an algebraic proof of the cyclic sum formula for multiple zeta values abstract we introduce an algebraic formulation of the cyclic sum formulas for multiple zeta values and for multiple zeta star values we also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta star values by reducing them to kawashima s relation

convergence of rank type equations abstract convergence results are presented for rank type difference equations whose evolution rule is defined at each step as the k th largest of p univariate difference equations if the univariate equations are individually contractive then the equation converges to a fixed point equal to the k th largest of the individual fixed points of the univariate equations examples are max type equations for k 1 and the median of an odd number p of equations for k p 1 2 in the non hyperbolic case conjectures are stated about the eventual periodicity of the equations generalizing long standing conjectures of g ladas

on families of bounded continuous solutions of some systems of functional difference equations on half intervals some sufficient conditions for the existence of families of bounded continuous solutions of some classes of systems of functional difference equations on the intervals 0 and 0 are given

geometry of the poincare compactification of a four dimensional food web system in this paper the behavior of dynamics at infinity of a four dimensional autonomous food web system has been investigated for this a topological method has been developed to understand the geometry of the poincare compactification which investigate the behavior of the vector field at infinity the global phase portrait has been shown on the poincare disc

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

the algebra of multiple harmonic series abstract recently there has been much interest in multiple harmonic series i 1 i 2 i k n 1 n 2 n k 1 1 n 1 i 1 n 2 i 2 n k i k which converge when the exponentsijare positive integers andi1 1 also known as multiple zeta values or euler zagier sums starting with the noncommutative polynomial algebraq x y we define a second multiplication which is commutative and associative and call the resulting structure the harmonic algebra h as a graded commutative algebra h turns out to be a free polynomial algebra with the number of generators in degreengiven by the witt formula for the numbern n of basis elements of degreenin the free lie algebra on two generators multiple harmonic series can be thought of as images under a map h 0 rwhich is a homomorphism with respect to the commutative multiplication where h 0is an appropriate subalgebra of h if we calli1 ikthe weight of the series i1 ik then forn 1 there are at mostn n series of weightnthat are irreducible in the sense that they are not sums of rational multiples of products of multiple harmonic series of lower weight in fact our approach gives an explicit set of algebraically irreducible multiple harmonic series we also show that there is a subalgebra h 0 h 1 h related to the shuffle algebra which contains the algebra s of symmetric functions in fact maps the elements of s h 0to algebraic combinations of zeta values i for integeri 2 the map is not injective we show how several results about multiple harmonic series can be recast as statements about the kernel of and propose some conjectures on the structure of the algebra h 0 ker

martingale dimensions for fractals we prove that the martingale dimensions for canonical diffusion processes on a class of self similar sets including nested fractals are always one this provides an affirmative answer to the conjecture of s kusuoka publ res inst math sci 25 1989 659 680

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

on functional equations of finite multiple polylogarithms abstract recently several people study finite multiple zeta values fmzvs and finite polylogarithms fps in this paper we introduce finite multiple polylogarithms fmps which are natural generalizations of fmzvs and fps and we establish functional equations of fmps as applications of these functional equations we calculate special values of fmps containing generalizations of congruences obtained by mestrovic z w sun l l zhao tauraso and j zhao

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

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

multivariable hoffman ihara operators and the operad of formal power series abstract we introduce a family of multilinear operators on a quasi shuffle algebra h which we call hoffman ihara operators it generalizes the linear operators constructed by hoffman and further studied by hoffman and ihara we prove the formula on the composition of these operators and show that this formula provides a unified and transparent understanding of some results on shuffle and quasi shuffle products an interpretation as an action of the operad of formal power series is also given

an alternative proof of a theorem of zagier abstract we give a new proof of zagier s theorem on evaluation of multiple zeta star values at 2 a 3 2 b the novelty of the proof consists in employing a hypergeometric identity of andrews which expresses a very well poised hypergeometric series in terms of a multiple series and the residue theorem

on a nonlinear second order system of difference equations we prove that all positive solutions of the next system of difference equationsx n 1 a y n px n 1 q y n 1 a x n py n 1 q email protected n 0 where parameters a p and q are positive numbers are bounded if one of the following conditions is satisfied p 2 4 or p 1 q q 1 completely describing the boundedness character of the system in this case

domains of undefinable solutions of some equations and systems of difference equations for some classes of difference equations and systems of difference equations we find the sets of initial values for which solutions of these equations and systems are not well defined so called domains of undefinable solutions

multiple q zeta values we introduce a q analog of the multiple harmonic series commonly referred to as multiple zeta values the multiple q zeta values satisfy a q stuffle multiplica tion rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values additionally multiple q zeta values can be viewed as special values of the multiple q polylogarithm which admits a multiple jackson q integral representation whose limiting case is the drinfel d simplex integral for the ordinary multiple polylogarithm when q 1 the multiple jackson q integral representation for multiple q zeta values leads to a second multiplication rule satisfied by them referred to as a q shuffle despite this it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting q extension for example a suitable q analog of broadhurst s formula for 3 1 n if one exists is likely to be rather complicated nevertheless we show that a number of infinite classes of relations inc luding hoffman s partition identities ohno s cyclic sum identities granville s sum formula euler s convolution formula ohno s generalized duality relation and the derivation relations of ihara and kaneko extend to multiple q zeta values

global stability of some classes of higher order nonlinear difference equations we give elegant and short proofs of some recent results on global stability of some classes of higher order nonlinear difference equations

more on a rational second order difference equation abstract here we give natural explanations for some recent relations for the solutions of the following difference equation z n 2 z n 1 3 z n 2 2 z n 1 2 n n 0 and extend them for the case of complex initial values we also present some other properties related to solutions of the equation as well as some properties of the solutions of an associate difference equation of first order

upper estimate of martingale dimension for self similar fractals we study upper estimates of the martingale dimension dm of diffusion processes associated with strong local dirichlet forms by applying a general strategy to self similar dirichlet forms on self similar fractals we prove that dm 1 for natural diffusions on post critically finite self similar sets and that dm is dominated by the spectral dimension for the brownian motion on sierpinski carpets

on a system of difference equations of odd order solvable in closed form it is shown that the next system of difference equationsx n c ny n 2 k 1 a n b ny n 2 k 1 i 1 k 1y n 2 i 1 x n 2 i y email protected nx n 2 k 1 a email protected nx n 2 k 1 i 1 k 1x n 2 i 1 y n 2 email protected n 0 where a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 2k 1 are real numbers is solved in closed form and the behavior of its well defined solutions when all the coefficients in the system are constant is described

on the eventual periodicity of xn 1 max 1xn anxn 1 with a period five parameter in this paper we will study the max difference equation x n 1 max 1x n a nx n 1 when a n is a period five sequence of real numbers greater than one we will prove that every solution is eventually periodic with period twenty

on schur multiple zeta functions a combinatoric generalization of multiple zeta functions abstract we introduce schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta star functions of the euler zagier type combinatorially we first study their basic properties including a region of absolute convergence and the case where all variables are the same then under an assumption on variables some determinant formulas coming from theory of schur functions such as the jacobi trudi giambelli and dual cauchy formula are established with the help of macdonald s ninth variation of schur functions moreover we investigate the quasi symmetric functions corresponding to the schur multiple zeta functions we obtain the similar results as above for them and furthermore describe the images of them by the antipode of the hopf algebra of quasi symmetric functions explicitly finally we establish iterated integral representations of the schur multiple zeta values of ribbon type which yield a duality for them in some cases

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

on the difference equation xn 1 pxn s xn tqxn s xn t abstract in this paper we address the difference equation x n 1 px n s x n t qx n s x n t n 0 1 with positive initial conditions we establish a sufficient condition for the global asymptotic stability of the positive equilibrium of this equation

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

quasi shuffle products revisited abstract quasi shuffle products introduced by the first author have been useful in studying multiple zeta values and some of their analogues and generalizations the second author together with kajikawa ohno and okuda significantly extended the definition of quasi shuffle algebras so it could be applied to multiple q zeta values this article extends some of the algebraic machinery of the first author s original paper to the more general definition and demonstrates how various algebraic formulas in the quasi shuffle algebra can be obtained in a transparent way some applications to multiple zeta values interpolated multiple zeta values multiple q zeta values and multiple polylogarithms are given

evaluation of the multiple zeta values zeta 2 dots 2 3 2 dots 2 a formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products m p 2n with m odd the existence of such a formula had been proved using motivic arguments by francis brown but the explicit formula more precisely certain 2 adic properties of its coefficients were needed for his proof in cite1 of the conjecture that all periods of mixed tate motives over z are q 2pi 1 linear combinations of multiple zeta values the formula is proved indirectly by computing the generating functions of both sides in closed form one as the product of a sine function and a 3 f 2 hypergeometric function and one as a sum of 14 products of sine functions and digamma functions and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality we also show that the space spanned by the multiple zeta values in question coincides with the space of double zeta values of odd weight and find a relation between this space and the space of cusp forms on the full modular group

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

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

eventual periodicity of some systems of max type difference equations the periodicity of the next system of max type difference equations xn 1 max1 i1 m1f1i1xn ki1 1 1 1 xn ki1 2 1 2 xn ki1 l 1 l n xn t1s 1 xn 2 max1 i2 m2f2i2xn ki2 1 2 1 xn ki2 2 2 2 xn ki2 l 2 l n xn t2s 2 xn l max1 il mlflilxn kil 1 l 1 xn kil 2 l 2 xn kil l l l n xn tls l n n0n n0 where s l mj tj kij h j n j h 1 l j h 1 l 1 2 l 1 2 l is a permutation of 1 2 l 1 2 l and fjij 0 l n0 0 fjij 0 l n0 0 j 1 l j 1 l ij 1 mj ij 1 mj is studied it is shown that under some conditions posed on functions fjijfjij all positive solutions of the system are eventually periodic with period ts for some t nt n

on the difference equation xn xn kb cxn 1 xn k abstract we describe the behavior of well defined solutions of the difference equation x n x n k b cx n 1 x n k n n 0 where k n parameters b c and the initial values x k x 1 are real numbers these results among others improve and extend our results in s stevic more on a rational recurrence relation appl math e notes 4 2004 80 85

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

multiple zeta values vs multiple zeta star values abstract we discuss an algebraic connection between two kinds of multiple zeta values or their q analogues the q multiple zeta values and the q multiple zeta star values these two classes of values generate the same algebra but in this paper we show that the translation map between these two classes has a quite interesting algebraic property in a general setting for example the compatibility with the harmonic product we also study several applications of the result

existence of bounded solutions of a class of neutral systems of functional differential equations some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations which extend and unify numerous results in the literature are proved

brownian motion on a random recursive sierpinski gasket we introduce a random recursive fractal based on the sierpinski gasket and construct a diffusion upon the fractal via a dirichlet form this form and its symmetrizing measure are determined by the electrical resistance of the fractal the effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion the main result is to obtain uniform upper and lower bounds for the transition density of the brownian motion on the fractal in terms of this metric the bounds are not tight as there are logarithmic corrections due to the randomness in the environment and the behavior of the shortest paths in the effective resistance metric is not well understood the results are deduced from the study of a suitable general branching process

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

relations of multiple zeta values and their algebraic expression abstract we establish a new class of relations which we call the cyclic sum identities among the multiple zeta values k1 kl n1 nl 11 n1k1 nkkl these identities have an elementary proof and imply the sum theorem for multiple zeta values they also have a succinct statement in terms of cyclic derivations as introduced by rota sagan and stein in addition we discuss the expression of other relations of multiple zeta values via the shuffle and harmonic products on the underlying vector space h of the noncommutative polynomial ring q x y and also using an action of the hopf algebra of quasi symmetric functions on q x y

on a difference equation with maximum abstract this paper studies the asymptotic behavior of positive solutions to the difference equation x n max 1 x n 1 a x n 2 n 0 1 where 0 0 we prove that every positive solution of this equation approaches x 1 or is eventually periodic with period 4 according as a 1 or a 1

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

stability of the zero solution of a family of functional differential equations abstract we give some sufficient conditions under which the zero solution of the next functional differential equation x t ax t bx 0 t j 1 k c j x j t f x t x k 1 t x k l t where a b c j j 1 k are real numbers f r l 1 r is a continuous function such that f 0 0 0 and j t c 2 0 j 0 k l is asymptotically stable

multiple series connected to hoffman s conjecture on multiple zeta values abstract recent results of zlobin and cresson fischler rivoal allow one to decompose any suitable p uple series of hypergeometric type into a linear combination over the rationals of multiple zeta values of depth at most p in some cases only the multiple zeta values with 2 s and 3 s are involved as in hoffman s conjecture in this text we study the depth p part of this linear combination namely the contribution of the multiple zeta values of depth exactly p we prove that it satisfies some symmetry property as soon as the p uple series does and make some conjectures on the depth p 1 part of the linear combination when p 3 our result generalizes the property that very well poised univariate hypergeometric series involve only zeta values of a given parity which is crucial in the proof by rivoal and ball rivoal that 2 n 1 is irrational for infinitely many n 1 the main feature of the proof is an algebraic approach based on representations of z 2 z p s p

transition density estimates for brownian motion on affine nested fractals a class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals for these fractals estimates on the transition density for the brownian motion are obtained using the associated dirichlet form an upper bound for the diagonal can be found using a nash type inequality then probabilistic techniques are used to obtain the off diagonal bound the approach differs from previous treatments as it uses only the dirichlet form and no estimates on the resolvent the bounds obtained are expressed in terms of an intrinsic metric on the fractal

new explicit global asymptotic stability criteria for higher order difference equations new explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained these criteria can be applied to autonomous and nonautonomous equations the celebrated clark asymptotic stability criterion is improved also applications to models from mathematical biology and macroeconomics are given

interpolation of multiple zeta and zeta star values we define polynomials of one variable t whose values at t 0and1 are the multiple zeta values and the multiple zeta star values respectively we give an application to the two one conjecture of ohno zudilin and also prove the sum formula and the cyclic sum formula for these polynomials

on a system of difference equations which can be solved in closed form we present an elegant method for solving the system of difference equationsx n 1 1x email protected n y n email protected n 1y n email protected n 0 where parameters a and b and the initial values x 0 and y 0 are positive numbers and by using obtained formulae we show that every solution of the system converges to a two periodic solution

duality and q multiple zeta values abstract following bachmann s recent work on bi brackets and multiple eisenstein series zudilin introduced the notion of multiple q zeta brackets which provides a q analog of multiple zeta values possessing both shuffle as well as quasi shuffle relations the corresponding products are related in terms of duality in this work we study zudilin s duality construction in the context of classical multiple zeta values as well as various q analogs of multiple zeta values regarding the former we identify the derivation relation of order two with a hoffman ohno type relation then we describe relations between the ohno okuda zudilin q multiple zeta values and the schlesinger zudilin q multiple zeta values

on the asymptotic behavior and periodic nature of a difference equation with maximum in this paper we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation x n max ax n 1 1xn 2 a n 0 where a 0 and 0 a 1 we prove that every positive solution of this difference equation approaches x 1 or is eventually periodic with a period 2 3 or 4

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

parabolic type equations associated with the dirichlet form on the sierpinski gasket by using analytic tools from stochastic analysis we initiate a study of some non linear parabolic equations on sierpinski gasket motivated by modellings of fluid flows along fractals which can be considered as models of simplified rough porous media unlike the regular space case such parabolic type equations involving non linear convection terms must take a different form due to the fact that convection terms must be singular to the linear part which defines the heat semigroup in order to study these parabolic type equations a new kind of sobolev inequalities for the dirichlet form on the gasket will be established these sobolev inequalities which are interesting on their own and in contrast to the case of euclidean spaces involve two l p norms with respect to two mutually singular measures by examining properties of singular convolutions of the associated heat semigroup we derive the space time regularity of solutions to these parabolic equations under a few technical conditions the burgers equations on the sierpinski gasket are also studied for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations and the existence uniqueness and regularity of its solutions are obtained

heat kernels on metric measure spaces and an application to semilinear elliptic equations we consider a metric measure space m d p and a heat kernel p t x v on m satisfying certain upper and lower estimates which depend on two parameters a and we show that under additional mild assumptions these parameters are determined by the intrinsic properties of the space m d namely a is the hausdorff dimension of this space whereas called the walk dimension is determined via the properties of the family of besov spaces w 2 on m moreover the parameters a and are related by the inequalities 2 a 1 we prove also the embedding theorems for the space w 2 2 and use them to obtain the existence results for weak solutions to semilinear elliptic equations on m of the form u f x u g x where is the generator of the semigroup associated with p t the framework in this paper is applicable for a large class of fractal domains including the generalized sierpinski carpet in r n

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

asymptotic behavior of solutions of a system of q functional difference equations abstract the asymptotic behavior and the existence of a bounded lipschitz continuous solution of a system of nonlinear q functional difference equations on a half interval of the real line are studied here the main results compare the behavior of such a solution of the system with bounded solutions of a linear q difference equation

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 a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results 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

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied