polynomial maps with strongly nilpotent jacobian matrix and the jacobian conjecture abstract let h k n k n be a polynomial map it is shown that the jacobian matrix jh is strongly nilpotent if and only if jh is linearly triangularizable if and only if the polynomial map f x h is linearly triangularizable furthermore it is shown that for such maps f sf is linearizable for almost all s k except a finite number of roots of unity

the equivalence problem of multitape finite automata abstract using a result of b h neumann we extend eilenberg s equality theorem to a general result which implies that the multiplicity equivalence problem of two nondeterministic multitape finite automata is decidable as a corollary we solve a long standing open problem in automata theory namely the equivalence problem for multitape deterministic finite automata the main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group

on dol power series abstract we study d0l power series we show how elementary morphisms introduced by ehrenfeucht and rozenberg can be used in connection with power series characterize the sequences of rational numbers and integers which can appear as coefficients in d0l power series and establish various decidability results

on the faber transform and efficient numerical rational approximation algorithms are presented for numerical type m n rational approximation m geqq n 1 which can be effectively applied either in general simply connected regions of the complex plane or as a special case on intervals of the real line in the former case particularly the methods are much more efficient than those previously proposed but they are also competitive even for intervals the approximations are based on the observation that the faber transform of a rational function is itself rational and are of two types a class of pade like approximants and approximations of rational caratheodory fejer type based on the singular value decomposition of a hankel matrix of faber coefficients which generalise those recently introduced for the unit disc by trefethen numer math 37 1981 pp 297 320 these latter particularly give rise to approximations which are sufficiently near to best for practical purposes

quadratic linear keller maps abstract it is proved that a quadratic linear keller map c n c n is linearly triangularizable if its rank is at most 2 or corank at most 1

complex chebyshev polynomials on circular sectors with degree six or less let tna denote the nth chebyshev polynomial on the circular sector sa z i z i this paper contains numerical values of ii tnaii o and the corresponding coefficients of t7 for n 1 1 6 and a 0 5 180 also all critical angles for t7 n 1 1 6 are listed where an angle is called critical when the number of absolute maxima of i t a changes at that angle all figures are given to six places the positions and hence the number of extremal points of t7a n 1 1 6 are presented graphically the method consists of a combination of semi infinite linear programming finite linear programming and newton s method

chebyshev series approximations for the bessel function y n z of complex argument we employ the truncated chebyshev series to approximate the bessel function of the second kind y n z for z 8 detailed manipulations and discussions for y 0 z and y 1 z are given results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the chebyshev series approximations advantages and disadvantages of the chebyshev series approximation compared with other polynomial approximation methods e g the tau method approximations are discussed

a note on the tau method approximations for the bessel functions y0 z and y1 z abstract this paper is to complete and improve the work reported in 1 2 using the lanczos method in coleman s version to approximate the bessel functions y 0 z and y 1 z we introduce symbolic representations of the scaled faber polynomials on any fan shaped section of the complex plane these faber polynomials are used as the perturbation terms in the method numerical comparison among the power series the chebyshev series and the method are conducted to show the accuracy improvement achieved by this new version of the method some concluding remarks and suggestions on future research are given

quadratic linear keller maps of nilpotency index three let f x h cn cn be a homogeneous polynomial map of degree d 2 and fa y ay d cn cn a power linear map such that f and fa are a generalized gorni zampieri pair we discuss the relation between the nilpotency indices of jh and j ay d and we show that f is linearly triangularizable if and only if fa is linearly triangularizable as a consequence we show that a quadratic linear keller map fa y ay 2 with nilpotency index three i e j ay 2 3 0 is linearly triangularizable

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

polynomial approximations in the complex plane abstract good polynomial approximations for analytic functions are potentially useful but are in short supply a new approach introduced here involves the lanczos method with perturbations proportional to faber or chebyshev polynomials for specific regions of the complex plane the results show that suitable forms of the method which are easy to use can produce near minimax polynomial approximations for functions which satisfy linear differential equations with polynomial coefficients in particular some accurate approximations of low degree for bessel functions are presented an appendix describes a simple algorithm which generates polynomial approximations for the bessel function j z of any given order

the faber polynomials for m fold symmetric domains abstract the faber polynomials for a region of the complex plane are of interest as a basis for polynomial approximations to analytic functions in this paper we study the faber polynomials associated with m fold symmetric domains explicit formulae of faber polynomials both for symmetric and nonsymmetric lemniscates are derived in addition we use a new determinant representation which relates the zeros of faber polynomials to the eigenvalues of a certain matrix to compute the zeros of faber polynomials for lemniscates

a hybrid arnoldi faber iterative method for nonsymmetric systems of linear equations we present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations say of the formax b wherea n n witha nonsingular andb n are given this hybrid method begins with a limited number of steps of the arnoldi method to obtain some information on the location of the spectrum ofa and then switches to a richardson iterative method based on faber polynomials for a polygonal domain the faber polynomials can be constructed recursively from the parameters in the schwarz christoffel mapping function in four specific numerical examples of non normal matrices we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid gmres or restarted versions of the gmres algorithm it is however sensitive as other hybrid methods also are to the amount of information on the spectrum ofa acquired during the first arnoldi phase of this procedure

the conformal bratwurst mapsand associated faber polynomials summary conformal maps from the exterior of the closed unit disk onto the exterior of bratwurst shape sets in the complex plane are constructed using these maps coefficients for the computation of the corresponding faber polynomials are derived a bratwurst shape set is the result of deforming an ellipse with foci on the real axis by conformally mapping the real axis onto the unit circle such sets are well suited to serve as inclusion sets for sets associated with a matrix for example the spectrum field of values or a pseudospectrum hence the sets can be applied in the construction and analysis of a broad range of iterative methods for the solution of linear systems the main advantage of the approach is that the conformal maps are derived from elementary transformations allowing an easy computation of the associated transfinite diameter asymptotic convergence factor and faber polynomials numerical examples are given

supplement to the faber polynomials for circular sectors degree of faber polynomial 6 p 0 6 u p l 3 sa2 2 s u 7 ua2 4 2 p 2 4 sa3 11 sa2 u 14 s ua2 16 s 11 ua 3 32 u 2 p 3 39 sa4 76 s 3 u 74 s 2 ua2 192 sa2 44 s ua3 256 s u 7 ua4 128 ua2 144 16 p 4 46 sa5 31 s 4 u 34 sa3 ua2 248 s 3 22 sa2 ua3 120 sa2u 20 s u 4 136 s ua2 256 s 7 ua5 24 ua3 104 u 20 p 5 21 s 6 34 sa5 u 65 s 4 u 2 1o sa4 20 sa3 ua3 112 s 3 u 65 s 2 u 4 312 s 2 ua 2 72 sa2 70 s ta5 304 s u 3 144 s u 21 u 6 100 u 4 72 u 2 32 16

the linear triangularizability of some keller maps abstract in this paper we prove that keller maps are linearly triangularizable under some conditions as a consequence we show that non homogeneous power linear keller maps with jh symmetric are linearly triangularizable

symbolic and numerical computation on bessel functions of complex argument and large magnitude the lanczos t method with perturbations proportional to faber polynomials is employed to approximate the bessel functions of the first kind j v z and the second kind y v z the hankel functions of the first kind h v 1 z and the second kind h v 2 z of integer order v for specific outer regions of the complex plane i e z r for some r the scaled symbolic representation of the faber polynomials and the appropriate automated t method approximation are introduced both symbolic and numerical computation are discussed in addition numerical experiments are employed to test the proposed t method computed accuracy for j 0 z and y 0 z for z 8 are presented the results are compared with those obtained from the truncated chebyshev series approximations and with those of the t method approximations on the inner disk z 8 some concluding remarks and suggestions on future research are given

the nagata automorphism is shifted linearizable abstract a polynomial automorphism f is called shifted linearizable if there exists a linear map l such that lf is linearizable we prove that the nagata automorphism n x 2 y z 2 y z z where x z y 2 is shifted linearizable more precisely defining l a b c as the diagonal linear map having a b c on its diagonal we prove that if a c b 2 then l a b c n is linearizable if and only if b c 1 we do this as part of a significantly larger theory for example any exponent of a homogeneous locally finite derivation is shifted linearizable we pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms and explain why this is a natural question

chebyshev expansions for the bessel function _ in the complex plane polynomial based approximations for jo z and j1 z are presented the first quadrant of the complex plane is divided into six sectors and separate approximations are given for i z i 8 can also be used to evaluate the bessel functions yo z and y z and the hankel functions of the first and second kinds

on lindenmayerian algebraic sequences abstract we define and study lindenmayerian algebraic sequences these sequences are a generalization of algebraic sequences k regular sequences and automatic sequences

quadratic homogeneous keller maps of rank two abstract let h be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its jacobian jh is nilpotent meisters and olech have shown that jh is strongly nilpotent if n 4 they also proved that it is not true when n 5 we show that if rank j h 2 and n arbitrary then jh is strongly nilpotent we also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2

the faber polynomials for annular sectors a conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the faber polynomials for that region these polynomials are of interest in providing near optimal polynomial approximations in a variety of contexts including the construction of semi iterative methods for linear equations the relevant conformai map for an annular sector z r z 1 0 argz it with 0 9 it is derived here and a recurrence relation is established for the coefficients of its laurent expansion about the point at infinity the recursive evaluation of scaled faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii properties of the coefficients of the scaled faber polynomials are established and those for polynomials of degree 15 are tabulated in a supplement at the end of this issue a simple closed form is obtained for the coefficients of the faber series for 1 z known results for an interval a circular arc and a circular sector are reproduced as special cases

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

computation of faber series with application to numerical polynomial approximation in the complex plane kovari and pommerenke 19 and elliott 8 have shown that the truncated faber series gives a polynomial approximation which for practical values of the degree of the polynomial is very close to the best approximation in this paper we discuss efficient fast fourier transform fft and recursive methods for the computation of faber polynomials and point out that the fft method described by geddes 13 for computing chebyshev coefficients can be generalized to compute faber coefficients we also give a corrected bound for the norm of the faber projection that given in elliott 8 being unfortunately slightly in error and very briefly discuss a possible extension of the method to the case when the mapping function which is required to compute the faber series is not known explicitly

on clenshaws s method and a generalisation to faber series the method of expanding the solution of linear ordinary differential equations in power or laurent series is classical and is usually associated with the name of frobenius in the early days of electronic computation it was appreciated that expansions in chebyshev series are often of more practical use and the necessary techniques developed by clenshaw this is usually carried out in order to approximate special functions defined by ordinary differential equations rather than as a technique for actually solving such equations in general for which finite difference methods are to be preferred in this paper we show that by means of a conformal map the problem of expansion in chebyshev series can in fact be reduced to that of expansion in a laurent series yielding a method which is usually computationally simpler moreoveer the method can be generalised to the case of faber expansions on regions of the complex plane a non trivial example is explored in order to illustrate the method and we also use the technique to generalise an identity relating chebyshev polynomials to the faber case

tau method approximations for the bessel function y1 z abstract we present method approximations for the bessel function of the second kind of order one y 1 z with some numerical results

tau method approximations for the bessel function y0 z we exploit the lanczos method with perturbations proportional to faber polynomials to approximate the bessel function of the second kind y0 z for specific regions of the complex plane accuracy comparisons are made among the power series direct and integrated forms of the method and the chebyshev series

accelerated polynomial approximation of finite order entire functions by growth reduction let f be an entire function of positive order and finite type the subject of this note is the convergence acceleration of polynomial approximants of f by incorporating information about the growth of f z for z we consider near polynomial approximation on a compact plane set k which should be thought of as a circle or a real interval our aim is to find sequences f n n of functions which are the product of a polynomial of degree n and an easy computable second factor and such that f n n converges essentially faster to f on k than the sequence p n n of best approximating polynomials of degree n the resulting method which we call reduced growth method rg method is introduced in section 2 in section 5 numerical examples of the rg method applied to the complex error function and to bessel functions are given

recognizable formal power series on trees trees are a very basic object in computer science they intervene in nearly a ly domain and they are studied for their own or used to represent conveniently a given situation there are at least three directions where investigations on trees themselves are motivated and this for different reasons first the notion of tree is the basis of algebraic semantics nivat 19 rosen 2 2 etc in this context the study of special languages of trees i e forests their classification and their behaviour under various types of transformations are of great importance arnold fl dauchet 8 j l ilin 1161 mongy 18 by essence work in this area is an extension of the algebraic theory of languages trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar thatcher 25 a second topic heavily related to trees concerns dzra structures trees mainly binary trees and its variants constitute one of the most widely known data structures see e g hlnuth 15 the analysis of the worst cale expected or average running time t ehzvgrr nfi certain algorithms requires sometimes long and delicate computations flajolet lo kemp 14 flajolet and steyaert 12 finally trees occupy a distinguished place in the enumeration of graphs and maps both because of the simplicity of their structure and for the relationship between their el lcodings and aigebraic languages the nature of the enumerating series and especially the question whether they are algebraic or not is one of the central problems in this domain cori 7 chottin 4 we propose here a theory of formal power series on trees and present some of their basic properties together with various examples of applications which as we hope will show the interest of its development within the framework we just sketched a formal power series on trees is a function which gssociates a number to esch tree thus we could also have called them tree functions ii in analogy with the term word function used by several authors paz and salomaa zo cobham 5 as an equivalent denomination for formal power series on words the main goal of a formal power series is to count or to represent the result of some computation on

the faber polynomials for circular lunes abstract we study the faber polynomials f n z generated by a circular lune symmetric about both axes with vertices at the points z 0 2 and exterior angle an explicit expression of f n z was obtained by computing the coefficients via a cauchy integral formula we also illustrate the location of the zeros of faber polynomial and of its derivative our results include a circle and a segment as special cases when 1 2 respectively

on semiiterative methods generated by faber polynomials consider a nonsingular system of linear algebraic equationsa x b or in fixed point formx tx c where the eigenvalues oft are contained in some compact subset of the complex plane using faber polynomials iteration methods of the form y m mu m 0 left c ty m 1 right mu m 1 y m 1 ldots mu m k y m k m 0 m j 1 for allm 1 are constructed which include the chebyshev iterative method as well as the stationaryk step methods when is a rectangle we also develop four step methods which yield fast convergence in cases when the speed of convergence of the optimized chebyshev method is poor

a faber series approach to cardinal interpolation for a compactly supported function 9 in rd we study quasiinterpolants based on point evaluations at the integer lattice we restrict ourselves to the case where the coefficient sequence af for given data f is computed by applying a univariate polynomial q to the sequence i zd and then convolving with the data flzd such operators appear in the well known neumann series formulation of quasi interpolation a criterion for the polynomial q is given such that the corresponding operator defines a quasi interpolant since our main application is cardinal interpolation which is well defined if the symbol of p does not vanish we choose q as the partial sum of a certain faber series this series can be computed recursively by this approach we avoid the restriction that the range of the symbol of p must be contained in a disk of the complex plane excluding the origin which is necessary for convergence of the neumann series furthermore for symmetric a we prove that the rate of convergence to the cardinal interpolant is superior to the one obtainable from the neumann series

on lindenmayerian algebraic power series abstract we define lindenmayerian algebraic power series and study their properties these series generalize both algebraic series and etol languages

complex polynomial approximation by the lanczos t method dawson s integral abstract the lanczos method with perturbations proportional to faber polynomials is used to obtain polynomial approximations for dawson s integral on circular sectors an upper bound on the truncation error is established for one form of the method which gives near minimax polynomial approximations and it is found that this bound provides a useful estimate of the truncation error the faber series for dawson s integral on a circular sector is also investigated numerical results show that the method can produce polynomial approximations as accurate as the truncated faber series with much less effort than is involved in computing the faber coefficients

power linear keller maps with ditto triangularizations abstract we show that power linear keller maps f x 1 a 1 x d x 2 a 2 x d x n a n x d are linearly triangularizable if 1 rk a 2 or 2 cork a 2 and d 3 or 3 cork a 3 d 5 and the diagonal of a is nonzero furthermore we show that the triangularizations can be chosen power linear as well

automata and languages generalized to ω continuous semirings abstract we generalize the following two language and automata theoretic results to continuous semirings i the family of languages accepted by finite automata is the smallest class containing all finite languages and closed under union product and star kleene s theorem ii the family of languages accepted by pushdown automata is the family ofcontext free languages

on additive nilpotency of jacobian matrices of polynomial maps abstract let h c n c n be a polynomial map jh the jacobian matrix of h and v h the linear subspace of m n c spanned by j h c n the set of evaluated jacobian matrices of h we describe the dimension of v h through a power linear gorni zampieri mate of h and give a numerical algorithm for computing this dimension and determining whether jh is additive nilpotent i e whether v h is a nilpotent subspace