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  • 1.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ssembatya, Vincent
    Makerere University.
    Tumwesigye, Alex
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Crossed Product Algebras for Piece-Wise Constant Functions2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016Chapter in book (Refereed)
    Abstract [en]

    In this paper we consider algebras of functions that are constant on the sets of a partition. We describe the crossed product algebras of the mentioned algebras with Z. We show that the function algebra is isomorphic to the algebra of all functions on some set. We also describe the commutant of the function algebra and finish by giving an example of piece-wise constant functions on a real line.

  • 2.
    Richter, Johan
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Tumwesigye, Alex
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Commutants in Crossed Product Algebras for Piece-Wise Constant Functions2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and OptimizationEditors / [ed] Sergei Silvestrov; Milica Rančić, Springer, 2016, p. 95-108Chapter in book (Refereed)
    Abstract [en]

    In this paper we consider crossed product algebras of algebras of piecewiseconstant functions on the real line with Z. For an increasing sequence of algebras (in which case the commutants form a decreasing sequence), we describe the set difference between the corresponding commutants.

  • 3.
    Tumwesigye, Alex Behakanira
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Dynamical Systems and Commutants in Non-Commutative Algebras2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis work is about commutativity which is a very important topic in Mathematics, Physics, Engineering and many other fields. In Mathematics, it is well known that matrix multiplication (or composition of linear operators on a finite dimensional vector space) is not always commutative. Commuting matrices or more general linear or non-linear operators play an essential role in Mathematics and its applications in Physics and Engineering. Many important relations in Mathematics, Physics and Engineering are represented by operators satisfying a number of commutation relations. Such commutation relations are key in areas such as representation theory, dynamical systems, spectral theory, quantum mechanics, wavelet analysis and many others.

    In Chapter 2 of this thesis we treat commutativity of monomials of operators satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain onedimensional dynamical systems.

    In Chapter 3, we treat the crossed product algebra for the algebra of piecewise constant functions on given set and describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra.

    In Chapters 4 and 5, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non-decreasing sequence of algebras.

    In Chapter 6 we give a description of the centralizer of the coefficient algebra in the Ore extension of the algebra of functions on a countable set with finite support.

  • 4.
    Tumwesigye, Alex Behakanira
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    On one-dimensional dynamical systems and commuting elements in non-commutative algebras2016Licentiate thesis, monograph (Other academic)
    Abstract [en]

    This thesis work is about commutativity which is a very important topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the order of "operation" of these processes does not matter. A typical example of two processes in real life that are not commutative is the process of opening the door and the process of going through the door. In mathematics, it is well known that matrix multiplication is not always commutative. Commutating operators play an essential role in mathematics, physics engineering and many other fields. A typical example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged. Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations.

    In chapter two of this thesis we treat commutativity of monomials of operatos satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. In chapter three, we treat the crossed product algebra for the algebra of piecewise constant functions on given set, describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. In chapter four, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras.

  • 5.
    Tumwesigye, Alex Behakanira
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Ore extensions of function algebras2018Conference paper (Other (popular science, discussion, etc.))
  • 6.
    Tumwesigye, Alex
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Richter, Johan
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    Commutants in crossed products for algebras of piecewise constant functions on the real lineManuscript (preprint) (Other (popular science, discussion, etc.))
  • 7.
    Tumwesigye, Alex
    et al.
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Makerere University.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    On monomial commutativity of operators satisfying commutation relations and periodic points for one-dimensional dynamical systems2014In: 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014 Conference date: 15–18 July 2014 Location: Narvik, Norway ISBN: 978-0-7354-1276-7 Editor: Seenith Sivasundaram Volume number: 1637 Published: 10 december 2014 / [ed] Seenith Sivasundaram, American Institute of Physics (AIP), 2014, p. 1110-1119Conference paper (Refereed)
    Abstract [en]

    T. Persson and S. D. Sivestrov investigated representations of operators satisfying the relation XX* = F(X*X) in connection with periodic points and orbits of the map F. In particular they derived commutativity conditions for two monomials in operators A and B on a Hilbert space satisfying the relation AB = BF(A). In this article we shall apply their results to special one-dimensional dynamical systems and and give an explicit description of the interplay between periodic orbits of one-dimensional piecewise polynomial maps and commutativity of monomials for special operators A and B. Furthermore we shall apply our results to derive conditions on β for the special case when F β is the β–shift dynamical system.

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