A NECESSARY CONDITION FOR THE ABSOLUTE CONTINUITY OF INVARIANT MEASURE OF CIRCLE MAPS WITH COUNTABLY INFINITE NUMBER OF BREAK POINTS

2017 ◽  
Vol 101 (3) ◽  
pp. 675-688
Author(s):  
Habibulla Akhadkulov ◽  
Azizan Bin Saaban ◽  
Mohd Salmi Md Noorani ◽  
Sokhobiddin Akhatkulov
Keyword(s):  
Invariant Measure ◽  
Infinite Number ◽  
Absolute Continuity ◽  
Necessary Condition ◽  
Circle Maps ◽  
Break Points ◽  
The Absolute ◽  
Countably Infinite

scholarly journals On Absolute Continuity of Conjugations between Circle Maps with Break Points

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Habibulla Akhadkulov ◽  
Mohd Salmi Md Noorani
Keyword(s):  
Absolute Continuity ◽  
Irrational Rotation ◽  
Piecewise Smooth ◽  
Necessary Condition ◽  
Circle Maps ◽  
Rotation Numbers ◽  
Break Points ◽  
The Absolute ◽  
Sufficient And Necessary Condition

LetT1andT2be piecewise smooth circle homeomorphisms with break points and identical irrational rotation numbers. We provide one sufficient and necessary condition for the absolute continuity of conjugation map betweenT1andT2.

Conjugation between circle maps with several break points

2015 ◽  
Vol 36 (8) ◽  
pp. 2351-2383 ◽  
Author(s):  
ABDELHAMID ADOUANI
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Function ◽  
Absolutely Continuous ◽  
Circle Maps ◽  
Almost Everywhere ◽  
Rotation Numbers ◽  
Break Points

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.

scholarly journals A Variant of the Necessary Condition for the Absolute Continuity of Symmetric Multivariate Mixture

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1505
Author(s):  
Evgeniy Anatolievich Savinov
Keyword(s):  
Absolute Continuity ◽  
Joint Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Necessary Condition ◽  
Conditional Distributions ◽  
One Dimensional ◽  
The Absolute

Sufficient conditions are given under which the absolute continuity of the joint distribution of conditionally independent random variables can be violated. It is shown that in the case of a dimension n>1 this occurs for a sufficiently large number of discontinuity points of one-dimensional conditional distributions.

Living Mirrors

2019 ◽  
Author(s):  
Ohad Nachtomy
Keyword(s):  
Infinite Number ◽  
Historical Background ◽  
The Absolute

This work presents Leibniz’s view of infinity and the central role it plays in his theory of living beings. Chapter 1 introduces Leibniz’s approach to infinity by presenting the central concepts he employs; chapter 2 presents the historical background through Leibniz’s encounters with Galileo and Descartes, exposing a tension between the notions of an infinite number and an infinite being; chapter 3 argues that Leibniz’s solution to this tension, developed through his encounter with Spinoza (ca. 1676), consists of distinguishing between a quantitative and a nonquantitative use of infinity, and an intermediate degree of infinity—a maximum in its kind, which sheds light on Leibniz’s use of infinity as a defining mark of living beings; chapter 4 examines the connection between infinity and unity; chapter 5 presents the development of Leibniz’s views on infinity and life; chapter 6 explores Leibniz’s distinction between artificial and natural machines; chapter 7 focuses on Leibniz’s image of a living mirror, contrasting it with Pascal’s image of a mite; chapter 8 argues that Leibniz understands creatures as infinite and limited, or as infinite in their own kind, in distinction from the absolute infinity of God; chapter 9 argues that Leibniz’s concept of a monad holds at every level of reality; chapter 10 compares Leibniz’s use of life and primitive force. The conclusion presents Leibniz’s program of infusing life into every aspect of nature as an attempt to re-enchant a view of nature left disenchanted by Descartes and Spinoza.

On the Absolute Continuity of Elliptic Measures

1986 ◽  
Vol 108 (5) ◽  
pp. 1119 ◽  
Author(s):  
Bjorn E. J. Dahlberg
Keyword(s):  
Absolute Continuity ◽  
The Absolute

Free-surface flows with two stagnation points

1996 ◽  
Vol 324 ◽  
pp. 393-406 ◽  
Author(s):  
J.-M. Vanden-Broeck ◽  
F. Dias
Keyword(s):  
Free Surface ◽  
Infinite Number ◽  
Wave Breaking ◽  
Free Surface Flows ◽  
Number Of Solutions ◽  
Surface Flows ◽  
Stagnation Points ◽  
Surface Profiles ◽  
Countably Infinite

Symmetric suction flows are computed. The flows are free-surface flows with two stagnation points. The configuration is related to the modelling of wave breaking at the bow of a ship. It is shown that there is a countably infinite number of solutions and that the free-surface profiles are characterized by waves.

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