Elementary volume and measurability properties of additive functions

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Tamar Kasrashvili ◽  
Aleks Kirtadze
Keyword(s):  
Functional Equation ◽  
Invariant Measure ◽  
Real Line ◽  
Elementary Volume ◽  
Additive Functions ◽  
Translation Invariant ◽  
One Dimensional ◽  
The One ◽  
Dimensional Lebesgue Measure ◽  
Theoretical Standpoint

AbstractThe paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

A coupling of finite particle systems

1993 ◽  
Vol 30 (01) ◽  
pp. 258-262 ◽  
Author(s):  
T. S. Mountford
Keyword(s):  
Invariant Measure ◽  
Branching Process ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Initial Configuration ◽  
Limit Measure ◽  
One Dimensional ◽  
Interacting Particle ◽  
The One ◽  
Finite Particle

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

The Unruh effect without space–time

2020 ◽  
Vol 17 (04) ◽  
pp. 2050057
Author(s):  
Michele Arzano
Keyword(s):  
Real Line ◽  
Thermal Effects ◽  
Thermal State ◽  
Space Time ◽  
Unruh Effect ◽  
Affine Transformations ◽  
One Dimensional ◽  
The Real ◽  
Simple Quantum ◽  
The One

We show how the characteristic thermal effects found for a quantum field in space–time geometries admitting a causal horizon can be found in a simple quantum system living on the real line. The analysis we present is essentially group theoretic in nature: a thermal state emerges naturally when comparing representations of the group of affine transformations of the real line. The freedom in the choice of different notions of translation generators is the key to the one-dimensional Unruh effect we describe.

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

A coupling of finite particle systems

1993 ◽  
Vol 30 (1) ◽  
pp. 258-262 ◽  
Author(s):  
T. S. Mountford
Keyword(s):  
Invariant Measure ◽  
Branching Process ◽  
Interacting Particle Systems ◽  
Particle Systems ◽  
Initial Configuration ◽  
Limit Measure ◽  
One Dimensional ◽  
Interacting Particle ◽  
The One ◽  
Finite Particle

We show that for a large class of one-dimensional interacting particle systems, with a finite initial configuration, any limit measure , for a sequence of times tending to infinity, must be invariant. This result is used to show that the one-dimensional biased annihilating branching process with parameter > 1/3 converges in distribution to the upper invariant measure provided its initial configuration is almost surely finite and non-null.

scholarly journals Reconstruction of the One-Dimensional Lebesgue Measure

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Noboru Endou
Keyword(s):  
Lebesgue Measure ◽  
Real Space ◽  
Outer Measure ◽  
Finite Additivity ◽  
One Dimensional ◽  
The Real ◽  
Measurable Set ◽  
The One ◽  
System 1 ◽  
Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

On absolutely nonmeasurable homomorphisms of commutative groups

2013 ◽  
Vol 50 (3) ◽  
pp. 287-295
Author(s):  
A. Kharazishvili
Keyword(s):  
Real Line ◽  
Quotient Group ◽  
Torsion Subgroup ◽  
Dimensional Torus ◽  
One Dimensional ◽  
The Real ◽  
The One

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.

scholarly journals The one-dimensional heat equation in the Alexiewicz norm

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Initial Data ◽  
Heat Equation ◽  
Real Line ◽  
Distributional Derivative ◽  
One Dimensional ◽  
The Real ◽  
Space Of Distributions ◽  
The One

AbstractA distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock–Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let Θ

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