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The Geometric Theory of Holomorphic Functions
Published in Steven G. Krantz, Complex Variables, 2019
Let C be the set of subsets of ℂ ∪ {∞} consisting of (i) circles and (ii) sets of the form L ∪ {∞} where L is a line in ℂ. We call the elements of C “generalized circles.” Then every linear fractional transformation φ takes elements of C to elements of C. One verifies this last assertion by noting that any linear fractional transformation is the composition of dilations, translations, and the inversion map z ↦ 1/z and each of these component maps clearly sends generalized circles to generalized circles.
Complex numbers
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Definition 6.1. Let a, b, c, d ∈ ℂ and assume ad – bc ≠ 0. Put f(z)=az+bcz+d. The function z ↦ f(z) is called a linear fractional transformation.
qLPV modeling and mixed-sensitivity
Published in International Journal of General Systems, 2023
Luiz Benício Degli Esposte Rosa, Matheus Senna de Oliveira, Renan Lima Pereira
The notation used in this article is standard. denotes the set of real matrices; I is an identity matrix and represents the transpose of a matrix. For square matrices, is the determinant and the trace of A. If , means that A is positive (semi-)definite and that it is negative (semi-)definite. Symmetric blocks in matrices are occasionally indicated by ★ and stands for the induced- norm. The notation is used to represent a state-space realization. For a concise notation, the dependence of the signals on k will be dropped wherever it does not confuse the reader. Furthermore, consider two dynamical systems and with appropriate dimensions. Then, the upper linear-fractional transformation (LFT) is defined as and the lower LFT is given by
Minimality criteria for convergent power series over Z p and rational maps with good reduction on the projective line over Q p
Published in Dynamical Systems, 2022
Sangtae Jeong, Dohyun Ko, Yongjae Kwon, Youngwoo Kwon
For a study object of ϕ on we consider a rational map of degree of the form such that and , where for all . From this assumption, it is evident that is nonzero in and . For a rational map of at least degree 2, the number of periodic points of a fixed period must be finite. Hence, there exists an element such that and are distinct. Let be the conjugation of ϕ by the linear fractional transformation g of the form, Then we directly check that and because and . By replacing with ϕ, we take a rational map ϕ in (6) of the desired form but note that it does not necessarily have good reduction.
Stability and stabilisation of a class of networked dynamic systems
Published in International Journal of Systems Science, 2018
The following symbols and notations are adopted. (*,#) stands for the upper linear fractional transformation. diag{Xi|Li = 1} represents a block-diagonal matrix with its ith diagonal block being Xi, while col{Xi|Li = 1} the vector/matrix stacked by Xi|Li = 1 with its ith row block being Xi. {Xij|i = M, j = Ni = 1, j = 1} denotes a matrix with M × N blocks and its ith row jth column block matrix being Xij. 0m and 0m × n, respectively denote the m dimensional zero column vector and the m × n dimensional zero matrix, while the subscript with respect to dimensions is omitted under the premise of no ambiguity. Similarly, Im, identity matrix with m × m dimension, is abbreviated as I. R♯ means real column vectors set with appropriate dimensions. The superscripts T and H are used to denote, respectively the transpose and conjugate transpose of a matrix/vector and XTWX is sometimes abbreviated as (*)TWX or XTW(*), especially when the term X has a complicated expression. For symmetrical matrices A and B, A < (≤, >, ≥)B means A − B is negative definite (negative semi-definite, positive definite, positive semi-definite).