two meromorphic mappings sharing 2n 2 hyperplanes regardless of multiplicity nevanlinna showed that two non constant meromorphic functions on c must be linked by a mobius transformation if they have the same inverse images counted with multiplicities for four distinct values after that this result is generalized by gundersen to the case where two meromorphic functions share two values ignoring multiplicity and share other two values with multiplicities truncated by 2 previously the first author proved that for n 2 there are at most two linearly non degenerate meromorphic mappings of cm into pn c sharing 2n 2 hyperplanes ingeneral position ignoring multiplicity in this article we will show that if two meromorphic mappings f and g of cm into pn c share 2n 1 hyperplanes ignoring multiplicity and another hyperplane with multiplicities truncated by n 1 then the map f g is algebraically degenerate

value distribution of meromorphic solutions of certain difference painleve equations abstract the borel exceptional value and the exponents of convergence of poles zeros and fixed points of finite order transcendental meromorphic solutions for difference painleve i and ii equations are estimated and the forms of rational solutions of the difference painleve ii equation and the autonomous difference painleve i equation are also given it is also proved that the non autonomous difference painleve i equation has no rational solution

further results on the asymptotic agreement problem the problem of reaching a concensus between two decision makers provided with different information is considered the problem in which the decision makers may have different underlying probability models is studied results are developed to characterize the likelihood of an agreement being reached eventually in terms of the nature of the inter decision maker communications the problem in which the decision makers are aware of the possibility that they may have different models is treated it is found that in this case a deadlock can be reached where neither decision maker can learn additional information from the concensus process and they cannot reach a concensus decision this result indicates that incorporating human uncertainty in probability assessment into the asymptotic agreement problem can lead to outcomes not anticipated in the general theory previously developed

on the uniqueness problems of entire functions and their derivatives in this paper we obtain some uniqueness theorems for entire functions and their derivatives sharing the same fixed points with the same multiplicities

growth and zeros of meromorphic solution of some linear difference equations abstract in this paper we study growth and zeros of linear difference equations p n z f z n p 1 z f z 1 p 0 z f z f z where f z p n z p 0 z are polynomials with f p n p 0 0 and satisfy deg p n p 0 max deg p j j 0 n 1 the corresponding homogeneous equation of the above equation is also investigated

entire functions that share one value with one or two of their derivatives we prove two uniqueness theorems for entire functions of finite order that share one finite value with one or two of their derivatives

the uniqueness of an entire function sharing a small entire function with its derivatives in this paper we prove a theorem on the growth of a solution of a linear differential equation from this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function the results in this paper improve many known results some examples are provided to show that the results in this paper are the best possible

correction to meromorphic functions that share four values it has been brought to my attention by norbert steinmetz that the symmetry reasoning that is used in the proof of theorem 2 in 1 to conclude that d1 is symmetric in a and b in the third line from the bottom on page 564 and also to conclude that 123 holds on page 565 is not valid we note that the symmetry reasoning that is used in the sentence in lines 8 and 9 from the bottom on page 564 is valid because this sentence is meant in the sense of the following sentence by considering a b point that is simple for f and double for 9 a similar argument to the above argument with 61 instead of 60 will produce

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

further results on entire functions that share one value with their derivatives abstract this paper deals with problems of the uniqueness of entire functions that share one value with their derivatives the results in this paper generalize a result of jank mues and volkmann and answer a question posed by h zhong and a question of yi and yang

a 3im 1cm result for periodic meromorphic functions abstract let f and g be two periodic meromorphic functions of hyper order less than 1 such that f and g share three values im and one value cm if f and g have a same period and the lower order of f is not equal to 1 then f g examples are given to illustrate the necessity of the conditions

non simultaneous blow up for a multi coupled reaction diffusion system abstract this paper deals with a reaction diffusion system multi coupled via inner sources and nonlinear boundary flux the necessary and sufficient condition for the existence of non simultaneous blow up is proposed some regions of parameters are determined where both simultaneous and non simultaneous blow up can occur in addition sufficient conditions for non simultaneous blow up under any positive initial data are obtained

nonexistence of positive solutions to a semilinear elliptic system and blow up estimates for a reaction diffusion system abstract we get simple conditions under which the elliptic system u u p 1 v q 1 v u p 2 v q 2 in r n n 3 with p i q i 1 i 1 2 has no positive radially symmetric solutions then by using this nonexistence result we establish blow up estimates for reaction diffusion system of fujita type u t u u p 1 v q 1 v t v u p 2 v q 2 with the homogeneous dirichlet boundary value condition

meromorphic functions sharing a set with applications to difference equations this paper is devoted to proving some uniqueness type results for an entire function f z that shares a common set with its shift f z c or its difference operator cf we also give some applications to solutions of non linear difference equations related to a conjecture proposed by c c yang

periodicity and unicity of meromorphic functions with three shared values abstract this research investigates the periodicity and unicity of meromorphic functions with two shared values cm and one im 2cm 1im which improves some results obtained by brosch and zheng concerning three shared values cm 3cm moreover examples show that some conditions are necessary

holomorphic curves with shift invariant hyperplane preimages if f c p n is a holomorphic curve of hyper order less than one for which 2n 1 hyperplanes in general position have forward invariant preimages with respect to the translation z z c then f is periodic with period c 2 c this result which can be described as a difference analogue of m green s picard type theorem for holomorphic curves follows from a more general result presented in this paper the proof relies on a new version of cartan s second main theorem for the casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives both of which are proved here finally an application to the uniqueness theory of meromorphic functions is given and the sharpness of the obtained results is demonstrated by examples

meromorphic functions that share four values an old theorem of r nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities then the functions are mbbius transformations of each other two of the shared values are picard values for both functions and the cross ratio of a particular permutation of the shared values equals 1 in this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities then the functions share all four values counting multiplicities

on properties of meromorphic solutions for difference equations concerning gamma function abstract in this paper we mainly consider the properties of differences of meromorphic solutions for the difference equation a 1 z f z 1 a 0 z f z 0 concerning a gamma function where a 1 z and a 0 z are nonzero polynomials by these properties we deduce that a gamma function satisfies that for every n n n z z 0 n z has only n zeros z z z j z 1 j 0 1 n 1 z 1 z 1 n 1 z and 1 z have same zeros at most except n exceptional zeros where g denotes the order of growth of a meromorphic function g and g and g denote the exponents of convergence of fixed points and zeros of g respectively

non simultaneous blow up for n componential parabolic systems abstract this paper deals with the non simultaneous and simultaneous blow up for some parabolic systems u i t u i u i p i coupled via nonlinear boundary flux u i u i 1 q i 1 i 1 2 n for radially symmetric solutions we obtain that one component can blow up by itself and may provide sufficient help to the blow up of the other k 0 1 n 2 ones under suitable initial data in particular such phenomena happen for every initial data in some exponent regions it is interesting that there exist initial data such that any two components blow up simultaneously either of which blows up depending on itself and also can give sufficient help to the other components blowing up simultaneously a necessary and sufficient condition is obtained on the simultaneous blow up of at least two components for all initial data moreover the non simultaneous and simultaneous blow up rates and sets are determined

optimal and robust memoryless discrimination from dependent observations discrimination is considered between two possible sources based on dependent observations of their output the discrimination problem is modeled by means of a general binary hypothesis test the main emphasis being on situations that cannot be modeled as signals in additive noise the observations are modeled as stationary m dependent or rho mixing processes the structure of the discriminator is such that the observations are passed through a memoryless nonlinearity summed up to form a test statistic which is then compared to a threshold only fixed sample size tests are considered four different performance measures which resemble the signal to noise ratios encountered in the signal in additive noise problems are derived under different problem formulations the optimal nonlinearities for each of the performance measures are derived as solutions to various integral equations for three of the four performance measures the authors have successfully obtained robust nonlinearities for uncertainty in the marginal and the joint probability density functions of the observations computer simulation results that demonstrate the advantage of using these nonlinearities over the i i d nonlinearity under the probability of error criterion are presented

solutions of nonlinear difference equations abstract the non vanishing finite order entire solutions of nonlinear difference equation f z n a n 1 f z n 1 a 1 f z q z e q z f z c p z are studied in this paper for n 2 and n 1 respectively where q z q z p z are polynomials and a n 1 a 1 c c we classified these solutions according to growth and zero distribution particularly we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms when n 2 and a 1 0 our results are improvements and complements of wen et al 13

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

a note on partial sharing of values of meromorphic functions with their shifts abstract we introduce the notion of partial value sharing as follows let e a f be the set of zeros of f z a z where each zero is counted only once and a is a meromorphic function small with respect to f a meromorphic function f is said to share a partially with a meromorphic function g if e a f e a g we show that partial value sharing of f z and f z c involving 3 or 4 values combined with an appropriate deficiency assumption is enough to guarantee that f z f z c provided that f z is a meromorphic function of hyper order strictly less than one and c c

distributed multisensor parameter estimation in dependent noise the problem of distributed estimation of a weak nonrandom location parameter theta in additive stationary dependent noise is addressed multisensor configurations with and without a coordinator are considered dependence in the sensor observations is described by m dependent phi mixing and p mixing models two cases of interest are addressed one in which sensor observations are dependent across time but independent across sensors and one in which sensor observations are dependent across both time and sensors numerical results on the performance evaluation of the various estimation schemes derived are presented and the relative performances of the various schemes are compared

finite order solutions of second order linear differential equations on considere l equation differentielle f a z f b z f 0 ou a z et b z sont des fonctions entieres on trouve des conditions sur a z et b z qui garantissent que chaque solution f 0 de l equation a un ordre fini

two entire functions sharing one small function cm and two other small functions im abstract in this paper we prove a result that if two entire functions share one small function cm and two other small functions im then f is a quasi mobius transformation of g which generalizes g g gundersen s 2cm 2im theorem see g g gundersen trans am math soc 277 1983 545 567 and g g gundersen trans am math soc 304 1987 847 850 from sharing values to sharing small functions in the case where both f and g are entire functions

non simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system abstract this paper deals with finite time quenching for the nonlinear parabolic system with coupled singular absorptions u t u v p v t v u q in 0 t subject to positive dirichlet boundary conditions where p q 0 is a bounded domain in r n with smooth boundary we obtain the sufficient conditions for global existence and finite time quenching of solutions and then determine the blow up of time derivatives and the quenching set for the quenching solutions as the main results of the paper a very clear picture is obtained for radial solutions with b r the quenching is simultaneous if p q 1 and non simultaneous if p 1 q or q 1 p if p q 1 with r 2 n then both simultaneous and non simultaneous quenching may happen depending on the initial data in determining the non simultaneous quenching criteria of the paper some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous dirichlet boundary value conditions in addition to techniques frequently used in the literature

combining and updating of local estimates and regional maps along sets of one dimensional tracks in this paper we consider the problem of combining and updating estimates that may have been generated in a distributed fashion or may represent estimates generated at different times of the same process sample path the first of these cases has applications in decentralized estimation while the second has applications in updating maps of spatially distributed random quantities given measurements along several tracks the method of solution for the second problem uses the result of the first and the similarity in the formulation and solution of these problems emphasizes the conceptual similarity between many problems in decentralized control and in the analysis of random fields

on results of h ueda and g brosch concerning the unicity of meromorphic functions by emplying the main ideas from g brosch two results of r nevanlinna s four value theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values and a fourth value or a pair of values weakly are proved which extend and generalize previous results of r nevanlinna g g gundersen h ueda and g brosch

meromorphic functions that share fixed points abstract in this paper we prove the following result let f z and g z be two nonconstant meromorphic entire functions n 11 n 6 a positive integer if f n z f z and g n z g z have the same fixed points then either f z c 1 e cz 2 g z c 2 e cz 2 where c 1 c 2 and c are three constants satisfying 4 c 1 c 2 n 1 c 2 1 or f z tg z for a constant t such that t n 1 1

distributed estimation algorithms for nonlinear systems in this paper we consider the problem of combining the local conditional distributions of a random variable which have been generated by local observers having access to their private information sufficient statistics for the local distributions are communicated to a coordinator who attempts to reconstruct the global centralized distribution using only the communicated statistics we obtain a distributed processing algorithm which recovers exactly the centralized conditional distribution the results can be applied in designing distributed hypothesis testing algorithms for event driven systems

a maximum variance model for robust detection and estimation with dependent data a model that accounts for uncertain data dependency is developed by generating a large class of stationary stochastic processes each with the same univariate distribution this class can be considered to be a contamination class about the nominal independent and identically distributed i i d process distribution the class is developed specifically for application to robust detector and estimator design based on asymptotic variance application of this dependency class leads to an intuitively pleasing result the minimax variance estimators and the maximin efficacy detectors are the same as obtained using i i d asymptotic estimation and detection theory thus our technique generalizes previously obtained robust design results for i i d data to this dependent data case

asymptotic behavior at infinity of the stationary solution to a semilinear heat equation in this paper we investigate the asymptotic behavior at infinity of the backward self similar solution of the differential equation u t u e u x t 0 where is a ball with the dirichlet boundary or r n 3 n we prove that under some reasonable condition at infinity every radial symmetric nontrivial bounded above solution of the equation y y n 1 y y 2 y e 1 0 tends to minus infinity as y this equation comes from the scaled ignition model furthermore log y 2 converges to a constant for sufficiently large y this result extends the similar one in lacey 1993 for an arbitrary solution which is bounded above and for dimension 3 n in space

on the value distribution of some difference polynomials abstract in this paper we continue to investigate the value distribution of some difference polynomials which can be considered as difference counterparts of some classic results of hayman

the role of non linear diffusion in non simultaneous blow up we study a parabolic system of two non linear reaction diffusion equations completely coupled through source terms and with power like diffusivity under adequate hypotheses on the initial data we prove that non simultaneous blow up is sometimes possible i e one of the components blows up while the other remains bounded the conditions for non simultaneous blow up rely strongly on the diffusivity parameters and significant differences appear between the fast diffusion and the porous medium case surprisingly flat homogeneous in space solutions are not always a good guide to determine whether non simultaneous blow up is possible

non simultaneous blow up in a parabolic system with three components abstract this paper deals with heat equations coupled via nonlinear boundary flux u 1 u 1 p 11 u 2 p 12 u 2 u 2 p 22 u 3 p 23 u 3 u 3 p 33 u 1 p 31 a necessary and sufficient condition for the existence of only one component blowing up for nondecreasing in time and radially symmetric solutions three kinds of exponent regions are obtained as follows i only one component blows up for every initial data ii the existence of two components blowing up simultaneously while the third one remains bounded also with two kinds of blow up rates 1 2 p 22 1 1 2 p 33 1 and p 23 1 p 33 2 p 33 1 1 2 p 33 1 iii e g u 1 remains bounded and u 2 u 3 blow up simultaneously with blow up rate p 23 1 p 33 2 p 33 1 1 2 p 33 1 for every initial data moreover the eight kinds of simultaneous blow up rates and blow up sets are discussed

normal families and uniqueness theorems for entire functions abstract there exists a set s with 3 elements such that if f is a non constant entire function satisfying e s f e s f then f f the number 3 is best possible the proof uses the theory of normal families in an essential way

elementary exact calculations of degree growth and entropy for discrete equations second order discrete equations are studied over the field of rational functions c z where z is a variable not appearing in the equation the exact degree of each iterate as a function of z can b

value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity abstract this research is a continuation of a recent paper due to the first four authors shared value problems related to a meromorphic function f z and its shift f z c where c c are studied it is shown for instance that if f z is of finite order and shares two values cm and one value im with its shift f z c then f is a periodic function with period c the assumption on the order of f can be dropped if f shares two shifts in different directions leading to a new way of characterizing elliptic functions the research findings also include an analogue for shifts of a well known conjecture by bruck concerning the value sharing of an entire function f with its derivative f

multiple absorption related blow up rates to a coupled heat system this paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi nonlinearities it is known that although absorption mechanisms may affect such as blow up criteria blow up time and initial data required for blow up solutions they cannot change blow up rates of solutions in general it has been reported in the current literature that blow up rates for scalar equations with absorptions are all absorption independent in a previous paper of the authors four absorption independent simultaneous blow up rates were obtained already for the same problem under weak absorptions the present paper will furthermore prove that if the absorptions are unbalanced in the model i e the absorption is stronger for one component and weaker for another then there are in addition eight possible absorption related blow up rates for the model besides the four absorption independent ones this exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions

second main theorem in the tropical projective space abstract tropical nevanlinna theory introduced by halburd and southall as a tool to analyze integrability of ultra discrete equations studies the growth and complexity of continuous piecewise linear real functions the purpose of this paper is to extend tropical nevanlinna theory to n dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves and by proving a tropical analogue of cartan s second main theorem it is also shown that in the 1 dimensional case this result implies a known tropical second main theorem due to laine and tohge

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

a nonlocal approach for asymptotic memoryless detection theory corresp previous literature on detector design has presented many tractable results using efficacy as a performance functional which is a small signal or local approach some nonlocal issues of memoryless detection are considered we argue that the asymptotic exponential rate of decrease of error probabilities or bayes risk provides the appropriate asymptotic measure of performance a main contribution is the development of an approximation for nonlocal design that retains the tractability of the efficacy approach we have proved the local consistency of our approximation also as a consequence of the analysis new proofs for pitman noether like theorems have been obtained

value sharing of meromorphic functions on a riemann surface we present some results on two meromorphic functions from s to c sharing a number of values where s is a riemann surface of one of the following types compact compact minus finitely many points the unit disk a torus the complex plane

value sharing results for shifts of meromorphic functions abstract in this paper we deal with the value distribution of difference products of entire functions and present some result on two difference products of entire functions sharing one value with the same multiplicities the research findings also include an analogue for shift of a well known conjecture by bruck our theorems improve the results of i laine and c c yang i laine c c yang value distribution of difference polynomials proc japan acad ser a math sci 83 2007 148 151 k liu and l z yang k liu l z yang value distribution of the difference operator arch math 92 2009 270 278 and j heittokangas et al j heittokangas r korhonen i laine j rieppo j l zhang value sharing results for shifts of meromorphic function and sufficient conditions for periodicity j math anal appl 355 2009 352 363 moreover we show by illustrating a number of examples that our results are best possible in certain senses

critical non simultaneous blow up exponents for a reaction diffusion system the paper deals with the critical non simultaneous blow up exponents for the coupled reaction diffusion equations ut u u m v p and vt v u q v n subject to homo geneous dirichlet boundary conditions for the radial blow up solutions we propose precise criteria to completely identify the four cases for non simultaneous and simultaneous blow up situations under admissible initial data there exist initial data such that non simultaneous or simultaneous blow up of solutions occurs or any blow up must be non simultaneous or simultaneous the necessary sufficient conditions are established for each case with a complete classification for all the parameters there moreover all possible simultaneous blow up rates are obtained these extend the results in the previous works for the same model

difference analogue of the lemma on the logarithmic derivative with applications to difference equations the lemma on the logarithmic derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa tions in this paper a dierence analogue of the logarithmic derivative lemma is presented and then applied to prove a number of results on meromorphic solutions of complex dierence equations these results include a dierence analogue of the clunie lemma as well as other results on the value distribution of solutions

from continuous to discrete painleve equations abstract discrete analogues of the painleve equations have recently been derived using either a method based on orthogonal polynomials or the singularity confinement method here we analyze additional methods for deriving integrable non autonomous discrete equations these methods are based on the discrete akns approach on the use of backlund and schlesinger transforms of the continuous painleve equations and on discrete analogues of miura transformations

uniqueness and periodicity of meromorphic functions concerning the difference operator in this paper we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point we also obtain several results concerning the shifts of meromorphic functions and the sufficient conditions for periodicity which improve some recent results in heittokangas et al 2009 10 and liu 2009 11

blow up rate estimates for a doubly coupled reaction diffusion system abstract this paper deals with a reaction diffusion system with coupled nonlinear inner sources and a nonlinear boundary flux blow up rates are determined for four different blow up situations the so called characteristic algebraic system is introduced to get a very simple and clear description for the desired blow up rate estimates it is pointed out that one cannot directly use super and sub solutions to establish blow up rate estimates since they do not share the same blow up time in general

unitary matrix models as exactly solvable string theories models of unitary matrices are solved exactly in a double scaling limit using orthogonal polynomials on a circle exact differential equations are found for the scaling functions of these models for the simplest model k 1 the painlev e ii equation with constant 0 is obtained there are possible nonperturbative phase transitions in these models the scaling function is of the form mathit n mathrm ensuremath 1 2 mathit k 1 ifmmode times else texttimes fi f mathit n 2 mathit k 2 mathit k 1 ensuremath lambda mathit x ensuremath lambda for the kth multicritical point the specific heat is mathit f 2 and is therefore manifestly positive equations are given for k 2 and 3 with a discussion of asymptotic behavior

nevanlinna theory of the askey wilson divided difference operator abstract this paper establishes a version of nevanlinna theory based on askey wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane c a second main theorem that we have derived allows us to define an askey wilson type nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero pole scarce that is their zeros poles are indeed deficient in the sense of difference nevanlinna theory a natural consequence is a version of askey wilson type picard theorem we also give an alternative and self contained characterisation of the kernel functions of the askey wilson operator in addition we have established a version of unicity theorem in the sense of askey wilson this paper concludes with an application to difference equations generalising the askey wilson second order divided difference equation

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

on a result of g brosch abstract we prove a uniqueness theorem for non constant meromorphic functions f g which share three values 0 1 and f a g b share the value 0 for a b 0 1 our theorem improves a result of g brosch

non simultaneous blow up of n components for nonlinear parabolic systems this paper deals with non simultaneous and simultaneous blow up for radially symmetric solution u1 u2 un to heat equations coupled via nonlinear boundary ui uipiui 1qi 1 i 1 2 n it is proved that there exist suitable initial data such that ui i 1 2 n blows up alone if and only if qi 1 pi all of the classifications on the existence of only two components blowing up simultaneously are obtained we find that different positions different values of k i n of ui k and ui leads to quite different blow up rates it is interesting that different initial data lead to different blow up phenomena even with the same requirements on exponent parameters we also propose that ui k ui k 1 ui i 1 2 n k 0 1 2 n 1 blow up simultaneously while the other ones remain bounded in different exponent regions moreover the blow up rates and blow up sets are obtained

an improvement of the nevanlinna gundersen theorem a well known result of nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane c and for four distinct values aj c if f aj g aj for all 1 j 4 then g is a mobius transformation of f in 1983 gundersen generalized the result of nevanlinna to the case where the above condition is replaced by min f aj 1 min g aj 1 for j 1 2 and f aj g aj for j 3 4 in this paper we prove that the theorem of gundersen remains valid to the case where min f aj 1 min g aj 1 for j 1 2 and min f aj 2 min g aj 2 for j 3 4 furthermore we work on the case where aj are small functions

unique range sets and uniqueness polynomials for algebraic curves we study unique range sets and uniqueness polynomials for algebraic functions on a smooth projective algebraic curve over an algebraically closed field of characteristic zero

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

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

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

meromorphic functions that share four small functions in this paper we prove that if two nonconstant meromorphic functions f and g share two small functions cm and share other two small functions im then f must be a quasi mobius transformation of g this result is a generalization of several results obtained by g g gundersen and li yang

an extension of picard s theorem for meromorphic functions of small hyper order abstract a version of the second main theorem of nevanlinna theory is proved where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper order as a corollary of this result it is shown that if n n and three distinct values of a meromorphic function f of hyper order less than 1 n 2 have forward invariant pre images with respect to a fixed branch of the algebraic function z z n 1 z 1 1 n 1 z 1 n 0 with constant coefficients then f f this is a generalization of picard s theorem for meromorphic functions of small hyper order since the empty pre images of the usual picard exceptional values are special cases of forward invariant pre images

growth of meromorphic solutions of linear difference equations abstract in this paper the authors continue to study the growth of meromorphic solutions of homogeneous or non homogeneous linear difference equations with entire coefficients and obtain some results which are improvement and extension of previous results in chiang and feng 2008 7 and laine and yang 2007 19 examples are also given to illustrate the sharpness of our results

on results of g brosch and t c alzahary we prove a uniqueness theorem for two non constant meromorphic functions sharing three values which improves a recent result of t c alzahary as a consequence of our main result we also improve a theorem of g brosch

meromorphic solutions of difference painleve equations we show that under certain conditions some difference painleve equations have nontrivial meromorphic solutions in the whole complex plane these meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle the existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers

blow up analysis for a quasilinear parabolic system with multi coupled nonlinearities this paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux as the results of the interaction among the multi coupled nonlinearities in the system some appropriate conditions for global existence and global nonexistence of solutions are determined respectively

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

critical exponents for non simultaneous blow up in a localized parabolic system abstract the paper deals with the radially symmetric solutions of u t u u m x t v n 0 t v t v u p 0 t v q x t subject to null dirichlet boundary conditions for the blow up classical solutions we propose the critical exponents for non simultaneous blow up by determining the complete and optimal classification for all the non negative exponents i there exist initial data such that u v blows up alone if and only if m p 1 q n 1 which means that any blow up is simultaneous if and only if m p 1 q n 1 ii any blow up is u v blowing up with v u remaining bounded if and only if m p 1 q n 1 m p 1 q n 1 iii both non simultaneous and simultaneous blow up may occur if and only if m p 1 q n 1 moreover we consider the blow up rate and set estimates which were not obtained in the previously known work for the same model

finite order meromorphic solutions and the discrete painleve equations let w z be a finite order meromorphic solution of the second order difference equation w z 1 w z 1 r z w z where r z w z is rational in w z and meromorphic in z then either w z satisfies a difference linear or riccati equation or else equation can be transformed to one of a list of canonical difference equations this list consists of all known difference painleve equation of the form together with their autonomous versions this suggests that the existence of finite order meromorphic solutions is a good detector of integrable difference equations

correction to meromorphic functions that share four values trans amer math soc 277 1983 no 2 545 567 mr0694375 84g 30028