Absolute Value Function Research Articles

The Influence of Noise Uncertainty and SNR Wall on the Performance of Hybrid Sensing Method

The paper discusses the hybrid sensing method and presents the hybrid detector (HD) which improves the sensing performance. The proposed HD takes advantage of the energy detection (ED) and a method based on the Covariance Absolute Value (CAV) or Cyclic Autocorrelation Function (CAF). The paper chara

Critical current of layered superconductor with defects in tilted magnetic field

The simulation of vortex lattice in layered high-temperature superconductor (HTSC) has been done using Monte Carlo method. The case of highly anisotropic superconductor with crossing lattices of Abrikosov and Josephson vortices under applying of tilted magnetic field has been examined. The critical current as a function of absolute value of dc magnetic field, parallel and perpendicular to the anisotropy axis c, has been calculated. The simulations have been done for different anisotropy parameter of HTSC and different concentrations of defects. Configurations of the Josephson vortices in presence of the Abrikosov vortices have been obtained.

Correction to: Representation of light pressure resultant force and moment as a tensor series

In the paper Nerovny et al. (2017), the commentaries about a convergence of series which represent the absolute value function and corresponding equations contain several mistakes (Sect. 2, from Eqs. (4) to (6)).

Fixed points of multiplicative contraction mappings on multiplicative metric spaces

In this paper, we first discussed multiplicative metric mapping by giving some topological properties of the relevant multiplicative metric space. As an interesting result of our discussions, we observed that the set of positive real numbers R_+ is a complete multiplicative metric space with respect to the multiplicative absolute value function. Furthermore, we introduced concept of multiplicative contraction mapping and proved some fixed point theorems of such mappings on complete multiplicative metric spaces.

Three bounded proofs for nonlinear multi‐input multi‐output approximate dynamic programming based on the Lyapunov stability theory

SummaryThe approximate dynamic programming needs 2 prerequisites to be an effective optimal control method. Firstly, it must be assured to be stable and convergent before application. Secondly, the control system should mainly be a nonlinear multi‐input multi‐output form. Thus, this paper introduces a nonlinear multi‐input multi‐output approximate dynamic programming and proves that it is stable in Lyapunov sense, therefore it is convergent. Besides, the Lyapunov function design is also analyzed. These proofs are based on the Lyapunov stability theory in the form of the utility function of quadratic, square‐weighted sum, and absolute value. Thereafter, 3 typical control examples of nonlinear multi‐input multi‐output approximate dynamic programming are offered to show their applications and verify the proofs. The proof overcomes the complex derivation, and the results contain 3 practical and systematic bounded proofs. It is for the first time that the proof focuses on nonlinear multi‐input multi‐output approximate dynamic programming from the view of utility function. What is more, the results can also serve as an effective analysis and guide for the utility function design and the stability criterion of nonlinear multi‐input multi‐output approximate dynamic programming as well.

Jacobian-free approximate solvers for hyperbolic systems: Application to relativistic magnetohydrodynamics

We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function, and compare them with Chebyshev-based PVM solvers. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Another important feature of the proposed methods is that they are suitable to be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems for which the Jacobians involve complex expressions, e.g., the relativistic magnetohydrodynamics (RMHD) equations. On the other hand, the proposed Jacobian-free solvers have also been extended to the case of approximate DOT (Dumbser–Osher–Toro) methods, which can be regarded as simple and efficient approximations to the classical Osher–Solomon method, sharing most of it interesting features and being applicable to general hyperbolic systems. To test the properties of our schemes a number of numerical experiments involving the RMHD equations are presented, both in one and two dimensions. The obtained results are in good agreement with those found in the literature and show that our schemes are robust and accurate, running stable under a satisfactory time step restriction. It is worth emphasizing that, although this work focuses on RMHD, the proposed schemes are suitable to be applied to general hyperbolic systems.

A Switchable Chaotic Oscillator with Two Amplitude–Frequency Controllers

A simple chaotic system with a single nonquadratic term is developed to be a switchable chaotic signal generator in this paper. Additional nonlinearity of the absolute value function is introduced for reforming the structure without damaging the basic dynamics but yielding a new independent amplitude–frequency controller. A switchable chaotic experimental oscillator is designed afterwards, where two coefficients corresponding to two independent rheostats rescale the amplitude and frequency of the chaotic signals smoothly. To our knowledge, this has never been found in other chaotic oscillators.

Minimization of transformed $l_1$ penalty: Closed form representation and iterative thresholding algorithms

The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm ($p \in (0,1)$). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the $l_p$ thresholding functions ($p \in [0,1]$) are in closed form only for $p=0,1,1/2,2/3$, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds consistently performs the best in sparse signal recovery with and without measurement noise.

Stochastic response of a fractional vibroimpact system

The paper proposes a method to investigate the stochastic dynamics of a vibroimpact single-degree-of-freedom fractional system under a Gaussian white noise input. It is assumed that the system has a hard type impact against a one-sided motionless barrier, which is located at the system’s equilibrium position; furthermore, the system under study is endowed with an element modeled with fractional derivative. The proposed method is based on stochastic averaging technique and overcome the particular difficulty due to the presence of fractional derivative of an absolute value function; particularly an analytical expression for the system’s mean squared response amplitude is presented and compared with results obtained by numerical simulations.

<p>Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: two theoretical perspectives</p>

This paper presents a study of networking of theories between the theory of registers of semiotic representation (TRSR) and the onto-semiotic approach of mathematical cognition and instruction (OSA). The results obtained show complementarities between these two theoretical perspectives, which might allow more detailed analysis of the students’ performance.Análisis de la actividad cognitiva subyacente en la resolución de una tarea sobre la derivabilidad de la función valor absoluto: dos perspectivas teóricas En este artículo se presenta un estudio de networking of theories, entre la teoría de los registros de representación semióticos (TRRS) y el enfoque onto-semiótico de la cognición e instrucción matemáticos (OSA). Los resultados obtenidos revelan complementariedades entre estas dos perspectivas teóricas cuya aplicación simultánea permitiría hacer análisis más pormenorizados de las producciones de los estudiantes.Handle: http://hdl.handle.net/10481/44148WOS-ESCIScopus record and citations

Diabatic Definition of Geometric Phase Effects.

Electronic wave functions in the adiabatic representation acquire nontrivial geometric phases (GPs) when corresponding potential energy surfaces undergo conical intersection (CI). These GPs have profound effects on the nuclear quantum dynamics and cannot be eliminated in the adiabatic representation without changing the physics of the system. To define dynamical effects arising from the GP presence, the nuclear quantum dynamics of the CI containing system is compared with that of the system with artificially removed GP. We explore a new construction of the system with removed GP via a modification of the diabatic representation for the original CI containing system. Using an absolute value function of diabatic couplings, we remove the GP while preserving adiabatic potential energy surfaces and CI. We assess GP effects in dynamics of a two-dimensional linear vibronic coupling model both for ground and excited state dynamics. Results are compared with those obtained with a conventional removal of the GP by ignoring double-valued boundary conditions of the real electronic wave functions. Interestingly, GP effects appear similar in two approaches only for the low energy dynamics. In contrast with the conventional approach, the new approach does not have substantial GP effects in the ultrafast excited state dynamics.

Quasi-Newton Method for Absolute Value Equation Based on Upper Uniform Smoothing Approximation

In this paper, an upper uniform smooth approximation function of absolute value function is proposed, and some properties of uniform smooth approximation function are studied. Then, absolute value equation (AVE), Ax - |x| = b, where A is a square matrix whose singular values exceed one, is transformed into smooth optimization problem by using the upper uniform smooth approximation function, and solved by quasi-Newton method. Numerical results in solving given AVE problems demonstrated that our algorithm is valid and superior to lower uniform smooth approximation function.

Scheduling with time-dependent discrepancy times

In time-dependent scheduling, various processing time functions are studied, yet absolute value functions have surprisingly been omitted from the discussion. Such a processing time function increases linearly with a job's discrepancy from its ideal midtime. The objective is to find a schedule that minimizes the makespan, introducing the discrepancy time minimization problem. This single-machine scheduling problem with time-dependent processing times is motivated by optimization of walking times on a car assembly line. Its decision version is NP hard, as we show by reduction of the even---odd partition problem. For the variant with known start time, we develop several heuristics. Further insights form lower bounds and dominance rules for a branch-and-bound search. Numerical experiments show the performance of our algorithms on problem instances of up to 60 jobs. For the variant with common ideal midtime and flexible start time, we present a polynomial-time algorithm.

Rousseau Among the Moderns: Music, Aesthetics, Politics by Julia Simon (review)

Reviewed by: Rousseau Among the Moderns: Music, Aesthetics, Politics by Julia Simon Mira Morgenstern Julia Simon, Rousseau Among the Moderns: Music, Aesthetics, Politics (University Park: Pennsylvania State University Press, 2013. Pp. 215. $64.95. Julia Simon’s study presents a new take on Rousseau. Unlike conventional thematic treatments emphasizing his literary or political writings, Simon takes the time to focus on Rousseau’s work in a field that he always claimed as primary to his identity, but that scholars have traditionally overlooked: his writings on music. While there have been collections of essays, even entire conferences, devoted to this theme (few of which receive critical mention in this volume), Simon argues that understanding key concepts in Rousseau’s musical oeuvre yields a new understanding of his political ideas that unifies his intellectual production in new and important ways. Simon embarks on this enterprise with a multifaceted analysis of how musical performance can create a sense of community. She writes from within her experience both as a musical performer and as a deep reader of Rousseau, tackling in a coherent manner his writings on music as a way of reading his theoretical writings and concerns. In so doing, Simon interweaves a complex understanding of how the ever-changing challenges of performance—including issues of tempo, harmony, and measure—inevitably presage the tensions between the individual and the community as highlighted in Rousseau’s political writings. She goes on to analyze the different musical approaches taken by Rousseau and Rameau as functions of relative and absolute value, while exploring the social and political issues implicated in these themes. Simon then treats Rousseau’s approaches to “authentic” or “folk” music, exploring some of the tensions implicit within the hierarchy of music established by its systematization. Finally, she has something to say about Rousseau’s notion of the redemptive power of music, which she links to what she terms its “aesthetic modernity”: that is, music’s perpetual ability to adjust to new challenges and emotions. [End Page 425] While there have been books and articles that analyze Rousseau’s contribution to music, this is one of the first to locate his approach within the controversies regarding music taking place in the eighteenth century. Simon’s book also links Rousseau’s broader aesthetic themes in his writings on music to the questions raised by different approaches to the modernisms of the nineteenth and twentieth centuries. Simon’s strongest chapter is her second one, on “Singing Democracy.” Here she makes the connection between Rousseau’s dynamic concept of music as a performative art and the more complex notion of the General Will, particularly the challenges of its articulation. Some of these themes are carried over into the fourth chapter, centering on what Simon terms “folk music.” She analyzes the possibility of the management of the audience as they are seduced into thinking that their responses are allowed to develop naturally, when in fact they are being subtly but effectively manipulated. Interestingly, though, it is here that Simon pulls her punches. While she does take note of the implicit hypocrisy of the grape-harvest festival at Clarens, with its self-proclaimed “equality” of all the denizens of that estate, Simon fails to probe the ironic quality of Rousseau’s texts. As it turns out, the author of these letters in La Nouvelle Héloïse is Saint-Preux, who throughout the pages of this novel, is not distinguished for his perspicacity. Simon also fails to consider other texts dealing with the notion of manipulated performativity, perhaps because these texts do not deal directly with music per se. But these texts—including the famous distinction between the Actor and the Orator in Lettre à d’Alembert, and the evocative description of the “matinée à l’Anglaise” at Clarens in La Nouvelle Héloïse—do treat notions of theatricality, a concept Simon does not hesitate to consider in other chapters. In this context, the omission is puzzling. More baffling is the lack of attention to Rousseau’s presentation in Lettre à d’Alembert of his notion of the perfect festival, which includes what might well be considered a manipulative, even sinister, version of a “transparent” celebration at a social dance...

Periodically forced system with symmetric motion limiting constraints: Dynamic characteristics and equivalent electronic circuit realization

A two-degree-of-freedom periodically-forced system with symmetric motion limiting constraints is considered in this paper. The incidence relation between dynamics and constraint parameters (clearance and constraint stiffness) is studied and some novel results are obtained by double-parameter simulation analysis. The fundamental group of impact motions having the excitation period and differing by the number of impacts is given special consideration for analyzing low frequency vibration characteristics of the system. Dynamics of the system are studied with emphasis on the mutual transition characteristics between neighboring regions of fundamental impact motions. An electronic circuit is designed for physical implementation of dynamics of the periodically-forced system with symmetric constraints. The non-linear terms of the system are replaced by using an absolute-value function and can be fully implemented with simple electronic elements (resistors and operational amplifiers). The electronic circuit is realized and studied. The oscilloscope outputs of electrical waveforms of various non-smooth oscillations, generated by the circuit itself, are experimentally observed. A good agreement among the numerical results of the mechanical model, the electronic design simulation of the circuit and the real oscilloscope outputs of hardware implementation is confirmed.

Smooth Newton method to absolute value equation based on lower uniform smoothing approximation function

We investigate the NP-hard absolute value equation (AVE) Ax − |x|=b, where A is an arbitrary square matrix whose singular values exceed one. The significance of the absolute value equations arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear complementarity problem that in turn is equivalent to the absolute value equation. In this paper, after a lower uniform smooth approximation function of absolute value function is established, absolute value equation is transformed into smooth optimization problem, and solved by smoothing Newton method. The convergence of method is shown, and the preliminary numerical experiments are given.

Polar Metric-Weighted Norm-Based Scan Matching for Robot Pose Estimation

A novel point-to-point scan matching approach is proposed to address pose estimation and map building issues of mobile robots. Polar Scan Matching (PSM) and Metric-Based Iterative Closest Point (Mb-ICP) are usually employed for point-to-point scan matching tasks. However, due to the facts that PSM considers the distribution similarity of polar radii in irrelevant region of reference and current scans and Mb-ICP assumes a constant weight in the norm about rotation angle, they may lead to a mismatching of the reference and current scan in real-world scenarios. In order to obtain better match results and accurate estimation of the robot pose, we introduce a new metric rule, Polar Metric-Weighted Norm (PMWN), which takes both rotation and translation into account to match the reference and current scan. For robot pose estimation, the heading rotation angle is estimated by correspondences establishing results and further corrected by an absolute-value function, and then the geometric property of PMWN called projected circle is used to estimate the robot translation. The extensive experiments are conducted to evaluate the performance of PMWN-based approach. The results show that the proposed approach outperforms PSM and Mb-ICP in terms of accuracy, efficiency, and loop closure error of mapping.

The Root of the Problem

The absolute value function opens the door to a better way to solve problems involving square roots.

Switching behavior of solutions of ordinary differential equations with abs-factorable right-hand sides

We consider nonsmooth dynamic systems that are formulated as the unique solutions of ordinary differential equations (ODEs) with right-hand side functions that are finite compositions of analytic functions and absolute-value functions. Various non-Zenoness results are obtained for such solutions: in particular, any absolute-value function in the ODE right-hand side can only switch between its two linear pieces finitely many times on any finite duration, even when a discontinuous control input is included. These results are extended to obtain numerically verifiable necessary conditions for the emergence of “valley-tracing modes”, in which the argument of an absolute-value function is identically zero for a nonzero duration. Such valley-tracing modes can create theoretical and numerical complications during sensitivity analysis or optimization. We show that any valley-tracing mode must begin either at the initial time, or when another absolute-value function switches between its two linear pieces.

A generalized Newton method for absolute value equations associated with circular cones

In this paper, we study a class of absolute value equations associated with circular cone (CCAVE for short), which is a generalization of the absolute value equations discussed recently in the literature, analogously to the fact that circular cone is the very generalization of the second-order cone. We show that the CCAVE is equivalent to a class of circular cone linear complementarity problems, and hence generalize the well-known equivalence between absolute value equations and linear complementarity problems. Useful properties of the generalized differential of the absolute value function over the circular cone are investigated, which helps to propose a generalized Newton method for solving the CCAVE. The convergence of this method is established under mild conditions, as well as the efficiency of which is illustrated by some preliminary numerical results.