computing the hypergeometric function the hypergeometric function of a real variable is computed for arbitrary real parameters the transformation theory of the hypergeometric function is used to obtain rapidly convergent power series the divergences that occur in the individual terms of the transformation for integer parameters are removed using a finite difference technique

numerically satisfactory solutions of hypergeometric recursions each family of gauss hypergeometric functions fn 2f1 a 1n b 2n c 3n z n z for fixed j 0 1 not all j equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions many of the 26 cases for different j values can be transformed into each other in this way only with four basic difference equations all other cases can be obtained for each of these recurrences we give pairs of numerically satisfactory solutions in the regions in the complex plane where t1 6 t2 t1 and t2 being the roots of the characteristic equation

exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables abstract we present a numerical study of some billiard tables depending on a perturbative parameter 0 and a hyperbolicity parameter h 0 these tables are ellipses for 0 and circumferences in the limit h 0 elliptic billiard tables are integrable and have four separatrices which break up when 0 we conjecture based on numerical experiments that as h 0 the area of the main lobes of the resulting turnstile which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories behaves like an exponential term e 2 h times an asymptotic series j 0 j h 2 j such that 0 0 this series is gevrey 1 of type 1 2 2 so that its borel transform is convergent on a disk of radius 2 2 in the limit 0 the series j 0 j 0 h 2 j is an analytic function which can be explicitly computed with a discrete melnikov method the asymptotic series j 0 j h 2 j associated to the second exponential term e 2 2 h has the same properties finally we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space our computations have been performed in multiple precision arithmetic namely with several thousands decimal digits and rely strongly on the expansion of the local invariant curves up to very high orders namely with several hundreds taylor coefficients our programs have been written using the pari system

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

dynamics of 4d symplectic maps near a double resonance abstract we study the dynamics of a family of 4 d symplectic mappings near a doubly resonant elliptic fixed point we derive and discuss algebraic properties of the resonances required for the analysis of a takens type normal form in particular we propose a classification of the double resonances adapted to this problem including cases of both strong and weak resonances around a weak double resonance a junction of two resonances of two different orders both being larger than 4 the dynamics can be described in terms of a simple in general non integrable hamiltonian model the non integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder we use a 4 d generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 d symplectic map we evaluate numerically the volume of a 4 d parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively in good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 d analog different numerical studies point out that double resonances play a key role to understand arnold diffusion this paper has to be seen also as a first step in this direction

the abc of hyper recursions each member of the family of gauss hypergeometric functions fn 2f1 a e1n b e2n c e3n z where a b c and z do not depend on n and ej 0 1 not all ej equal to zero satisfies a second order linear difference equation of the form anfn 1 bnfn cnfn 1 0 because of symmetry relations and functional relations for the gauss functions the set of 26 cases for different ej values can be reduced to a set of 5 basic forms of difference equations in this paper the coefficients an bn and cn of these basic forms are given in addition domains in the complex z plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified the determination of such a pair asks for a detailed study of the asymptotic properties of the gauss functions fn for large values of n and of other gauss functions outside this group this will be done in a later paper

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

an analytical investigation of shape control of large space structures by applied temperatures an analytical procedure for the static shape control of flexible space structures subjected to thermal distortions is developed which is based on prescribing temperatures in control elements having much higher coefficients of thermal expansion than the main structure the temperatures at the control elements are defined so as to minimize the overall thermal distortion of the structure from its ideal shape and a matrix equation is obtained which can be solved for the set of optimum control temperatures a formulation of the procedure for continuous structures governed by differential equations and a formulation for discrete finite element modeled structures governed by matrix equations are presented the equations from the continuous formulation are employed for the shape control of a simple beam distorted by nonuniform heating and the discrete formulation is applied in a general purpose finite element structural analysis computer program for the shape control of a 750 m radiometer antenna reflector dish subjected to orbital heating a reduction in thermal distortion by a factor of nearly 50 was obtained with the use of only seven control elements results for four different sets of control locations for the antenna are presented in which reductions in distortion of up to a factor of four were obtained

new properties of hypergeometric series derivable from feynman integrals ii a generalisation of the h function for pt i see ibid vol 20 p 4109 1987 further examples of the use of feynman integrals enable the derivation of new properties of hypergeometric series including new analytic continuation formulae for a generalised hypergeometric series and for a kampe de feriet function this motivates the derivation of two new summation formulae for a generalised hypergeometric series and furthermore leads to a natural generalisation of the h function while the latter as is well known contains as particular cases most of the special functions of applied mathematics it does not contain some of importance for instance the riemann zeta function nor indeed any polylogarithm our generalisation of the h function does contain the polylogarithm it also contains the exact partition function of the gaussian model from statistical mechanics another new result is the simple summation formula 3f2 1 1 3 2 2 2 x 1 ln 1 1 x 12 2

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

on the mutual information and low snr capacity of memoryless noncoherent rayleigh fading channels the memoryless noncoherent single input single output siso rayleigh fading channel is considered closed form expressions are derived for the mutual information between the output and the input of this channel when the input magnitude distribution is discrete and is restricted to having two mass points it is subsequently shown how these expressions can be used to obtain closed form expressions for the capacity of this channel for signal to noise ratio snr values of up to approximately 0 db and a tight capacity lower bound for snr values between 0 db and 10 db the expressions for the channel capacity and its lower bound are given as functions of a parameter which can be obtained via numerical root finding algorithms

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

melnikov method and exponentially small splitting of separatrices abstract we study the splitting of separatrices for high frequency perturbations of a planar hamiltonian system it is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period 2 e and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane on the other hand in many examples the melnikov function can be calculated explicitly and is exponentially small with an exponent very close but larger than the exponent from the upper estimates so the melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error we found a class of hamiltonians such that the melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude is less than 0ep for some p 0

two point taylor expansions in the asymptotic approximation of double integrals application to the second and fourth appell functions abstract the main difficulty in laplace s method of asymptotic expansions of double integrals is originated by a change of variables we consider a double integral representation of the second appell function f 2 a b b c c x y and illustrate over this example a variant of laplace s method which avoids that change of variables and simplifies the computations essentially the method only requires a taylor expansion of the integrand at the critical point of the phase function we obtain in this way an asymptotic expansion of f 2 a b b c c x y for large b b c and c we also consider a double integral representation of the fourth appell function f 4 a b c d x y we show in this example that this variant of laplace s method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain we obtain in this way an asymptotic approximation of f 4 a b c d x y for large values of a b c and d in this second example the method requires a taylor expansion of the integrand at two points simultaneously for this purpose we also investigate in this paper taylor expansions of two variable analytic functions with respect to two points giving cauchy type formulas for the coefficients of the expansion and details about the regions of convergence

bounds for the appell function f2 abstract the exact matrix elements for certain second order radiation processes in hydrogenic atoms can all be expressed in terms of a certain generalized hypergeometric function of two variables in this paper we obtain bounds for this function as the two variables tend to the singular point z 1 from the upper half plane the bounds are shown to he uniform in the hypergeometric parameters in the ranges of interest and may he used to bound the exact matrix elements in both the non relativistic and relativistic cases

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

rapidly forced planar vector fields and splitting of separatrices abstract we prove that for analytic rapidly forced two dimensional systems the melnikov function is exponentially small with respect to the period we provide an example of non validity of melnikov s method and we obtain upper bounds for conservative and analytic planar systems for big amplitude periodic rapidly forcing

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

new properties of hypergeometric series derivable from feynman integrals i transformation and reduction formulae a study in the statistical mechanics of phase transitions has involved certain seemingly simple feynman integrals which have yielded surprisingly significant mathematical results the integrals enable the derivation of new transformation summation and reduction formulae for some single and multiple variable hypergeometric series the main result reported in this paper is a transformation formula a special case of which is a summation formula for a symmetric combination of two non terminating clausen s hypergeometric series of unit argument

capacity of multiple antenna systems with both receiver and transmitter channel state information the capacity of multiple antenna systems operating in rayleigh flat fading is considered under the assumptions that channel state information csi is available at both transmitter and receiver and that the transmitter is subjected to an average power constraint first the capacity of such systems is derived for the special case of multiple transmit antennas and a single receive antenna the optimal power allocation scheme for such a system is shown to be a water filling algorithm and the corresponding capacity is seen to be the same as that of a system having multiple receive antennas with a single transmitter antenna whose outputs are combined via maximal ratio combining a suboptimal adaptive transmission technique that transmits only over the antenna having the best channel is also proposed for this special case it is shown that the capacity of such a system under the proposed suboptimal adaptive transmission scheme is the same as the capacity of a system having multiple receiver antennas with a single transmitter antenna combined via selection combining next the capacity of a general system of multiple transmitter and receiver antennas is derived together with an equation that determines the cutoff value for such a system the optimal power allocation scheme for such a multiple antenna system is given by a matrix water filling algorithm in order to eliminate the need for cumbersome numerical techniques in solving the cutoff equation approximate expressions for the cutoff transmission value are also provided it is shown that compared to the case in which there is only receiver csi large capacity gains are available with optimal power and rate adaptation schemes the increased capacity is shown to come at the price of channel outage and bounds are derived for this outage probability

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

exact three dimensional analysis for static torsion of piezoelectric rods based on the theory of elasticity exact analytical and numerical solutions of piezoelectric rods under static torsion are studied in this paper direct solution method is used the main scope is to check the extension of validity of assumptions in previous papers that had been made based on linear distribution of electric potential through the cross section and their influences on deflection and the angle of rotation stress and electric induction functions are employed to obtain the exact solution of the static and electrostatic equilibrium equations under torsional loading it is shown that previous assumptions are valid only in some types of piezoelectric materials while in other types these assumptions lead to considerable deviations from accurate modeling the present analytical solutions are compared with three dimensional finite element analysis results and absolute agreements are found at the end of this article torsional rigidity shape effects on induced piezoelectric deformation and the range of valid region for linear distribution of electric potential assumption have been studied

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

computing the kummer function u a b z for small values of the arguments we describe methods for computing the kummer function u a b z for small values of z with special attention to small values of b for these values of b the connection formula that represents u a b z as a linear combination of two 1f1 functions needs a limiting procedure we use the power series of the 1f1 functions and consider the terms for which this limiting procedure is needed we give recursion relations for higher terms in the expansion and we consider the derivative u a b z as well we also discuss the performance for small z of an asymptotic approximation of the kummer function in terms of modified bessel functions

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

the analytic continuation of the gaussian hypergeometric function 2 f 1 a b c z for arbitrary parameters abstract our objective is to provide a complete table of analytic condinuation formulas for the gaussian hypergeometric function 2f1 a b c z which allow its fast and accurate computation for arbitrary values of z and of the parameters a b c to this end we distinguish 12 basis sets of the two dimensional space of the solutions of the hypergeometric differential equation representing 2f1 in each of them yields 12 analytic continuation formulas each two of them are series in one of the arguments z z z 1 1 z 1 1 z 1 z 1 1 z respectively such that any given argument z with the exception of two single points in the complex plane lies in the convergence domain of at least one of them we present rapidly converging series representations of 2f1 for all possible constellations of parameters for minimizing the effort for the derivation of these series we have extensively used the symmetry group of the hypergeometric equation which is shown to be isomorphic to the cubic group oh

saint venant end effects in piezoceramic materials the stress decay rate and characteristic decay length for anisotropic materials and fiber composites have been intensively investigated however those for piezoelectric materials have not been studied in this paper we examine the saint venant end effects of piezoceramic materials by considering the problem of a semi infinite piezoceramic strip polarized in the thickness direction and with applied voltages on the upper and lower surfaces it is assumed that the gradient of electric potential in the axial direction is much smaller than that in the thickness direction thus the governing equations in terms of the airy stress function and electric potential function can be uncoupled the governing equation in terms of the airy stress function involves only two non dimensional parameters after non dimensionalization finally the stress decay rates and characteristic decay lengths for a variety of piezoceramic materials are computed numerically and their variation with the two non dimensional parameters is presented

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

splitting of a small separatrix loop near the saddle center bifurcation in area preserving maps abstract when the saddle center bifurcation occurs in an analytic family of area preserving maps a parabolic fixed point first appears at the origin and then this point bifurcates creating an elliptic fixed point and a hyperbolic one separatrices of the hyperbolic fixed point form a small loop around the elliptic point in general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter we derive an asymptotic formula which describes the splitting and study the properties of the pre exponential factor

study of the double mathematical pendulum ii investigation of exponentially small homoclinic intersections we consider the double mathematical pendulum in the limit when the ratio of pendulum masses is close to zero and the ratio of pendulum lengths is close to infinity we found that the limit system has a hyperbolic periodic trajectory whose invariant manifolds intersect transversally and the intersections are exponentially small in this case we obtain an asymptotic formula of the homoclinic invariant for the limit system

on the closed form solutions for non extensive value at risk abstract we examine possible closed form solutions for the cumulative distribution function for systems where the probability density function can be adequately described by the generalized non extensive statistics framework application to financial time series as a possible value at risk technique indicates reasonable agreement with the data under consideration including all possible extremes and asymmetries of the returns numerical results to illustrate the efficiency of the method are presented

exponentially small splitting a direct approach abstract in this paper we go beyond what was proposed in theory by melnikov 15 to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time periodic equations not only we derive integral formula for splitting distances of all orders but also we develop an analytic theory to evaluate the acquired multiple integrals we reveal that the dominance of the exponentially small poincare melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order melnikov integrals this symmetry is beheld by many non hamiltonian equations

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

identifying minimal and dominant solutions for kummer recursions we identify minimal and dominant solutions of three term recurrence relations for the confluent hypergeometric functions 1f1 a 1n c 2n z and u a 1n c 2n z where i 0 1 not both equal to 0 the results are obtained by applying perron s theorem together with uniform asymptotic estimates derived by t m dunster for whittaker functions with large parameter values the approximations are valid for complex values of a c and z with ph z