Almost contact Hom-Lie algebras and Sasakian Hom-Lie algebras

In this paper, we consider Hom-Lie groups and introduce left invariant almost contact structures on them (almost contact Hom-Lie algebras). On such Hom-Lie groups, we construct the almost contact metrics and the contact forms. We give the notion of normal almost contact Hom-Lie algebras and describe [Formula: see text]-contact and Sasakian structures on Hom-Lie algebras. Also, we study some of their properties. In addition, it is shown that any Sasakian Hom-Lie algebra is a [Formula: see text]-contact Hom-Lie algebra. Finally, we present examples of Sasakian Hom-Lie algebras and in particular, we show that the skew symmetric matrix [Formula: see text] carries a Sasakian structure.

Minors of a skew symmetric matrix: a combinatorial approach

Knuth's combinatorial approach to Pfaffians is used to reprove and clarify a century-old formula, due to Brill. It expresses arbitrary minors of a skew symmetric matrix in terms of Pfaffians.

Retinal Fundus Detection Using Skew Symmetric Matrix

The retina is the light sensitive tissue, lining the back of our eye. Light rays are focused onto the retina through our cornea, pupil and lens. The retina converts the light rays into impulses that travel through optic nerve to our brain, where they are interpreted as the images. The task of manually segmenting fundus from retina images is generally time-consuming and difficult. In most settings, the task is done by marking the fundus regions slice-by-slice, which limits the human rater’s view and generates distorted images. Manual segmentation is also typically done largely based on a single image with intensity enhancement provided by an injected contrast agent. In the current research the fundus is detected and extracted in retinal image. Fundus is distinguished from normal tissues by their image intensity, threshold-based or region growing techniques. The fundus in this approach is detected with the help of geometric features. Skew symmetric matrix is used to avoid any angular orientation. In this approach the accuracy on fundus is quite promising. Accuracy of fundus detection is improved according to the area and the acceptance rate .In this approach ,once the image is loaded, it is filtered and normalized. Then superpixels are generated using linear iterative clustering approach and the features are generated. From the available set of features, some of the features are selected using sequential forward selection approach .Classifier is constructed in order to determine different classes in a test image. Proposed work is two class problem in which algorithm is applied that consists of skew symmetric matrix .Experimental results show substantial improvements in the accuracy and the performance of fundus detection as well as in false acceptance rate and false rejection rate.

Characterizing isoclinic matrices and the Cayley factorization

Quaternions, particularly the double and dual forms, are important for the representation rotations and more general rigid-body motions. The Cayley factorization allows a real orthogonal 4 × 4 matrix to be expressed as the product of two isoclinic matrices and this is a key part of the underlying theory and a useful tool in applications. An isoclinic matrix is defined in terms of its representation of a rotation in four-dimensional space. This paper looks at characterizing such a matrix as the sum of a skew symmetric matrix and a scalar multiple of the identity whose product with its own transpose is diagonal. This removes the need to deal with its geometric properties and provides a means for showing the existence of the Cayley factorization.

Lorentz transformation from an elementary point of view

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

Cayley formula in Minkowski space-time

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).