scholarly journals A switch convergence for a small perturbation of a linear recurrence equation

2021 ◽  
Vol 35 (2) ◽  
Author(s):  
Gerardo Barrera ◽  
Shuo Liu
Keyword(s):  
Small Perturbation ◽  
Recurrence Equation ◽  
Linear Recurrence

scholarly journals Generalized Alcuin's Sequence

10.37236/2796 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Daniel Panario ◽  
Murat Sahin ◽  
Qiang Wang
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Recurrence Equation ◽  
Linear Recurrence ◽  
Fixed Integer ◽  
New Family ◽  
Special Case

We introduce a new family of sequences $\{t_k(n)\}_{n=-\infty}^{\infty}$ for given positive integer $k$. We call these new sequences asgeneralized Alcuin's sequences because we get Alcuin's sequence which has several interesting properties when $k=3$. Also, $\{t_k(n)\}_{n=0}^{\infty}$ counts the number of partitions of $n-k$ with parts being $k, \left(k-1\right), 2\left(k-1\right),$ $3\left(k-1\right)$, $\ldots, \left(k-1\right)\left(k-1\right)$. We find an explicit linear recurrence equation and the generating function for $\{t_k(n)\}_{n=-\infty}^{\infty}$. For the special case $k=4$ and $k=5$, we get a simpler formula for $\{t_k(n)\}_{n=-\infty}^{\infty}$ and investigate the period of $\{t_k(n)\}_{n=-\infty}^{\infty}$ modulo a fixed integer. Also, we get a formula for $p_{5}\left(n\right)$ which is the number of partitions of $n$ into exactly $5$ parts.

Recurrence Contribution to Determining the Wear of Slide Bearings

2015 ◽  
Vol 220-221 ◽  
pp. 301-306
Author(s):  
Krzysztof Wierzcholski ◽  
Andrzej Miszczak
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Operating Time ◽  
Variable Coefficients ◽  
Recurrence Equation ◽  
Hydrodynamic Theory ◽  
Linear Recurrence ◽  
Recurrence Equations ◽  
Constant Coefficients

Numerous problems of numerical calculations occurring in power-train tribology and transport concerning the process of determining the problems of worn out bearings demand more and more information referring to the anticipation of the wear of slide bearings in the succeeding years of machine operations. Therefore, the paper presents the methods for working towards solutions to some specific classes of second and higher order non-homogeneous recurrence equations with variable coefficients occurring in the hydrodynamic theory of the problems dealing with bearing wear. Contrary to linear recurrence equations with constant coefficients, linear recurrence equations with variable coefficients rarely have analytical solutions. Numerical answers to such equations are always practicable. With reference to a large number of analytical methods for solutions to linear recurrence equations with variable coefficients, three research directions are usually followed.The first one depends upon the successive determination of particular linear independent solutions to the considered recurrence equation. The second direction can be characterized by a reduction in the order of the recurrence equation for obtaining the first order always solved recurrence equation. The third direction of solutions to recurrence equations with variable coefficients contains the methods for analytical solutions by means of a summation factor.The majority of the general methods of analytical solutions to linear recurrence equations with variable coefficients constitute the adaptation of the methods applied in solutions to suitable differential equations. As regards the final conclusions, the application of the theory presented in this paper contains numerical solutions referring to the wear values of the slide journal-bearing system in the indicated period of operating time.

scholarly journals Shift Equivalence of P-finite Sequences

10.37236/1126 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Manuel Kauers
Keyword(s):  
Equivalence Problem ◽  
Recurrence Equation ◽  
The Other ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Shift Equivalence ◽  
Finite Sequences ◽  
Homogeneous Linear

We present an algorithm which decides the shift equivalence problem for P-finite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shifting one of the sequences $s$ times makes it identical to the other, for some integer $s$. Our algorithm computes, for any two P-finite sequences, given via recurrence equation and initial values, all integers $s$ such that shifting the first sequence $s$ times yields the second.

scholarly journals Some Special Types of Orbits around Jupiter

Aerospace ◽  
2021 ◽  
Vol 8 (7) ◽  
pp. 183
Author(s):  
Yongjie Liu ◽  
Yu Jiang ◽  
Hengnian Li ◽  
Hui Zhang
Keyword(s):  
Gravity Model ◽  
Small Perturbation ◽  
Equilibrium Points ◽  
Major Axis ◽  
Semi Major Axis ◽  
Ground Track ◽  
The Earth ◽  
Degenerate Equilibrium ◽  
The Mean ◽  
Element Theory

This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

scholarly journals The Concepts of Well-Posedness and Stability in Different Function Spaces for the 1D Linearized Euler Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Stefan Balint ◽  
Agneta M. Balint
Keyword(s):  
Euler Equation ◽  
Function Space ◽  
Euler Equations ◽  
Small Perturbation ◽  
Function Spaces ◽  
Well Posedness ◽  
Small Solution ◽  
Null Solution ◽  
Linearized Euler Equations ◽  
The Stability

This paper considers the stability of constant solutions to the 1D Euler equation. The idea is to investigate the effect of different function spaces on the well-posedness and stability of the null solution of the 1D linearized Euler equations. It is shown that the mathematical tools and results depend on the meaning of the concepts “perturbation,” “small perturbation,” “solution of the propagation problem,” and “small solution, that is, solution close to zero,” which are specific for each function space.

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