The idea of using p-adic numbers in Turing machines and finite automata to describe random branching of the process of computation was recently introduced. In the last two years some advantages of ultrametric algorithms for finite automata and Turing machines were explored. In this paper advantages of ultrametric automata with one head versus multihead deterministic and nondeterministic automat...

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach’s Fixed Point Theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not involving any metrics. We demonstrate its applications to the metric, ultrametric and topological cases, and to ordered abelian groups and fields.

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but complexity of probabilistic automata and complexity of ultrametric automata can differ very much.

The analysis of objects living on ultrametric trees, in particular the block-diagonalization of 4?replica matrices M ; , is shown to be dramatically simpliied through the introduction of properly chosen operations on those objects. These are the Replica Fourier Transforms on ultrametric trees. Those transformations are deened and used in the present work.

We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $\sigma$-compact locally compact Hausdorff spaces. As its application, we an extension metrics, which states that if $X$ is space, $A$ closed subset $X$, and $d$ metric generates the same topology $A$, then there exists such $D$ $D|_{A^{2}}=d$. Moreover, retraction, ca...

We initiate and examine integer powers of the (possibly unbounded) diagonal operators on the so-called p-adic Hilbert space Eω (see [10], [11], and [3]). For that, we first give and recall the required background on author’s recent work related to the formalism of unbounded linear operators in the p-adic setting [5]. Next, we shall be dealing with integer powers of the diagonal operators, their...

We introduce a notion of ultrametric automata and Turing machines using p-adic numbers to describe random branching of the process of computation. These automata have properties similar to the properties of probabilistic automata but complexity of probabilistic automata and complexity of ultrametric automata can differ very much.

The analysis of objects living on ultrametric trees, in particular the block-diagonalization of 4−replica matrices M, is shown to be dramatically simplified through the introduction of properly chosen operations on those objects. These are the Replica Fourier Transforms on ultrametric trees. Those transformations are defined and used in the present work.

We study hierachical segmentation in the framework of edgeweighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical edgesegmentations. We end this paper by showing how the proposed framework allows to see constrained connectivity as a classical water...