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Conservation Rules of Direct Sum Decomposition of Groups

100%
Nakasho K., Yamazaki H., Okazaki H., Shidama Y.
Formalized Mathematics
|
2016
|
tom 24
|
nr 1
81-94
EN
In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization.
2
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Equivalent Expressions of Direct Sum Decomposition of Groups1

100%
Nakasho K., Okazaki H., Yamazaki H., Shidama Y.
Formalized Mathematics
|
2015
|
tom 23
|
nr 1
67-73
EN
In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization.
3
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Definition and Properties of Direct Sum Decomposition of Groups1

100%
Nakasho K., Yamazaki H., Okazaki H., Shidama Y.
Formalized Mathematics
|
2015
|
tom 23
|
nr 1
15-27
EN
In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.
4
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Internal Symmetries and Additional Quantum Numbers for Nanoparticles

70%
Yarzhemsky V. G.
Nanoscale Systems: Mathematical Modeling, Theory and Applications
|
2013
|
tom 2
96-106
EN
Wavefunctions of symmetrical nanoparticles are considered making use of induced representation method. It is shown that when, at the same total symmetry, the order of local symmetry group decreases, additional quantum numbers are required for complete labelling of electron states. It is shown that the labels of irreducible representations of intermediate subgroups can be used for complete classification of states in the case of repeating IRs in symmetry adapted linear combinations. The intermediate symmetry approach is extended to singlet and triplet two-electron states making use of Mackey theorem on symmetrized squares of induced representations.
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