On completely multiplicative functions whose values are roots of unity

Summary We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = U_ℓ, where U_ℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = U_ℓ.     

高阶常系数线性微分方程的一种积分形式特解

利用微分算子及n阶常系数非齐次线性微分方程的特征方程根与系数的关系给出其特解的逐次积分形式,并由此给出自由项f(x)=Pm(x)eλx(其中Pm(x)为m次多项式)时特解的简单递推公式.     

植物根土界面摩擦力的影响因素研究

为探索影响林木根系与土壤界面摩擦力的因素,采用施加拔出荷载的方式对林木根系进行单根拔出试验.结果表明:根系与土壤界面的摩擦力随根系直径和埋深的增加而呈线性规律增大;平均摩擦力的大小随土壤含水率的增加呈现先增大后减小的规律;摩擦力随加载速率的增加而减小;不同树种的根系与土壤摩擦力和摩擦系数差异显著,平均摩擦力大小表现为白桦落叶松蒙古栎油松,平均摩擦系数表现为落叶松蒙古栎白桦油松.     

Extended Jury criterion

In this paper,algebraic criteria are established to determine whether or not a real coefficient polynomial has one or two pairs of conjugate complex roots whose moduli are equal to 1 and the other roots have moduli less than 1 directly from its coefficients.The form and the function of the criteria are similar to those of the Jury criterion which can be used to determine whether or not all the moduli of the roots of a real coefficient polynomial are less than 1.     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

银行存款模型及应用分析

近年来,越来越多的国际知名企业都认识到了从原始数据中寻找规律对决策管理的重要性.一些顶尖级的银行产品企业,象IBM,Oracle,Informix和Sybase等,已经开发了许多用于银行数据挖掘的软件产品,并为国际上的一些著明银行建立了高精度的统计模型以支持银行管理.存款是银行评价业绩的一项重要指标.建立高精度的存款模型有利于银行的日常资金管理,能提高银行的资金利用率,降低成本等.本文以国内某大城市两大银行的实际业务数据为背景,给出了银行存款模型的建立过程,并分析了模型的应用.本文的一些有趣结果对时间序列建模有一定的启示.     

Chemical Composition,Antifungal and Anti-Biofilm Activities of Volatile Fractions of Convolvulus althaeoides L. Roots from Tunisia

The antifungal drugs currently available and mostly used for the treatment of candidiasis exhibit the phenomena of toxicity and increasing resistance. In this context, plant materials might represent promising sources of antifungal agents. The aim of this study is to evaluate for the first time the chemical content of the volatile fractions (VFs) along with the antifungal and anti-biofilm of Convolvulus althaeoides L. roots. The chemical composition was determined by gas chromatography coupled to a flame ionization detector and mass spectrometry. In total, 73 and 86 chemical compounds were detected in the n-hexane (VF1) and chloroform (VF2) fractions, respectively. Analysis revealed the presence of four main compounds: n-hexadecenoic acid (29.77%), 4-vinyl guaiacol (12.2%), bis(2-ethylhexyl)-adipate (9.69%) and eicosane (3.98%) in the VF extracted by hexane (VF1). n-hexadecenoic acid (34.04%), benzyl alcohol (7.86%) and linoleic acid (7.30%) were the main compounds found in the VF extracted with chloroform (VF2). The antifungal minimum inhibitory concentrations (MICs) of the obtained fractions against Candida albicans, Candida glabrata and Candida tropicalis were determined by the micro-dilution technique and values against Candida spp. ranged from 0.87 to 3.5 mg/mL. The biofilm inhibitory concentrations (IBF) and sustained inhibition (BSI) assays on C. albicans, C. glabrata and C. tropicalis were also investigated. The VFs inhibited biofilm formation up to 0.87 mg/mL for C. albicans, up to 1.75 mg/mL against C. glabrata and up to 0.87 mg/mL against C. tropicalis. The obtained results highlighted the synergistic mechanism of the detected molecules in the prevention of candidosic biofilm formation.     

Double roots of power series and related matters

For a given collection of distinct arguments , multiplicities and a real interval containing zero, we are interested in determining the smallest for which there is a power series with coefficients in , and roots of order respectively. We denote this by . We describe the usual form of the extremal series (we give a sufficient condition which is also necessary when the extremal series possesses at least non-dependent coefficients strictly inside , where is 1 or 2 as is real or complex). We focus particularly on , the size of the smallest double root of a power series lying on a given ray (of interest in connection with the complex analogue of work of Boris Solomyak on the distribution of the random series ). We computed the value of for the rationals in of denominator less than fifty. The smallest value we encountered was . For the one-sided intervals and the corresponding smallest values were and .

     

On roots and squares – estimation,intuition and creativity

The paper presents findings of a small scale study of a few items related to problem solving with squares and roots, for different teacher groups (pre-service and in-service mathematics teachers: elementary and junior high school). The research participants were asked to explain what would be the units digit of a natural number to be squared in order to obtain a certain units digit as a result. They were also asked to formulate a rule – an algorithm for calculating the square of a 2-digit number which is a multiple of 5. Based on this knowledge and estimation capability, they were required to find, without using calculators, the square roots of given natural numbers. The findings show that most of the participants had only partial intuition regarding the units’ digit of a number which is squared when the units’ digit of the square is known. At the same time, the participants manifested some evidence of creativity and flow of ideas in identifying the rule for calculating the square of a natural number whose units digit is 5. However, when asked to identify, by means of estimation and based on knowledge from previous items, the square roots of three natural numbers, only few of them managed to find the three roots by estimation.