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Recognizing the sensitivity of vegetation to precipitation is fundamental to monitoring and assessing ecosystem health and predicting ecosystem response to climate change. However, there is a lack of detailed, large-scale assessment of the sensitivity of temperate vegetation to climate variability. We analyzed the spatiotemporal trends of NDVI in Inner Mongolia (IM) for the period from 2000 to 2020, evaluated the sensitivity of NDVI to precipitation by constructing a linear mixed-effects model, and assessed the relative importance of an array of biotic and abiotic factors in affecting the vegetation sensitivity to precipitation using random forest models. We found that (1) the NDVI in IM had a significant or slight increase in 80.86% of the region during the studied two decades, though it showed a large inter-annual fluctuation, and the areas with stable or decreasing NDVI were mainly in the desert areas. Precipitation and wind speed were the two major drivers for the NDVI changes. (2) The typical steppes demonstrated the highest sensitivity to precipitation, followed by desert steppes and forest steppes, broad-leaved forests, and deserts and coniferous forests. (3) The major factor affecting the vegetation sensitivity to precipitation varied with vegetation types; it is temperature in the coniferous forests, forest steppes and steppe deserts, but it is the sunshine hours in the deciduous broad-leaved forests, the typical steppes, and the desert steppes. The high sensitivity of the typical steppe vegetation to precipitation is related with its location on climate gradient, it has a high production potential but limited by precipitation under a semiarid climate. Our analysis deepens the understanding of the NDVI sensitivity to precipitation and its influencing factors in different vegetation regions, and is valuable for predicting vegetation changes and developing vegetation management strategies to adapt to climate changes.

Abstract The research of resistive network will become the basis of many fields. At present, many exact potential formulas of some complex resistor networks have been obtained. Computer numerical simulation is the trend of computing, but written calculation will limit the time and scale. In this paper, the potential formulas of a $$m\times n$$ m × n scale cobweb resistor network and fan resistor network are optimized. Chebyshev polynomial of the second class and the absolute value function are used to express the novel potential formulas of the resistor network, and described in detail the derivation process of the explicit formula. Considering the influence of parameters on the potential formulas, several idiosyncratic potential formulas are proposed, and the corresponding three-dimensional dynamic images are drawn. Two numerical algorithms of the computing potential are presented by using the mathematical model and DST-VI. Finally, the efficiency of calculating potential by different methods are compared. The advantages of new potential formulas and numerical algorithms by the calculation efficiency of the three methods are shown. The optimized potential formulas and the presented numerical algorithms provide a powerful tool for the field of science and engineering.

In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.

The establishment of a resistor network model has become a sharp edge to solve complex scientific problems. In this paper, we introduce Chebyshev polynomials to express the potential formula of the hammock resistor network and improve the general solution of the hammock resistor network. Moreover, through the change in different parameters, special potential formulas are proposed and displayed in 3D dynamic view. A fast algorithm of the calculating potential is given by using the matrix equation model, discrete cosine transform-II, and the fast matrix-vector multiplication. Finally, we show the advantages of our improved potential formula and fast algorithm by the calculation efficiency of the three methods. The modified potential formula and the presented fast algorithm provide a new tool for the field of science and engineering.

Abstract Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an $$m\times n$$ m × n globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

Desertification is of significant concern as one of the world’s most serious ecological and environmental problems. China has made great achievements in afforestation and desertification control in recent years. The climate varies greatly across northern China. Using a long-time series of remote sensing data to study the effects of desertification will further the understanding of China’s desertification control engineering and climate change mechanisms. The moist index was employed in this research to determine the climate type and delineate the potential occurrence range of desertification in China. Then, based on the Google Earth Engine platform, MODIS data were used to construct various desertification monitoring indicators and applied to four machine learning models. By comparing different combinations of indicators and machine learning models, it was concluded that the random forest model with four indicator combinations had the highest accuracy of 86.94% and a Kappa coefficient of 0.84. Therefore, the random forest model with four indicator combinations was used to monitor desertification in the study area from 2000 to 2020. According to our studies, the area of desertification decreased by more than 237,844 km2 between 2000 and 2020 due to the impact of human activities and in addition to climatic factors such as the important role of precipitation. This research gives a database for the cause and control of desertification as well as a reference for national-scale desertification monitoring.

Abstract In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n × n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n × n Foeplitz matrix is sparse and can be expressed by the nth and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n × n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n × n ( n > 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.

In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, and g-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaan g-circulant matrix.

It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant, g-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci g-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.

Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant and g-circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant and g-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and g-circulant matrices by utilizing the relation between left circulant, g-circulant matrices, and circulant matrix, respectively.

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and g-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant and g-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and g-circulant matrices by utilizing the relation between left circulant, and g-circulant matrices and circulant matrix, respectively.

We first give the block style spectral decomposition of arbitrary block skew circulant matrix with skew circulant blocks. Secondly, we obtain the singular value of block skew circulant matrix with skew circulant blocks as well. Finally, based on the block style spectral decomposition, we deal with the optimal backward perturbation analysis for the block skew circulant linear system with skew circulant blocks.

Circulant matrices may play a crucial role in solving various differential equations. In this paper, the techniques used herein are based on the inverse factorization of polynomial. We give the explicit determinants of the RFPrLrR circulant matrices and RLPrFrL circulant matrices involving Fibonacci, Lucas, Pell, and Pell-Lucas number, respectively.

Circulant matrices play an important role in solving delay differential equations. In this paper, circulant type matrices including the circulant and left circulant and g-circulant matrices with any continuous Fibonacci and Lucas numbers are considered. Firstly, the invertibility of the circulant matrix is discussed and the explicit determinant and the inverse matrices by constructing the transformation matrices are presented. Furthermore, the invertibility of the left circulant and g-circulant matrices is also studied. We obtain the explicit determinants and the inverse matrices of the left circulant and g-circulant matrices by utilizing the relationship between left circulant, g-circulant matrices and circulant matrix, respectively.

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra of n×n complex skew-circulant matrices are displayed in this paper.

Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS R-factor block circulant (retrocirculant) matrix over field F. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field F is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of AX=b in the class of the level-2 FLS (R,r)-circulant(retrocirculant) matrix of type (m,n) over field F are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

Circulant matrices have important applications in solving various differential equations. The level-k scaled factor circulant matrix over any field is introduced. Algorithms for finding the minimal polynomial of this kind of matrices over any field are presented by means of the algorithm for the Gröbner basis of the ideal in the polynomial ring. And two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for computing the inverse of partitioned matrix with level-k scaled factor circulant matrix blocks over any field is given by using the Schur complement, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo prime number.

Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with the k-Fibonacci and k-Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

Circulant matrices have become important tools in solving integrable system, Hamiltonian structure, and integral equations. In this paper, we prove that Gaussian Fibonacci circulant type matrices are invertible matrices for n>2 and give the explicit determinants and the inverse matrices. Furthermore, the upper bounds for the spread on Gaussian Fibonacci circulant and left circulant matrices are presented, respectively.

Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degree n, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

In this paper, we first prove a relation between the critical points of the skew-circulant matrix and the eigenvalues of its principal matrix. Furthermore, we reprove the inequality about the zeros of a polynomial and its critical points by using the properties of skew-circulant matrix, which is to show that we can not only find the skew-circulant matrix, but also give more structure matrices to prove this inequality. [ABSTRACT FROM AUTHOR]