scholarly journals On the Universal Regularity of the Numbers of Generalized Recurrence Sequence and Solutions to Its Characteristic Equation of Second Order

2020 ◽  
pp. 27-33
Author(s):  
P. Kosobutskyy
Keyword(s):  
Characteristic Equation ◽  
Quadratic Equation ◽  
Second Order ◽  
Recurrence Sequence ◽  
Phase Coordinates ◽  
Fibonacci Sequences

In this work shows that the classical oscillations of the ratio of neighboring members of the Fibonacci sequences are valid for arbitrary directions on the plane of the phase coordinates, approaching, to a maximum, the solutions to the characteristic quadratic equation at a given point. The values of the solutions to the characteristic equation along the satellites are asymptotically close to their integer values of the corresponding root lines.

scholarly journals Some Convolution Formulae Related to the Second-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 788 ◽  
Author(s):  
Zhuoyu Chen ◽  
Lan Qi
Keyword(s):  
Power Series Expansion ◽  
Second Order ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Symmetry Properties ◽  
Elementary Methods ◽  
Linear Recurrence Sequence ◽  
Convolution Formula ◽  
The Relationship

The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.

A Comment on the Approximate Calculation of Eigenvalues for Certain Second-Order Linear Differential Equations

1962 ◽  
Vol 66 (623) ◽  
pp. 722-722
Author(s):  
William Squire
Keyword(s):  
Approximate Calculation ◽  
Quadratic Equation ◽  
Second Order ◽  
Interesting Aspect ◽  
Approximate Evaluation ◽  
Order Linear ◽  
Combined Use ◽  
Linear Differential ◽  
Lowest Eigenvalue ◽  
Variational Expression

In a note by Goodey on the combined use of the WKB solution and Rayleigh's principle to estimate the lowest eigenvalue of a second-order linear differential equation, a numerical error concealed an interesting aspect of the result. The exact value λ = 2·062 corresponds to 0·6564π; and not 0·654π as stated. The approximate value 0·6559π is therefore lower than the exact value.The possibility of approximate evaluation of the integral in a variational expression affecting the direction of approach to the exact value was pointed out by the author in a recent paper on the application of quadrature by differentiation. The application of the method described there to the example considered by Goodey may be of some interest as it gives the eigenvalue to within 1/2 per cent by the solution of a quadratic equation.

scholarly journals Netted matrices

2003 ◽  
Vol 2003 (39) ◽  
pp. 2507-2518 ◽  
Author(s):  
Pantelimon Stănică
Keyword(s):  
Second Order ◽  
Recurrence Sequence ◽  
The Matrix

We prove that powers of4-netted matrices (the entries satisfy a four-term recurrenceδai,j=αai−1,j+βai−1,j+γai,j−1) preserve the property of nettedness: the entries of theeth power satisfyδeai,j(e)=αeai−1,j(e)+βeai−1,j−1(e)+γeai,j−1(e), where the coefficients are all instances of the same sequencexe+1=(β+δ)xe−(βδ+αγ)xe−1. Also, we find a matrixQn(a,b)and a vectorvsuch thatQn(a,b)e⋅vacts as a shifting on the general second-order recurrence sequence with parametersa,b. The shifting action ofQn(a,b)generalizes the known property(0111)e⋅(1,0)t=(Fe−1,Fe)t. Finally, we prove some results about congruences satisfied by the matrixQn(a,b).

scholarly journals ABOUT ONE FAST-CHANGING CHARACTERISTIC EQUATION SPECTRAL PROBLEMS FOR DIFFERENTIAL EQUATIONS OF SECOND ORDER

2014 ◽  
Vol 19 (11) ◽  
pp. 1-6
Author(s):  
Alexandr Nikolayevich Shevtsov ◽  
◽  
Vladimir Nikolayevich Kestelman ◽  
Abdizhahan Manapovich Sarsenbi ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Characteristic Equation ◽  
Second Order ◽  
Spectral Problems

On the subject of influence of dissipative and permanent momentums on stability of uniform rotations of two elastically connected free gyroscopes of Lagrange

2019 ◽  
Vol 33 ◽  
pp. 132-141
Author(s):  
Yuri Kononov ◽  
Yaroslav Sviatenko
Keyword(s):  
Asymptotic Stability ◽  
Characteristic Equation ◽  
Joint Stiffness ◽  
Quadratic Equation ◽  
Stability Conditions ◽  
Spherical Joint ◽  
Fourth Degree ◽  
Left Hand ◽  
Resisting Medium ◽  
Complex Coefficients

In many works, there are studies of the asymptotic stability of rotation of a free Lagrange gyroscope in a resisting medium. This article generalizes this problem to the case of uniform rotations of two free Lagrange gyroscopes connected by an elastic restoring spherical hinge. The rotation of each gyroscope is maintained by a constant moment in an inertial coordinate system. The characteristic equation of the perturbed motion is presented in the form of an algebraic equation of the fourth degree with complex coefficients. Based on the innor approach, conditions of asymptotic stability are obtained in the form of a system of three inequalities. The left-hand side of these inequalities is represented, respectively, in the form of determinants of the third, fifth, and seventh orders. Up to first-order values of smallness, relative to the reciprocal of the stiffness coefficient, a study is made of the effect of the joint stiffness on stability conditions. From the conditions of positivity of the highest coefficients in three inequalities, it is shown that for a sufficiently large rigidity, the stability conditions are determined by only one inequality. Cases of degeneration of an elastic spherical joint into a spherical inelastic, cylindrical, and universal elastic joint (Hooke's joint) are considered. In the case of an inelastic spherical joint, the system of three inequalities is slightly simplified. The greatest simplification arises in the case of a cylindrical hinge. In this case, the characteristic equation is represented as a quadratic equation with complex coefficients. According to the innoric approach, the conditions of asymptotic stability are written in the form of a single inequality, the left side of which is presented in the form of third-order determinants. It is shown that this inequality coincides with the inequality obtained earlier for the case of a sufficiently large rigidity of the hinge. If the angular velocities of the proper rotations of the gyroscopes coincide, the inequality obtained for the cylindrical hinge coincides with the well-known inequality for one gyroscope. In the case of a universal elastic hinge (Hooke's hinge), the first inequality is represented as a square inequality with respect to the angular velocity of proper rotation.

scholarly journals Analytical modeling of the binary dynamic circuit motion

2021 ◽  
Vol 9 (5) ◽  
pp. 23-32
Author(s):  
Anatolii Alpatov ◽  
Victor Kravets ◽  
Volodymyr Kravets ◽  
Erik Lapkhanov
Keyword(s):  
Characteristic Equation ◽  
Initial Phase ◽  
Analytical Modeling ◽  
Phase State ◽  
Translational Motion ◽  
Elastic Interaction ◽  
Dissipative Function ◽  
Fourth Degree ◽  
Phase Coordinates ◽  
Particular Solution

The binary dynamic circuit, which can be a design scheme for a number of technical systems is considered in the paper. The dynamic circuit is characterized by the kinetic energy of the translational motion of two masses, the potential energy of these masses’ elastic interaction and the dissipative function of energy dissipation during their motion. The free motion of a binary dynamic circuit is found according to a given initial phase state. A mathematical model of the binary dynamic circuit motion in the canonical form and the corresponding characteristic equation of the fourth degree are compiled. Analytical modeling of the binary dynamic circuit is carried out on the basis of the proposed particular solution of the complete algebraic equation of the fourth degree. A homogeneous dynamic circuit is considered and the reduced coefficients of elasticity and damping are introduced. The dependence of the reduced coefficients of elasticity and damping is established, which provides the required class of solutions to the characteristic equation. An ordered form of the analytical representation of a dynamic process is proposed in symmetric determinants, which is distinguished by the conservatism of notation with respect to the roots of the characteristic equation and phase coordinates.

Synthesis of Spherical Four-Bar Function Generators to Match Two Prescribed Velocity Ratios

1983 ◽  
Vol 105 (4) ◽  
pp. 631-636 ◽  
Author(s):  
C. H. Chiang
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Characteristic Equation ◽  
Quadratic Equation ◽  
Present Problem ◽  
Crank Angle ◽  
Function Generators ◽  
Special Case

The present problem is a variation of synthesizing spherical four-bar function generators to coordinate three pairs of finitely separated crank-angle displacements. It is to be stressed here that, since the technique is on the basis of equations of three relative poles, the characteristic equation is simply a quadratic equation in the unknown tan ψo (initial output crank angle). It is not necessary, as in the case of a characteristic equation derived from displacement equations, to resort to techniques such as iterative methods for solving nonlinear equations. Equations are so presented as to facilitate computer programing. The synthesis of spherical crank-rockers is a special case of the present problem.

scholarly journals The Solutions of Second-Order Linear Matrix Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Kefeng Li ◽  
Chao Zhang
Keyword(s):  
Time Scales ◽  
Characteristic Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Existence Of A Solution ◽  
Order Linear ◽  
The Matrix ◽  
Linear Matrix Equations

This paper studies the solutions of second-order linear matrix equations on time scales. Firstly, the necessary and sufficient conditions for the existence of a solution of characteristic equation are introduced; then two diverse solutions of characteristic equation are applied to express general solution of the matrix equations on time scales.

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