Displaying similar documents to “Conservation Rules of Direct Sum Decomposition of Groups”

Definition and Properties of Direct Sum Decomposition of Groups1

Kazuhisa Nakasho, Hiroshi Yamazaki, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

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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...

Equivalent Expressions of Direct Sum Decomposition of Groups1

Kazuhisa Nakasho, Hiroyuki Okazaki, Hiroshi Yamazaki, Yasunari Shidama (2015)

Formalized Mathematics

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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...

Decompositions of local rigid ACD groups

Adolf Mader, Otto Mutzbauer (2001)

Colloquium Mathematicae

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We study direct decompositions of extensions of rigid completely decomposable groups by finite primary groups. These decompositions are unique and can be found by finite procedures. By passing to certain quotients the determination of the direct decompositions is made more efficient.

Componentwise and Cartesian decompositions of linear relations

S. Hassi, H. S. V. de Snoo, F. H. Szafraniec

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Let A be a, not necessarily closed, linear relation in a Hilbert space ℌ with a multivalued part mul A. An operator B in ℌ with ran B ⊥ mul A** is said to be an operator part of A when A = B +̂ ({0} × mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for an operator part are established...

The generalized equations of Riccati and their applications to the theory of linear differential equations

T. Iwiński

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CONTENTSIntroduction.............................................................................................................................. 31. Definition of the Riccati equation of the n-th order....................................................... 62. Theorems on the existence of solutions of R equations. Relations between the solutions of linear differential equations and the solutions of the corresponding R equations..................................................................................................

Butler groups splitting over a base element

Clorinda De Vivo, Claudia Metelli (2007)

Colloquium Mathematicae

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We characterize a particular kind of decomposition of a Butler group that is the general case for Butler B(1)-groups; and exhibit a decomposition of a B(2)-group which is not of that kind.

A Simple Proof of the Polar Decomposition Theorem

Paweł Wójcik (2017)

Annales Mathematicae Silesianae

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In this expository paper, we present a new and easier proof of the Polar Decomposition Theorem. Unlike in classical proofs, we do not use the square root of a positive matrix. The presented proof is accessible to a broad audience.