feigenvalues for mandelsets the authors discuss generalizations of feigenbaum s constants alpha and delta to complex polynomial maps of degree higher than 2 and present some numerical estimates universality classes are found to depend on the nature of the critical points of the polynomial

implicit vector quasi equilibrium problems with applications to variational inequalities abstract in this paper we introduce and study a class of implicit vector quasi equilibrium problems which includes implicit vector equilibrium problems implicit vector quasi variational inequalities and implicit vector quasi complementarity problems as special cases we establish some existence results under compact and noncompact settings by using one person game theorems as an application of our results we have obtained existence results for variational inequalities our results extends and unify the well known earlier works of many authors

existence of equilibria for generalized games and generalized social systems with coordination the general existence of equilibria for non compact generalized games in which the l majorized preference mappings may not have lower semicontinuity is established it is shown that the constraint mappings may only be either lower of upper semicontinuous and the set of players may not be countable the underlying spaces are also shown to be either infinite dimensional locally convex topological vector spaces

the spectrum of the period doubling operator in terms of cycles the eigenvalue spectrum of the period doubling operator is evaluated both in terms of unstable periodic orbits and by a diagonalization at the operator cycles up to length ten yield the twelve leading eigenvalues direct diagonalization yields some 50 eigenvalues the feigenbaum 6 the hausdorff dimension and the escape rate for the period doubling repeller are evaluated to high accuracy in this letter we apply the technique for the extraction of correlation exponents of i to evaluation of the spectrum of the period doubling renormalization operator the basic idea of relating the feigenbaum constant 6 to the scaling or the presentation function is due to sullivan 2 and is implemented as an eigenvalue spectrum calcula tion in 3 51 we refer the reader to 3 41 for an introduction to the cycle expansions in general and their application to the evaluation of the stability the dimension and the escape rate of the period doubling repeller in particular the fredholm determinant evaluation method and the transfer operator diagonalization 11 used here are superior to the method of locating zeros of finite products of dynamical 6 functions 6 of 3 4 7 81 and in that sense the present letter supersedes the above references the repeller we study here is the non wandering set of the period doubling presenta tion function 9

the ƒ a formalism applied to the crossover from conservative to dissipative scaling behaviour in period doubling systems abstract we use the a spectral dimension formalism to give a global description of the crossover from conservative to dissipative scaling behaviour in period doubling systems as a generic model we consider the two dimensional henon map for which the jacobian 0 b 1 measures the amount of dissipation and we calculate the successive approximants k a b to the generalized spectral dimension for small i e experimentally feasible values of k the crossover properties from the extreme dissipative case b 0 to the conservative case b 1 are discussed

scaling factors associated with m furcations of the 1 μ x z map a numerical study is made of the scaling behavior associated withm furcations m 3 4 5 in the mapx t 1 1 x t z z 1 the scaling constants and are calculated as functions ofz as well as the more general scaling functions andf a

crossover from dissipative to conservative behaviour in period doubling systems abstract we investigate the crossover properties between the conservative and dissipative limit for period doubling bifurcations in two dimentional iterative maps as a generic example we use the standard form of the henon map the approximants to the feigenbaum constant lie on a universal crossover curve which turns out to be non monotonic a similar curve is found for the scaling factor more generally we discuss the crossover of the trajectory scaling function 1 and its properties in the conservative limit

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above

dynamics and universality of unimodal mappings with infinite criticality we consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization we study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity it is shown that a limiting dynamics exists with a critical point that is flat but still having a well behaved analytic continuation to a neighborhood of the real interval pinched at the critical point we study the dynamics of limiting maps and prove their rigidity in particular the sequence of fixed points of renormalization for finite criticalities converges uniformly on the real domain to a mapping of the limiting type

effect of very short impulses on nonlinear dissipative kinetics abstract delayed nonautonomous kinetics is considered on the example of the kinetic equation t ax t del b t x 2 t del with b t representing a train of separate impulses dependence on impulse width is discussed for the case of short delay times sufficiently short impulses lead to significant changes in x t appearance of chaotic behaviour absent for longer impulses abrupt drop of the average value of x etc which enables the experimental determination of del it is suggested that there exists a time scale on which the normal macroscopic kinetic description breaks down due to the natural inertia of physical processes related e g to the presence of fast variables averaged out in the macroscopic description hence the effects described in this paper may be used for the phenomenological definition and experimental determination of a new characteristic of the macroscopic dissipative system related to that time scale the inertial relaxation time inertial

generalized abstract economy and systems of generalized vector quasi equilibrium problems the present paper is in two fold the first fold is devoted to the existence theory of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact nonparacompact strategy set in the second fold we consider systems of generalized vector quasi equilibrium problems for multivalued maps for short sgvqeps which contain systems of vector quasi equilibrium problems systems of generalized mixed vector quasi variational inequalities and debreu type equilibrium problems for vector valued functions as special cases by using the results of first fold we establish some existence results for solutions of sgvqeps

on general best proximity pairs and equilibrium pairs in free abstract economies abstract in this paper using lassonde s fixed point theorem for kakutani factorizable multifunctions and park s fixed point theorem for acyclic factorizable multifunctions we will prove new existence theorems for general best proximity pairs and equilibrium pairs for free abstract economies which generalize the previous best proximity theorems and equilibrium existence theorems due to srinivasan and veeramani p s srinivasan p veeramani on best approximation pair theorems and fixed point theorems abstr appl anal 2003 1 2003 33 47 p s srinivasan p veeramani on existence of equilibrium pair for constrained generalized games fixed point theory appl 2004 1 2004 21 29 and kim and lee w k kim k h lee existence of best proximity pairs and equilibrium pairs j math anal appl 316 2006 433 446 in several aspects

random matrix theory of parametric correlations in the spectra of disordered metals and chaotic billiards abstract a random matrix theory is developed for the adiabatic response to an external perturbation of the energy spectrum of a mesoscopic system the basic assumption is that spectral correlations are governed by level repulsion following dyson the dependence of the energy levels on the perturbation parameter is modeled by a brownian motion process in a fictitious viscous fluid a fokker planck equation for the evolution of the distribution function is solved to yield the correlation of level densities at different energies and different parameter values an approximate solution is obtained by asymptotic expansion and an exact solution by mapping onto a free fermion model a generalization to multiple parameters is also considered corresponding to brownian motion in a fictitious world with multiple temporal dimensions complete agreement is obtained with microscopic theory

self similarity of period doubling branching 3 d reversible mappings abstract we consider 3 dimensional 3 d mappings which are reversible i e possess a time reversal symmetry in the mappings studied here the reversibility is such that it guarantees the existence of one parameter families curves of symmetric periodic orbits in the phase space it is found that such curves can often intersect one another in particular a curve of n cycles can intersect a curve of 2n cycles which in turn can intersect a curve of 4n cycles etc we show that the tree of branching curves of successively doubled periods in the 3 d phase space possesses some geometric self similarity in particular we identify the scaling factors 4 018 16 363 and 8 721 previously studied 1 parameter 2 d area preserving reversible mappings in which these scalings also occur are special cases of 3 d reversible mappings in that they have an integral of motion the more general mappings studied here have no such integral we discuss the reasons for our result in terms of normal forms

an extension of mehta theorem with applications abstract a remarkable fundamental theorem established by mehta plays an important role in proving existence of fixed points maximal elements and equilibria in abstract economies in this paper we extend himmelberg s measure of precompactness to the general setting of l c spaces and obtain a generalization of mehta s theorem as an application we develop some new fixed point theorems involving a kind of condensing mappings

generalized fuzzy games and fuzzy equilibria abstract in this paper we define the concepts of generalized fuzzy game and fuzzy equilibrium next we shall prove new existence theorems of equilibrium in generalized fuzzy games with uncountable number of agents our results generalize the corresponding results due to arrow debreu kim lee and yannelis prabhakar in several aspects

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

universality of parametric spectral correlations local versus extended perturbing potentials we explore the influence of an arbitrary external potential perturbation v on the spectral properties of a weakly disordered conductor in the framework of a statistical field theory of a nonlinear ensuremath sigma model type we find depending on the range and the profile of the external perturbation two qualitatively different universal regimes of parametric spectral statistics i e cross correlations between the spectra of hamiltonians h and h v we identify the translational invariance of the correlations in the space of hamiltonians as the key indicator of universality and find the connection between the coordinate system in this space which makes the translational invariance manifest and the physically measurable properties of the system in particular in the case of localized perturbations the latter turn out to be the eigenphases of the scattering matrix for scattering off the perturbing potential v they also have a purely statistical interpretation in terms of the moments of the level velocity distribution finally on the basis of this analysis a set of results obtained recently by the authors using random matrix theory methods is shown to be applicable to a much wider class of disordered and chaotic structures

existence of best proximity pairs and equilibrium pairs in this paper using the fixed point theorem for kakutani factorizable multifunctions we shall prove new existence theorems of best proximity pairs and equilibrium pairs for free abstract economies which include the previous fixed point theorems and equilibrium existence theorems

universality in quantum parametric correlations we investigate the universality of correlation functions of chaotic and disordered quantum systems as an external parameter is varied a general scaling procedure is introduced which makes the theory invariant under reparametrizations under certain general conditions we show that this procedure is unique the approach is illustrated with the particular case of the distribution of eigenvalue curvatures we also derive a semiclassical formula for the nonuniversal scaling factor and give an explicit expression valid for arbitrary deformations of a billiard system

equilibria and maximal elements of abstract fuzzy economies and qualitative fuzzy games the purpose of this paper is to introduce the concepts of abstract fuzzy economies and qualitative fuzzy games and to use the connectedness of topological spaces without linear structures to study the existence problems of fuzzy equilibrium and fuzzy maximal elements of abstract fuzzy economies and qualitative fuzzy games the results presented in this paper are new in the cases of ordinary abstract economies and ordinary qualitative games

on the approximation of upper semi continuous correspondences and the equilibriums of generalized games in section 1 we explain some of the definitions and terminology that we use in section 2 we prove several theorems concerning the approximation of upper semi continuous correspondences having for range a locally convex space theorems 1 and 2 and corollary 1 generalize certain approximation theorems by g haddad 16 pp 1352213541 g haddad and j m lasry 17 pp 299 3001 and j p aubin and a cellina 2 pp 86891 see also f s de blasi lo and j m lasry and r robert 21 although some of the details of the proofs of theorems 1 and 2 are new the basic ideas are taken from the above mentioned papers theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular see section 1 this theorem see the remark following its proof contains a classical result of m hukuhara 18 pp 56573 theorem 3 is proved using theorem 2 and propositions 1 and 4 variants of propositions 2 3 and 4 were given in 193 in section 3 we give among others theorem 5 which concerns the existence of equilibriums of generalized games abstract economies the main purpose of this result is to replace the continuity hypothesis in the w shafer h sonnenschein equilibrium theorem for generalized games by an upper semi continuity one the proof of theorem 5 is based on the results on the approximation of upper semi continuous correspondences obtained in section 2 theorem 5 us used in section 4 in section 4 we establish theorems 6 and 7 these theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of k fan type theorems 1 2 3 5 6 and 7 are the main results of this paper 267 0022 247x 88 3 00

edge effects in some perturbations of the gaussian unitary ensemble a bordering of gaussian unitary ensemble matrices is considered in which the bordered row consists of zero mean complex gaussians n 0 2 in 0 2 off the diagonal and the real gaussian n 2 on the diagonal we compute the explicit form of the eigenvalue probability function for such matrices as well as that for matrices obtained by repeating the bordering the correlations are in general determinantal and in the single bordering case the explicit form of the correlation kernel is computed in the large n limit it is shown that and or can be tuned to induce a separation of the largest eigenvalue this effect is shown to be controlled by a single parameter universal correlation kernel

morphology of the period doubling universal function results are presented concerning the structure on the real line of the universal function which is the fixed point solution of the feigenbaum cvitanovic renormalization group equation associated with period doubling chaos in quadratic maps it is shown that the values which the function takes at its turning points can be algebraically characterized by relation to the infinite cycle associated with the original turning point of the map on the interval these extrema become increasingly numerous as the argument increases and their locations can be found progressively using knowledge of the previously determined extrema as well as providing a simple understanding of the structure of the universal function these simple observations may be of assistance in investigating the convergence or asymptotic behavior of approximations to the universal function and perhaps in providing a corrector step to some of these schemes

on generalized operator quasi equilibrium problems abstract in this paper we will introduce the generalized operator equilibrium problem and generalized operator quasi equilibrium problem which generalize the operator equilibrium problem due to kazmi and raouf k r kazmi a raouf a class of operator equilibrium problems j math anal appl 308 2005 554 564 into multi valued and quasi equilibrium problems using a fan browder type fixed point theorem in s park foundations of the kkm theory via coincidences of composites of upper semicontinuous maps j korean math soc 31 1994 493 519 and an existence theorem of equilibrium for 1 person game in x p ding w k kim k k tan equilibria of non compact generalized games with l majorized preferences j math anal appl 164 1992 508 517 as basic tools we prove new existence theorems on generalized operator equilibrium problem and generalized operator quasi equilibrium problem which includes operator equilibrium problems

period doubling in maps with a maximum of order z abstract we consider the scaling behaviour in period doubling systems exemplified by the one dimensional map n 1 1 n z which has a maximum of order z z 1 the feigenbaum scaling factors and are studied as functions of z and more generally the scaling functions 1 and a in particular using the universal functions g x and h x we establish the inequality

parametric level statistics in random matrix theory exact solution an exact solution to the problem of parametric level statistics in non gaussian ensembles of n by n hermitian random matrices with either soft or strong level confinement is formulated within the framework of the orthogonal polynomial technique being applied to random matrices with strong level confinement the solution obtained leads to emergence of a new connection relation that makes a link between the parametric level statistics and the scalar two point kernel in the thermodynamic limit

siegel disc singularity spectra the author numerically determines the generalized dimensions and spectrum of singularities of a critical point orbit on the boundary of a siegel disc the local scaling exponent alpha mn and spectrum dependence on the degree d of the critical point are also considered he finds that limd to infinity mod alpha mn mod d approximately 0 523 53 and hence asymptotically the lower limit of the spectrum is constant whilst the upper limit is proportional to d

best proximity pairs and equilibrium pairs for kakutani multimaps new existence theorems of best proximity pairs for a finite family of kakutani multimaps are proved in a normed space setting as applications new existence theorems of equilibrium pairs for free n person and 1 person games are obtained our results generalize and improve several well known results 2008 elsevier ltd all rights reserved

maximal elements and equilibria of generalized games for condensing correspondences in this paper we first obtain some fixed point theorems in locally convex topological vector spaces which in turn imply some existence theorems of maximal elements for condensing correspondences then by employing approximation methods we prove some existence theorems of equilibria for generalized games in which the constraint correspondences are condensing and lower or upper semicontinuous instead of having open lower sections or open graphs in infinite dimensional locally convex topological vector spaces finally an existence theorem for the n person game is also given in particular the corresponding results in the literature have been improved

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

existence and stability of solutions for maximal element theorem on hadamard manifolds with applications abstract in this paper maximal element theorem on hadamard manifolds is established first we prove the existence of solutions for maximal element theorem on hadamard manifolds further we prove that most of problems in maximal element theorem on hadamard manifolds in the sense of baire category are essential and that for any problem in maximal element theorem on hadamard manifolds there exists at least one essential component of its solution set as applications we study existence and stability of solutions for variational relation problems on hadamard manifolds and existence and stability of weakly pareto nash equilibrium points for n person multi objective games on hadamard manifolds

asymptotics of scaling parameters for period doubling in unimodal maps with asymmetric critical points the universal period doubling scaling of a unimodal map with an asymmetric critical point is governed by a period 2 point of a renormalization operator the period 2 point is parametrized by the degree of the critical point and the asymmetry modulus in this paper we study the asymptotics of period 2 points and their associated scaling parameters in the singular limit of degree tending to 1

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

multifractality and nonextensivity at the edge of chaos of unimodal maps we examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity z 1 first we determine analytically the sensitivity to initial conditions t then we consider a renormalization group operation on the partition function z of the multifractal attractor that eliminates one half of the multifractal points each time it is applied invariance of z fixes a length scale transformation factor 2 in terms of the generalized dimensions d there exists a gap in the values of equal to q 1 1 q d 1 d 1 where q is the q generalized lyapunov exponent and q is the nonextensive entropic index we provide an interpretation for this relationship previously derived by lyra and tsallis between dynamical and geometrical properties

random matrix theory in semiclassical quantum mechanics of chaotic systems the statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles the author explains this correspondence including the fact that long ranged spectral statistics show deviations from the results of random matrix theory the method depends on the ambiguity of quantisation of a given classical system although the energy levels depend on the particular quantisation used the spectral statistics are assumed to depend only on the classical motion the ambiguity of quantisation can be represented by a small perturbation acting on the hamiltonian this perturbation executes a random walk which causes its matrix elements to undergo a random walk if the matrix elements are completely uncorrelated the spectral statistics are those of random matrix theory in the case of classically chaotic systems the matrix elements are constrained by sum rules related to classical periodic orbits this leads to deviations of the long ranged spectral statistics from the predictions of random matrix theory

asymptotic and essentially singular solutions of the feigenbaum equation for suitably defined largen we express feigenbaum s equation as a singular schroder functional equation whose solution is obtained using a scaling ansatz in the limit of infiniten certain self consistency conditions on the scaled schroder solution lead to an essentially singular solution of feigenbaum s equation with a length scale factor of 0 0333 and a limiting feigenvalue of 30 50 in agreement with eckmann and wittwer s value of 0 0333831 and their conjectured estimate of 30

windows in one dimensional maps abstract the window structure in the chaotic part of the parameter space of one dimensional maps is described in detail explicit expressions are derived for both the internal structure and the width of the periodic windows by considering a universal local submap both features are found to depend only on the order of the extremum of this submap moreover the windows can be ordered in trees the various branches of these trees define accumulating families of windows for which exact scaling relations can be derived finally we discuss an extension of our description of window phenomena into two dimensions

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

equilibria of non compact generalized games with l majorized preference correspondences an existence theorem of maximal elements in a non compact set for l majorized correspondences is obtained next an existence theorem of an equilibrium in a qualitative game is proved and is then applied to achieve an existence theorem of equilibrium in a non compact abstract economy with l majorized preference correspondences these results are either very closely related to or generalizations of those recent results of borglin keiding yannelis prabhakar toussaint and tulcea

parametric correlations of the energy levels of ray splitting billiards parameter dependent statistical properties of the spectra of ray splitting billiards are studied experimentally and theoretically the autocorrelation functions c x and c omega x of level velocities as well as the generalized conductance c 0 are calculated for two different classically chaotic ray splitting billiards experimentally a modified sinai ray splitting billiard is studied consisting of a thin microwave rectangular cavity with two quarter circle shaped teflon inserts the length of the cavity serves as the experimentally adjustable parameter for the theoretical estimates of the parametric correlations we compute the quantum spectrum of a scaling triangular ray splitting billiard our experimental and numerical results are compared with each other and with the predictions of random matrix theory

existence of equilibria for generalized games without paracompactness in this paper we first prove an existence theorem of maximal elements without any paracompactness then we establish an existence theorem of equilibrium points of generalized games with infinite many players and without any paracompactness by applying our maximal element existence theorem our equilibrium existence theorem answers the two questions raised by kim 6 in the affirmative

phase length distributions in intermittent band switching the distribution of phase lengths t for intermittent band switching is investigated its form is observed to deviate from a exponential function a minimal phase length is seen to exist and the probabilities for the first few occuring phase lengths are often strongly enhanced or suppressed these deviations are analyzed and described explicitly in terms of the parameters of a model map

an exploration of the henon quadratic map abstract the two dimensional mapping introduced by henon in 1976 which yields a strange attractor for the parameter values a 1 4 b 0 3 has been investigated over the extended region 2 a 6 1 b 1 analytical solutions have been obtained for the onset of real stable cycles of periods 1 through 6 the associated bifurcations to stable cycles of twice the period 2 to 12 have also been determined all the resulting boundary curves are displayed in the a b parameter plane finally an interesting three dimensional generalization of the henon map is announced

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

on constructing complete solution classes of the cvitanovic feigenbaum equation abstract the cvitanovic feigenbaum equation is studied with the distant purpose of determining under what auxiliary conditions the solution is unique to this end several complete solution classes are constructed with the aim of studying how these depend on the conditions imposed those imposed in this paper always include strict unimodality and for the most part continuity other conditions which figure are piecewise linearity even ness and concavity under the latter condition with and without the preceding one two sharp lower bounds are derived on feigenbaum s scaling factor for any value of the exponent z or the unique solutions corresponding to these lower bounds are exhibited they are piecewise linear also shown is the unique solution describing the dynamics on the attractor it is piecewise linear but neither even nor continuous

universality in the transition to chaos of dissipative systems the period doubling bifurcation process for two dimensional transforms exhibits a new class of universality when a small dissipation is taken into account the effective jacobian is then defined as a function of both the dissipation and the rank n of the cascade cycle 2n numerical simulations of a simple mechanical system and numerical calculations on the henon mapping show that the decrement lies on a continuous curve as function of the effective jacobian a method using this result to understand experimental data is explained and a first order approximation of the renormalization process yields an analytic expression of the curve among the different transitions to chaos the period doubling bifurcation cascade 1 2 has been extensively studied this transition is characterized by an experimental convergence rate of the bifurcation threshold sequence to the accumulation point the threshold of chaos it is well known that the decrement of this bifurcation cascade can take different values each value corresponds to a specific class of systems which can be characterized by some general features of the system undergoing the transition 3 4 5 we are concerned here with the two values i 4 699 the decrement of the well known one dimensional transform with a quadratic maximum 2 and ii 8 721 the decrement of a two dimensional non dissipative transforms 3 these two classes of systems are generic in physics and the two values i and ii are therefore relevant values of the decrement however these two exponents stand for the infinite dissipation case and the conservative one thus leaving out the general physical situation of a finite dissipation only hints of the effect of a small dissipation in a two dimensional mapping have been given 6 before the work of zisook 7 a thorough study of the effect of dissipation is set forth here the first two sections deal with the physical model used to perform the numerical investigation and the experimental data thus obtained a study of the renormalization process enables to generalise the relation n j n j2 first given by zisook in 7 to all transforms where the jacobian does not depend on the linearization point in the phase space furthermore a first order approximation gives an excellent analytic expression of the universal function displaying the crossover of the decrement between ii and i

universality of correlation functions of hermitian random matrices in an external field the behavior of correlation functions is studied in a class of matrix models characterized by a measure exp s containing a potential term and an external source term s n tr v m ma in the large n limit the short distance behavior is found to be identical to the one obtained in previously studied matrix models thus extending the universality of the level spacing distribution the calculation of correlation functions involves finite n determinant formulae reducing the problem to the large n asymptotic analysis of a single kernel k this is performed by an appropriate matrix integral formulation of k multi matrix generalizations of these results are discussed

feigenbaum theory for unimodal maps with asymmetric critical point we extend the renormalization analysis for period doubling in unimodal maps to the case of asymmetric critical points universal scaling phenomena are governed by period two points of a renormalization operator

parametric spectral statistics in unitary random matrix ensembles from distribution functions to intra level correlations we establish a general framework to explore parametric statistics of individual energy levels in unitary random matrix ensembles for a generic confinement potential tr w h we i find the joint distribution functions of the eigenvalues of h and h h v for an arbitrary fixed v both for finite matrix size n and in the thermodynamic n limit ii derive many point parametric correlation functions of the two sets of eigenvalues and show that they are naturally parametrized by the eigenvalues of the reactance matrix for scattering off the potential v iii prove the universality of the correlation functions in unitary ensembles with non gaussian non invariant confinement potential tr w h v iv establish a general scheme for exact calculation of level number dependent parametric correlation functions and apply the scheme to the calculation of intra level velocity autocorrelation function and the distribution of parametric level shifts

persistent current flux correlations calculated by quantum chaology we consider classically chaotic systems with the topology of a ring threaded by quantum flux 4 using semiclassical asymptotics we calculate the rwaveraged autocorrelation function c 4 of slopes of the energy levels persistent currents normalized by the mean level spacing for flux values differing by 4 our result furnishes the uniform approximation here w the rms winding number of the classical periodic orbits whose period is connected by heisenberg s relation to the mean level spacing is a large semiclassical parameter of order l li d for a system with d freedoms

generalized games and non compact quasi variational inequalities abstract in this paper by developing an approximation approach which is originally due to tuleca in 1986 we prove the existence of equilibria for generalized games in which constraint mappings correspondences are lower resp upper semicontinuous instead of having lower resp upper open sections or open graphs in infinite dimensional topological spaces then existence theorems of solutions for quasi variational inequalities and non compact generalized quasi variational inequalities are also established finally existence theorems of constrained games with non compact strategy sets are derived our results unify and generalize many well known results given in the existing literature in particular we answer the question raised by yannelis and prabhakar in 1983 in the affirmative under more weaker conditions