dehn surgery on arborescent links this paper studies dehn surgery on a large class of links called arborescent links it will be shown that if an arborescent link l is sufficiently complicated in the sense that it is composed of at least 4 rational tangles t pl qi with all qj 2 and none of its length 2 tangles are of the form t 1 2ql 1 2q2 then all complete surgeries on l produce haken manifolds the proof needs some result on surgery on knots in tangle spaces let t r 2s p 2q b ti u t2 li k be a tangle with k a closed circle and let m b int n ti u t2 we will show that if s 1 and p 1 mod 2q then om remains incompressible after all nontrivial surgeries oln k two bridge links are a subclass of arborescent links for such a link l p q most dehn surgeries on it are non haken however it will be shown that all complete surgeries yield manifolds containing essential laminations unless p q has a partial fraction decomposition of the form 1 r 1 s in which case it does admit non laminar surgeries

derivation of a general mixed interpolation formula a general procedure is developed to derive a mixed interpolation formula for approximating any n 1 times differentiable function f x for x in 0 nh by a function f n x of the type f n x au 1 kx bu 2 kx sum n 2 i 0 cix i interpolating points being the ones given by x j jh h 0 j o 1 n where u 1 kx and u 2 kx are the two linearly independent solutions of a suitable second order ordinary differential equation ode and k 0 is an appropriately chosen parameter the results for the particular case when u 1 kx and u 2 kx represent the trigonometric functions follow easily an analysis of the error is also discussed and specific numerical examples are included for the sake of comparison with the known interpolation formulae tables showing the comparison of the maximum errors occuring in the use of the various interpolation formulae has also been presented for some specially chosen functions

exponential fitting collocation methods for a class of volterra integral equations abstract in this paper we propose a collocation method for a class of volterra integral equations whose solutions contain periodic functions since the exponential fitting interpolation has an advantage in approximating periodic functions we consider employing it with collocation method to construct our scheme the global convergence analysis of the scheme is also presented based on the interpolation error the theoretical results as well as the superiority of the method are verified in the numerical part

on a class of modified newton cotes quadrature formulae based upon mixed type interpolation abstract we present quadrature rules which integrate exactly not only polynomials up to a certain degree but also the functions sin kx and cos kx where k is a free parameter the formulae we obtain are modified newton cotes formulae they are derived by replacing the integrand by an interpolation function of the form a cos kx b sin kx n 2j 0cjxj based on equally spaced nodes the total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified simpson s 3 8 rule numerical examples show the efficiency of the modified rules and the importance of the error term

a three point formula for numerical quadrature of oscillatory integrals with variable frequency abstract a simple three point formula is constructed for the evaluation of general oscillatory integrals a rigorous derivation of the local error term is presented and the implications to high frequency oscillations are discussed simple examples given include integrals with variable frequency for which the usual filon formula would be inappropriate for cases where filon s formula is appropriate the new formula appears to be computationally more efficient the main application of the formula is to an example chosen from a class of integrals arising in the theory of water waves on a sloping beach comparison with exact results is possible from the work of stoker 16 for a case which whilst special in the physical sense does not simplify the integral involved in all cases the implementation of the formula is as straightforward as the implementation of the ordinary simpson rule

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

exponentially fitted singly diagonally implicit runge kutta methods it is the purpose of this paper to derive diagonally implicit exponentially fitted ef runge kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations whose solutions are supposed to exhibit an exponential behaviour in addition to the standard approach for the derivation of ef methods we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages the derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages as classically done in the standard derivation of multistage ef based methods standard and revised ef methods are then compared in terms of linear stability and numerical performances

unlinking via simultaneous crossing changes given two distinct crossings of a knot or link projection we consider the question under what conditions can we obtain the unlink by changing both crossings simultaneously more generally for which simultaneous twistings at the crossings is the genus reduced though several examples show that the answer must be complicated they also suggest the correct necessary conditions on the twisting numbers let l be an oriented link in s3with a generic projection onto the plane r let a be a short arc in r2 transverse to both strands of l at a crossing so that the strands pass through a in opposite directions then the inverse image of a contains a disk punctured twice with opposite orientation by l define a crossing disk d for a link l in s3 to be a disk which intersects l in precisely two points of opposite orientation it is easy to see that any crossing disk arises in the manner described twisting the link q times as it passes through d is equivalent to doing llq surgery on dd in s3 and adds 29 crossings to this projection of l we say that this new link l q is obtained by adding q twists at d call dd a crossing circle for l a crossing disk d and its boundary do are essential if dd bounds no disk disjoint from l this is equivalent to the requirement that l cannot be isotoped off d in s3 od here we examine how a link can be turned into the unlink via simultaneous twists on disjoint crossing disks in particular we prove an analogue for pairs of crossing disks of the following theorem a much more general version of which is proven in st 1 41 see also gal

extended exponentially fitted interpolation formulas for oscillatory functions we consider exponentially fitted interpolation formulas that use not only the pointwise values of an w dependent function f but also of its derivatives up to the nth order at the two boundary nodes of a closed interval a b the function f is of the form f x f 1 x cos wx f 2 x sin wx email protected a b where the functions f 1 and f 2 are smooth on the interval as n increases numerical results about the errors for the formulas are illustrated and compared some properties of the coefficients of the formulas at the two nodes are presented and extended to the case with three nodes we investigate if the properties of the coefficients can be generalised to the cases with more than three nodes finally some other numerical results regarding the errors are given

on the error and its control in a two parameter generalised newton cotes rule a quadrature formula containing two free phase parameters k k and recently written by van daele et al 13 is rederived using an extension of lagrange s identity by using this method a closed form expression is determined for the local error term e f and the relevant peano kernel is given explicitly sufficient conditions are established under which this kernel remains definite thus allowing a particularly simple expression to be deduced for e f which reduces to the classical result for boole s rule in the limit k k 0 recent work on optimisation of global error is extended to include the new rule it is shown that this error can be reduced by a factor o h 2 on a certain curve in the phase space a further reduction by a factor o h 2 is sometimes possible by choosing the phase parameters on the intersection of this curve and another such when these curves intersect only at complex values the reduction is still achieved a closed form expression for the global error is also derived under these circumstances and this is seen to be asymptotically o h 1 0 as h 0 a second limiting form of the five point formula is found to reduce to the generalised boole s rule written by vanden berghe et al 14 and a third limiting form is also written this third rule which is a special case of a one parameter family of generalised boole s rules is seen to perform better than the other two in the two examples studied several numerical examples are given with extensive diagrams to illustrate all uses of the techniques proposed

mixed interpolation methods with arbitrary nodes abstract previous work on interpolation by linear combinations of the form ac x bs x i 0 n 2 i x i where c and s are given functions and the coefficients a b and j are determined by the interpolation conditions was restricted to uniformly spaced interpolation nodes here we derive both newtonian and lagrangian formulae for the interpolant for arbitrarily chosen distinct nodes in the newtonian form the interpolating function is expressed as the sum of the interpolating polynomial based on the given nodes and two correction terms involving an auxiliary function for which a recurrence relation is obtained each canonical function for the lagrangian form may be expressed as a product of the corresponding lagrange polynomial and a function which depends on divided differences of c x and s x

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

on an unsupervised learning rule for scalar quantization following the maximum entropy principle in this paper we observe that a particular class of rational function rf approximations may be viewed as feedforward networks like the radial basis function rbf network the training of the rf network may be performed using a linear adaptive filtering algorithm we illustrate the application of the rf network by considering two nonlinear signal processing problems the first problem concerns the one step prediction of a time series consisting of a pair of complex sinusoid in the presence of colored non gaussian noise simulated data were used for this problem in the second problem we use the rf network to build a nonlinear dynamic model of sea clutter radar backscattering from a sea surface here real life data were used for the study

topographic map formation by maximizing unconditional entropy a plausible strategy for online unsupervised competitive learning and nonparametric density estimation an unsupervised competitive learning rule called the vectorial boundary adaptation rule vbar is introduced for topographic map formation since vbar is aimed at producing an equiprobable quantization of the input space it yields a nonparametric model of the input probability density function furthermore since equiprobable quantization is equivalent to unconditional entropy maximization we argue that this is a plausible strategy for maximizing mutual information shannon information rate in the case of online learning we use mutual information as a tool for comparing the performance of our rule with kohonen s self organizing feature map algorithm

comments on on a novel unsupervised competitive learning algorithm for scalar quantization this note propose an alternative to a neural network for designing scaler quantizers proposed by van hulle and martinez ibid vol 5 p 498 501 may 1994 it also points out that the performance measure used is of limited applicability

on a novel unsupervised competitive learning algorithm for scalar quantization this letter presents a novel unsupervised competitive learning rule called the boundary adaptation rule bar for scalar quantization it is shown both mathematically and by simulations that bar converges to equiprobable quantizations of univariate probability density functions and that in this way it outperforms other unsupervised competitive learning rules

on the error estimation for a mixed type of interpolation we analyse the error induced by the approximation of a function by an interpolation function which is a combination of algebraic and first order trigonometric polynomials we prove that under certain conditions this error can be expressed in a similar form as in the purely polynomial case as an application we establish in closed form the local truncation error for a class of extended linear multistep methods of the adams type

error analysis for frequency dependent interpolation formulas using first derivatives abstract we call attention to the entire errors which result when frequency dependent interpolation formulas are utilized to approximate oscillatory functions f x at some x on the domain of interest with a frequency of the form f x f 1 x cos x f 2 x sin x where the functions f 1 and f 2 are smooth enough to be approximated by polynomials the interpolation formulas to be considered utilize not only the pointwise values of the function f but also of its derivative f at two or three nodes on a closed and bounded interval in particular investigations about the interpolation formulas i or i using three equally spaced nodes or three unequally spaced nodes enable us to construct i or i related composite formulas which are obtained from applying the formulas i or i onto subintervals where the union of all the subintervals is the domain of interest numerical results show that newly constructed composite formulas are superior in their accuracy to other approximations to interpolate the oscillatory functions finally the entire errors with respect to the interpolation formulas using the derivative information at two or three nodes are obtained

the optimal exponentially fitted numerov method for solving two point boundary value problems second order boundary value problems are solved by means of exponentially fitted numerov methods these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe l gr ixaru g vanden berghe exponential fitting kluwer academic publishers dordrecht 2004 special attention is paid to the expression of the error term of such methods an algorithm concerning the choice of the best suited method and its parameter is discussed several numerical examples are given to sustain the theory

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

exponentially fitted methods for solving time fractional nonlinear reaction diffusion equation abstract in this article we will develop a new numerical scheme with the second order in time and a class of fourth order or sixth order in space based on the exponential fitting techniques to approximate the nonlinear time fractional reaction diffusion equation with fixed order and distributed order derivatives these techniques depend on a parameter which will be used to annihilate the error and increase the order of accuracy the proposed methods are proved to be unconditionally stable and convergent by fourier analysis also the theoretical results and the effectiveness of the numerical scheme are confirmed by numerical test problems and a comparison with other methods is presented

quadrature rules using first derivatives for oscillatory integrands we consider the integral of a function y x i y x 11 y x dx and its approximation by a quadrature rule of the form qn y x k 1n wky xk k 1n ky xk i e by a rule which uses the values of both y and its derivative at nodes of the quadrature rule we examine the cases when the integrand is either a smooth function or an dependent function of the form y x f1 x sin x f2 x cos x with smoothly varying f1 and f2 in the latter case the weights wk and k are dependent we establish some general properties of the weights and present some numerical illustrations

exponentially fitted methods applied to fourth order boundary value problems fourth order boundary value problems are solved by means of exponentially fitted methods of different orders these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe special attention is paid to the expression for the error term and to the choice of the parameter in order to make the error as small as possible some numerical examples are given to illustrate the practical implementation issues of these methods

exponentially fitted two step runge kutta methods construction and parameter selection abstract we derive exponentially fitted two step runge kutta methods for the numerical solution of y f x y specially tuned to the behaviour of the solution such methods have nonconstant coefficients which depend on a parameter to be suitably estimated the construction of the methods is shown and a strategy of parameter selection is presented some numerical experiments are provided to confirm the theoretical expectations

rate distortion theory of finite point processes we study the compression of data in the case where the useful information is contained in a set rather than a vector i e the ordering of the data points is irrelevant and the number of data points is unknown our analysis is based on rate distortion theory and the theory of finite point processes we introduce fundamental information theoretic concepts and quantities for point processes and present general lower and upper bounds on the rate distortion function to enable a comparison with the vector setting we concretize our bounds for point processes of fixed cardinality in particular we analyze a fixed number of unordered gaussian data points and show that we can significantly reduce the required rates compared to the best possible compression strategy for gaussian vectors as an example of point processes with variable cardinality we study the best possible compression of poisson point processes for the specific case of a poisson point process with uniform intensity on the unit square our lower and upper bounds are separated by only a small gap and thus provide a good characterization of the rate distortion function

asymptotic level density for a class of vector quantization processes it is shown that for a class of vector quantization processes related to neural modeling that the asymptotic density q x of the quantization levels in one dimension in terms of the input signal distribution p x is a power law q x c p x sup alpha where the exponent alpha depends on the number n of neighbors on each side of a unit and is given by alpha 2 3 1 3n sup 2 3 n 1 sup 2 the asymptotic level density is calculated and monte carlo simulations are presented

truncation errors in exponential fitting for oscillatory problems a generalization of peano s kernel theorem due to ghizzetti and ossicini quadrature formulae birkhauser basel switzerland 1970 provides expressions in the form of integrals for the truncation errors in a variety of exponential fitting formulae for oscillatory problems in some circumstances this leads to an expression analogous to the lagrange form of remainder more generally the error can be expressed as a sum of two terms of lagrange type our examples include formulae for quadrature and numerical differentiation and linear multistep methods for ordinary differential equations two families of exponential fitting quadrature formulae are investigated one with evenly spaced abscissas and the other based on the philosophy of gaussian quadrature in particular the integral representation can be used to determine the asymptotic rate of decay of the error with increasing frequency for a class of oscillatory integrands

dehn surgery on knots in 3 manifolds it has been known for over 30 years that every closed connected orientable 3manifold is obtained by surgery on a link in s3 8 however a classification of such 3 manifolds in terms of this surgery construction has remained elusive this is due primarily to the lack of uniqueness of the surgery description in 5 kirby gave us a calculus of surgery diagrams however the lack of a canonical surgical method of getting from one 3 manifold to another has hampered further progress in this paper we show that if one restricts attention to the case where a surgery curve is homotopically trivial in a 3 manifold then one has the following uniqueness theorem

the least squares fit of highly oscillatory functions using eta based functions abstract in this paper we examine the possibility of using the eta functions as a new base for high quality approximations of oscillatory functions with slowly varying weights we focus on the least squares and piecewise least squares approximation of such functions and compare the results obtained by using eta based sets of functions with those obtained by means of the legendre polynomials and fourier series we find out that the accuracies from these are more or less equivalent for small frequencies but they exhibit different behaviors when the frequency is increased the accuracy worsens for the legendre polynomials and fourier series base but it remains bounded for the new base in accordance with the known properties of the eta functions such an advantage makes the new base quite attractive for being used in many other mathematical contexts where highly oscillatory functions are involved