Maximum modulus principle estimates for one dimensional fractional diffusion equation

We present scheme I for solving one-dimensional fractional diffusion equation with variable coefficients based on the maximum modulus principle and two Grunwald approxima- tions. Scheme II is obtained by using classic Crank-Nicolson approximations in order to improve the time convergence. Schemes are proved to be unconditionally stable and second-order accuracy in spatial grid size for the problem with order of fractional derivative belonging to [（√17- 1）/2, 2] using the maximum modulus principle. A numerical example is given to confirm the theoretical analysis result.

A PRIORI ESTIMATE FOR MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS

Let G he a hounded domain in E Consider the following quasi-linear elliptic equationAlthough the houndedness of generalized solutions of the equation is proved for very general structural conditions, it does not supply a priori estimate for maximum modulus of solutions. In this paper an estimate to the maximum modulus is made firstly for a special case of quasi-linear elliptic equations, i.e. the A and B satisfy the following structural conditions

Maximum Modulus of Polynomials

The result is best possible. In this paper we shall present a refinement of this result and some other related results.

Effect of consolidation ratios on maximum dynamic shear modulus of sands

The dynamic shear modulus (DSM) is the most basic soil parameter in earthquake or other dynamic loading conditions and can be obtained through testing in the field or in the laboratory. The effect of consolidation ratios on the maximum DSM for two types of sand is investigated by using resonant column tests. And, an increment formula to obtain the maximum DSM for cases of consolidation ratio κc>1 is presented. The results indicate that the maximum DSM rises rapidly when κc is near 1 and then slows down, which means that the power function of the consolidation ratio increment κc-1 can be used to describe the variation of the maximum DSM due to κc>1. The results also indicate that the increase in the maximum DSM due to κc>1 is significantly larger than that predicted by Hardin and Black's formula.

Some Results Concerning Growth of Polynomials

Let P（z） be a polynomial of degree n having no zeros in ｜z｜〈 1, then for every real or complex number β with ｜β｜≤ 1, and ｜z｜ = 1, R ≥ 1, it is proved by Dewan et al. [4] that｜P（Rz）＋β（R＋1/2）^nP（z）｜≤1/2{（｜R^n＋β（R＋1/2）^n｜＋｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜ -（｜R^n＋β（R＋1/2）^n｜-｜1＋β（R＋1/2）^n｜max｜z｜=1｜P（z）｜}.In this paper we generalize the above inequality for polynomials having no zeros in }z} 〈 k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.

ON THE GENERALIZED ORDER OF DIRICHLET SERIES

By the method of Knopp-Kojima, the generalized order of Dirichlet series is studied and some interesting relations on the maximum modulus, the maximum term and the coefficients of entire function defined by Dirichlet series of slow growth are obtained, which briefly extends some results of paper [1].

THE PITS PROPERTY OF ENTIRE FUNCTIONS DEFINED BY DIRICHLET SERIES

The value distribution of entire functions defined by Dirichlet series are studied in this present article.It is proved that entire functions defined by Dirichlet series have the pits property,which improve the relative results on lacunary Taylor series obtained by Littlewood J.E.and Offord A.C.

On an Inequality of Pual Turan Concerning Polynomials-Ⅱ

Let P（z） be a polynomial of degree n and for any complex number α, let D;P（z） = nP（z）+（α- z） P′（z） denote the polar derivative of the polynomial P（z） with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.

Some Inequalities for the Polynomial with S-Fold Zeros at the Origin

Let p（z） be a polynomial of degree n, which has no zeros in ｜z｜〈 1, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, pp. 38-41] established｜zp＇（z）＋nβ/2p（z）｜≤n/2{（｜β/2｜＋｜1＋β/2｜）｜z｜=1max｜p（z）｜-（｜1＋β/2｜-｜β/2｜）｜z｜=1min｜p（z）｜},for any ｜β｜≤ 1 and ｜z｜ = 1. In this paper we improve the above inequality for the polynomial which has no zeros in ｜z｜ 〈 k, k≥ 1, except s-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.

Some Generalization for an Operator Which Preserving Inequalities Between Polynomials

For a polynomial p（z） of degree n which has no zeros in ｜z｜ 〈 1, Dewan et al., （K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 （2010）, 38-41） establishedfor any complex number β with ｜β｜≤ and｜z｜ = 1. In this paper we consider the operator B, which carries a polynomial p（z） into

A Note on a Theorem of T.J.Rivlin

In this paper, we obtain a result that improves the results of Govil and Nwaeze, Qazi and the classical result of Rivlin.

On Growth of Polynomials with Restricted Zeros

Let P（z） be a polynomial of degree n which does not vanish in ｜z｜ 〈k, k≥ 1. It is known that for each 0 〈 s 〈 n and 1 ≤ R ≤ k,M（P（s）,R）≤（1/（R^s＋k^s）[{d（s）/dx（s）（1＋x^n）}x=1]（（R＋k）/（1＋k）^nM（P,1）. In this paper, we obtain certain extensions and refinements of this inequality by in- volving binomial coefficients and some of the coefficients of the polynomial P（z）.

全平面上Laplace-Stieltjes变换所表示整函数的增长性

In the paper, the α-order of the Laplace-Stieltjes Transform is introducedfirstly, then we get the relationship between α-order represented by the maximum modulus and α-order represented by A^*n, λn. Lastly, we obtain the relationship between type τrepresented by the maximum modulus and type τ represented by A^*n, λn.

Inequalities for the Polar Derivatives of a Polynomial

Let P（z） be a polynomial of degree n, having all its zeros in ｜z｜≤ 1. In this paper, we estimate kth polar derivative of P（z） on ｜z｜ = 1 and thereby obtain compact generalizations of some known results which among other things yields a refinement of a result due to Paul Tura＇n.

STUDY ON DAMAGE BIFURCATION AND INSTABILITY OF ROCK-LIKE MATERIALS

The critical bifurcation orientation and its corresponding hardening modulus for rock-like geomaterials are derived by considering the effect of stiffness degradation and volumetric dilatancy under the assumption of isotropic damage. The dependency of the localized orientation on the degree of damage and initial Poisson＇s ratio of rock is examined and the bifurcation behavior of the uniaxial compression sample under the plane-stress condition is compared with that under plane-strain condition. It is shown that the localization orientation angle intimately depends on both the initial Poisson＇s ratio and degree of damage for the rock sample under the uniaxial compression condition. As the initial Poisson＇s ratio or degree of damage increases, the orientation angle of the plane on which localization tends to be initiated gets to decrease. At the same time, the localization orientation angle of a rock sample under the plane-stress condition is larger than that under the plane-strain condition.

Digital discrimination of neutron and γ ray using an organic scintillation detector based on wavelet transform modulus maximum

A novel algorithm for the discrimination of neutron and y-ray events with wavelet transform modulus maximum（WTMM） in an organic scintillation has been investigated.Voltage pulses arising from a BC501A organic liquid scintillation detector in a mixed radiation field have been recorded with a fast digital sampling oscilloscope.The WTMM method using frequency-domain features exhibits a strong insensitivity to noise and can be used to discriminate neutron and y-ray events based on their different asymptotic decay trend between the positive modulus maximum curve and the negative modulus maximum curve in the scale-space plane.This technique has been verified by the corresponding mixed-field data assessed by the time-of-flight（TOF） method and the charge comparison（CC）method.It is shown that the characterization of neutron and y ray achieved by the discrimination method based on WTMM is consistent with that afforded by the TOF method and better than the CC method.Moreover,the WTMM method itself has presented its ability to eliminate the noise without any pretreatment to the pulses.

Performance Improvement of Raman Distributed Temperature System by Using Noise Suppression

In Raman distributed temperature system, the key factor for performance improvement is noise suppression, which seriously affects the sensing distance and temperature accuracy. Therefore, we propose and experimentally demonstrate dynamic noise difference algorithm and wavelet transform modulus maximum （WTMM） to de-noising Raman anti-Stokes signal. Experimental results show that the sensing distance can increase from 3 kin to 11.5 km and the temperature accuracy increases to 1.58 ℃ at the sensing distance of 10.4kin.