Terms Of Known Constants Research Articles

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.

Calculation of the Characteristics of Traveling Waves in Layered Media

A new method is considered for the calculation of traveling-wave characteristics in a layered medium. It is mainly based on the transfer-matrix method [2–10] and the generalized reflection-transmission coefficient (GRTC) method [11–14] for the calculation of characteristics. Introducing an impedance tensor of rank 2, Ẑ (z), in the boundary-value stress problem and utilizing in this way the boundary conditions and the Lame equation, we obtain a system of differential equations which is solved by the fourth-order Runge-Kutta method. Specifying a model of the layered medium in terms of known constants, we find the traveling-wave characteristic γ as a function of frequency ω . Given γ, we easily calculate the dispersion curves, which closely fit the GRTC method. Compared with the GRTC method, the impedance-tensor method fully solves the dispersion curve of the normal-mode traveling waves, and it is applicable to both a homogeneous planar Earth model and a model with a slow layer.

More infinite products: Thue–Morse and the gamma function

Letting $$(t_n)$$ denote the Thue–Morse sequence with values 0, 1, we note that the Woods–Robbins product $$\begin{aligned} \prod _{n \ge 0}\left( \frac{2n+1}{2n+2}\right) ^{(-1)^{t_n}} = 2^{-1/2} \end{aligned}$$ involves a rational function in n and the ± 1 Thue–Morse sequence $$((-1)^{t_n})_{n \ge 0}$$ . The purpose of this paper is twofold. On the one hand, we try to find other rational functions for which similar infinite products involving the ± 1 Thue–Morse sequence have an expression in terms of known constants. On the other hand, we also try to find (possibly different) rational functions R for which the infinite product $$\prod R(n)^{t_n}$$ also has an expression in terms of known constants.

Design of Contractive Output Feedback Controllers For Filtered Backstepping

This paper proposes a novel observer based filtered backstepping controller for a class of nonlinear systems. For this purpose, the contraction theory based tools are exploited to design an exponentially convergent linear time varying observer. The proposed observer satisfies the sufficient observability condition and the resulting output feedback controller assures ultimate boundedness of closed loop trajectories. The use of partial contraction concept for convergence analysis serves a dual purpose. It relaxes the choice of high gain filter parameter, whose magnitude needs to be constrained inside a conservative range in Lyapunov based approaches. Moreover, the exact dependency between closed loop system performance and design parameters are quantified in terms of known constants. The developed technique is successfully simulated for a D.C. motor driven manipulator system.

Ns−TcCorrelations in Granular Superconductors

Following a short discussion of the granular model for an inhomogeneous superconductor, we review the Uemura and Homes correlations and show how both follow in two limits of a simple granular superconductor model. Definite expressions are given for the almost universal coefficients appearing in these relationships in terms of known constants.

Nonstoichiometry of zirconia

Phenomenological equations are suggested for describing the processes of disordering in electron and ion subsystems. Dependences of the concentration of point defects on the partial pressure of oxygen for each kind of defect are obtained in the course of the solution of the equations of defect formation due to the Schottky, Frenkel, and anti-Frenkel mechanisms. Equilibria of zirconia with the ambient at high temperatures are considered together with the processes of defect ionization in an electron subsystem. To the obtained systems of quasichemical equations a chosen condition for electric neutrality is added and the system is transformed into systems of algebraic equations. Linearization of these equations and their solution made it possible to describe the number densities of electrons, holes, vacancies, and interstices in terms of known constants and the partial pressure of oxygen in the general form. The solutions to these systems were chosen in the form of the product of the equilibrium constant of the investigated reaction of defect formation and the exponential function of the partial pressure of oxygen.

Closed-Form Summation of Some Trigonometric Series

The problem of numerical evaluation of the classical trigonometric series ... (formule)... where ν > 1 in the case of S 2n (α) and C 2n+1 (α) with n = 1, 2, 3,... has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when α is equal to a rational multiple of 2π, these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving C ν (α) and S ν (α) in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established

Closed-form summation of some trigonometric series

The problem of numerical evaluation of the classical trigonometric series \[ S ν ( α ) = ∑ k = 0 ∞ sin ⁡ ( 2 k + 1 ) α ( 2 k + 1 ) ν and C ν ( α ) = ∑ k = 0 ∞ cos ⁡ ( 2 k + 1 ) α ( 2 k + 1 ) ν , {S_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\sin (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}}\quad {\text {and}}\quad } {C_\nu }(\alpha ) = \sum \limits _{k = 0}^\infty {\frac {{\cos (2k + 1)\alpha }}{{{{(2k + 1)}^\nu }}},} \] where ν > 1 \nu > 1 in the case of S 2 n ( α ) {S_{2n}}(\alpha ) and C 2 n + 1 ( α ) {C_{2n + 1}}(\alpha ) with n = 1 , 2 , 3 , … n = 1,2,3, \ldots has been recently addressed by Dempsey, Liu, and Dempsey; Boersma and Dempsey; and by Gautschi. We show that, when α \alpha is equal to a rational multiple of 2 π 2\pi , these series can in the general case be summed in closed form in terms of known constants and special functions. General formulae giving C ν ( α ) {C_\nu }(\alpha ) and S ν ( α ) {S_\nu }(\alpha ) in terms of the generalized Riemann zeta function and the cosine and sine functions, respectively, are derived. Some simpler variants of these formulae are obtained, and various special results are established.

Formulae concerning the computation of the Clausen integral Cl 2(Θ)

The main problem under study concerns the expression of the Clausen integral Cl 2( Θ) in closed form in terms of known constants and special functions when Θ is equal to a rational multiple of π belonging to [0, 2π]. A general formula giving Cl 2( pπ q ) in terms of the derivative of the di-gamma function and the sine function is deduced from an appropriate Fourier series expansion. Some variants of this formula are obtained. In further sections, the formulae expressing Cl 2(2 Θ) and, more generally, Cl 2( mΘ)( m=2,3,4,…) as linear combinations of terms of the form Cl 2( Θ+ α) (α: const.) are established. The various results are illustrated by means of typical examples of practical application. The last section contains two simple approximations enabling the computation of Cl( Θ) for any Θ in [0,π] with a relative error smaller than 0.63% and 0.003%, resp.. The paper ends with an appendix in which, among other things, a peculiar trigonometric identity is established as a by-product.

Forced Vibrations of a Body on an Infinite Elastic Solid

Abstract The paper considers the forced vibration of a rigid body resting on a homogeneous elastic medium of infinite surface area and constant depth which may be finite or infinite. Four modes of vibration for a body with a circular base are investigated; (a) vertical translation, (b) torsion, (c) horizontal translation, (d) rocking. For a semi-infinite medium the amplitude response can be obtained for any mass in terms of known constants of the system and two fundamental functions f1 and f2, which depend only on the exciting frequency and the properties of the medium. Close approximations to these functions have been evaluated for each mode. Experiments on an elastic model are described, the results of which are in good agreement with theoretical prediction. The behavior of a stratum of infinite area is more complex since functions f1 and f2 also depend on stratum depth. These functions have been evaluated for the torsional mode only, experimental results being given for the other modes.