Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: What is dimension?   This chapter explores the notion of "dimension" of a set. Various power laws
by which an Euclidean space can be characterized are used to define dimensions,
which then explore different aspects of the set. Also discussed are the
generalization to multifractals, and discrete and continuous scale invariance
with the emergence of complex dimensions. The idea of renormalization group
flow equations can be introduced in this framework, to show how the power laws
determined by dimensional analysis (engineering dimensions) get modified by
extra anomalous dimensions. As an example of the RG flow equation, the scaling
of conductance by disorder in the context of localization is used. A few
technicalities, including the connection between entropy and fractal dimension,
can be found in the appendices.

27
The categories of this paper are: comp-gas
2
The categories of this paper are: math
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin