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Traditional subspace clustering methods usually replace the matrix rank function by the matrix kernel norm to recover the original low rank matrices. However, in the process of minimizing the matrix kernel norm, these algorithms pay too much attention to the calculation of the large singular values of the matrix, resulting in inaccurate estimation of the matrix rank. To this end, this paper analyzed the long tail distribution of matrix singular values and proposed a low rank subspace clustering model based on truncated Schatten-p norm. The proposed model fitted the long tail distribution of matrix singular va-lues well and toke full account of the contribution of small singular values to the process of low rank matrix recovery. The mo-del could make full use of small singular values to fit the long tail distribution of matrix singular values, ultimately achieved an accurate estimation of matrix rank function and improved the performance of subspace clustering. The experimental results show that, compared with the WNNM-LRR and BDR subspace clustering algorithms, the proposed method improves the clustering accuracy by 11% and 8% on Extended Yale B dataset, respectively. The proposed method can better fit the distribution of data singular values and construct the similarity matrices more accurately. [ABSTRACT FROM AUTHOR]

Sensitivity analysis of random systems may convey important information on the dynamical properties of the system. In this paper, we determine the effects of parameters' perturbation on two dynamic poverty indexes: the headcount ratio and the income gap ratio. This is achieved by perturbing the generator of the Markov process governing the evolution in time of the economic agents among three classes of income, the initial distribution of individuals and the vector of mean income for poverty class. The paper presents two bounds on the aforementioned poverty indexes which show how the perturbations on the model parameters propagate on the poverty indexes. The paper contributes to the literature presenting for the first time effective bounds for dynamic poverty indicators and exploring the applicability of the perturbation approach to real data. [ABSTRACT FROM AUTHOR]

We define row path norm and column path norm of a matrix and relate path norms with other standard matrix norms. A row (resp. column) path norm gives a path that maximizes relative row (resp. column) distances starting from the first row (resp. column). The comparison takes place from the last row (resp. column) to the first row (resp. column), tracing the path. We categorize different versions of path norms and provide algorithms to compute them. We show that brute-force methods to compute path norms have exponential running time. We give dynamic programming algorithms, which, in contrast, take quadratic running time for computing the path norms. We define path norms on Church numerals and Church pairs. Finally, we present applications of path norms in computing condition number, and ordering the solutions of magic squares and Latin squares [ABSTRACT FROM AUTHOR]

This paper considers the perturbation analysis of a class of fully fuzzy linear systems in which the coefficient matrix is a positive fuzzy matrix. The original fuzzy linear systems is extended into a brand new and simple crisp matrix equation using an embedding method. By discussing the perturbation of the extended crisp linear equation, the paper completes the perturbation analysis of the original fuzzy linear system. There are three cases of perturbation are analysed and the respective relative error bounds for solutions of fuzzy linear system are derived. Some numerical examples are given to illustrated our obtained results. [ABSTRACT FROM AUTHOR]

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory. [ABSTRACT FROM AUTHOR]

In the present work, we define harmonic complex balancing numbers by considering well-known balancing numbers and inspiring harmonic numbers. Mainly, we investigate some of their basic characteristic properties such as the Binet formula and Cassini identity, etc. In addition, one type of symmetric matrix family whose entries are harmonic complex balancing numbers is constructed. Additionally, some linear algebraic properties are obtained. Furthermore, some inequalities are stated by exploiting the well-known inequalities between various matrix norms. Finally, we illustrate the results with some numerical examples. [ABSTRACT FROM AUTHOR]

We characterize the infimum of a matrix norm of a square matrix A induced by an absolute norm, over the fields of real and complex numbers. Usually this infimum is greater than the spectral radius of A. If A is sign equivalent to a nonnegative matrix B then this infimum is the spectral radius of B. [ABSTRACT FROM AUTHOR]

This paper considers the minimax estimation of the high-dimensional sparse covariance matrices in the presence of missing observations. Based on random missing data, the upper bounds of convergence rate about the data-driven thresholding estimator are constructed over a large class of sparse covariance matrices under the l 1 norm and the Frobenius norm. In addition, we use Le Cam's lemma and the relation between the total variation affinity and the Kullback–Leibler divergence to establish the lower bounds which illustrate the desired upper bounds cannot be improved. It is worth mentioning that the approach we adopt to get the lower bounds is simpler than the existing ones. [ABSTRACT FROM AUTHOR]

For G ∈ R m × n and g ∈ R m , the minimization min ∥ G ψ − g ∥ 2 , with ψ ∈ R n , is known as the Tykhonov regularization. We transport the Tykhonov regularization to an infinite-dimensional setting, that is min ∥ T (h) − k ∥ , where T : H → K is a continuous linear operator between Hilbert spaces H , K and h ∈ H , k ∈ K . In order to avoid an unbounded set of solutions for the Tykhonov regularization, we transform the infinite-dimensional Tykhonov regularization into a multiobjective optimization problem: min ∥ T (h) − k ∥ and min ∥ h ∥ . We call it bounded Tykhonov regularization. A Pareto-optimal solution of the bounded Tykhonov regularization is found. Finally, the bounded Tykhonov regularization is modified to introduce the precise Tykhonov regularization: min ∥ T (h) − k ∥ with ∥ h ∥ = α . The precise Tykhonov regularization is also optimally solved. All of these mathematical solutions are optimal for the design of Magnetic Resonance Imaging (MRI) coils. [ABSTRACT FROM AUTHOR]

In this paper, we give the formula for the Euclidean norm of circulant matrix with the Guassian Pell numbers. Moreover, we find the upper and lower bounds of the spectral norm of this matrix. Furthermore, we obtain the silver ratio in Gaussian Pell numbers. [ABSTRACT FROM AUTHOR]

In this article, a new concept of the Complex Neutrosophic Matrix is introduced to solve different problems related to uncertainties. Based on the proposed matrix, we have provided various algebraic operations like addition, subtraction, union many others which will be of great help in estabalishing the fundamental concepts. The results obtained through the above operations will be consequently utilized in solving the higher-dimensional problems due to their matrix properties. This novel concept lays the foundation of various research solutions in the field of the complex neutrosophic matrix, which are yet to be considered by researchers.The matrix norm convergence of the proposed matrix has also beenstudied for the necessary foundation of the complex neutrosophic matrix. Also, the future studies of the proposed concept have been proposed. [ABSTRACT FROM AUTHOR]

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics. [ABSTRACT FROM AUTHOR]

In this study, we have defined a Max matrix as ... and have examined its some properties, such as determinant, inverse and norm, where Ln denotes the nth Lucas number. First, we have given the determinant and inverse of matrix matrix C by using known results for general Max matrix. Then we have established equality for Euclidean norm and an upper bound for the spectral norm of matrix C. Finally, we have computed the determinant and inverse of Hadamard inverse of matrix C. [ABSTRACT FROM AUTHOR]

Fan, Wang, and Zhong estimate the difference between the singular vectors of a matrix and those of a perturbed matrix in terms of the maximum norm. Their estimations are used effectively to establish the asymptotic properties of robust covariance estimators (see Journal of Machine Learning Research, 2018;18:1‐42). In this paper, we give the corresponding lower bound estimates, which show Fan‐Wang‐Zhong's estimations optimal. [ABSTRACT FROM AUTHOR]

The problem is, given a set of n vectors in a d -dimensional normed space, find a subset with the largest length of the sum vector. We prove that, in the case of the ℓ p norm, the problem is APX-complete for any p ∈ [ 1 , 2 ] and is not in APX if p ∈ (2 , ∞). In the case of an arbitrary norm, we propose an algorithm which finds an optimal solution in time O (n d − 1 (d + log ⁡ n)) , improving previously known algorithms. In particular, the two-dimensional problem can be solved in nearly linear time. We also present an improved algorithm for the cardinality-constrained version of the problem with running time O (d n d + 1). In the two-dimensional case, this version is shown to be solvable in nearly quadratic time. [ABSTRACT FROM AUTHOR]

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations AX≈B and AXC≈B. This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory. [ABSTRACT FROM AUTHOR]

Let ϕ be a subadditive weight on a C* -algebra A, and let Mϕ+ be the set of all elements x in A+ with ϕ(x) < +00. A seminorm ‖ • ‖ is introduced on the lineal Mϕsa = linRMϕ+, and a sufficient condition for the seminorm to be a norm is given. Let I be the unit of the algebra A, and let ϕ(I) = 1. Then, for every element x of Asa, the limit ρϕ(x) = limt→0+(ϕ(I + tx) - 1)/t exists and is finite. Properties of ρϕ are investigated, and examples of subadditive weights on C* -algebras are considered. On the basis of Lozinskii's 1958 results, specific subadditive weights on Mn(C) are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained. [ABSTRACT FROM AUTHOR]

Given multiple univariate or multivariate real polynomials f 1 , f 2 , ... , f m and a desired real zero z , we study the problem of computing a real polynomial f ˜ such that f ˜ (z) = 0 and the distance between f ˜ and f 1 , f 2 , ... , f m is minimal, where the distance is measured by a pair of norms (‖ ⋅ ‖ p , ‖ ⋅ ‖ q). As the difficulty of finding solutions to this problem relies on selection of the norm pair (p , q) , previous works just discussed some special cases, e.g., (p , q) = (2 , 2) , (2 , ∞) and (∞ , ∞) , where the corresponding problems were translated in geometric views. In this paper, we revisit this problem from an optimization view for all the cases of (2 , q) with q ∈ (0 , 2 ]. That is, instead of directly finding the nearest polynomial, we first transform the problem into solving a constrained minimization problem based on l 2 , q -norm minimization. For all q in (0 , 2 ] , then a unified iterative algorithm is proposed to solve the constrained optimization problem and also the convergence is uniformly demonstrated. Finally, a simple example is presented to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this study, Symmetric-Skew Symmetric Splitting (SSS) method is introduced to solve the system of linear equations. Then, the solution of AXB + CXD + EXF = M generalized Sylvester transpose matrix equation is established by using this method. Lastly, an example is given to illustrate the performance of the SSS method. [ABSTRACT FROM AUTHOR]

Bounding the number of morphisms between compact Riemann surfaces is a long standing problem coming from complex and algebraic geometry. We show that linear algebra techniques allow to improve the known results when we assume a kind of condition number bound for the period matrix. [ABSTRACT FROM AUTHOR]

This paper presents a systematic study of the space complexity of estimating the Schatten p-norms of an n x n matrix in the turnstile streaming model. Both kinds of space complexities, bit complexity and sketching dimension, are considered. Furthermore, two sketching models, general linear sketching and bilinear sketching, are considered. When p is not an even integer, we show that any one-pass algorithm with constant success probability requires near-linear space in terms of bits. This lower bound holds even for sparse matrices, i.e., matrices with O(1) nonzero entries per row and per column. However, when p is an even integer, we give for sparse matrices an upper bound which, up to logarithmic factors, is the same as estimating the pth moment of an n-dimensional vector. These results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices. Similar near-linear lower bounds are obtained for Ky Fan norms, SVD entropy, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to this work. The results for general linear sketches give separations in the sketching complexity of Schatten p-norms with the corresponding vector p-norms, and rule out a table-lookup nearest-neighbor search for p = 1, making progress on a question of Andoni. The results for bilinear sketches are tight for the rank problem and nearly tight for p ≥2; the latter is the first general subquadratic upper bound for sketching the Schatten norms. [ABSTRACT FROM AUTHOR]

In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectral radius and the norms of the related matrices, respectively. Then, by means of the argument principle, stability criteria are presented which are necessary and sufficient conditions for the stability of matrix polynomials. Furthermore, a numerical algorithm is provided for checking the stability of matrix polynomials. Numerical examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Using matrix norm techniques, we give a unified convergence analysis of a general projection method for computing stationary density functions of Frobenius-Perron operators associated with piecewise C 2 and stretching transformations of the unit interval. [ABSTRACT FROM AUTHOR]

• The controllability of the reactor point kinetics equations is analyzed. • There exists inconsistency between the results from two kinds of approaches. • Degree of controllability is defined by matrix norm and singular value decomposition. The controllability of linearized equations of the equivalent normalized version of the reactor point kinetics equations with six delayed neutron groups and without neutron source at an equilibrium power is analyzed. There exists inconsistency between the results from two kinds of approaches. The equations analyzed are controllable according to the controllability evaluation by analytical derivation while they are uncontrollable by numerical calculation. The reason of this inconsistency is that the large difference between the time effect of prompt neutrons and that of delayed neutrons in reactors reduces the controllability matrix to become a rank-deficient matrix under the conditions of using finite-precision arithmetic. In order to improve the analysis approach and address the divergence in outcomes, a novel definition of degree of controllability is proposed by means of applying matrix norm and singular value decomposition (SVD). This definition resolves the inconsistent conclusions, i.e., the equations analyzed are controllable but their degree of controllability is very low. The calculation of the degree of controllability shows that in the multiple representations of point reactor kinetics, the representations of few-group models are more propitious to analyze and design the control system of nuclear reactor than the representation of six-group model. [ABSTRACT FROM AUTHOR]

A new dynamic multi-attribute group decision making method based on matrix grey target decision model is proposed. The attributes' data of alternatives about decision makers in different stages are represented by matrices, and they are considered as performance values in 4-dimensional space. The best, worst attributes' values in other 3-dimensions formed the new matrices, which are defined as expected bull's-eye, unexpected bull's-eye, and then the deviations of alternatives and expected, unexpected bull's eye are presented using matrix norm. The alternatives are ranked by the deviations. Finally, the examples are provided to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

Experimentally, the generalized phase shift can be realized by a simple phase shift introduction device which adjusting the angle of glass accurately, unlike the traditional method of constant phase shift introduced" is changed to "Experimentally, the generalized phase shift can be realized by a simple device which adjusting the angle of glass accurately. [ABSTRACT FROM AUTHOR]

In this paper, an iterative algorithm is constructed to improve the solution of a system of linear equations, of the form Ax = b, when it is solved using the Gauss-Jordan method and by using finite arithmetic. Understanding the functioning of the algorithm, showing its scope and analyzing how it is deduced, is achieved through the concept of matrix norm, together with some of its properties. The concept of the condition number of a matrix is introduced, and are found bounds for it by using the matrix norms. Finally, it is explained the iterative algorithm of the Refinement, showing the power of this one, when it is solved a system of linear equations ill conditioned. [ABSTRACT FROM AUTHOR]

Controlling switched systems is a difficult task, even when dealing with piecewise linear systems (CPWLs), which consist of a set of linear differential equations and a set of switching conditions. This difficulty is largely due to the loss of linearity in the entire system, and it is necessary to solve differential and algebraic equations to determine the solution. In this paper, a new method to tune the parameters of the controllers applied to switched systems is derived using information from the saltation matrix, particularly its induced norm. First, the parameters are tuned using classical methods, and then, after analyzing the norm of the saltation matrix, a new set of parameters that guarantees the stability of the period-1 orbit is obtained. The method is validated using analytical solutions for two different systems (boost and boost-flyback power converters) and is also experimentally validated for the boost-flyback power converter. [ABSTRACT FROM AUTHOR]

Some optimal choices for a parameter of the Dai–Liao conjugate gradient method are proposed by conducting matrix analyses of the method. More precisely, first the $\ell _{1}$ and $\ell _{\infty }$ norm condition numbers of the search direction matrix are minimized, yielding two adaptive choices for the Dai–Liao parameter. Then we show that a recent formula for computing this parameter which guarantees the descent property can be considered as a minimizer of the spectral condition number as well as the well-known measure function for a symmetrized version of the search direction matrix. Brief convergence analyses are also carried out. Finally, some numerical experiments on a set of test problems related to constrained and unconstrained testing environment, are conducted using a well-known performance profile. [ABSTRACT FROM AUTHOR]

Abstract Let S = { x 1 , x 2 , … , x n } be a set of distinct positive integers, and let f be an arithmetical function. The GCD matrix (S) f on S associated with f is defined as the n × n matrix having f evaluated at the greatest common divisor of x i and x j as its ij entry. The LCM matrix [ S ] f is defined similarly. We consider inertia, positive definiteness and ℓ p norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions. [ABSTRACT FROM AUTHOR]

We propose an alternative proof of Pellet's theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouché's theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet's theorem. [ABSTRACT FROM AUTHOR]

The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between the supporting vectors of a real matrix A and the Tikhonov regularization minx∊Rn ǁAx - bǁ + αǁXǁ reaching the conclusion that, by an appropriate choice of b and , the supporting vectors of A can be obtained via solving the Tikhonov regularization minx∊Rn ǁAx - bǁ + αǁXǁ [ABSTRACT FROM AUTHOR]

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L2 on the interval (-1, 1) with the weight (1-t2)α and (1-t2)β, respectively. Under the assumption that β -α is larger than the order of the derivative, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, first, we show that the following optimization problem always has an optimal solution which does not depend on the matrix norm, and give a sufficient condition for the uniqueness of optimal solutions:(Section.Display) whereis any unitarily invariant norm,, andor. Then, we characterize the optimal solutions and also give sufficient conditions for the uniqueness and the rank-one property of optimal solutions for the following optimization problem:(Section.Display) where, is the inner product of matrices;,,, and’s,are different arbitrary norms. [ABSTRACT FROM AUTHOR]

We consider a problem of determining the component-wise distance (called the radius) of a linear system of equations or inequalities to a system that is either solvable or unsolvable. We propose explicit characterization of these radii and show relations between them. Then the radii are classified in the polynomial vs. NP-hard manner. We also present a generalization to an arbitrary linear system consisting from both equations and inequalities with both free and nonnegative variables. Eventually, we extend the concept of the component-wise distance to a non-uniform one. [ABSTRACT FROM AUTHOR]

In this paper, we compute the norms of circulant matrices with the complex Fibonacci and Lucas numbers. Moreover, we give golden ratio in complex Fibonacci numbers. [ABSTRACT FROM AUTHOR]

By means of the properties of structured matrices from the design of Hopfield neural networks, we establish the necessary and sufficient conditions for the solvability of the inverse eigenvalue problem A X = X Λ in structured matrix set SAR J n . In the case where A X = X Λ is solvable in SAR J n , we derive the generalized representation of the solutions. In addition, in corresponding solution set of the equation, we provide the explicit expression of the nearest matrix to a given matrix in the Frobenius norm. [ABSTRACT FROM AUTHOR]

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is concerned with finite sums of reciprocals of Fibonacci numbers. We obtain the spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this paper, we study norms of circulant and r-circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study norms of some circulant matrices and some special matrices, which entries consist of harmonic Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this study we investigate the monotonic behavior of the smallest eigenvalue t n and the largest eigenvalue T n of the n × n matrix E n T E n , where the ij -entry of E n is 1 if j | i and 0 otherwise. We present a proof of the Mattila–Haukkanen conjecture which states that for every n ∈ Z + , t n + 1 ≤ t n and T n ≤ T n + 1 . Also, we prove that lim n → ∞ ⁡ t n = 0 and lim n → ∞ ⁡ T n = ∞ and we give a lower bound for t n . [ABSTRACT FROM AUTHOR]

Identifying important patents helps to drive business growth and focus investment. In the past, centrality measures such as degree centrality and betweenness centrality have been applied to identify influential or important patents in patent citation networks. How such a complex process like technological change can be analyzed is an important research topic. However, no existing centrality measure leverages the powerful graph kernels for this end. This paper presents a new centrality measure based on the change of the node similarity matrix after leveraging graph kernels. The proposed approach provides a more robust understanding of the identification of influential nodes, since it focuses on graph structure information by considering direct and indirect patent citations. This study begins with the premise that the change of similarity matrix that results from removing a given node indicates the importance of the node within its network, since each node makes a contribution to the similarity matrix among nodes. We calculate the change of the similarity matrix norms for a given node after we calculate the singular values for the case of the existence and the case of nonexistence of that node within the network. Then, the node resulting in the largest change ( i.e. , decrease) in the similarity matrix norm is considered to be the most influential node. We compare the performance of our proposed approach with other widely-used centrality measures using artificial data and real-life U.S. patent data. Experimental results show that our proposed approach performs better than existing methods. [ABSTRACT FROM AUTHOR]

This paper investigates the convergence and error evaluation of the distributed analytical representation and iterative technique (DARIT-field) which is a new approach for the analysis of field coupling to multiconductor transmission lines. The DARIT-field method has the advantage of high computational efficiency over the other methods. In order to know the convergence speed and the error at each iteration step of DARIT-field, an analytic expression of iterative error is derived by combining the two telegrapher's equations into a matrix equation and using the euclidean norm to explore its upper bound. The expression shows that the convergence speed is mainly influenced by three parameters, namely coupling factor (CF), terminal loads, and the line length to excitation field wavelength ratio $(d/\lambda)$. The convergence speed is a function of CF, terminal loads, and the line length to excitation field wavelength ratio $(d/\lambda)$. These results allow the users to make a compromise between computational cost and accuracy by selecting the number of iterations. [ABSTRACT FROM AUTHOR]

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L² on the half-line with the weight tαe-t and tβe-t, respectively. Under the assumption that β-α is larger than the order of the derivative, we determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity, thus proving a conjecture raised by Böttcher and Dörfler in 2010 [4]. [ABSTRACT FROM AUTHOR]

We consider correspondence analysis (CA) and taxicab correspondence analysis (TCA) of relational datasets that can mathematically be described as weighted loopless graphs. Such data appear in particular in network analysis. We present CA and TCA as relaxation methods for the graph partitioning problem. Examples of real datasets are provided. [ABSTRACT FROM AUTHOR]

Abstract: Text of abstract For some reflexive spaces of matrices we calculate or estimate hyperreflexivity constants with respect to different operator norms on the space of matrices. It is known that the hyperreflexivity constant of every one-dimensional space of operators on a Hilbert space is 1. We show that for Banach spaces this does not hold in general. Namely, if matrices are considered as operators on , then the hyperreflexivity constant of the one-dimensional subspace which is spanned by the identity matrix is . [Copyright &y& Elsevier]

In this study, a new modified group iterative scheme for solving the two-dimensional (2D) fractional hyperbolic telegraph differential equation with Dirichlet boundary conditions is obtained from the 2h-spaced standard and rotated Crank–Nicolson FD approximations. The findings of new four-point modified explicit group relaxation method demonstrates the rapid rate of convergence of proposed method as compared to the existing schemes. Numerical tests are performed to test the capability of the group iterative scheme in comparison with the point iterative scheme counterparts. The stability of the derived modified group method is proven by the matrix norm algorithm. The obtained results are tabulated and concluded that exact solutions are exactly symmetric with approximate solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard (Hilbert) space. [Copyright &y& Elsevier]

The thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices, but it often has negative eigenvalues when used in real data analysis. To fix this drawback of thresholding estimation, we develop a positive-definite ℓ1-penalized covariance estimator for estimating sparse large covariance matrices. We derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties. Under weak regularity conditions, nonasymptotic statistical theory is also established for the proposed estimator. The competitive finite-sample performance of our proposal is demonstrated by both simulation and real applications. [ABSTRACT FROM PUBLISHER]