Linearly Induced Mappings between Cones of Quadratic Forms
This paper deals with mappings between cones of positive quadratic forms which are induced by linear mappings between the underling vector spaces, i.e., the spaces which are the domains of the forms. Three fundamental results are proved, two of which were previously announced by the second author. The first result states that there is an inclusion-reversing one-to-one correspondence between the lattice of subspaces of a given space and the lattice of faces of the cone of quadratic forms on that space. The second result states that all cone-isomorphisms between cones of quadratic forms are induced by linear isomorphisms between the underlying spaces. The third result states that a given cone-linear mapping F between cones of quadratic forms is induced by some linear mapping between the underlying spaces if and only if both F and its transpose preserve faces.
Order-isomorphisms between cones of continuous functions
SummaryThis paper studies the structure of order-isomorphisms between the cones of strictly-positive-valued elements of subspaces of two spaces of continuous functions; a complete description is obtained in some cases. Conditions are also found that ensure that such an order-isomorphism is linear provided it is homogeneous along at least one ray.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.