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In recent years, the reduced biquaternion algebras have been widely used in color image processing problems and in the field of electromagnetism. This paper studies eigen-problems of reduced biquaternion matrices by means of a complex representation of a reduced biquaternion matrix and derives new algebraic algorithms to find the eigenvalues and eigenvectors of reduced biquaternion matrices. This paper also concludes that the number of eigenvalues of an n × n reduced biquaternion matrix is infinite. In addition, the proposed algebraic algorithms are shown to be effective in application to a color face recognition problem. • The eigen-problems of reduced biquaternion matrices are further studied based on the complex representation form. • Propose new algebraic algorithms for finding the eigenvalues and the eigenvectors of a reduced biquaternion matrix. • An n × n reduced biquaternion matrix has infinite eigenvalues. • There are multiple eigenvalues corresponding to the same eigenvector of a reduced biquaternion matrix. • The proposed method is more comprehensive and can find more eigenvalues of a reduced biquaternion matrix. [ABSTRACT FROM AUTHOR]

This Editorial paper provides an overview of a selection of contributions from researchers working in the fields of Subdivision, Applied Geometric and Algebraic Methods, Isogeometric Analysis, Refinability and Geometric Deep Learning, with special emphasis on numerical computation, advanced algorithms, their analysis, and their applications. [ABSTRACT FROM AUTHOR]

• The algorithm proposed in this paper is simpler and easier to understand compared to the algorithms in references [3] and [8]. • The paper focuses on the power domination problem on trees and presents a linear-time algorithm to solve it using the labeling method. In this paper, the power domination problem on trees is studied and we give a linear-time algorithm to it by using the labeling method. Our algorithm is simpler and easier to understand than those in [3,7]. [ABSTRACT FROM AUTHOR]

In the theory and applications of graphs, it is a basic problem to compute the numbers of independent sets and matchings of given sizes. Since the problem of computing the total number of independent sets and that of matchings of graphs is #P-complete, it is unlikely to give efficient algorithms to find the numbers of independent sets and matchings of given sizes. In this paper, for graphs with order n and treewidth at most p , we present two dynamic algorithms to compute the numbers of independent sets of all sizes with runtime O (2 p  ·  pn 3 ) and the numbers of matchings of all sizes with runtime O (2 2 p  ·  pn 3 ), respectively. By the algorithms presented in this paper, for graphs with small treewidths, the numbers of independent sets and matchings of all possible sizes can be computed efficiently. [ABSTRACT FROM AUTHOR]

The n -dimensional HSDC with the logic graph H n is one of the most attractive server-centric data center networks for high incremental scalability. The routing design in network topology is very important for information transmission. In particular, the application of disjoint path covers can solve various problems such as program code optimization and mapping parallel programs to parallel structures. H n consists of 2 n cliques and each two vertices in a clique are adjacent. For any vertex sets { x , u } and { y , v } in W and B , respectively, and any two disjoint vertex sets A 1 , A 2 in some clique, where (W , B) is a bipartition of H n , in this paper, we prove that, when | A i | ≠ 1 and A i ∩ { x , y , u , v } = ∅ for i = 1 , 2 , if | A 1 ∪ A 2 | ≤ n − 3 , then there exists an (x , y) -path P 1 and a (u , v) -path P 2 in H n passing through A 1 and A 2 , respectively, such that P 1 and P 2 have no common vertices and P 1 ∪ P 2 contains all vertices of H n. Then P 1 ∪ P 2 is called the 2-disjoint path cover in H n and we propose the efficient algorithms for the path design. Furthermore, we prove that our main result is optimal. • In the problem of data harvest and information transmission, it is very important to design the disjoint path cover in the logic graph of Data Center Network. • We consider the construction of 2-disjoint path cover such that each path should contain all prescribe vertices which have low probability of failure. • The efficient algorithms have been given and we prove that the number of prescribed vertices is not more than n − 3 and explain that this upper bound is sharp. [ABSTRACT FROM AUTHOR]

Analogs of balls in n -dimensional Euclidean space are defined in the monoid of strings composed of letters of an alphabet if a metric is defined on the monoid. These analogs are not only balls in the monoid of strings but also regular languages. Since the Levenshtein distance, a reasonable metric between strings, was introduced in 1965, researchers have aimed to determine a formula for computing the volumes of balls of strings under this distance and their growth rate. However, this problem remains open. In this paper, we first derive a formula for the volumes of balls of strings under a metric obtained by extending the Hamming distance as a metric on the whole monoid and subsequently reveal the growth rate of balls of strings under the Levenshtein distance using this formula. Furthermore, we describe a conjecture on the formula for the volumes of spheres of strings under the Levenshtein distance. Next, we construct an efficient randomized algorithm that computes estimates of the volumes of the balls whose estimation errors are less than a given value with a probability greater than a specified value. Finally, we examine the validity of the proposed conjecture, using the (true) volumes obtained by exhaustive searches for small spheres and estimates of the volumes obtained by this algorithm for large ones. [ABSTRACT FROM AUTHOR]

In this paper, two new algorithms for computing the Weighted Moore–Penrose inverse A M , N † of a general matrix A for weights M and N which are based on elementary row and column operations on two appropriate block partitioned matrices are introduced and investigated. The computational complexity of the introduced two algorithms is analyzed in detail. These two algorithms proposed in this paper are always faster than those in Sheng and Chen (2013) and Ji (2014), respectively, by comparing their computational complexities. In the end, an example is presented to demonstrate the two new algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with the stability analysis of Linear Time-Invariant (LTI) systems in the presence of a bounded and time-invariant delay parameter. The paper extends the theory of characteristic quasi-polynomial to derive equivalent stability conditions. Based on the quasi-polynomial, a two-variate polynomial is defined and it is proved that system stability is equivalently guaranteed if all roots of the new polynomial are not allocated in a certain region of the right half plane (RHP). Moreover, the algorithm uses the proposed equivalent stability conditions to develop a set of implementable stability conditions based on the exposed edges theorem. The algorithm is applied to three simulation examples that admit some novel results for these systems. The first example shows the step-by-step procedure for the proposed algorithm. The second and the third ones compare various aspects of the proposed algorithm with some existing methods. Two hundred and fifty delay systems are randomly generated and the proposed algorithm is compared with them in terms of conservativeness and computational burden. The results reveal the superiority of the proposed method over the existing ones. [ABSTRACT FROM AUTHOR]

• Two new computational algorithms are proposed for estimating the Lyapunov exponents from time series data in order to test the null hypothesis of a chaotic behavior. • The results obtained provide robust evidence that the Jacobian indirect methods provide better estimates than traditional direct methods in all the experiments we have conducted. • We have shown empirically that the algorithms proposed are robust to the presence of (small) measurement errors because the results obtained are comparable to those which are noise free. • The empirical size of the algorithms proposed decreased and the empirical power increased as the sample size increased which means that our hypothesis tests are consistent and reliable. • This paper opens a new research line where new contributions may appear considering new machine learning methods and deep learning algorithms for estimating the Lyapunov exponents from time series data. Most of the existing methods and techniques for the detection of chaotic behaviour from empirical time series try to quantify the well-known sensitivity to initial conditions through the estimation of the so-called Lyapunov exponents corresponding to the data generating system, even if this system is unknown. Some of these methods are designed to operate in noise-free environments, such as those methods that directly quantify the separation rate of two initially close trajectories. As an alternative, this paper provides two nonlinear indirect regression methods for estimating the Lyapunov exponents on a noisy environment. We extend the global Jacobian method, by using local polynomial kernel regressions and local neural net kernel models. We apply such methods to several noise-contaminated time series coming from different data generating processes. The results show that in general, the Jacobian indirect methods provide better results than the traditional direct methods for both clean and noisy time series. Moreover, the local Jacobian indirect methods provide more robust and accurate fit than the global ones, with the methods using local networks obtaining more accurate results than those using local polynomials. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph. The weak rupture degree of G is defined as r w (G) = max { ω (G − X) − | X | − m e (G − X) : ω (G − X) > 1 } , where the maximum is taken over all X , the subset of V (G), ω (G − X) is the number of components in G − X , and m e (G − X) is the size (edge number) of a largest component in G − X. This is an important parameter to quantitatively describe the invulnerability of networks. In this paper, based on a study of relationship between network structure and the weak rupture degree, a polynomial algorithm for computing the weak rupture degree of trees is given. [ABSTRACT FROM AUTHOR]

In this paper we employ the Euclidean division for polynomials to recover uniquely a Jacobi matrix in terms of the mixed spectral data consisting of its partial entries and the information given on its full spectrum together with a subset of eigenvalues of its truncated matrix obtained by deleting the last row and last column, or its rank-one modification matrix modified by adding a constant to the last element. A necessary and sufficient condition is provided for the existence of the inverse problem. A numerical algorithm and a numerical example are given. [ABSTRACT FROM AUTHOR]

This paper proposes numerical quadrature rules for two-dimensional Cauchy principal-value integrals of the forms ∫ ∫ Ω f (x , y) (x − s) 2 + (y − t) 2 d y d x and ∫ ∫ Ω f (x , y) (x − s) (y − t) d y d x. The derivation of these quadrature rules is based on the Euler–Maclaurin error expansion of a modified trapezoidal rule for one-dimensional Cauchy singular integrals. The corresponding error estimations are investigated, and the convergence rates O (h m 2 μ + h n 2 μ) are obtained for the proposed quadrature rules, where h m and h n are partition sizes in x and y directions, μ is a positive integer determined by integrand. To further improve accuracy, a splitting extrapolation algorithm is developed based on the asymptotic error expansions. Several numerical tests are performed to verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In computer aided geometric design a polynomial is usually represented in Bernstein form. This paper presents a family of compensated algorithms to accurately evaluate a polynomial in Bernstein form with floating point coefficients. The principle is to apply error-free transformations to improve the traditional de Casteljau algorithm. At each stage of computation, round-off error is passed on to first order errors, then to second order errors, and so on. After the computation has been "filtered" (K − 1) times via this process, the resulting output is as accurate as the de Casteljau algorithm performed in K times the working precision. Forward error analysis and numerical experiments illustrate the accuracy of this family of algorithms. [ABSTRACT FROM AUTHOR]

Highlights • To fill a low-rank Toeplitz matrix from a small subset of its entries. • Definition of "Toeplitz structure smoothing operator". • The augmented Lagrange multiplier algorithm with smoothing for TMC problem. • Approximation matrices may hold on Toeplitz structure throughout the iteration. Abstract Toplitz matrix completion (TMC) is to fill a low-rank Toeplitz matrix from a small subset of its entries. Based on the augmented Lagrange multiplier (ALM) algorithm for matrix completion, in this paper, we propose a new algorithm for the TMC problem using the smoothing technique of the approximation matrices. The completion matrices generated by the new algorithm are of Toeplitz structure throughout iteration, which save computational cost of the singular value decomposition (SVD) and approximate well the solution. Convergence results of the new algorithm are proved. Finally, the numerical experiments show that the augmented Lagrange multiplier algorithm with smoothing is more effective than the original ALM and the accelerated proximal gradient (APG) algorithms. [ABSTRACT FROM AUTHOR]

• A novel fractional gradient descent method with adaptive momentum is presented for training BP neural networks. • The convergence of the proposed algorithm is proved. • The proposed fractional algorithm can effectively improve the training performance and expand the selection range of learning rate. A novel fractional gradient descent method with adaptive momentum is presented in this paper to improve the convergence speed and stability for BP neural network training. The fractional Grünwald-Letnikov derivative is used for the fractional gradient. The coefficient of the momentum term is set as an adaptive variable, depending on the fractional gradient of the current step and the weight change of the previous step. We give a detailed convergence proof of the proposed method. Experiments on MNIST data sets and XOR problem demonstrate that the fractional gradient descent method with adaptive momentum term can effectively improve convergence speed, maintain stability of BP neural network training, help escape from local minimum points, and enlarge the selection range of the learning rate. [ABSTRACT FROM AUTHOR]

In this paper, we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval. Based on some matrix computation theories, a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results. Furthermore, the ideas are spread to general symmetric interval matrices. Finally, numerical examples are presented which show the validity of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cramér-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding the trajectories in an appropriate family of occupation measures, we provide the linearisation of such problems in which the continuous control is given by reinsurance policies and the singular one by dividends and capital injections. The linearisation translates into a dual dynamic programming (DDP) algorithm. An important part of the paper is dedicated to structural considerations allowing reasonable implementation. We also hint connections to methods relying on moment sum of squares and LMI (linear matrix inequality)-relaxations to approximate the optimal candidates. [ABSTRACT FROM AUTHOR]

• Algebra matrix method integrates the structural factor and the state information of social networks. • Monte Carlo simulation experiments verify the effectiveness of the algebra matrix evaluation. • The evaluation deviations of diffusion threshold are shown best performance comparing with three popular mean field methods. • The weighted average strategy is proposed as applications. Tacit knowledge is the knowledge existing in human brain which is not easy to be recorded or quantified, and often is learned in the face-to-face interactions. The tacit knowledge diffusion depends on the decision-making of tacit knowledge owners, and the expression of explicit knowledge carriers. However, the comprehensive influence of the tacit knowledge owners, explicit knowledge carriers and the relations of them were not attracted enough attention. In this paper, an algebra matrix method is used to integrate the multidimensional information of network structures and the nodes' states. By the algebra matrix method, the diffusion threshold of the tacit knowledge is calculated, which is called algebra matrix evaluation. This evaluation method is proven to be effective by comparing with Monte Carlo simulations on three types of artificial networks and five reals. With applications of the algebra matrix evaluation, we construct a co-author network according the data of the academic papers from 1980 to 2017 on Aminer platform, and define states of tacit knowledge owners and the explicit knowledge carriers by the scholar's career lengths and the paper's cited quantities respectively. It is found that the thresholds of tacit knowledge diffusion are decreasing with the expansions of the scale of the largest connected components, whether tacit knowledge diffuses in the co-author networks or in the largest connected components. And with the evolution of cumulative co-author network, the diffusion thresholds of tacit knowledge in the largest connected component decrease in ladder-like with unequal steps. Furthermore, it is find ignoring the state factor will lead to the deviation in the evaluation of tacit knowledge diffusion thresholds, which is 16.33% in the largest connected components and 45.07% in the whole network. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a nonlinear edge-preserving diffusion equation with an adaptive source term for binarization of degraded document images. The role of nonlinear diffusion term is to smooth images with preservation of text edges and corners, while the source term is responsible for the desired binarization. Unlike other binarization techniques (such as clustering-based and threshold-based), the idea behind the proposed method is that a sequence of gradually binarized images is obtained by solving the evolution equation starting with the image to be binarized, and tends to the slightly smoothed version of the desired binary image at infinity. A semi-implicit parallel splitting-up method is developed for solving the proposed model effectively. The proposed model with algorithm is tested on the DIBCO (Document Image Binarization Competitions) series datasets. The results show that it has generally the best performance, compared to four PDE (partial differential equation)-based binarization models, and six recent and benchmark binarization algorithms (non-PDE based). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present functional definitions of all types of generalized inverses related to the {1}-inverse, which is a continuation of the work of Campbell and Meyer (2009). According to these functional definitions, we further derive novel representations for all types of generalized inverses related to the {1}-inverse in terms of the bases for R (A *), N (A) and N (A *). Based on these representations, we present the corresponding algorithm for computing various generalized inverses related to the {1}-inverse of a matrix and analyze the computational complexity of our algorithm for a constant matrix. Finally, we implement our algorithm and several known algorithms for symbolic computation of the Moore-–Penrose inverse in the symbolic computational package MATHEMATICA and compare their running times. Numerical experiments show that our algorithm outperforms these known algorithms when applied to compute the Moore–Penrose inverse of one-variable rational matrices, but is not the best choice for two-variable rational matrices in practice. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the supervised backpropagation training of multilayer neural networks from a dynamical systems point of view. We discuss some links with the qualitative theory of differential equations and introduce the overfly algorithm to tackle the local minima problem. Our approach is based on the existence of first integrals of the generalised gradient system with build-in dissipation. [ABSTRACT FROM AUTHOR]

Abstract The shift and invert Arnoldi (SIA) method is a numerical algorithm for approximating the product of Toeplitz matrix exponential with a vector. In this paper, we extend the SIA method to chemical master equation (CME) and propose a SIA algorithm based on the strategy of reorthogonalization (SIRA). We establish a theoretical error of the resulting approximation of SIRA algorithm. Numerical experiments show that the SIRA algorithm is more efficient than the Krylov FSP algorithm in terms of finite models, and the error estimate can be used to determine whether this result obtained by SIRA algorithm is acceptable or not. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the theory on direct and inverse spectral problems for Jacobi matrices is revisited in a kind of pseudo-Jacobi matrices J (n , r , β) with a mixed path as its graph in the non-self-adjoint setting. In this context, a sign change in one of the nondiagonal entries of the matrix yields strong perturbations in its spectral properties. The reconstruction of a pseudo-Jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated. An algorithm for the reconstruction of matrices from prescribed spectral data is provided and illustrative numerical experiments are performed. [ABSTRACT FROM AUTHOR]

Abstract We are interested by an alternating method for an inverse Cauchy problem. Our main results of this paper consists in exhibiting local error indicators, based on a posteriori analysis, with respect to both space approximation and iterative algorithms. In contrast to existing results, the choice of adaptive finite element meshes will be considered after a comparison between the two kinds of indicators. This lead to an efficient method which have the ability to dramatically reduce the computational cost comparing with uniform grid refinement. Some numerical experiments support these results. [ABSTRACT FROM AUTHOR]

Abstract Uncertain fractional differential equation (UFDE) is of importance tool for the description of uncertain dynamic systems. Generally we may not obtain its analytic solutions in most cases. This paper focuses on proposing a numerical method for solving UFDE involving Caputo derivative. First, the concept of α -path to an UFDE with initial value conditions is introduced, which is a solution of the corresponding fractional differential equation (FDE) involving with the same initial value conditions. Then the relations between its solution and associate α -path are investigated. Besides, a formula is derived for calculating expected value of a monotonic function with respect to solutions of UFDEs. Based on the established relations, numerical algorithms are designed. Finally, some numerical experiments of nonlinear UFDEs are given to demonstrate the effectiveness of the numerical algorithms. [ABSTRACT FROM AUTHOR]

Abstract The Laplacian of a graph mathematically formalizes the interactions occurring between nodes/agents connected by a link. Its extension to account for the indirect peer influence through longer paths, weighted as a function of their length, is represented by the notion of transformed d -path Laplacians. In this paper, we propose a second-order consensus protocol based on these matrices and derive criteria for the stability of the error dynamics, which also consider the presence of a communication delay. We show that the new consensus protocol is stable in a wider region of the control gains, but admits a smaller maximum delay than the protocol based on the classical Laplacian. We show numerical examples to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Highlights • Two new Taylor algorithms for the computation of the matrix exponential are proposed. • They are based on matrix polynomial evaluation methods more efficient than Paterson–Stockmeyer method. • In tests they showed higher efficiency and accuracy than state-of-the-art Padé based methods. • They also showed a higher efficiency than state-of-the-art Taylor methods, and higher accuracy in some tests. Abstract This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson–Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances. [ABSTRACT FROM AUTHOR]

Abstract Let S be a set of at least two vertices in a graph G. A subtree T of G is a S-Steiner tree if S ⊆ V (T). Two S -Steiner trees T 1 and T 2 are edge-disjoint (resp. internally disjoint) if E (T 1) ∩ E (T 2) = ∅ (resp. E (T 1) ∩ E (T 2) = ∅ and V (T 1) ∩ V (T 2) = S). Let λ G (S) (resp. κ G (S)) be the maximum number of edge-disjoint (resp. internally disjoint) S -Steiner trees in G , and let λ k (G) (κ k (G)) be the minimum λ G (S) (resp. κ G (S)) for S ranges over all k -subsets of V (G). Clearly, λ 2 (G) (resp. κ 2 (G)) is the classical edge-connectivity λ (G) (resp. connectivity κ (G)). In this paper, we study the λ 3 -connectivity and κ 3 -connectivity of a recursive circulant G , determine λ 3 (G) = δ (G) − 1 for each recursive circulant G , and κ 3 (G) = δ (G) − 1 for each recursive circulant G except G ≅ G (2 m , 2). [ABSTRACT FROM AUTHOR]

Abstract Estimation distribution algorithm (EDA) is an evolution technique that uses sampling to generate the offspring. Most developed EDAs focus on solving the optimization problems which only have the constraints of variable boundaries. In this paper, EDAs are proposed for solving the constrained optimization problems (COPs) involving various types of constraints. In particular, a modified extreme elitism selection method is designed for EDAs to handle the constraints. This selection extrudes the role of some top best solutions to pull the mean vector of the Gaussian distribution towards these best solutions and makes EDAs form a primary evolutionary direction. The EDAs based on five different Gaussian distribution with this selection are evaluated using a set of benchmark functions and some engineering design problems. It is found that for solving these problems, the EDA that is based on a single multivariate Gaussian distribution model with the modified extreme elitism selection outperforms the other EDAs and some state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

Abstract In this paper, two iterative algorithms are proposed to solve stochastic algebraic Riccati matrix equations arising in the linear quadratic optimal control problem of linear stochastic systems with state-dependent noise. In the first algorithm, a standard Riccati matrix equation needs to be solved at each iteration step, and in the second algorithm a standard Lyapunov matrix equation needs to be solved at each iteration step. In the proposed algorithms, a weighted average of the estimates in the last and the previous steps is used to update the estimate of the unknown variable at each iteration step. Some properties of the sequences generated by these algorithms under appropriate initial conditions are presented, and the convergence properties of the proposed algorithms are analyzed. Finally, two numerical examples are employed to show the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Diversity is an important measure that according to the context, can describe different concepts of general interest: competition, evolutionary process, immigration, emigration and production among others. It has been extensively studied in different areas, as ecology, political science, economy, sociology and others. The quality of spatial context of the city can be gauged through this measure. The spatial context with its corresponding dataset can be modelled using spatial networks. Consequently, this allows us to study the diversity of data present in this specific type of networks. In this paper we propose an algorithm to measure diversity in spatial networks based on the topology and the data associated to the network. In the experiments developed with networks of different sizes, it is observed that the proposed index is independent of the size of the network, but depends on its topology. [ABSTRACT FROM AUTHOR]

The solution of discrete ill-posed problems has been a subject of research for many years. Among the many methods described in the literature, the Bregman algorithm has attracted a great deal attention and been widely investigated. Recently, a nonstationary preconditioned version of this algorithm, referred to as the nonstationary modified linearized Bregman algorithm, was proposed. The aim of this paper is to discuss numerical aspects of this algorithm and to compare computed results with known theoretical properties. We also discuss the effect of several parameters required by the algorithm on the computed solution. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples. [ABSTRACT FROM AUTHOR]

The Wiener polarity index (usually denoted by W p ) of a graph G is defined as the number of unordered pairs of the vertices of G which are at distance 3. Denote by CT n the family of all n -vertex chemical trees. In a recent paper, Ashrafi and Ghalavand [1] determined the first three minimum W p values of n -vertex chemical trees for n  ≥ 7 and characterized the chemical trees attaining the first two minimum W p values among all the members of CT n for n  ≥ 4. In this note, the chemical trees with the third minimum W p value are characterized from the graph family CT n for n  ≥ 7, and the chemical trees from the family CT n , n  ≥ 4, with the first two minimum W p values are also obtained in an alternative but shorter way. [ABSTRACT FROM AUTHOR]

In this paper, we propose an improved empirical mode decomposition (EMD) method, termed IEMD, with modified envelope algorithm based on C 2 piecewise rational and C 2 monotone piecewise rational cubic spline interpolations, for the decomposition of nonlinear and non-stationary EMI signals. In the sifting procedure, we first construct the upper and lower envelopes employing C 2 piecewise rational cubic spline interpolation (PRCSI) technique. Considering the existence of undershoots, we further modify the original envelopes iteratively with C 2 monotone piecewise rational cubic spline interpolation (MPRCSI) technique, for the elimination of undershoots as accurate as possible. Experiments on synthetic and real signals, compared with three improved versions of EMD, three for ensemble EMD (EEMD) and TL-SVD, demonstrate that the proposed method is more valid and flexible when applied to the decomposition of synthesis harmonic signals and real EEG signals, signifying it also can be applied to the decomposition of EMI signals which are known to be extremely nonlinear and non-stationary. [ABSTRACT FROM AUTHOR]

Ensemble clustering has attracted much attention for its robustness, stability, and accuracy in academic and industry communities. In order to yield base clusterings with high quality and diversity simultaneously in ensemble clustering, many efforts have been done by exploiting different clustering models and data information. However, these methods neglect correlation between different base clusterings during the process of base clusterings generation, which is important to obtain a quality and diverse clustering decision. To overcome this deficiency, a sequential ensemble clusterings generation algorithm for mixed data is developed in this paper based on information entropy. The first high quality base clustering is yield by maximizing the entropy-based criterion. Afterward, a sequential paradigm is utilized to incrementally find more base clusterings, in which the diversity between a new base clustering and the former base partitions is measured by the normalized mutual information. Extensive experiments conducted on various data sets have demonstrated the superiority of our proposal as compared to several existing base clusterings generation algorithms. [ABSTRACT FROM AUTHOR]

The fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique, Sumudu transform approach and homotopy polynomials to obtain the approximate solution of a nonlinear fractional Drinfeld–Sokolov–Wilson equation. The nonlinear Drinfeld–Sokolov–Wilson equation naturally occurs in dispersive water waves. The uniqueness and convergence analysis are shown for the suggested technique. The convergence of the solution is fixed and managed by auxiliary parameter ℏ. The numerical results are shown graphically. Results obtained by the application of the technique disclose that the suggested scheme is very accurate, flexible, effective and simple to use. [ABSTRACT FROM AUTHOR]

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Finding efficient algorithms for computing characteristic polynomial of graphs is an active area of research and for some graph classes, like threshold graphs, there exist very fast algorithms which exploit combinatorial structure of the graphs. In this paper, we put forward some novel ideas based on divisor technique to obtain fast algorithms for computing the characteristic polynomial of threshold and chain graphs. [ABSTRACT FROM AUTHOR]

This paper aims to study the stochastic stabilization problem of complex networks without strong connectedness (CNSC). To deal with large-scale complex networks which are not strongly connected, a hierarchical method and a hierarchical algorithm are given, respectively. Meanwhile, we construct a logarithmic Lyapunov function for CNSC. Next, based on the theory of asymptotically autonomous systems, the Lyapunov method and the graph theory, the whole complex network can be stabilized by stabilizing a part of nodes. Then, stability criteria of CNSC are given, whose conditions reflect the relationship between dynamic properties and topology structure clearly. Furthermore, the theoretical results are applied to coupled oscillators on CNSC to ensure the stability. Finally, a numerical example is provided to illustrate the effectiveness and practicability of the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider several scheduling problems on a serial-batch machine for scheduling jobs with or without precedence relations. Under the serial-batch setting, the jobs in a batch are processed in succession and are removed until the last job in this batch finishes its processing. Thus, the processing time of a batch is equal to the sum of processing times of jobs in the batch. When a new batch starts, a constant setup time is required for the machine. The objectives of the problems involve minimizing makespan and a maximum cost. For these problems, we either present polynomial-time algorithms to generate all Pareto optimal points and find a corresponding Pareto optimal schedule for each Pareto optimal point, or give the strong NP-hardness proof. Experimentation results show that the proposed algorithms for the considered problems are very efficient. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the static output feedback set stabilization for context-sensitive probabilistic Boolean control networks (CS-PBCNs) via the semi-tensor product of matrices. An algorithm for finding the largest control invariant set with probability one is obtained by the algebraic representations of logical dynamics. Based on the analysis of the set stabilization, necessary and sufficient conditions for S -stabilization are obtained. Static output feedback controllers are designed to achieve S -stabilization for a CS-PBCN. At last, examples to study metastatic melanoma are given to show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, a class of differential variational inequalities are studied, and a new approach is introduced to relax the convexity condition. Firstly, an existence theorem of Carathéodory weak solution for the differential variational inequalities is established. Secondly, an algorithm for solving the problem is developed and the convergence analysis for the algorithm is given. Finally, a numerical example is reported to illustrate the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new numerical algorithm for solving the Riesz tempered space fractional diffusion equation. The stability and convergence of the numerical scheme are discussed via the technique of matrix analysis. Finally, numerical experiments are performed to confirm the effectiveness of our numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to estimate the biophysical parameters in an improved version of Morris–Lecar (M–L) neuron model in a noisy environment. To observe the influence of noisy stimulation in estimation procedure, a Gaussian white noise has been added to the membrane voltage of the model system. Estimation of the parameters has been investigated by a proposed algorithm. The denoising technique (local projection method) has been applied to reduce the influence of noisy stimuli and the effectiveness of the method is reported. The proposed scheme performs well for an excitable neuron model and provides good estimates between the estimated parameters and the actual values in a reasonable way. This approach can be used for parameter estimation for other nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we present an adaptive method for the TV-based model of three norms L q ( q = 1 2 , 1 , 2 ) for the image restoration problem. The algorithm with the L 2 norm is used in the smooth regions, where the value of |∇ u | is small. The algorithm with the L 1 2 norm is applied for the jumps, where the value of |∇ u | is large. When the value of |∇ u | is moderate, the algorithm with the L 1 norm is employed. Thus, the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted. The numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images. Specifically, in contrast to the L 1 norm as in the Rudin–Osher–Fatemi model, the L 2 norm yields better results in the smooth and flat regions, and the L 1 2 norm is more suitable in regions with strong discontinuities. Therefore, our adaptive algorithm is efficient and robust even for images with large noises. [ABSTRACT FROM AUTHOR]

In this paper, we provide a novel fractional particle swarm optimization (FPSO) algorithm. The traditional PSO is one of the most well-known bio-inspired algorithms used in optimization problems, which basically consists of a number of particles that collectively move in search of the global optimum. Nevertheless, despite its success over the past 20 years, the PSO is also known to be unable to converge, and even stagnate, in many complex problems with multiple local optima. In order to overcome this drawback, this paper proposes a modified version of the PSO algorithm, considering a fractional calculus approach. Stability results evaluation is carried out to analytically prove the convergence of the fractional extensions. This is naturally followed by simulation results to test the fractional-based PSOs under several well-known objective functions, thus highlighting the relationship between the fractional order velocity and position of particles with the convergence of the algorithm. Experimental results show that the FPSO and its variants significantly outperform the traditional PSO. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iterative method that is the combination of the proximal point algorithm, viscosity approximation method and alternating resolvent method for finding the common zeros of two accretive operators in Banach spaces. And we will prove the strong convergence theorems for the iterative algorithms and give the example of the main theorems. The results of this paper are improvements and extensions of the corresponding ones announced by many others. [ABSTRACT FROM AUTHOR]

This paper investigates the Pareto optimization scheduling problem on a single machine with two competing agents A and B in which agent A wants to minimize the number of tardy A -jobs and agent B wants to minimize the maximum cost of B -jobs. In the literature, the constrained optimization problem of minimizing the number of tardy A -jobs under the restriction that the maximum cost of B -jobs is bounded is solved in polynomial time. This implies that the corresponding Pareto optimization scheduling problem can be solved in a weakly polynomial time. In this paper, by presenting a new algorithm for the constrained optimization problem, we provide a strongly polynomial-time algorithm for the corresponding Pareto optimization scheduling problem. Experimentation results show that the proposed algorithm for the considered problem is efficient. [ABSTRACT FROM AUTHOR]

Air pollution models plays a critical role in atmospheric environment research. Chemical kinetic equations is an important component of air pollution models. The chemical equations is numerically sticky because of its stiffness, nonlinearity, coupling and nonnegativity of the exact solutions. Over the past decades, numerous papers about chemical equation solvers have been published. However, these solvers cannot preserve the nonnegativity of the exact solutions. Therefore, in the calculation, the negative numerical concentration values are usually set to zero artificially, which may cause simulation errors. To obtain real nonnegative numerical concentration values, very small step-size has to be adopted. Then enormous amount of CPU time is consumed to solve the chemical equations. In this paper, we revisit this topic and derive a new algorithm. Our algorithm Modified-Backward-Euler (MBE) Method can unconditionally preserve the nonnegativity of the exact solutions. MBE is a simple, robust and efficient solver. It is much faster and more precise than the traditional solvers such as LSODE and QSSA. The numerical results and parameter suggestions are shown at the end of the paper. MBE is based on the P-L structure of the chemical equations and a deeper view into the nature of Euler Methods. It cannot only be used to solve chemical equations, but can also be applied to conquer ordinary differential equations (ODEs) with similar P-L structure. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations A 1 X = C 1 , Y B 1 = D 1 , A 2 W = C 2 , Z B 2 = D 2 , A 3 V = C 3 , V B 3 = C 4 , A 4 V B 4 = C 5 , A 5 X + Y B 5 + C 6 W + Z D 6 + E 6 V F 6 = G 6 , and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper. [ABSTRACT FROM AUTHOR]