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1

Symbolic integration with respect to the Haar measure on the unitary groups

Puchała Z., Miszczak J. A.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2017

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Vol. 65, nr 1

21--27

EN

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

2

Center of the algebra of functions on the quantum group SUq(2) and related topics

Krajczok J., Sołtan P.

Commentationes Mathematicae

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2016

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Vol 56, No. 2

251--272

EN

The center of the algebra of continuous functions on the quantum group SUq(2) is determined as well as centers of other related algebras. Several other results concerning this quantum group are given with direct proofs based on concrete realization of these algebras as algebras of operators on a Hilbert space.

3

Construction of a Fourier transform based on Haar measure in structure (Rn, +, T)

Żukowicz M

Zeszyty Naukowe / Akademia Morska w Szczecinie

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2014

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nr 39 (111)

181--184

EN

An important aspects of learning are the theoretical elements of mathematics and the methods of obtaining the function or designbecause it allows for further development of the issues and the ability to apply this problem in practice. The Fourier transform is very useful in science and technology. The article shows how and in what structures is possible to use the Fourier transform in the set Rn. The article describes the project of compact-open topology, which is essential to the construction of the Haar measure and integral. It was also described the concept of the nature of a locally compact topological group, because a group of characters is also essential to the construction of the Fourier transform and inverse Fourier transform. It has been proven that a group of characters is a locally compact topological group, which allows the introduction of the Haar measure. It has given an example application of the theory of Haar integral when calculating the Fourier transform for some of the electronics.

4

On the factorization of the Haar measure on finite Coxeter groups

Urban R.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 1

141--148

EN

Let W be a finite Coxeter group and let λW be the Haar measure on W; i.e., λW(ω) = |W|−1 for every ω ∈ W: We prove that there exist a symmetric set T ̸= W of generators of W consisting of elements of order not greater than 2 and a finite set of probability measures {μ1..., μk} with their supports in T such that their convolution product μ1 ∗ ...∗ μk = λW:

5

Finite partitions of topological groups into congruent thick subsets

Weber H., Zoli E.

Journal of Applied Analysis

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2008

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Vol. 14, nr 1

1-12

EN

We establish fairly general sufficient conditions for a locally compact group (a Baire topological group) to admit partitions into finitely many congruent μ-thick (everywhere of second category) subset.

6

Note on the factorization of the Haar measure on finite Coxeter groups

Urban R.

Probability and Mathematical Statistics

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2004

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Vol. 24, Fasc. 1

173--180

EN

In this note we show that for the finite Coxeter groups of types An, Bn, Dn, F4, G2 and I2 (m) it is possible to choose an appropriate set S of generators of order not greater than 2 and a finite set of probability measures {μ1,…, μk} with their supports in S such that μ1∗…∗ μk = λ, where λ (g) = |G|−1 for every g ∈ G.