A representational scheme under which the ranking between represented dissimilarities is iso-morphic to the ranking between the corresponding shape dissimilarities can support perfect shape classiication, because it preserves the clustering of shapes according to the natural kinds prevailing in the external world. We discuss the computational requirements of rank-preserving representation, and ...

We prove that PSL2(Z[ 1 p ]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL2(Z[ 1 p ]) and PSL2(Z[ 1 q ]) are not quasi-isometric unless p = q, and, independent of p, the quasi-isometry group of PSL2(Z[ 1 p ]) is PSL2(Q). In addition, we characterize PSL2(Z[ 1 p ]) uniquely among all finitely generated groups by ...

In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm solving the problem is found. A result more or less supporting the conjecture of decidability of the general problem is given.

Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be word composed of $k$ pairwise distinct, concatenated words equal length. Berger Defant conjecture that for any sufficiently well-behaved aperiodic morphic $w$, there exists constant $c$ such index $i$, with block length at most $ck$ starts the $i$th position $w$. They prove their in case binary words, we extend result alphabets arbi...

Let b ≥ 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.

The graph isomorphism problem is to determine whether two given graphs are iso-morphic or not. In this paper, we present a new graph invariant, called the probability propagation matrix. By means of this graph invariant, we present a heuristic algorithm for the problem. The algorithm is easy to implement and highly parallelizable.