DECOMPOSITION OF STOCHASTIC FLOWS AND ROTATION MATRIX

2002 ◽  
Vol 02 (01) ◽  
pp. 93-107 ◽  
Author(s):  
PAULO R. C. RUFFINO
Keyword(s):  
Asymptotic Behaviour ◽  
Constant Curvature ◽  
Symmetric Matrix ◽  
Rotation Matrix ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Affine Transformations ◽  
Simply Connected ◽  
Skew Symmetric Matrix

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φt = ξt°Ψt° Θt, where ξt and Ψt have the same properties of Liao's decomposition and (ξt°Ψt) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξt via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.

Cayley formula in Minkowski space-time

2015 ◽  
Vol 12 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Time ◽  
Rotation Matrix ◽  
Symmetric Matrices ◽  
Rotation Matrices ◽  
Minkowski Space Time ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in [Formula: see text]. For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi-skew-symmetric matrices A1 and A2 satisfying the properties [Formula: see text] and [Formula: see text] Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay (A).

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals Overlapping Pfaffians

10.37236/1263 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Donald E. Knuth
Keyword(s):  
Symmetric Matrix ◽  
History Of ◽  
Combinatorial Construction ◽  
Skew Symmetric Matrix

A combinatorial construction proves an identity for the product of the Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices. Several applications of this identity are followed by a brief history of Pfaffians.

A Further Extension of Cayley's Parameterization

1959 ◽  
Vol 11 ◽  
pp. 48-50 ◽  
Author(s):  
Martin Pearl
Keyword(s):  
Symmetric Matrix ◽  
Arbitrary Field ◽  
Complex Field ◽  
Result Theorem ◽  
Characteristic 2 ◽  
Image Position ◽  
Row Space ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In a recent paper (3)* the following theorem was proved for real matrices.Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then1is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.

KUNITA-TYPE STOCHASTIC FLOWS OF HOMEOMORPHISMS IN EUCLIDEAN SPACE

2007 ◽  
Vol 10 (04) ◽  
pp. 523-538 ◽  
Author(s):  
ZONGXIA LIANG
Keyword(s):  
Differential Equation ◽  
Euclidean Space ◽  
Stochastic Differential Equation ◽  
Local Characteristic ◽  
Stochastic Flow ◽  
Stochastic Flows ◽  
Continuous Semimartingale ◽  
Lipschitz Conditions

In this paper we prove Kunita-type stochastic differential equation (SDE) [Formula: see text], t > s, driven by a spatial continuous semimartingale F (x, t) =(F1 (x, t), …, Fm (x, t)), x ∈ ℜm, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜm into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works12 by the author.

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