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A formula expressing the coefficients of an expansion of ultraspherical polynomials which has been differentiated an arbitrary number of times in terms of the coefficients of the original expansion is proved. The particular examples of Chebyshev and Legendre polynomials are considered.
Expressions for the number of truss components per truss division are presented along with expressions for the area and dimensions of mosaic hexagonal panel arrangements. The expressions were developed by substituting the number of truss components in specific truss divisions into associated polynomial equations and solving for the coefficients of the polynomials. To assist in automated or astronaut truss/panel assembly operations, a concept for assembling a tetrahedral truss with hexagonal panels is presented. The assembly concept minimizes the exchange of truss assembly devices and panel attachment devices, assuming that the number of exchanges is a driving assembly concern.
A classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same foci is examined. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not too close, sharp estimates of this type were derived and the corresponding extremal polynomials were determined. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. Some new results were also presented for a weighted approximation problem of this type.
The type of interconnection network employed has a profound effect on the performance of a multicomputer and multiprocessor design. Adequate models are needed to aid in the design and development of interconnection networks. A novel modeling approach using statistical and optimization techniques is described. This method represents an attempt to compare diverse interconnection network designs in a way that allows not only the best of existing designs to be identified but to suggest other, perhaps hybrid, networks that may offer better performance. Stepwise linear regression is used to develop a polynomial surface representation of performance in a (k+1) space with a total of k quantitative and qualitative independent variables describing graph-theoretic characteristics such as size, average degree, diameter, radius, girth, node-connectivity, edge-connectivity, minimum dominating set size, and maximum number of prime node and edge cutsets. Dependent variables used to measure performance are average message delay and the ratio of message completion rate to network connection cost. Response Surface Methodology (RSM) optimizes a response variable from a polynomial function of several independent variables. Steepest ascent path may also be used to approach optimum points.
One of the most important functions of any air traffic management system is the assignment of ground-holding times to flights, i.e., the determination of whether and by how much the take-off of a particular aircraft headed for a congested part of the air traffic control (ATC) system should be postponed in order to reduce the likelihood and extent of airborne delays. An analysis is presented for the fundamental case in which flights from many destinations must be scheduled for arrival at a single congested airport; the formulation is also useful in scheduling the landing of airborne flights within the extended terminal area. A set of approaches is described for addressing a deterministic and a probabilistic version of this problem. For the deterministic case, where airport capacities are known and fixed, several models were developed with associated low-order polynomial-time algorithms. For general delay cost functions, these algorithms find an optimal solution. Under a particular natural assumption regarding the delay cost function, an extremely fast (O(n ln n)) algorithm was developed. For the probabilistic case, using an estimated probability distribution of airport capacities, a model was developed with an associated low-order polynomial-time heuristic algorithm with useful properties.
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
The rigid body motions and low frequency surface errors of grazing incidence Wolter telescopes are studied. The analysis is based on surface error descriptors proposed by Paul Glenn. In his analysis, the alignment and surface errors are expressed in terms of Legendre-Fourier polynomials. Individual terms in the expression correspond to rigid body motions (decenter and tilt) and low spatial frequency surface errors of mirrors. With the help of the Legendre-Fourier polynomials and the geometry of grazing incidence telescopes, exact and approximated first order equations are derived in this paper for the components of the ray intercepts at the image plane. These equations are then used to calculate the sensitivities of Wolter type I and II telescopes for the rigid body motions and surface deformations. The rms spot diameters calculated from this theory and OSAC ray tracing code agree very well. This theory also provides a tool to predict how rigid body motions and surface errors of the mirrors compensate each other.
The trellis structure of linear block codes (LBCs) is discussed. The state and branch complexities of a trellis diagram (TD) for a LBC is investigated. The TD with the minimum number of states is said to be minimal. The branch complexity of a minimal TD for a LBC is expressed in terms of the dimensions of specific subcodes of the given code. Then upper and lower bounds are derived on the number of states of a minimal TD for a LBC, and it is shown that a cyclic (or shortened cyclic) code is the worst in terms of the state complexity among the LBCs of the same length and dimension. Furthermore, it is shown that the structural complexity of a minimal TD for a LBC depends on the order of its bit positions. This fact suggests that an appropriate permutation of the bit positions of a code may result in an equivalent code with a much simpler minimal TD. Boolean polynomial representation of codewords of a LBC is also considered. This representation helps in study of the trellis structure of the code. Boolean polynomial representation of a code is applied to construct its minimal TD. Particularly, the construction of minimal trellises for Reed-Muller codes and the extended and permuted binary primitive BCH codes which contain Reed-Muller as subcodes is emphasized. Finally, the structural complexity of minimal trellises for the extended and permuted, and double-error-correcting BCH codes is analyzed and presented. It is shown that these codes have relatively simple trellis structure and hence can be decoded with the Viterbi decoding algorithm.
The planetary ephemerides approximation for radar astronomy is discussed, and, in particular, the effect of this approximation on the performance of the programmable local oscillator (PLO) used in Goldstone Solar System Radar is presented. Four different approaches are considered and it is shown that the Gram polynomials outperform the commonly used technique based on Chebyshev polynomials. These methods are used to analyze the mean square, the phase error, and the frequency tracking error in the presence of the worst case Doppler shift that one may encounter within the solar system. It is shown that in the worst case the phase error is under one degree and the frequency tracking error less than one hertz when the frequency to the PLO is updated every millisecond.
A class of projection operators with values in a subspace of polynomials is analyzed. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthoronality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.
This paper considers the l1-optimal control problem for general rational plants. It is shown that, for plants with no poles or zeros on the unit circle, an optimal compensator exists and that the resulting closed loop transfer function is polynomial whenever there are at least as many controls as regulated outputs and at least as many measurements as exogeneous inputs. Exactly or approximately, optimal rational compensators can be obtained by solving a sequence of finite linear programs for the coefficients of a polynomial closed-loop transfer function. No assumptions on plant poles or zeros are required to obtain at least approximately optimal compensators. It is shown that constrained problems in which a set of outputs is regulated subject to l(infinity)-norm constraints on another set of outputs can be solved using a slight modification of the same algorithm.
Formulas analogous to the frequency response functions for commonly used filters in orbit error removal are analytically derived to devise observational strategies for the large-scale oceanic variability and to decipher the signal contents of previous results. These include the polynomial orbit error approximations, i.e., the linear, bias-only and quadratic corrections, and the sinusoidal orbit error approximations (the purely sinusoidal correction, and the sinusoid-and-bias correction). It is shown that the frequency response function for a polynomial correction is a function of the ratio of wavelength/track length and to retain 90 percent or more of the signal at a certain wavelength, the ratio must be less than 0.65 (for the quadratic case), 0.90 (linear), and 1.54 (bias-only).
Fast, efficient Fermat number transform used to compute F'(x) analogous to computation of syndrome in conventional decoding scheme. Eliminates polynomial multiplications and reduces number of multiplications in reconstruction of F'(x) to n log (n). Euclidean algorithm used to evaluate F(x) directly, without going through intermediate steps of solving error-locator and error-evaluator polynomials. Algorithm suitable for implementation in very-large-scale integrated circuits.
The Analytic Predictor Corrector (APC) and Energy Controller (EC) atmospheric guidance concepts were adapted to control an interplanetary vehicle aerobraking in the Martian atmosphere. Changes are made to the APC to improve its robustness to density variations. These changes include adaptation of a new exit phase algorithm, an adaptive transition velocity to initiate the exit phase, refinement of the reference dynamic pressure calculation and two improved density estimation techniques. The modified controller with the hybrid density estimation technique is called the Mars Hybrid Predictor Corrector (MHPC), while the modified controller with a polynomial density estimator is called the Mars Predictor Corrector (MPC). A Lyapunov Steepest Descent Controller (LSDC) is adapted to control the vehicle. The LSDC lacked robustness, so a Lyapunov tracking exit phase algorithm is developed to guide the vehicle along a reference trajectory. This algorithm, when using the hybrid density estimation technique to define the reference path, is called the Lyapunov Hybrid Tracking Controller (LHTC). With the polynomial density estimator used to define the reference trajectory, the algorithm is called the Lyapunov Tracking Controller (LTC). These four new controllers are tested using a six degree of freedom computer simulation to evaluate their robustness. The MHPC, MPC, LHTC, and LTC show dramatic improvements in robustness over the APC and EC.
During this research period we have made significant progress in the four proposed areas: (1) design of robust controllers via H infinity optimization; (2) design of robust controllers via mixed H2/H infinity optimization; (3) M-delta structure and robust stability analysis for structured uncertainties; and (4) a study on controllability and observability of perturbed plant. It is well known now that the two-Riccati-equation solution to the H infinity control problem can be used to characterize all possible stabilizing optimal or suboptimal H infinity controllers if the optimal H infinity norm or gamma, an upper bound of a suboptimal H infinity norm, is given. In this research, we discovered some useful properties of these H infinity Riccati solutions. Among them, the most prominent one is that the spectral radius of the product of these two Riccati solutions is a continuous, nonincreasing, convex function of gamma in the domain of interest. Based on these properties, quadratically convergent algorithms are developed to compute the optimal H infinity norm. We also set up a detailed procedure for applying the H infinity theory to robust control systems design. The desire to design controllers with H infinity robustness but H(exp 2) performance has recently resulted in mixed H(exp 2) and H infinity control problem formulation. The mixed H(exp 2)/H infinity problem have drawn the attention of many investigators. However, solution is only available for special cases of this problem. We formulated a relatively realistic control problem with H(exp 2) performance index and H infinity robustness constraint into a more general mixed H(exp 2)/H infinity problem. No optimal solution yet is available for this more general mixed H(exp 2)/H infinity problem. Although the optimal solution for this mixed H(exp 2)/H infinity control has not yet been found, we proposed a design approach which can be used through proper choice of the available design parameters to influence both robustness and performance. For a large class of linear time-invariant systems with real parametric perturbations, the coefficient vector of the characteristic polynomial is a multilinear function of the real parameter vector. Based on this multilinear mapping relationship together with the recent developments for polytopic polynomials and parameter domain partition technique, we proposed an iterative algorithm for coupling the real structured singular value.
The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.
A novel domain decomposition method based on a hybrid variational principle is presented. Prior to any computation, a given finite element mesh is torn into a set of totally disconnected submeshes. First, an incomplete solution is computed in each subdomain. Next, the compatibility of the displacement field at the interface nodes is enforced via discrete, polynomial and/or piecewise polynomial Lagrange multipliers. In the static case, each floating subdomain induces a local singularity that is resolved very efficiently. The interface problem associated with this domain decomposition method is, in general, indefinite and of variable size. A dedicated conjugate projected gradient algorithm is developed for solving the latter problem when it is not feasible to explicitly assemble the interface operator. When implemented on local memory multiprocessors, the proposed methodology requires less interprocessor communication than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory and compares favorably with factorization based parallel direct methods.
The energy contained in the highly excited vibrational and rotational states in a diatomic gas in a thermochemical nonequilibrium state during expansion is estimated. The estimation is made on the assumption that the populations of the vibrational and rotational states, when normalized by their respective equilibrium values, are describable by simple functions containing no more than four arbitrary parameters. A cubic polynomial, a logarithmic-cubic polynomial, and a bimodal step function are used for this purpose. The four parameters are determined by imposing conditions known at the ground state and the dissociation limit and the mass conservation law. The energy in excess of that accounted for by assuming a Boltzmann distribution of these states, defined here as excess excitation energy, is calculated for N2, O2, NO, CO, OH, and H2. A calculation made for a typical nozzle flow shows that the excess energy may reach 6 percent of the total enthalpy of the flow, and that the flow velocity may decrease by as much as 4 percent due to the nonequilibrium excitation phenomenon.