Truncated Power Series Research Articles

Wave propagation in a prestressed elastic tube filled with a viscous fluid

For a better analysis of the effects of initial stresses on flow in elastic tubes, the propagation of harmonic waves in an initially inflated and axially stretched thick cylindrical shell is studied. Considering the initial field is static, the fluid is treated as an incompressible Newtonian fluid while the tube wall is taken to be an incompressible, isotropic, elastic material. Employing the theory of small deformation superimposed on large initial deformations, for an axially symmetric perturbed motions, the governing differential equations are obtained in the cylindrical polar coordinates. Because of the variable character of the coefficients of the resulting differential equations of the solid body, the field equations are solved by the method of truncated power series. After using the boundary conditions, the dispersion relations are obtained as a function of inner pressure, axial stretch and the thickness ratio. Various special cases, as well as the general case are discussed by numerical means.

Multiparameter estimation in truncated power series distributions under the stein's loss

LetX 1,…,X p be p(≥2)independent random variables, where each X.has a distribution belonging to a one parameter truncated power series distribution. The problem is to estimate simultaneously the unknown parameters under asymmetric loss developed by James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, 361-380). Several new classes of dominating estimators are obtained by solving a certain difference inequality.

An analysis of power series operators

An analysis of some common power series operations is given in terms of the length of the truncated power series. It is shown that functions on power series can be efficiently computed. This can be the basis of accurate, efficient integrators for ordinary differential, integral, and integro-differential equations.

Pulse waves in a prestressed elastic tube

For a better understanding of the effects of initial stresses on flow in elastic tubes, the propagation of time harmonic waves in an initially inflated and axially stretched thick cylindrical shell is studied. To simplify the mathematical analysis, the fluid is assumed to be inviscid and incompressible while the tube wall is taken to be an incompressible elastic material. Utilizing the theory of small deformations superimposed on large initial deformations, for an axially symmetric perturbed motion the governing differential equations are obtained in cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of solid body, the field equations are solved by a truncated power series method. After employing the boundary conditions the dispersion relations are obtained as a function of inner pressure, axial stretch and thickness ratio.

Nonlinear vibrations of in-plane loaded, imperfect, orthotropic plates using the perturbation technique

A perturbation technique is used to study the effects of in-plane inertia, rotary inertia, and shear deformation on the nonlinear free vibration response of an imperfect, in-plane loaded orthotropic plate. The von Karman type governing equilibrium equations of the plate correspond to those of a recently proposed shear deformation theory which employs parabolic shear strain variation across the thickness. The perturbation parameter is taken as the thickness to side length ratio of the plate. By expressing the generalized displacements in the form of a truncated power series of the perturbation parameter, the five governing equations of the problem under consideration are reduced to a single second order ordinary differential equation in terms of the transverse displacement. The solution of this equation is obtained by the method of multiple scales. Numerical results illustrate the influence of various parameters under consideration.

Multisingularity and scaling in partial differential approximants. I

A partial differential approximant (or PDA),F(x, y) approximates a functionf(x, y), specified by its truncated power series, in terms of a solution of a defining linear partial differential equation with polynomial coefficients. The intrinsic multisingularities of a PDA, which may approximate corresponding singularities off(x, y), are analysed formally and shown to obey, in general, asymptotic scaling (as familiar in the theory of critical phenomena), i. e.F(x, y) ≈C|x͠|-γZ(y͠/|x͠|ϕ) +B, wherex͠andy͠are linear combinations of the deviations, (x-xc) and (y-yc), from the multisingular point (xc,yc). Explicit formulae, suitable for numerical computation, are derived for the characteristic exponents,γandϕ, for the scaling functionZ( • ), for its expansion coefficients and for the related coefficientsCandB, in the case when the crossover exponent,ϕ, lies in the interval (½, 2). (Part II extends these results to general values ofϕ, which requires the introduction of the nonlinear scaling fields associated with the PDA.)

Review of the stirling numbers, their generalizations and Statistical Applications

The basic properties of the Stirling numbers and their generalizations are reviewed. Statistical applications of these numbers in expressing (i) the distrib u t l o n ot successes In Poisson (generalized Bernoulli) trials, (ii) the occupancy distributions, (iii) the convolutions of truncated power series distributions and related minimun variance unbiased estimators and (iv) the generalized discrete distributions and factorial moments are presented.

Optimum sections for given torsional and flexural rigidity

The paper presents an exploration by numerical means of the problem of finding the optimum section of a beam designed to have given torsional ( J 0 ) and flexural rigidity ( I 0 ). The section is defined by its conformal transformation on to a unit circle and the corresponding equations for the optimization problem are derived. The special case when the conformal transformation is given as a truncated power series ( N terms of the form 3 n - 2, n = 1, 2,...), leads to a problem of nonlinear programming. This is solved numerically for 1.2 < J 0 / I 0 < 2, between the extremes of J 0 / I 0 = 1.2, which is known to give an equilateral triangle and J 0 / I 0 = 2, which is solved by a circle. Results for N = 24 give solutions that are convex for 1.5 < J 0 / I 0 < 2, but show concavity for J 0 / I 0 = 1.4 and lower values. Results for N = 36 give confirmation and almost eliminate concavity at J 0 / I 0 = 1.4. However, because this last result required 18 h of computer time, it was not possible to extend the calculations to larger values of N . The results obtained are satisfactory down to J 0 / I 0 = 1.5 and suggest that the solutions may be convex shapes for the complete range. An approximate solution based on the lines of shearing stress for the torsion of an equilateral triangle is shown to give close agreement with the numerical results and thus to constitute a good practical upper-bound solution of the problem.

Poloidal and toroidal fields in geomagnetic field modeling

The theory of surface operators is described and applied to four surface operators on spheres: the dimensionless surface gradient, ▽1=r▽‐r∂r; the dimensionless surface curl, ; the dimensionless surface Laplacian, ▽1²=▽1 · ▽1; and the Funk‐Hecke operators, integral operators with axisymmetric kernels. Three methods are given for solving ▽1²g=f as g=▽1−2f; one method works numerically when f has a rapidly convergent expansion in spherical harmonics, the second works when f is smooth in longitude but not latitude, and the third (a Funk‐Hecke operation) works when f is rough in all directions. With this apparatus, a complete proof is given of the Helmholtz representation of an arbitrary vector field v S (r), the spherical surface of radius r centered on the origin: there are unique scalar fields f, g, h on S(r) such that and 〈g〉r=〈h〉r=0. Here 〈g〉r is the average value of g on S (r). From the Helmholtz representation on spherical surfaces, the Mie or poloidal‐toroidal representation in spherical shells is deduced. Suppose S(a,c) is the spherical shell whose inner and outer boundaries are S(a) and S(c). Suppose B is solenoidal in S (a,c), i.e., ▽·B=0 and 〈Br〉a=0. Then there are unique scalar fields P and Q in S(a,c) such that B=▽ × Λ1P + Λ1Q and 〈P〉r=〈Q〉r=0 for a ⪕ r ⪕ c. The fields P = ▽ × Λ1P and Q=Λ1Q are the poloidal and toroidal parts of B. Applications of this formalism to geomagnetic field modeling are discussed. Gauss's resolution of the geomagnetic field B on S(b) into internal and external parts is generalized; if the radial current Jr does not vanish on S (b), then to Gauss's expression must be added a toroidal field on S(b) due entirely to Jr on S(b). A simple proof is given of Runcorn's theorem that to first order in susceptibility no external magnetic field results from magnetization in a horizontally homogeneous spherical shell polarized by sources inside the shell. A Funk‐Hecke‐based method of modeling ionospheric currents is described, which may be more accurate than truncated spherical harmonic expansions and easier to use than Biot‐Savart integrals. Finally, the formalism makes possible the modeling of satellite samples of the geomagnetic field in a spherical shell S (a,c) where electric currents cannot be neglected. Two approximation schemes are described. One is a truncated power series expansion in (c‐a)/H, where H is the radial length scale of the currents. The other assumes that most of B in S(a,c) is not due to the currents between S (a) and S(c), and that the currents in S(a,c) are field‐aligned. Then the collection of physically possible magnetic fields in S(a,c) is only 50% larger, in a well‐defined sense, than the collection of vacuum fields there. Methods of calculation are given explicitly.

Asymptotically minimum variance unbiased estimation for a class of power series distributions

The problem of finding an asymptotically minimum variance unbiased estimator (A.M.V.U.E.) for the parameter of certain truncated power series distributions, is discussed, when the generating function of their coefficients are i) polynomials of binomial type ii) generalized ascending factorials iii) polynomials with coefficients the well known Eulerian numbers.

Triple Power Law for Concrete Creep

An improved creep law for concrete at constant temperature and water content is proposed. It gives the creep rate as a product of power functions of the load duration, the age at loading and the current age of concrete. This law exhibits a gradual smooth transition from the double power law for very short load durations to the logarithmic law for very long load durations. The higher the age at loading, the longer the load duration at the transition. Determination of creep compliance requires evaluation of a binomial integral, which can be carried out either with the help of a truncated power series or by replacement of certain integrals with sums. A table of values from which interpolation is possible is also given. Extensive fitting of creep data from the literature reveals only a modest improvement in the overall coefficient of variation of the deviations from test data; however, the terminal slopes of creep curves are significantly improved, which is especially important for extrapolation of creep measurements. Compared to the previous double power‐logarithmic law, the present formulation has an advantage of continuity in curvature, and compared to the log‐double power law, the present formulation has a greater range of applicability involving also very short creep durations, including the dynamic range. The new formulation also significantly limits the occurrence of divergence of creep curves, and permits even a complete suppression of this property, although at the cost of a distinct impairment in data fits.

A Computer Aided Design Approach to the Development of Simplified Control Models for Non-Linear Distributed Parameter Processes

A method is presented for the generation of simplified models useful in process control applications. The process may comprise e.g. distributed heat and mass transfer phenomena as well as lumped parameter subsystems. Non-linearities represented by continuously differentiable functions of the dependent variables are allowed.The distributed phenomena are approximated by truncated power series in the space variables with time dependant coefficients. These will be used as state variables in the resulting simplified model. The partial differential equations are integrated over a finite number of space regions, meaning that the mass and energy balances are satisfied in average in each region. The procedure will result in a system of ordinary differential equations with time as independent variable. If the original partial differential equations are non-linear (linear) then the simplified system will be non-linear (linear).The method as such has been tested recently in various applications. However, a considerable amount of algebraic work is required, except in very simple applications. A computerized procedure has therefore been developed to generate the coefficient matrices of the ordinary differential equations in a standard format. The basic subroutines can be organized to fit various partial differential equations for arbitrary polynomial degrees.The method is applied to a packed distillation column, a process that is fundamentally non-linear. A low order control model is worked out and examined. Conclusions are drawn regarding the use of the model, e.g. in control system design and for measuring point selection.

A method of three-dimensional strain measurement on non-ideal objects using holographic interferometry

Holographic interferometry has been applied to determine the components of tensile, compressive and shear strain in the surface layer of a non-ideal object subjected to high values of stress. The method consists of determining the three components of the vector displacement at a sufficient number of discrete points on the surface of the object. Functions in the form of truncated power series are fitted to the fringe order-distance data using the method of least squares. Interpolation of these functions is then carried out to obtain the fringe-order numbers at closely spaced equal intervals. The displacement in each interval is calculated and the displacement-distance relationship is then numerically differentiated to obtain strain. Components of stress at a particular point are then estimated by multiplying the strain value by the appropriate modulus. These stresses result from the application of considerable forces that are likely to cause large rigid-body displacements unless the object can be satisfactorily restrained. The difficulty of providing the required restraining forces has been avoided by mounting the object on a kinematic mount from which it can be removed before stressing and replaced afterwards. The accuracy of relocation of the object is sufficient to ensure the satisfactory formation of holographic interference fringes. As the use of the two-exposure holographic method results in a direction ambiguity of the displacement vector at any point of the surface, this method is complemented by electrical strain measurement. The strain is sampled in two convenient orthogonal directions at some part of the surface. The application of the uniqueness theorem of elasticity then permits the direction of displacement to be uniquely specified at any point of the surface. The electrical strain measurement also provides a check of the strain values determined by interferometry at various positions on the surface. This check is necessary since precise knowledge of the surface geometry of a non-ideal object may be lacking. Incorrect assumptions made about the geometry are shown to result in serious errors in the evaluation of strain. The method described has been used to estimate the stresses in the vicinity of a threaded hole in a sample of pipe for representative values of torque applied in screwing a connecting member into the pipe. Possible reasons for failure of the pipe in service are then deduced.

Carbon monoxide mixing ratio inference from gas filter radiometer data

A new algorithm has been developed which permits, for the first time, real-time data reduction of nadir measurements taken with a gas-filter correlation radiometer to determine tropospheric carbon-monoxide concentrations. The algorithm significantly reduces the complexity of the equations to be solved while providing accuracy comparable to line-by-line calculations. The method is based on a regression analysis technique using a truncated power series representation of the primary instrument output signals to infer directly a weighted average of trace gas concentration. The results produced by a microcomputer-based implementation of this technique are compared with those produced by the more rigorous line-by-line methods. This algorithm has been used in the reduction of MAPS Shuttle and aircraft data.

Differential Dynamic Programming as a Tool for Dynamic Optimization in a Travelling-Grate Pelletizing Plant

Induration in a travelling-grate pelletizing plant is a nonlinear distributed parameter process. A high-order dynamic model of the induration zone has been simplified into a model of low order using the method of subdomains. The Differential Dynamic Programming (DDP) method is used for solving an optimal control problem containing the simplified model. The problem of minimizing the inlet gas energy necessary for the induration is considered. The purpose is to examine the feasibility of the DDP method as a tool for dynamic optimization in this kind of process.The approach of the method of subdomains used in the simplification is briefly that the distributed variables are approximated by truncated power series, in the space variable, with time-dependent coefficients. The approximations are inserted into the partial differential equations, forming residuals which are integrated over certain subdomains. The result of this procedure is a non-linear model of low order with time as the independent variable. The optimal control problem treated here has the inlet gas temperature as the control variable. The greater loss of energy at high temperatures is covered by using a quadratic function of the inlet gas temperature in the cost functional. Furthermore, the derivative of pellet temperature with respect to time is limited by a specified value in order to avoid heat shock. The influence of the energy obtained through the exothermal reaction of magnetite into hematite is considered. Variations in the model parameters bed height and pressure drop are performed to determine a defined efficiency of the energy of the inlet gas.Results of the optimization would be to use high beds and as high as possible a gas temperature. The results also show that the method of subdomains in combination with DDP forms a powerful tool for the discovery and verification of control strategies in processes of this type.

Infinite power series

AbstractThe creation, manipulation and evaluation of univariate infinite power series is discussed. Unlike truncated power series, which store the first n terms of an expansion, infinite power series create a procedure for calculating a general term, and are thus a formal representation of the entire expansion. No term is calculated until it is actually needed. All results may be saved along with the environment in which they were evaluated in order to prevent repeated calculation. Laurent expansion is used, and expansion in rational powers is allowed to expand expressions containing poles and branch points. The formulas applicable to the manipulation of these extended series forms are presented. The system is implemented in MODE‐REDUCE with infinite power series a new data type, or mode, which allows interfacing with the usual algebraic operators including substitution, differentiation, and integration. A series may also be inverted. Examples of the capabilities and use of the system are presented.

A mixture theory for elastic laminated composites

A theory is developed which governs the dynamic response of a homogeneous, elastic, dispersive material. The material is used as a model of a two phase layered material in which each of the layers is isotropically elastic. The theory is derived from a general theory for all two phase periodic materials which was developed earlier [1].The general theory was derived using the theory of mixtures. Dispersion is accommodated through the use of elastodynamic operators and for the layered material a micro model is used to establish the forms of the operators appropriate to the material. These specific operators are simplified by replacing them with truncated power series before introducing them into the equations of linear momentum. The theory for layered materials contains nineteen model constants and equations are developed from which these constants can be derived from the layer constants. The equations are derived partly using micro model analysis and partly by matching specific dynamic behaviors of the model and prototype. The ability with which the model predicts the dynamic response of the layered material is assessed in two ways. Both compare spectra reflecting the behavior of infinite trains of the principal kinds of waves. The first compares spectral lines from the model with those derived from the exact theory for layered materials. The second compares lines from the model with those obtained from experiments. Predictions from the model prove to be quite accurate.

On Curves on Formal Groups

The structure of the group of curves on a formal group is determined when the formal group is on a truncated power series algebra over a commutative ring. The resulting curve functor is faithful but not full. Applications to the Lie algebra of the formal group are given.

Series description of small oscillations in non-linear autonomous systems

Autonomous bivariate systems in which the non-linear terms are not multiplied by a small coefficient are investigated. Attention is restricted to systems for which the real part of the characteristic values of the locally linearized system is small. Expressions for the variation of the radius of the trajectory in the phase plane at a given angle, and for the variation of the number of cycles with time, are derived. These expressions take the form of truncated power series in the radius, with coefficients given explicitly in terms of the derivatives through the third order of the functions characterizing the system. Both expressions are integrated. The exponential approach of a system to a limit cycle is worked out. Numerical illustrations of the use of the expressions for the derivatives and for the approach to a limit cycle are presented.

On curves on formal groups

The structure of the group of curves on a formal group is determined when the formal group is on a truncated power series algebra over a commutative ring. The resulting curve functor is faithful but not full. Applications to the Lie algebra of the formal group are given.