Displaying similar documents to “Whitney arcs and 1-critical arcs”

A differential equation related to the l p -norms

Jacek Bojarski, Tomasz Małolepszy, Janusz Matkowski (2011)

Annales Polonici Mathematici

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Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the l p -distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

Continua with unique symmetric product

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Illanes (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a metric continuum. Let F n ( X ) denote the hyperspace of nonempty subsets of X with at most n elements. We say that the continuum X has unique hyperspace F n ( X ) provided that the following implication holds: if Y is a continuum and F n ( X ) is homeomorphic to F n ( Y ) , then X is homeomorphic to Y . In this paper we prove the following results: (1) if X is an indecomposable continuum such that each nondegenerate proper subcontinuum of X is an arc, then X has unique hyperspace F 2 ( X ) , and (2) let X be an arcwise...

Decompositions of nearly complete digraphs into t isomorphic parts

Mariusz Meszka, Zdzisław Skupień (2009)

Discussiones Mathematicae Graph Theory

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An arc decomposition of the complete digraph Kₙ into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class ( Kₙ-R)/t and the ceiling tth class ( Kₙ+S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest possible. The existence of cyclically 1-generated decompositions of Kₙ into cycles → C n - 1 and into paths...

A note on arc-disjoint cycles in tournaments

Jan Florek (2014)

Colloquium Mathematicae

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We prove that every vertex v of a tournament T belongs to at least m a x m i n δ ⁺ ( T ) , 2 δ ⁺ ( T ) - d ⁺ T ( v ) + 1 , m i n δ ¯ ( T ) , 2 δ ¯ ( T ) - d ¯ T ( v ) + 1 arc-disjoint cycles, where δ⁺(T) (or δ¯(T)) is the minimum out-degree (resp. minimum in-degree) of T, and d ⁺ T ( v ) (or d ¯ T ( v ) ) is the out-degree (resp. in-degree) of v.

Disconnections of plane continua

Bajguz, W.

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The paper deals with locally connected continua X in the Euclidean plane. Theorem 1 asserts that there exists a simple closed curve in X that separates two given points x , y of X if there is a subset L of X (a point or an arc) with this property. In Theorem 2 the two points x , y are replaced by two closed and connected disjoint subsets A , B . Again – under some additional preconditions – the existence of a simple closed curve disconnecting A and B is stated.

On Whitney pairs

Marianna Csörnyei (1999)

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    l i m x ↦ x 0 ( | f ( x ) - f ( x 0 ) | ) / ( | ϕ ( x ) - ϕ ( x 0 ) | ) = 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    l i m t ↦ t 0 ( | t - t 0 | ) / ( | ϕ ( t ) - ϕ ( t 0 ) | ) = 0 . We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.