computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

generalized confluent cauchy vandermonde matrices displacement structures inversion formulas and tangential interpolations generalized confluent cauchy and cauchy vandermonde matrices are introduced these two kinds of matrices generalize the ordinary cauchy and cauchy vandermonde matrices with multiple nodes studied earlier by various authors by using displacement structure theory fast inversion formulas for these matrices are derived the tangential interpolation interpretations for associated linear systems with such matrices are given the fast algorithm for solving this kind of linear systems are also considered

displacement structure approach to cauchy and cauchy vandermonde matrices inversion formulas and fast algorithms the confluent cauchy and cauchy vandermonde matrices are considered which were studied earlier by various authors in different ways in this paper we use another way called displacement structure approach to deal with matrices of this kind we show that the cauchy and cauchy vandermonde matrices satisfy some special type of matrix equations this leads quite naturally to the inversion formulas and fast algorithms for matrices of this kind

the autocorrelation function of a pseudointegrable system abstract the autocorrelation function of a pseudointegrable system is considered the system consists of billiards on plane surface formed out of three squares arranged in an l shape this system has the important property of being constructed from copies of an integrable subsystem the single square the motion can be decomposed into a continuous and a discrete part the unpredictability in the system being associated with the latter a discrete autocorrelation function is calculated and its decay properties investigated structure found in this autocorrelation function is associated with the continued fraction expansion of the ratio of velocity components for repeating continued fractions such as the golden mean the autocorrelation function exhibits a selfsimilar structure for the general case of a randomly chosen velocity ratio we derive the time dependence of the number of occurences of large autocorrelation values which differs from the behavior in integrable and chaotic systems

qualitative properties of systems of two complex homogeneous ode s a connection to polygonal billiards a correspondence between the orbits of a system of two complex homogeneous polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed this correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind this allows us to transfer results well known from the theory of polygonal billiards such as ergodicity the existence of periodic orbits the absence of exponential divergence the existence of additional conservation laws and the presence of discontinuities in the dynamics to the corresponding systems of ordinary differential equations it also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations

interpolation by cauchy vandermonde systems and applications abstract cauchy vandermonde systems consist of rational functions with prescribed poles they are complex ect systems allowing hermite interpolation for any dimension of the basic space a survey of interpolation procedures using cv systems is given some equipped with short new proofs which generalize the well known formulas of lagrange neville aitken and newton for interpolation by algebraic polynomials the arithmetical complexitiy is o n 2 for n hermite data also inversion formulas for the cauchy vandermonde matrix are surveyed moreover a new algorithm solving the system of n linear cauchy vandermonde equations for multiple nodes and multiple poles recursively is given which does not require additional partial fraction decompositions as an application construction of rational b splines with prescribed poles is discussed

level statistics and eigenfunctions of square torus billiards manifesting the transition from regular to chaotic behaviors we thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards we confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards furthermore we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards but also develop an analytical distribution to excellently fit the numerical level statistics finally we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories as the effective wavelengths are considerably shorter than the size of internal hole

schrodinger equation for convex plane polygons a tiling method for the derivation of eigenvalues and eigenfunctions motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem motivated by a recently advanced conjecture on the ergodic properties of quantum systems the problem of solving the schrodinger equation for a free particle in a plane polygonal enclosure is revisited it will be shown that two elementary lemmas suffice to give a complete characterization of the polygons for which a solution can be found in terms of a finite superposition of plane waves without making use of advanced group theoretical techniques it turns out inter alia that these polygons considered as classical billiards are all and only those which are completely integrable in the sense of arnold s theorem

local instability of orbits in polygonal and polyhedral billiards we classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra we sharpen a theorem of boldrighini keane and marchetti we show that polygonal and polyhedral billiards have zero topological entropy we also prove that billiards in polygons are positive expansive when restricted to the set of non periodic points the methods used are elementary geometry and symbolic dynamics

nodal domains of the equilateral triangle billiard we characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains the number of nodal domains has a quadratic form in terms of the quantum numbers with a non trivial number theoretic factor the patterns of the eigenfunctions follow a group theoretic connection in a way that makes them predictable as one goes from one state to another extensive numerical investigations bring out the distribution functions of the mode number and signed areas the statistics of the boundary intersections is also treated analytically finally the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula we believe that the results belong generically to non separable systems thus extending the previous works which are concentrated on separable and chaotic systems

on hermite interpolation by cauchy vandermonde systems the lagrange formula the adjoint and the inverse of a cauchy vandermonde matrix for a given cauchy vandermonde system and for given multiple nodes a lagrange type formula for the interpolant is derived interpolating a given function in the sense of hermite we give explicit analytic representations of the basic functions in terms of the nodes and prescribed poles they are used to derive formulas for the entries of the adjoint of the confluent cauchy vandermonde matrix corresponding to the interpolation problem thus providing an explicit representation of its inverse

arithmetical degeneracies in simple quantum systems the authors examine the accidental level degeneracies occurring in the quantum mechanical problem of a free particle moving in a polyhedral box when the problem is integrable some remarkable properties of the distribution of degeneracies are studied in several two three and four dimensional examples and are related to well known problems of number theory the numerical results of exact enumerations are compared with analytical predictions or with conjectured expressions in some cases they consider in particular the asymptotic scaling form of the degeneracy distribution up to some maximal energy e and the maximal degeneracy occurring for energies less than some given e

non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons this paper proves the existence of non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons for anyn 3 there exists a corresponding convexn agon forn 3 this will be a right triangle with a small acute angle while in three dimensional space it will be a prism then agon serving as the base

dual polygonal billiards and necklace dynamics we study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics our main result is that for a certain class of tables called the quasi rational polygons the dual billiard orbits are bounded this implies that for the subset of rational tables i e polygons with rational vertices the dual billiard orbits are periodic

correlations and spectra of triangular billiards abstract we consider the motion of a point particle in right triangular billiards for which acute angles are not rationally connected to we consider in particular the behaviour of correlations for a particular phase space variable the behaviour is compatible with the mixing character of the dynamics even though large fluctuations are exhibited by fourier transforming correlation functions we may as well analyse scaling properties of the spectral measure both dynamical and spectral features are quite similar to what observed in a class of quantum kicked systems

zeta function of the bessel operator on the negative real axis we investigate the pole structure of the zeta function zeta anu s sigma k lambda k s built out of the eigenvalues lambda k of a bessel operator anu subject to dirichlet boundary conditions at one end of the domain this leads us to the study of zeta anu on the negative real axis where most of the singularities occur

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

periodic orbit theory and spectral rigidity in pseudointegrable systems we calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers g that arise from rectangular systems with one or two salient corners from the periodic orbits we calculate the spectral rigidity delta3 l using semiclassical quantum mechanics with l reaching up to quite large values we find that the diagonal approximation is applicable when averaging over a suitable energy interval comparing systems of various shapes we find that our results agree well with delta3 calculated directly from the eigenvalues by spectral statistics therefore additional terms such as e g diffraction terms seem to be small in the case of the systems investigated in this work by reducing the size of the corners the spectral statistics of our pseudointegrable systems approaches that of an integrable system whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large both types of behavior can be well understood by the properties of the periodic orbits in the system

on the quantum zeta function it is remarkable that the quantum zeta function defined as a sum over energy eigenvalues e admits of exact evaluation in some situations for which not a single e be known herein we show how to evaluate instances of z s and of an associated parity zeta function y s for various quantum systems for some systems both z n y n can be evaluated for infinitely many integers n such z y values can be used for example to effect sharp numerical estimates of a system s ground energy the difficult problem of evaluating the analytic continuation z s for arbitrary complex s is discussed within the contexts of perturbation expansions path integration and quantum chaos

exponential energy growth in a fermi accelerator an unbounded energy growth of particles bouncing off two dimensional 2d smoothly oscillating polygons is observed notably such billiards have zero lyapunov exponents in the static case for a special 2d polygon geometry a rectangle with a vertically oscillating horizontal bar we show that this energy growth is not only unbounded but also exponential in time for the energy averaged over an ensemble of initial conditions we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type we demonstrate numerically that the ensemble averaged energy indeed grows exponentially at a close to the analytically predicted rate namely the process is controllable

isospectral membranes a connection between shape and density a connection is established via conformal maps between vibrating membranes that are isospectral with respect to shape and those that are isospectral with respect to density in particular inhomogeneous circular membranes are constructed that are isospectral to polygonal membranes of uniform density via the schwarz christoffel mapping although some corners of the polygons lead to singularities in the constructed densities the densities are shown to be integrable

on the multiple nevanlinna pick matrix interpolation in the class cp and the caratheodory matrix coefficient problem abstract in the present paper an intrinsic one to one correspondence between the finite multiple nevanlinna pick matrix interpolation np problem in the caratheodory class and the caratheodory matrix coefficient cc problem or equivalently the truncated trigonometric matrix moment tm problem is established thanks to this correspondence the np problem is reduced to what amounts to solving the cc problem or the tm problem associated with the so called toeplitz block vector of the former in both the nondegenerate and degenerate cases simultaneously

level statistics and eigenfunctions of pseudointegrable systems dependence on energy and genus number we study the level statistics second half moment i 0 and rigidity ensuremath delta 3 and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g we find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval at low enough energies the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows poisson like behavior at higher energies the level statistics of most systems moves from poisson like toward wigner like behavior with increasing g on investigating the wave functions we find many chaotic functions that can be described as a random superposition of regular wave functions the amplitude distribution p ensuremath psi of these chaotic functions was found to be gaussian with the typical value of the localization volume v mathrm loc ensuremath approx 0 33 for systems with periodic boundaries we find several additional energy regimes where i 0 is relatively close to the poisson limit in these regimes the eigenfunctions are either regular or localized functions where p ensuremath psi is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case an interesting intermediate case between chaotic and localized eigenfunctions also appears

slow relaxation in weakly open rational polygons the interplay between the regular piecewise linear and irregular vertex angle boundary effects in nonintegrable rational polygonal billiards of m equal sides is discussed decay dynamics in polygons of perimeter p m and small opening delta is analyzed through the late time survival probability s m approximately equal t delta two distinct slow relaxation channels are established the primary universal channel exhibits relaxation of regular sliding orbits with delta 1 the secondary channel is given by delta 1 and becomes open when m p m delta it originates from vertex order disorder dual effects and is due to relaxation of chaoticlike excitations

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

displacement structures and fast inversion formulas for confluent polynomial vandermonde like matrices in the present paper confluent polynomial vandermonde like matrices with general recurrence structure are introduced three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method a relationship between confluent polynomial vandermonde like matrices and confluent cauchy like matrices is pointed out

generic spectral properties of right triangle billiards this article presents a new method to calculate eigenvalues of right triangle billiards its efficiency is comparable to the boundary integral method and more recently developed variants its simplicity and explicitness however allow new insight into the statistical properties of the spectra we analyse numerically the correlations in level sequences at high level numbers 10 5 for several examples of right triangle billiards we find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow surprisingly the genus plays an important role at the quantum level also based on this observation a mechanism is discussed which may explain the particular quantum classical correspondence in right triangle billiards though this class of systems is rather small it contains examples for integrable pseudo integrable and non integrable ergodic weakly mixing dynamics so that the results might be relevant in a more general context

on rational definitions in complete algebras without rank abstract algebras without rank are suggested in order to simplify the algebraic treatment of control structures the regular and equational normal forms of rational schemes are proved without using any representation of finite or infinite trees

radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box in the present work we introduce a methodology for solving a quantum billiard with dirichlet boundary conditions the procedure starts from the exactly known solutions for the particle in a circular disk which are subsequently radially rescaled in such a way that they obey the new boundary conditions in this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy

the polygon quantum billiard problem the present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates the polygon ground state theorem is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the elemental polygons defined in the text the polygon first excited state theorem is established which states that for every n sided regular polygon n equivalent first excited states exist each of which contains a nodal curve that is a line of mirror symmetry of the related polygon a vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the n sided regular polygon is spanned by any two of these n nodal diagonal eigenstates i e the first excited state is twofold degenerate at various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations thus for example correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive nonoverlapping classes of polygons stemming from the preceding conclusions a discussion is included on the possibility of employing transverse magnetic tm modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic

on the ground state of regular polygonal billiards the helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector

the io and oi hierarchies abstract an analysis of recursive procedures in algol 68 with finite modes shows that a denotational semantics of this language can be described on the level of program schemes using a typed calculus with fixed point operators in the first part of this paper we derive classical schematological theorems for the resulting class of level n schemes in part two we investigate the language families obtained by call by value and call by name interpretation of level n schemes over the algebra of formal languages it is proved that differentiating according to the functional level of recursion leads to two infinite hierarchies of recursive languages the io and oi hierarchies which can be characterized as canonical extensions of the regular context free and io and oi macro languages respectively sufficient conditions are derived to establish strictness of io like hierarchies finally we derive that recursion on higher types induces an infinite hierarchy of control structures by proving that level n schemes are strictly less powerful than level n 1 schemes

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

dynamics of kaleidoscopic maps abstract first we introduce a certain class of piecewise affine elliptic rotation maps on t 2 called the kaleidoscopic maps and describe its importance and then we concentrate our efforts on a special case when the rotation angle of a kaleidoscopic map is n n z for the special case we answer the conjectures regarding the periodicity and the singularity structure of such kaleidoscopic dynamics in the process we prove the partial riddling of the regular orbits that gives rise to the classification of periodic sets and estimate the hausdorff dimension of the singular set finally we study the dynamics of such kaleidoscopic maps restricted within the singular set and answer conjectures concerning the chaos the local chaos and the ergodicity with respect to the normalized hausdorff measure of the singular set

rational interpolation and recursive solution of lowner vandermonde systems of equations abstract rational interpolation problems for functions having numerator degree higher than denominator degree are connected with solutions of systems of equations the coefficient matrix of which has a mixed structure the first columns are of vandermonde type whereas the last columns form a lowner matrix here three term recursions for the rational interpolants are developed which can be translated into recurrence formulas for the solutions of homogeneous systems with such a coefficient matrix on this base an o n2 algorithm for the solution of n n nonhomogeneous lowner vandermonde systems is obtained

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

velocity dependent invariants for hard particles in one dimension the relative motion of n impenetrable mass points in one dimension is shown to be equivalent to a billiard in a simplex of n 1 dimension the orientation of the confining hyperplanes being fixed by the mass ratios the discrete group generated by reflections in these planes and its closure in so n 1 are used to show that the only invariant functions of the particle velocities are total momentum and total energy if n 3 and at least one mass is different from the others

multidimensional extension of a wentzel kramers brillouin improvement for spherical quantum billiard zeta functions some insight is offered into the dimensional dependence of the wentzel kramers brillouin wkb and improved wkb approximations introduced by steiner in connection with hyperspherical originally circular quantum billiards the accuracy of every new d dimensional picture is estimated by examining the residues of the spectral zeta function poles

sum rules for zeros of bessel functions and an application to spherical aharonov bohm quantum bags the authors study the sum zeta h s sigma jej s over the eigenvalues ej of the schrodinger equation in a spherical domain with dirichlet walls threaded by a line of magnetic flux rather than using green function techniques they tackle the mathematically non trivial problem of finding exact sum rules for the zeros of bessel functions jv which are extremely helpful when seeking numerical approximations to ground state energies these results are particularly valuable if v is neither an integer nor half an odd integer

on functional determinants of laplacians in polygons and simplicial complexes the functional determinant of an elliptic operator with positive discrete spectrum may be defined ase z 0 wherez s the zeta function is the sum sum limits n lambda n delta analytically continued ins in this paperz 0 is calculated for the laplace operator with dirichlet boundary conditions inside polygons with the topology of a disc in the euclidean plane our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries our expression can be viewed as the energy for a system of static point particles corresponding to the corners of the polygon with self energy and pair interaction energy we have completely explicit closed expressions for triangles and regular polygons with an arbitrary number of sides among these there are five special cases three triangles the square and the circled where thez 0 are known by other means one special case fixes an integration constant and the other provide four independent analytical checks on our calculation

precise bounds and asymptotics for the first dirichlet eigenvalue of triangles and rhombi abstract we study the asymptotic expansion of the first dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached in certain cases which include isosceles and right triangles we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero plus the constant term and estimates on the remainder for rhombi and other triangle families such as isosceles triangles where now the angle opening approaches we have the first two terms plus bounds on the remainder these results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned apart from being accurate near the singular limits considered our lower bounds for the rhombus improve upon the bound by hooker and protter for angles up to approximately 22 and in the range 31 54 these results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it

spectral zeta functions for q bessel equations explicit formulae for the zeta functions of the zeros of hahn exton and jackson s q bessel functions are derived they can be regarded as spectral sum rules for some discrete quantum billiards

comment on sum rules for quantum billiards based on work of itzykson et al see ibid vol 19 no 3 p l111 16 1986 for a d dimensional rectangular box the function zeta t s considered by itzykson et al is related to epstein s zeta function and can be written in a form which exhibits the analytic structure explicitly

sum rules for quantum billiards the authors write explicit integral expressions for sums of inverse powers of the eigenvalues of the laplacian with dirichlet boundary conditions in a simply connected bounded two dimensional domain

schrodinger equation for convex plane polygons ii a no go theorem for plane waves representation of solutions in the present article we discuss whether or not ansatzes more general than those considered hitherto for the representation in terms of plane waves of eigensolutions of the schrodinger equation for a free particle in a plane convex polygonal domain lead to the construction of a complete set for a class of domains larger than the well known one containing the polygons for which such a complete set can be obtained in terms of a superposition of a finite number of plane waves our conclusion is in the negative comments are also made on some features of a representation in terms of analytic functions of two complex variables

zeta functions of nearly circular domains we study the zeta functions of negative integer powers of eigenvalues of the laplacian on two dimensional domains which are close to the unit disc d for closed form expressions are obtained which describe when the technique developed is applied to derive an asymptotic expansion for zeta functions of regular n sided polygons in the limit

a rational billiard flow is uniquely ergodic in almost every direction on demontre qu un flot de billard rationnel est uniquement ergodique dans presque chaque direction

unfoldings and the rank one approximation of the tensor abstract the rank one approximation plays an important role in the study of large scale tensors and the existent finite methods for computing the rank one approximation are generally based on the matricization of the given tensor in this paper exploiting the segmentation of indices of the tensor we present a dynamical strategy for the matricization of the tensor and design a numerical method to compute the rank one approximation of the given tensor more importantly different from the common techniques of regarding a tensor as a matrix and using the matrix singular value decomposition as the approximation tool we transform the given tensor into an order 3 tensor and introduce the projection technique to the calculation of the rank one approximation of the given tensor the complexity analysis and numerical experiments show that the computational cost of our algorithm is no more than that of the existing algorithms numerical results show that compared with the existent finite algorithms our algorithm can provide a more accurate rank one approximation for more than 93 examples

e optimal designs for rational models in this paper experimental designs for a rational model y p x q x e are investigated where p x o 1 x p x p and q x 1 p 1 x p q x q are polynomials and e is a random error two approaches bayesian d optimal and bayesian optimal design for extrapolation are examined the first criterion maximizes the expected increase in shannon information provided by the experiment asymptotically and the second minimizes the asymptotic variance of the maximum likelihood estimator mle of the mean response at an extrapolation point x e corresponding locally optimal designs are also discussed conditions are derived under which a p q 1 point design is a locally d optimal design the bayesian d optimal design is shown to be independent of the parameters in p x and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model the existence and uniqueness of the locally optimal design for extrapolation are proven the number of support points for the locally optimal design for extrapolation is exactly p q 1 these p q 1 design points are proved to be independent of the extrapolation point x e and the parameters in p x the corresponding weights are also independent of the parameters in p x but depend on x e and are not equal

on interpolation by rational functions with prescribed poles with applications to multivariate interpolation abstract this paper is concerned with interpolation in the sense of hermite by certain rational functions of one or several complex variables in the univariate setting the interpolants are generalized polynomials of a cauchy vandermonde space whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of cauchy vandermonde spaces a newton type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than o m 2 arithmetical operations where m is the number of nodes it is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic the algorithm will be extended to the multivariate setting for subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of cauchy vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is o m 2 n n 2 m where m and n are the numbers of interpolation points in the x and y direction respectively

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces