Browsing by Subject "Measure theory."

Now showing 1 - 6 of 6

Open Access
Approximation of equilibrium measures by discrete measures
(2012) Alpan, Gökalp
Basic notions of potential analysis are given. Equilibrium measures can be approximated by discrete measures by means of Fekete points and Leja sequences. We give the sets for which exact locations of Fekete points and Leja sequences are known. An open problem about the location of Fekete points for a Cantor-type set K(γ) is presented.
Open Access
Conditions for uniqueness of limit Gibbs states
(1998) Şahin, Mehmet Arafat
In this work we studied the problem of phase transitions in one-dirnensional models with unique ground state. A model ha\dng two spins, one ground state and exhibiting phase transition is constructed.
Open Access
Gibbs measures and phase transitions in one-dimensional models
(2000) Mallak, Saed
In this thesis we study the problem of limit Gibbs measures in one-dimensional models. VVe investigate uniqueness conditions for the limit Gibbs measures for one-dimensional models. VVe construct a one-dimensional model disproving a uniqueness conjecture formulated before for one-dimensional models. It turns out that this conjecture is correct under some natural regularity conditions. VVe also apply the uniqueness theorem to some one-dimensional models.
Open Access
Gibbs measures and phase transitions in various one-dimensional models
(2013) Şensoy, Ahmet
In the thesis, limiting Gibbs measures of some one dimensional models are investigated and various criterions for the uniqueness of limiting Gibbs states are considered. The criterion for models with unique ground state formulated in terms of percolation theory is presented and some applications of this criterion are discussed. A one-dimensional long range Widom-Rowlinson model with periodic and biased particle activities is explored. It is shown that if the spin interactions are sufficiently large versus particle activities then the Widom-Rowlinson model does not exhibit a phase transition at low temperatures. Finally, an interdisciplinary approach is followed. A financial application of the theory of phase transition is considered by applying the Ising model to understand the role of herd behavior on stock market crashes. Accordingly, model suggests a criteria to detect the existence of herd behavior in financial markets under certain assumptions.
Open Access
Lebesgue-radon-nikodym decompositions for operator valued completely positice maps
(2014) Danış, Bekir
We discuss the notion of Radon-Nikodym derivatives for operator valued completely positive maps on C*-algebras, first introduced by Arveson [1969], and the notion of absolute continuity for completely positive maps, previously introduced by Parthasarathy [1996]. We begin with the definition and basic properties of positive and complete positive maps and we study the Stinespring dilation theorem which plays an essential role in the theory of Radon-Nikodym derivatives for completely positive maps, based on Poulsen [2002]. Then, the Radon-Nikodym derivative definition and basic properties belonging to Arveson is recorded and finally, we study the Lebesgue type decompositions defined by Parthasarathy in the light of the article Gheondea and Kavruk [2009].
Open Access
A measure disintegration approach to spectral multiplicity for normal operators
(2012) Ay, Serdar
In this thesis we studied the notion of direct integral Hilbert spaces, first introduced by J. von Neumann, and the closely related notion of decomposable operators, as defined in Kadison and Ringrose [1997] and Abrahamse and Kriete [1973]. Examples which show that some of the most familiar spaces in analysis are direct integral Hilbert spaces are presented in detail. Then we give a careful treatment of the notion of disintegration of a probability measure on a locally compact separable metric space, and using the machinery we obtain, a proof of the Spectral Multiplicity Theorem for Normal Operators employing the notion of disintegration of measures is given, based on Abrahamse and Kriete [1973], Arveson [1976], Arveson [2002]. In Chapter 5 the notion of essential preimage is presented in the sense of the article Abrahamse and Kriete [1973], and its relation with the spectral multiplicity function is discussed.