This paper systematically investigates the properties and characterization of interval B-tensors and interval double B-tensors. We propose verifiable necessary and sufficient conditions that allow for determining whether an entire interval tensor family belongs to these classes based solely on its extreme point tensors. The study elucidates profound connections between these interval tensors and other structured ones such as interval Z-tensors and P-tensors, while also providing simplified criteria for special cases like circulant structures. Furthermore, under the condition of even order and symmetry, we prove that interval B-tensors (double B-tensors) ensure the property of being an interval P-tensor. This work extends interval matrix theory to tensors, offering new analytical tools for fields such as polynomial optimization and complementarity problems involving uncertainty.

As is well-known, a symmetric matrix with +/- 1 entries must not be positive definite. For a 4th order 3-dimensional symmetric tensor with its some entries 1 or-1, we show the analytic sufficient and necessary conditions of its positive definiteness. Moreover, a sufficient and necessary condition of a positive semidefinite tensor is given. By applying these conclusions, several strict inequalities are bulit for ternary quartic homogeneous polynomials.

This paper introduces interval B_π^R^I-tensors as a natural extension of B_π^R-tensors to the interval setting. We provide two practical verifiable criteria for an interval tensor to be an interval B_π^R^I-tensor, one based on endpoint inequalities and another constructing an explicit vector π. Connections with interval P-tensors, positive definite interval tensors, and interval Z-tensors are established. Applications in polynomial optimization and interval tensor complementarity problems are briefly discussed.

In this paper, we mainly discuss the non-negativity conditions for quartic homogeneous polynomials with 3 variables, which is the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries 1 or -1 , an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. Moreover, a necessary and sufficient condition is established for copositivity of such a tensor also. Several (strict) inequalities of ternary quartic homogeneous polynomial are established by means of these analytic conditions.

The effective second-harmonic generation (SHG) coefficient is a crucial data that quantifies the efficiency of transforming fundamental frequency light into its second harmonic. With the help of the symmetry of nonlinear optical susceptibility tensors, we mainly discuss the computability of such a effective SHG coefficient in uniaxial crystals. For one thing, the calculation of effective SHG coefficient is converted into the optimization models with some geometric constraints by means of the peculiarity of fundamental frequency light. Secondly, the number of variables of such maximum models are cutted in half to $2$ to calculate it easier, and a comparison between the effective SHG coefficient and C-eigenvalue of susceptibility tensor is given also. Finally, some examples of typical crystal classes are presented to verify the correctness and broader applicabilities of the theoretical results.

In recent years, tensor robust principal component analysis (TRPCA) has been extensively applied in the field of image processing. The traditional tensor nuclear norm (TNN) can only be applied to third-order tensors, and identical penalty weights are imposed on all singular values. To adaptively assign distinct weights to singular values of different magnitudes, a tensor arctangent norm (TAN) is proposed in this work. This norm is capable of better distinguishing the magnitudes of singular values, as smaller weights are assigned to large singular values and larger weights to small ones. Benefiting from the excellent derivative properties of the arctangent function, when the first-order Taylor expansion of the arctangent function is employed for approximation, TAN can clearly delineate the boundaries between large and small singular values. Subsequently, the TAN-TRPCA model is proposed. To extend the applicability of TAN-TRPCA to higher-order tensors, a tensor mode-$(1\times 4)$ unfolding operation is introduced. Finally, excellent experimental results are achieved on color images, hyperspectral images, and color video datasets.

M-eigenvalues of fourth order hierarchically symmetric tensors play a significant role in nonlinear elastic material analysis and quantum entanglement problems. This paper focuses on computing extreme M-eigenvalues for such tensors. To achieve this, we first reformulate the M-eigenvalue problem as a sequence of unconstrained optimization problems by introducing a shift parameter. Subsequently, we develop a memory gradient method specifically designed to approximate these extreme M-eigenvalues. Under this framework, we establish the global convergence of the proposed method. Finally, comprehensive numerical experiments demonstrate the efficacy and stability of our approach.

. It is well-known that a symmetric matrix with its entries +/- 1 can not be positive definite. But this is not true for symmetric tensors (hypermatrix). In this paper, we mainly discussed the positive (semi-)definiteness criterion of a class of fourth-order, three-dimensional symmetric tensors with entries tijkl is an element of {-1, 0,1}. Through theoretical derivations and detailed classification discussions, the criteria for determining the positive (semi-)definiteness of such a class of tensors were provided based on the relationships and number values of its entries. This has established some unique properties of higher symmetric tensors that are distinct from the ones of matrices.

This paper focuses on the strict copositivity of fourth-order three-dimensional symmetric tensors. A necessary and sufficient condition is provided for the strict copositivity of a fourth-order symmetric tensor. Subsequently, we discuss the strict copositivity of fourth-order three-dimensional symmetric tensors with their entries ± 1, 0 and further build their necessary and sufficient conditions. Utilizing these theorems, we can effectively verify the strict copositivity of general fourth-order three-dimensional symmetric tensors.

The most general scalar potential of two real scalar fields and a Higgs boson is a quartic homogeneous polynomial about 3 variables, which defines a 4th order 3 dimensional symmetric tensor. Hence, the boundedness from below of such a scalar potential involves the positive (semi-)definiteness of the corresponding tensor. So, we mainly discuss analytical expressions of positive (semi-)definiteness for such a special 4th order 3-dimension symmetric tensor in this paper. Firstly, an analytically necessary and sufficient condition is given to test the positive (semi-)definiteness of a 4th order 2 dimensional symmetric tensor. Furthermore, by means of such a result, the 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.

In this paper, we focus on the positive definiteness and Hurwitz stability of interval tensors. First, we introduce auxiliary tensors 𝒜^z and establish equivalent conditions for the positive (semi-)definiteness of interval tensors. That is, an interval tensor is positive definite if and only if all 𝒜^z are positive (semi-)definite. For Hurwitz stability, it is revealed that the stability of the symmetric interval tensor 𝒜_s^I can deduce the stability of the interval tensor 𝒜^I, and the stability of symmetric interval tensors is equivalent to that of auxiliary tensors 𝒜̃^z. Finally, taking 4th order 3-dimensional interval tensors as examples, the specific sufficient conditions are built for their positive (semi-)definiteness.

For a fourth order three-dimensional cyclic symmetric tensor, a sufficient and necessary condition is bulit for its positive semi-definiteness. A sufficient and necessary condition of positive definiteness is showed for a fourth order n-dimensional symmetric tensor. With the help of such a condition, the positive definiteness of a class of fourth order three-dimensional cyclic symmetric tensors is given. Moreover, the positive definiteness of a class of non-cyclic symmetric tensors is showed also. By applying these conclusions, several (strict) inequalities are erected for ternary quartic homogeneous polynomials.

It is well-known that a symmetric matrix with its entries ±1 is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of 4th order 3-dimensional symmetric tensors with entries t_ijkl∈{-1,0,1}. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher symmetric tensors that distinct from ones of matrces

The image noise level indicates the degree to which an image is contaminated bynoise, and serving as a crucial parameter in the image processing process. For the third-order tensor corresponding to a color image, traditional methods for estimatingthe noise level will disrupt the data structure of the tensor. In order to solve the aforementioned problems, we directly select the tensor using a sliding block of size  and then re-arrange it. The main innovations of this paper are as follows:        (1) A new noise level estimation model based on tensor decomposition is proposed. The third-order tensor is decomposed into the form of a covariance matrix according to the definition of the T-product, and then the noise level of the colorimage is estimated, and good experimental results were achieved in the experiment.       (2) Through theoretical analysis, it has been proved that the multiple eigenvalues of the matrix obtained by T-product of the third-order tensor have a direct relationship with the noise level of the color image.

Наиболее общий скалярный потенциал, зависящий от двух вещественных скалярных полей и бозона Хиггса, является однородным полиномом четвертой степени от трех переменных, который определяет трехмерный симметричный тензор 4-го порядка. Ограниченность такого скалярного потенциала снизу означает положительную (полу)определенность соответствующего тензора. В связи с этим представлены аналитические необходимые и достаточные условия положительной (полу)определенности такого специального тензора. С помощью этого результата получены необходимые и достаточные условия ограниченности снизу для общего скалярного потенциала, зависящего от двух вещественных скалярных полей и бозона Хиггса.

In this paper, we mainly dicuss the non-negativity conditions for quartic homogeneous polynomials with 3 variables, which is the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries 1 or -1, an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. And by means of usual maxi-min theory, a necessary and sufficient condition is established for copositivity of such a tensor also. Applying these conclusions to a general 4th order 3-dimensional symmetric tensor, the analytic conditions are successfully obtained for verifying the (strict) copositivity, and these conditions can be very easily parsed and validated. Moreover, several (strict) inequalities of ternary quartic homogeneous polynomial are established by means of these analytic conditions.

In this article, we mainly give the strictly copositive conditions of a special class of third order three dimensional symmetric tensors. More specifically, by means of the polynomial decomposition method, the analytic sufficient and necessary conditions are established for checking the strict copositivity of a 3rd order 3-dimensional symmetric tensor with its entries in {-1,0,1}. Several strict inequalities of cubic ternary homogeneous polynomials are presented by applying these conclusions. Some criteria which ensure the strict copositivity of a general 3rd order 3-dimensional tensor are obtained

This paper focuses on the strict copositivity analysis of 4th-order 3-dimensional symmetric tensors. A necessary and sufficient condition is provided for the strict copositivity of a fourth-order symmetric tensor. Subsequently, building upon this conclusion, we discuss the strict copositivity of fourth-order three-dimensional symmetric tensors with its entries ± 1, 0, and further build their necessary and sufficient conditions. Utilizing these theorems, we can effectively verify the strict copositivity of a general fourth-order three-dimensional symmetric tensors.

The purpose of this letter is to point out that some conclusions in the paper (Eur. Phys. J. C {\bf 76}, 324(2016)) are incomplete, and to give complete and improved conclusions. The analytic necessary and sufficient conditions are given for the boundedness-from-below conditions of general scalar potentials of two real scalar fields $\phi_1$ and $\phi_2$ and the Higgs bonson $\mathbf{H}$.

In this paper, we mainly dicuss the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries $1$ or $-1$, an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. And by means of usual maxi-min theory, a necessary and sufficient condition is established for copositivity of such a tensor also. Applying these conclusions to a general 4th order 3-dimensional symmetric tensor, the analytic conditions are successfully obtained for verifying the (strict) copositivity, and these conditions can be very easily parsed and validated.