By Ishiguro M., Sakamoto Y.
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Extra resources for A Bayesian approach to binary response curve estimation
2 the choice of the side factors X and Y is somewhat arbitrary in the presence of zero divisors. 8 shows that it is possible to make that choice in a continuous fashion. Before we go into the Rees–Suschkewitsch structure theory for locally compact semigroups with 0, let us look at a specific example in some detail. 2. Let S be the set of real 2 2 matrices of rank Ä 1. S is a locally compact second-countable semigroup with 0 under ordinary matrix multiplication. The zero is not an isolated point of S .
Given a left ideal I , define the family of sets F Á fJ jJ a closed left ideal of S; J I g. 16(i)] and a left ideal, hence a member of F . The usual inclusion relation is a partial order on F . Furthermore, any linearly ordered subfamily of F has a minimal element, since sets in F are compact. By Zorn’s lemma, there exists at least one minimal element (with respect to inclusion) in F . Call this minimal closed left ideal I0 . It remains to prove that I0 is a minimal left ideal of S . Suppose that I1 I0 , I1 ideal of S .
Needless to say, there may be many different descriptions, each useful for dealing with particular problems. The second part of Sect. 6 is devoted to the theory of nonnegative matrices. In this special case, more concrete results are obtained. , always means that all the entries of the matrix (vector) are 0. Let Md be the set of d d real matrices. Under ordinary matrix multiplication, 2 this set is a semigroup. If we use the euclidean distance on
A Bayesian approach to binary response curve estimation by Ishiguro M., Sakamoto Y.