Remarks on fixed points of automorphisms and higher-order Weierstrass points in prime characteristic

1990 ◽  
Vol 69 (1) ◽  
pp. 301-303 ◽  
Author(s):  
Arnaldo Garcia
Keyword(s):  
Fixed Points ◽  
Higher Order ◽  
Prime Characteristic ◽  
Weierstrass Points

On Higher Order Weierstrass Points

1972 ◽  
Vol 95 (2) ◽  
pp. 357 ◽  
Author(s):  
Bruce A. Olsen
Keyword(s):  
Higher Order ◽  
Weierstrass Points

scholarly journals Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts

2019 ◽  
Vol 17 (1) ◽  
pp. 677-688 ◽  
Author(s):  
Hai-Ying Chen ◽  
Xiu-Min Zheng
Keyword(s):  
Fixed Points ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Order Difference ◽  
First Order ◽  
Special Points ◽  
And Shifts

Abstract In this paper, we investigate the relationships between fixed points of meromorphic functions, and their higher order differences and shifts, and generalize the case of fixed points into the more general case for first order difference and shift. Concretely, some estimates on the order and the exponents of convergence of special points of meromorphic functions and their differences and shifts are obtained.

scholarly journals Fast growth and fixed points of solutions of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 6 (10) ◽  
pp. 10833-10845
Author(s):  
Yu Chen ◽  
◽  
Guantie Deng ◽  
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Exponent Of Convergence ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential

<abstract><p>In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.</p></abstract>

A syntactic method for finding least fixed points of higher-order functions over finite domains

1997 ◽  
Vol 7 (4) ◽  
pp. 357-394
Author(s):  
TYNG-RUEY CHUANG ◽  
BENJAMIN GOLDBERG
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Normal Forms ◽  
Abstract Interpretation ◽  
Higher Order ◽  
Semantic Domain ◽  
Recent Developments ◽  
Strictness Analysis ◽  
Finite Domains ◽  
Higher Order Functions

This paper describes a method for finding the least fixed points of higher-order functions over finite domains using symbolic manipulation. Fixed point finding is an essential component in the calculation of abstract semantics of functional programs, providing the foundation for program analyses based on abstract interpretation. Previous methods for fixed point finding have primarily used semantic approaches, which often must traverse large portions of the semantic domain even for simple programs. This paper provides the theoretical framework for a syntax-based analysis that is potentially very fast. The proposed syntactic method is based on an augmented simply typed lambda calculus where the symbolic representation of each function produced in the fixed point iteration is transformed to a syntactic normal form. Normal forms resulting from successive iterations are then compared syntactically to determine their ordering in the semantic domain, and to decide whether a fixed point has been reached. We show the method to be sound, complete and compositional. Examples are presented to show how this method can be used to perform strictness analysis for higher-order functions over non-flat domains. Our method is compositional in the sense that the strictness property of an expression can be easily calculated from those of its sub-expressions. This is contrary to most strictness analysers, where the strictness property of an expression has to be computed anew whenever one of its subexpressions changes. We also compare our approach with recent developments in strictness analysis.

Fixed points and frontiers: a new perspective

1991 ◽  
Vol 1 (1) ◽  
pp. 91-120 ◽  
Author(s):  
Sebastian Hunt ◽  
Chris Hankin
Keyword(s):  
Fixed Points ◽  
Abstract Interpretation ◽  
Higher Order ◽  
Upper And Lower Bounds ◽  
Elegant Method ◽  
Computation Of Fixed Points ◽  
New Perspective ◽  
New Formulation ◽  
Intractable Problem ◽  
Higher Order Functions

AbstractAbstract interpretation is the collective name for a family of semantics-based techniques for compile-time analysis of programs. One of the most costly operations in automating such analyses is the computation of fixed points. The frontiers algorithm is an elegant method, invented by Chris Clack and Simon Peyton Jones, which addresses this issue.In this article we present a new approach to the frontiers algorithm based on the insight that frontiers represent upper and lower subsets of a function's argument domain. This insight leads to a new formulation of the frontiers algorithm for higher-order functions, which is considerably more concise than previous versions.We go on to argue that for many functions, especially in the higher-order case, finding fixed points is an intractable problem unless the sizes of the abstract domains are reduced. We show how the semantic machinery of abstract interpretation allows us to place upper and lower bounds on the values of fixed points in large lattices by working within smaller ones.

scholarly journals Growth and fixed points of solutions and their arbitrary-order derivatives of higher-order linear differential equations in the unit disc

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu Chen ◽  
Guan-Tie Deng ◽  
Zhan-Mei Chen ◽  
Wei-Wei Wang
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Unit Disc ◽  
Arbitrary Order ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Iterated Order ◽  
Linear Differential ◽  
Derivatives Of

AbstractIn this paper, we investigate the growth and fixed points of solutions of higher-order linear differential equations in the unit disc. We extend the coefficient conditions to a type of one-constant-control coefficient comparison and obtain the same estimates of iterated order of solutions. We also obtain better estimates by providing a precise value of iterated order of solution instead of a range of that in the case of coefficient characteristic function comparison. Moreover, we utilize iteration to investigate and estimate the fixed points of solutions’ arbitrary-order derivatives with higher-order equations $f^{(k)}+A_{k-1}(z)f^{(k-1)}+{\cdots }+A_{1}(z)f'+A_{0}(z)f=0$ f ( k ) + A k − 1 ( z ) f ( k − 1 ) + ⋯ + A 1 ( z ) f ′ + A 0 ( z ) f = 0 and provide a concise method to judge if the items generated by the iteration do not vanish identically and ensure the iteration proceeds. Our results are an improvement over those by B. Belaïdi, T. B. Cao, G. W. Zhang and A. Chen.

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