For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^＊; A is a hyper-generalized n. idempotent if A^n = A^＋. The set of all n-idempotents, all gen...For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^＊; A is a hyper-generalized n. idempotent if A^n = A^＋. The set of all n-idempotents, all generalized n-idempotents and all hyper-generalized n-idempotents are denoted by In（H）, gIn（H） and HgIn（H）, respectively. In this note, we obtain a chain of proper inclusions gIn（H） belong to HgIn（H） belong to In＋2（H）.展开更多
This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.
In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinator...In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinatorics. The number of them is also given.展开更多
In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to...In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.展开更多
文摘For an integer n ≥2, we say that an operator A is an n-idempotent if A^n = A; A is a generalized n-idempotent if A^n = A^＊; A is a hyper-generalized n. idempotent if A^n = A^＋. The set of all n-idempotents, all generalized n-idempotents and all hyper-generalized n-idempotents are denoted by In（H）, gIn（H） and HgIn（H）, respectively. In this note, we obtain a chain of proper inclusions gIn（H） belong to HgIn（H） belong to In＋2（H）.
基金supported by the National Natural Science Foundation of China under grant No.11171222the Doctoral Program of the Ministry of Education under grant No.20094407120001
文摘This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.
文摘In this paper, we describe the canonical partial order on the idempotent set of the strong endomorphism monoid of a graph, and using this we further characterize primitive idem potenes from the viewpoint of combinatorics. The number of them is also given.
文摘In this article, I consider projection groups on function spaces, more specifically the space of polynomials P<sub>n</sub>[x]. I will show that a very similar construct of projection operators allows us to project into the subspaces of P<sub>n</sub>[x] where the function h ∈P<sub>n</sub>[x] represents the closets function to f ∈P<sub>n</sub>[x] in the least square sense. I also demonstrate that we can generalise projections by constructing operators i.e. in R<sup>n+1</sup> using the metric tensor on P<sub>n</sub>[x]. This allows one to project a polynomial function onto another by mapping it to its coefficient vector in R<sup>n+1</sup>. This can be also achieved with the Kronecker Product as detailed in this paper.
