## Displaying similar documents to “On the Group of Piecewise Linear Monotone Bijections of an Arc.”

### On Whitney pairs

Fundamenta Mathematicae

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A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $li{m}_{x↦{x}_{0}}\left(|f\left(x\right)-f\left({x}_{0}\right)|\right)/\left(|\varphi \left(x\right)-\varphi \left({x}_{0}\right)|\right)=0$ for every ${x}_{0}$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $li{m}_{t↦{t}_{0}}\left(|t-{t}_{0}|\right)/\left(|\varphi \left(t\right)-\varphi \left({t}_{0}\right)|\right)=0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

### Über monotone Matrixfunktionen

Mathematische Zeitschrift

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### Characterizing the arc by composition of functions

Colloquium Mathematicae

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### Whitney arcs and 1-critical arcs

Fundamenta Mathematicae

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A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that $li{m}_{y\to x,y\in \gamma }|f\left(y\right)-f\left(x\right)|/|y-x|=0$ for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney...

### On Continuous Curves which are Homogeneous except for a Finite Number of Points

Fundamenta Mathematicae

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### A Remark on a Class of Linear Monotone Operators.

Mathematische Zeitschrift

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### Imbedding collections of compact 0-dimensional subsets of ${E}^{2}$ in continuous collections of mutually exclusive arcs

Fundamenta Mathematicae

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### The restricted arc-width of a graph.

The Electronic Journal of Combinatorics [electronic only]

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### On monotone solutions of linear advanced equations.

Memoirs on Differential Equations and Mathematical Physics

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### Two-to-one maps on solenoids and Knaster continua

Fundamenta Mathematicae

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It is shown that 2-to-1 maps cannot be defined on certain solenoids, in particular on the dyadic solenoid, and on Knaster continua.

### Monotone homogeneity of dendrites

Commentationes Mathematicae Universitatis Carolinae

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Sufficient as well as necessary conditions are studied for a dendrite or a dendroid to be homogeneous with respect to monotone mappings. The obtained results extend ones due to H. Kato and the first named author. A number of open problems are asked.