Determination of Grassmann Manifolds Which are Boundaries

1991 ◽  
Vol 34 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Parameswaran Sankaran
Keyword(s):  
Grassmann Manifold ◽  
Vector Subspace ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Grassmann Manifolds

AbstractLetFGn,kdenote the Grassmann manifold of allk-dimensional (left)F-vector subspace ofFnfor F = ℝ, the reals,C, the complex numbers, orHthe quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the caseF= ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝGn,kto bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

BREAKING OF LINEAR SYMMETRIES AND MICHEL'S THEORY: GRASSMANN MANIFOLDS, AND INVARIANT SUBSPACES

2008 ◽  
Vol 23 (03n04) ◽  
pp. 547-565
Author(s):  
GIUSEPPE GAETA
Keyword(s):  
Symmetry Breaking ◽  
Group Action ◽  
Invariant Manifolds ◽  
Invariant Subspaces ◽  
Geometric Approach ◽  
Symmetry Type ◽  
Original Formulation ◽  
Grassmann Manifolds ◽  
Nonlinear Flows

Michel's theory of symmetry breaking in its original formulation has some difficulty in dealing with problems with a linear symmetry, due to the degeneration in the symmetry type implied by the linearity of group action. Here we propose a fully geometric, approach to the problem, making use of Grassmann manifolds. In this way Michel theory can also be applied to the determination of dynamically invariant manifolds for equivariant nonlinear flows.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

On the Modulii of Analytic Functions

1970 ◽  
Vol 13 (3) ◽  
pp. 325-327 ◽  
Author(s):  
Malcolm J. Sherman
Keyword(s):  
Analytic Functions ◽  
Point Of View ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Necessary And Sufficient Condition ◽  
Concrete Form ◽  
Norm Equality ◽  
Image Position ◽  
Necessary And Sufficient

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

GEOMETRY OF UNIVERSAL GRASSMANN MANIFOLD FROM ALGEBRAIC POINT OF VIEW

1989 ◽  
Vol 01 (01) ◽  
pp. 1-46 ◽  
Author(s):  
KANEHISA TAKASAKI
Keyword(s):  
Lie Algebra ◽  
Grassmann Manifold ◽  
Point Of View ◽  
Simple Application ◽  
Grassmann Manifolds ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Component Theory ◽  
Infinitesimal Transformations ◽  
Algebraic Formulation

An algebraic formulation of the geometry of the universal Grassmann manifold is presented along the line sketched by Sato and Sato [32]. General issues underlying the notion of infinite-dimensional manifolds are also discussed. A particular choice of affine coordinates on Grassmann manifolds, for both the finite- and infinite-dimensional case, turns out to be very useful for the understanding of geometric structures therein. The so-called “Kac-Peterson cocycle”, which is physically a kind of “commutator anomaly”, then arises as a cocycle of a Lie algebra of infinitesimal transformations on the universal Grassmann manifold. The action of group elements for that Lie algebra is also discussed. These ideas are extended to a multi-component theory. A simple application to a non-linear realization of current and Virasoro algebras is presented for illustration.

scholarly journals Rational homotopy of maps between certain complex Grassmann manifolds

2018 ◽  
Vol 68 (1) ◽  
pp. 181-196 ◽  
Author(s):  
Prateep Chakraborty ◽  
Shreedevi K. Masuti
Keyword(s):  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Rational Homotopy ◽  
Grassmann Manifolds ◽  
Continuous Map ◽  
Complex Grassmann Manifold ◽  
Vector Subspaces

AbstractLetGn,kdenote the complex Grassmann manifold ofk-dimensional vector subspaces of ℂn. Assumel,k≤ ⌊n/2⌋. We show that, for sufficiently largen, any continuous maph:Gn,l→Gn,kis rationally null homotopic if (i) 1 ≤k<l, (ii) 2 < l <k< 2(l− 1), (iii) 1 < l <k,ldividesnbutldoes not dividek.

scholarly journals A measure of non-immersability of the Grassmann manifolds in some euclidean spaces

1998 ◽  
Vol 41 (1) ◽  
pp. 197-205
Author(s):  
Cornel Pintea
Keyword(s):  
Critical Points ◽  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Differentiable Mapping ◽  
Euclidean Spaces ◽  
Grassmann Manifolds ◽  
Vector Subspaces

LetGk, n, be the Grassmann manifold consisting in all non-orientedk-dimensional vector subspaces of the spaceRk+n. In this paper we will show that any differentiable mappingf:Gk, n→Rm, has infinitely many critical points for suitable choices of the numbersm,n,k.

scholarly journals When Does a Dual Matrix Have a Dual Generalized Inverse?

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1386
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Explicit Expressions

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

Lp Behavior of the Eigenfunctions of the Invariant Laplacian

1993 ◽  
Vol 36 (4) ◽  
pp. 458-465
Author(s):  
E. G. Kwon
Keyword(s):  
Harmonic Functions ◽  
Maximum Modulus ◽  
Open Unit ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximum Modulus Principle ◽  
Open Unit Ball ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet be the invariant Laplacian on the open unit ball B of Cn and let Xλ denote the set of those f € C2(B) such that counterparts of some known results on X0, i.e. on M-harmonic functions, are investigated here. We distinguish those complex numbers λ for which the real parts of functions in Xλ belongs to Xλ. We distinguish those λ for which the Maximum Modulus Priniple remains true. A kind of weighted Maximum Modulus Principle is presented. As an application, setting α ≥ ½ and λ = 4n2α(α — 1), we obtain a necessary and sufficient condition for a function f in Xλ to be represented asfor some F ∊ LP(∂B).

scholarly journals On the Irreducibility of Artin's Group of Graphs

2015 ◽  
Vol 7 (2) ◽  
pp. 117
Author(s):  
Malak M. Dally ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Complex Numbers ◽  
Sufficient Condition ◽  
Artin Group ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We consider the graph $E_{3,1}$ with three generators $\sigma_1, \sigma_2, \delta$, where $\sigma_1$ has an edge with each of $\;\sigma_2$ and  $\;\delta$. We then define the Artin group of the graph $E_{3,1}$ and consider its  reduced  Perron representation of degree three. After we specialize the indeterminates used in defining the representation to  non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation.<br />

Determination of the Minimum Departure Interval between Trains with same High-Speed at the Relatively Initial Station

2014 ◽  
Vol 587-589 ◽  
pp. 1737-1740
Author(s):  
Jing Jing Shao ◽  
Lei Shan Zhou ◽  
Zi Xi Bai ◽  
Yong Feng Shang
Keyword(s):  
High Speed ◽  
Operational Efficiency ◽  
High Density ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Research Focus ◽  
High Speed Railway ◽  
High Speed Trains ◽  
Necessary And Sufficient

Chinese high-speed railway is in a boom and making train diagram with high-density trains to relieve capacity intense and improve operational efficiency has become the research focus. Different kinds of station intervals between adjacent trains are the basis for the train diagram. According to the situation in which trains with different speed run on the same line and the proportion of high-speed trains is much larger than that of middle-speed trains, this paper raises principles and methods to determine the minimum departure interval between trains with same speed at the relatively initial station. The minimum departure interval between trains with same speed at the relatively initial station is a necessary and sufficient condition for making train diagram since there is no overtaking between same-speed trains.

Sign in / Sign up

Export Citation Format

Share Document