Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made

Preparation of linear hydroxy substituted polyphosphazenes

The synthesis of partially hydroxy-substituted phosphazene prepolymers amenable to processing into cellular, flexible polyurethane foams was investigated. Factors determined include (1) the environment of the hydroxyl group; (2) the ease of the hexachlorocyclotriphosphazene polymerization; (3) the nature of the nonreactive substituents; and (4) the mode of introduction of the hydroxyl entity. The specific approaches taken, the rationale of the selections made, and the results are discussed

Preparation of perfluorinated 1,2,4-oxadiazoles

Fluorinated alkyl or alkylether 1,2,4 oxadiazole compounds are prepared by cyclizing the corresponding alkyl or alkylether imidoyl amidoximes in vacuo or in an inert atmosphere at a temperature within the range of 40 C to 100 C. for a period of 8 to 144 hours in the presence of an acid compound which can accept ammonia to form a salt. The imidoyl amidoximes usable in this process are either polymeric or nonpolymeric. The products, when polymeric, have excellent heat, chemical and solvent resistance

Oscillator strengths and line widths of dipole-allowed transitions in ¹⁴N₂ between 89.7 and 93.5 nm

Line oscillator strengths in the 20 electric dipole-allowed bands of ¹⁴N₂ in the 89.7–93.5nm (111480–106950cm⁻¹) region are reported from photoabsorptionmeasurements at an instrumental resolution of ∼6mÅ (0.7cm⁻¹) full width at half maximum. The absorptionspectrum comprises transitions to vibrational levels of the 3pσᵤc′₄¹Σᵤ⁺, 3pπᵤc³Πᵤ, and 3sσgo₃¹ΠᵤRydberg states and of the b′¹Σᵤ⁺ and b¹Πᵤ valence states. The J dependences of band f values derived from the experimental line f values are reported as polynomials in J′(J′+1) and are extrapolated to J′=0 in order to facilitate comparisons with results of coupled Schrödinger-equation calculations. Most bands in this study are characterized by a strong J dependence of the band f values and display anomalous P-, Q-, and R-branch intensity patterns. Predissociation line widths, which are reported for 11 bands, also exhibit strong J dependences. The f value and line width patterns can inform current efforts to develop comprehensive spectroscopic models that incorporate rotational effects and predissociation mechanisms, and they are critical for the construction of realistic atmospheric radiative-transfer models.This work was supported in part by NASA Grant No. NNG05GA03G to Wellesley College and Australian Research Council Discovery Program Grant No. DP0558962

Power-law tails from multiplicative noise

We show that the well-known Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution at large enough momenta. At finite ratio of the correlation strength for the multiplicative and additive noise the stationary energy distribution becomes exactly the Tsallis distribution.Comment: 4 pages, LaTeX, revtex4 style, 2 figure