scholarly journals On relationships between two linear subspaces and two orthogonal projectors

2019 ◽  
Vol 7 (1) ◽  
pp. 142-212 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Vector Space ◽  
Matrix Decomposition ◽  
Complex Vector ◽  
Linear Subspaces ◽  
Survey Paper ◽  
Finite Dimensional ◽  
Generic Position ◽  
Ep Matrix ◽  
Orthogonal Projectors

Abstract Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of the m-dimensional complex column vector space 𝔺m with respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

Linear Symmetry Classes

1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson
Keyword(s):  
Vector Space ◽  
Permutation Group ◽  
Cyclic Group ◽  
Symmetry Class ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Linear Character ◽  
Finite Permutation Group ◽  
Symmetry Classes ◽  
Finite Dimensional

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

EQUIVARIANT HOLOMORPHIC HERMITIAN PRINCIPAL BUNDLES OVER A EUCLIDEAN SPACE

2013 ◽  
Vol 05 (03) ◽  
pp. 345-360
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Euclidean Space ◽  
Vector Space ◽  
Algebraic Group ◽  
Inner Product ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Principal Bundles ◽  
Isomorphism Classes ◽  
Finite Dimensional

Let V be a finite dimensional complex vector space equipped with an inner product. Let G denote the group of all affine automorphisms of V preserving the metric defined by the inner product. Let H be a connected reductive affine algebraic group defined over ℂ. We give an explicit classification of the isomorphism classes of G-equivariant holomorphic hermitian principal H-bundles over V.

scholarly journals Classification of regular dicritical foliations

2016 ◽  
Vol 37 (5) ◽  
pp. 1443-1479 ◽  
Author(s):  
GABRIEL CALSAMIGLIA ◽  
YOHANN GENZMER
Keyword(s):  
Vector Space ◽  
Normal Forms ◽  
Blow Up ◽  
The Other ◽  
Holomorphic Foliations ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Dimensional ◽  
Birational Transformation

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

scholarly journals Piecewise Linear Sheaves

2019 ◽  
Author(s):  
Masaki Kashiwara ◽  
Pierre Schapira
Keyword(s):  
Vector Space ◽  
Piecewise Linear ◽  
Building Blocks ◽  
Convex Polyhedra ◽  
Triangulated Category ◽  
Real Vector ◽  
Proper Cone ◽  
Finite Dimensional ◽  
Higher Dimensional

Abstract On a finite-dimensional real vector space, we give a microlocal characterization of (derived) piecewise linear sheaves (PL sheaves) and prove that the triangulated category of such sheaves is generated by sheaves associated with convex polyhedra. We then give a similar theorem for PL $\gamma $-sheaves, that is, PL sheaves associated with the $\gamma $-topology, for a closed convex polyhedral proper cone $\gamma $. Our motivation is that convex polyhedra may be considered as building blocks for higher dimensional barcodes.

scholarly journals Left ideals in the near-ring of affine transformations

1991 ◽  
Vol 43 (1) ◽  
pp. 115-122
Author(s):  
Wolfgang Mutter
Keyword(s):  
Vector Space ◽  
Affine Transformations ◽  
Finitely Generated ◽  
Linear Subspaces ◽  
Affine Subspaces ◽  
Finite Dimensional ◽  
Galois Correspondence

In this paper we determine the left ideals in the near-ring Aff(V) of all affine transformations of a vector space V. It is shown that there is a Galois correspondence between the filters of affine subspaces of V and those left ideals of Aff(V) which are not left invariant. In particular, the not left invariant finitely generated left ideals of Aff(V) are precisely the annihilators of the affine subspaces of V. A similar correspondence exists between the filters of linear subspaces of V and the left invariant left ideals of Aff (V). If V is finite-dimensional, then all left ideals of Aff(V) are finitely generated.

Some Results on the Schur Index of a Representation of a Finite Group

1970 ◽  
Vol 22 (3) ◽  
pp. 626-640 ◽  
Author(s):  
Charles Ford
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Complex Field ◽  
Complex Vector Space ◽  
Linear Transformations ◽  
Complex Vector ◽  
Finite Dimensional ◽  
A Finite Group

Let ℭ be a finite group with a representation as an irreducible group of linear transformations on a finite-dimensional complex vector space. Every choice of a basis for the space gives the representing transformations the form of a particular group of matrices. If for some choice of a basis the resulting group of matrices has entries which all lie in a subfield K of the complex field, we say that the representation can be realized in K. It is well known that every representation of ℭ can be realized in some algebraic number field, a finitedimensional extension of the rational field Q.

Synthesis and characterization of flame-retardant-wrapped carbon nanotubes and its flame retardancy in epoxy nanocomposites

2021 ◽  
pp. 096739112110245
Author(s):  
Jiangbo Wang
Keyword(s):  
Carbon Nanotubes ◽  
Heat Release ◽  
Flame Retardant ◽  
Flame Retardancy ◽  
Total Heat Release ◽  
Pristine Cnts ◽  
Peak Heat Release Rate ◽  
Ep Matrix ◽  
Better Than

A novel phosphorus-silicon containing flame-retardant DOPO-V-PA was used to wrap carbon nanotubes (CNTs). The results of FTIR, XPS, TEM and TGA measurements exhibited that DOPO-V-PA has been successfully grafted onto the surfaces of CNTs, and the CNTs-DOPO-V-PA was obtained. The CNTs-DOPO-V-PA was subsequently incorporated into epoxy resin (EP) for improving the flame retardancy and dispersion. Compared with pure EP, the addition of 2 wt% CNTs-DOPO-V-PA into the EP matrix could achieve better flame retardancy of EP nanocomposites, such as a 30.5% reduction in peak heat release rate (PHRR) and 8.1% reduction in total heat release (THR). Furthermore, DMTA results clearly indicated that the dispersion for CNTs-DOPO-V-PA in EP matrix was better than pristine CNTs.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

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