Search

Numerical range is the subject of study for over a century. Its extension to matricial range is defined in terms of completely positive maps. A linear map ϕ between matrix algebras is completely positive if and only if its Choi matrix $ C(\phi) $ C (ϕ) is positive semidefinite. In this paper, we extend the study to some joint matricial ranges defined by those ϕ's where $ C(\phi) $ C (ϕ) is Hermitian with specified spectrum. We prove some convexity theorems which extend previous results on generalized joint numerical ranges and matricial ranges. We also extend Bohnenblust's result on joint positive definiteness of Hermitian matrices and Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of Hermitian matrices. [ABSTRACT FROM AUTHOR]

The most general scalar potential of two real scalar fields and a Higgs boson is a quartic homogeneous polynomial in three variables, which defines a th-order three-dimensional symmetric tensor. Hence, the boundedness of such a scalar potential from below involves the positive (semi-)definiteness of the corresponding tensor. In this paper, we therefore mainly discuss analytic expressions of positive (semi-)definiteness for such a special tensor. First, an analytically necessary and sufficient condition is given to test the positive (semi-)definiteness of a th-order two-dimensional symmetric tensor. Furthermore, by means of such a result, the analytic necessary and sufficient conditions of the boundedness from below are obtained for a general scalar potential of two real scalar fields and the Higgs boson. [ABSTRACT FROM AUTHOR]

Let A be a positive semidefinite matrix. It is known that the Hadamard exponential of A is positive semidefinite; it is positive definite if and only if no two columns of A are identical. We give an alternative proof of the latter part with an application to Hadamard inverses. [ABSTRACT FROM AUTHOR]

This paper presents an analytical study for short-term voltage stability based on a novel indicator. First, it is shown that the reduced Jacobian matrix of network equations can be transformed to obtain an approximately symmetric matrix which is usually positive definite when the system under study is stable. The minimum eigenvalue of the matrix is suggested as a short-term Voltage Solvability Indicator (VSI). Then, properties of the reduced conductive matrix and susceptance matrix which have significant impacts on VSI are examined. The focus of the study is power systems with multiple constant power injections, while VSI can also be applied to analyze the impact of generic power system models. It is shown that as the penetration level of constant power injections increases, VSI decreases and the system becomes more vulnerable to voltage collapse. Finally, the relationship between the eigenvalues of the reduced Jacobian matrix and the structure-preserving Jacobian matrix is established. As a result, the suggested VSI can be efficiently computed by exploring the sparsity of structure-preserving Jacobian matrix of network equations. Case studies of several test systems including a simplified 575-machine 8117-bus East China power system are reported, demonstrating the effectiveness and practicability of the proposed method.

In magnetic resonance imaging, high angular resolution diffusion imaging (abbr. HARDI) is used to characterize non-Gaussian diffusion processes. One approach to analyzing HARDI data is to model the apparent diffusion coefficient with higher order diffusion tensors from a diffusivity function. An intrinsic property of the diffusivity function is positive semi-definite, which reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. In this paper, we provide a workable criterion for judging the positive semi-definiteness of a diffusivity function and shows that it is effective via two numerical examples. [ABSTRACT FROM AUTHOR]

When analyzing multivariate longitudinal binary data, we estimate the effects on the responses of the covariates while accounting for three types of complex correlations present in the data. These include the correlations within separate responses over time, cross‐correlations between different responses at different times, and correlations between different responses at each time point. The number of parameters thus increases quadratically with the dimension of the correlation matrix, making parameter estimation difficult; the estimated correlation matrix must also meet the positive definiteness constraint. The correlation matrix may additionally be heteroscedastic; however, the matrix structure is commonly considered to be homoscedastic and constrained, such as exchangeable or autoregressive with order one. These assumptions are overly strong, resulting in skewed estimates of the covariate effects on the responses. Hence, we propose probit linear mixed models for multivariate longitudinal binary data, where the correlation matrix is estimated using hypersphere decomposition instead of the strong assumptions noted above. Simulations and real examples are used to demonstrate the proposed methods. An open source R package, BayesMGLM, is made available on GitHub at https://github.com/kuojunglee/BayesMGLM/ with full documentation to produce the results. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators. In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step L1-type time discretizations, we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy by using the properties of discrete orthogonal convolution kernels and discrete complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way. [ABSTRACT FROM AUTHOR]

In this paper, we consider the positive definiteness of fourth-order partially symmetric tensors. First, two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dimensional partially symmetric tensors. Then, we obtain several sufficient conditions for rank-one positive definiteness of fourth-order three dimensional partially symmetric tensors. [ABSTRACT FROM AUTHOR]

In the conventional finite element method, only a few types of elements can directly use the weight functions to generate a proper, diagonally lumped mass matrix (LMM), which satisfies mass conservation and positive definiteness. This study proposes a new set of weight functions that can be efficiently constructed and used for generating a proper LMM based on classical 6‐, 9‐, and 10‐node triangular elements. Furthermore, we prove that these new elements are all complete and consistent, which ensures the convergence of the numerical solution. Finally, some modal and forced/free vibration analyses are conducted to demonstrate the computational accuracy and validation of the proposed LMM. [ABSTRACT FROM AUTHOR]

Sparse tensors play fundamental roles in hypergraph data, sensor node network data and remote sensing data. In this paper, we establish new H-eigenvalue inclusion sets for sparse tensors by their majorization matrix’s digraph and representation matrix’s digraph. Numerical examples are proposed to verify that our conclusions are more accurate and less computations than existing results. As applications, we provide some checkable sufficient conditions for the positive definiteness of even-order sparse tensors, and propose lower and upper bounds of H-spectral radius of nonnegative sparse tensors. [ABSTRACT FROM AUTHOR]

A new parametric class of covariance functions of wide-sense homogeneous and isotropic random fields on Euclidean spaces is presented. While the new class possesses uncoupled fractal and Hurst effects, it is shown to encompass the Generalized Cauchy and Dagum models. On the basis of the new functions, product-based covariance models are developed, along with parametric conditions guaranteeing a decoupling of fractal and Hurst effects. [ABSTRACT FROM AUTHOR]

H-tensors play a key role in identifying the positive definiteness of even-order real symmetric tensors. Some criteria have been given since it is difficult to judge whether a given tensor is an H-tensor, and their range of judgment has been limited. In this paper, some new criteria, from an increasing constant k to scale the elements of a given tensor can expand the range of judgment, are obtained. Moreover, as an application of those new criteria, some sufficient conditions for judging positive definiteness of even-order real symmetric tensors are proposed. In addition, some numerical examples are presented to illustrate those new results.

Let A be a real (p, q)-th order m × n dimensional partially symmetric rectangular tensor with p and q even. Firstly, in order to judge the positive definiteness of A , an l p / 2 , q / 2 -singular value inclusion interval with parameters is constructed. Subsequently, by finding the optimal values of parameters, the optimal parameter inclusion interval of l p / 2 , q / 2 -singular values is derived, which provides a sufficient condition for the positive definiteness of A . Secondly, when A is a nonnegative tensor, lower and upper bounds for its l p / 2 , q / 2 -spectral radius are given. Thirdly, a direct method for finding all l p / 2 , q / 2 -singular value/vectors pairs of A with p = q = 4 and m = n = 2 is presented. Finally, a numerical example is given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

A positive definite homogeneous multivariate form plays an important role in the field of optimization, and positive definiteness of the form can be identified by a special structured tensor. In this paper, based on the equivalence between the form and the corresponding tensor, and the links of the positive definiteness of a tensor with ℋ-tensor, we propose an ℋ-tensor-based criterion for identifying the positive definiteness of multivariate homogeneous forms. Some numerical examples are provided to illustrate the efficiency and validity of our results.

A new Brauer-type Z-eigenvalue inclusion set for an even-order real tensor is presented. It is proved that it is tighter than the existing inclusion sets. As an application, a sufficient condition for the positive definiteness of an even-order real symmetric tensor (also a homogeneous polynomial form) and asymptotically stability of time-invariant polynomial systems is given. [ABSTRACT FROM AUTHOR]

• A Voltage Solvability Indicator (VSI) has been proposed for short-term voltage stability studies. • VSI can be used to characterize the singular surface of power system DAE model, and to accurately describe the Voltage Instability Regions (VIRs) • VSI not only facilitates theoretical analysis, thereby elucidating the mechanisms of voltage collapse, but also finds practical applications in large-scale power systems. • Both theoretical analysis and case studies consistently suggest that as the penetration of constant power injections increases, power systems become more vulnerable to voltage collapse. • For large-scale power systems, the sparse computational method proposed in this study can save significant amount of time in computing VSI. This paper presents an analytical study for short-term voltage stability based on a novel indicator. First, it is shown that the reduced Jacobian matrix of network equations can be transformed to obtain an approximately symmetric matrix which is usually positive definite when the system under study is stable. The minimum eigenvalue of the matrix is suggested as a short-term Voltage Solvability Indicator (VSI). Then, properties of the reduced conductive matrix and susceptance matrix which have significant impacts on VSI are examined. The focus of the study is power systems with multiple constant power injections, while VSI can also be applied to analyze the impact of generic power system models. It is shown that as the penetration level of constant power injections increases, VSI decreases and the system becomes more vulnerable to voltage collapse. Finally, the relationship between the eigenvalues of the reduced Jacobian matrix and the structure-preserving Jacobian matrix is established. As a result, the suggested VSI can be efficiently computed by exploring the sparsity of structure-preserving Jacobian matrix of network equations. Case studies of several test systems including a simplified 575-machine 8117-bus East China power system are reported, demonstrating the effectiveness and practicability of the proposed method. [ABSTRACT FROM AUTHOR]

An optimal Z-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.

In this paper we present a short overview of results that provide relationships among scalar, matrix-valued and certain operator-valued positive definite kernels. We refine and extend some of them in order that they may be applied for strict positive definiteness as well. This is a topic not well explored in the literature but that has potential usefulness in the characterization of several classes of positive definite and strictly positive definite kernels. This is ratified in the paper with the inclusion of a number of applications and examples.

Existing literature shows that the strong ellipticity of an elasticity tensor can be equivalently transformed into the positive definiteness of even-order symmetric tensors constructed from the components of this elasticity tensor, and that an even-order symmetric tensor is positive definite if and only if all of its Z-eigenvalues are positive. In order to judge the strong ellipticity of an elasticity tensor, we first construct a Z-eigenvalue inclusion interval with parameters for an even-order symmetric tensor, and then by finding the optimal value of parameters we derive the optimal parameter interval, which is used to obtain a sufficient condition for the positive definiteness of the even-order symmetric tensor. Additionally, we prove that the new interval and sufficient condition are preferable than some existing ones. Finally, by using the positive definiteness of even-order symmetric tensors, we derived a necessary and sufficient condition and a checkable criterion for the strong ellipticity of an elasticity tensor. [ABSTRACT FROM AUTHOR]

Managing a large-scale portfolio with many assets is one of the most challenging tasks in the field of finance. It is partly because estimation of either covariance or precision matrix of asset returns tends to be unstable or even infeasible when the number of assets p exceeds the number of observations n. For this reason, most of the previous studies on portfolio management have focused on the case of p < n . To deal with the case of p > n , we propose to use a new Bayesian framework based on adaptive graphical LASSO for estimating the precision matrix of asset returns in a large-scale portfolio. Unlike the previous studies on graphical LASSO in the literature, our approach utilizes a Bayesian estimation method for the precision matrix proposed by Oya and Nakatsuma (Japanese J Stat Data Sci, 2022.) so that the positive definiteness of the precision matrix should be always guaranteed. As an empirical application, we construct the global minimum variance portfolio of p = 100 for various values of n with the proposed approach as well as the non-Bayesian graphical LASSO approach, and compare their out-of-sample performance with the equal weight portfolio as the benchmark. We also compare them with portfolios based on random matrix theory filtering and Ledoit-Wolf shrinkage estimation which were used by Torri et al. (Comput Manage Sci 16:375–400, 2019). In this comparison, the proposed approach produces more stable results than the non-Bayesian approach and the other comparative approaches in terms of Sharpe ratio, portfolio composition and turnover even if n is much smaller than p. [ABSTRACT FROM AUTHOR]

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with H -tensors. Then, based on the notion of diagonally dominant tensors, some criteria for H -tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

The seminal works by George Matheron provide the foundations of walks through dimensions for positive definite functions defined in Euclidean spaces. For a d-dimensional space and a class of positive definite functions therein, Matheron called montée and descente two operators that allow for obtaining new classes of positive definite functions in lower and higher dimensional spaces, respectively. The present work examines three different constructions to dimension walks for continuous positive definite functions on hyperspheres. First, we define montée and descente operators on the basis of the spectral representation of isotropic covariance functions on hyperspheres. The second approach provides walks through dimensions following Yadrenko’s construction of random fields on spheres. Under this approach, walks through unit dimensions are not permissible, while it is possible to walk under ± 2 dimensions from a covariance function that is valid on a d-dimensional sphere. The third construction relies on the integration of a given isotropic random field over latitudinal arcs. In each approach, we provide spectral representations of the montée and descente as well as illustrative examples. [ABSTRACT FROM AUTHOR]

Diversity smoothing has been widely used for angle estimation with multiple input multiple output (MIMO) radar in the presence of coherent or correlated targets, and the parameter identifiability is an interesting issue. Previous works have shown sufficient conditions for some special cases, such as the coherent targets with conventional MIMO radar, and the array size are derived as a function of the target number and target structure. In this article, we further introduce the interelement spacings to build a complete parameter identifiability scheme. For monostatic MIMO radars, the antenna numbers and interelement spacings of transmit and receive arrays are derived as functions of the target number and the target structure. The optimization models are built to calculate the maximum number of detectable targets for a given target structure. Additionally, the conditions for the bistatic MIMO radar are derived from two-dimension viewpoint. It is shown that the new results improve upon previous spatial smoothing or diversity smoothing methods and recover them in special cases. Simulation results are presented that corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

Positive definite polynomials are important in the field of optimization. H-tensors play an important role in identifying the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose some new criterion for identifying H-tensor. As applications, we give new conditions for identifying positive definiteness of the even-order homogeneous multivariate form. At last, some numerical examples are provided to illustrate the efficiency and validity of new methods.

Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.

A number of consistent models describing stationary positive definite functions on R m × R n stem from Bochner's celebrated theorem characterizing continuous and stationary positive definite functions on R m. If S m denotes the unit sphere in R m + 1 , the same is true of positive definite functions on S m × R n which are radial with respect to the S m component and stationary with respect to the R n component. In this paper, we summarize results on these topics, mainly those that somehow characterize the positive definiteness of the function through Fourier transforms of the sections of the function itself. We present a new perspective on the existent results in the literature along with new characterizations and applications. [ABSTRACT FROM AUTHOR]

In the spectral analysis of operators associated with Sturm–Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L 2 (0 , 1) . Generally speaking, it should be noted that this very important problem turned out to be very difficult. [ABSTRACT FROM AUTHOR]

The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix ℓ ∞ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix ℓ ∞ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the ℓ ∞ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector ℓ ∞ error in the estimation of the optimal portfolio weight is bounded by the matrix ℓ ∞ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014. [ABSTRACT FROM AUTHOR]

In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are fourth-order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of fourth-order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for fourth-order 2-dimensional symmetric tensors. For fourth-order 3-dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) copositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for fourth-order 2-dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for fourth-order 2-dimensional symmetric tensors. Finally, these results may be applied to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson. [ABSTRACT FROM AUTHOR]

Firstly, the optimal Z -eigenvalue inclusion interval for the interval in Theorem 3 of (Acta Appl. Math. 169:323–339, 2020) for even order tensors is given. Secondly, by making full use of the information of Z -eigenvectors, new Geršgorin-type Z -eigenvalue inclusion intervals with n parameters for fourth-order and sixth-order tensors are constructed. Thirdly, by selecting appropriate parameters, two optimal intervals for fourth-order and sixth-order tensors are presented. Finally, as applications, several sufficient conditions for the positive definiteness of even order real symmetric tensors (also homogeneous polynomial forms) as well as the asymptotically stability of time-invariant polynomial systems are obtained. Moreover, bounds of the Z -spectral radius of weakly symmetric nonnegative tensors are obtained, which are used to estimate the convergence rate of the greedy rank-one update algorithm and derive bounds of the geometric measure of entanglement of symmetric pure state with nonnegative amplitudes. [ABSTRACT FROM AUTHOR]

This paper concerns the interconnection between the positive definiteness of limiting coderivative and the local strong maximal monotonicity of set-valued mappings suspected in Mordukhovich and Nghia (SIAM J. Optim. 26, 1032–1059, 2016, Conjecture 3.6). We disprove the conjecture by a counterexample and provide some special classes at which it is true. However, the positive definiteness of limiting coderivative characterizes a new property called nearly strong monotonicity. Consequently, we show that the strong metric regularity of set-valued mappings could be obtained under the positive definiteness of limiting coderivative. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhengzhou University (Natural Science Edition) is the property of Journal of Zhengzhou University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract The positive definiteness of rectangular tensors has wide applications in solid mechanics and quantum physics. By modifying the existing definition of singular value for rectangular tensors, some V-singular value inclusion sets for rectangular tensors with positive diagonal entries are established to provide verifiable sufficient conditions for the positive definiteness of rectangular tensors. In addition, an upper bound for the largest V-singular value of nonnegative rectangular tensors is provided.

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with H-tensors. Then, based on the notion of diagonally dominant tensors, some criteria for H-tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples.

This paper is devoted to the spectral analysis of one class of integral operators, associated with the boundary value problems for differential equations of fractional order. Approximation matrices are also investigated. In particular, the positive definiteness of the studied operators is shown, which makes it possible to select areas in the complex plane where there are no eigenvalues of these operators.

Partially specified correlation matrices (not to be confused with matrices with missing data or EM-correlation matrices) can appear in research settings such as integrative data analyses, quantitative systematic reviews or whenever the study design only allows for the collection of certain variables. Although approaches to fill in these missing entries have been considered for special cases of low-dimensional matrices, a general approach that can handle correlation matrices of arbitrary size and number of missing entries is needed. The present article relies on the theory of convex optimization and semidefinite programming to derive a semidefinite program that can offer researchers a mathematically principled approach to fill in the missing entries. An easy-to-use function in the R programming language is also presented that implements the theory derived herein. [ABSTRACT FROM AUTHOR]

In this paper, two Brualdi-type eigenvalue inclusion sets are presented, which are characterized by the girth of the digraph associated with a tensor and are proved weaker than the conditions of Brualdi-type eigenvalue inclusion sets given by Bu et al. (2015) [7]. As applications of the above results, the conditions of nonsingularity and positive definiteness of tensors are given. [ABSTRACT FROM AUTHOR]

In this paper, two new classes of rectangular tensors called rectangular M-tensors and strong rectangular M-tensors are introduced. It is shown that an even-order partially symmetric rectangular M-tensor is positive semidefinite and an even-order partially symmetric strong rectangular M-tensor is positive definite. As a generalization of rectangular M-tensors, we introduce the rectangular H-tensors. In addition, some properties of (strong) rectangular M-tensors are established. [ABSTRACT FROM AUTHOR]

First, we introduce the two-Motzkin-like number as the weight of vertically constrained Motzkin-like path with no leading vertical steps from (0, 0) to (n, 0) consisting of up steps, down steps, horizontal steps, vertical steps in the down direction and vertical steps in the up direction. Secondly, we provide sufficient conditions under which the two-Motzkin-like numbers (resp. the q-analogue of the two-Motzkin-like numbers) are Stieltjes moment sequences (resp. are q-Stieltjes moment sequences) and therefore infinitely log-convex sequences. As applications, on the one hand, we show that many well-known counting coefficients, including the central trinomial 2 n 2 n 2 and pentanomial 2 n 4 n 4 numbers of even indices respectively are Stieltjes moment sequences and, therefore, infinitely log-convex sequences in a unified approach. On the other hand, we prove that the sequence of polynomials of square trinomials ∑ k = 0 2 n n k 2 2 q k are q-Stieltjes moment sequence of polynomials. Finally, we provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences in more generalized triangular array. [ABSTRACT FROM AUTHOR]

Covariance matrix carries a big weight in the domain of portfolio. Most researchers spend more time on model construction, risk measurement and algorithm analysis, but paid little attention to the deeper theory research and rigorous proof on the positive definiteness of the covariance matrix. This paper aims to obtain the equivalent condition about the positive definite of covariance matrix. In theory, some innovative and significant results are identified by matrix theory on risk value range of equal weight portfolio and optimal portfolio. These results obtained by using rigorous mathematical proof in this paper can explain some actual portfolio problems about risk. In practice, four numerical examples are shown to verify the validity of the proposed lemmas and theorems. Finally, the results of this paper show that the positive definiteness of the covariance matrix will no longer be an assumption, but a condition of the rigorous theoretical basis, which provides theoretical support for the research of portfolio theory. [ABSTRACT FROM AUTHOR]

In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results. [ABSTRACT FROM AUTHOR]

In the spectral analysis of operators associated with Sturm–Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green’s function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L2(0,1). Generally speaking, it should be noted that this very important problem turned out to be very difficult.