|z|~2=z·的妙用

:·三=}:}’是复数绝对值运算的主要工具,而当1:}~l时,灵活运用!:}’一:·三一1可以达到化难为易,化繁为简的目的,使解题过程简捷巧妙. 例1设复数a、刀、y有.al二l刀,二l,,I~解设。~ }a!一1 1且:井一生.a二工畔a十之1十“名一—-一a, 口:并一茂(刀 下)(y a)(a十刀) a声夕要使。     

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.     

Hard Squares for z = �C1

The hard square model in statistical mechanics has been investigated for the case when the activity z is −1. For cyclic boundary conditions, the characteristic polynomial of the transfer matrix has an intriguingly simple structure, all the eigenvalues x being zero, roots of unity, or solutions of x ³ = 4cos²(πm/N). Here we tabulate the results for lattices of up to 12 columns with cyclic or free boundary conditions and the two obvious orientations. We remark that they are all unexpectedly simple and that for the rotated lattice with free or fixed boundary conditions there are obvious likely generalizations to any lattice size.     

The Diophantine Equations x2± y4=±z6 and x2+y8= z3

In this article we determine all solutions to the equationx ^p+y ^q=z ^r, (p,q,r){(2,4,6), (2,6,4), (4,6,2), (2,8,3)} in coprime integers x,y,z. First we determine a set of curves of genus 2, such that every solution corresponds to a rational point on one of these curves. Then we determine the rational points on these curves using either covers of rank 0 elliptic curves or a method known as effective Chabauty which works if theMordell–Weil rank of the jacobian is smaller than the dimension.     

On Univalence of the Power Deformation z（f（z）/z）c

The authors mainly concern the set U _f of c ?? ? such that the power deformation $ z(\frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a ₂ z ² + ?? in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ? unless f is the identity function. In particular, the interior of U _f is simply connected. This fact enables us to apply various versions of the ??-lemma for the holomorphic family $ z(\frac{{f(z)}} {z})^c $ of injections parametrized over the interior of U _f . The necessary or sufficient conditions for U _f to contain 0 or 1 as an interior point are also given.     

On the left side derivative of the Hausdorff dimension of the Julia sets for z 2 + c at c = −3/4

Let d(c) denote the Hausdorff dimension of the Julia set J _c of the polynomial f _c (z) = z ² + c. The function c ? d(c) is real-analytic on the interval (?5/4,?3/4), which is included in the 1/2 bulb of the Mandelbrot set. The number c = ?3/4 is the parameter at which, going from the right, the attracting fixed point bifurcates to an orbit of period two. Recently [13], we studied d′(c) when c ? ?3/4 from the right. Here, under numerically verified assumption d(?3/4) < 4/3, we will show that there exists K _?3 ^4/? > 0 such that d′(c)(?3/4 ? c)^{?3d(?3/4)/2+2} → ?K _?3 ^4/? , when c tends to ?3/4 from the left. In particular we obtain d′(c) ? ?∞. This case is much harder than that considered in [13].     

Pell方程组x2-ay2=1和 z2-by2=1的解数

设a,b是不同的正整数,本文证明了当max(a,b)＞10126时,Pell方程组x2-ay2=1和z2-by2=1至多有2组正整数解(x,y,z).     

Die Unmöglichkeit von x6+y6=z6 und x9+y9=z9 in quadratischen Körpern

Ohne Zusammenfassung     

Zeros of the partial sums of cos (z) and sin (z). I

We extend results of Szeg (1924) and Kappert (1996) on the location of the zeros of the normalized partial sums of cos(z) and sin(z), and their rates of convergence to the associated Szeg curves.     

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.     

SOLUTIONS OF THE EQUATION x^m＋y^m=z^m IN SL2z

This paper solves an open problem of Vaserstein. The main result is: the equation x^m y^m=z^m has a solution in SL2z if and only if m is not divisible by 4 or 6 and m is not congruent to 3 modulo 6.     

关于方程f″+Af=F(z)的复振荡

本文讨论了当A(z)为多项式,F(z)为具有无穷多个零点的整函数时,微分方程 f″+A(z)f=F(z)的解f(z)的复振荡的性质.     

The pluripolar hull of {w=e −1/z }

In this paper we show that the pluripolar hull ofE={(z, ω)∈C²:ω=e ^−1/z,z≠0} is equal toE. This implies thatE is plurithin at 0, which answers a question of Sadullaev. The result remains valid ife ^−1/z is replaced by certain other holomorphic functions with an essential singularity at 0.     

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) .