On an Extension of an Theorem of Hayman

Hayman proved the following Theorem A Let φ=f'-αf~n, α≠0,α∈C,1 ) If f be transcendental meromorphic and n≥5, then φ assune all finite valuesand infinitely often.2 ) If f be transcendental integral and n≥3, then φ assume all finite values andinfinitely often.     

an＋1=anq与an＋1／an=q等价吗

在证明等比数列的操作中常会看到这样一种现象:在一定条件下,推得递推公式a_(n 1)=a_nq(q为常数)(1)那么就认为数列{a_n}是等比数列了.为对照说明方便,下面列出等比数列的定义:a_(n 1)/a_n=q(n∈N~*)(2)以下以新教材高中数学第一册(上)参考题三一习题为线索来探讨(1)式与(2)式的等价性问题.     

On an Arithmetic Function

Ｌｅｔｆ（ｎ）ｄｅｎｏｔｅｔｈｅｎｕｍｂｅｒｏｆｆａｃｔｏｒｉｚａｔｉｏｎｓｏｆｔｈｅｎａｔｕｒａｌｎｕｍｂｅｒｎｉｎｔｏｆａｃｔｏｒｓｌａｒｇｅｒｔｈａｎ１，ｗｈｅｒｅｔｈｅｏｒｄｅｒｏｆｔｈｅｆａｃｔｏｒｓｄｏｅｓｎｏｔｃｏｕｎｔ．Ａｌｓｏｌｅｔ...     

The canonical embedding of an unramified morphism in an ��tale morphism

We show that every unramified morphism ${X\to Y}$ has a canonical and universal factorization ${X\hookrightarrow E_{X/Y}\to Y}$ where the first morphism is a closed embedding and the second is étale (but not separated).     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

How to Trisect an Angle

Two of the first compass-and-straightedge constructions a child learns in plane geometry are the bisection of an angle and the division of a line segment into any desired number of equal parts. Both are so easy to do that many pupils find it hard to believe there is no way in which the two instruments can be used to trisect an angle Indeed.it     

唇齿相依的an与Sn

从所周知,如何由数列|an|的递推关系式求数列的通项公式an,如何由数列的通项公式an求数列的前n项和公式Sn是<数列>这一章中我们要解决的两大基本问题,当已知数列的前n项和公式Sn时,则通过an={S1=(n=1),Sn-Sn-1(n≥2),很容易求得数列通项公式an,an与Sn可谓"唇齿相依";在高考与竞赛中还时常出现由an与Sn的关系式,求数列的通项公式的问题,本文通过一个典型题目的多种解法,介绍解决这类问题的几何常用策略,供大家参考.……     

Can an a priori error estimate for an approximate solution of an ill-posed problem be comparable with the error in data?

For a linear operator equation of the first kind with perturbed data, it is shown that the global (on typical sets) a priori error estimate for its approximate solution can have the same order as that for the approximate data only if the operator of the problem is normally solvable. If the operator of the problem is given exactly, this is possible only if the problem is well-posed (stable).     

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.     

Corrigendum to “New algebraic properties of an amalgamated algebra along an ideal”

At some point, after publication, we realized that Proposition 4.1(2) and Theorem 4.4 in [2 D’Anna, M., Finocchiaro, C. A., Fontana, M. (2016). New algebraic properties of an amalgamated algebra along an ideal. Commun. Algebra 44(5):1836–1851.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] hold under the assumption (not explicitly declared) that B = f(A)+J. Furthermore, we provide here the exact value for the embedding dimension of A?^fJ, also when B≠f(A)+J, under the hypothesis that J is finitely generated as an ideal of the ring f(A)+J.     

Statistical Prediction under an Order Restriction

In this paper, a statistical prediction problem under ordered location and scale parameters are considered. Double-shrinkage predictors are given which use all the available data and improve on single-shrinkage predictors, and hence the best equivariant predictors.     

Concerning an approach to identifying the Lamé parameters of an elastic functionally graded cylinder

Based on the general linear elasticity relations, an axisymmetric problem on the steady-state oscillations of a functionally graded hollow cylinder is formulated. The Lamé parameters are considered variable in radial coordinate. Oscillations are caused by the distributed load applied to the outer part of the cylinder boundary. Using the variable separation method, the direct problem on determining the radial and longitudinal components of the displacement field is investigated. The influence of the laws of variation for the Lamé parameters on acoustic characteristics is analysed. The inverse coefficient problem on the identification of the variable Lamé parameters from the data on the amplitude-frequency characteristic is stated. Based on the weak formulation of the problem for an elastic inhomogeneous body, a general linearised relation for the desired and given characteristics is obtained. A system of the Fredholm integral equations of the first kind is formulated with respect to two unknown corrections to the restored laws of the Lamé parameters change. The solution is built by means of an iterative process. A reconstruction of various laws of changing the Lamé parameters is carried out. The accuracy of the presented algorithm is estimated, and recommendations for the most efficient implementation of the reconstruction procedure are proposed.     

What makes an investigation difficult?

The national school-leaving examination in mathematics in England and Wales includes both formal timed written examinations, and a school-based component. Although originally conceived in quite broad terms, the school-based component has been operationalised in terms of quite a narrow range of problems, typically combinatoric in nature, and generally called ‘investigations’. The marking guides given to teachers for the assessment of these problems are generic, in that they are meant to apply to all such problems. In this article it is argued that generic marking guides are unlikely to be successful, and that teachers engaged in assessing school-based work need to take into account features of the specific activity in which the student engages. A corpus of 20,000 student responses to 80 investigative tasks were analyzed to produce a framework of task-factors. The framework characterises tasks in terms of seven factors arranged in four categories. The first category involves the match between the task metaphor and the intended task and the extent to which the task metaphor is likely to be shared by students. The second category concerns the complexity of the task structure, both in terms of the search-space of the task and the relationship between the dependent and independent variables. The third category involves the complexity of the generalisation, both as a term-to-term and as a position-to-term rule. The fourth category is related to whether the kinds of proof required (or possible) are inductive or deductive.     

On an Inequality of Jenkins Type

ＯｎａｎＩｎｅｑｕａｌｉｔｙｏｆＪｅｎｋｉｎｓＴｙｐｅ￥ＷｅｉＨａｎｂａｉ（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ，２００４３３，Ｐ．Ｒ．Ｃ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐｅｐｅｒ，ｔｈｅａｕ...     

Ball launched against an inclined plane – an example of using recurrent sequences in school physics

This note gives the solution to the motion of a perfectly hard ball that was launched against a perfectly hard inclined plane. The solution comprises recurrent formulas that give the parts of the trajectory between the impacts, which are parabolas. In the upward part of the motion, the parabolas become higher and narrower, in the downward part they become lower and wider. Using an interactive spreadsheet model, the law of energy conservation is verified, and the effect of inputs on the shape of the trajectory is examined. Two special cases of the motion are of particular interest when the upward and downward parts of the trajectory merge into one. The solution illustrates the use of recurrent sequences in physics, and computer investigation into a dynamic physical system.     

递推数列an＋1-p＋q／an的通项公式及其性质

文 [1]中给出并证明了以下命题 .命题　设ａ ,ｂ∈Ｒ ,且实数ｘ1,ｘ2 ,… ,ｘｎ 满足ｘ1=ａ ｂｘｎ,ｘ2 =ａ ｂｘ1,…ｘｎ=ａ ｂｘｎ - 1.则当ｎ为奇数时 ,ａ1=ａ2 =… =ａｎ;当ｎ为偶数时 ,ａ1=ａ3 =… =ａｎ- 1,ａ2 =ａ4 =…=ａｎ.上述命题实际上给出了满足ａｎ 1=ｐ ｑａｎ 的数列 {ａｎ}的一个性质 .经研究 ,本文拟给出满足ａｎ 1=ｐ ｑａｎ的数列 {ａｎ}的通项公式 ,并以此为基础给出比文 [1]更为深刻的结论 .现在记 ｆ(ｋ) =Ｃ1ｋｂｋ- 1 Ｃ3 ｋ 1ｂｋ - 2 … Ｃ2ｋ- 3 2ｋ- 2 ｂ1 Ｃ2ｋ- 12ｋ- 1ｂ0 (ｋ≥ 1,ｋ∈…     

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

“Quasiphotons” in an active medium

We consider the propagation of concentrated wave packets along space-time rays for the case of frequency-dependent velocity: c=c_o(x,t)+c₁(x,t,), 1. The complex ray method is applied to construct formal asymptotic expansions. It is shown that these packets may propagate only at certain medium-determined frequencies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 104–115, 1985.In conclusion, I would like to thank I. A. Molotkov for proposing the problem and for useful comments and A. P. Kachalov for fruitful discussion.     

Positive Entire Solutions to an Elliptic System

Ｉｎａｒｅｃｅｎｔｐａｐｅｒ [1],ＹｅｈａｓｄｅｍｏｎｓｔｒａｔｅｄｔｈａｔｔｈｅｓｉｎｇｕｌａｒｎｏｎｌｉｎｅａｒｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎΔｕ =ｆ(ｕ , ｕ ,ｘ ) /ｕλ,　ｘ∈Ｒ2 ,0 <λ<1ｈａｓａｐｏｓｉｔｉｖｅｅｎｔｉｒｅｓｏｌｕｔｉｏｎｕｎｄｅｒｓｏｍｅｃｏｎｄｉｔｉｏｎｓｉｍｐｏｓｅｄｏｎｆ.Ｐｒｏｂｌｅｍｓｏｆｆｉｎｄｉｎｇｐｏｓｉｔｉｖｅｅｎｔｉｒｅｓｏｌｕｔｉｏｎｓｔｏｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎｓｈａｖｅｂｅｅｎｓｔｕｄｉｅｄｂｙｍａｎｙａｕｔｈｏｒｓｓｉｎｃｅｔｈｅｙｈ…