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

numerical solution of volterra equations based on mixed interpolation abstract modified gregory formulae of order 3 and 4 associated with composite modified trapezium and simpson quadrature rules are used to correct in an iterative way the solution of volterra integral equations obtained by the classical polynomial interpolation approach several numerical experiments are performed and discussed

path integral representation for fractional brownian motion fractional brownian motion fbm is a generalization of the usual brownian motion a path integral representation that has recently been suggested for it is shown to be not for the fbm but for a different generalization of the brownian motion a new path integral representation is given and its measure has fractional derivatives of the path in it the measure shows that the process is gaussian but is in general non markovian even though brownian motion itself is markovian it is shown how the propagator for the motion of free fbm may be evaluated this is somewhat more complex than for the usual path integrals due to the occurrence of fractional derivatives we also find the propagator in the presence of a linear absorption potential and for fbm on a ring

community response to noise a review of factors influencing the relationship between noise exposure and reaction social surveys of the relationship between noise exposure and the subjective reactions to it were reviewed this review indicated that remarkably similar results have been obtained across different nationalities with different measurement techniques only a small percentage typically less than 20 of the variation in individual reaction is accounted for by noise exposure analysis of potential errors in both measurement of noise and reaction suggests that elimination of errors would only slightly increase the observed correlations variables such as attitude to the noise source and sensitivity to noise account for more variation in reaction than does noise exposure the weaker relationship between noise exposure and attitude than between reaction and attitude suggests that the attitude reaction relationship is not entirely due to noise exposure causing a change in attitude itself noise reaction correlations based on individual data are significantly lower in studies of impulsive noise than nonimpulsive noise this may be caused in part by the restricted range of noise exposure studied in some socioacoustic investigations of impulsive noise however the significantly higher correlations of attitude and reaction in impulsive noise studies suggest that attitude plays an even larger part while noise exposure plays a lesser part in determining reaction to impulsive noise relative to nonimpulsive noise

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

on a new type of mixed interpolation abstract we approximate every function f by a function f n x of the form a cos kx b sin kx n 2 i 0 c i x i so that f jh f n jh for the n 1 equidistant points jh j 0 n that interpolation function f n x is proved to be unique and can be written as the sum of the n th degree interpolation polynomial based on the same points and two correction terms the error term is also discussed the results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0 this new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations

five diagonal finite difference methods based on mixed type interpolation for a certain fourth order two point boundary value problem abstract modified finite difference methods based on mixed type interpolation are derived and applied for fourth order linear ordinary differential equations subject to a special kind of boundary conditions in this paper we present and analyse methods of orders 2 4 and 6 several numerical experiments are performed to illustrate the theoretical background

the schultz curve 25 years later a research perspective the contemporary technical rationale for assessing effects impacts of transportation noise on communities rests in large part on a purely descriptive dosage effect relationship of the sort first synthesized by schultz j acoust soc am 64 377 405 1978 although u s federal adoption of an annoyance based rationale for regulatory policy has made this approach a familiar one it is only one of several historical perspectives and not necessarily the most useful for all purposes last reviewed by the u s federal interagency committee on noise ficon 10 years ago the accuracy and precision of estimates of the prevalence of a consequential degree of noise induced annoyance yielded by functions of noise exposure leave much to be desired this tutorial article traces the development of the dosage effect relationship on which ficon currently relies in a wider historical context of efforts to understand and predict community response to transportation noise it also identifies areas in which advances in genuine understanding might lead to improved means for predicting community response to transportation noise

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

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

on gregory and modified gregory type corrections to newton cotes quadrature abstract gregory type formulae associated with the class of composite newton cotes quadrature rules of the closed type are established furthermore it is shown how these formulae can be extended by introducing mixed interpolation functions which contain a polynomial and a trigonometric part the case of the modified gregory rules associated with the composite simpson quadrature rule is worked out in detail also the error term is analysed and the obtained rules are numerically tested

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

use of hamiltonian mechanics in systems driven by colored noise the evaluation of the path integral representation for stochastic processes in the weak noise limit shows that these systems are governed by a set of equations which are those of a classical dynamics we show that even when the noise is colored these may be put into a hamiltonian form which leads to better insights and improved numerical treatments we concentrate on solving hamilton s equations over an infinite time interval in order to determine the leading order contribution to the mean escape time for a bistable potential the paths may be oscillatory and inherently unstable in which case one must use a multiple shooting numerical technique over a truncated time period in order to calculate the infinite time optimal paths to a given accuracy we look at two systems in some detail the underdamped langevin equation driven by external exponentially correlated noise and the overdamped langevin equation driven by external quasi monochromatic noise we deduce that caustics focusing and bifurcation of the optimal path are general features of all but the simplest stochastic processes

hamilton jacobi and fokker planck equations for the harmonic oscillator in the inertial regime abstract in this work we use feynman s path integral formalism to show the strict equivalence between the hamilton jacobi hj and fokker planck fp equations for a brownian harmonic oscillator characterized by a langevin equation within the inertial regime in this case the lagrangian function is gaussian and then the path integration which defines the conditional probability density can be replaced by the extremal path this extremal action principle allows us to derive in a straightforward way not only the hj differential equation but also its solution the extremal action the probability density related to the extremal action is also the solution of the fp equation and shown to be exactly the same as those obtained by chandrasekhar in his celebrated paper chandrasekhar 1943 and wang and uhlenbeck 1945

modified quadrature rules based on a generalised mixed interpolation formula abstract in the present paper based on a recently developed generalised mixed interpolation formula which integrates exactly any linear combination of polynomials up to n 2 degree and two other functions u 1 kx and u 2 kx representing two linearly independent solutions of a general second order linear differential equation of the form y x kq kx y x k 2 p kx y x 0 where k is a free parameter various quadrature rules have been derived the formulae that we have derived can be called the generalised modified newton cotes formulae gmncf of the closed type they are obtained by replacing the integrand by an interpolation function of the form au 1 kx bu 2 kx i 0 n 2 c i x i used for equally spaced nodes x j jh the truncation errors involved in the present quadrature formulae are also examined several numerical examples are handled by the generalised modified rules and the utility of the error formulae is also tested in these examples

characteristic functions and higher order lagrangians in the context of the extension of the hamilton jacobi theory to include lagrangians involving higher derivatives the characteristic function seems not to have been considered this omission is here rectified

exact path integral evaluation of the heat distribution function of a trapped brownian oscillator using path integrals we derive an exact expression valid at all times t for the distribution p q t of the heat fluctuations q of a brownian particle trapped in a stationary harmonic well we find that p q t can be expressed in terms of a modified bessel function of zeroth order that in the limit t infinity exactly recovers the heat distribution function obtained recently by imparato et al phys rev e 76 050101 r 2007 from the approximate solution to a fokker planck equation this long time result is in very good agreement with experimental measurements carried out by the same group on the heat effects produced by single micron sized polystyrene beads in a stationary optical trap an earlier exact calculation of the heat distribution function of a trapped particle moving at a constant speed v was carried out by van zon and cohen phys rev e 69 056121 2004 however this calculation does not provide an expression for p q t itself but only its fourier transform which cannot be analytically inverted nor can it be used to obtain p q t for the case v 0

a five diagonal finite difference method based on mixed type interpolation for computing eigenvalues of fourth order two point boundary value problems a modified finite difference method based on mixed type interpolation is derived and applied for computing eigenvalues of fourth order two point boundary value problems a parallelized version of the resulting algorithm is discussed and implemented on a helios based parallel processing transputer system some numerical experiments are performed

path integral approach to fractional levy motion we give an explicit and simple derivation of the propagator for fractional levy motion using path integral methods some known forms of propagators like for brownian motion are recovered in limiting cases we also discuss the associated kinetic equations as well as some extensions

extremal trajectories for stochastic equations obtained directly from the langevin differential operator shows that the differential operator for the extremal trajectory of a stochastic process can be connected directly to the systematic part of the differential operator that defines the stochastic equation by assuming linearity in this operator and gaussianity for the fluctuation the author is able to write these relations for markovian as well as non markovian processes

on extremal paths for stochastic processes that involve step potentials and the generalized hamilton jacobi equation abstract from the behavior of extremal paths across a boundary defined by a finite step potential we obtain by means of a least action principle a first integral of the euler lagrange equation a conservation principle this first integral gives the geometrical properties which are refraction and reflexion laws for extremal paths across the boundary and also it leads to a generalized hamilton jacobi equation for the extremal action

self reported sleep disturbances due to railway noise exposure response relationships for nighttime equivalent and maximum noise levels the objective of the present survey was to study self reported sleep disturbances due to railway noise with respect to nighttime equivalent noise level lp a eq night and maximum noise level lp a max a sample of 1349 people in and around oslo in norway exposed to railway noise was studied in a cross sectional survey to obtain data on sleep disturbances sleep problems due to noise and personal characteristics including noise sensitivity individual noise exposure levels were determined outside of the bedroom facade the most exposed facade and inside the respondents bedrooms the exposure response relationships were analyzed by using logistic regression models controlling for possible modifying factors including the number of noise events train pass by frequency lp a eq night and lp a max were significantly correlated and the proportion of reported noise induced sleep problems increased as both lp a eq night and lp a max increased noise sensitivity type of bedroom window and pass by frequen

hamilton jacobi and fokker planck equations for the harmonic oscillator abstract using feynman s path integral formalism applied to stochastic classical processes we show the equivalence between the hamilton jacobi hj and fokker planck fp equations associated with an overdamped brownian harmonic oscillator in this case the langevin equation leads to a gaussian lagrangian function and then the path integration which defines the conditional probability density can be replaced by the extremal path due to this fact and following the classical dynamics formalism we prove the strict equivalence between the hj and fp equations we do this first for an ordinary brownian harmonic oscillator and then the proof is extended to an electrically charged brownian particle under the action of force fields magnetic field and additional time dependent force fields we observe that this extremal action principle allows us to derive in a straightforward way not only the hj differential equation but also its solution the extremal action

on modified gregory rules based on a generalised mixed interpolation formula a mixed interpolation function in its generalised form is used to derive the generalised modified gregory formulae these formulae are expressed in the form of the classical rules along with two correction terms the error terms are briefly discussed the newly derived quadrature formulae are tested with certain numerical examples which shows the efficiency of the generalised modified rules over classical gregory rules as well as the modified rules based on the mixed trigonometric interpolation the importance of the error terms are also discussed