for a symmetric group $g:=sym(n)$ and a conjugacy class $mathcal{x}$ of involutions in $g$, it is known that if the class of involutions does not have a unique fixed point, then - with a few small exceptions - given two elements $a,x in mathcal{x}$, either $angbrac{a,x}$ is isomorphic to the dihedral group $d_{8}$, or there is a further element $y in mathcal{x}$ such that $angbrac{a,y} cong ang...

A Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the ...

A simple and effective method based on Haar wavelets is proposed for the solution of Pocklington's integral equation. The properties of Haar wavelets are first given. These wavelets are utilized to reduce the solution of Pocklington's integral equation to the solution of algebraic equations. In order to save memory and computation time, we apply a threshold procedure to obtain sparse algebraic ...

In this paper we study the problem of template matching in signals, specifically focusing on images. First we present the solution to the case where the template is translated in the image, then discuss how to locate a template if it appears scaled in the image. The tools used for this scaling algorithm are wavelets, specifically the Haar wavelet. Wavelets naturally encode scaling information, ...

In applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. Thes...

Let A be a C*-algebra with an identity. Consider the completed tensor product A®A of A with itself with respect to the minimal or the maximal C*-tensor product norm. Assume that A: A —>A®A is a non-zero •-homomorphism such that (A ® t)A = (i ® A)A where / is the identity map. Then A is called a comultiplication on A . The pair (A, A) can be thought of as a 'compact quantum semi-group'. A left i...

Excessive number of Haar-like features and the complex threshold calculation of covariance matrix feature are two key issues of Adaboost face detection. In this paper, an efficient feature named covariance feature is proposed. The novel method divides the face image into several regions and it calculate covariance feature of any two regions. Then optimal weak classifiers will be picked out by A...

The complicated moments of the traces of unitary and orthogonal Haar distributed random matrices are studied. The exact formulas for different values of moment and matrix orders are obtained.