Concave Minimization for Sparse Solutions of Absolute Value Equations

Based on concave function, the problem of finding the sparse solution of absolute value equations is relaxed to a concave programming, and its corresponding algorithm is proposed, whose main part is solving a series of linear programming. It is proved that a sparse solution can be found under the assumption that the connected matrixes have range space property(RSP). Numerical experiments are also conducted to verify the efficiency of the proposed algorithm.

Five New MVL Current Mode Differential Absolute Value Circuits Based on Carbon Nano-tube Field Effect Transistors(CNTFETs)

Carbon Nano-Tube Field Effect Transistors(CNTFETs) are being widely studied as possible successors to silicon MOSFETs.Using current mode has many advantages such as performing sum operation by means of a simple wired connection.Also,direction of the current can be used to exhibit the sign of digits.It is expected that the advantages of current mode approaches will become even more important with increased speed requirements and decreased supply voltage.In this paper,we present five new circuit designs for differential absolute value in current mode logic which have been simulated by CNTFET model.The considered base current for this model is 2 μA and supply voltage is 0.9 V.In all of our designs we used N-type CNTFET current mirrors which operate as truncated difference circuits.The operation of Differential Absolute Value circuit calculates the difference between two input currents and our circuit designs are operate in 8 logic levels.

The AOR-Base Splitting Modified Fixed Point Iteration for Solving Absolute Value Equations

Recently,Yu et al.presented a modified fixed point iterative(MFPI)method for solving large sparse absolute value equation(AVE).In this paper,we consider using accelerated overrelaxation(AOR)splitting to develop the modified fixed point iteration(denoted by MFPI-JS and MFPI-GSS)methods for solving AVE.Furthermore,the convergence analysis of the MFPI-JS and MFPI-GSS methods for AVE are also studied under suitable restrictions on the iteration parameters,and the functional equation between the parameter T and matrix Q.Finally,numerical examples show that the MFPI-JS and MFPI-GSS are efficient iteration methods.

A Novel Inverse-Free Neurodynamic Approach for Solving Absolute Value Equations

We propose a novel inverse-free neurodynamic approach (NIFNA) for solving absolute value equations (AVE). The NIFNA guarantees global convergence and notably improves convergence speed by achieving fixed-time convergence. To validate the theoretical findings, numerical simulations are conducted, demonstrating the effectiveness and efficiency of the proposed NIFNA.

A New Fixed-Time Dynamical Systemfor Absolute Value Equations

A novel dynamical model with fixed-time convergence is presented to solve the system of absolute value equations(AVEs).Under a mild condition,it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs.Moreover,in contrast to the existing inversion-free dynamical system(C.Chen et al.,Appl.Numer.Math.168(2021),170–181),a conservative settling-time of the proposed method is given.Numerical simulations illustrate the effectiveness of the new method.

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.

SUMMATION OF ABSOLUTE VALUE TEST FOR MULTIPLE OUTCOME COMPARISON WITH MODERATE EFFECT

In biomedical research,in order to evaluate the effect of a drug,investigators often need to compare the differences between one treatment group and another one by using multiple outcomes.The rank-sum tests can handle the case where the outcome differences between two groups are in the same direction.If they are not,MAX can handle it and is very useful when one/some of the differences is/are relatively larger than the others.When the individual outcome difference between two groups is moderate,a new method,summation of the absolute value of rank-based test for each outcome,is proposed in this work.Power comparison with the existing methods based on simulation studies and a real example show that the proposed test is a robust test,and works well when the difference for each outcome is moderate.The authors also derive some theoretical results for comparing the power between MAX and the the proposed method.

Further study on tensor absolute value equations

In this paper,we consider the tensor absolute value equations(TAVEs),which is a newly introduced problem in the context of multilinear systems.Although the system of the TAVEs is an interesting generalization of matrix absolute value equations(AVEs),the well-developed theory and algorithms for the AVEs are not directly applicable to the TAVEs due to the nonlinearity(or multilinearity)of the problem under consideration.Therefore,we first study the solutions existence of some classes of the TAVEs with the help of degree theory,in addition to showing,by fixed point theory,that the system of the TAVEs has at least one solution under some checkable conditions.Then,we give a bound of solutions of the TAVEs for some special cases.To find a solution to the TAVEs,we employ the generalized Newton method and report some preliminary results.

The Sparsest Solution to the System of Absolute Value Equations

On one hand,to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas;and on the other hand,the system of absolute value equations(AVEs)has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs.Motivated by the development of these two aspects,we consider the problem to find the sparsest solution to the system of AVEs in this paper.We first propose the model of the concerned problem,i.e.,to find the solution to the system of AVEs with the minimum l0-norm.Since l0-norm is difficult to handle,we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem.Then,we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs.When the concerned system of AVEs reduces to the system of linear equations,the obtained results reduce to those given in the literature.The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.

GOAL PROGRAMMING ALGORITHM FOR A TYPE OF LEAST ABSOLUTE VALUE REGRESSION PROBLEM

This paper develops goal programming algorithm to solve a type of least absolute value (LAV) problem. Firstly, we simplify the simplex algorithm by proving the existence of solutions of the problem. Then, we present a goal programming algorithm on the basis of the original techniques. Theoretical analysis and numerical results indicate that the new method contains a lower number of deviation variables and consumes less computational time as compared to current LAV methods.

An Introduction to the Theory of Field Extensions

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

Source location error analysis and optimization methods

The efficiency of an optimization method for acoustic emission/microseismic（AE/MS） source location is determined by the compatibility of its error definition with the errors contained in the input data.This compatibility can be examined in terms of the distribution of station residuals.For an ideal distribution,the input error is held at the station where it takes place as the station residual and the error is not permitted to spread to other stations.A comparison study of two optimization methods,namely the least squares method and the absolute value method,shows that the distribution with this character constrains the input errors and minimizes their impact,which explains the much more robust performance by the absolute value method in dealing with large and isolated input errors.When the errors in the input data are systematic and/or extreme in that the basic data structure is altered by these errors,none of the optimization methods are able to function.The only means to resolve this problem is the early detection and correction of these errors through a data screening process.An efficient data screening process is of primary importance for AE/MS source location.In addition to its critical role in dealing with those systematic and extreme errors,data screening creates a favorable environment for applying optimization methods.

Dual Quaternions and Dual Quaternion Vectors

We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.

A modeling approach for fuzzy programming with echelon form membership functions

In order to solve fuzzy mathematical programming with soft constraints,the initial models must first be converted into crisp models.Membership functions are employed to describe the fuzzy right-hand side parameters needed to achieve this conversion.In some cases,echelon form membership functions(EFMFs)are required to depict the actual fuzzy situation.However,due to their discrete properties,fuzzy programming problems with such membership functions cannot be modeled by traditional methods.Motivated by these challenges,this paper introduces a novel absolute value representation modeling approach to formulate fuzzy programming using EFMFs.This approach can translate a discrete model to a continuous one which can then be easily solved.Finally,by means of a numerical example,the effectiveness of our new approach is demonstrated.

Bohr Inequality for Multiple Operators

An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].

Method to Quantifying the Logical Node Importance for IEC 61850 Based Substation Automation Systems

The problem of logical node(LN)importance quantification in an IEC 61850 based substation automation system(SAS)is investigated in this paper.First,a weighted and directed static complex network model is established by analyzing the characteristics of SAS,according to IEC 61850.Then,we propose a method,which combines topology value and information adjunction value by introducing a first-order linear feedback controller to quantify the value of LNs.On this basis,some definitions for equivalent network conversion are proposed to greatly reduce the complexity of the original network topology.Also,the absolute value and relative value are introduced to quantify LN importance from the perspective of the node’s necessity and influence,respectively.Finally,simulation results of the case study demonstrate that the proposed method is effective and provides a broader and clearer perspective for viewing the logical node importance for IEC61850 based SAS.

Spectral Representation of Proton NMR Spectroscopy for the Pattern Recognition of Complex Materials

Proton nuclear magnetic resonance(NMR)spectroscopy provides a powerful tool for chemical profiling,also known as spectral fingerprinting,because of its inherent reproducibility.NMR is now increasing in use for authentication of complex materials.Typically,the absorbance spectrum is used that is obtained as the phase-corrected real component of the Fourier transform(FT)of the free induction decay(FID).However,the practice discards half the information that is available in the dispersion spectrum obtained as the imaginary component from the FT.For qualitative analysis or quantitative analysis of small sets of absorbance peaks,the symmetric and sharp peaks of the real spectra work well.However,for pattern recognition of entire spectra,trading peak resolution for peak reproducibility is beneficial.The absolute value of the complex spectrum gives the length or magnitude of magnetization vector in the complex plane;therefore,the magnitude relates directly to the signal(i.e.,induced magnetization).The magnitude spectrum is obtained as the absolute value from the real and imaginary spectral components after the FT of the FID.By breaking with tradition and using the magnitude spectrum the reproducibility of the spectra and consequent recognition rates can be improved.This study used a 500-MHz 1H NMR instrument to obtain spectra from 4 diverse datasets;12 tea extracts,8 liquor samples,9 hops extracts,and 25 Cannabis extracts.Six classifiers were statistically evaluated using 100 bootstrapped Latin partitions.The classifiers were a fuzzy rule-building expert system(FuRES)tree,support vector machine trees(SVMTreeG and SVMTreeH),a regularized linear discriminant analysis(LDA),super partial least squares discriminant analysis(sPLS-DA),and a one against all support vector machine(SVM).All classifiers gave better or equivalent results for the magnitude spectral representation than for the real spectra,except for one case of the 24 evaluations.In addition,the enhanced reproducibility of the absolute value spectra is demonstrated by comparisons of the pooled within sample standard deviations.For pattern recognition of NMR spectra,the magnitude spectrum is advocated.

Enhanced asymmetrically clipped DC biased optical OFDM for intensity-modulated direct-detection systems

In this paper,we propose an EADO-OFDM(Enhanced Asymmetrically Clipped DC Biased Optical Orthogonal Frequency Division Multiplexing)method for IM/DD(Intensity-Modulated DirectDetection)optical systems,in which the AV-DCO-OFDM(Absolute Valued DC Biased Optical OFDM)symbols on the even subcarriers and ACO-OFDM(Asymmetrically Clipped Optical OFDM)symbols on the odd subcarriers are combined for simultaneous transmission.Moreover,we discuss the PDF(Probability Density Function)and electrical SNR(Signal to Noise Ratio)of the symbols,which are utilized to estimate the BER(Bit Error Ratio)performance and overall performance of EADO-OFDM.The Monte Carlo simulation results have validated the theoretical analysis and have also confirmed the EADO-OFDM is attractive considering the following benefits.Firstly,EADO-OFDM is more energy efficient compared to the power-efficient DCO-OFDM(DC Biased Optical OFDM),since the required DC bias is smaller when appropriate constellation size combinations are chosen.In addition,EADO-OFDM performs better than the conventional ADO-OFDM(Asymmetrically Clipped DC Biased Optical OFDM),because the absolute value operation causes no clipping distortion.