Transformations preserving normality and Wishart-ness

Some Observations on Dirac Measure-Preserving Transformations and their Results

Dirac measure is an important measure in many related branches to mathematics. The current paper characterizes measure-preserving transformations between two Dirac measure spaces or a Dirac measure space and a probability measure space. Also, it studies isomorphic Dirac measure spaces, equivalence Dirac measure algebras, and conjugate of Dirac measure spaces. The equivalence classes of a Dirac ...

Arc- and Tree-preserving Transformations

1. Introduction. In an earlier paper(2) by one of us, referred to hereafter as A.P.T., arc-preserving transformations were defined and studied in connection with an irreducibility condition on the transformation. It was shown, for example, that if A and B are compact locally connected metric continua which are cyclic (that is, without cut points) any single valued continuous arc-preserving and ...

Sufficmrcy and Model-preserving Transformations

Measure Preserving Transformations

This gives us a new probability measure on (Ω,F), so we may define expectations with respect to this conditioned probability measure. Thus for F measurable Y : Ω → R we define the conditional expectation E[Y | X = x] by taking the expectation of Y with respect to the measure (1.1). Consider now how to generalize the idea of conditional probability to the case when P (X = x) = 0. We wish to do t...

Union-preserving Transformations of Space

1. General statement. Soph us Lie studied transformations from lineal-elements into lineal-elements, and also transformations from surface-elements into surface-elements of space. The contact group is obtained by requiring all unions to be converted into unions. Lie's fundamental theorems may be stated as follows. All the contact linealelement transformations form the group of extended point tr...