Linear Fractional Transformation Research Articles

On a Question of Constructing Mӧbius Transformations via Spheres and Rigid Motions

A Mӧbius Transformation or a Fractional Linear Transformation is a complex-valued function that maps points in the extended complex plane into itself either by translations, dilations, inversions, or rotations or even as a combination of the four mappings. Such a mapping can be constructed by a stereographic projection of the complex plane on to a sphere, followed by a rigid motion of the sphere, and a projection back onto the plane. Both Mӧbius transformations and Stereographic projections are abundantly used in diverse fields such as map making, brain mapping, image processing etc. In 2008, Arnold and Rogness created a short video named as Mӧbius Transformation Revealed and made it available on YouTube which became an instant hit. In answering a question posted in the accompanied paper by the same name, Siliciano in 2012 showed that for any given Mӧbius transformation and an admissible sphere, there is exactly one rigid motion of the sphere with which the transformation can be constructed. The present work is prepared on a suggestion posted by Silciano in characterizing rigid motions in constructing a specific Mӧbius transformation. We show that different admissible spheres under a unique Mӧbius transformation would require different rigid motions.

Composition operators on Hardy-Smirnov spaces

We investigate composition operators CΦ on the Hardy-Smirnov space H2(Ω) induced by analytic self-maps Φ of an open simply connected proper subset Ω of the complex plane. When the Riemann map τ:U→Ω used to define the norm of H2(Ω) is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on H2(Ω). These descriptions are particular cases of the results obtained by Gunatillake-Jovovic-Smith in 2015 and Gunatillake in 2017. Additionally, if the coefficients of τ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on H2(Ω). We finish this paper showing that if Ω is unbounded and Φ is a non-automorphic self-map of Ω with a fixed point, then CΦ is never complex symmetric on H2(Ω).

Finite-time reliable nonfragile control for fractionalorder nonlinear systems with asymmetrical saturation and structured uncertainties

This paper investigates the finite-time stabilization problem of fractional-order nonlinear differential systems via an asymmetrically saturated reliable control in the sense of Caputo’s fractional derivative. In particular, an asymmetrical saturation control problem is converted to a symmetrical saturation control problem by using a linear matrix inequality framework criterion to achieve the essential results. Specifically, in this paper, we obtain two sets of sufficient conditions under different scenarios of structured uncertainty, namely, norm-bounded parametric uncertainty and linear fractional transformation uncertainty. The uncertainty considered in this paper is a combination of polytopic form and structured form. With the help of control theories of fractional-order system and linear matrix inequality technique, some sufficient criteria to ensure reliable finite-time stability of fractional-order differential systems by using the indirect Lyapunov approach are derived. As a final point, the derived criteria are numerically validated by means of examples based on financial fractional-order differential system and permanent magnet synchronous motor chaotic fractional-order differential system.

Recurrence of multiples of composition operators on weighted Dirichlet spaces

A bounded linear operator T acting on a Hilbert space $$\mathcal {H}$$ is said to be recurrent if for every non-empty open subset $$U\subset \mathcal {H}$$ there is an integer n such that $$T^n (U)\cap U\ne \emptyset$$ . In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces $$\mathcal {S}_\nu$$ ; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete previous work of Costakis, Manoussos, and Parissis on the recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $$(\lambda ,\nu ,\phi )\in {\mathbb {C}}\times \mathbb {R}\times \mathrm{LFM}(\mathbb {D})$$ for which the scalar multiple of composition operator $$\lambda C_\phi$$ acting on $$\mathcal {S}_\nu$$ fails to be recurrent.

Model spaces invariant under composition operators

AbstractGiven a holomorphic self-map $\varphi $ of $\mathbb {D}$ (the open unit disc in $\mathbb {C}$ ), the composition operator $C_{\varphi } f = f \circ \varphi $ , $f \in H^2(\mathbb {\mathbb {D}})$ , defines a bounded linear operator on the Hardy space $H^2(\mathbb {\mathbb {D}})$ . The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb {\mathbb {D}})$ , which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.

Determining allowable parametric uncertainty in an uncommon quadrotormodel for closed loop stability

In this article, control oriented uncertainty modeling of an uncommon quadrotor in hover is discussed. This quadrotor consists of two counter-rotating big rotors on longitudinal axis and two counter-rotating small tilt rotors on lateral axis. Firstly, approximate linear model of this vehicle around hover is obtained by using Newton--Euler formulation. Secondly, specific uncertainty is assigned to each parameter. Resulting uncertain model is converted into a linear fractional transformation framework for robustness analysis. Next, the most critical uncertain parameters in terms of robust stability in a proposed quadrotor model are investigated using $ \mu $ sensitivities. Finally, skewed-$ \mu $ analysis determines maximum possible uncertainty bounds for model parameters that are difficult to identify accurately.

Lyapunov function computation for autonomous systems with complex dynamic behavior

A computational approach is presented in this paper to construct local Lyapunov functions for autonomous dynamical systems with multiple isolated locally asymptotically stable (such as point-like, periodic, or strange) attractors. We consider systems of nonlinear ODEs, where the right-hand-side of the dynamic equations is given in the form of rational functions (i.e., fraction of polynomials). The Lyapunov function is searched in a parameterized quadratic form of rational terms of the state variables. The quadratic decomposition of the rational state-dependent inequalities is performed using the linear fractional transformation (LFT) and further algebraic/numeric simplification steps. Unlike the sum of squares (SOS) approach, the sufficient linear matrix inequality (LMI) conditions for the Lyapunov function are formulated only locally on a compact polytopic subset of the state space, which allows indefinite matrix solutions for the quadratic decomposition. The local solution is enforced using affine annihilators with matrix Lagrange multipliers. Alongside the typical Lyapunov conditions, further boundary LMI constraints are prescribed using Finsler’s lemma to ensure the required geometric properties of the Lyapunov function. The results are illustrated on four planar benchmark models having either multiple locally stable equlibria or a limit cycle, and on the Lorenz system.

Complex symmetric weighted composition operators on Bergman spaces and Lebesgue spaces

In the paper, we investigate weighted composition operators on Bergman spaces of a half-plane. We characterize weighted composition operators which are hermitian and those which are complex symmetric with respect to a family of conjugations. As it turns out, weighted composition operators enhanced by a symmetry must be bounded. Hermitian, and unitary weighted composition operators are proven to be complex symmetric with respect to an adapted and highly relevant conjugation. We classify which the linear fractional functions give rise to the complex symmetry of bounded composition operators. We end the paper with a natural link to complex symmetry in Lebesgue space.

Modeling thermal systems with fractional models: human bronchus application

System thermal modeling allows heat and temperature simulations for many applications, such as refrigeration design, heat dissipation in power electronics, melting processes and bio-heat transfers. Sufficiently accurate models are especially needed in open-heart surgery where lung thermal modeling will prevent pulmonary cell dying. For simplicity purposes, simple RC circuits are often used, but such models are too simple and lack of precision in dynamical terms. A more complete description of conductive heat transfer can be obtained from the heat equation by means of a two-port network. The analytical expressions obtained from such circuit models are complex and nonlinear in the frequency $$\omega $$ . This complexity in Laplace domain is difficult to handle when it comes to control applications and more specifically during surgery, as heat transfer and temperature control of a tissue may help in reducing necrosis and preserving a greater amount of a given organ. Therefore, a frequency-domain analysis of the series and shunt impedances will be presented and different techniques of approximations will be explored in order to obtain simple but sufficiently precise linear fractional transfer function models. Several approximations are proposed to model heat transfers of a human middle bronchus and will be quantified by the absolute errors.

A new deterministic global computing algorithm for solving a kind of linear fractional programming

This paper investigates a class of linear fractional programming (LFP) problem, which minimizes the sum of a finite number of linear fractional functions over a polyhedral region. Firstly, the equivalence problem (EP) of the LFP problem is given by a new two-stage transformation method. Secondly, considering the characteristics that the branch-and-bound algorithm can guarantee the global optimality of the solution to an optimization problem, and then based on the EP, we discuss the bounding operation, branching operation, pruning operation and rectangle-region reduction technique of this algorithm. After that, the convergence of the algorithm is proved and its computational complexity is deduced from the worst case. Finally, some experiments are reported to verify the effectiveness, feasibility and other performance of the proposed algorithm.

Invariant rational functions, linear fractional transformations and irreducible polynomials over finite fields

For a subgroup of PGL(2,q) we show how some irreducible polynomials over Fq arise from the field of invariant rational functions. The proofs rely on combining two actions of PGL(2,F), one on the projective line over a field F and the other on the rational function field F(x). The invariant functions in F(x) are used to show that regular patterns exist in the factorization of certain polynomials into irreducible polynomials. We use some results about group actions and the orbit polynomial, whose proofs are included.

A Cryptographic Scheme for Construction of Substitution Boxes Using Quantic Fractional Transformation

In the new era of cryptography, Substitution Boxes (S-Boxes) are very important to raise confusion in cipher text and the security of encryption directly depends on the algebraic strength of S-box. To avoid a hacker attack, researchers are focusing on creating dynamic S-boxes that are much stronger. The dominating concept for developing S-boxes is linear fractional transformation. In this article, we proposed a novel technique to generate cryptographically strong S-box by using fractional transformation based on finite field. The substitute box is constructed in two phases. Firstly, general form of dynamic fractional transformation designed which work for odd exponents in the range [1-255]. The S-box is then constructed using quantic fractional transformation as an example. Secondly, in order to increase the unpredictability of proposed S-box, we use the symmetric group’s S256 permutation. The usefulness of the constructed S-box was also tested using several criteria such as nonlinearity, differential uniformity, strict avalanche criteria, linear approximation probability and bit independence criteria. To assess the reliability of S-box, its performance outcomes are compared to those of previously developed S-boxes. Furthermore, we utilized the suggested S-Box to the image encryption approach. Then, to determine the effectiveness of the encryption scheme, use several tests such as contrast, correlation, homogeneity, entropy, and energy. We have compared our results with different algorithms which ensured that the proposed strategy for ciphered image is excellent.

SSD – Software for Systems with Delays: Reproducible Examples and Benchmarks on Model Reduction and H2 Norm Computation

We present SSD – Software for Systems with Delays, a de novo MATLAB package for the analysis and model reduction of retarded time delay systems (RTDS). Underneath, our delay system object bridges RTDS representation and Linear Fractional Transformation (LFT) representation of MATLAB. This allows seamless use of many available visualizations of MATLAB. In addition, we implemented a set of key functionalities such as H2 norm and system gramian computations, balanced realization and reduction by direct integral definitions and utilizing sparse computation. As a theoretical contribution, we extend the frequency-limited balanced reduction to delay systems first time, propose a computational algorithm and give its implementation. We collected two sets of benchmark problems on H2 norm computation and model reduction. SSD is publicly available in GitHub∗. Our reproducible paper and two benchmark collections are shared as executable notebooks.∗https://github.com/gumussoysuat/ssd

Одна геометрическая модель дробно-линейных преобразований

A model of linear fractional transformations of the complex plane in the form of points of the complex three-dimensional projective space without a linear “forbidden” quadric is presented. A model of real linear fractional transformations of the complex plane in the form of points of the real three-di­mensional projective space without a linear “forbidden” quadric is presented. A geometric separation of points corresponding to parabolic, hyperbolic and elliptic real linear fractional transformations by a “parabolic” cone touching the forbidden quadric is found. Some pro­per­ties of model points corresponding to real linear fractional transfor­ma­tions are found. Some properties of model points corresponding to fun­da­men­tal groups transformations of biconnected domains of the complex pla­ne are found.

Modelling and Parameter Identification of Ex-Situ Biological Biogas Upgrading

In this paper, a model of ex-situ biological biogas upgrading process is first developed. Then, parameter sensitivity analysis is performed, in order to determine the most relevant parameters for subsequent parameter identification, based on a linear fractional transformation (LFT) reformulation of the model. Biogas composition and volumetric production have been well predicted by the calibrated model, allowing its adoption as a designing tool for start-up operation of experimental pilot-scale activity.

О структурных формах проективной структуры

A projective structure on a smooth manifold is a maximal atlas such that all its transition maps are the fractional linear transformations. Our aim is to interpret this notion in terms of the higher order frame bundles and their structure forms. It is shown that the projective structure gener­ates the sequence of differential geometric structures. The construction is following: Step 1. For a smooth manifold the so-called quotient frame bundle as­sociated to the 2nd order frame bundle on the manifold is constructed. Step 2. Given projective structure on the manifold, the mappings from the quotient frame bundle to the higher order frame bundles are con­structed. These mappings are the differential geometric structures. Step 3. The pullbacks of the structure forms of the frame bundles via the mappings are considered. These are called structure forms of the pro­jective structure. The expressions of their exterior differentials in terms of the forms themselves are found. These expressions coincide with the structure equations of a projective space.

Robust integrated control/structure co-design for stratospheric balloons

Stratospheric balloons offer cost-effective platforms for optical payloads in the context of astronomy missions. During the 2018 flight of the Faint Intergalactic medium Redshifted Emission Balloon (FIREBall) experiment, the moon light was scattered from the surface of the balloon and re-directed into the telescope which resulted in degraded optical performance. To reduce this parasite effect, it is sought to increase the length of the fight train. However, this change in the mechanical design significantly modifies the dynamics of the system and the pointing performance must not be altered. In this purpose, a robust integrated control/structure co-design method is proposed. After deriving a Linear Fractional Transformation (LFT) model of the system, the co-design is tackled as a multi-objective, structured, robust H2/H∞ problem that is solved with a non-smooth optimization algorithm to maximize the train's length under constraints of pointing performance. By optimizing in a single iteration the controllers along with the structural parameter with regard to the worst-case configurations of the uncertain parameters, time-consuming procedures requiring not only to iterate between control and mechanical design, but also to analyze the robustness based on Monte-Carlo simulations, are avoided.

Enhanced AOCS verification techniques for Euclid's high-pointing performance*

This paper presents an enhanced Verification and Validation (V&V) framework accompanied by dedicated tools that allows to analytically evaluate pointing error performance of high-pointing accuracy missions in the presence of disturbances and uncertainties. The proposed analysis approach poses the V&V problem in the robust control framework using linear fractional transformation modelling theory (to include model uncertainties) and frequency dependent weighting functions (to capture the spectral properties of the disturbances and the pointing error performance metrics), and is based on the structured singular value approach as well as the integral quadratic constraint (IQC) framework. The validity of this formal approach is exemplified through the verification of the stringent pointing performance of the Euclid mission during the science observation phase. The results show that the proposed enhanced V&V approach is capable of providing certificates for robust stability and performance for all modelled uncertainties.

Some Characters of a Generalized Rational Difference Equation

Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2^r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results.

Taming the Load Current of Identical Cells in Matrix

We explore the load current [Formula: see text] for a rectangular array (matrix) of [Formula: see text] identical cells where [Formula: see text] strings (columns) of [Formula: see text] serial cells (rows) are arrayed in parallel. [Formula: see text] is equal to [Formula: see text] with the internal resistance of the cell and the load resistance exchanged. By treating a linear fractional function as a translated inversely-proportional function, we can easily capture the properties of [Formula: see text] and the relative magnitude of [Formula: see text] and [Formula: see text] via their ratio. The limiting behaviors of the load current are discussed beyond the ideal-cell and short-circuit limits. For the given total number of cells, we graphically verify the recent findings on the matrix of cells that produces the maximum load current. Finally, we analyze the possibility of a car starting with lemon cells or AA dry cells in matrix. This work would be useful in creating a high school or university curriculum that unifies identical cells in series, parallel, or matrix.