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There are various kinds of fuzzy inner products. It was shown (Byun et al. in Mathematics 8:571, 2020) that a fuzzy inner product on a vector space, producing a fuzzy real number as its value, is merely the embedding of a real-valued inner product into the fuzzy number systems over the set of tuples consisting of elements in the space because of its linear property. Thus, the linear property is needed to be weakened. In this paper, a new concept, called a fuzzy real weak inner product on a vector space, is introduced and some nontrivial examples are given. Then, a fuzzy real weak inner product in a suitable condition is deeply studied, which may be either almost positive-definite or positive-definite. Furthermore, it satisfies both Cauchy inequality and Parallelogram law even though it is not linear. And some properties of a general type of a fuzzy real weak inner product, especially the approximation related to the Parallelogram law, are investigated in analytic viewpoint.

We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the torus.

We introduce$M$ -tensors. This concept extends the concept of$M$ -matrices. We denote$Z$ -tensors as the tensors with nonpositive off-diagonal entries. We show that$M$ -tensors must be$Z$ -tensors and the maximal diagonal entry must be nonnegative. The diagonal elements of a symmetric$M$ -tensor must be nonnegative. A symmetric$M$ -tensor is copositive. Based on the spectral theory of nonnegative tensors, we show that the minimal value of the real parts of all eigenvalues of an$M$ -tensor is its smallest H $^+$ -eigenvalue and also is its smallest H-eigenvalue. We show that a$Z$ -tensor is an$M$ -tensor if and only if all its H $^+$ -eigenvalues are nonnegative. Some further spectral properties of$M$ -tensors are given. We also introduce strong$M$ -tensors, and some corresponding conclusions are given. In particular, we show that all$H$ -eigenvalues of strong$M$ -tensors are positive. We apply this property to study the positive definiteness of a class of multivariate forms associated with$Z$ -tensors. We also propose an algorithm for testing the positive definiteness of such a multivariate form. [PUBLICATION ABSTRACT]

Studying the positive definiteness of the covariance matrix of discrete samples helps to determine whether the dimensionality of the samples can be reduced, which is beneficial for optimizing the number of samples and designing optimal plans for sampling surveys. This paper aims to provide a method to determine the variable numbers of the sample subjecting to Poisson distribution. Methods . It is based on the theory of I -linear combination and its properties which are the author’s previous studying results. Results . study shows the covariance matrix of multi-Poisson distribution is positively defined and the probability of the sample covariance matrix of multi-poisson distribution is about 1 when the sample capacity is very large. Conclusion . The dimension size of the sample data matrix of multi-poisson distribution can be reduced when the sample capacity n is no more than the dimension size p .

For the recently introduced isotropic-relaxed micromorphic generalized continuum model, we show that, under the assumption of positive-definite energy, planar harmonic waves have real velocity. We also obtain a necessary and sufficient condition for real wave velocity which is weaker than the positive definiteness of the energy. Connections to isotropic linear elasticity and micropolar elasticity are established. Notably, we show that strong ellipticity does not imply real wave velocity in micropolar elasticity, whereas it does in isotropic linear elasticity.

This article introduces covariance regression analysis for a p-dimensional response vector. The proposed method explores the regression relationship between the p-dimensional covariance matrix and auxiliary information. We study three types of estimators: maximum likelihood, ordinary least squares, and feasible generalized least squares estimators. Then, we demonstrate that these regression estimators are consistent and asymptotically normal. Furthermore, we obtain the high dimensional and large sample properties of the corresponding covariance matrix estimators. Simulation experiments are presented to demonstrate the performance of both regression and covariance matrix estimates. An example is analyzed from the Chinese stock market to illustrate the usefulness of the proposed covariance regression model. Supplementary materials for this article are available online.

The positive definite homogeneous multivariate forms play an important role in the automatic control and medical imaging, and the definiteness of the forms can be identified by special structured tensors. In this paper, we first state the equivalence between the positive definite multivariate forms and the corresponding tensors and account for the links between the positive definite tensors with ℋ-tensors. Then based on diagonal dominance, some iterative criterions are presented to test ℋ-tensors. Furthermore, we establish new iterative schemes for testing the positive definite multivariate homogeneous forms. The efficiency and validity of new methods are illustrated by numerical examples.

The linear theory of coupled gradient elasticity has been considered for hemitropic second gradient materials, specifically the positive definiteness of the strain and strain gradient energy density, which is assumed to be a quadratic form of the strain and of the second gradient of the displacement. The existence of the mixed, fifth-rank coupling term significantly complicates the problem. To obtain inequalities for the positive definiteness including the coupling term, a diagonalization in terms of block matrices is given, such that the potential energy density is obtained in an uncoupled quadratic form of a modified strain and the second gradient of displacement. Using orthonormal bases for the second-rank strain tensor and third-rank strain gradient tensor results in matrix representations for the modified fourth-rank and the sixth-rank tensors, such that Sylvester’s formula and eigenvalue criteria can be applied to yield conditions for positive definiteness. Both criteria result in the same constraints on the constitutive parameters. A comparison with results available in the literature was possible only for the special case that the coupling term vanishes. These coincide with our results.

High-dimensional covariance matrix estimation plays a central role in multivariate statistical analysis. It is well-known that the sample covariance matrix is singular when the sample size is smaller than the dimension of the variable, but the covariance estimate must be positive-definite. This motivates some modifications of the sample covariance matrix to preserve its efficient estimation of pairwise covariance. In this paper, we modify the sample correlation matrix using the Bagging technique. The proposed Bagging estimator is flexible for general continuous data. Under some mild conditions, we show theoretically that the Bagging estimator can ensure positive-definiteness with probability one in finite samples. We also prove the consistency of the bootstrap estimator of Pearson correlation and the consistency of our Bagging estimator when the dimension p is fixed. Simulation results and a real application are provided to demonstrate that our method strikes a better balance between RMSE and likelihood, and is more robust, than other existing estimators.

A time-fractional diffusion problem with a Caputo time-fractional derivative of order α ∈ ( 0 , 1 ) is considered, the solution of which is typically weakly singular at the initial time. For this problem, we give an H 1 -norm analysis of the stability and convergence of an integral-averaged L1 method on nonuniform time meshes. The averaging of the L1 scheme that we use is known as the L1 + or L 1 ¯ scheme. A new positive definiteness result for the integral-averaged L1 fractional-derivative operator is established. It improves the previous positive definiteness results in the literature and plays an important role in the analysis of H 1 -norm error of the integral-averaged L1 method. The H 1 -norm stability holds for the general nonuniform time meshes, while the H 1 -norm convergence is proved for the time graded meshes and the H 1 -norm convergence order in time is min { 3 + α , γ α } / 2 for all α ∈ ( 0 , 1 ) , where γ ≥ 1 is the mesh grading parameter. Two full discretization methods using finite differences and finite elements in space are considered. The theoretical results are illustrated by numerical results.