convergence of dynamical zeta functions i study poles and zeros of zeta functions in one dimensional maps numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function this describes convergence of the calculations of the first zero which is generally the physically interesting quantity some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow

fast summation at nonequispaced knots by nfft we develop a new algorithm for the fast computation of discrete sums f y j sum k 1 n alpha k k y j x k j 1 m based on the recently developed fast fourier transform fft at nonequispaced knots our algorithm in particular our regularization procedure is simply structured and can be easily adapted to different kernels k our method utilizes the widely known fft and can consequently incorporate advanced fft implementations in summary it requires cal o n log n m arithmetic operations we prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions

chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

computing periodic orbits with arbitrary precision this paper deals with the computation of periodic orbits of dynamical systems up to any arbitrary precision these very high requirements are useful for example in the studies of complex pole location in many physical systems the algorithm is based on an optimized shooting method combined with a numerical ordinary differential equation ode solver tides that uses a taylor series method nowadays this methodology is the only one capable of reaching precision up to thousands of digits for odes the method is shown to be quadratically convergent some numerical tests for the paradigmatic lorenz model and the henon heiles hamiltonian are presented giving periodic orbits up to 1000 digits

the generalized milnor thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters this paper presents an answer to an open problem in the dynamical systems of three letters the generalized milnor thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two dimensional parameter plane the concept of equal topological entropy class is introduced by the dual star product which is a generalization of the derrida gervois pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval the algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the milnor thurston characteristic polynomials moreover the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space 3 of three letters

on the organisation of transient chaos application to irregular scattering it is shown how the local structure of chaotic repellers being responsible for transient chaotic behaviour is deduced from the properties of hyperbolic periodic orbits relations between static and dynamical multifractal spectra with respect to the natural invariant measure on the repeller are derived for invertible maps of the plane the results obtained for maps with unit jacobian apply to hamiltonian systems with two degrees of freedom which exhibit the phenomenon of irregular scattering and are characterised by an exponential decay of trapping probability

determining the appropriate sample size for nonparametric tests for location shift for the case in which two independent samples arc to be compared using a nonparametric test for location shift we propose a bootstrap technique for estimating the sample sizes required to achieve a specified power the estimator called boot uses information from a small pilot experiment for the special case of the wilcoxon test a simulation study is conducted to compare boot to two other sample size estimators one method called anpv is based on the assumption that the underlying distribution is normal with a variance estimated from the pilot data the other method called noether adapts the sample size formula of noether for use with a location shift alternative the boot and noether sample size estimators are particularly appropriate for this nonparametric setting because they do not require assumptions about the shape of the underlying continuous probability distribution the simulation study shows that a sample size estimates can have large uncertainty b boot is at least as accurate as an

counting the number of periods in one dimensional maps with multiple critical points abstract the problem of counting the number of different periodic orbits in continuous maps of an interval is solved a map with m monotone pieces laps and m 1 critical points may have at most m 1 independent kneading sequences which provide the most convenient parameters for the map when one or more kneading sequences are kept constant or bound to vary simultaneously various degenerated cases of the map arise the number of period n orbits of the general m lap map as various degenerated cases is given by combinations of nink n with k m where nm n is the number of period n orbits of a particular kind of degenerated m lap map with only one kneading sequence the quantity nm n may be calculated or enumerated by many different methods which are all discussed in this paper we also give tabulated values for nm n for m 2 to 7

repeller structure in a hierarchical model ii metric properties we study the renormalization dynamics deriving from a hierarchical tight binding schrodinger equation in the part i of this work we analyzed the topological structure of the recurrent set a chaotic repeller and its relation with the spectral problem in this part we turn our attention to the metric properties of the repeller we first study periodic orbits and their bifurcation unfolding and we organize them on a binary tree we then apply a thermodynamic formalism which provides a complete characterization of the scaling properties of the energy spectrum the distributionf of local dimensions is determined by computing both a generalized function through the periodic orbits and the bandwidths of periodic approximants of the schrodinger operator when the growth rater of the potential is smaller than 1 we find evidence of a phase transition implying that two different classes of states coexist in the spectrum the asymptotic behavior of the lebesgue measure of the spectrum is also studied a linear scaling of to 0 is observed forr 1 while forr 1 the measure of the periodic approximants goes to 0 as r h with the hierarchical orderh finally we show that the localized state present for r 1 is characterized by a superexponential scaling of the bandwidth

a devil s staircase into chaotic scattering abstract we discuss analytically the transition from a regular to a chaotic classical dynamics of a sample scattering model introduced earlier the transition arises as a homoclinic bifurcation of an atypical type in phase space as a discontinuity for the grammatical complexity and as an onset of a devil s staircase for the topological entropy

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

a semiclassical initial value approximation for the trace of green s function a semiclassical initial value approximation for the trace of green s function is derived in contrast to the well known formula of gutzwiller applicability of the present expression does not require knowledge of the system s periodic orbits but constructs the trace from classical trajectories originating from all points on a poincare surface a given trajectory provides a contribution to the trace each time it returns to the surface with a weight based in part on the inner product on this surface of coherent states associated with the initial and returning points the treatment is generalized to obtain a version of the initial value formula that is useful for systems having discrete symmetries the initial value trace expression is shown to be semiclassically valid for chaotic systems by a stationary phase treatment that demonstrates its reduction to gutzwiller s formula in the classical limit numerical calculations of energy eigenvalues verify the applicability of the approximation not only to chaotic systems but to integrable systems and systems with mixed phase space the approximation presented here has numerical advantages over methods for determining the trace based on initial value treatments of the time dependent propagator especially for systems with homogeneous potential energy functions

fast ewald summation based on nfft with mixed periodicity in this paper we develop new fast fourier based methods for the coulomb problem we combine the ewald summation formulas and the fast summation approach based on the nonequispaced fast fourier transform nfft in order to develop efficient methods for calculating the coulomb energies as well as the acting forces in charged particle systems subject to mixed periodic boundary conditions therewith we extend the applicability of nfft based methods which already exist for open as well as for 3d periodic boundary conditions to arbitrary combinations of periodic and open boundary conditions we reconsider the derivation of the ewald formulas for 2d and 1d periodic systems introduce the new algorithms and present high precision numerical results

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

indices in classical mechanics the authors discuss the linearization of the equations of motion around a periodic orbit in a conservative two degree of freedom hamiltonian system with an emphasis on computationally useful formulae indices are defined using singularities in projections conjugate points self conjugate points and the possibility of assigning a symbolic code are discussed

queues stacks and transcendentality at the transition to chaos abstract we examine the one humped map at the period doubling transition to chaos and ask whether its long term memory is stack like last in first out or queue like first in first out we show that it can be recognized by a real time automaton with one queue or two stacks and give several new grammatical characterizations of it we argue that its memory has a queue like character since a single stack does not suffice we also show that its dynamical zeta function generating function and growth function are transcendental the same results hold for any period multiplying cascade we suggest that transcendentality might be a sign of dynamical phase transitions in other systems as well

generalized redundancies for time series analysis abstract extensions to various information theoretic quantities such as entropy redundancy and mutual information are discussed in the context of their role in nonlinear time series analysis we also discuss linearized versions of these quantities and their use as benchmarks in tests for nonlinearity many of these quantities can be expressed in terms of the generalized correlation integral and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics such as the bds and green savit statistics further numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation finally we consider several local versions of these quantities including a local kolmogorov sinai entropy which gives an estimate of variability of the short term predictability

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

classical chaotic scattering periodic orbits symmetries multifractal invariant sets the infinite set of periodic orbits of a chaotic system is investigated their globally exact symbolic dynamics is related to birkhoff s method of iterated symmetry lines which produces symmetric periodic orbits of various classes a complete picture emerges as to what part of the periodic orbits can be obtained by means of the birkhoff lines in addition the thermodynamic formalism is applied to the multifractal structure created by the horseshoe arrangement of the periodic orbits

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

fast approximate fourier transforms for irregularly spaced data several algorithms for efficiently evaluating trigonometric polynomials at irregularly spaced points are presented and analyzed the algorithms can be viewed as approximate generalizations of the fast fourier transform fft and they are compared with regard to their accuracy and their computational efficiency

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

the topological skeleton of cellular automaton dynamics abstract we have developed statistical techniques to study the structure of the state transition graphs of cellular automata with periodic boundary conditions in the limit of large system size we organize our results around the concept of a topological skeleton the topological skeleton is the set of physically relevant states covering this skeleton is a surface typically thin and dense which contains the bulk of the set of states states in the skeleton have some long histories states on the surface by contrast have only short histories they are reached only near the beginning of cellular automaton evolution we study in detail a sequence of rules which exhibit mostly skeletal to mostly surface structure

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

controlling transient chaos the method of ott grebogi and yorke 1990 is extended to control transient chaos the controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor the time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime the number of controlled trajectories however decreases the maximum perturbation according to a power law applicability to experimental situations and comparison with permanent chaotic cases are discussed

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

propagation of periodic and chaotic action potential trains along nerve fibers abstract in this paper we report the findings that the action potential trains transmitted along the nerve fibers are encoded not only periodically but also chaotically first spontaneous action potentials along a single fiber of injured sciatic nerves in the anesthetized rat were recorded then the data were divided into two groups and analyzed with different methods phase space representation spectral analysis and the calculation of correlation dimension were used for the first group of data sampled with constant frequency due to the serious influence of the measurement noise no reliable conclusion can be drawn from them for the second group of data of the interspike intervals isi which seem to convey more rich and important information nonlinear forecasting method the surrogate data and the plot of isi n 1 vs isi n were used in the analysis good results have been obtained which confirm with those from the cell model the largest lyapunov exponent lle were calculated not only to further support our findings of chaos but also to quantitatively determine the degree of chaos

periodic orbits as the skeleton classical and quantum chaos a description of a low dimensional deterministic chaotic system in terms of unstable periodic orbits cycles is a powerful tool for theoretical and experimental analysis of both classical and quantum deterministic chaos comparable to the familiar perturbation expansions for nearly integrable systems the infinity of orbits characteristic of a chaotic dynamical system can be resummed and brought to a selberg product form dominated by the short cycles and the eigenvalue spectrum of operators associated with the dynamical flow can then be evaluated in terms of unstable periodic orbits methods for implementing this computation for finite subshift dynamics are introduced

a numerical method for locating stable periodic orbits in chaotic systems abstract a numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented the method is based on the selection of trajectory segments which are characterized by a relatively low positive local lyapunov exponent once such a selection is made convergence to stable or weakly unstable periodic orbits is obtained by a newton method the algorithm is rather general and can be used for systems with more than two degrees of freedom the proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field in the latter case new stable periodic orbits are found in the region of strong chaotic motion the possible quantum localization on these orbits is discussed briefly

sample size determination for some common nonparametric tests abstract the article discusses the problem of determining the number of observations required by some common nonparametric tests so that the tests have power at least 1 against alternatives that differ sufficiently from the hypothesis being tested it is shown that the number of observations depends on certain simple probabilities a method is suggested for fixing the value of the appropriate probability when determining sample size

hamiltonian transport on unstable periodic orbits transport in hamiltonian systems is approached on the basis of the unstable periodic orbits upos embedded in the chaotic region correlation functions and diffusion rates are defined for upo ensembles of two dimensional periodic maps global diffusion in these maps is interpreted in terms of the ensembles ut containing all the upos of period t as t the relative populations of closed orbits and accelerator modes in ut approach a gaussian distribution characterized by the diffusion coefficient d evidence for the validity of this mechanism is provided for the cat and sawtooth maps prime lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion these ensembles are generically small subsets of ut for some t it is shown however that the corresponding diffusion rates assume for t odd precisely the known value of d in the entire time interval t t for t even all orbits are closed and d is reproduced only in the interval t 12 t the diffusion rates associated with ensembles ut of the sawtooth maps appear to approximate well the value of d if t is of the order of the diffusion time the case of local transport rates associated with ensembles that are localized in phase space is illustrated

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

kaiser bessel basis for particle mesh interpolation in this work we introduce the kaiser bessel interpolation basis for the particle mesh interpolation in the fast ewald method a reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems and is shown to be effective in optimizing the shape parameter of the kaiser bessel basis in terms of accuracy by comparing the optimized kaiser bessel basis with the traditional b spline basis we demonstrate that the former is more accurate than the latter in part of the working parameter space say a relatively small real space cutoff a relatively small reciprocal space mesh and a relatively large truncation of basis in some cases the kaiser bessel basis is found to be more than one order of magnitude more accurate

entropy of hidden markov processes via cycle expansion hidden markov processes hmp is one of the basic tools of the modern probabilistic modeling the characterization of their entropy remains however an open problem here the entropy of hmp is calculated via the cycle expansion of the zeta function a method adopted from the theory of dynamical systems for a class of hmp this method produces exact results both for the entropy and the moment generating function the latter allows to estimate via the chernoff bound the probabilities of large deviations for the hmp more generally the method offers a representation of the moment generating function and of the entropy via convergent series

universal encoding for unimodal maps the standard encoding procedure to describe the chaotic orbits of unimodal maps is accurately investigated we show that the grammatical rules of the underlying language can be easily classified in a compact form by means of a universal parameter two procedures to construct finite graphs which approximate non markovian cases are discussed showing also the intimate relation with the corresponding construction of approximate markov partitions the convergence of the partial estimates of the topological entropy is discussed proving that the error decreases exponentially with the length of the sequences considered the rate is shown to coincide with the topological entropyh itself finally a superconvergent method to estimateh is introduced

differentiability implies continuity in neuronal dynamics recent work has identified nonlinear deterministic structure in neuronal dynamics using periodic orbit theory troublesome in this work were the significant periods of time where no periodic orbits were extracted dynamically dark regions tests for periodic orbit structure typically require that the underlying dynamics are differentiable since continuity of a mathematical function is a necessary but insufficient condition for differentiability regions of observed differentiability should be fully contained within regions of continuity we here verify that this fundamental mathematical principle is reflected in observations from mammalian neuronal activity first we introduce a null jacobian transformation to verify the observation of differentiable dynamics when periodic orbits are extracted second we show that a less restrictive test for deterministic structure requiring only continuity demonstrates widespread nonlinear deterministic structure only partially appreciated with previous approaches 2001 published by elsevier science b v

a note on sample size determination for a nonparametric test of location sample size determination methods are considered for hypothesis testing about location parameters of two continuous distributions using independent random samples and the mann whitney wilcoxon mww test for a specified shift size and power these methods are based on the so called noether s formula derived from a normal approximation to the power of the mww test and a pilot sample from each of the distributions compared with alternate methods including the one using bootstrap similar to that of hamilton and collings 1991 the new methods are shown to be at least as good in terms of accuracy and substantially more efficient in terms of speed and variability the simpler method using linearly smoothed empirical cdf s of the pilot samples and requiring no bootstrapping is recommended for practical use extensions and adaptations for obtaining an upper confidence bound on the sample size and to general linear rank tests are indicated

on sample size of the kruskal wallis test with application to a mouse peritoneal cavity study summary as the nonparametric generalization of the one way analysis of variance model the kruskal wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown although the kruskal wallis test has been widely used for data analysis power and sample size methods for this test have been investigated to a much lesser extent this article proposes new power and sample size calculation methods for the kruskal wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model no assumption is made on the shape of the underlying population distributions simulation results show that in terms of sample size calculation for the kruskal wallis test the proposed methods are more reliable and preferable to some more traditional methods a mouse peritoneal cavity study is used to demonstrate the application of the methods

optimal sample size planning for the wilcoxon mann whitney test there are many different proposed procedures for sample size planning for the wilcoxon mann whitney test at given type i and type ii error rates and respectively most methods assume very specific models or types of data to simplify calculations eg ordered categorical or metric data location shift alternatives etc we present a unified approach that covers metric data with and without ties count data ordered categorical data and even dichotomous data for that we calculate the unknown theoretical quantities such as the variances under the null and relevant alternative hypothesis by considering the following synthetic data approach we evaluate data whose empirical distribution functions match the theoretical distribution functions involved in the computations of the unknown theoretical quantities then well known relations for the ranks of the data are used for the calculations in addition to computing the necessary sample size n for a fixed allocation proportion t n1 n where n1 is the sample size in the first group and n n1 n2 is the total sample size we provide an interval for the optimal allocation rate t which minimizes the total sample size n it turns out that for certain distributions a balanced design is optimal we give a characterization of such distributions furthermore we show that the optimal choice of t depends on the ratio of the two variances which determine the variance of the wilcoxon mann whitney statistic under the alternative this is different from an optimal sample size allocation in case of the normal distribution model