A compact f-f model of high-dimensional piecewise-linear function over a degenerate intersection

A set of piecewise-linear (PL) basis functions for signal or functional decomposition is introduced. These basis functions provide a PL approximation to the signal and an a priori determination of the required number for a finite term expansion to achieve a certain pointwise approximation error. Moreover, the expansion coefficients are linear combinations of samples of the function to be expanded and are virtually trivial to determine.

Nonlinear effects of a power amplifier (PA) in communications systems include spectral spreading, which causes adjacent channel interference. For predistorter design, it is crucial to employ a model that can sufficiently model the intermodulation distortion (IMD) caused by the PA. The order of IMD generated by a polynomial is determined by its highest order basis. For a piecewise linear function, the order of nonlinearity cannot be directly deduced from its basis function. This paper investigates how IMD is introduced by the simplicial canonical piecewise linear (SCPWL) function. We examine the operation imposed by the SCPWL basis function on sinusoid and multi-tone signal in the time domain and frequency domain. Our analysis results in expressions that can be used to predict the output spectrum of the SCPWL basis. The SCPWL basis operation involves spreading of the input signal spectrum. The output spectrum of each SCPWL basis includes infinite IMD components. The IMD order of the SCPWL model is not determined by the highest order basis. Instead, the number of SCPWL parameters, i.e., the number of linear affine segments that defines the function, determines how accurately it can model the IMD components.

The moments produced by the aerodynamic control effectors on aircraft are typically non-linear functions of displacement with a region of linearity about zero displacement. In this paper we briefly review a novel approach that was developed at AFRL for solving the control allocation problem where the control moments are piecewise linear functions of the control displacement. The advantages of this approach are that it avoids cumbersome, high-order polynomial curve fits of the aerodynamic data and thus avoids solution of a non-linear programming problem. Stability and control derivatives are almost always stored in multi-dimensional look-up tables where it is assumed that the data is piecewise linear. The approach that is presented utilizes this piecewise linear assumption for the control effector moment data and accounts for non-linearities in the moment-effector relationships and position constraints on the effectors as well. The control allocation problem using piecewise linear functions is posed as a mixed-integer linear program (MILP) and solved using a freely available GNU-licensed mixed-integer linear programming code. We demonstrate two applications of the piecewise linear control allocation (PLCA) approach. The first application to be considered is to use the PLCA approach in the inner-loop control law of a re-usable launch vehicle on approach and landing. The second application is constraint estimation for trajectory reshaping and re-targeting. In order to successfully re-target a trajectory when an aircraft has experienced degraded performance due to a failure or damage to the vehicle, the effects of the failure or damage on the lift, drag, and trimmability of the vehicle must be known a priori over the entire flight envelope

This paper investigates the performance of current-mode chaotic neural networks that are optimized for travelling salesman problem (TSP). Piece-wise linear (PWL) functions used as the output functions in chaotic neurons are found to ease the design of neural network circuits. Three different PWL output functions are compared by solving 10-city TSP. Simulation results showed excellent performance of the chaotic neural networks with PWL output function. The design parameters to implement the PWL output function suitable for TSP chaotic neural networks are obtained.

The infinite relaxations in integer programming were introduced by Gomory and Johnson to provide a general framework for the theory of cutting planes: the so-called valid functions, and in particular the minimal and extreme functions, can be seen as automatic rules for the generation of cuts. However, while many extreme functions are piecewise linear and therefore easy to describe, the set of extreme functions turns out to have a very complicated mathematical structure, as several extreme functions are known that exhibit a somewhat pathological behavior. In this paper we show that if some smoothness assumption is imposed on an extreme function $$\pi $$, then $$\pi $$ is necessarily piecewise linear. More precisely, we show that if a continuous extreme function for the Gomory–Johnson one-dimensional infinite group relaxation is a piecewise $${\mathcal {C}}^2$$ function, then it is a piecewise linear function.

We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of functions: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As a result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.

The electric current volume integral equation formulation is discretized with piecewise linear basis and testing functions. The resulting weak formulation is more complicated, and contains additional terms compared with the previous discretization utilizing piecewise constant approximations. However, the use of fully linear functions provides superior solution accuracy while keeping good convergence properties of the original integral equation.

The paper studies a second order nonlinear differential equation whose right hand side is a piecewise linear function. It proves the coexistence of a countably set of periodic solutions and an uncountable set of bounded non-periodic solutions. The Weierstrass approximation theorem is applied so that the piecewise linear function can be approximated by a polynomial on the finite interval. The result can be also used to explain the chaos on some smooth nonlinear dynamical systems.

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

In this paper we apply one of the main results from the theory of noninvertible maps to predict the oscillations amplitude of a chaotic attractor, using only the lines where the piece-wise linear (PWL) map is not differentiable, their first iterates (called critical lines or curves), and their finite-rank iterates. This approach is also valid for smooth differentiable noninvertible maps, where the critical lines are defined as the first iterates of the set of points for which the Jacobian determinant of the map cancels. Moreover, a novel bifurcation encountered in the case of PWL map is presented: chaotic orbit arising from the collision of a typical bifurcation for smooth maps (the Neimark-Hopf, or center bifurcation) with the bifurcation typical only for piece-wise smooth and non differentiable maps (the border-collision bifurcation). Finally, it is shown that this novel bifurcation can give birth to two (or more) chaotic attractors, and the conditions allowing to predict the number of chaotic attractors which could coexist are proposed and discussed in the context of possible applications.

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

We present the first combinatorial algorithm for efficiently computing the Reeb space in all dimensions. The Reeb space is a higher‐dimensional generalization of the Reeb graph, which is standard practice in the analysis of scalar fields, along with other computational topology tools such as persistent homology and the Morse‐Smale complex. One significant limitation of topological tools for scalar fields is that data often involves multiple variables, where joint analysis is more insightful. Generalizing topological data structures to multivariate data has proven challenging and the Reeb space is one of the few available options. However, none of the existing algorithms can efficiently compute the Reeb space in arbitrary dimensions and there are no available implementations which are robust with respect to numerical errors. We propose a new algorithm for computing the Reeb space of a generic piecewise linear map over a simplicial mesh of any dimension called arrange and traverse. We implement a robust specialization of our algorithm for tetrahedral meshes and evaluate it on real‐life data.

Research has identified several students’ misinterpretations of the scientific underpinnings of the photoelectric effect. Studies pointed out that the graphical representation of kinetic energy of the ejected electrons versus frequency of incident photons is unclear when a blend of mathematical and scientific reasoning is applied. Students cannot find a scientific sense of the formula using the graph despite the fundamental linear dependence inherited in it. While research suggests various ways to help improve the understanding, little attention is given to the algebraic structure of the equation KMax = hf − Wo and its corresponding graphical representation. Detailed scrutiny revealed that some phases of the photoelectric effect’s algebraic expression do not support its visual representation. This paper suggests a more consistent alignment of the rules of mathematics with the photoelectric effect process to allow coherence. This suggestion emerged as an alternative algebraic representation of the law of conservation of energy in the form of a piecewise linear function. It is hypothesized that such covariational relation between photons’ frequency and the ejected electrons’ kinetic energy is more consistent with observable phenomena behaviour and students’ math knowledge. Mathematical modelling of the photoelectric effect that concluded with a linear piecewise function was supported by a physics simulation The Photoelectric Effect (http://phet.colorado.edu) and delivered to a group of twenty (N = 20) high school physics students. A didactical outline of the instructional unit and a general evaluation of the student’s responses are included. This study in its ealier version was published in Physics Education Journal (Sokolowski 2021).