract: This paper studies the properties of solutions of first-order multi-delay differential inequalities, and obtains the largest existence interval of positive solution of differential inequality. On this basis, the paper discusses the zero distance of solutions of first-order multi-delay differential equations, and gives the corresponding theorems. The theorems are proved by studying the nature of delay differential inequality and two important conclusions are obtained. The conclusions using lemma play an important role in the proof of the theorems. Finally, to illustrate the correctness of these theorems, the paper gives two examples. The examples are calculated by using Matlab and some important data are obtained, which show the practicality of the theorem.
Keywords: applied mathematics; first-order; delay; zero; distance estimation

Title: Influence of infinitesimals’ order on limit of exponent function with both base and exponent being infinitesimals

Authors: GONG Yicheng, LI Shougui
PP: 1-3
Abstract: This paper discusses the reasons why the limit is indefinite when the base and exponent are both infinitesimals. By comparing the order of the infinitesimal base and infinitesimal exponent, two theorems are put forward and proved and then illustrated. When the base is of the same order as the exponent, limu(x)v(x) is definitely 1, and so when the base is of the lower order than the exponent. Finally some examples are given to illustrate that limu(x)v(x) is indefinite when the base is of the higher order than the exponent.
Keywords: fundamental mathematics; limit; infinitesimal; order; indefiniteness

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