Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {F_n(z)} is studied, where F₁(z) = f₁(z) and F_n(z) = F_n−1(f_n(z)), with f_n → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.     

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.     

微分方程f″ e~(az)f′ Q(z)f=F(z)的复振荡

该文研究了线性微分方程f″ eazf′ Q( z) f=F( z)的复振荡问题,其中Q( z)、F( z) ( 0 )是整函数,且σ( Q) =1 ,σ( F) < ∞,Q( z) =h( z) ebz,h( z)是多项式,b≠- 1是复常数,那么上述线性微分方程的所有解f( z)满足λ( f) =λ( f) =σ( f) =∞,　λ2 ( f) =λ2 ( f) =σ2 ( f) =1 .至多除去两个例外复数a及一个可能的有穷级例外解f0 ( z) .     

Rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ f ( x ) 2 ± f ( y ) 2 = z 2

Periodica Mathematica Hungarica - We discuss the rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ . This problem can be solved either by the theory of elliptic curves or by...     

The Value Regions of the Systems {f(z 1), f′(z 1)} and {f(z 1), f(z 2)} in the Class of Typically Real Functions

In the class T consisting of regular and typically real functions in the disk |x| < 1, the value regions of the system {f(z ₁), f(z ₁)} and {f(z ₁), f(z ₂)} are found for fixed z ₁ and z ₂. As an application, the value regions of f(z ₁) and f(z ₂) are found for f T with fixed value f(z ₁). Bibliography: 11 titles.     

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n ² ? 1 and 12n ² + 1 are odd primes, then the general elliptic curve y ² = x ³+(36n ²?9)x?2(36n ²?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y ² = x ³ + 27x ? 62 really is an unusual elliptic curve which has large integral points.     

关于代数微分方程(f')~n=R(z,f)的亚纯解

在本文中,我们首先考虑了具有理系数的代数微分方程（ｆ＇）ｎ＝Ｒ（ｚ,ｆ）亚 纯解的个数估计问题,并举例说明所得结果是精确的．其次,我们运用 Ｎｅｖａｎｌｉｎｎａ值 分布论,讨论了具亚纯系数的典型代数微分方程（ｆ＇）３＝ａ０（ｆ－ τ１）２（ｆ－ τ２）２（ｆ－ τ３）２ 的可分解亚纯解．文中的结果推广或改进了高仕安［１］,Ｇｕｎｄｅｒｓｅｎ Ｇ．和ＬａｉｎｅＩ［２］以 及何育赞, ＬａｉｎｅＩ．［３－５］等人的工作．     

关于代数微分方程(f')~n=R(z,f)的亚纯解

在本文中,我们首先考虑了具有理系数的代数微分方程（ｆ＇）ｎ＝Ｒ（ｚ,ｆ）亚 纯解的个数估计问题,并举例说明所得结果是精确的．其次,我们运用 Ｎｅｖａｎｌｉｎｎａ值 分布论,讨论了具亚纯系数的典型代数微分方程（ｆ＇）３＝ａ０（ｆ－ τ１）２（ｆ－ τ２）２（ｆ－ τ３）２ 的可分解亚纯解．文中的结果推广或改进了高仕安［１］,Ｇｕｎｄｅｒｓｅｎ Ｇ．和ＬａｉｎｅＩ［２］以 及何育赞, ＬａｉｎｅＩ．［３－５］等人的工作．     

Sur l'intégration de l'équationdz=f(x,y, z)dx+g(x,y, z)dy

Résumé On étudie le système (2), f et g satisfaisant certaines conditions analogues à celles de Carathéodory pour les équations différentielles ordinaires et on énonce des conditions suffisantes pour que ce système admette une intégrale et, de même, des conditions suffisantes pour que l'équation (13) admette une intégrale. à M. Enrico Bompiani pour son Jubilé scientifique.     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.     

Regular functions f(z) for which zf'(z) is θ-spiral-like of order α

Siberian Mathematical Journal -     

The periodic points and the invariant set of an ϵ-contractive map

It is shown that the invariant set of an ϵ-contractive map f on a compact metric space X is the same as the set of periodic points of f. Furthermore, the set of periodic points of f is finite and, only assuming that X is locally compact, there is at most one periodic point in each component X. The theorems are applied to prove a known fixed-point theorem, a result concerning inverse limits, a result about periodic points of compositions, and a result showing that ϵ-contractive maps on continua are really contraction maps with a change in metric. It is shown that all our results hold for locally contractive maps on compact metric spaces.     

A family of quadriphase sequences of period 4(2 n − 1) with low correlation and large linear span

In this paper, a new family of quadriphase sequences with period 4(2 ⁿ ? 1) is proposed for n = me, where m is an odd integer. The correlation values of the family, their distribution, and the linear spans of the proposed sequences are completely determined under two situations.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).