For each integer [Formula: see text], we define a category whose objects are discriminant algebra functors in rank namely, choices of how to attach functorially rank-[Formula: text] quadratic with the same discriminant. We show that defined by Loos, Rost, and present authors all isomorphic this category, prove furthermore ranks unique up isomorphism.

Summary . In this article we further develop field theory in Mizar [1], [2]: prove existence and uniqueness of splitting fields. We define the a polynomial p ∈ F [ X ] as smallest extension , which splits into linear factors. From follows, that for E have = ( A ) where is set ’s roots. Splitting fields are unique, however, only up to isomorphisms; be more precise -isomorphims i.e. isomorphisms ...

one of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. in this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. we also devote to characterizing bijective morphisms on quantum effects leaving the witness...

A notion of simulation of one datatype by another is deened as a constructive preorder. A calculus of datatype simulation is then developed by formulating constructive versions of least-xed-point theorems in lattice theory. The calculus is applied to the construction of several isomorphisms between classes of datatypes. In particular constructive adaptations of theorems in lattice theory about ...

We present a theory of isomorphisms between typed sets in Isabelle/HOL. Those isomorphisms can serve to link a shallow embedding with a theory that defines certain concepts directly in HOL. Thus, it becomes possible to use the advantage of a shallow embedding that it allows for efficient proofs about concrete terms of the embedded formalism with the advantage of a deeper theory that establishes...

It is shown, that ordinary isomorphisms (associativity and commuta-tivity of tensor, isomorphisms for tensor unit and currying) provide a complete axiom system for the isomorphism of types in Symmetric Monoidal Closed categories. This gives also a decision algorithm. The problem originally arises from computer science, as the isomorphism of types in SMC categories corresponds to the isomorphism...

it is shown that every  almost linear bijection $h : arightarrow b$ of a unital $c^*$-algebra $a$ onto a unital$c^*$-algebra $b$ is a $c^*$-algebra isomorphism when $h(3^n u y) = h(3^n u) h(y)$ for allunitaries  $u in a$, all $y in a$, and all $nin mathbb z$, andthat almost linear continuous bijection $h : a rightarrow b$ of aunital $c^*$-algebra $a$ of real rank zero onto a unital$c^*$-algebra...