a three dimensional boundary element scheme for acoustic radiation in low mach number flows a boundary element scheme is proposed for acoustical radiation in moving flows the scheme relies upon a transformation valid at low mach numbers which permits the case with flow to be restated as an equivalent problem in the absence of flow but with modified boundary conditions this restatement of the original problem is then amenable to boundary integral representation and boundary element solution results obtained by using this approach are presented for the test case of a pulsating or juddering sphere of finite dimensions in a uniform low mach number flow a fully three dimensional boundary element model is used the validity of the results is established by comparison with an analytic perturbation solution where applicable and by comparison with results from an alternative axisymmetric numerical scheme

a numerical study of some radial basis function based solution methods for elliptic pdes abstract during the last decade three main variations have been proposed for solving elliptic pdes by means of collocation with radial basis functions rbfs in this study we have implemented them for infinitely smooth rbfs and then compared them across the full range of values for the shape parameter of the rbfs this was made possible by a recently discovered numerical procedure that bypasses the ill conditioning which has previously limited the range that could be used for this parameter we find that the best values for it often fall outside the range that was previously available we have also looked at piecewise smooth versus infinitely smooth rbfs and found that for pde applications with smooth solutions the infinitely smooth rbfs are preferable mainly because they lead to higher accuracy in a comparison of rbf based methods against two standard techniques a second order finite difference method and a pseudospectral method the former gave a much superior accuracy

accuracy and stability of finite element schemes for the duct transmission problem a one dimensional numerical model is used to investigate the characteristics of finite element computational schemes for linearized acoustical transmission in ducts with flow primitive variables and coupled first order equations are used the relative performances of lagrangian and hermitian elements with galerkin and residual least squares formulations are assessed results of the numerical study are shown to correlate with the characteristics of analytic solutions for the equivalent regular grid difference equations galerkin solutions are shown to introduce spurious nonphysical modes which must be eliminated by careful attention to the local resolution requirements of the finite element mesh residual least squares formulations do not introduce spurious numerical modes but result in significant numerical damping this is particularly severe if lagrangian elements are used

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

moments in infinite channel queues we consider a new approach to the problem of evaluating the moments of the steady state number of customers in system b for an infinite channel queue having general input our new approach uses the rate conservation law as a consequence we show that for a variety of g gi queues var b as long as service times have finite first moment in addition we give some ordering relations for the variance of b when comparing two alternative systems with convex ordered inter arrival time distributions

an efficient method for sound transmission in non uniform circular ducts an efficient method is presented in this paper for predicting the reflection and transmission coefficients of acoustic waves in ducts at the location of a continuous variation in cross sectional area conventional numerical methods yield high dimensionality in the resulting numerical problem due to segmentation of the non uniform section into a number of subsections this paper shows that an ingenious application of spherical coordinates according to the geometry of the non uniform section enables us to determine analytically the system of characteristic functions for various profiles of the non uniform portion this system of characteristic functions can be used as a complete set to express the solution over the non uniform region so that the numerical problem to be solved is of a much lower dimension than that yielded by conventional numerical methods a technique to render the characteristic functions independent of the frequency is also explained to keep the computation time short the present method is applicable to non plane wave modes of propagation by virtue of the use of standard three dimensional curvilinear co ordinates

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

a high order multivariate approximation scheme for scattered data sets we present a high order multivariate approximation scheme for scattered data sets each data point is represented as a taylor series and the high order derivatives in the taylor series are treated as random variables the approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares the approximation is an interpolation when the data points are given as exact or a nonlinear regression function when nonzero measurement errors are associated with the data points using this formulation the gradient information on each data point can be used to significantly reduce the approximation error all parameters of the approximation scheme can be computed automatically from the data points an uncertainty bound of the approximation function is also produced by the scheme numerical experiments demonstrate that although this method is more computationally intensive than traditional methods it produces more accurate approximation functions

an integral formulation for wave propagation on weakly non uniform potential flows abstract an integral formulation for acoustic radiation in moving flows is presented it is based on a potential formulation for acoustic radiation on weakly non uniform subsonic mean flows this work is motivated by the absence of suitable kernels for wave propagation on non uniform flow the integral solution is formulated using a green s function obtained by combining the taylor and lorentz transformations although most conventional approaches based on either transform solve the helmholtz problem in a transformed domain the current green s function and associated integral equation are derived in the physical space a dimensional error analysis is developed to identify the limitations of the current formulation numerical applications are performed to assess the accuracy of the integral solution it is tested as a means of extrapolating a numerical solution available on the outer boundary of a domain to the far field and as a means of solving scattering problems by rigid surfaces in non uniform flows the results show that the error associated with the physical model deteriorates with increasing frequency and mean flow mach number however the error is generated only in the domain where mean flow non uniformities are significant and is constant in regions where the flow is uniform

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

preconditioned conjugate gradients radial basis functions and toeplitz matrices abstract radial basis functions provide highly useful and flexible interpolants to multivariate functions further they are beginning to be used in the numerical solution of partial differential equations unfortunately their construction requires the solution of a dense linear system therefore much attention has been given to iterative methods in this paper we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data thus our method applies to the corresponding toeplitz matrices the number of iterations required to achieve a given tolerance is independent of the number of variables

wave propagation and attenuation in herschel venturi tubes wave propagation and attenuation in a classical flow metering device the herschel venturi tube is investigated analytically and experimentally the analytical results are shown to compare well with the experimental observations for anechoically terminated herschel venturi tubes in the absence of flow the taylor series approximations to the governing differential equations and the stepwise expansion and contraction method are shown to be identical the effect of mean flow on the wave attenuation performance of herschel venturi tubes is also studied in terms of a solution obtained by the runge kutta method and the applicability of the stepwise expansion and contraction method is discussed with the mean flow through the duct

application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations a novel approach to the numerical solution of weakly singular volterra integral equa tions is presented using the cm multiquadric mq radial basis function rbf expansion rather than the more traditional finite difference finite element or polynomial spline schemes to avoid the collocation procedure that is usually ill conditioned we used a global minimization procedure combined with the method of successive approximations that utilized a small finite set of mq basis functions accurate solutions of weakly singular volterra integral equations are obtained with the minimal number of mq basis functions the expansion and optimization procedure was terminated whenever the global errors were less than 5 lo 2002 elsevier science ltd all rights reserved

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

algebraic trace functions over the primes we study sums over primes of trace functions of l adic sheaves using an extension of our earlier results on algebraic twists of modular forms to the case of eisenstein series and bounds for type ii sums based on similar applications of the riemann hypothesis over finite fields we prove general estimates with power saving for such sums we then derive various concrete applications

acoustics in variable area duct finite element and finite difference comparisons to experiment plane wave sound propagation without flow in a rectangular duct with a converging diverging area variation is studied experimentally and theoretically the area variation was of sufficient magnitude to produce large reflections and induce modal scattering the rms pressure and phase angle on both the flat and curved surface were measured and tabulated the steady state finite element theory of astley and eversman and the transient finite difference theory of white are in good agreement with the data it is concluded that numerical finite difference and finite element theories appear ideally suited for handling duct propagation problems which encounter large area variations f h j k l p p p0 s t u

algebraic and transcendental formulas for the smallest parts function abstract building on work of hardy and ramanujan rademacher proved a well known formula for the values of the ordinary partition function p n more recently bruinier and ono obtained an algebraic formula for these values here we study the smallest parts function introduced by andrews spt n counts the number of smallest parts in the partitions of n the generating function for spt n forms a component of a natural mock modular form of weight 3 2 whose shadow is the dedekind eta function using automorphic methods in particular the theta lift of bruinier and funke we obtain an exact formula and an algebraic formula for its values in contrast with the case of p n the convergence of our expression is non trivial and requires power savings estimates for weighted sums of kloosterman sums for a multiplier in weight 1 2 these are proved with spectral methods following an argument of goldfeld and sarnak

acoustic transmission in non uniform ducts with mean flow part i the method of weighted residuals 3n this and a companion paper the problem of transmission of sound through non uniform ducts carrying a high speed subsonic compressible flow is approached using the method of weighted residuals mwr in the form of a modified galerkin method and the finite element method fem the intent of the investigation is to carry out a careful evaluation of these methods in this computationally difficult problem to this end both mwr and fem have been limited to in core computer implementations to generate useful results with relatively modest computational requirements this paper part i details the mwr formulation and presents numerical results establishing the degree of accuracy of mwr as compared to exact eigenvalue calculations and approximate one dimensional transmission calculations the comparison of mwr and fem results is carried out in the companion paper part ii

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

impossibility of fast stable approximation of analytic functions from equispaced samples it is shown that no stable procedure for approximating functions from equally spaced samples can converge exponentially for analytic functions to avoid instability one must settle for root exponential convergence the proof combines a bernstein inequality of 1912 with an estimate due to coppersmith and rivlin in 1992

acoustic transmission in non uniform ducts with mean flow part ii the finite element method this second paper in a two part series describes the implementation of the finite element method for the solution of the problem of acoustic transmission through a non uniform duct carrying a high speed subsonic compressible flow a finite element scheme based on both the galerkin method and the residual least squares method and with eight noded isoparametric elements is described multi modal propagation is investigated by coupling of the solution in the duct non uniform section to modal expansions in uniform sections the accuracy of the finite element results for both the eigenvalue and transmission problems is assessed by comparison with exact solutions and with results from the method of weighted residuals in the form of a modified galerkin method as introduced in part i of this pair of papers the results of calculations show that modal interactions particularly in transmitted modes become increasingly important with increasing duct flow mach number power transmission coefficient calculations for the geometries studied reveal no indication of a linear basis for the phenomenon of subsonic acoustic choking

the asymptotic cardinal function of the multiquadratic ϕ r r2 c2 12as c a radial basis function approximation has the form s x j rd yj x xj 2 x rd where 0 r is some given function yj j zd are real coefficients and the centres xj j zd are points in rd it is known that radial basis function approximations using the multiquadric r r2 c2 12 possess many useful and interesting properties when the centres form an infinite regular lattice we analyse the limiting case as c and identify a class of functions that arise as uniform limits of the multiquadric interpolants in the univariate case we observe that the cardinal function for the multiquadratic becomes the sinc function as c the limit of the multivariate cardinal function is also identified

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

comparison of a finite element model with a multiple scales solution for sound propagation in varying ducts with swirling flows a multiple scales ms solution is proposed to study sound propagation in slowly varying ducts with mean swirling flows instead of the standard linearized euler equations the ms method is applied to the so called galbrun s equation this equation is based on an eulerian lagrangian description and corresponds to a wave equation written only in terms of the lagrangian perturbation of the displacement this yields simpler differential equations to solve for the ms model as well as simpler boundary conditions in this paper galbrun s equation is also solved by a mixed pressure displacement finite element method fem the proposed fem model has already been tested in authors previous papers this model is quite general and is extended here to arbitrary mean flows including compressibility and swirling flow effects some ms and fem solutions are then compared in order to validate both models

the time dependent finite difference procedure for propagation of sound in a non uniform lined duct a comparison with experiments abstract a time dependent finite difference procedure is developed for the calculation of acoustic fields which incorporate finite impedance and rigid wall boundaries a numerical mapping technique is evoked to transform the solution domain into a rectangular geometry and the procedure is used to solve for frequencies where cross modes can propagate in the straight sections of duct numerical results are corroborated with experiments

a mixed finite element method for acoustic wave propagation in moving fluids based on an eulerian lagrangian description a nonstandard wave equation established by galbrun in 1931 is used to study sound propagation in nonuniform flows galbrun s equation describes exactly the same physical phenomenon as the linearized euler s equations lee but is derived from an eulerian lagrangian description and written only in term of the lagrangian perturbation of the displacement this equation has interesting properties and may be a good alternative to the lee only acoustic displacement is involved even in nonhomentropic cases it provides exact expressions of acoustic intensity and energy and boundary conditions are easily expressed because acoustic displacement whose normal component is continuous appears explicitly in this paper galbrun s equation is solved using a finite element method in the axisymmetric case with standard finite elements the direct displacement based variational formulation gives some corrupted results instead a mixed finite element satisfying the inf sup condition is proposed to avoid this problem a first set of results is compared with semianalytical solutions for a straight duct containing a sheared flow obtained from pridmore brown s equation a second set of results concerns a more complex duct geometry with a potential flow and is compared to results obtained from a multiple scale method which is an adaptation for the incompressible case of rienstra s recent work

on longitudinal acoustic propagation in convergent and divergent nozzle flows abstract the longitudinal propagation of sound in quasi one dimensional low mach number nozzle flow is considered in section 1 the solution in the ray approximation section 2 1 is used to transform the wave equation into a schrodinger form which is studied for the family of power law ducts including as a particular case the conical nozzle it is shown that the coincidence of flow sources sinks with sound sources can lead to appearance of essential singularities section 2 2 which can be removed by using a riccati transformation section 2 4 the exact solutions of the acoustic equations for the parabolic figure 1 and hyperbolic figure 2 nozzles are obtained in terms of bessel functions section 2 3 respectively of complex order and argument the general formulas together with limiting forms in the compactness ray and asymptotic approximations section 3 1 are used to establish properties of the acoustic velocity and pressure section 3 2 kinetic and compression energies and energy flux and wave action section 3 3 for example it is shown that the equipartition of energies for moderate variations in cross section gives way table 2 to a predominance of kinetic compression energies respectively near blockages openings the effects of non uniform mean flow table 1 are discussed by comparing horns with nozzles section 3 4 e g it is shown that the duality principle in three alternative forms does not extend from horn to nozzles and the acoustic equations have no elementary solutions for the latter in contrast with the former

phase shift and attenuation characteristics of acoustic waves in a flowing gas confined by cylindrical walls abstract wave propagation in flowing ideal gases confined by cylindrical waveguides is described in the low frequency range using an iterative frobenius series expansion method the primary concern is to present a mathematical model enabling any radial dependent flow profile to be analyzed in contrast to previous analytical results the present model is applicable in the general case where cubic and higher order terms in the axial acoustic velocity become important and to examine the influence of a non vanishing radial velocity term as a numerical test case it is found that a gas flow velocity w r for simplifying reasons assumed to be a linear combination of a flat flow profile and a parabolic flow profile corresponding to a mean flow equal to w is well approximated by a flat flow profile of the same mean flow value w at low shear wavenumbers and at higher shear wavenumbers calculations were done for shear wavenumbers up to 8 in actual fact the error introduced by making this mean flow approximation is smaller than the error introduced by neglecting the radial velocity term

a comparison of measured and computed sound pressure levels in a non uniform acoustically lined duct a comparison of measured and numerically calculated acoustical fields is presented for a non uniform lined duct in the absence of appreciable mean flow the frequency range investigated includes the cut on frequencies of several transverse modes in certain portions of the duct measured pressure fields are compared to those predicted by one and two dimensional numerical models the validity of the one dimensional model is confirmed for frequencies below cut on of the first transverse mode for higher frequencies the one dimensional model is clearly unsatisfactory as might be expected the two dimensional model gives reasonable results for frequencies below and above cut on of the transverse modes although it indicates a very strong sensitivity of acoustical fields to small variations in local wall impedance data

on the convergence of cardinal interpolation by parameterized radial basis functions abstract we consider cardinal interpolation on gridded data by using various radial basis functions associated with one or two parameters one of which leads asymptotically to so called flat limits previously it had been shown that the classical paley wiener functions can be recovered by such cardinal interpolations as the parameter tends to infinity in this article we extend the results by relaxing the requirements on the approximand functions from several points of view the radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics inverse multiquadrics and shifted thin plate spline radial basis functions for instance we also generalise the classes of admitted approximands as well as the radial basis functions to generalised multiquadrics in place of the well known ordinary or for example inverse multiquadrics an interesting analytical aspect of this work is that unlike the classical whittaker shannon theorem functions approximands may be reproduced for the parameter c in the generalised multiquadrics cardinal approximands where the usual shannon series does not converge with theses approximands due to the slow decay of the sinc function which does not allow e g polynomials as approximands in contrast to the latter the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter c that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity

divisor problem in arithmetic progressions modulo a prime power abstract we obtain an asymptotic formula for the average value of the divisor function over the integers n x in an arithmetic progression n a mod q where q p k for a prime p 3 and a sufficiently large integer k in particular we break the classical barrier q x 2 3 e with an arbitrary e 0 for such formulas and using some new arguments generalise and strengthen a recent result of r khan 2015 making it uniform in k

plane wave analysis of conical and exponential pipes with incompressible mean flow the general equation for one dimensional wave propagation at low flow mach numbers m less than or equals slant0 2 is derived and is solved analytically for conical and exponential shapes the transfer matrices are derived and shown to be self consistent comparison is also made with the relevant data available in the literature the transmission loss behaviour of conical and exponential pipes and mufflers involving these shapes are studied analytical expressions of the same are given for the case of a stationary medium the mufflers involving conical and exponential pipes are shown to be inferior to simple expansion chambers of similar dimensions at higher frequencies from the point of view of noise abatement as was observed earlier experimentally

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

fully meshless solution of the one dimensional multigroup neutron transport equation with the radial basis function collocation method abstract in this paper a fully meshless method is proposed for the numerical solution of the one dimensional multigroup neutron transport equation with anisotropic scattering both first order and even parity forms of the transport equation are studied the radial basis function collocation method is chosen for the spatial treatment and legendre polynomials are used to approximate the angular variable the selection of the legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables multiquadric is utilized as the radial function seven problems are considered to evaluate the performance of the method the results show that the method converges exponentially and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains for the one group isotropic benchmark problem discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinates finite element scheme when the shape parameter is properly chosen

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

new sum product type estimates over finite fields abstract let f be a field with positive odd characteristic p we prove a variety of new sum product type estimates over f they are derived from the theorem that the number of incidences between m points and n planes in the projective three space p g 3 f with m n o p 2 is o m n k m where k denotes the maximum number of collinear planes the main result is a significant improvement of the state of the art sum product inequality over fields with positive characteristic namely that 1 a a a a a 1 1 5 for any a such that a p 5 8

regular families of kernels for nonlinear approximation abstract this article studies sufficient conditions on families of approximating kernels which provide n term approximation errors from an associated nonlinear approximation space which match the best known orders of n term wavelet expansion these conditions provide a framework which encompasses some notable approximation kernels including splines cardinal functions and many radial basis functions such as the gaussians and general multiquadrics examples of such kernels are given to justify the criteria additionally the techniques involved allow for some new results on n term greedy interpolation of sobolev functions via radial basis functions

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

the influence of a liquid flow on sound fields confined by conical walls abstract sound propagation in conical waveguides is analyzed for the case where a background flow is maintained through the cone and points along the radial co ordinate r only a general expression for the acoustic pressure is derived and the perturbations of the acoustic field accommodated by the presence of a background flow are found by use of the green function method in the second part of this paper flow measurement properties are discussed the designation of propagation constants such as the wave number and phase speed lose much of their intuitive meaning in a conical waveguide instead the so called pseudo guide wave number and pseudo phase speed are introduced as they are well defined in the cone case and simplifies to the wave number and phase speed in the cylinder case respectively it is shown that changes due to a background flow in pseudo guide wave number pseudo phase speed and zero point crossing times all exhibit an oscillatory behavior as a function of the r co ordinate this is in contrast to the case of a cylinder where such changes become a linear function of the distance from the transmitter the oscillatory behavior in the changes in zero point crossing times as a function of r does not hamper flow measurement in the cone case since the only requirement to be fulfilled is in principle that a one to one correspondence between measured output changes in zero point crossing times and actual flow determined by v 0 at a specific receiver location exists it is shown that this requirement is fulfilled as changes in zero point crossing times depend linearly on the flow coefficient v 0 at a given r co ordinate there are however certain discrete r co ordinate values where this is not the case namely those where changes in zero point crossing times become zero for any value of v 0 in other words if the receiver is positioned near r co ordinates where zero point crossing times are at a maximum or a minimum for a given value of v 0 the truncated cone flow meter sensor is able to measure flow unambiguiosly by detection of changes in zero point crossing times induced by a background flow

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2