conditionally solvable path integral problems ii natanzon potentials new classes of exactly solvable potentials are discussed within the path integral formalism they are constructed from the hypergeometric and confluent natanzon potentials respectively it is found that they allow incorporation of four free parameters which give rise to fractional power behaviour long range and strongly anharmonic terms we find six different classes of such potentials

nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

singularity formation for prandtl s equations abstract we consider prandtl s equations for an impulsively started disk and follow the process of the formation of the singularity in the complex plane using the singularity tracking method we classify van dommelen and shen s singularity as a cubic root singularity we introduce a class of initial data uniformly bounded in h 1 which have a dipole singularity in the complex plane these data lead to a solution blow up whose time can be made arbitrarily short within the class this is numerical evidence of the ill posedness of the prandtl equations in h 1 the presence of a small viscosity in the streamwise direction changes the behavior of the singularities they stabilize at a distance from the real axis which depends on the amount of viscosity we show that the van dommelen and shen singularity and the singularity predicted by e and engquist in w e b engquist blowup of the solutions to the unsteady prandtl s equations comm pure appl math 50 1997 1287 1293 have different complex structures

sublayer of prandtl boundary layers the aim of this paper is to investigate the stability of prandtl boundary layers in the vanishing viscosity limit nu to 0 in grenier commun pure appl math 53 9 1067 1091 2000 one of the authors proved that there exists no asymptotic expansion involving one of prandtl s boundary layer with thickness of order sqrt nu which describes the inviscid limit of navier stokes equations the instability gives rise to a viscous boundary sublayer whose thickness is of order nu 3 4 in this paper we point out how the stability of the classical prandtl s layer is linked to the stability of this sublayer in particular we prove that the two layers cannot both be nonlinearly stable in l infty that is either the prandtl s layer or the boundary sublayer is nonlinearly unstable in the sup norm

quasilinearization method and its verification on exactly solvable models in quantum mechanics the proof of the convergence of the quasilinearization method of bellman and kalaba whose origin lies in the theory of linear programming is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems this powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones and is not based unlike perturbation theories on existence of some kind of small parameter the general properties of the method particularly its uniform and quadratic convergence which often also is monotonic are analyzed and verified on exactly solvable models in quantum mechanics namely application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions the method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist

minority game with peer pressure abstract to study the interplay between global market choice and local peer pressure we construct a minority game like econophysical model in this so called networked minority game model every selfish player uses both the historical minority choice of the population and the historical choice of one s neighbors in an unbiased manner to make decision results of numerical simulation show that the level of cooperation in the networked minority game differs remarkably from the original minority game as well as the prediction of the crowd anticrowd theory we argue that the deviation from the crowd anticrowd theory is due to the negligence of the effect of a four point correlation function in the effective hamiltonian of the system

on the stability of boundary layers of incompressible euler equations abstract in this paper we investigate the stability and instability of boundary layers of incompressible euler equations copyright 2000 academic press dans cet article on etudie la stabilite et l instabilite de couches limites des equations d euler

blowup of solutions of the unsteady prandtl s equation we prove that for certain class of compactly supported c initial data smooth solutions of the unsteady prandtl s equation blow up in nite time

on the minority game analytical and numerical studies abstract we investigate further several properties of the minority game we have recently introduced we explain the origin of the phase transition and give an analytical expression of 2 n in the n 2 m region the ability of the players to learn a given payoff is also analyzed and we show that the darwinian evolution process tends to a self organized state in particular the lifetime distribution is a power law with exponent 2 furthermore we study the influence of identical players on their gain and on the systems performance finally we show that large brains always take advantage of small brains

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

mapping of shape invariant potentials under point canonical transformations the authors give explicit point canonical transformations which map twelve types of shape invariant potentials which are known to be exactly solvable into two potential classes the eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively

evolution in minority games i games with a fixed strategy space in this paper we study the minority game in the presence of evolution in particular we examine the behavior in games in which the dimension of the strategy space m is the same for all agents and fixed for all time we find that for all values of m not too large evolution results in a substantial improvement in overall system performance we also show that after evolution results obey a scaling relation among games played with different values of m and different numbers of agents analogous to that found in the non evolutionary adaptive games best system performance still occurs for a given number of agents at mc the same value of the dimension of the strategy space as in the non evolutionary case but system performance is now nearly an order of magnitude better than the non evolutionary result for m mc the system evolves to states in which average agent wealth is better than in the random choice game despite and in some sense because of the persistence of maladaptive behavior by some agents as m gets large overall systems performance approaches that of the random choice game

failsafe modes in incomplete minority game abstract we make a failsafe extension to the incomplete minority game model give a brief analysis on how incompleteness will effect system efficiency simulations that limited incompleteness in strategies can improve the system efficiency among three failsafe modes the back to best mode brings most significant improvement and keeps the system efficiency in a long range of incompleteness a simple analytic formula has a trend which matches simulation results the immg model is used to study the effect of distribution and we find that there is one junction point in each series of curves at which system efficiency is not influenced by the distribution of incompleteness when p i p i c the concentration of incompleteness weakens the effect on the other side of p i c concentration will be helpful when p i is close to zero agents using incomplete strategies have on average better profits than those using standard strategies and the back to best agents have a wider range of p i to win

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

the full strategy minority game abstract the full strategy minority game f s m g is an instance of the minority game m g which includes a single copy of every potential agent in this work we explicitly solve the f s m g thanks to certain symmetries of this game furthermore by considering the m g as a statistical sample of the f s m g we compute approximated values of the key variable 2 n in the symmetric phase for different versions of the m g as another application we prove that our results can be easily modified in order to handle certain kinds of initial biased strategy scores in particular when the bias is introduced at the agents level we also show that the f s m g verifies a strict period two dynamics i e period two dynamics satisfied with probability 1 giving to the best of our knowledge the first example of an instance of the m g for which this feature can be analytically proved thanks to this property it is possible to compute in a simple way the probability that a general instance of the m g breaks the period two dynamics for the first time in a given simulation

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

el farol revisited this article is concerned with the global behavior of agents in the el farol bar problem in particular we discuss global attendance in terms of its mean and variance and show that there is a strong dependence of both on the externally imposed comfort level we present a possible interpretation for the observed behavior and propose that the mean attendance converges to the perceived threshold value as opposed to the actually imposed one

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

conditionally solvable path integral problems some specific conditionally exactly solvable potentials are discussed within the path integral formalism they generalize the usually known potentials by the incorporation of fractional power behaviour and strongly anharmonic terms we find four different kinds of such potential the first is related to the coulomb potential the second is an anharmonic confinement potential and the third and fourth are related to the manning rosen potential

bound states and the potential parameter spectrum in this article we answer the following question if the wave equation possesses bound states but it is exactly solvable for only a single non zero energy can we find all bound state solutions energy spectrum and associated wavefunctions to answer this question we use the tridiagonal representation approach to solve the wave equation at the given energy by expanding the wavefunction in a series of energy dependent square integrable basis functions in configuration space the expansion coefficients satisfy a three term recursion relation which is solved in terms of orthogonal polynomials depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the potential parameter spectrum pps this discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite set inverting the relation between the energy and the pps gives the bound state energy spectrum therefore the answer to the above question is affirmative

a well posedness theory for the prandtl equations in three space variables abstract the well posedness of the three space dimensional prandtl equations is studied under some constraint on its flow structure together with the instability result given in 28 it gives an almost necessary and sufficient structural condition for the stability of the three dimensional prandtl equations it reveals that the classical burgers equation plays an important role in determining this type of flow with special structure that avoids the appearance of the secondary flow an unstabilizing factor in the three dimensional prandtl boundary layers and the sufficiency of the monotonicity condition on the tangential velocity field for the existence of solutions to the prandtl boundary layer equations is illustrated in the three dimensional setting moreover it is shown that this structured flow is linearly stable for any smooth three dimensional perturbation

singularity formation for complex solutions of the 3d incompressible euler equations abstract moore s approximation method first formulated for vortex sheets is generalized and applied to axi symmetric flow with swirl and with smooth initial data the approximation preserves the forward cascade of energy but neglects any backflow of energy it splits the euler equations into two sets of equations one for u u r z t containing all non negative wavenumbers in z and the second for u u the equations for u are exactly the euler equations but with complex initial data traveling waves solutions u u r z i t with imaginary wave speed are found numerically for this problem the asymptotic properties of the resulting fourier coefficients show a singularity forming in finite time at which the velocity blows up

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

almost global existence for the prandtl boundary layer equations we consider the prandtl boundary layer equations on the half plane with initial datum that lies in a weighted h1 space with respect to the normal variable and is real analytic with respect to the tangential variable the boundary trace of the horizontal euler flow is taken to be a constant we prove that if the prandtl datum lies within varepsilon of a stable profile then the unique solution of the cauchy problem can be extended at least up to time t varepsilon geqq rm exp varepsilon 1 rm log varepsilon 1

complex space singularities of 2d euler flow in lagrangian coordinates we show that for 2d space periodic incompressible flow the solution can be evaluated numerically in lagrangian coordinates with the same accuracy that is achieved in standard eulerian spectral methods this allows the determination of complex space lagrangian singularities lagrangian singularities are found to be closer to the real domain than eulerian singularities and seem to correspond to fluid particles which escape to complex infinity by the current time various mathematical conjectures regarding eulerian lagrangian singularities are presented

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

energy spectrum of a generalized scarf potential using the asymptotic iteration method and the tridiagonal representation approach the well known trigonometric scarf potential is generalized by adding a sinusoidal term and then treated using the asymptotic iteration method aim and the tridiagonal representation approach tra the energy spectrum of the associated bound states is computed for the aim we have improved convergence of the quantization condition that terminates the iterations asymptotically this is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results plateau of convergence we have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point the value of this point or points may depend on the physical parameters the numerical results have been compared favorably with those resulting from the tra the well known trigonometric scarf potential is generalized by adding a sinusoidal term and then treated using the asymptotic iteration method aim and the tridiagonal representation approach tra the energy spectrum of the associated bound states is computed for the aim we have improved convergence of the quantization condition that terminates the iterations asymptotically this is accomplished by looking for the range of initial values of the space variable in the terminating condition that produces stable results plateau of convergence we have shown that with increasing iteration this plateau of convergence grows up rapidly to an optimal iteration number and then shrinks slowly to a point the value of this point or points may depend on the physical parameters the numerical results have been compared favorably with those resulting from the tra

zero viscosity limit for analytic solutions of the primitive equations the aim of this paper is to prove that the solutions of the primitive equations converge in the zero viscosity limit to the solutions of the hydrostatic euler equations we construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic euler equation and boundary layer correctors as the first order term and an error that we show to be o sqrt nu the main assumption is spatial analyticity of the initial datum

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

trading model with pair pattern strategies abstract a simple trading model based on pair pattern strategy space with holding periods is proposed power law behavior is observed for the return variance 2 the price impact h and the predictability k for both models with linear and square root impact functions the sum of the traders wealth displays a positive value for the model with a square root price impact function and a qualitative explanation is given based on the observation of the conditional excess demand a u the cumulative wealth distribution also obeys a power law behavior with an exponent close to that of real markets an evolutionary trading model is further proposed the elimination mechanism effectively changes the behavior of traders and a power law behavior is observed in the measure of zero return distribution p r 0 the trading model with other types of traders e g traders with the mg s strategies and producers are also carefully studied

boolean game on scale free networks abstract inspired by the local minority game we propose a network boolean game and investigate its dynamical properties on scale free networks the system can self organize to a stable state with better performance than the random choice game although only the local information is available to each agent by introducing the heterogeneity of local interactions we find that the system will achieve the best performance when each agent s interaction frequency is linearly correlated with its information capacity generally the agents with more information gain more than those with less information while in the optimal case each agent almost has the same average profit in addition we investigate the role of irrational factor and find an interesting symmetrical behavior

incomplete strategy effect in minority game abstract we perform simulations with the extended incomplete minority game model and find that limited incompleteness in strategies can improve the system efficiency greatly there is a critical p i c such that agents in a system can get the best cooperation the critical strategy density value c decreases monotonously with the increase of p i we also mix incomplete strategy agents with standard strategy agents to study the effect of incompleteness distribution on system properties and group average profit concentration of incompleteness weakens its effect when p i is close to zero agents using incomplete strategies averagely have better profits than those using standard strategies

comprehensive analysis of conditionally exactly solvable models we study a quantum mechanical potential introduced previously as a conditionally exactly solvable ces model besides an analysis following its original introduction in terms of the point canonical transformation we also present an alternative supersymmetric construction of it we demonstrate that from the three roots of the implicit cubic equation defining the bound state energy eigenvalues there is always only one that leads to a meaningful physical state finally we demonstrate that the present ces interaction is in fact an exactly solvable natanzon class potential

the van dommelen and shen singularity in the prandtl equations abstract in 1980 van dommelen and shen provided a numerical simulation that predicted the spontaneous generation of a singularity in the prandtl boundary layer equations from a smooth initial datum for a nontrivial euler background in this paper we prove this numerical conjecture by rigorously establishing the finite time blowup of the boundary layer thickness

on the global existence of solutions to the prandtl s system abstract in this paper we establish a global existence of weak solutions to the two dimensional prandtl s system for unsteady boundary layers in the class considered by oleinik j appl math mech 30 1966 951 provided that the pressure is favourable this generalizes the local well posedness results due to oleinik 1966 mathematical models in boundary layer theory chapman hall london 1999 for the proof we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations

local in time well posedness for compressible mhd boundary layer abstract in this paper we are concerned with the motion of electrically conducting fluid governed by the two dimensional non isentropic viscous compressible mhd system on the half plane with no slip condition on the velocity field perfectly conducting wall condition on the magnetic field and dirichlet boundary condition on the temperature on the boundary when the viscosity heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate there is a boundary layer which is described by a prandtl type system under the non degeneracy condition on the tangential magnetic field instead of monotonicity of velocity by applying a coordinate transformation in terms of the stream function of magnetic field as motivated by the recent work 27 we obtain the local in time well posedness of the boundary layer system in weighted sobolev spaces

a blow up criterion for classical solutions to the prandtl equations in this paper we obtain a blow up criterion for classical solutions to the two dimensional prandtl equations under a monotonicity assumption in weighted sobolev spaces our proof is based on nonlinear energy estimates inspired by masmoudi and wong commun pure appl math 68 1683 1741 2015 and maximum principles for parabolic equations

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

multifractal features of financial markets abstract we study the tick dynamical behavior of three assets in financial markets the kospi the won dollar and yen dollar exchange rates using the rescaled range r s analysis the multifractal hurst exponents with long run memory effect can be obtained from those assets and we discuss whether there exists the crossover or not for the hurst exponents at characteristic time scales particularly we find that the probability distribution of returns approaches to a lorentz distribution different from gaussian properties

new class of conditionally exactly solvable potentials in quantum mechanics motivated by an idea of dutra 1993 we obtain a new class of one dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value it is shown that using shape invariant potentials as input one may generate different classes of such potentials even in more than one dimension we also illustrate that wkb and supersymmetry inspired wkb methods provide very good approximations for these potentials with the latter doing comparatively better

from turbiner s quasi exactly soluble potentials in n dimensions to analytic solutions of the combined coulomb plus oscillator system turbiner s one dimensional quasi exactly soluble potentials can be interpreted as central potentials in n dimensions using a particular transformation one shows how these potentials convert to seemingly unrelated isolated solutions of other schrodinger equations including solutions of the combined coulomb plus harmonic oscillator system

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

two new solvable potentials we succeeded in finding two new solvable potentials by working in a complete square integrable basis that carries a tridiagonal jacobi matrix representation for the wave operator the conventional methods do not lead to exact solutions for these potentials hence a new approach has been adopted for these potentials first we obtain the potential parameter spectrum the set of values of the potential parameters that lead to an exact solution at a given energy then the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum for a given potential parameter this procedure has been applied in 3d to obtain the energy spectrum for a special screened coulomb potential with a barrier and in 1d for a single wave potential

evolution in minority games ii games with variable strategy spaces abstract we continue our study of evolution in minority games by examining games in which agents with poorly performing strategies can trade in their strategies for new ones from a different strategy space in the context of the games discussed in this paper this means allowing for strategies that use information from different numbers of time lags m we find in all the games we study that after evolution wealth per agent is high for agents with strategies drawn from small strategy spaces small m and low for agents with strategies drawn from large strategy spaces large m in the game played with n agents wealth per agent as a function of m is very nearly a step function the transition is at m mt where mt mc 1 here mc is the critical value of m at which n agents playing the game with a fixed strategy space fixed m have the best emergent coordination and the best utilization of resources we also find that overall system wide utilization of resources is independent of n furthermore although overall system wide utilization of resources after evolution varies somewhat depending on some other aspects of the evolutionary dynamics in the best cases utilization of resources is on the order of the best results achieved in evolutionary games with fixed strategy spaces simple explanations are presented for some of our main results

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved