Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The four-group Z_2 x Z_2 as a discrete invariance group of effective
  neutrino mass matrix   Two sets of four 3x3 matrices 1^(3), varphi_1, varphi_2, varphi_3 and 1^(3),
mu_1, mu_2, mu_3 are constructed, forming two unitarily isomorphic reducible
representations 3 of the group Z_2 x Z_2 called often the four-group. They are
related to each other through the effective neutrino mixing matrix U with
s_{13} = 0, and generate four discrete transformations of flavor and mass
active neutrinos, respectively. If and only if s_{13} = 0, the generic form of
effective neutrino mass matrix M becomes invariant under the subgroup Z_2 of
Z_2 x Z_2 represented by the matrices 1^(3) and varphi_3. In the approximation
of m_1 = m_2, the matrix M becomes invariant under the whole Z_2 x Z_2
represented by the matrices 1^(3), varphi_1, varphi_2, varphi_3. The effective
neutrino mixing matrix U with s_{13} = 0 is always invariant under the whole
Z_2 x Z_2 represented in two ways, by the matrices 1^(3), varphi_1, varphi_2,
varphi_3 and 1^(3), mu_1, mu_2, mu_3.

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The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat