Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An introduction to hyperholomorphic spectral theories and fractional
  powers of vector operators   The aim of this paper is to give an overview of the spectral theories
associated with the notions of holomorphicity in dimension greater than one. A
first natural extension is the theory of several complex variables whose Cauchy
formula is used to define the holomorphic functional calculus for $n$-tuples of
operators $(A_1,...,A_n)$. A second way is to consider hyperholomorphic
functions of quaternionic or paravector variables. In this case, by the
Fueter-Sce-Qian mapping theorem, we have two different notions of
hyperholomorphic functions that are called slice hyperholomorphic functions and
monogenic functions. Slice hyperholomorphic functions generate the spectral
theory based on the $S$-spectrum while monogenic functions induce the spectral
theory based on the monogenic spectrum. There is also an interesting relation
between the two hyperholomorphic spectral theories via the $F$-functional
calculus. The two hyperholomorphic spectral theories have different and
complementary applications. Here we also discuss how to define the fractional
Fourier's law for nonhomogeneous materials, such definition is based on the
spectral theory on the $S$-spectrum.

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The categories of this paper are: comp-gas
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess
5
The categories of this paper are: gr-qc