In the present paper the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states there is uncountably infinite number of maximizing quantum operations. Conversely, for any operation of described type there is uncountably infinite number of...

Toeplitz CAR flows are a class of E0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always e...

Let be a semigroup of words over an alphabet . Let consist of those elements of for which every prefix and suffix of belongs to . We show that is a free semigroup. Moreover, is called separative if also the complement is a semigroup. There are uncountably many separative semigroups over , if has at least two letters. We prove that if is separative, then every word has a unique minimum factoriza...

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the ima...

In this paper we study the number of minimal subvarieties of subvariety lattice. Since we can see that the subvariety lattice is dually isomorphic to the lattice of logics, the number of minimal subvarieties correspond to the number of maximal consistent logics. The maximal consistent logic over intuitionistic logic Int and FLew is only classical logic CL. For weaker logic number of minimal sub...

We classify C-orderable groups admitting only finitely many C-orderings. We show that if a C-orderable group has infinitely many C-orderings, then it actually has uncountably many C-orderings, and none of these is isolated in the space of C-orderings. As a relevant example, we carefully study the case of Baumslag-Solitar’s group B(1, 2). We show that B(1, 2) has four C-orderings, each of which ...

We give a complete description of the possible Hausdorff dimensions escaping sets for meromorphic functions with finite number singular values. More precisely, any given $d\in [0,2]$ we show that there exists such function which dimension set is equal to $d$. The main ingredient glue together suitable by using quasiconformal mappings. Moreover, are uncountably many quasiconformally equivalent h...