Showing papers on "Closed-form expression published in 1996"

Optimal planar interception with fixed end conditions: closed-form solution

Abstract: A planar constant-speed interception with prescribed end conditions is analyzed. The performance index is the time of capture penalized by the control energy. For this problem, the optimal control of the pursuer is obtained in closed form, based on solving a set of nonlinear algebraic equations involving elliptic integrals. The construction of the solution is inspired by the singularly perturbed structure of the nondimensional equations of motion.

Analytical expressions for correlation functions and Kirchhoff integrals for Gaussian surfaces with ocean-like spectra

Abstract: We present closed-form expressions for the correlation functions associated with Gaussian surfaces with certain ocean-like spectra. These correlation functions may be used to derive asymptotic expansions for the Kirchhoff integral with a wide range of validity. Some comparisons with numerical simulations are also presented. Our analysis establishes that the Kirchhoff integral is well approximated by a Bragg scattering model at very high incidence angles (that is, well away from normal incidence) even when conditions for perturbation theory do not apply. We can compute the nonlinear corrections to Bragg explicitly-these corrections grow as one approaches normal incidence. A novel feature of our analysis is the use of the computer mathematics system Mathematica to construct the relevant asymptotic series. These results eliminate the need for extensive amounts of numerical fast Fourier transform (FFT) computation, and may also be used to simplify computations of scattering cross sections from more complex surfaces with spectra that are perturbations of those we have considered.

A Graphical Approach to College Algebra

Abstract: 1. Linear Functions, Equations, and Inequalities 1.1. Real Numbers and the Rectangular Coordinate System 1.2. Introduction to Relations and Functions 1.3. Linear Functions 1.4. Equations of Lines and Linear Models 1.5. Linear Equations and Inequalities 1.6. Applications of Linear Functions 2. Analysis of Graphs of Functions 2.1. Graphs of Basic Functions and Relations Symmetry 2.2. Vertical and Horizontal Shifts of Graphs 2.3. Stretching, Shrinking, and Reflecting Graphs 2.4. Absolute Value Functions 2.5. Piecewise-Defined Functions 2.6. Operations and Composition 3. Polynomial Functions 3.1. Complex Numbers 3.2. Quadratic Functions and Graphs 3.3. Quadratic Equations and Inequalities 3.4. Further Applications of Quadratic Functions and Models 3.5. Higher-Degree Polynomial Functions and Graphs 3.6. Topics in the Theory of Polynomial Functions (I) 3.7. Topics in the Theory of Polynomial Functions (II) 3.8. Polynomial Equations and Inequalities Further Applications and Models 4. Rational, Power, and Root Functions 4.1. Rational Functions and Graphs 4.2. More on Rational Functions and Graphs 4.3. Rational Equations, Inequalities, Models, and Applications 4.4. Functions Defined by Powers and Roots 4.5. Equations, Inequalities, and Applications Involving Root Functions 5. Inverse, Exponential, and Logarithmic Functions 5.1. Inverse Functions 5.2. Exponential Functions 5.3. Logarithms and Their Properties 5.4. Logarithmic Functions 5.5. Exponential and Logarithmic Equations and Inequalities 5.6. Further Applications and Modeling with Exponential and Logarithmic Functions 6. Analytic Geometry 6.1. Circles and Parabolas 6.2. Ellipses and Hyperbolas 6.3. Summary of Conic Sections 6.4. Parametric Equations 7. Systems of Equations and Inequalities Matrices 7.1. Systems of Equations 7.2. Solution of Linear Systems in Three Variables 7.3. Solution of Linear Systems by Row Transformations 7.4. Matrix Properties and Operations 7.5. Determinants and Cramer's Rule 7.6. Solution of Linear Systems by Matrix Inverses 7.7. Systems of Inequalities and Linear Programming 7.8. Partial Fractions 8. Further Topics in Algebra 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 Counting Theory 8.5 The Binomial Theorem 8.6 Mathematical Induction 8.7 Probability R. Reference: Basic Algebraic Concepts R.1. Review of Exponents and Polynomials R.2. Review of Factoring R.3. Review of Rational Expressions R.4. Review of Negative and Rational Exponents R.5. Review of Radicals Appendix: Geometry Formulas

Modal theory for the two-frequency mutual coherence function in random media

Abstract: Short pulse propagation in random media is mainly determined by the two-frequency mutual coherence function which is governed in the multiple scattering regime by a parabolic equation. In this paper the modal expansion theory is presented as a new analytical approach for media which are statistically isotropic and homogeneous. By performing a separation of variables, the problem of the 3D partial differential equation is reduced to solving a one-dimensional eigenvalue problem. The full expansion theorem is presented applicable for any initial source configuration. For media characterized by a quadratic structure function, the eigenvalue problem is exactly solvable. The two-frequency coherence function is obtained as a modal series for the three most important source configurations, namely the plane wave, the point source and the beam wave. By Poisson's theorem, the series is summed up into a closed form expression and is shown to yield the known solutions in the literature. In this paper, we only present the general modal expansion theorem and the exact solution for a beam in a quadratic medium.

Closed-form expression for the coupling matrix elements related to TM- scattering from a symmetric double strip grating

Abstract: This paper treats the problem of TM plane-wave scattering from a planar periodic symmetric double-strip grating placed at a dielectric interface. The formulation is based on a multimode equivalent network representation and the relevant integral equation defined on two separate symmetrically spaced intervals is rigorously solved by reducing to two simpler equations with known solutions. From this a new simple closed form expression is obtained for the coupling matrix elements. Some computations based on this new expression are carried out and the results are compared to those obtained by the Reimann-Hilbert method and also to some previously obtained single-strip results in the limiting case.

A closed-form analytic solution for the - equations of neutron transport in slab geometry

Abstract: A closed-form analytic solution for the discrete-ordinates equations of neutron transport is obtained without recourse to integral transforms. The method is applied to the slab-geometry problem in the one-speed approximation using a one-sided set of boundary conditions. The solution is valid for any number of discrete ordinates and all possible orders of anisotropic scattering, and it does not require a priori knowledge of the particular solution. A set of algebraic expressions for the neutron balance at the boundaries of a slab is obtained and used as a basis for one iterative and one non-iterative numerical algorithm that are valid for all homogeneous and heterogeneous slabs. The numerical solutions are error free other than the roundoff of the computing system. The roundoff error is minimized by the Effective Albedo and Transmittance method.