Linear Fractional Transformation Research Articles

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions.

Molding robust S-box design based on linear fractional transformation and multilayer Perceptron: Applications to multimedia security

This study introduces a novel and refined approach for generating exceptionally efficient S-boxes. The proposed methodology employs a hybrid approach that combines linear fractional transformation (LFT) with a multilayer perceptron (MLP) architecture. This method makes use of a perceptron with three layers: input, hidden, and output. Each layer's neuron count is fine-tuned to conform to the S-box layout. In addition, a threshold nonlinear transformation is utilized to increase nonlinearity, and a novel algorithm for boosting nonlinearity is introduced. The utilization of both LFT and MLP approaches has led to the development of S-boxes that possess nearly ideal average nonlinearity values, surpassing those that have been presented in literature. Notably, one S-box achieved an exceptional nonlinearity score of 114.50. Furthermore, to demonstrate how well the S-box works, this study also employs it in an image encryption application.

Rank-two programs involving linear fractional functions

AbstractThe aim of this paper is to deepen the study of solution methods for rank-two nonconvex problems with polyhedral feasible region, expressed by means of equality, inequality and box constraints, and objective function in the form of $$\phi \left( c^Tx+c_0,\frac{d^Tx+d_0}{b^Tx+b_0}\right) $$ ϕ c T x + c 0 , d T x + d 0 b T x + b 0 or $$\bar{\phi }\left( \frac{\bar{c}^Ty+\bar{c}_0}{a^Ty+a_0}, \frac{d^Ty+d_0}{b^Ty+b_0}\right) $$ ϕ ¯ c ¯ T y + c ¯ 0 a T y + a 0 , d T y + d 0 b T y + b 0 . These problems arise in bicriteria programs, quantitative management science, data envelopment analysis, efficiency analysis and performance measurement. Theoretical results are proved and applied to propose a solution algorithm. Computational results are provided, comparing various splitting criteria.

Comparing the Performance of Robust Controllers for Vibration Suppression in Long Rotor Systems

In this paper, the vibration control of the multivariable model of rotor bearing systems is considered for investigation. Some simply structured controllers that can suppress vibrational disturbances are tested for their robustness via the H∞ optimality criteria. Initially, intelligent optimisation techniques are used to minimize the H∞ mixed-sensitivity norm of the Linear Fractional Transformation (LFT) of the simple two-term PI controllers acting on the rotor system models. This results in some controllers that can suppress the vibration but with a slow oscillatory response. After this, an appropriate interpretation of the Bode plot singular values of the combined sensitivity and control effort matrix is used to explain the performance shortcomings of this controller. Moreover, the existing simply structured controllers in the literature exhibiting a faster performance are examined by using singular value plots. It is shown that when the maximum singular value of the control effort matrix drops below the 0 db line, the performance will be boosted. Finally, the H∞ controllers are designed by using the robust control toolbox in MATLAB. This resulted in rapid disturbance rejection, with the vibration amplitude diminishing to zero after 0.3 s due to double-step disturbances. However, these controllers in the frequency domain have an order of eight and may not be realizable to be implemented in practice. It is concluded that examining the Bode plot of the maximum singular value of the control effort matrix is a useful tool for evaluating performance in the frequency domain. However, designing robust controllers by toolboxes in the time domain can lead to superb performance with higher-order controllers.

Analytical and numerical modelling of the electrostatic behaviour of highly insulating materials in the time domain

The usual response of an insulator subjected to a voltage step involves time power laws. We present here mathematical tools allowing to calculate this time domain response in open circuit after an initial charge deposit, within the framework of linear systems theory, using linear fractional transfer functions. In the time domain, the inverse Laplace transform of the data taken from the frequency domain leads to Mittag-Leffler functions, generalizing the Debye exponential response to an extended fractional α-order response. The open-circuit boundary conditions are different from the closed-circuit ones. We nevertheless demonstrate that using a transfer function deduced from the Cole-Cole response in closed-circuit, a precise analytical formula of the potential decay after an initial charge deposit may be established, and a numerical computation of this decay may be performed using easily available software. Applying the superposition principle, the voltage return following a brief short circuit may also be deduced. Experimental results are presented and the limits of the superposition principle applied to real materials are discussed.

Image Encryption based on Elliptic Curve Points and Linear Fractional Transformation

Image Encryption based on Elliptic Curve Points and Linear Fractional Transformation

8×8 S-boxes over Klein four-group and Galois field $ {GF}\left({2}^{4}\right) $: AES redesign

<abstract> <p>This research paper is supplemented with a unique formation to design state-of-the-art S-boxes. The invented approach is simple but has the capability of creating confusion in our newly proposed algorithm. Our core planned work refined the method of already designed S-boxes to accomplish more compact ones. Various structures were merged here, namely affine transformation, fractional linear transformation, structure of Klein four-group, and the algebraic structures of the Galois fields, $ GF\left({2}^{4}\right) $ and $ GF\left({2}^{8}\right). $ These structures were utilized to synthesize newly $ 1600 $ robust S-boxes. Besides, we discussed encryption steps of AES with these newly generated S-boxes. We highlighted some specific characteristics, performance of parameter's improvement, and their utilization. Nonlinear properties were mainly set to inspect the behavior of I/O bits and could apply image encryption. Then, the performance of proposed S-boxes and newly structured AES was tested in comparison with other prevailing S-boxes.</p> </abstract>

Linear fractional optimization over the efficient set of multi-objective integer quadratic problem

In this paper, we introduce an exact algorithm for optimizing a linear fractional utility function over the efficient set of a multi objective integer quadratic problem. The algorithm is based on the “Branch and Cut” principle, which combines the branching process to ensure decision variables’ integrity and efficient cuts built off the non-increasing gradients’ directions of objective functions to eliminate inefficient integer solutions. The proposed approach accelerates the convergence to the efficient solution that optimizes the utility function. After presenting and describing the algorithm, a detailed didactic example is illustrated, followed by an experimental study to validate our approach and show computational costs.

Transient response feedback control for horizontal vibration of high-speed elevator car with prescribed performance

The high-speed elevator car is subject to violent horizontal vibration due to the wind pressure generated by the ventilation hole of the elevator rapid operation, improper adjustment of the gap and steps at the joint of the elevator car guide rail, pitting corrosion or wear on the guide surface of the guide shoe, inadequate installation and adjustment of the guide shoe or guide rail and so on. Aiming at the problems of poor transient convergence and real-time performance, low control accuracy and stability of the existing active control algorithm for the horizontal vibration control variables of the elevator car system, a transient response feedback control strategy with specified performance is proposed in this paper. Firstly, a six degree of freedom elevator car system model is established, and the state space equation is constructed based on the linear fractional transformation algorithm; Secondly, a class of preset transient performance function is introduced to construct the convergence boundary in advance to ensure that the system control variables are strictly constrained within this boundary, and the stability of the transformed variables is ensured by equivalent transformation; Finally, four different control methods including passive control, H2/H∞ control, PID control and transient response feedback control proposed in this paper are compared and analyzed by using MATLAB under two typical guideway excitation. The simulation results show that the proposed transient response feedback control strategy reduces the root mean square value and maximum value of the horizontal displacement acceleration of the elevator car system by more than 81%, effectively suppresses the horizontal vibration of the elevator car, and ensures the transient and steady-state performance of the elevator car system.

A medical image encryption scheme based on Mobius transformation and Galois field

Data security and privacy are considered to be the biggest problems faced by service providers who have worked with public data for a long time. A key element of modern encryption that is utilized to increase textual confusion is the Substitution box (S-box) and the algebraic strength of the S-box has a significant impact on how secure the encryption method is. In this article, we present a unique method that uses a linear fractional transformation on a finite field to produce cryptographically robust S-boxes. Firstly, we choose a specific irreducible polynomial of degree 8 in Z2[x] to construct GF(28). Later, we used the action of PGL(2,GF(28)) on GF(28) to generate a robust S-box. The effectiveness of the built-in S-box was evaluated using several criteria including non-linearity, differential uniformity, strict avalanche criteria, linear approximation probability, and bit independence criterion. The proposed S-box's characteristics are compared to those of most recent S-boxes to confirm the higher performance. Additionally, the S box was used to encrypt images to show its usefulness for multimedia security applications. We performed several tests, including contrast, correlation, homogeneity, entropy, and energy, to evaluate the success of the encryption technique. The proposed method for ciphering an image is very effective, as proven by its comparison with several S boxes.

An algorithm for construction of substitution box based on subfield of galois field $$GF({2}^{16})$$ and dynamic linear fractional transformation

An algorithm for construction of substitution box based on subfield of galois field $$GF({2}^{16})$$ and dynamic linear fractional transformation

A study of text-theoretical approach to S-box construction with image encryption applications

Data protection is regarded as one of the biggest issues facing companies that have been using public data for a long time. Numerous encryption techniques have been used to address these issues and safeguard data from any malicious attempts and assaults. A substitution box (S-box) is the basic component for modern block ciphers, which helps to ensure robust security of plain data while encryption and permits its lawful decipherment. The goal of this paper is to recommend an effective, original, and straightforward technique for the creation of robust S-boxes. A sample S-box is generated in the proposed work using the word “UNITY” but other words can also be used to generate many powerful S-boxes. The ASCII code is used to translate the word “UNITY” into binary form, after which a distinct matrix is constructed for each character of the word. In the next phase, a linear fractional transformation is constructed using these matrices, which is then utilized to generate the S-box. The constructed S-box was then evaluated against typical security standards to support its high cryptographic authority. The generated S-box's statistical and algebraic resilience is demonstrated by its very low linear probability and differential probability scores of 0.125 and 0.039, respectively, and a high non-linearity score of 111.5. To evaluate the effectiveness of the image encryption scheme, digital images are encoded using the created S-box. The performance and comparative research demonstrate that the suggested S-box is a real candidate for applications in the field of image encryption and has a stronger performance base.

Linear fractional self-maps of the unit ball

Abstract Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit ball. In this paper, we discuss a class of maps in $\mathbb {C}^N$ that generalize linear fractional maps. We then proceed to determine precisely when such a map is a self-map of the unit ball. In particular, we take a novel approach, obtaining numerous new results about this class of maps along the way.

Frequency domain identifiability and sloppiness of descriptor systems with an LFT structure

Identifiability and sloppiness are investigated in this paper for the parameters of a descriptor system based on its frequency responses. Two metrics are suggested respectively for measuring absolute and relative sloppiness of the parameter vector. In this descriptor system, system matrices are assumed to depend on its parameters through a linear fractional transformation, which enables it to include a large class of networked dynamic systems as a special case. When an associated transfer function matrix is of full normal row rank, a matrix rank based necessary and sufficient condition is derived for parameter identifiability. This condition can be verified recursively. An algorithm is suggested to find a set of frequencies, at which system frequency responses are capable to uniquely determine its parameter values. An ellipsoid approximation is given for the set consisting of all the parameter values of the descriptor system, whose frequency response deviates within a prescribed distance from that associated with a globally identifiable parameter vector value. Explicit formulas are also derived for the suggested absolute and relative sloppiness metrics for some application significant cases.

Finite-Horizon Robustness Analysis of an Automatic Landing System Under Probabilistic Uncertainty

This paper demonstrates the utility of novel probabilistic robustness analysis tools on an industry-sized autolanding system of a civil transport aircraft. The closed-loop aircraft model is representative of an Airbus A330 during final approach and landing. It is modeled as a linear time-varying system by linearizing with respect to a nominal reference trajectory. Disturbances due to random parametric uncertainty, atmospheric turbulence, and static wind shear are considered. Probabilistic uncertainty in the aerodynamic coefficients and mass parameters is quantified efficiently by applying a polynomial chaos series expansion. This generalized Fourier series is applied after employing a linear fractional transformation. The computational benefits of expanding uncertain interconnections in linear fractional representation build an appealing advantage of the applied analysis methods. Probabilistic and worst-case perturbations around the reference trajectory are quantified. The analysis results are compared with nonlinear Monte Carlo simulations of the complete closed-loop system.

Some Practical Applications of Discretely Tunable Couplers’ Matrix Model

The findings of a research of the practical application of a previously developed model of a decametric waves discrete coupler are presented. Due to the high accuracy achieved in describing the transforming properties of Coupler in the operating frequency range, together with the apparatus of the theory of complex variables, in particular with the method of conformal maps of fractional linear functions, it was possible to solve a number of applied problems. A method for calculating the area of consistent loads of discretely tunable power circuits of arbitrary structure has been developed. For completeness of the result, the case of matching a generator with a complex value of internal resistance with a given quality (with an acceptable value of the traveling wave coefficient) has been considered. The proposed approach to matching assumes that for any possible antenna load there must be appropriate combinations of discretely tunable elements of the Coupler power circuit that ensure matching with a given quality. The design ratio for images of arbitrary circles and straight lines conformally mapped by a matching quadrilateral to a given complex domain (i.e., expressions for calculating the coordinates of the circle center and its radius, as well as coefficients of the equation of straight lines) has been obtained. Studies of the matching properties of various circuits implemented on discrete elements have been carried out. The results of this study are important when choosing the structural design of the Coupler power circuit. Calculation formulas are obtained that allow us to estimate the relative losses of active power in the Coupler power circuit. This contributes to the reasonable choice of the best configuration option from several, similar in quality matching, thereby avoiding complicated temperature settings of the circuit elements. An exhaustive search method of tuning the discrete contour of the Coupler has been proposed, which is extremely simple and does not require heavy calculations with matrices and complex numbers.

On fractional order computational solutions of low-pass electrical transmission line model with the sense of conformable derivative

In this study, we solve fractional order PDEs with free parameters that regulate wave motion in a low-pass electrical transmission line equation (LPET) using the unified technique. To convert the governing equation into an ordinary differential equation (ODE), we have used a simple linear fractional transformation. The unified technique can therefore be used to find various single and extracting wave solutions to the equation of attention. Using Maple-18 software, all of the established solutions are validated and verified. By selecting appropriate values for the fractional order and free parameters, the two dimensional (2D) and three dimensional (3D) profiles of some of the assimilation solutions demonstrating kink, singular kink, cross kink, bright-dark kink, lump type kink, anti-kink, dark-anti kink, singular periodic, periodic, singular soliton solutions are displayed. The effect of free parameters and fractionality on the dynamical properties of the precise and exploited solutions are graphically illustrated and discussed in detail. We have also compared our results to those found in the literature review. Wherever possible, the judgment demonstrates no significant differences between our realized and published solutions. The results of this investigation show that the adopting technique is both productive and powerful in extracting wave solutions to nonlinear evolution problems that arise in the areas of applied mathematics, mathematical physics, and engineering.

Continued fractions related to a group of linear fractional transformations

Abstract There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=z+2 . This group is the unique subgroup of the modular group PSL ( 2 , Z ) {\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.

Robust Model Predictive Control for Two-DOF Flexible-Joint Manipulator System

This paper presents a practical study on how to improve the ℋ∞ performance and meet the input–output constraints of the two-degrees-of-freedom (DOF) flexible-joint manipulator system (FJMS) with parameter uncertainties and external disturbances. For this reason, a robust constrained moving-horizon ℋ∞ controller is designed to improve the system ℋ∞ performance while still satisfying the input–output constraints of the uncertain system. First, the uncertain controlled system model of the two-DOF FJMS is established via the Lagrange equation method, Spong’s assumption, and the linear fractional transformation (LFT) technique. Then, the control requirements and input–output constraints of the uncertain system are transformed into the linear matrix inequality (LMI) via the theory of ℋ∞ control and the full-block multiplier technique. Next, the LMI optimization problem refreshed by the current state is addressed at each sample moment with the idea of the moving-horizon control of the model predictive control (MPC), and the calculated gain is implemented to the nonlinear closed-loop system under the state feedback structure. The validity and feasibility of the designed control scheme is finally verified via the results of simulation experiments.

Projection theorems for linear-fractional families of projections

AbstractMarstrand’s theorem states that applying a generic rotation to a planar setAbefore projecting it orthogonally to thex-axis almost surely gives an image with the maximal possible dimension$\min(1, \dim A)$. We first prove, using the transversality theory of Peres–Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in$PSL(2,\mathbb{C})$or a generic real linear-fractional transformation in$PGL(3,\mathbb{R})$. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of$PSL(2,\mathbb{C})$or$PGL(3,\mathbb{R})$. Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.