the last remake of the gambier mapping abstract we present a special discrete analogue of the gambier coupled riccati system where the coupling is introduced in an additive way we show that there exist choices of the parameters for which the singularities of the mapping are confined computing the continuous limit of the mapping we show that the resulting differential equation never possesses the painleve property we present an explanation of this apparent paradox

mtl algebras arising from partially ordered groups after dropping the implication like operation and reversing the order an mtl algebra becomes a partially ordered structure l 0 1 based on the single addition like operation furthermore may be restricted to a partial but cancellative addition without loss of information we deal in this paper with the case that the resulting partial algebra l 0 1 is embeddable into the positive cone of a partially ordered group it turns out that most of the examples of mtl algebras known at present can be represented in this way so the systematisation of a considerable class of mtl algebras is achieved and the methods of their construction are brought onto a common line

alternate discrete painleve equations from limits of q pvi and d pv abstract we study the special limits of discrete painleve equations belonging to the q p vi and d p v families when the independent variable goes to 1 or 0 respectively we obtain discrete systems which are shown to be either contiguities of solutions of continuous painleve equations usually of p vi but also of the other painleve equations except p i which has no parameters and so no possibility of contiguity or linearisable mappings

a family of strict and discontinuous triangular norms answers to some questions about triangular norms are given precisely a procedure of construction of infinitely many triangular norms is obtained one for each operation n2 n such that n is a strictly ordered commutative semigroup each of them is discontinuous at each point of a dense subset of 0 1 2 and it satisfies the condition t kx y t x ky for infinitely many k 0 1

multiplicative equations related to the affine weyl group e8 we derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated with the affine weyl group e8 1 five such systems are obtained three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions in the case of the linearisable mappings we derive non autonomous forms which contain a free function of the independent variable and we present the linearisation in each case the two remaining systems are deautonomised to new discrete painleve equations we show that these equations are in fact special forms of much richer systems associated with the affine weyl groups e7 1 and e8 1 respectively

on the non autonomous form of the q4 mapping and its relation to elliptic painleve equations we apply the deautonomization procedure to mappings obtained by one dimensional reductions of the q4 and q3 integrable lattice systems of adler bobenko and suris we show that in the case of the q4 mapping the non autonomous forms are elliptic discrete painleve equations while for q3 the deautonomization leads to linearizable systems

on some classes of discrete polynomials and ordinary difference equations we introduce two classes of discrete polynomials and use them to construct ordinary difference equations admitting a lax representation in terms of these polynomials we also construct lattice integrable hierarchies in their explicit form and show some examples

discretising systematically the painleve equations we apply a method that we have already proposed for the systematic discretisation of differential systems to the construction of discrete analogues of the painleve equations the important point is that in the case of the painleve equations one needs to preserve the integrability we show that this is indeed possible using our systematic procedure applied to the equations obtained from the hamiltonian description of the painleve equations

higher linearizable mappings and their explicit integration we present results on linearizable mappings which can be obtained from discrete painleve equations associated with the affine weyl groups e 1 8 and e 1 7 hence the monicker higher of the sakai classification through limiting procedures we derive their non autonomous forms which involve an arbitrary function of the independent variable and present their explicit integration we also obtain a relation between a sub class of the mappings studied here and mappings of the gambier type we also show that the gambier mappings which are obtained as limits of discrete painleve equations can be reduced to a single master form

algebraic entropy for semi discrete equations we extend the definition of algebraic entropy to semi discrete difference differential equations calculating the entropy for a number of integrable and non integrable systems we show that its vanishing is a characteristic feature of integrability for this type of equation

deautonomizing integrable non qrt mappings we examine a class of previously derived integrable mappings which do not belong to the qrt family and show that they can be extended to non autonomous forms without loss of the integrable character we derive more non qrt integrable mappings obtain their non autonomous forms and show how they can be integrated

a study of the antisymmetric qrt mappings we study a particular class of the mappings introduced by quispel roberts and thompson that come from a two component system but can be naturally reduced to a one component one we classify all these mappings on the basis of the canonical forms of the qrt matrices we also present the extension of these systems to nonautonomous forms which are usually discrete painleve equations

an iterative method for the computation of nonlinear wide angle pulsed acoustic fields of medical diagnostic transducers the development and optimization of medical ultrasound transducers and imaging modalities require a computational method that accurately predicts the nonlinear acoustic pressure field a prospective method should provide the wide angle pulsed field emitted by an arbitrary planar source distribution and propagating in a three dimensional large scale domain holding a nonlinear acoustic medium in this paper a method is presented that is free of any assumed wavefield directionality the nonlinear acoustic wave equation is solved by treating the nonlinear term as a contrast source this formulation leads to an iterative scheme that involves the repetitive solution of a linear wave problem through green s function method it is shown that accurate field predictions may be obtained within a few iterations moreover by employing a dedicated numerical convolution technique the method allows for a discretization down to two points per wavelength or period of the highest frequency of interest the performance of the method is evaluated through a number of nonlinear field predictions for pulsed transducers with various geometries the results demonstrate the directional independence of the method moreover comparison with results from several existing methods shows that the method accurately predicts the nonlinear field for weak to moderate nonlinearity

multiple scale analysis of discrete nonlinear partial difference equations the reduction of the lattice potential kdv we consider multiple lattices and functions defined on them we introduce slow varying conditions for functions defined on the lattice and express the variation of a function in terms of an asymptotic expansion with respect to the slow varying lattices we use these results to perform multiple scale reduction of the lattice potential korteweg de vries equation

how to discretize differential systems in a systematic way we present a systematic approach to the construction of discrete analogues for differential systems our method is tailored to first order differential equations and relies on a formal linearization followed by a pade like rational approximation of an exponential evolution operator we apply our method to a host of systems for which there exist discretization results obtained by what we call the intuitive method and compare the discretizations obtained a discussion of our method as compared to one of the mickens is also presented finally we apply our method to a system of coupled riccati equations with emphasis on the preservation of the integrable character of the differential system

limits and degeneracies of discrete painleve equations a sequel abstract we present the discrete systems which result from the discrete painleve equations q p vi and d p v associated to the affine weyl group e 7 1 two different procedures limits and degeneracies are used giving rise to a host of new discrete painleve equations but also to some equations which are integrable through linearisation

from discrete painleve equations to mappings with periodic coefficients and back we investigate relations between two families of discrete painleve equations obtained respectively from degenerate forms of q piii and q pvi our approach is based on the recently derived extensions of qrt mappings to ones with periodic coefficients we establish the relationships that exist between the two families by deautonomizing the corresponding mappings with periodic coefficients in the process we obtain mappings that cannot be extended to q painleve forms and show that these mappings are not of infinite order

unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation various numerical algorithms have been developed to solve the khokhlov kuznetsov zabolotskaya kzk parabolic nonlinear wave equation in this work a generalized time domain numerical algorithm is proposed to solve the diffraction term of the kzk equation this algorithm solves the transverse laplacian operator of the kzk equation in three dimensional 3d cartesian coordinates using a finite difference method based on the five point implicit backward finite difference and the five point crank nicolson finite difference discretization techniques this leads to a more uniform discretization of the laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term in addition a new empirical algorithm based on the lu decomposition technique is proposed to solve the system of linear equations obtained from this discretization the proposed empirical algorithm improves the calculation speed and memory usage while the order of computational complexity remains linear in calculation of the diffraction term in the kzk equation for evaluating the accuracy of the proposed algorithm two previously published algorithms are used as comparison references the conventional 2d texas code and its generalization for 3d geometries the results show that the accuracy efficiency performance of the proposed algorithm is comparable with the established time domain methods

linearisable ultradiscrete systems with sign variables and the confinement of singularities we present the singularity analysis of the ultradiscrete analogue of linearisable mappings of quispel roberts thompson qrt type the ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach we show that in all cases the mappings possess confined singularities the same is true for two non autonomous equations which are equally linearisable in the discrete case we construct explicitly the solutions of the ultradiscrete mappings analysed here

discrete mappings with an explicit discrete lyapunov function related to integrable mappings abstract we propose discrete mappings of second order that have a discrete analogue of lyapunov function the mappings are extensions of the integrable quispel roberts thompson qrt mapping and a discrete lyapunov function of the mappings is identical to an explicit conserved quantity of the qrt mapping moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the existence of lyapunov function we also give applications of a mapping with an adjusted parameter a probabilistic mapping and coupled mappings

epidemic dynamics discrete time and cellular automaton models abstract we present a simple model of population dynamics in the presence of an infection the model is based on discrete time equations for sane and infected populations in interaction and correctly describes the dynamics of the epidemic we find that for some choices of the parameters the model can possess conserved quantities we also propose an ultra discrete cellular automaton version of the model which despite its extremely simple structure still captures the essence of the epidemic dynamics

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

nonlinear effects in the farfield of a directive sound source nonlinear propagation of a periodic sound beam in a dissipative fluid is considered using fourier series expansion and numerical methods to solve the governing equation of motion in the parabolic approximation the nearfield was considered in a previous paper aanonsen et al j acoust soc am 75 749 768 1984 the analysis is now extended to the farfield numerical and asymptotic results are derived and used to explain the development of the fundamental and harmonic components from the nearfield into the farfield a discussion is also given of some earlier models for the farfield of directional waves emphasis is put on the importance of imposing the proper matching conditions between the nearfield solution and the spherical solution in the farfield in order to obtain a good approximation propagation and saturation curves are calculated as well as beam patterns for various harmonic components the results are compared with available experimental observations nonlinear effects although generated

existence of finite order meromorphic solutions as a detector of integrability in difference equations abstract the existence of sufficiently many finite order in the sense of nevanlinna meromorphic solutions of a difference equation appears to be a good indicator of integrability it is shown that out of a large class of second order difference equations the only equation that can admit a sufficiently general finite order meromorphic solution is the difference painleve ii equation the proof given relies on estimates obtained by arguments related to singularity confinement the existence of meromorphic solutions of a general class of first order difference equations is also proven by a simple method based on banach s fixed point theorem

singularity confinement and full deautonomisation a discrete integrability criterion abstract we present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector our method is based on the full deautonomisation procedure which consists in analysing non autonomous extensions of a given discrete system obtained by adding terms that are initially absent but whose presence does not alter the singularity pattern a justification for this approach is given through an algebro geometric analysis we also introduce the notions of early and late confinement while the former is a confinement that may exist already for the autonomous system the latter corresponds to a singularity pattern longer than that of the autonomous case late confinement will be shown to play an important role in the singularity analysis of systems with non trivial gauge freedom for which the existence of an undetected gauge in conjunction with a sketchy analysis might lead to erroneous conclusions as to their integrability an algebro geometric analysis of the role of late confinement in this context is also offered this novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

growth and integrability in discrete systems we introduce a new discrete integrability criterion inspired from the recent findings of ablowitz and collaborators this criterion is based on the study of the growth of some characteristic of the solutions of a mapping using nevanlinna theory since the practical implementation of the latter does not always lead to a clear cut answer we complement the growth criterion by the singularity confinement property this combination turns out to be particularly efficient its application allows us to recover the known forms of the discrete painleve equations and to show that no new ones may exist within a given parametrization

the discrete chazy iii system of labrunie conte is not integrable we analyse the discrete form of the chazy iii equation proposed by labrunie and conte with the help of two different integrability criteria singularity confinement and algebraic entropy we show that for all values of the free parameter this third order mapping fails both criteria and thus cannot be integrable

discretising the painleve equations a la hirota mickens we present a systematic method for discretising the painleve equations inspired by the method of hirota while extending it and by that of mickens by specifying it to the case at hand we derive various discrete analogues of painleve i and ii we obtain forms that have been previously derived as well as new ones in particular equations with a geometry described by the affine weyl group e8 1 as a by product we obtain also linearisable equations some of which are new

a classification of two dimensional integrable mappings and rational elliptic surfaces we classify two dimensional integrable mappings by investigating the actions on the fiber space of rational elliptic surfaces while the qrt mappings can be restricted on each fiber there exist several classes of integrable mappings which exchange fibers we also show an equivalent condition when a generalized halphen surface becomes a halphen surface of index m

direct method to construct integrals for nth order autonomous ordinary difference equations a direct method to construct integrals for an nth order autonomous ordinary difference equation oe wnnf wn wnn1 is presented as an illustration we first consider third order autonomous oe wn3

singularity confinement and algebraic integrability two important notions of integrability for discrete mappings algebraic integrability and singularity confinement have been used for discrete mappings algebraic integrability is related to the existence of sufficiently many conserved quantities and singularity confinement is associated with the local analysis of singularities in this article the relationship between these two notions is explored for birational autonomous mappings the main result of this article is that algebraically integrable mappings are shown to have the singularity confinement property using this result the proof of the nonexistence of algebraic conserved quantities for a class of discrete systems is given

modeling of nonlinear ultrasound propagation in tissue from array transducers a computationally efficient model capable of simulating finite amplitude ultrasound beam propagation in water and in tissue from phased linear arrays and other transducers of arbitrary quasiplanar geometry is described it is based on a second order operator splitting approach tavakkoli et al j acoust soc am 104 2061 2072 1998 with a fractional step marching scheme whereby the effects of diffraction attenuation and nonlinearity can be computed independently over incremental steps this approach is an extension to that of christopher and parker j acoust soc am 90 507 521 90 488 499 1991 wherein linear and nonlinear effects are propagated separately over incremental steps and the computation of the diffractive substeps are based on an angular spectrum technique with a modified sampling scheme for accurate and efficient implementation of diffractive propagation from nonradially symmetric sources results of the model are compared with published data predicted field profiles for nonlinear propagation in tissue from realistic array transducers using the pulse inversion method are presented

a general method for constructing left continuous t norms a new method for constructing left continuous t norms is introduced and analyzed in this paper we shall construct via embedding a left continuous t norm from any countable residuated totally and densely ordered commutative integral monoid moreover we can construct a left continuous t norm from any countable totally ordered commutative integral monoid which is not necessarily densely ordered and residuated a special case the embedding of such monoids on lexicographic product spaces is investigated in detail and several examples are demonstrated the results shed some light to chang s mv algebra to a recently proposed extraordinary t norm and to the standard semantics of the recently introduced logic ii mtl of hajek

observations on the monoidal t norm logic the paper is a contribution to the theory of the monoidal t norm based logic mtl and stronger logics analogous to the well known extensions of the basic fuzzy logic lukasiewicz godel and product logic main attention is payed to the logic iimtl mtl plus the characteristic axioms ii1 ii2 of product logic

mappings of hirota kimura yahagi type can have periodic coefficients too we present an extension of mappings of hirota kimura yahagi type where the coefficients instead of being constant are periodic functions of the independent variable several examples of mappings with periodic coefficients of various periods are produced we prove the integrability of all the examples by constructing the corresponding conserved quantities

integrable mappings of the plane preserving biquadratic invariant curves we provide a general framework to construct integrable mappings of the plane that preserve a one parameter family b x y k of biquadratic invariant curves where parametrization by k is very general these mappings are reversible by construction i e they are the composition of two involutions and can be shown to be measure preserving they generalize integrable maps previously given by mcmillan and quispel roberts and thompson by considering a transformation of the case of the symmetric biquadratic to a canonical form we provide a normal form for the symmetric integrable map acting on each invariant curve we give a lax pair for a large subclass of our symmetric integrable maps including at least a 10 parameter subfamily of the 12 parameter symmetric quispel roberts thompson maps

some properties of the k dimensional lyness map this paper is devoted to study some properties of the k dimensional lyness map f x1 xk x2 xk a ki 2xi x1 our main result presents a rational vector field that gives a lie symmetry for f this vector field is used for k 5 to give information about the nature of the invariant sets under f when k is odd we also present a new as far as we know first integral for ff which allows us to deduce in a very simple way several properties of the dynamical system generated by f in particular for this case we prove that except on a given codimension one algebraic set none of the positive initial conditions can be a periodic point of odd period

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

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

integrability criteria for differential difference systems a comparison of singularity confinement and low growth requirements a new approach for the study of the integrability of differential difference systems is introduced given a differential difference system of such a form that it can be iterated without necessitating the integration of a differential equation at any step a two stage strategy is used first the singularity confinement necessary integrability criterion is used in order to limit the possible choices once the system is sufficiently reduced the much stronger requirement of nonexponential growth of the degree of the iterates of some initial condition is implemented this method turns out to be powerful and practical the investigation of a given class of differential difference systems has resulted in some well known integrable systems but also to two promising integrability candidates one of which is reduced to a known integrable case

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

a novel nth order difference equation that may be integrable we derive an nth order difference equation as a dual of a very simple periodic equation and construct n 1 2 explicit integrals and integrating factors of this equation in terms of multi sums of products we also present a generating function for the degrees of its iterates exhibiting polynomial growth in conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of levi and yamilov these three facts combine to suggest the integrability of the nth order difference equation

integrable mappings and polynomial growth abstract we describe birational representations of discrete groups generated by involutions having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics these involutions correspond respectively to two kinds of transformations on q q matrices the inversion of the q q matrix and an involutive permutation of the entries of the matrix we concentrate on the case where these permutations are elementary transpositions of two entries in this case the birational transformations fall into six different classes for each class we analyze the factorization properties of the iteration of these transformations these factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties some mappings yield a polynomial growth of the complexity of the iterations for three classes the successive iterates for q 4 actually lie on elliptic curves this analysis also provides examples of integrable mappings in arbitrary dimension even infinite moreover for two classes the homogeneous polynomials are shown to satisfy non trivial non linear recurrences the relations between factorizations of the iterations the existence of recurrences on one or several variables as well as the integrability of the mappings are analyzed

a new class of integrable discrete systems we derive a new class of integrable autonomous mappings which possess invariants biquartic in the dependent variables we link these mappings to discrete painleve equations we show how mappings with biquartic invariants can be naturally built up from the requirement that their solutions be parametrized in terms of elliptic functions

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

nuclei and conuclei on residuated lattices in this paper we investigate some properties of nuclei and conuclei on residuated lattices and their concrete structures by means of nuclei and conuclei we give the respective characterizations for a residuated lattice reducing to a generalized mv algebra and a generalized bl algebra based on studying regular commutative bounded integral residuated lattices we introduce the concept of a pseudo dual quantale and investigate some properties of it and prove that the set of fuzzy sets of a nonempty set equipped with multiplication and negation operations is a commutative girard quantale finally we discuss the relationship between quantic nuclei and quantic conuclei on a pseudo dual quantale and show that the quantic nuclei and the ideal conuclei on a pseudo girard quantale are in one to one correspondence

do ultradiscrete systems with parity variables satisfy the singularity confinement criterion ultradiscrete singularity confinement test which is an integrability detector for ultradiscrete equations with parity variables is applied to various ultradiscrete equations the ultradiscrete equations exhibit singularity structures analogous to those of the discrete counterparts exact solutions to linearisable ultradiscrete equations are also constructed to explain the singularity structures

discrete painleve equations with maximal periodicities we present the discrete painleve equations the coefficients of which have the maximal periodicity allowed by the geometric description in terms of affine weyl groups for each equation of the degeneration pattern obtained by sakai and which starts at the group we give the corresponding discrete painleve equation some of these systems were already known but here we proceed in a systematic way filling in all the gaps in particular we derive the equations with maximal periodicity associated to the affine weyl group which is the richest system in the sakai classification and which has yet to be studied in detail

algebraic entropy of an extended hietarinta viallet equation we introduce a series of discrete mappings which is considered to be an extension of the hietarinta viallet mapping with one parameter we obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping for some parameter values the mapping has a confined singularity in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials for other parameter values the mapping does not pass the singularity confinement test the properties of irreducibility and co primeness of the terms play crucial roles in the discussion

how to find the conserved quantities of nonlinear discrete equations a method of finding conserved quantities of nonlinear ordinary difference equations is briefly discussed the method is based on the hypothesis that the conserved quantities are expressed by a ratio of two polynomials of dependent variables we obtain the conserved quantities of the discrete equation of motion of an anharmonic oscillator and a nonlinear difference equation of third order

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

full deautonomisation of a lattice equation in this letter we report on the unexpected possibility of applying the full deautonomisation approach we recently proposed for predicting the algebraic entropy of second order birational mappings to discrete lattice equations moreover we show on two examples that the full deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2 in particular we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable and we show that our approach accurately predicts this nonintegrable character finally we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings

growth complexity spectrum of some discrete dynamical systems we first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices and consider the degree d n of the numerators or denominators of the corresponding successive rational expressions for the nth iterate the growth of this degree is generically exponential with n d n n is called the growth complexity we introduce a semi numerical analysis which enables to compute these growth complexities for all the 9 possible birational transformations these growth complexities correspond to a spectrum of eighteen algebraic values we then drastically generalize these results replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters again it is shown that the associated birational or even rational transformations yield algebraic values for their growth complexities 1999 elsevier science b v all rights reserved

on the integrability of a new lattice equation we examine a lattice equation recently derived by one of us cs together with heredero and levi we show that this equation has confined singularities possesses integrable one and two dimensional reductions can be bilinearized and possesses multisoliton solutions all these arguments together with the fact that as shown in the initial publication the equation has zero algebraic entropy plead in favour of its integrable character

on the autonomous limit of discrete painleve equations abstract we consider the autonomous limit of the known families of discrete painleve equations we show how these autonomous systems can be written as special cases of the qrt mapping for the latter we present in detail its integration in both the symmetric and asymmetric cases

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

the q painleve v equation and its geometrical description we study the q painleve v equation which can be obtained from the degeneration of the q pvi in the form of the asymmetric q piii equation and present its geometrical description based on the bilinear formulation we obtain the equations for the multi dimensional functions of q pv in the form of nonautonomous hirota miwa systems which lives in the weight lattice of the a4 affine weyl group this geometrical approach furnishes in a straightforward way the miuras and the schlesingers of q pv

on the fuzzy logical content of cadiag 2 cadiag 2 is a successful expert system assisting in the differential diagnosis in internal medicine with its aid conjectures about present diseases are derived from possibly vague information about a patient s symptoms in this paper we provide a mathematical formalisation of the inferential mechanism of cadiag 2 a gentzen style calculus for the resulting logic is introduced and used to compare the system s behaviour with t norm based fuzzy logics

the discrete first second and thirty fourth painleve hierarchies the discrete first and second painleve equations dp and dp are integrable difference equations which have classical first second or third painleve equations p p or p as continuum limits dp and dp are believed to be integrable because they are discrete isomonodromy conditions for associated single valued linear problems an infinite hierarchy of integrable difference equations that share the same linear deformation problem as dp was shown to exist by cresswell and joshi in this paper we recall the results shown for dp and show how to deduce a hierarchy for dp each member of the respective hierarchies is shown to be generated by difference recursion operators furthermore we show that continuum limits of these difference hierarchies lead to the p p and p hierarchies finally we construct miura transformations of the dp hierarchy and show that these lead to the hierarchy of the discrete thirty fourth painleve equation

higher dimensional integrable mappings derived from coupled discrete nonlinear schrodinger equations a systematic investigation to derive higher dimensional coupled autonomous integrable mappings from scalar and coupled discrete nonlinear schordinger equations is presented the integrability in the sense of liouville of the obtained higher dimensional mappings is ensured by showing that each of them is symplectic and possesses sufficient number of integrals of motion

do integrable cellular automata have the confinement property we analyse a criterion introduced by joshi and lafortune for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure we show that while this criterion can be used in order to single out integrable ultradiscrete systems there do exist cases where the system is nonintegrable and still the criterion is satisfied conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied we investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be finally we comment on the growth properties of ultradiscrete systems

a systematic method for constructing discrete painleve equations in the degeneration cascade of the e8 group we present a systematic and quite elementary method for constructing discrete painleve equations in the degeneration cascade for e8 1 starting from the invariant for the autonomous limit of the e8 1 equation one wishes to study the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms these new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping we explain the method on three examples two symmetric mappings and an asymmetric one and we discuss the link between our results and the known geometric structure of these mappings