Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Applications of Automata and Graphs: Labeling-Operators in Hilbert Space
  I   We show that certain representations of graphs by operators on Hilbert space
have uses in signal processing and in symbolic dynamics. Our main result is
that graphs built on automata have fractal characteristics. We make this
precise with the use of Representation Theory and of Spectral Theory of a
certain family of Hecke operators. Let G be a directed graph. We begin by
building the graph groupoid G induced by G, and representations of G. Our main
application is to the groupoids defined from automata. By assigning weights to
the edges of a fixed graph G, we give conditions for G to acquire fractal-like
properties, and hence we can have fractaloids or G-fractals. Our standing
assumption on G is that it is locally finite and connected, and our labeling of
G is determined by the "out-degrees of vertices". From our labeling, we arrive
at a family of Hecke-type operators whose spectrum is computed. As
applications, we are able to build representations by operators on Hilbert
spaces (including the Hecke operators); and we further show that automata built
on a finite alphabet generate fractaloids. Our Hecke-type operators, or
labeling operators, come from an amalgamated free probability construction, and
we compute the corresponding amalgamated free moments. We show that the free
moments are completely determined by certain scalar-valued functions.

18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat