mock modular forms as p adic modular forms in this paper we consider the question of correcting mock modular forms in order to obtain p adic modular forms in certain cases we show that a mock modular form m is a p adic modular form furthermore we prove that otherwise the unique correction of m is intimately related to the shadow of m

a comparison of companion matrix methods to find roots of a trigonometric polynomial abstract a trigonometric polynomial is a truncated fourier series of the form f n t j 0 n a j cos jt j 1 n b j sin jt it has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the fourier coefficients the ccm method however previous work provided no examples so one goal of this new work is to experimentally test the ccm method a second goal is introduce a new alternative the elimination chebyshev algorithm and experimentally compare it with the ccm scheme the elimination chebyshev matrix ecm algorithm yields a companion matrix with real valued elements albeit at the price of usefulness only for real roots the new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables c s where c cos t and s sin t the elimination method next reduces the system to a single univariate polynomial p c we show that this same polynomial is the resultant of the system and is also a generator of the groebner basis with lexicographic ordering for the system both methods give very high numerical accuracy for real valued roots typically at least 11 decimal places in matlab ieee 754 16 digit floating point arithmetic the ccm algorithm is typically one or two decimal places more accurate though these differences disappear if the roots are newton polished by a single newton s iteration the complex valued matrix is accurate for complex valued roots too though accuracy decreases with the magnitude of the imaginary part of the root the cost of both methods scales as o n 3 floating point operations in spite of intimate connections of the elimination chebyshev scheme to two well established technologies for solving systems of equations resultants and groebner bases and the advantages of using only real valued arithmetic to obtain a companion matrix with real valued elements the ecm algorithm is noticeably inferior to the complex valued companion matrix in simplicity ease of programming and accuracy

computing eigenvalues of real symmetric matrices with rational filters in real arithmetic powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals best known are the works of sakurai and coauthors and polizzi and coauthors even if the matrices are real symmetric most such methods rely on complex arithmetic leading to expensive linear systems to solve an appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems for which there is no need to leave the real axis investigation shows that using rational interpolation per se suffers from instability however related techniques involving real rational filters can be very effective this article presents a technique of this kind that is related to previous work published in japanese by murakami

numerical experiments on the accuracy of the chebyshev frobenius companion matrix method for finding the zeros of a truncated series of chebyshev polynomials for a function f x that is smooth on the interval email protected a b but otherwise arbitrary the real valued roots on the interval can always be found by the following two part procedure first expand f x as a chebyshev polynomial series on the interval and truncate for sufficiently large n second find the zeros of the truncated chebyshev series the roots of an arbitrary polynomial of degree n when written in the form of a truncated chebyshev series are the eigenvalues of an nxn matrix whose elements are simple explicit functions of the coefficients of the chebyshev series this matrix is a generalization of the frobenius companion matrix we show by experimenting with random polynomials wilkinson s notoriously ill conditioned polynomial and polynomials with high order roots that the chebyshev companion matrix method is remarkably accurate for finding zeros on the target interval yielding roots close to full machine precision we also show that it is easy and cheap to apply newton s iteration directly to the chebyshev series so as to refine the roots to full machine precision using the companion matrix eigenvalues as the starting point lastly we derive a couple of theorems the first shows that simple roots are stable under small perturbations of magnitude e to a chebyshev coefficient the shift in the root x is bounded by e df dx x o e 2 for sufficiently small e second we show that polynomials with definite parity only even or only odd powers of x can be solved by a companion matrix whose size is one less than the number of nonzero coefficients a vast cost saving

finding the zeros of a univariate equation proxy rootfinders chebyshev interpolation and the companion matrix when a function f x is holomorphic on an interval x in a b its roots on the interval can be computed by the following three step procedure first approximate f x on a b by a poly

balanced3ϕ2summation theorems foru n basic hypergeometric series abstract in this paper we begin the theory and application of the u n 1 generalization of the classical bailey transform and bailey lemma we work in the setting of multiple basic hypergeometric series very well poised on unitary groups u n 1 the u n 1 bailey transform is obtained from a suitably modified u n 1 terminating very well poised 4 3 summation theorem and termwise transformations it is then interpreted as a matrix inversion result for two infinite lower triangular matrices this provides a higher dimensional generalization of andrews matrix inversion formulation of the bailey transform as in the classical case the concept of a u n 1 bailey pair is introduced and then inverted this u n 1 inversion enables us to derive u n 1 terminating balanced 3 2 summations directly from suitable u n 1 terminating very well poised 6 5 summations and vice versa these pairs of u n 1 dual identities extend the classical one variable case of andrews special limiting cases of the u n 1 terminating balanced 3 2 summation yield u n 1 q gauss summations several u n 1 q chu vandermonde summations u n 1 q binomial theorems and a u n 1 cauchy identity many other consequences of the u n 1 6 2 and 3 2 summations and the u n 1 bailey transform are further developed in our subsequent papers these include a derivation of the u n 1 bailey lemma several terminating u n 1 q whipple transformations u n 1 10 9 transformations and nonterminating u n 1 q whipple transformations the classical case of all this work corresponding to a 1 or equivalently u 2 contains a substantial amount of the theory and application of one variable basic hypergeometric series

representations of mock theta functions abstract motivated by the works of liu we provide a unified approach to find appell lerch series and hecke type series representations for mock theta functions we establish a number of parameterized identities with two parameters a and b specializing the choices of a b we not only give various known and new representations for the mock theta functions of orders 2 3 5 6 and 8 but also present many other interesting identities we find that some mock theta functions of different orders are related to each other in the sense that their representations can be deduced from the same a b parameterized identity furthermore we introduce the concept of false appell lerch series we then express the appell lerch series false appell lerch series and hecke type series in this paper using the building blocks m x q z and f a b c x y q introduced by hickerson and mortenson as well as m x q z and f a b c x y q introduced in this paper we also show the equivalences of our new representations for several mock theta functions and the known representations

bailey pairs and indefinite quadratic forms we construct classes of bailey pairs where the exponent of q in n is an indefinite quadratic form as an application we obtain families of q hypergeometric mock theta multisums

truncated hecke rogers type series abstract the recent work of george andrews and mircea merca on the truncated version of euler s pentagonal number theorem has opened up a new study on truncated theta series since then several papers on the topic have followed the main purpose of this paper is to generalize the study to hecke rogers type double series which are associated with some interesting partition functions our proofs heavily rely on a formula from the work of zhi guo liu on the q partial differential equations and q series

a pseudospectral method for two point boundary value problems pseudospectral collocation is employed for the numerical solution of nonlinear two point boundary value problems with separated end conditions second order finite difference schemes are used as preconditioners for the spectral calculation and a solution of the discretized equations is obtained using versions of the defect correction principle the method and a variant based on an adaptive grid technique are tested on a variety of sample problems and are shown to provide high accuracy with low storage requirements

computing the zeros of a fourier series or a chebyshev series or general orthogonal polynomial series with parity symmetries in recent years good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated chebyshev polynomial or legendre polynomial series in each case the roots can be found from the eigenvalues of a generalized frobenius companion matrix whose elements are trivial functions of the fourier coefficients or chebyshev coefficients however the qr method for computing the companion matrix eigenvalues has a cost that grows proportionally to n 3 where n is the polynomial degree by exploiting the special structure of the companion matrices the cost can be reduced to o n 2 but only for large n here we show that if the polynomial has definite parity such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of chebyshev polynomials of odd degree one can exploit these symmetries to halve the size of the problem this reduces costs in the companion matrix method by a factor ranging between four and eight for trigonometric polynomials we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure we further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty four in the companion matrix algorithm special functions such as spherical harmonics mathieu functions prolate spheroidal wavefunctions and hough functions all represented by truncated fourier series with double parity are a rich source of applications

computing zeros on a real interval through chebyshev expansion and polynomial rootfinding robust polynomial rootfinders can be exploited to compute the roots on a real interval of a nonpolynomial function f x by the following i expand f as a chebyshev polynomial series ii convert to a polynomial in ordinary series of powers form and iii apply the polynomial rootfinder complex valued roots and real roots outside the target interval are discarded the expansion is most efficiently done by adaptive chebyshev interpolation with n equal to a power of two where n is the degree of the truncated chebyshev series all previous evaluations of f can then be reused when n is increased adaption stops when n is sufficiently large so that further increases produce no significant change in the interpolant we describe two conversion strategies the convert to powers method uses multiplication by mildly ill conditioned matrices to create a polynomial of degree n the degree doubling strategy defines a polynomial of larger degree 2n but is very well conditioned the convert to powers method although faster restricts n to moderate values this can always be accomplished by subdividing the target interval both these strategies allow simultaneous approximation of many roots on an interval whether simple or multiple for nonpolynomial f x

from sheaves on p 2 to a generalization of the rademacher expansion moduli spaces of stable coherent sheaves on a surface are of much interest for both mathe matics and physics yoshioka computed generating functions of poincare polynomials of such moduli spaces if the surface is p 2 and the rank of the sheaves is 2 motivated by physical arguments this pa per investigates the modular properties of these generating functions it is shown that these functions can be written in terms of the lerch sum and theta function based on this we prove a conjecture by vafa and witten which expresses the generating functions of euler numbers as a mixed mock mod ular form moreover we derive an exact formula of rademacher type for the fourier coefficients of this function this formula requires a generalization of the classical circle method this is the first example of an exact formula for the fourier coefficients of mixed mock modular forms which is of independent mathematical interest 1 introduction and statement of results in the past interactions between physics and mathematics have led to many interesting results motivated by strong weak coupling duality or s duality in physics this article considers various gen erating functions which appear in the study of moduli spaces of stable coherent sheaves on the projective plane p 2 we express the generating functions of poincare polynomials of moduli spaces of rank 2 sheaves in terms of the lerch sum and theta function which we will recall later using these expressions we prove a conjec ture by vafa and witten 45 for the generating functions of euler numbers these functions appear to be related to ramanujan s mock theta functions and therefore transform almost as weakly holomorphic modular forms i e meromorphic mod ular forms whose poles if there are any may only lie in cusps our second main result is an exact formula for the fourier coefficients of these generating functions that formally resembles the rademacher expansion for the coefficients of weakly holomorphic modular forms moduli spaces of coherent sheaves on a complex surface s receive much attention see for example 28 for an extensive work on such moduli spaces more specifically one is interested in the moduli space m r c1 c 2 of semi stable sheaves of rank r with first chern class c1 and second chern class c2 w e will consider topological invariants of m in particular the poincare polynomial p m s 2d imc m i 0 bi m s i and the euler number m p m 1 w here

ramanujan s lost notebook vi the mock theta conjectures 1 introduction in this paper we shall consider only ramanujan s two families of fifth order mock theta functions these functions were briefly described in ramanujan s last letter to g h hardy 11 pp 354 3551 and g n watson subsequently 12 131 proved all the assertions about these functions contained in the letter subsequently the identities watson proved were greatly generalized in 2 also the fifth order mock theta functions were shown to have double series expansions involving indefinite quadratic forms 7 however there remains a profound mystery about these functions namely no one including ramanujan has ever proved that these functions are indeed mock theta functions and not just some clever combination of theta functions this problem was described in detail by watson 12 p 2741 and redescribed in 7 pp 113 l 141 in s p 97 eqs 3 5 and 3 6 a formula from the lost notebook was pointed out that would if valid lead to the establishment of at least some of the fifth order mock theta functions as truly mock theta functions in the sense of watson 12 p 78 footnote there are indeed ten such identities live for each of the two families we have two objects in this paper first we wish to show that the identities in each family are equivalent i e if one is true all five are true and if one is false all five are false to introduce our second objective we recall some basic notions from partition theory 4 p 142 8 p 841 the rank of a partition is the largest

tenth order mock theta functions in ramanujan s lost notebook ramanujan s lost notebook contains many results on mock theta functions in particular the lost notebook contains eight identities for tenth order mock theta functions previously the author proved the first six of ramanujan s tenth order mock theta function identities it is the purpose of this paper to prove the seventh and eighth identities of ramanujan s tenth order mock theta function identities which are expressed by mock theta functions and a definite integral l j mordell s transformation formula for the definite integral plays a key role in the proofs of these identities also the properties of modular forms are used for the proofs of theta function identities

complex coordinate methods for hydrodynamic instabilities and sturm liouville eigenproblems with an interior singularity makes a wide detour around the singularity specific guidelines for choosing f x for different problems are given in the text results are impressive for an eigenvalue problem with a pole in the middle of the original real interval a sturm liouville problem of the fourth kind just six basis functions suffice to calculate the real and imaginary parts of the lowest eigenvalue to within 1 4 for strong instability i e modes whose phase speeds have large imaginary parts the complex mapping is unnecessary because the critical latitudes are complex and distant from the real axis even so the mapping is useful for instability problems because it can be used to make calculations for very slowly growing modes to follow the changes in c right up to the neutral curve where the imaginary part of c 0 although especially valuable for spectral algorithms the same trick can be applied with finite difference methods also the main disadvantage of the algorithm is that the eigenfunction must be calculated in a second separate step but this is usually a minor flaw in comparison to the complex mapping s virtues for coping with singular eigenvalue problems

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

numerical perturbative and chebyshev inversion of the incomplete elliptic integral of the second kind abstract the incomplete elliptic integral of the second kind e sin t m 0 t 1 m sin 2 t dt where m 0 1 is the elliptic modulus can be inverted with respect to angle t by solving the transcendental equation e sin t m z 0 we show that newton s iteration t n 1 t n e sin t m z 1 m sin 2 t always converges to t z m e 1 z m within a relative error of less than 10 10 in three iterations or less from the first guess t 0 z m 2 r 2 where defining 1 z e 1 m r 1 m 2 2 and atan 1 m we briefly discuss three alternative initialization strategies homotopy initialization t 0 z m 1 m z 0 mt z m m 1 perturbation series in powers of m and inversion of the chebyshev interpolant of the incomplete elliptic integral although all work well and are general strategies applicable to a very wide range of problems none of these three alternatives is as efficient as the empirical initialization which is completely problem specific this illustrates tai tsun wu s maxim usefulness is often inversely proportional to generality

a regular characterization of graph languages definable monadic second order logic abstract the class of monadic second order definable graph languages is the smallest class containing certain elementary graph languages and closed under the boolean set operations and under projections this holds for many restricted types of graphs

on the seventh order mock theta functions in a recent paper h we proved the mock theta conjectures these are identities stated by ramanujan in his lost notebook jr2 pp 19 20 involving two of the 5th order mock 0 functions in fact ramanujan gave one such identity for each of the ten 5th order functions as shown in a g1 these ten identities are all equivalent to the two proved in i hi in this paper we prove the corresponding identities for ramanujan s 7th order mock 0 functions and show how they are related to rank differences of partitions to state our results we first introduce some standard notation if q and x are complex numbers with ql 1 and n is a nonnegative integer then

on ramanujan s definition of mock theta function in his famous deathbed letter ramanujan defined the notion of a mock theta function and offered some examples of functions he believed satisfied his definition very recently griffin et al established for the first time that ramanujan s mock theta functions actually satisfy his own definition on the other hand zwegers 2002 doctoral thesis zwegers s 2002 mock theta functions phd thesis univ utrecht utrecht the netherlands showed that all of ramanujan s examples are holomorphic parts of harmonic maass forms this has led to an alternate definition of a mock theta function this paper shows that ramanujan s definition of mock theta function is not equivalent to the modern definition

automatic spectral collocation for integral integro differential and integrally reformulated differential equations automatic chebyshev spectral collocation methods for fredholm and volterra integral and integro differential equations have been implemented as part of the chebfun software system this system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization the same objects can be used in matrix free iterative methods in linear algebra in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients as a further application of the ability to implement operator equations a method of greengard 1 for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems in the integral form large condition numbers associated with differentiation matrices in high order problems are avoided the ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun the integral method also can be extended to first order systems although chebfun syntax does not currently allow easy implementation in this case

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

an algorithm for the convolution of legendre series an mathcal o n 2 algorithm for the convolution of compactly supported legendre series is described the algorithm is derived from the convolution theorem for legendre polynomials and the recurrence relation satisfied by spherical bessel functions combining with previous work yields an mathcal o n 2 algorithm for the convolution of chebyshev series numerical results are presented to demonstrate the improved efficiency over the existing algorithm

an extension of matlab to continuous functions and operators an object oriented matlab system is described for performing numerical linear algebra on continuous functions and operators rather than the usual discrete vectors and matrices about eighty matlab functions from plot and sum to svd and cond have been overloaded so that one can work with our chebfun objects using almost exactly the usual matlab syntax all functions live on 1 1 and are represented by values at sufficiently many chebyshev points for the polynomial interpolant to be accurate to close to machine precision each of our overloaded operations raises questions about the proper generalization of familiar notions to the continuous context and about appropriate methods of interpolation differentiation integration zerofinding or transforms applications in approximation theory and numerical analysis are explored and possible extensions for more substantial problems of scientific computing are mentioned

a chebyshev polynomial interval searching method lanczos economization for solving a nonlinear equation with application to the nonlinear eigenvalue problem to search a given real interval for roots our algorithm is to replace f by fn its n term chebyshev expansion on the search interval min max and compute the roots of this proxy this strategy is efficient if and only if f itself is expensive to evaluation such as when f is the determinant of a large matrix whose elements depend nonlinearly on for such expensive functions it is much cheaper to search for zeros of fn which can be evaluated in o n operations than to iterate or look for sign changes on a fine grid with f itself it is possible to systematically increase n until the chebyshev series converges acceptably fast without wasting previously computed values of f by imitating the clenshaw curtis quadrature our strategy of replacing f by fn is similar to lanczos economization of power series which also replaces an expensive function by a chebyshev approximation that is more rapidly evaluated the errors induced by the chebyshev approximation can be eliminated by a final step of one or two iterations with f itself using the zeros of fn as initial guesses we show through numerical examples that the algorithm works well the only sour note is that it is sometimes necessary to split the search interval into subintervals each with a separate chebyshev expansion when f varies by many orders of magnitude on the search interval

computing the real roots of a fourier series plus linear polynomial a chebyshev companion matrix approach abstract fourier series often need to be generalized by appending a linear polynomial to the usual series of sines and cosines the integral of a trigonometric polynomial is one example another is a time series of climate data where the periodic oscillations of the diurnal and annual cycles are accompanied by a non periodic trend global warming stock market averages fluctuate about a generally upward trend such non periodic variations with time are commonly called secular trends we borrow secular to label a truncated fourier series plus a linear trend as a linear secular trigonometric polynomial standard fourier rootfinding methods are wrecked by the extra nonperiodic term therefore we introduce a new algorithm for computing the zeros of a fourier polynomial with secular trend first we expand the linear secular trigonometric polynomial f n t as a truncated chebyshev series because of the special structure it is easy to calculate a problem dependent truncation m such that the error of the truncated chebyshev series is guaranteed to be less than a user specified tolerance we then find the roots of the truncated chebyshev series as the eigenvalues of the chebyshev companion matrix this computes all roots but we explain why the method is not reliable for complex valued roots unless these are close to the real axis no a priori information is required of the user except the coefficients of the linear secular trigonometric polynomial numerical examples show that 13 decimal place accuracy for real roots is typical

roots of polynomials expressed in terms of orthogonal polynomials a technique is presented for determining the roots of a polynomial p x that is expressed in terms of an expansion in orthogonal polynomials the roots are expressed as the eigenvalues of a nonstandard companion matrix b n whose coefficients depend on the recurrence formula for the orthogonal polynomials and on the coefficients of the orthogonal expansion some questions on the numerical stability of the eigenvalue problem to which they give rise are discussed the problem of finding the roots of a transcendental function f x can be reduced to the problem considered by approximating f x by a chebyshev polynomial we illustrate the effectiveness of this convert to chebyshev strategy by solving several transcendental equations using this plus our new algorithm we analyze the numerical stability through both linear algebra theory and numerical experiments and find that this method is very well conditioned

computing real roots of a polynomial in chebyshev series form through subdivision with linear testing and cubic solves abstract an arbitrary polynomial of degree n f n x can always be represented as a truncated chebyshev polynomial series chebyshev form this representation is much better conditioned than the usual power form of a polynomial we describe a new method for finding the real roots of f n x in chebyshev form the canonical interval x 1 1 is subdivided into n s subintervals each is tested for zeros using the error bound for linear interpolation on zero possible intervals f n is approximated by a cubic polynomial whose roots are then found by the usual sixteenth century formulas