If C ≃ 2 is the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from C into [0, 1] sends Haar measure to Lebesgue measure on the interval. In fact, C admits many distinct topological group structures. In this note, we show that the Haar measures induced by these distinct group structures all share this property. We prove this by showin...

This paper develops an on-line identification algorithm to estimate system parameters of non-linear continuous time systems. The algorithm, based on Haar wavelet expansions, uses the observed input-output data to estimate unknown system parameters. A recursive formula augmented with indirect matrix-inversion schemes is proposed to substantially reduce computation requisites of an existed offlin...

In a previous paper [1] an Euler angle parameterization for SU(4) was given. Here we present the derivation of a generalized Euler angle parameterization for SU(N). The formula for the calculation of the Haar measure for SU(N) as well as its relation to Marinov’s volume formula for SU(N) [2,3] will also be derived. As an example of this parameterization’s usefulness, the density matrix paramete...

We propose a superfast discrete Haar wavelet transform (SFHWT) as well as its inverse, using the QTT representation for the Haar transform matrices and input-output vectors. Though the Haar matrix itself does not have a low QTT-rank approximation, we show that factor matrices used at each step of the traditional multilevel Haar wavelet transform algorithm have explicit QTT representations of lo...

In this paper, an operational matrix of integration based on Haar wavelets (HW) is introduced, and a procedure for applying the matrix to solve space and time fractional telegraph equations is formulated. The space and time fractional derivatives are considered in the Caputo sense. The accuracy and effectiveness of the proposed method is demonstrated by the five test problems. Approximate solut...

Recall that a locally compact group G is called unimodular if the left Haar measure on G is equal to the right one. It is proved in this paper that G is unimodular iff it is approximable by finite quasigroups (Latin squares).

Take a generic subgroup G, endowed with its Haar measure, from U(n,K), the unitary group of dimension n over the field K of real, complex or quaternion numbers. We give some equalities in law for Z := det(Id − G), G ∈ G : under some general conditions, Z can be decomposed as a product of independent random variables, whose laws are explicitly known (Section 2). Consequently G, endowed with a ge...

Independent Haar-unitary random matrices and independent Haar-orthogonal are known to be asymptotically liberating ensembles, they give rise asymptotic free independence when used for conjugation of constant matrices. G. Anderson B. Farrel showed that a certain family discrete unitary can actually the same end. In this paper, we investigate fluctuation moments higher-order induced on by with en...

The aim of this project is to analyze and analytical solution of different optimal control problems are difficult to obtain because they involve differential equations with single or multiple boundary conditions. In this dissertation, numerical methods using Haar Wavelet a represented to overcome this difficulty. The method reduces the differential equations into a set of line a matrix algebrai...

In a previous paper [1] an Euler angle parameterization for SU(4) was given. Here we present the derivation of a generalized Euler angle parameterization for SU(N). The formula for the calculation of the Haar measure for SU(N) as well as its relation to Marinov’s volume formula for SU(N) [2,3] will also be derived. As an example of this parameterization’s usefulness, the density matrix paramete...