Normal families related to shared sets

We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a positive integer. If for every f∈ F, i） the zeros of f（z） have a multiplicity of at least k＋ 1, and ii） E^-f（k）（S） lohtain in E^-f（S）, then F is normal on .4. At the same time, the corresponding results of normal function are also proved.

Normal Families of Meromorphic Functions and Shared Set

In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f have multiplicity at least 2 and there exist nonzero complex numbers b_f,c_f satisfying(i) b_f/c_f is a constant;(ii) min{σ(0,b_f),σ(0,c_f),σ(b_f,c_f)} ≥m for some m > 0;(iii) E_f'(S_f)■ E_f(S_f),where S_f = {b_f,c_f}.Then F is normal in A.At the same time,the corresponding results are also proved.The results in this paper improve and generalize the related results

Uniqueness of Entire Function Related to Shared Set

In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such that if f and its derivative f（k）satisfy E（S,f）= E（S,f（k））,and the zeros of f（z）-d are of multiplicity ≥ k ＋ 1,then f = f（k）.

Shared Sets and Normal Families

This paper deals with the problem of normal families concerning share sets. Moreover, the examples show that the conditions of theorem are necessary.

Meromorphic Functions That Share Three Sets

In this paper we study the problem of meromorphic functions sharing three sets and obtain some theorems which are the extension and complement of some known relative results given by Yi Hongxun and others.

The Uniqueness of Meromorphic Functions with Common Range Sets and Deficient Values

In this paper, the uniqueness of meromorphic functions with common range sets and deficient values are studied. This result is related to a question of Gross.

A Note on Normal Families of Meromorphic Functions and Sharing Set

In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.

UNIQUENESS THEOREMS OF L-FUNCTIONS IN THE EXTENDED SELBERG CLASS

We establish uniqueness theorems of L-functions in the extended Selberg class,which show how an L-function and a meromorphic function are uniquely determined by their shared values in two finite sets.This can be seen as a new solution of a problem proposed by Gross.

URBAN FORM FOR A SUSTAINABLE FUTURE: How Sustainable Is Distributed Working in the Networked City?

Our understanding of urban form depends on how we perceive the city.Much of the literature on urban form(1)has focused on the pre-industrial and industrial city,celebrating its compact form,contiguous functions and single dominant centre.More recently writings by Castels(2)and Soja(3)have described the dispersed,city of the post industrial era.This networked city triggered by the freedom afforded by the new technology(4)has exposed a new dimension to urban form.The model of the compact city advocated by those such as Lord Rogers Task Force for delivering the Urban Renaissance(5)is being questioned(6)and a new model of“high density nodes,in a high density landscape resulting in a low density city,”as in the Deltametropolis,described by Dirk Frieling(7).Compactness,cramming more development into the city and making public spaces of a higher density and quality,Rogers and Burdett argue(8)will make“urban living attractive,ecologically sustainable,economically strong and socially inclusive.”The alternative argument is that the economic success of cities is reliant on the networking of resources across a metropolitan region.Echenique argues(9)that cities disperse in their search for mobility and space.“Mobility increases the effi ciency of households and fi rms which in turn generates more income and profi ts.As income increases,so does the demand for space,residential and commercial alike.”Sustainability has become the current banner of political correctness.Sustainability however is a slippery word.It is easy to focus on one aspect and lose the value of its holistic meaning.For many architects“green buildings”equals a sustainable future.However,clever design solutions single-mindedly pursued with little regard to the wider exploration of the potential environmental savings that may be achieved through organisational innovation are only half the answer.A holistic approach concerned with both building and organisational design and focused on“lean thinking”(10)could make considerable inroads into reducing the ecological footprint.The paper draws on DEGW’s experience of advising major corporations and cities on strategies for managing the process of intensifi cation and change(11).It explores how major improvements might be gained in meeting our goals for the sustainable city through reconsidering the way we work and allocate space.The underlying proposition is that technology has offered us new opportunities which have changed our paradigm of living and working.This in turn has provided us with a new perception of the city,as a distributed series of high density centres connected by good public and private transport,within a low density landscape.The paper argues that considerable improvements in workplace sustainability can be achieved by applying a holistic approach.These may include a combination of strategies,from rethinking the organisation of work processes and the locations and time work is undertaken,to reducing the need for resources by a more intensive use of land and fl oor space.Disjointed,dispersed“urban sprawl”can be wasteful.The alternative emerging urban form is a planned,dispersed,“networked”city with well integrated public and private transport that yields greater choice of location and lifestyles so supporting social,economic and environmental sustainability.

On Unicity of Meromorphic Functions and Their Derivatives

In this paper, we prove the following theorem: Suppose that k , n be two positive integers, and a , b , w be three finite complex numbers with a n≠b n , w n=1 . If a meromorphic function f(z) and its k-th derivative f (k) (z) share two finite set S 1={aw i | i=1,2,…, n} , S 2={bw i | i=1,2,…,n} , then f(z)=tf (k) (z) , where t n=1.