normal criteria for families of meromorphic functions by using nevanlinna theory we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family

density of states and dynamic scaling on the vicsek snowflake fractal real space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the heisenberg ferromagnet and antiferromagnet on the infinite vicsek snowflake fractal the authors have calculated the associated dynamic exponents zf and za by linearising about the respective fixed points of the ferromagnetic antiferromagnetic recursion relations and find them to satisfy the relation za zf 2 in addition using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho f omega and rho a omega have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha omega approximately bz alpha df rho lambda alpha omega near their respective fixed points alpha a f df denoting the fractal dimension finally the amplitudes galpha omega appearing in the solution to the scaling equation rho alpha omega approximately galpha omega omega dfz alpha 1 for the density of states rho alpha omega were found to be periodic functions of ln omega with period ln lambda alpha where lambda alpha is the eigenvalue associated with ferromagnetic alpha f or antiferromagnetic alpha a fixed point

normality criteria of meromorphic functions with multiple zeros abstract take positive integers n k 2 let f be a family of meromorphic functions in a domain d c such that each f f has only zeros of multiplicity at least k if for each pair f g in f f f k n and g g k n share a non zero complex number a ignoring multiplicity then f is normal in d

normal families new perspectives this paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains these include short and efficient proofs of generalizations of i the picard theorems ii gol dberg s theorem a meromorphic function on c which is the solution of a first order algebraic differential equation has finite order and iii the fatou julia theorem the julia set of a rational function of degree d 2 is the closure of the repelling periodic points we also discuss bloch s principle and provide simple solutions to some problems of hayman connected with this principle over twenty years ago on the way to a partial explication of the phenomenon known as bloch s principle i proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains 68 over the years the lemma has grown and in dextrous hands proved amazingly versatile with applications to a wide variety of topics in function theory and related areas with the renewed interest in normal families arising largely from the important role they play in complex dynamics it seems sensible to survey some of the most striking of these applications to the one variable theory with the aim of making this technique available to as broad an audience as possible that is the purpose of this report one pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity in such cases we have made no effort to resist the temptation to write out complete proofs hardly anything beyond a basic knowledge of function theory is required to understand what follows so the reader is urged to take courage and plough on through and now we turn to our tale 1 let d be a domain in the complex plane c we shall be concerned with analytic maps i e meromorphic functions f d r2 received by the editors october 15 1997 and in revised form may 26 1998 1991 mathematics subject classification primary 30d45 secondary 30d35 34a20 58f23

uniqueness theorems of entire functions sharing one value im in this paper we study the uniqueness problems of entire functions sharing one value with ignoring the multiplicities and give some results which extend the theorems given by zhang and lin and also generalize some previous results of fang fang and hua yang and hua

x fem in isogeometric analysis for linear fracture mechanics the extended finite element method x fem has proven to be an accurate robust method for solving problems in fracture mechanics x fem has typically been used with elements using linear basis functions although some work has been performed using quadratics in the current work the x fem formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics in comparison with x fem with conventional finite elements of equal degree the nurbs based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom dof results for linear through quartic nurbs basis functions are presented for a multiplicity of one or a multiplicity equal the degree

a rescaling approach to two and three dimensional lattice spectra an exact real space rescaling transformation is used to calculate the spectral properties of tight binding hamiltonians with an arbitrary range of interaction of two and three dimensional lattices the intrinsic problem of the proliferation of initial short range interactions under rescaling is shown to be tractable numerically this method provides a direct way of obtaining the local green functions and local density of states singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation band edge fixed points and exponents have been found for the square triangular and cubic lattices in two and three dimensions there is rapid convergence to accurate numerical results for the green function outside the band while good results have been obtained for the in band e g density of states spectra in two dimensions by suitable choices of cluster size and attenuation factor insight into the relationship between these factors follows from an investigation of finite cluster approximations to the exact transformation it is found that the zeros of the green functions in the complex plane are julia sets which have a fractal structure for a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent

uniqueness and value sharing of entire functions abstract in this paper we study the uniqueness problems on entire functions sharing one value with the same multiplicities we generalize and unify some previous results of fang and hua m l fang x h hua entire functions that share one value j nanjing univ math biquarterly 13 1 1996 44 48 yang and hua c c yang x h hua uniqueness and value sharing of meromorphic functions ann acad sci fenn math 22 2 1997 395 406 and fang m l fang uniqueness and value sharing of entire functions comput math appl 44 2002 828 831

normal families of meromorphic functions concerning shared values abstract in this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family f of meromorphic functions in a domain d is normal if for each pair of functions f and g in f f a f n and g a g n share a value b in d where n is a positive integer and a b are two finite constants such that n 4 and a 0 this result is not true when n 3

uniqueness and value sharing of meromorphic functions in this paper we study the uniqueness of meromorphic functions and prove the following theorem let f z and g z be two non constant meromorphic functions n k two positive integers with n 3k 8 if f n z k and g n z k share 1 cm then either f z c 1e c z g z c 2e c z where c 1 c 2 and c are three constants satisfying 1 k c 1c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1 our results improves the results of fang m l fang uniqueness and value sharing of entire functions comput math appl 44 2002 823 831 7 fang and hong m l fang w hong a unicity theorem for entire functions concerning differential polynomials indian j pure appl math 32 9 2001 1343 1348 8 and lin and yi w c lin h x yi uniqueness theorems for meromorphic function indian j pure appl math 32 9 2004 121 132 9

mapped b spline basis functions for shape design and isogeometric analysis over an arbitrary parameterization abstract it is well known that b spline surfaces are defined by a regular array of control vertices in case of models with arbitrary topology it is extremely difficult to maintain continuity conditions among neighboring surfaces the scenario is also true for applications in isogeometric analysis iga this paper presents a novel method for shape design and isogeometric analysis from a quadrilateral control mesh of arbitrary topology using mapped b spline basis functions based on an arbitrary input quadrilateral control mesh a global parameterization of the final surface is first defined through a gravity center method gcm a re parameterization method is then applied to map a b spline basis function to others that are explicitly defined and are tailored to each of the control vertices which can be either regular or extraordinary ones the final surface is defined by all the input control vertices with their corresponding mapped basis functions for practical implementation the surface can be evaluated patch by patch depending on the order of the b spline basis function used for mapping to others the global continuity of the resulting surface including at extraordinary points can be arbitrary higher order in the present paper a uniform cubic b spline basis function is used and the resulting surface is globally c 2 continuous several numerical examples are provided to demonstrate the proposed method for both shape design and isogeometric analysis

the method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities babuska and oh have introduced a new approach called the method of auxiliary mapping mam to deal with elliptic boundary value problems with singularities they showed that for the laplace equation with corner singularities in the context of the p version of the finite element method mam yielded an exponential rate of convergence at virtually no extra cost in this paper those results are extended by showing the exponential convergence of mam for homogeneous laplace equations with boundary singularities and helmholtz equations with both corner and boundary singularities in addition a convergence result is developed for mam as applied to the h p version of the finite element method to clarify the power of mam a series of benchmark runs are made for three examples using our implementation mapfem comparisons are made with two h version finite element codes pltmg6 and fesop both of which use adaptive meshes the finite difference code ellpack and the recent singular element code isbfm the examples include the well known motz problem and both homogeneous and nonhomogeneous helmholtz equations

normality criteria of meromorphic functions sharing one value abstract let k be a positive integer and f be a family of meromorphic functions in a domain d c such that each f f has only zeros of multiplicity at least k 1 if for each pair f g in f f f k and g g k share a non zero complex number a ignoring multiplicity then f is normal in d

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

uniqueness theorems for entire functions concerning fixed points this paper is devoted to studying the uniqueness problem of entire functions sharing one value or fixed points we improve some results given by fang and extend some results given by fang and qiu and by lin and yi

fixed points and uniqueness of meromorphic functions in the paper we study the uniqueness and the shared fixed points of meromorphic functions and prove two main theorems which improve the results of fang and fang and qiu

uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a polynomial in this paper we get two uniqueness theorems of meromorphic functions whose certain nonlinear differential polynomials share a polynomial the results in this paper extend the corresponding results given by fang 2002 in 7 our reasoning in this paper will correct a defective reasoning in the proof of theorem 4 in bhoosnurmath and dyavanal 2007 8 an example is provided to show that some conditions of the main results in this paper are necessary

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

uniqueness and value sharing of entire functions abstract in this paper we study the uniqueness of entire functions and prove the following theorem let f z and g z be two nonconstant entire functions n k two positive integers with n 2 k 4 if f n z k and g n z k share 1 with counting the multiplicity then either f z c 1 e cz g z c 2 e cz where c 1 c 2 and c are three constants satisfying 1 k c 1 c 2 n nc 2 k 1 or f z tg z for a constant t such that t n 1

mesh grading in isogeometric analysis this paper is concerned with the construction of graded meshes for approximating so called singular solutions of elliptic boundary value problems by means of multipatch discontinuous galerkin isogeometric analysis schemes such solutions appear for instance in domains with re entrant corners on the boundary of the computational domain in problems with changing boundary conditions in interface problems or in problems with singular source terms making use of the analytic behavior of the solution we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach we prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in isogeometric analysis

parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology abstract isogeometric analysis iga over a mesh of arbitrary topology is a challenging task since the concept of iga was proposed in the literature one can find some results in iga over an irregular mesh based on approximating schemes while little work is reported so far using interpolatory schemes this paper presents a parametric mesh regularization pmr method for iga using mapped interpolatory basis functions over a mesh of arbitrary topology given a geometric model defined over a mesh of arbitrary topology a global parametric mesh is first defined from the given control mesh using a gravity center method gcm yuan and ma 2014 the resulting parametric mesh is further regularized such that it ensures absolute interpolatory modeling with mapped interpolatory basis functions necessary geometric rules are developed for the regularization process while maintaining the same mesh topological structure the interpolatory basis function used in this paper is a kind of truncated interpolatory basic function tibf studied in our early work yuan and ma 2013 it can also be applied to other mapped interpolatory basis functions in addition to the capability of interpolatory modeling the proposed pmr method can also be applied to approximating schemes which further enriches the capability of such schemes in both modeling and isogeometric analysis of some models in a more appropriate and convenient manner the proposed pmr method has also the potential in achieving higher order continuity at extraordinary vertices using higher order mapped interpolatory basis functions

graded parametric cutfem and cutiga for elliptic boundary value problems in domains with corners we develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neig

extraction formulas of stress intensity factors for the biharmonic equations containing crack singularities abstract we derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped or simply supported or free boundary conditions along the crack faces each of these formulas can be written in terms of the integral of the given source function multiplied by the cut off dual singularity and the integral of the unknown true solution multiplied by the cut off dual singularity over the unit disk the unknown true solution in the extraction formulas is calculated by either the implicitly enriched galerkin method or the iteration methods the former was developed by the authors kim oh palta kim and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients we show the iteration methods quickly converge and the proposed enrichment method yields highly accurate stress intensity factors we also demonstrate that for a known true solution the extraction formulas yield exact stress intensity factors

on the quadratic mapping z z2 μ for complex μ and z the fractal structure of its m set and scaling abstract for each complex denote by f the largest bounded set in the complex plane that is invariant under the action of the mapping z z2 mandelbrot 1980 1982 chap 19 reported various remarkable properties of the m set the set of those values of the complex for which f contains domains and of the closure m of m the goals of the present work are as follows a to restate some previously reported properties of f m and m in new ways and to report new observations b to deduce some known properties of the mapping f for real and z with f 1 4 2 and z 1 2 1 2 1 4 1 2 1 2 1 4 in many ways the properties of the transformation f are easier to grasp in the complex plane than in an interval this exemplifies the saying that when one wishes to simplify a theory one should complexify the variables c to serve as introduction to some recent pure mathematical work triggered by mandelbrot 1980 further pure mathematical work is strongly urged

uniqueness of entire functions that share one value using nevanlinna s value distribution theory we study the uniqueness of entire functions that share only one value and prove some theorems which are related to one famous problem of hayman w k hayman research problems in functions theory athlone press london 1967

uniqueness of meromorphic functions and differential polynomials in this paper we study the uniqueness theorem of meromorphic functions concerning differential polynomials and obtain two theorems which improve and generalize the related results of fang s s bhoosnurmath and r s dyavanal and xiao yu zhang and wei chuan lin

uniqueness of meromorphic functions sharing one value in this paper we study the uniqueness of meromorphic resp entire functions sharing one value and obtain some results which improve and generalize the related results due to fang bhoosnurmath dyavanal yang hua fang hua bai han etc

uniqueness of meromorphic functions sharing a small function and its applications in this paper we shall study the uniqueness problems of meromorphic functions sharing a small function our results improve or generalize many previous results on value sharing of meromorphic functions

the numerical methods for oscillating singularities in elliptic boundary value problems the singularities near the crack tips of homogeneous materials are monotone of type r and r log r depending on the boundary conditions along nonsmooth domains however the singularities around the interfacial cracks of the heterogeneous bimaterials are oscillatory of type r sin e log r the method of auxiliary mapping mam introduced by babuska and oh was proven to be successful in dealing with r type singularities however the effectiveness of mam is reduced in handling oscillating singularities this paper deals with oscillating singularities as well as the monotone singularities by extending mam through introducing the power auxiliary mapping and the exponential auxiliary mapping

enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and or corners summary the mapping method was introduced in jeong et al 2013 for highly accurate isogeometric analysis iga of elliptic boundary value problems containing singularities the mapping method is concerned with constructions of novel geometrical mappings by which push forwards of b splines from the parameter space into the physical space generate singular functions that resemble the singularities in other words the pullback of the singularity into the parameter space by the novel geometrical mapping a non uniform rational basis spline nurbs surface mapping becomes highly smooth one of the merits of iga is that it uses nurbs functions employed in designs for the finite element analysis however push forwards of rational nurbs may not be able to generate singular functions moreover the mapping method is effective for neither the k refinement nor the h refinement in this paper highly accurate stress analysis of elastic domains with cracks and or corners are achieved by enriched iga in which push forwards of nurbs via the design mapping are combined with push forwards of b splines via the novel geometrical mapping the mapping technique in a similar spirit of x fem or gfem we propose three enrichment approaches enriched iga for corners enriched iga for cracks and partition of unity iga for cracks copyright 2013 john wiley sons ltd

entire or meromorphic functions sharing one value in this paper we deal with the uniqueness problems on entire or meromorphic functions concerning differential polynomials that share one value with the same multiplicities moreover we greatly generalize some results obtained by fang lin and yi fang and fang

the h p version of the finite element method for problems with interfaces regularities of the solutions of interface problems in two dimensions are described in the frame of the weighted sobolev spaces and countably normed spaces based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h p version of the finite element method for interface problems can lead to the exponential rate of convergence numerical results on an elliptic equation with interfaces are presented the optimal mesh factor optimal degree factors and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view

reproducing polynomial singularity particle methods and adaptive meshless methods for two dimensional elliptic boundary value problems abstract oh et al h s oh j g kim j w jeong the closed form reproducing polynomial particle shape functions for meshfree particle methods comput methods appl mech engrg 196 2007 3435 3461 introduced the reproducing polynomial particle rpp shape functions that are piecewise polynomial and satisfy the kronecker delta property in this paper we introduce reproducing polynomial particle methods rppm that is the galerkin approximation method associated with the use of the rpp approximation space planting particles in the computation domain in a patchwise uniform manner we also introduce patchwise rppm furthermore constructing partition of unity functions with flat top adaptively we introduce adaptive patchwise rppm and compare it with reproducing singularity particle methods rspm that is the galerkin approximation method associated with the use of reproducing singularity particle rsp shape functions

value distribution and shared sets of differences of meromorphic functions we investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions in particular we show that for a finite order transcendental meromorphic function f with 1 f f and a non zero complex constant c if n 2 then f z nf z c assumes every non zero value a c infinitely often this research also shows that there exist two sets s1 with 9 resp 5 elements and s2 with 1 element such that for a finite order nonconstant meromorphic resp entire function f and a non zero complex constant c ef z sj ef z c sj j 1 2 imply f z f z c this gives an answer to a question of gross concerning a finite order meromorphic function f and its shift

uniqueness of meromorphic functions sharing one small function in this paper we study the uniqueness problem of nonlinear differential polynomials of meromorphic functions that share one small function a z our results extend the corresponding result obtained by zhang et al 2008

extension of the method of auxiliary mapping for three dimensional elliptic boundary value problems summary in this paper we extend the method of auxiliary mapping mam introduced by babuska and oh to three dimensions so that the extended mam 3 d mam can eectively handle three dimensional elliptic problems containing the singularities caused by the non smooth domains there are three type of singularities caused by non smoothness of domains in r 3 the vertex the edge and the vertex edge combined singularities to deal with the singularities of these types we present three auxiliary mappings and formulas for the transformed bilinear forms and the transformed linear functionals by these auxiliary mappings then we present 3 d mam and constructions of the blending type elemental mappings for elements containing singularities numerical experiments that show the eectiveness of 3 d mam are provided copyright 2001 john wiley sons ltd

anisotropic nurbs approximation in isogeometric analysis abstract we investigate the approximation error of mapped nurbs spaces such as those used in isogeometric analysis in accordance to the isoparametric paradigm we extend the results of 6 to the case of anisotropic meshes such as those needed in the presence of boundary layers the theoretical results cover anisotropic h refinement and are supported by numerical tests involving different geometries

mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities abstract the method of auxiliary mapping mam introduced by babuska and oh 2 is an effective method for dealing with singularities in elasticity 16 mam was extended to boundary element method bem in the framework of mesh free particle methods 17 reproducing polynomial particle methods 20 and also to infinite domain problems 18 similarly we consider nurbs geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems however the mapping techniques proposed in this paper are different from mam in order to generate singular shape functions mam uses conformal mappings that locally change the physical domain whereas the nurbs mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis moreover unlike mam the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned we prove error estimates in sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework