on the communication complexity of distributed algebraic computation abstract we consider a situation where two processors p sub 1 and p sub 2 ar e to evaluate a collection of functions function 1 function 8 of two vector variables x y under the assumption that processor p sub 1 respectively p sub 2 has access only to the value of the variable x respectively y and the functional form of function 1 function 8 we provide some new bounds on the communication complexity the amount of information that has to be exchanged between the processors for this problem an almost optimal bound is derived for the case of one way communication when the functions function 1 function 8 are polynomials we also derive some new lower bounds for the case of two way communication which improve on earlier bounds by abelson a 80 as an application we consider the case where x and y are n x n matrices and f x y is a particular entry of the inverse of x y under certain restriction on the class of allowed communication protocols we obtain an omega n squared lower bound in contrast to the omega n lower bound obtained by applying abelson s results our results are based on certain tools from classical algebraic geometry and field extension theory

on the communication complexity of solving a polynomial equation this paper considers the problem of evaluating a function f x y x in re m y in re n using two processors p 1 and p 2 assuming that processor p 1 respectively p 2 has access to input x respectively y and the functional form of f a new general lower bound is established on the communication complexity i e the minimum number of real valued messages that have to be exchanged the result is then applied to the case where f x y is defined as a root z of a polynomial equation sum i 0 n 1 x i y i z i 0 and a lower bound of n is obtained this is in contrast to the omega 1 lower bound obtained by applying earlier results of abelson

on the recursive sequence xn 1 α xn 1xn in this paper we study the global stability the permanence and the oscillation character of the recursive sequence xn 1 xn 1xn n 0 1 where is a negative number and the initial conditions x 1 and x0 are negative numbers

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

on the difference equation xn 1 α xn 1xn in this paper we study the behavior of the solutions of the difference equation x n 1 a x n 1x n n 0 1 where a is a negative number included are results which considerably improve and correct those in the recently published paper a e hamza on the recursive sequence x n 1 a x n 1x n j math anal appl 322 2006 668 674 we also refute conjecture 2 in g ladas open problems and conjectures j difference equ appl 7 2 2001 477 482

the difference equation xn 1 α xn k i 0k 1cixn i has solutions converging to zero abstract the aim of this note is to show that the following difference equation x n 1 x n k i 0 k 1 c i x n i n 0 1 where k n c i 0 i 0 k 1 i 0 k 1 c i 1 and 1 has solutions which monotonically converge to zero this result shows the existence of such solutions which was not shown in the recently accepted paper a e hamza on the recursive sequence x n 1 x n 1 x n j math anal appl in press

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

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

a metric property of cherry vector fields on the torus abstract in this paper we consider cherry vector fields on the torus with exactly two singularities one sink and one saddle and we prove that the stable manifold of the sink has full lebesgue measure whenever the divergence of the vector field at the saddle is less than or equal to zero this result follows from a result dealing with monotone circle mappings which are constant on an interval

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

communication complexity of computing the hamming distance let bf x bf y in 0 1 n persons a and b are given bf x and bf y respectively they communicate in order that both find the hamming distance d h n bf x bf y three communication models viz deterministic varepsilon error and varepsilon randomized are considered let c d h n c varepsilon d h n and d varepsilon d h n be the respective minimum number of bits that must be communicated under the three models it is shown that n log n 1 sqrt n leqq c left d h n right leqq n lceil log n 1 rceil it is also shown that both c varepsilon d h n and d varepsilon d h n are lower bounded by omega n thus solving an open problem posed by yao

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

normal forms for rational difference equations with applications to the global periodicity problem we propose a classification and derive the associated normal forms for rational difference equations with complex coefficients as an application we study the global periodicity problem for second order rational difference equations with complex coefficients we find new necessary conditions as well as some new examples of globally periodic equations

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

a control theory point of view on beverton holt equation in population dynamics and some of its generalizations abstract this paper is devoted to develop some ad hoc control theory formalism useful for the famous beverton holt equation arising in population dynamics in particular the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate for that purpose the environment carrying capacity gains are allowed to be time varying and designed for control purposes the controllability property is also investigated on this dynamic extended system as well as the stability equilibrium points and attractor oscillating trajectories the properties of the dynamic system associated with the beverton holt inverse equation allow extrapolate in a simple dual way the above properties to the standard beverton holt equation some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances in all cases a reference model being also of beverton holt type is proposed to be followed by the control system