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.

Biaccessibility in quadratic Julia sets

2000 ◽  
Vol 20 (6) ◽  
pp. 1859-1883 ◽  
Author(s):  
SAEED ZAKERI
Keyword(s):  
Fixed Point ◽  
Measure Zero ◽  
Julia Set ◽  
Basin Of Attraction ◽  
Countable Union ◽  
Common Theme ◽  
Chebyshev Map ◽  
Straight Line ◽  
The Common ◽  
The Basin Of Attraction

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.

On the “Essential Metrization” of Uniform Spaces

1966 ◽  
Vol 18 ◽  
pp. 1224-1236
Author(s):  
Elias Zakon
Keyword(s):  
Present Note ◽  
Measure Zero ◽  
Measure Theory ◽  
Usual Sense ◽  
Topological Properties ◽  
Uniform Spaces ◽  
Approximate Continuity ◽  
Measurable Functions

The notion of “essential metrization” was introduced in (8) where it was used to obtain some extensions of the theorems of Egoroff and Lusin on measurable functions. In the present note we shall further develop the theory of essential metrization, in its own right.As is well known, sets of measure zero (“null sets“) may be disregarded in many problems of measure theory. Hence the usual topological prerequisites of such problems are actually too restrictive and may be replaced by what could be called “topology modulo null-sets,” imitating such notions as “approximate continuity,” “essential supremum,” etc. Thus we consider spaces in which certain topological properties (such as metrizability, separability, etc.) hold not in the usual sense but only “essentially,” i.e. to within some null sets, as defined below.

scholarly journals Density topology and approximate continuity

1961 ◽  
Vol 28 (4) ◽  
pp. 497-505 ◽  
Author(s):  
Casper Goffman ◽  
C. J. Neugebauer ◽  
T. Nishiura
Keyword(s):  
Density Topology ◽  
Approximate Continuity

Catalogue reduction techniques

1978 ◽  
Vol 48 ◽  
pp. 389-390 ◽  
Author(s):  
Chr. de Vegt
Keyword(s):  
Adjustment Process ◽  
Proper Motions ◽  
Measuring Process ◽  
Aerial Photogrammetry ◽  
Reduction Techniques ◽  
Observational Technique ◽  
The Common ◽  
Astrometric Data

AbstractReduction techniques as applied to astrometric data material tend to split up traditionally into at least two different classes according to the observational technique used, namely transit circle observations and photographic observations. Although it is not realized fully in practice at present, the application of a blockadjustment technique for all kind of catalogue reductions is suggested. The term blockadjustment shall denote in this context the common adjustment of the principal unknowns which are the positions, proper motions and certain reduction parameters modelling the systematic properties of the observational process. Especially for old epoch catalogue data we frequently meet the situation that no independent detailed information on the telescope properties and other instrumental parameters, describing for example the measuring process, is available from special calibration observations or measurements; therefore the adjustment process should be highly self-calibrating, that means: all necessary information has to be extracted from the catalogue data themselves. Successful applications of this concept have been made already in the field of aerial photogrammetry.

Renilla Bioluminescence: A Morphologic Approach to the Location of Photocytes

1974 ◽  
Vol 32 ◽  
pp. 208-209
Author(s):  
Ben O. Spurlock ◽  
Milton J. Cormier
Keyword(s):  
Excited State ◽  
Ground State ◽  
Molecular Basis ◽  
Light Emission ◽  
Chemical Basis ◽  
Electronic Excited State ◽  
Colonial Form ◽  
The Common ◽  
Eighteenth And Nineteenth Centuries ◽  
Catalyzed Oxidation

The phenomenon of bioluminescence has fascinated layman and scientist alike for many centuries. During the eighteenth and nineteenth centuries a number of observations were reported on the physiology of bioluminescence in Renilla, the common sea pansy. More recently biochemists have directed their attention to the molecular basis of luminosity in this colonial form. These studies have centered primarily on defining the chemical basis for bioluminescence and its control. It is now established that bioluminescence in Renilla arises due to the luciferase-catalyzed oxidation of luciferin. This results in the creation of a product (oxyluciferin) in an electronic excited state. The transition of oxyluciferin from its excited state to the ground state leads to light emission.

Mammalian Bone Marrow Acetylcholinesterase

1982 ◽  
Vol 40 ◽  
pp. 44-45
Author(s):  
Ezzatollah Keyhani
Keyword(s):  
Central Nervous System ◽  
Bone Marrow ◽  
Nervous System ◽  
Common Property ◽  
Extracellular Medium ◽  
The Central Nervous System ◽  
The Common ◽  
Mammalian Bone

Acetylcholinesterase (EC 3.1.1.7) (ACHE) has been localized at cholinergic junctions both in the central nervous system and at the periphery and it functions in neurotransmission. ACHE was also found in other tissues without involvement in neurotransmission, but exhibiting the common property of transporting water and ions. This communication describes intracellular ACHE in mammalian bone marrow and its secretion into the extracellular medium.

Correlation analysis of radiation damage

1978 ◽  
Vol 36 (1) ◽  
pp. 214-215
Author(s):  
R. Hegerl ◽  
A. Feltynowski ◽  
B. Grill
Keyword(s):  
Electron Microscopy ◽  
Independent Random Variables ◽  
Optical Parameters ◽  
Bright Field Image ◽  
Bright Field ◽  
Expectation Value ◽  
All Optical ◽  
Image Element ◽  
Stochastic Series ◽  
The Common

Till now correlation functions have been used in electron microscopy for two purposes: a) to find the common origin of two micrographs representing the same object, b) to check the optical parameters e. g. the focus. There is a third possibility of application, if all optical parameters are constant during a series of exposures. In this case all differences between the micrographs can only be caused by different noise distributions and by modifications of the object induced by radiation.Because of the electron noise, a discrete bright field image can be considered as a stochastic series Pm,where i denotes the number of the image and m (m = 1,.., M) the image element. Assuming a stable object, the expectation value of Pm would be Ηm for all images. The electron noise can be introduced by addition of stationary, mutual independent random variables nm with zero expectation and the variance. It is possible to treat the modifications of the object as a noise, too.

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