Approximate Limit and Approximate Continuity

2020 ◽  
pp. 233-243
Keyword(s):  
Approximate Continuity ◽  
Approximate Limit

On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

scholarly journals Quasiconvexity and Density Topology

2014 ◽  
Vol 57 (1) ◽  
pp. 178-187 ◽  
Author(s):  
Patrick J. Rabier
Keyword(s):  
Open Subset ◽  
Measure Zero ◽  
Quasiconvex Functions ◽  
Density Topology ◽  
Approximate Continuity ◽  
The Common

AbstractWe prove that if f : ℝN → ℝ̄ is quasiconvex and U ⊂ ℝN is open in the density topology, then supU ƒ = ess supU f ; while infU ƒ = ess supU ƒ if and only if the equality holds when U = RN: The first (second) property is typical of lsc (usc) functions, and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of ƒ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

scholarly journals Exact and approximate limit behaviour of the Yule tree’s cophenetic index

2017 ◽  
Author(s):  
Krzysztof Bartoszek
Keyword(s):  
Limit Distribution ◽  
Heuristic Methods ◽  
Birth Process ◽  
Type Distribution ◽  
Continuous Branch ◽  
Branch Lengths ◽  
Limit Behaviour ◽  
Contraction Type ◽  
Approximate Limit ◽  
Balance Index

AbstractIn this work we study the limit distribution of an appropriately normalized cophenetic index of the pure–birth tree conditioned onncontemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and inL2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction–type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure–birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time–consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

DENJOY'S APPROXIMATE CONTINUITY FOR THE SOLUTIONS OF MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 10 (01) ◽  
pp. 119-141
Author(s):  
SIYAN XU
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximate Continuity

In this paper we prove Denjoy's approximate continuity for the solutions of multivalued stochastic differential equations.

scholarly journals Equivalence of A-approximate continuity for self-adjoint expansive linear maps

2008 ◽  
Vol 429 (7) ◽  
pp. 1504-1521 ◽  
Author(s):  
Sz.Gy. Révész ◽  
A. San Antolı´n
Keyword(s):  
Approximate Continuity ◽  
Linear Maps

Interaction between atomic hydrogen and YBa2Cu3O7

1992 ◽  
Vol 7 (11) ◽  
pp. 2908-2915 ◽  
Author(s):  
Angel L.G. Ponce ◽  
Jose J. Fripiat
Keyword(s):  
Induction Period ◽  
Atomic Hydrogen ◽  
Dissociative Chemisorption ◽  
Experimental Conditions ◽  
Reaction Proceeds ◽  
Order Of Magnitude ◽  
O Vacancy ◽  
The Creation ◽  
Approximate Limit

Atomic hydrogen obtained from dissociative chemisorption of molecular H2 on Pt particles deposited on the surface of YBa2Cu3O7 reacts with the oxide in producing O vacancies and intercalating H at 82 °C under a H2 pressure of about 400 Torr. An induction period which extends over 1 h is observed as long as the concentration in O vacancies is below 0.1. Above this approximate limit the reaction proceeds quickly until about 1 mol H2 has been consumed. It then slows down progressively, but it is not completed even after 27 h of reaction and ∼1.4 H2 consumed, under these experimental conditions. The enthalpy for the creation of the O vacancy is 143 kcal/g mol O, while the H intercalation enthalpy is −57.5 kcal/g mol H. It appears that the reaction of molecular H2 over YBa2Cu3O7 not coated with Pt proceeds similarly, but the rate is nearly one order of magnitude slower, under identical conditions. The stoichiometry of the reaction agrees with the earlier suggestion that O1 is a labile oxygen which can be replaced by intercalated H. This site and the interstitial vacant sites in the copper chains square plane would be the first ones to be occupied by intercalated H.

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