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| 2. |
- Djinja, Domingos, et al.
(författare)
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Linear integral operators on Lp spaces representing polynomial covariance type commutation relations
- 2024
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Ingår i: Afrika Matematika. - 1012-9405 .- 2190-7668. ; 35:1
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A) by linear integral operators in Banach spaces Lp. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on Lp. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.
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| 4. |
- Djinja, Domingos, 1985-, et al.
(författare)
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Representations of polynomial covariance type commutation relations by linear integral operators on Lp over measure spaces
- 2022
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Ingår i: Stochastic Processes, Statistical Methods, and Engineering Mathematics. - : Springer Nature. - 9783031178191 - 9783031178207 ; , s. 59-95
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Konferensbidrag (refereegranskat)abstract
- Representations of polynomial covariance type commutation relations by linear integral operators on Lp over measures spaces are constructed. Conditions for such representations are described in terms of kernels of the corresponding integral operators. Representation by integral operators are studied both for general polynomial covariance commutation relations and for important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions. Examples of integral operators on Lp spaces representing the covariance commutation relations are constructed. Representations of commutation relations by integral operators with special classes of kernels such as separable kernels and convolution kernels are investigated.
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| 5. |
- Djinja, Domingos, 1985-, et al.
(författare)
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Representations of polynomial covariance type commutation relations by linear integral operators with general separable kernels in Lp spaces
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Representations of polynomial covariance type commutation relations by linear integral operators on Lp over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance type commutation relations are obtained in terms of their kernels. For important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions, these conditions on the kernels are specified in terms of the coefficients of the monomials and affine functions. By applying these conditions, examples of integral operators on Lp spaces, with separable kernels representing covariance commutation relations associated to monomials, are constructed for the kernels involving multi-parameter trigonometric functions, polynomials and Laurent polynomials on bounded intervals. Commutators of these operators are computed and exact conditions for commutativity of these operators in terms of the parameters are obtained.
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| 7. |
- Djinja, Domingos, 1985-, et al.
(författare)
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Some algebraic properties of representations of polynomial covariance type commutation relations
- 2023
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Ingår i: Algebra without Borders – Classical and Constructive Nonassociative Algebraic Structures - Foundations and Applications. - : Springer. - 9783031393341 ; , s. 379-417
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Bokkapitel (refereegranskat)abstract
- In this work conditions for additivity property of representations of polynomial covariance commutation relations are derived for operator algebras. Some other properties that this kind of representations fulfill are described. A reduction degree of the polynomial property of representations of this kind of commutation relations is presented for operator algebras. Moreover, representations of polynomial covariance commutation relations are derived for linear operators acting on the space of bounded real infinite sequences lp.
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| 8. |
- Tumwesigye, Alex Behakanira, 1982-, et al.
(författare)
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Commutants in crossed product algebras for piecewise constant functions on the real line
- 2020
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer. - 9783030418496 ; , s. 427-444
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Konferensbidrag (refereegranskat)abstract
- In this paper we consider commutants in crossed product algebras, for algebras of piece-wise constant functions on the real line acted on by the group of integers Z. The algebra of piece-wise constant functions does not separate points of the real line, and interplay of the action with separation properties of the points or subsets of the real line by the function algebra become essential for many properties of the crossed product algebras and their subalgebras. In this article, we deepen investigation of properties of this class of crossed product algebras and interplay with dynamics of the actions. We describe the commutants and changes in the commutants in the crossed products for the canonical generating commutative function subalgebras of the algebra of piece-wise constant functions with common jump points when arbitrary number of jump points are added or removed in general positions, that is when corresponding constant value set partitions of the real line change, and we give complete characterization of the set difference between commutants for the increasing sequence of subalgebras in crossed product algebras for algebras of functions that are constant on sets of a partition when partition is refined. © Springer Nature Switzerland AG 2020.
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| 9. |
- Tumwesigye, Alex Behakanira, 1982-
(författare)
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Dynamical Systems and Commutants in Non-Commutative Algebras
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 10. |
- Tumwesigye, Alex Behakanira, 1982-
(författare)
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On one-dimensional dynamical systems and commuting elements in non-commutative algebras
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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