scholarly journals Equivalent Expressions of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 67-73
Author(s):  
Kazuhisa Nakasho ◽  
Hiroyuki Okazaki ◽  
Hiroshi Yamazaki ◽  
Yasunari Shidama
Keyword(s):  
Equivalent Form ◽  
Normal Subgroups ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Equivalent Expressions

Summary 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.

scholarly journals Definition and Properties of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 15-27
Author(s):  
Kazuhisa Nakasho ◽  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Equivalent Form ◽  
Fourth Section ◽  
Product Group ◽  
The Third ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Direct Product Group ◽  
Product Map

Summary 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.

scholarly journals On Direct Sum Decomposition of Integers and Y. Ito's Conjecture

1998 ◽  
Vol 21 (2) ◽  
pp. 433-440 ◽  
Author(s):  
Masahito DATEYAMA ◽  
Teturo KAMAE
Keyword(s):  
Direct Sum ◽  
Direct Sum Decomposition

scholarly journals Implicit Function Theorem. Part II

2019 ◽  
Vol 27 (2) ◽  
pp. 117-131
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Implicit Function Theorem ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Implicit Function ◽  
Continuous Linear ◽  
Lipschitz Continuous ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Continuous Linear Operators

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

scholarly journals The h-vector of a Gorenstein codimension three domain

1995 ◽  
Vol 138 ◽  
pp. 113-140 ◽  
Author(s):  
E. De Negri ◽  
G. Valla
Keyword(s):  
Hilbert Function ◽  
Additive Group ◽  
Dimensional Vector ◽  
Infinite Field ◽  
Homogeneous Ideal ◽  
Dimensional Vector Space ◽  
Direct Sum ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Direct Sum Decomposition

Let k be an infinite field and A a standard G-algebra. This means that there exists a positive integer n such that A = R/I where R is the polynomial ring R := k[Xv …, Xn] and I is an homogeneous ideal of R. Thus the additive group of A has a direct sum decomposition A = ⊕ At where AiAj ⊆ Ai+j. Hence, for every t ≥ 0, At is a finite-dimensional vector space over k. The Hilbert Function of A is defined by

Nilpotent Orbit Varieties and the Atomic Decomposition of the Q-Kostka Polynomials

1998 ◽  
Vol 50 (3) ◽  
pp. 525-537 ◽  
Author(s):  
William Brockman ◽  
Mark Haiman
Keyword(s):  
Atomic Decomposition ◽  
Jordan Block ◽  
Geometric Interpretation ◽  
Nilpotent Orbit ◽  
Cohomology Rings ◽  
Direct Sum ◽  
Kostka Polynomials ◽  
Orbit Closures ◽  
Direct Sum Decomposition ◽  
Coordinate Rings

AbstractWe study the coordinate rings of scheme-theoretic intersections of nilpotent orbit closures with the diagonal matrices. Here μ′ gives the Jordan block structure of the nilpotent matrix. de Concini and Procesi [5] proved a conjecture of Kraft [12] that these rings are isomorphic to the cohomology rings of the varieties constructed by Springer [22, 23]. The famous q-Kostka polynomial is the Hilbert series for the multiplicity of the irreducible symmetric group representation indexed by λ in the ring . Lascoux and Schützenberger [15, 13] gave combinatorially a decomposition of as a sum of “atomic” polynomials with non-negative integer coefficients, and Lascoux proposed a corresponding decomposition in the cohomology model.Our work provides a geometric interpretation of the atomic decomposition. The Frobenius-splitting results of Mehta and van der Kallen [19] imply a direct-sum decomposition of the ideals of nilpotent orbit closures, arising from the inclusions of the corresponding sets. We carry out the restriction to the diagonal using a recent theorem of Broer [3]. This gives a direct-sum decomposition of the ideals yielding the , and a new proof of the atomic decomposition of the q-Kostka polynomials.

scholarly journals Complex structures on the complexification of a real Lie algebra

2018 ◽  
Vol 5 (1) ◽  
pp. 150-157
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Algebra ◽  
Complex Structure ◽  
Complex Structures ◽  
Direct Sum ◽  
Direct Sum Decomposition

AbstractLet g = a+b be a Lie algebra with a direct sum decomposition such that a and b are Lie subalgebras. Then, we can construct an integrable complex structure J̃ on h = ℝ(gℂ) from the decomposition, where ℝ(gℂ) is a real Lie algebra obtained from gℂby the scalar restriction. Conversely, let J̃ be an integrable complex structure on h = ℝ(gℂ). Then, we have a direct sum decomposition g = a + b such that a and b are Lie subalgebras. We also investigate relations between the decomposition g = a + b and dim Hs.t∂̄J̃ (hℂ).

Sign in / Sign up

Export Citation Format

Share Document