scholarly journals Admissible Solutions of the Schwarzian Type Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Baoqin Chen ◽  
Sheng Li
Keyword(s):  
Difference Equation ◽  
Rational Function ◽  
Admissible Solution ◽  
Irreducible Polynomials ◽  
Polynomial Coefficients ◽  
Admissible Solutions ◽  
Special Case

This paper is to investigate the Schwarzian type difference equationΔ3f/Δf-3/2Δ2f/Δf2k=Rz,f=P(z,f)/Q(z,f),whereR(z,f)is a rational function infwith polynomial coefficients,P(z,f), respectivelyQ(z,f)are two irreducible polynomials infof degreep, respectivelyq. Relationship betweenpandqis studied for some special case. Denoted=max⁡p,q. Letf(z)be an admissible solution of(*)such thatρ2(f)<1; then fors (≥2) distinct complex constantsα1,…,αs ,q+2k∑j=1sδ(αj,f)≤ 8k.In particular, ifN(r,f)=S(r,f), thend+2k∑j=1sδ (αj,f)≤4k.

scholarly journals Admissible solutions of the Schwarzian differential equation

1991 ◽  
Vol 50 (2) ◽  
pp. 258-278 ◽  
Author(s):  
Katsuya Ishizaki
Keyword(s):  
Differential Equation ◽  
Rational Function ◽  
Admissible Solution ◽  
Möbius Transformation ◽  
Admissible Solutions ◽  
Mobius Transformation ◽  
Image Position

AbstractLet R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

scholarly journals Extremal behaviour of ± 1-valued completely multiplicative functions in function fields

2021 ◽  
Vol 56 (1) ◽  
pp. 79-94
Author(s):  
Nikola Lelas ◽  
Keyword(s):  
Rational Function ◽  
Finite Fields ◽  
Function Fields ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Irreducible Polynomials ◽  
Liouville Function ◽  
The Mean

We investigate the classical Pólya and Turán conjectures in the context of rational function fields over finite fields 𝔽q. Related to these two conjectures we investigate the sign of truncations of Dirichlet L-functions at point s=1 corresponding to quadratic characters over 𝔽q[t], prove a variant of a theorem of Landau for arbitrary sets of monic, irreducible polynomials over 𝔽q[t] and calculate the mean value of certain variants of the Liouville function over 𝔽q[t].

Admissible solutions for Dirac equations with singular and non-monotone nonlinearity

2011 ◽  
Vol 54 (2) ◽  
pp. 363-371 ◽  
Author(s):  
Yujun Dong ◽  
Jing Xie
Keyword(s):  
Shooting Method ◽  
Dirac Equations ◽  
Admissible Solutions ◽  
Image Position ◽  
Special Case

AbstractBy making use of Merle's general shooting method we investigate Dirac equations of the formHere it is possible that F(0) = −∞ and that F(s) defined on (0,+∞) is not monotonously nondecreasing. Our results cover some known ones as a special case.

scholarly journals Power convexity of a class of Hessian equations in the ball

2015 ◽  
Vol 3 (3) ◽  
pp. 134
Author(s):  
Yunhua Ye
Keyword(s):  
Lower Bound ◽  
Gaussian Curvature ◽  
Principal Curvature ◽  
Power Functions ◽  
Strictly Convex ◽  
Hessian Equations ◽  
Curvature Estimates ◽  
Admissible Solutions ◽  
Special Case

<p>Power convexities of a class of Hessian equations are considered in this paper. It is proved that some power functions of the smooth admissible solutions to the Hessian equations are strictly convex in the ball. For a special case of the equation, a lower bound principal curvature and Gaussian curvature estimates are given.</p>

Exact optical solitons of Radhakrishnan–Kundu–Lakshmanan equation with Kerr law nonlinearity

2019 ◽  
Vol 33 (06) ◽  
pp. 1950061 ◽  
Author(s):  
Behzad Ghanbari ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Mustafa Bayram
Keyword(s):  
Optical Fiber ◽  
Rational Function ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Kerr Nonlinearity ◽  
Optical Solitons ◽  
Kerr Law ◽  
Kerr Law Nonlinearity ◽  
Exponential Rational Function Method ◽  
Special Case

A new generalized exponential rational function method (GERFM) is used to acquire some new optical solitons of Radhakrishnan–Kundu–Lakshmanan (RKL) equation with Kerr nonlinearity. This equation is used to model propagation of solitons through an optical fiber. The well-known exponential rational function method is also a special case of the GERFM. The results reveal that the mentioned method is efficient and simple for solving different nonlinear partial differential equations.

scholarly journals Global behavior of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$

2011 ◽  
Vol 31 (1) ◽  
pp. 43 ◽  
Author(s):  
R. Abo-Zeid ◽  
Cengiz Cinar
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Admissible Solutions ◽  
The Difference

The aim of this work is to investigate the global stability, periodic nature, oscillation and the boundedness of all admissible solutions of the difference equation $x_{n+1}=\frac{Ax_{n-1}} {B-Cx_{n}x_{n-2}}$, n=0,1,2,... where A, B, C are positive real numbers.

scholarly journals Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

2017 ◽  
Vol 59 (1) ◽  
pp. 159-168
Author(s):  
Y. Zhang ◽  
Z. Gao ◽  
H. Zhang
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Necessary Conditions ◽  
Meromorphic Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Polynomial Coefficients ◽  
Difference Analogue ◽  
Borel Exceptional Values

AbstractWe study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

On Rational Functions Related to Algorithms for a Computation of Roots. I

2019 ◽  
Vol 7 (4) ◽  
pp. 17-25
Author(s):  
Mirosław Baran
Keyword(s):  
Rational Function ◽  
Iterative Algorithm ◽  
Chebyshev Polynomials ◽  
Direct Proof ◽  
Rational Functions ◽  
General Construction ◽  
Square Root ◽  
Essential Completion ◽  
Special Case ◽  
Surprising Fact

We discuss a less known but surprising fact: a very old algorithm for computing square root known as the Bhaskara-Brouncker algorithm contains another and faster algorithms. A similar approach was obtained earlier by A.K. Yeyios [8] in 1992. By the way, we shall present a few useful facts as an essential completion of [8]. In particular, we present a direct proof that k-th Yeyios iterative algorithm is of order k. We also observe that Chebyshev polynomials Tn and Un are a special case of a more general construction. The most valuable idea followed this paper is contained in applications of a simple rational function Φ(w; z) = z-w/z+w.

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