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Matrices
Published in Jamal T. Manassah, Elementary Mathematical and Computational Tools For Electrical and Computer Engineers Using Matlab®, 2017
DEFINITION A unimodular matrix has the defining property that its determinant is equal to one. In the remainder of this section, we restrict our discussion to 2 ⊗ 2 unimodular matrices, as these form the tools for the matrix formulation of ray optics and Gaussian optics, which are two of the major subfields of photonics engineering.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
Consider standard torus as the factor group of the abelian group with respect to its discrete subgroup of integer vectors. Denote the related group homomorphism being simultaneously a smooth covering map. The coordinates in the space will be denoted by Let A be an unimodular matrix with integer entries. Since the linear map , generated by the matrix A, transforms the subgroup onto itself, such a matrix generates diffeomorphism of the torus called the automorphism of the torus [1,2,11]. Topological properties of such maps are the classical object of research (see, for example, [1,11,18,23]). Because the torus automorphism also preserves the standard volume element on the torus carried over from , then its ergodic properties have also been the subject of research in many works [5,12,26]. The following classical Halmos theorem holds for automorphisms of a torus [12].
Series concatenation of 2D convolutional codes by means of input-state-output representations
Published in International Journal of Control, 2018
Joan-Josep Climent, Diego Napp, Raquel Pinto, Rita Simões
Note that the fact that two equivalent encoders differ by unimodular matrix multiplication also implies that the primeness properties of the encoders of a code are preserved, i.e. if admits a rFP (rZP) encoder then all its encoders are rFP (rZP). A 2D finite support convolutional code that admits rFP encoders is called non-catastrophic, and it is named basic if all its encoders are rZP. An encoder of the form up to a row permutation is called systematic. Not all 2D convolutional codes admit a systematic encoder. We call 2D systematic code to a 2D convolutional code that admits a systematic encoder. The class of 2D systematic codes is contained in the class of the 2D basic convolutional codes as the following lemma implies. The proof is straightforward and we omit it.
A pair of linear canonical Hankel transformations and associated pseudo-differential operators
Published in Applicable Analysis, 2018
The theory of linear canonical transformation (LCT) was motivated by the work of two different projects by Collins [1] on the field of paraxial optics, on the other hand, Moshinsky and Quesne [2] in the field of nuclear physics in early seventies. The LCT is a four parameter class of linear integral transformation for studying the behavior of many useful transformations and system responses in physics and engineering in general. Therefore, LCT is found as a powerful mathematical tool in many fields of physics and engineering, but its applications in the field of pure mathematics is still missing. In this correspondence, we have defined a pair of canonical Hankel transformations and corresponding pseudo-differential operators and also discussed on their theory. A general class of LCT has been studied by [3,4]. The conventional canonical transformation represents any affine linear transformation in the (x, y) plane and specified by a unimodular matrix (i.e. determinant is one):