University of Isfahan Mathematics and Society 2345-6493 9 1 2024 04 19 Continuous-attractor in Rabinovich-Fabricant system and its novel generalized model Continuous-attractor in Rabinovich-Fabricant system and its novel generalized model 1 32 28121 10.22108/msci.2024.138938.1601 FA Mohammad Hadi Moslehi Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697 Tehran, Iran Seyed Mohammad Amin Khatami, Department of Computer Science, Birjand University of Technology, Birjand, Iran Journal Article 2023 08 29 The aim of this work is the numerical study of the Rabinovich-Fabrikant system and its generalized model, which shows the occurrence of very rich dynamic behaviors with the interaction of three parameters of the generalized system. In particular, we observe the period-doubling bifurcation phenomenon leading to chaos, which has rarely been reported in previous works in the Rabinovich-Fabricant system. The complex dynamic behaviors of the system are investigated by using the Lyapunov spectrum, the parameters dependent bifurcation diagram and different sections of the phase space. This study is based on the numerical solution of differential equations and their numerical bifurcation analysis using Matlab software. The obtained results are new, because the generalization of the Rabinovich-Fabrikant system of the current study was proposed and studied for the first time. The generalized model describes the three-mode interaction. It can be used to simulate systems in radio and electronics engineering in which there is a three-mode interaction and which include cubic nonlinear terms. In addition, although the Rabinovich-Fabricant systems simulate systems of a physical nature, and in this regard, the coefficients embedded in them must be positive, its highly nonlinear and chaotic nature, due to the presence of third-order sentences, gives them a unique quality to be applied in secure communication. Resultantly, its artificial generalization with the use of negative parameters which adds to the complexity of its rich dynamics, is of particular importance. 1. Introduction A dynamic system is classified into two types, discrete and continuous, in terms of time evolution. In this paper, we will look at a particular continuous dynamical system known as the "Rabinovitch-Fabrikant system", which was invented in 1979 by two theoretical physicists, Mikhail Rabinovitch and Anatoly Fabrikant [1]. This system is actually a simplification of a complex nonlinear parabolic equation that models different physical systems such as Tollmien–Schlichting waves in hydrodynamic flows, wind waves on water, Langmuir waves in plasma. Having five equilibrium points, it is not topologically equivalent to many classical systems such as the Lorenz and Chen systems (with three equilibrium points), Rössler system (with two equilibrium points) and the like. In the following, we will call this system the RF system, for short. There are several reasons why this system has been noticed. One is the fact that it models a physical system and, therefore, is not a synthetic model and can be used to model numerous physical phenomena. Another reason is that due to its strong nonlinearity (due to the presence of third-order terms in its mathematical model), a rigorous mathematical analysis cannot be performed on it, hence, the system may show new and interesting properties that have not yet been reported. At the same time, it creates serious challenges for the numerical methods of solving ordinary differential equations. The mathematical model of the Rabinovich-Fabrikant (RF) system is described by the following equations: (1.1) \begin{equation} \label{RF1} \begin{aligned} & \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\ & \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+a x_2 \\ & \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right) \end{aligned} \end{equation} In the Rabinovitch-Fabricant (RF) system modeled by the system of ordinary differential equations (1.1), $a>0$ and $b \in \mathbb{R}$ is the bifurcation parameter. Since it is currently impossible to fully analyze this system mathematically, most of the research is based on numerical and computer analysis. Following this common practice, we will also, in this paper, have an approach based on numerical analysis. 2. Main Results Different dedicated numerical methods for ODEs, implemented in different software packages, might give different results for the same parameters values and initial conditions. system (1.1) by Danca et al. [2, 3, 4] has been studied. By numerical analysis, they showed that this system exhibits unusual and very rich dynamics, including multistability. Here, we obtain their results with a different numerical approach. They used the 3-step predictor–corrector Local Iterative Linearization (LIL) method, which is an implicit 3-step method [5]. In this paper, to calculate the Lyapunov exponents, we have used the MATDS software package applicable in Matlab software by adding codes to get the desired output, and to solve $\mathbf{ODE}$, the $ode45 $ solver implemented in this software with relative error $RelTol=10^{-7}$ and absolute error $AbsTol=10^{-7}$ are used. The step length used in numerical methods is also considered to be $h=10^{-6}$. 2.1. Generalized Rabinovitch-Fabricant system. Recently, a generalization of the system (1.1) has been presented and numerically investigated as follows [6,8]: \begin{equation*} \label{RF3} \begin{aligned} & \frac{dx_1}{dt}=\left[p\left(x_1^2+x_3\right)+q\left(-x_2^2+3 x_3\right)-1\right], x_2+a x_1 \\ & \frac{dx_2}{dt}=\left[p\left(-x_1^2+3 x_3\right)+q\left(x_2^2+x_3\right)+1\right], x_1+a x_2 \\ &\frac{dx_3}{dt}=-2 x_3(b+(p+q) x_1 x_2) . \end{aligned} \end{equation*} This system becomes the system (1.1) with $p=1$ and $q=0$. In this section, we introduce the following system as a generalization of the system (1.1): (2.1) \begin{equation} \label{RF2} \begin{aligned} & \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\ & \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+c x_2 \\ & \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right) \end{aligned} \end{equation} This system becomes system (1.1) with $c=a$. This generalization is simpler than generalization (2.1), but it includes the same results in the detection of chaos and especially the process of converting the bifurcation of periodic period to chaos. According to the relationship \[ \sum_{i=1}^3 \frac{\partial f_i}{\partial x_i} =(2x_1x_2+a)+(c)+(-2x_1x_2-2b)=a+c-2b, \] The system (2.1) is dissipative whenever $a+c-2b <0$. We fix the parameters $a=-1$ and $b=-0.1$ and consider $c$ as the branching parameter. Therefore, with $c<0.8$, the system will be dissipative and can be chaotic. Considering that system, (1.1) has a special application in encryption and secure communications, generalizing the system and adding to its complexity will achieve these goals and reduce the possibility of decryption and making communications unsafe. In the following, we examine the complex dynamics of the system for $a=-1$, $b=-0.1$ and some values of $c<0.8$. 2.1.1. Period-doubling bifurcation route to chaos. By keeping $a=-1$ and $b=-0.1$ and changing $c$ on the interval $[-0.82,-0.80]$, the bifurcation diagram and lyapunov exponents of the system (2.1), in the figure 1 are drawn. It can be seen that for $c \in [-0.8200,-0.8030]$ the system has two negative lyapunov exponents and one zero lyapunov exponent, and in this interval we have a Periodic Solutions, also for $c \in [-0.8030,-0.8000]$ We have a negative lyapunov exponent, a zero lyapunov exponent and a positive lyapunov exponent, so the system will be chaotic. The bifurcation diagram drawn in figure 1a shows a period doubling route to chaos. In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory. the new one having double the period of the original. This process is called periodic doubling, and its continuous repetition is one of the ways leading to chaos The bifurcation process of periodic period leading to chaos can be seen in figure 2. 2.1.2. transient chaos. A chaotic but transient behavior is observed in figure 3. As the diagram in figure 3a shows, this chaotic behavior lasts significantly, but after some time it disappears and leads to a fixed point. 2.1.3. Coexistence of attractors. Figure 4 depicts the coexistence of some absorbers. Relevant details are specified in the figure captions. 3. Conclusions In this article, we studied the Rabinovich-Fabricant system and its generalized model numerically. For the generalized system, we determined the occurrence of very rich and complex dynamic behaviors with the interaction of three parameters. In particular, we showed the bifurcation phenomenon of period doubling leading to chaos, which was not reported before in the Rabinovitch-Fabricant system. We have seen that by changing the bifurcation parameter in a very small interval, the evolution process of the system, with the continuous repetition of bifurcation bifurcation of periodic period, ends from periodic solution to chaos. The complex dynamic behaviors of the system were investigated using the Lyapunov exponents, the bifurcation diagram depending on the parameters and different sections of the phase space. This study was based on the numerical solution of differential equations and their numerical bifurcation analysis using Matlab software. The obtained results are new, as this generalization of the Rabinovitch-Fabrikant system is proposed for the first time. The generalized model can be used to simulate systems in radio and electronics engineering where there is a three-mode interaction and which include third-order nonlinear terms. The Rabinovitch-Fabricant system simulate systems of a physical nature, and in this regard, the coefficients embedded in it must be positive, but its highly nonlinear and chaotic nature, arising from the presence of third-order sentences, gives them a unique quality to be applied in secure communication. Therefore, its artificial generalization, using negative parameters, which itself adds to the complexity of its rich dynamics, is of particular importance. The aim of this work is the numerical study of the Rabinovich-Fabrikant system and its generalized model, which shows the occurrence of very rich dynamic behaviors with the interaction of three parameters of the generalized system. In particular, we observe the period-doubling bifurcation phenomenon leading to chaos, which has rarely been reported in previous works in the Rabinovich-Fabricant system. The complex dynamic behaviors of the system are investigated by using the Lyapunov spectrum, the parameters dependent bifurcation diagram and different sections of the phase space. This study is based on the numerical solution of differential equations and their numerical bifurcation analysis using Matlab software. The obtained results are new, because the generalization of the Rabinovich-Fabrikant system of the current study was proposed and studied for the first time. The generalized model describes the three-mode interaction. It can be used to simulate systems in radio and electronics engineering in which there is a three-mode interaction and which include cubic nonlinear terms. In addition, although the Rabinovich-Fabricant systems simulate systems of a physical nature, and in this regard, the coefficients embedded in them must be positive, its highly nonlinear and chaotic nature, due to the presence of third-order sentences, gives them a unique quality to be applied in secure communication. Resultantly, its artificial generalization with the use of negative parameters which adds to the complexity of its rich dynamics, is of particular importance. 1. Introduction A dynamic system is classified into two types, discrete and continuous, in terms of time evolution. In this paper, we will look at a particular continuous dynamical system known as the "Rabinovitch-Fabrikant system", which was invented in 1979 by two theoretical physicists, Mikhail Rabinovitch and Anatoly Fabrikant [1]. This system is actually a simplification of a complex nonlinear parabolic equation that models different physical systems such as Tollmien–Schlichting waves in hydrodynamic flows, wind waves on water, Langmuir waves in plasma. Having five equilibrium points, it is not topologically equivalent to many classical systems such as the Lorenz and Chen systems (with three equilibrium points), Rössler system (with two equilibrium points) and the like. In the following, we will call this system the RF system, for short. There are several reasons why this system has been noticed. One is the fact that it models a physical system and, therefore, is not a synthetic model and can be used to model numerous physical phenomena. Another reason is that due to its strong nonlinearity (due to the presence of third-order terms in its mathematical model), a rigorous mathematical analysis cannot be performed on it, hence, the system may show new and interesting properties that have not yet been reported. At the same time, it creates serious challenges for the numerical methods of solving ordinary differential equations. The mathematical model of the Rabinovich-Fabrikant (RF) system is described by the following equations: (1.1) \begin{equation} \label{RF1} \begin{aligned} & \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\ & \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+a x_2 \\ & \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right) \end{aligned} \end{equation} In the Rabinovitch-Fabricant (RF) system modeled by the system of ordinary differential equations (1.1), $a>0$ and $b \in \mathbb{R}$ is the bifurcation parameter. Since it is currently impossible to fully analyze this system mathematically, most of the research is based on numerical and computer analysis. Following this common practice, we will also, in this paper, have an approach based on numerical analysis. 2. Main Results Different dedicated numerical methods for ODEs, implemented in different software packages, might give different results for the same parameters values and initial conditions. system (1.1) by Danca et al. [2, 3, 4] has been studied. By numerical analysis, they showed that this system exhibits unusual and very rich dynamics, including multistability. Here, we obtain their results with a different numerical approach. They used the 3-step predictor–corrector Local Iterative Linearization (LIL) method, which is an implicit 3-step method [5]. In this paper, to calculate the Lyapunov exponents, we have used the MATDS software package applicable in Matlab software by adding codes to get the desired output, and to solve $\mathbf{ODE}$, the $ode45 $ solver implemented in this software with relative error $RelTol=10^{-7}$ and absolute error $AbsTol=10^{-7}$ are used. The step length used in numerical methods is also considered to be $h=10^{-6}$. 2.1. Generalized Rabinovitch-Fabricant system. Recently, a generalization of the system (1.1) has been presented and numerically investigated as follows [6,8]: \begin{equation*} \label{RF3} \begin{aligned} & \frac{dx_1}{dt}=\left[p\left(x_1^2+x_3\right)+q\left(-x_2^2+3 x_3\right)-1\right], x_2+a x_1 \\ & \frac{dx_2}{dt}=\left[p\left(-x_1^2+3 x_3\right)+q\left(x_2^2+x_3\right)+1\right], x_1+a x_2 \\ &\frac{dx_3}{dt}=-2 x_3(b+(p+q) x_1 x_2) . \end{aligned} \end{equation*} This system becomes the system (1.1) with $p=1$ and $q=0$. In this section, we introduce the following system as a generalization of the system (1.1): (2.1) \begin{equation} \label{RF2} \begin{aligned} & \frac{dx_1}{dt}=x_2\left(x_3-1+x_1^2\right)+a x_1 \\ & \frac{dx_2}{dt}=x_1\left(3 x_3+1-x_1^2\right)+c x_2 \\ & \frac{dx_3}{dt}=-2 x_3\left(b+x_1 x_2\right) \end{aligned} \end{equation} This system becomes system (1.1) with $c=a$. This generalization is simpler than generalization (2.1), but it includes the same results in the detection of chaos and especially the process of converting the bifurcation of periodic period to chaos. According to the relationship \[ \sum_{i=1}^3 \frac{\partial f_i}{\partial x_i} =(2x_1x_2+a)+(c)+(-2x_1x_2-2b)=a+c-2b, \] The system (2.1) is dissipative whenever $a+c-2b <0$. We fix the parameters $a=-1$ and $b=-0.1$ and consider $c$ as the branching parameter. Therefore, with $c<0.8$, the system will be dissipative and can be chaotic. Considering that system, (1.1) has a special application in encryption and secure communications, generalizing the system and adding to its complexity will achieve these goals and reduce the possibility of decryption and making communications unsafe. In the following, we examine the complex dynamics of the system for $a=-1$, $b=-0.1$ and some values of $c<0.8$. 2.1.1. Period-doubling bifurcation route to chaos. By keeping $a=-1$ and $b=-0.1$ and changing $c$ on the interval $[-0.82,-0.80]$, the bifurcation diagram and lyapunov exponents of the system (2.1), in the figure 1 are drawn. It can be seen that for $c \in [-0.8200,-0.8030]$ the system has two negative lyapunov exponents and one zero lyapunov exponent, and in this interval we have a Periodic Solutions, also for $c \in [-0.8030,-0.8000]$ We have a negative lyapunov exponent, a zero lyapunov exponent and a positive lyapunov exponent, so the system will be chaotic. The bifurcation diagram drawn in figure 1a shows a period doubling route to chaos. In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory. the new one having double the period of the original. This process is called periodic doubling, and its continuous repetition is one of the ways leading to chaos The bifurcation process of periodic period leading to chaos can be seen in figure 2. 2.1.2. transient chaos. A chaotic but transient behavior is observed in figure 3. As the diagram in figure 3a shows, this chaotic behavior lasts significantly, but after some time it disappears and leads to a fixed point. 2.1.3. Coexistence of attractors. Figure 4 depicts the coexistence of some absorbers. Relevant details are specified in the figure captions. 3. Conclusions In this article, we studied the Rabinovich-Fabricant system and its generalized model numerically. For the generalized system, we determined the occurrence of very rich and complex dynamic behaviors with the interaction of three parameters. In particular, we showed the bifurcation phenomenon of period doubling leading to chaos, which was not reported before in the Rabinovitch-Fabricant system. We have seen that by changing the bifurcation parameter in a very small interval, the evolution process of the system, with the continuous repetition of bifurcation bifurcation of periodic period, ends from periodic solution to chaos. The complex dynamic behaviors of the system were investigated using the Lyapunov exponents, the bifurcation diagram depending on the parameters and different sections of the phase space. This study was based on the numerical solution of differential equations and their numerical bifurcation analysis using Matlab software. The obtained results are new, as this generalization of the Rabinovitch-Fabrikant system is proposed for the first time. The generalized model can be used to simulate systems in radio and electronics engineering where there is a three-mode interaction and which include third-order nonlinear terms. The Rabinovitch-Fabricant system simulate systems of a physical nature, and in this regard, the coefficients embedded in it must be positive, but its highly nonlinear and chaotic nature, arising from the presence of third-order sentences, gives them a unique quality to be applied in secure communication. Therefore, its artificial generalization, using negative parameters, which itself adds to the complexity of its rich dynamics, is of particular importance.

https://math-sci.ui.ac.ir/article_28121_5312dac12d66328fb9beb567f8ba4fea.pdf

University of Isfahan Mathematics and Society 2345-6493 9 1 2024 03 01 Shape operator of an $ (n-1) $-dimensional distribution on an $ n $-dimensional manifold and their classification Shape operator of an $ (n-1) $-dimensional distribution on an $ n $-dimensional manifold and their classification 33 50 28080 10.22108/msci.2023.138841.1600 FA Mehran Aminian Department of Mathematics, Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran Mehran Namjoo Department of Mathematics, Faculty of Mathematics, Vali-e-Asr University, Rafsanjan, Iran 0000-0001-5949-6766 Journal Article 2023 08 21 This paper aims to study of shape operator of an $ (n-1) $-dimensional distribution on an $ n $-dimensional smooth manifold. In this study firstly we state formulae for the shape operator and its symmetric and anti-symmetric components and in continuation we show their relationships with some notions such as integrability, totally umbilic and totally geodesic. Finally, by considering at most two eigenvalues for the shape operator, we classify this distribution and their foliations in simply connected space forms. 1. Introduction The study of surfaces in ordinary three-dimensional space was expanded by Gauss in the early 19th century by introducing the concepts of first and second fundamental forms and curvature. This approach was generalized by studying the submanifolds of a Riemannian manifold. Details of this matter in [10, 7, 11] have been studied. But other aspects need to be studied and in this paper, we intend to examine them. In differential geometry, a distribution on a manifold is an association of vector subspaces that have special properties, and often this distribution is a subbundle of a tangent bundle. The distributions that have the integrability condition create a foliation on the manifold, that is, they separate the manifold into smaller submanifolds. These concepts have many applications in different fields of mathematics, such as integrable systems, Poisson geometry, differential topology, etc [17, 2, 6]. The study of the geometry of regular distributions is a natural extension of the study of the geometry of submanifolds. The mode of integrable distributions is in accordance with the study of foliations. For more details we refer the reader to [16]. If the distribution is not integrable, then the tensor field of its shape operator, it is not symmetrical and its decomposition into symmetric and antisymmetric parts is related to the geometric properties of the distribution. The second fundamental form as well as the shape operator, is the main and basic tool which this research aims to study. We prove different properties of symmetric and antisymmetric components. Finally, by considering at most two eigenvalues for the shape operator, we classify the distribution and its foliations in simply connected space forms. 2. Main Results Suppose $ (M^n,g) $ be a Riemannian manifold, $ {\mathcal H}$ an $ (n-1) $-dimensional regular distribution (horizontal distribution) on $ M $ and $ {\mathcal V} $ the distribution of its orthogonal complement to the Riemannian metric $ g $ (vertical distribution). So $ TM= {\mathcal H}\oplus{\mathcal V}$. Let $ H $ and $ V $ be smooth $ (1,1) $-tensor fields that attribute to a vector field (to a vector) its horizontal and vertical parts, i.e. $ {H}(E)=E-\left<E,N\right>N $ and $ {V}(E)=\left<E,N\right>N $, where $ N $ is an unit vector field may be locally defined and it is perpendicular to the horizontal distribution. Here the symbol $ \left<,\right> $ stands for the inner multiplication of the metric $ g $, and the second fundamental form of $ {\mathcal H} $ is defined in the following form \begin{equation*} B^{{\mathcal H}}(E,F)={ V}(\nabla_{{ H}(E)}{{ H}(F)}), \end{equation*} where $ \nabla $ denotes the Levi-Civita connection of metric $ g $ and $ E,F\in {\mathcal X}(M)$. Definition 2.1. [12] Suppose $ N $ be an unit vertical vector field. The shape operator $ S $ obtained from $ N $, is given by the following relation \begin{equation*} \left<SX,Y\right>=\left<B^{{\mathcal H}}(X,Y),N\right>, \end{equation*} where $ X $, and $ Y $, are horizontal vector fields. The following proposition is similar to that in [12]. Proposition 2.2. Shape operator specifies a linear operator $S:{\mathcal H}_p\rightarrow{\mathcal H}_p $, at any point $ p\in M $ and for each $ v\in {\mathcal H}_p $, $Sv=-\nabla_vN$. Remark 2.3. If the vertical vector field $ N $ (which may be locally defined) is replaced with $ -N $, then the sign of $ S $ changes. So even if $ M $ lacks a vector field that is vertical throughout, then the shape operator $ S $ is defined globally up to sign. The sign of ambiguity should be removed from the inherent formulas. Symmetric and antisymmetric components of $ B^{{\mathcal H}} $ which is denoted by $ B^{{\mathcal H}}_s $ and $ B^{{\mathcal H}}_a $, respectively are defined as follows: \begin{eqnarray*} B^{{\mathcal H}}_s(E,F)=\dfrac{1}{2}\left( B^{{\mathcal H}}(E,F)+ B^{{\mathcal H}}(F,E)\right), \\\;B^{{\mathcal H}}_a(E,F)=\dfrac{1}{2}\left( B^{{\mathcal H}}(E,F)- B^{{\mathcal H}}(F,E)\right), \end{eqnarray*} where$ E,F\in {\mathcal X}(M)$. Therefore $ B^{{\mathcal H}}=B^{{\mathcal H}}_s+B^{{\mathcal H}}_a $. Motivated by this, we offer the following definition. Definition 2.4. Symmetric and antisymmetric components of $ S $, which is denoted by $ S_s $ and $ S_a $, respectively are defined by \begin{eqnarray*} \left<S_sX,Y\right>=\left<B^{{\mathcal H}}_s(X,Y),N\right>, \\\left<S_aX,Y\right>=\left<B^{{\mathcal H}}_a(X,Y),N\right>, \end{eqnarray*} where $ X $ and $ Y $, are horizontal vector fields. As a result $ S=S_s+S_a $, where \begin{eqnarray*} S_s=\frac{1}{2}\left(S+S^t\right),\\ S_a=\frac{1}{2}\left(S-S^t\right), \end{eqnarray*} and $S ^t $ denotes the transpose of $S$. Proposition 2.5. For each $ v\in {\mathcal H}_p $, the following relationships are established: \begin{align*} S_sv=&-\dfrac{1}{2}\left(\nabla_vN+{ H}((\nabla N)^tv)\right) \\=&-\frac{1}{2}\left(\nabla_vN+(\nabla N)^tv-\left<v,\nabla_NN\right>N\right), \end{align*} and \begin{align*} S_av=&-\dfrac{1}{2}\left(\nabla_vN-{ H}((\nabla N)^tv)\right) \\=&-\frac{1}{2}\left(\nabla_vN-(\nabla N)^tv+\left<v,\nabla_NN\right>N\right). \end{align*} Similar to what was stated in [3,12], we arrive at the following proposition. Proposition 2.6. For any horizontal vector field $ X, Y $, we have \begin{align*} \left<S_sX,Y\right>&=\left<B^{{\mathcal H}}_s(X,Y),N\right> \\&=-\frac{1}{2}({\mathcal L}_Ng)(X,Y). \end{align*} Definition 2.7. We call $ \mu^{\mathcal{H}}$ mean curvature and it is defined in the following form \begin{align*} \mu^{{\mathcal H}}=&\frac{1}{n-1}\mathrm{tr}B^{{\mathcal H}} \\=& \frac{1}{n-1}\mathrm{tr}B^{{\mathcal H}}_s \\=&\frac{1}{n-1}\sum_{i=1}^{n-1}B^{{\mathcal H}}_s (e_i,e_i), \end{align*} where $ \{e_i\}_{i=1}^{n-1} $ is a local orthogonal frame for $ \mathcal{H} $. Corollary 2.8. Thee mean curvature $ \mu^{{\mathcal H}} $ has the following properties (a) \begin{align*} \mu^{{\mathcal H}}&= \frac{1}{n-1}(\mathrm{tr}S)N \\&=-\frac{1}{2(n-1)}(\mathrm{tr}{\mathcal L}_Ng)N. \end{align*} (b) $ \mu^{{\mathcal H}}=0 $, if and only if $ \mathrm{tr}S=0 $, also $ \mathrm{tr}S=0 $ if and only if $ \mathrm{tr}{\mathcal L}_Ng=0 $. \end{enumerate} The following definition is similar to the definition which was stated in [3] Definition 2.9. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution. (a) We call $ \mathcal{H} $ totally umbilic if for all horizontal vector fields $ X,Y $, $$B^{{\mathcal H}}_s(X,Y)=\left<X, Y\right>\mu^{\mathcal{H}}.$$ (b) We call $ \mathcal{H} $ totally geodesic if for all horizontal vector fields $ X,Y $, $$B^{{\mathcal H}}_s(X,Y)=0.$$ (c) We call $ \mathcal{H} $ minimal if $$ \mu^{{\mathcal H}}=0.$$ Based on the above definition, we conclude the following propositions. Proposition 2.10. The distribution $ {\mathcal H} $ is totally geodesic, if and only if it is totally umbilic and minimal. Proposition 2.11. The distribution $ {\mathcal H} $ is totally umbilic, if and only if for each selection $ N $, tensor $ S_s $ be a scalar. Especially $ {\mathcal H} $ it is totally geodesic If and only if for each selection $ N $, $ S_s=0 $. Proposition 2.12. Let $ W $ be a nowhere zero Killing vector field. Then the distribution $ {\mathcal H}=\left<W\right>^{\bot} $ is totally geodesic and its symmetric shape operator is zero. A discussion similar to the proposition 2.11 and knowing that the integrability of a distribution is equivalent to the symmetry of its second fundamental form (see [12] for more details), will lead to the following proposition. Proposition 2.13. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution. (a) The distribution $ {\mathcal H}$ is integrable if and only if distribution shape operator, for each selection $ N $, be symmetrical. Hence $ S_s=S $ and $ S_a=0 $. (b) The distribution $ {\mathcal H}$ is totally umbilic and integrable if and only if the distribution shape operator, for each choice $ N $, be a scalar and therefore each leaf of its foliation is a totally umbilic hypersurface. In the following, we assume that the vertical vector field $ N $, be defined throughout manifold and therefore the shape operator obtained from $ N $, without sign of ambiguously defined throughout manifold. Let $ R^n(c) $ be a simply connected Riemannian space form with fixed sectional curvature $ c $ which is the Euclidean space $ \mathbb{R}^n $ for $ c=0 $, is the Euclidean sphere $ \mathbb{S}^n $ for $ c=1 $, and is the Hyperbolic space $ \mathbb{H}^n $ for $ c=-1 $. A hypersurface in $ R^n(c) $ is called an isoparametric if its principal curvatures are constant everywhere, counting multiplicities. In this case, we have the following proposition. Proposition 2.14. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution on $ R^n(c) $ and $ S=\alpha I $, where $\alpha$ is an arbitrary constant. Then either $ c=0 $ or $ c=-1 $ and $ {\mathcal H}$ is totally umbilic and integrable. If $ c=0 $, then $ \alpha=0$, the distribution is totally geodesic, and each leaf of its foliation is a hyperplane. If $ c=-1 $, then $ \alpha=\pm1$, the distribution is non-minimal and each leaf of its foliation is a parabolic hypersurface. Corollary 2.15. There is no $ (n-1) $-dimensional distribution which is integrable and totally geodesic on $ \mathbb{S}^n $ or $ \mathbb{H}^n $. Proposition 2.16. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution on $ R^n(c) $, with $ n>2 $, and $ S=f I $, in which $ f $ is a smooth function on $ R^n(c) $. Then either $ c=0 $ or $ c=-1 $, and $ {\mathcal H}$ is totally umbilic and integrable. If $c=0 $, then the distribution is totally geodesic and all leaves are hyperplanes and the function $ f $ is zero. If $ c=-1 $, then the distribution is non-minimal and each leaf is a hyperbolic or parabolic hypersurface. Also for all $ x\in \mathbb{H}^n $, non-zero vector $ a\in \mathbb{R}_1^{n+1} $, it is found that $ \left<a,a\right>\in\{0,1\} $, and $ f(x)=-\frac{\left<a,x\right>}{\sqrt{\left<a,a\right>+\left<a,x\right>^2}}$. Corollary 2.17. There is no $ (n-1) $-dimensional distribution which is integrable and totally umbilic on $ \mathbb{S}^n $, $ n>2 $. Proposition 2.18. There is no $ (n-1) $-dimensional distribution which is integrable on $ R^n(c) $ that its shape operator has only two distinct constant eigenvalues. Isoparameter hypersurfaces of $ \mathbb{R}^n $ and $ \mathbb{H}^n $ have at most two constant principal curvatures, and so, according to the propositions 2.14-2.18, we obtain the following result. Corollary 2.19. The only $ (n-1) $-dimensional distribution on $ \mathbb{R}^n $, which is integrable and all eigenvalues of its shape operator are constant, is generated by $ n-1 $ linearly independent constant vector fields and so the distribution is totally geodesic and every leaf of its foliation is a hyperplane and the distribution shape operator is zero. Corollary 2.20. There is no $ (n-1) $-dimensional distribution on $ \mathbb{H}^n $, which is integrable and all eigenvalues of its shape operator be constant and has at least two eigenvalues. 3. Conclusion In this paper, we studied the shape operator of a distribution and its properties, especially its decomposition into symmetric and antisymmetric parts. We showed that a distribution is totally umbilic if and only if the symmetric component of its shape operator is scalar, we also proved that it is totally geodesic if and only if the symmetric component of its form operator is zero. Finally, we considered the classification of $ {(n-1)} $-dimensional distributions and studied simply connected space forms with a shape operator having at most two eigenvalues. This paper aims to study of shape operator of an $ (n-1) $-dimensional distribution on an $ n $-dimensional smooth manifold. In this study firstly we state formulae for the shape operator and its symmetric and anti-symmetric components and in continuation we show their relationships with some notions such as integrability, totally umbilic and totally geodesic. Finally, by considering at most two eigenvalues for the shape operator, we classify this distribution and their foliations in simply connected space forms. 1. Introduction The study of surfaces in ordinary three-dimensional space was expanded by Gauss in the early 19th century by introducing the concepts of first and second fundamental forms and curvature. This approach was generalized by studying the submanifolds of a Riemannian manifold. Details of this matter in [10, 7, 11] have been studied. But other aspects need to be studied and in this paper, we intend to examine them. In differential geometry, a distribution on a manifold is an association of vector subspaces that have special properties, and often this distribution is a subbundle of a tangent bundle. The distributions that have the integrability condition create a foliation on the manifold, that is, they separate the manifold into smaller submanifolds. These concepts have many applications in different fields of mathematics, such as integrable systems, Poisson geometry, differential topology, etc [17, 2, 6]. The study of the geometry of regular distributions is a natural extension of the study of the geometry of submanifolds. The mode of integrable distributions is in accordance with the study of foliations. For more details we refer the reader to [16]. If the distribution is not integrable, then the tensor field of its shape operator, it is not symmetrical and its decomposition into symmetric and antisymmetric parts is related to the geometric properties of the distribution. The second fundamental form as well as the shape operator, is the main and basic tool which this research aims to study. We prove different properties of symmetric and antisymmetric components. Finally, by considering at most two eigenvalues for the shape operator, we classify the distribution and its foliations in simply connected space forms. 2. Main Results Suppose $ (M^n,g) $ be a Riemannian manifold, $ {\mathcal H}$ an $ (n-1) $-dimensional regular distribution (horizontal distribution) on $ M $ and $ {\mathcal V} $ the distribution of its orthogonal complement to the Riemannian metric $ g $ (vertical distribution). So $ TM= {\mathcal H}\oplus{\mathcal V}$. Let $ H $ and $ V $ be smooth $ (1,1) $-tensor fields that attribute to a vector field (to a vector) its horizontal and vertical parts, i.e. $ {H}(E)=E-\left<E,N\right>N $ and $ {V}(E)=\left<E,N\right>N $, where $ N $ is an unit vector field may be locally defined and it is perpendicular to the horizontal distribution. Here the symbol $ \left<,\right> $ stands for the inner multiplication of the metric $ g $, and the second fundamental form of $ {\mathcal H} $ is defined in the following form \begin{equation*} B^{{\mathcal H}}(E,F)={ V}(\nabla_{{ H}(E)}{{ H}(F)}), \end{equation*} where $ \nabla $ denotes the Levi-Civita connection of metric $ g $ and $ E,F\in {\mathcal X}(M)$. Definition 2.1. [12] Suppose $ N $ be an unit vertical vector field. The shape operator $ S $ obtained from $ N $, is given by the following relation \begin{equation*} \left<SX,Y\right>=\left<B^{{\mathcal H}}(X,Y),N\right>, \end{equation*} where $ X $, and $ Y $, are horizontal vector fields. The following proposition is similar to that in [12]. Proposition 2.2. Shape operator specifies a linear operator $S:{\mathcal H}_p\rightarrow{\mathcal H}_p $, at any point $ p\in M $ and for each $ v\in {\mathcal H}_p $, $Sv=-\nabla_vN$. Remark 2.3. If the vertical vector field $ N $ (which may be locally defined) is replaced with $ -N $, then the sign of $ S $ changes. So even if $ M $ lacks a vector field that is vertical throughout, then the shape operator $ S $ is defined globally up to sign. The sign of ambiguity should be removed from the inherent formulas. Symmetric and antisymmetric components of $ B^{{\mathcal H}} $ which is denoted by $ B^{{\mathcal H}}_s $ and $ B^{{\mathcal H}}_a $, respectively are defined as follows: \begin{eqnarray*} B^{{\mathcal H}}_s(E,F)=\dfrac{1}{2}\left( B^{{\mathcal H}}(E,F)+ B^{{\mathcal H}}(F,E)\right), \\\;B^{{\mathcal H}}_a(E,F)=\dfrac{1}{2}\left( B^{{\mathcal H}}(E,F)- B^{{\mathcal H}}(F,E)\right), \end{eqnarray*} where$ E,F\in {\mathcal X}(M)$. Therefore $ B^{{\mathcal H}}=B^{{\mathcal H}}_s+B^{{\mathcal H}}_a $. Motivated by this, we offer the following definition. Definition 2.4. Symmetric and antisymmetric components of $ S $, which is denoted by $ S_s $ and $ S_a $, respectively are defined by \begin{eqnarray*} \left<S_sX,Y\right>=\left<B^{{\mathcal H}}_s(X,Y),N\right>, \\\left<S_aX,Y\right>=\left<B^{{\mathcal H}}_a(X,Y),N\right>, \end{eqnarray*} where $ X $ and $ Y $, are horizontal vector fields. As a result $ S=S_s+S_a $, where \begin{eqnarray*} S_s=\frac{1}{2}\left(S+S^t\right),\\ S_a=\frac{1}{2}\left(S-S^t\right), \end{eqnarray*} and $S ^t $ denotes the transpose of $S$. Proposition 2.5. For each $ v\in {\mathcal H}_p $, the following relationships are established: \begin{align*} S_sv=&-\dfrac{1}{2}\left(\nabla_vN+{ H}((\nabla N)^tv)\right) \\=&-\frac{1}{2}\left(\nabla_vN+(\nabla N)^tv-\left<v,\nabla_NN\right>N\right), \end{align*} and \begin{align*} S_av=&-\dfrac{1}{2}\left(\nabla_vN-{ H}((\nabla N)^tv)\right) \\=&-\frac{1}{2}\left(\nabla_vN-(\nabla N)^tv+\left<v,\nabla_NN\right>N\right). \end{align*} Similar to what was stated in [3,12], we arrive at the following proposition. Proposition 2.6. For any horizontal vector field $ X, Y $, we have \begin{align*} \left<S_sX,Y\right>&=\left<B^{{\mathcal H}}_s(X,Y),N\right> \\&=-\frac{1}{2}({\mathcal L}_Ng)(X,Y). \end{align*} Definition 2.7. We call $ \mu^{\mathcal{H}}$ mean curvature and it is defined in the following form \begin{align*} \mu^{{\mathcal H}}=&\frac{1}{n-1}\mathrm{tr}B^{{\mathcal H}} \\=& \frac{1}{n-1}\mathrm{tr}B^{{\mathcal H}}_s \\=&\frac{1}{n-1}\sum_{i=1}^{n-1}B^{{\mathcal H}}_s (e_i,e_i), \end{align*} where $ \{e_i\}_{i=1}^{n-1} $ is a local orthogonal frame for $ \mathcal{H} $. Corollary 2.8. Thee mean curvature $ \mu^{{\mathcal H}} $ has the following properties (a) \begin{align*} \mu^{{\mathcal H}}&= \frac{1}{n-1}(\mathrm{tr}S)N \\&=-\frac{1}{2(n-1)}(\mathrm{tr}{\mathcal L}_Ng)N. \end{align*} (b) $ \mu^{{\mathcal H}}=0 $, if and only if $ \mathrm{tr}S=0 $, also $ \mathrm{tr}S=0 $ if and only if $ \mathrm{tr}{\mathcal L}_Ng=0 $. \end{enumerate} The following definition is similar to the definition which was stated in [3] Definition 2.9. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution. (a) We call $ \mathcal{H} $ totally umbilic if for all horizontal vector fields $ X,Y $, $$B^{{\mathcal H}}_s(X,Y)=\left<X, Y\right>\mu^{\mathcal{H}}.$$ (b) We call $ \mathcal{H} $ totally geodesic if for all horizontal vector fields $ X,Y $, $$B^{{\mathcal H}}_s(X,Y)=0.$$ (c) We call $ \mathcal{H} $ minimal if $$ \mu^{{\mathcal H}}=0.$$ Based on the above definition, we conclude the following propositions. Proposition 2.10. The distribution $ {\mathcal H} $ is totally geodesic, if and only if it is totally umbilic and minimal. Proposition 2.11. The distribution $ {\mathcal H} $ is totally umbilic, if and only if for each selection $ N $, tensor $ S_s $ be a scalar. Especially $ {\mathcal H} $ it is totally geodesic If and only if for each selection $ N $, $ S_s=0 $. Proposition 2.12. Let $ W $ be a nowhere zero Killing vector field. Then the distribution $ {\mathcal H}=\left<W\right>^{\bot} $ is totally geodesic and its symmetric shape operator is zero. A discussion similar to the proposition 2.11 and knowing that the integrability of a distribution is equivalent to the symmetry of its second fundamental form (see [12] for more details), will lead to the following proposition. Proposition 2.13. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution. (a) The distribution $ {\mathcal H}$ is integrable if and only if distribution shape operator, for each selection $ N $, be symmetrical. Hence $ S_s=S $ and $ S_a=0 $. (b) The distribution $ {\mathcal H}$ is totally umbilic and integrable if and only if the distribution shape operator, for each choice $ N $, be a scalar and therefore each leaf of its foliation is a totally umbilic hypersurface. In the following, we assume that the vertical vector field $ N $, be defined throughout manifold and therefore the shape operator obtained from $ N $, without sign of ambiguously defined throughout manifold. Let $ R^n(c) $ be a simply connected Riemannian space form with fixed sectional curvature $ c $ which is the Euclidean space $ \mathbb{R}^n $ for $ c=0 $, is the Euclidean sphere $ \mathbb{S}^n $ for $ c=1 $, and is the Hyperbolic space $ \mathbb{H}^n $ for $ c=-1 $. A hypersurface in $ R^n(c) $ is called an isoparametric if its principal curvatures are constant everywhere, counting multiplicities. In this case, we have the following proposition. Proposition 2.14. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution on $ R^n(c) $ and $ S=\alpha I $, where $\alpha$ is an arbitrary constant. Then either $ c=0 $ or $ c=-1 $ and $ {\mathcal H}$ is totally umbilic and integrable. If $ c=0 $, then $ \alpha=0$, the distribution is totally geodesic, and each leaf of its foliation is a hyperplane. If $ c=-1 $, then $ \alpha=\pm1$, the distribution is non-minimal and each leaf of its foliation is a parabolic hypersurface. Corollary 2.15. There is no $ (n-1) $-dimensional distribution which is integrable and totally geodesic on $ \mathbb{S}^n $ or $ \mathbb{H}^n $. Proposition 2.16. Let $ \mathcal{H} $ be an $ (n-1) $-dimensional distribution on $ R^n(c) $, with $ n>2 $, and $ S=f I $, in which $ f $ is a smooth function on $ R^n(c) $. Then either $ c=0 $ or $ c=-1 $, and $ {\mathcal H}$ is totally umbilic and integrable. If $c=0 $, then the distribution is totally geodesic and all leaves are hyperplanes and the function $ f $ is zero. If $ c=-1 $, then the distribution is non-minimal and each leaf is a hyperbolic or parabolic hypersurface. Also for all $ x\in \mathbb{H}^n $, non-zero vector $ a\in \mathbb{R}_1^{n+1} $, it is found that $ \left<a,a\right>\in\{0,1\} $, and $ f(x)=-\frac{\left<a,x\right>}{\sqrt{\left<a,a\right>+\left<a,x\right>^2}}$. Corollary 2.17. There is no $ (n-1) $-dimensional distribution which is integrable and totally umbilic on $ \mathbb{S}^n $, $ n>2 $. Proposition 2.18. There is no $ (n-1) $-dimensional distribution which is integrable on $ R^n(c) $ that its shape operator has only two distinct constant eigenvalues. Isoparameter hypersurfaces of $ \mathbb{R}^n $ and $ \mathbb{H}^n $ have at most two constant principal curvatures, and so, according to the propositions 2.14-2.18, we obtain the following result. Corollary 2.19. The only $ (n-1) $-dimensional distribution on $ \mathbb{R}^n $, which is integrable and all eigenvalues of its shape operator are constant, is generated by $ n-1 $ linearly independent constant vector fields and so the distribution is totally geodesic and every leaf of its foliation is a hyperplane and the distribution shape operator is zero. Corollary 2.20. There is no $ (n-1) $-dimensional distribution on $ \mathbb{H}^n $, which is integrable and all eigenvalues of its shape operator be constant and has at least two eigenvalues. 3. Conclusion In this paper, we studied the shape operator of a distribution and its properties, especially its decomposition into symmetric and antisymmetric parts. We showed that a distribution is totally umbilic if and only if the symmetric component of its shape operator is scalar, we also proved that it is totally geodesic if and only if the symmetric component of its form operator is zero. Finally, we considered the classification of $ {(n-1)} $-dimensional distributions and studied simply connected space forms with a shape operator having at most two eigenvalues.

https://math-sci.ui.ac.ir/article_28080_dc37e53f978b233ca46a2f844ae2c837.pdf

University of Isfahan Mathematics and Society 2345-6493 9 1 2024 05 05 Existence of the solutions of an interval tensor complementarity problem Existence of the solutions of an interval tensor complementarity problem 51 75 28025 10.22108/msci.2023.137807.1579 FA Rozita Beheshti Department of Mathematics, Faculty of science, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran. Javad Fathi Department of Mathematics, Faculty of science, University of Hormozgan Mostafa Zangiabadi Department of Mathematics, Faculty of science, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran. Journal Article 2023 05 24 In this paper, we consider a general tensor complementarity problem with interval parameters, and study the conditions under which, the existence and uniqueness of the solution of the problem are guaranteed. Furthermore, we proved that the solution set of the interval tensor complementarity problem is not necessarily convex. 1. Introduction Interval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations. The result of an interval computation is an interval, a pair of numbers, an upper and a lower bound, and this pair of numbers guarantees to enclose the exact answer. Maybe we still don’t know the truth, but at least we know how much we don’t know! [4]. How is it possible for interval analysis to guarantee that a computational result is true? The answer is very simple. Using the interval analysis we estimate at each calculation step all kinds of errors: inputs errors, rounding errors and truncation errors. One of the most famous references on IA is probably Moore’s Interval Analysis book. Throughout the paper, vectors are written as $\left\{x, y, \ldots \right\}$, matrices are shown by $ \left\{A, B,\ldots \right\}$ and tensors are written as $ \left\{ \mathcal{A}, \mathcal{B},\ldots \right\}.$ Let $[n]$, $\mathbb{R} (\mathbb{C}) $, and $\mathbb{R}^n (\mathbb{C}^n)$ denote the set $\{ 1, 2,\ldots,n\}$, the set of all real (complex) numbers, and the set of all $n$-dimensional real (complex) vectors; respectively. $x \geq 0 \ \ \ (x > 0)$ means $x_i \geq 0 \ \ \ (x_i > 0)$ for all $i \in \left[ n \right]$. Let $\mathbb{R} _{+}^n = \lbrace x \in \mathbb{R}^n \mid x \geq 0\rbrace$ be the positive cone in $\mathbb{R}^n.$ An order $m$ dimension $n$ real tensor $\mathcal{A} = (a_{i_1 i_2\cdots i_m}),$ denoted by $ \mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } ,$ consists of $n^m$ entries: \[a_{i_1 i_2 \cdots i_m} \in \mathbb{R}, \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] If $n_1=\cdots=n_m = n$, then it is said $\mathcal{A}$ is an $m$-order $n$-dimensional cubical tensor or for simplicity just $m$-order $n$-dimensional tensor. A vector is a tensor of order $1$ and a matrix is a tensor of order $2$. A tensor $\mathcal{A} = (a_{i_1 i_2 \cdots i_m}) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is called nonnegative (positive) if \[a_{i_1 i_2 \cdots i_m} \ge 0 \; (a_{i_1 i_2 \cdots i_m}>0 ), \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] A tensor $ \mathcal{A}$ is said to be symmetric if its entries $ a_{i_1 i_2 \cdots i_m}$ are invariant under any permutation of $ m $ indices $ ( a_{i_1 i_2 \cdots i_m}). $ All the tensors discussed in this paper are real.\\ For any two tensors, $\mathcal{A} = (a_{i_1 \cdots i_m } ),$ and $ \mathcal{B} = (b_{i_1 \cdots i_m } ) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ of identical orders and dimensions, their inner product is defined as $$\left\langle {\mathcal{A},\mathcal{B}} \right\rangle = \sum\limits_{i_1 \cdots i_m } {a_{i_1 \cdots i_m } b_{i_1 \cdots i_m}}. $$ Definition 1.1. If $\mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is an $m$-order tensor and $ B \in \mathbb{R}^{J \times n_k }$ is a matrix, then $\mathcal{A} \times_k B $ denotes the mode-$k$ product of $\mathcal{A}$ with $B$, which is of size $ n_1 \times \cdots \times n_{k-1} \times J \times n_{k+1} \times \cdots \times n_m $, and each element of it is defined as follows \[(\mathcal{A} \times_k B)_{i_1,\ldots,i_{k-1}, j, i_{k+1},\ldots,i_m} = \sum\limits_{i_k = 1}^{n_k } a_{i_1 \cdots i_m } b_{j,i_k}.\] If we do the mode-$k$ product of $ \mathcal{A} $ and $B $ for all possible $k \in [m]$ as \[ \mathcal{A} \times_1 B \times_2 \cdots \times_m B,\] and $B $ is reduced to some row vector, say $x^T=\left( x_1,\ldots,x_n \right),$ the following notations are frequently used in this paper: \begin{align*} &\mathcal{A} x^m \equiv \mathcal{A} \times_1 x^T \times_2 \cdots \times_m x^T = \sum\limits_{i_1 \cdots i_m =1}^n {a_{i_1 \cdots i_m} x_{i_1} \cdots x_{i_m}} \in \mathbb{R}, \\ &\mathcal{A} x^{m-1 } \equiv \mathcal{A} \times_2 x^T \times_3 \cdots \times_m x^T = \sum\limits_{i_2 \cdots i_m =1}^n {a_{i, i_2 \cdots i_m} x_{i_2} \cdots x_{i_m}} \in \mathbb{R}^n. \end{align*} We call a number $\lambda \in \mathbb{C}$ an eigenvalue of $\mathcal{A}$ if it and a nonzero vector $x \in \mathbb{C}^n$ are solutions of the following homogeneous polynomial equations: (1.1) \begin{equation}\label{e3} \left( \mathcal{A} x^{m-1} \right)_i = \lambda x_i^{m - 1}, \ \ \ \forall i=1,\ldots,n, \end{equation} and call the solution $x$ an eigenvector of $\mathcal{A}$ associated with the eigenvalue $\lambda.$ If we denote $x^{[m-1]}$ as a vector in $\mathbb{C}^n$ such that its $i$th component is $x_i^{m - 1},$ then (1.1) can be simply expressed as $$\mathcal{A} x^{m-1} = \lambda x^{[m-1]}.$$ The set of all the eigenvalues of $\mathcal{A}$ is called the spectrum of $\mathcal{A}.$ The largest modulus of the elements in the spectrum of $\mathcal{A}$ is called the spectral radius of $\mathcal{A},$ denoted as $\rho(\mathcal{A}).$ Definition 1.2. Let $\mathcal{A}$ be an $m$-order and $n$-dimensional cubical tensor, then $(1)$ $\mathcal{A}$ is called an $\mathcal{Z}$-tensor if all of its non-diagonal elements are non-positive. This definition is equivalent to having $\mathcal{A}=s\mathcal{I}-\mathcal{B}$, where $s > 0$, $\mathcal{B}$ is a non-negative tensor and $\mathcal{I}=(I_{i_1 \cdots i_m })$, is the identity tensor with entries $$I_{i_1 \cdots i_m}=\left\{ \begin{gathered} 1, \ \ \ i_1=\cdots= i_m,\hfill \\ 0, \ \ \ otherwise. \hfill \\ \end{gathered} \right.$$ $(2)$ $\mathcal{A}$ is called an $\mathcal{M}$-tensor if $\mathcal{A}$ is an $\mathcal{Z}$-tensor and $\mathcal{A} = s\mathcal{I} - \mathcal{B}$, $s \geq \rho(\mathcal{B})$. If $s > \rho(B)$, then $\mathcal{A}$ is called a strong (nonsingular) $\mathcal{M}$-tensor. Definition 1.3. An $m$-order $n$-dimensional tensor $\mathcal{A}$ is said to be P-tensor, if for each nonzero $x \in \mathbb{R}^n $, there exists some index $i$ such that (1.2) \begin{equation}\label{e4} x_i \left( {\mathcal{A}x^{m - 1}} \right)_i >0. \end{equation} The tensor $\mathcal{M}(\mathcal{A} )=(m_{i_1 \cdots i_m})$ is called the comparison tensor of $\mathcal{A}$ if $$ m_{i_1 \cdots i_m}= \begin{cases}-\left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases} $$ Definition 1.4. A tensor $\mathcal{A}$ is called an $\mathcal{H}$-tensor, if its comparison tensor is an $\mathcal{M}$-tensor, and it is called a nonsingular $\mathcal{H}$ -tensor, if its comparison tensor is nonsingular. The tensor complementarity problem denoted by the $\operatorname{TCP}(\mathbf{q}, \mathcal{A})$, is to find a vector $\mathbf{x}$ such that: $$ \mathbf{x} \geq 0, ~\mathcal{A} \mathbf{x}^{m-1}+\mathbf{q} \geq 0,~\left\langle\mathbf{x}, \mathcal{A} \mathbf{x}^{m-1}+\mathbf{q}\right\rangle=0, $$ where $\left\langle , \right\rangle $ denotes the inner product. Theorem 1.5. [6] Tensor $\mathcal{A}$ is $P$-tensor if and only if TCP(q, $\mathcal{A}$) has a unique solution for every $q > 0.$ 2. Main Results Interval linear algebra is a mathematical field developed from classical linear algebra. The only difference is that we do not work with real numbers but we deal with the real closed intervals $x^I : = \left[ {\underline{x}, \overline{x}} \right]$, where $\underline{x} \leq \overline{x}.$ An interval tensor is a tensor which every of its elements is interval. An $m$-order $n$-dimensional cubical interval tensor is denoted by $\mathcal{A}^I : = \left[{\underline{\mathcal{A}}, \overline{\mathcal{A}} } \right]$, where ${\underline{ \mathcal{A}}}$ and ${\overline {\mathcal{A}}}$ are real tensors, and \[\mathcal{A}^I (i_1,\cdots,i_m ) = \left[ {\underline {\mathcal{A}} (i_1,\cdots,i_m), \overline {\mathcal{A}} (i_1,\cdots,i_m )} \right].\] The set of all interval tensors of size $ n_1 \times \cdots \times n_m$, is denoted by $\mathbb{I}\mathbb{R}^{n_1 \times \cdots \times n_m}. $ The parametric form of interval, interval vector and interval tensor can be expressed as follows, respectively. $${a^I=\left\{a(t) \in \mathbb{R}: a(t)=\underline{a}+t\left(\overline{a}-\underline{a}\right), \underline{a} \in \mathbb{R}, \overline{a} \in \mathbb{R}, t \in[0,1]\right\},}$$ $${x^I=\left\{x(t) \in \mathbb{R}^{n}: x(t)=\underline{x}+t\left(\overline{x}-\underline{x}\right), \underline{x} \in \mathbb{R}^{n}, \overline{x} \in \mathbb{R}^{n}, t \in[0,1]\right\},} $$ $$ {\mathcal{A}^I=\left\{\mathcal{A}(t) \in \mathbb{R}^{n \times n \cdots \times n}: \mathcal{A}(t)=\underline{\mathcal{A}}+t\left(\overline{\mathcal{A}}-\underline{\mathcal{A}}\right), \underline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, \overline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, t \in[0,1]\right\}.}$$ Definition 2.1. For $\mathcal{A}^I=[a_{i_1 \cdots i_m}] \in \mathbb{I R}^{n \times n \cdots \times n}$, we define the comparison tensor of $\mathcal{A}^I$ and it is represented as $\mathcal{M}(\mathcal{A}^I )=(m_{i_1 \cdots i_m}) \in \mathbb{R}^{n \times n \cdots \times n}$ by setting $$ m_{i_1 \cdots i_m}= \begin{cases}-\left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases}$$ Definition 2.2. An interval tensor $\mathcal{A}^I \in \mathbb{I R}^{n \times n \cdots \times n}$ is called an interval $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, if all $\mathcal{A} \in\mathcal{A}^I$ are $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, respectively. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval tensor and $q^I \in \mathbb{I R}^{n}$ be an $n$ dimensional interval vector. Then we consider the family of $T CP(\mathcal{A}, q)$'s $$q+\mathcal{A} x^{m - 1} \geq 0, \quad x \geq 0, x^ T ( q+\mathcal{A} x^{m - 1} ) = 0, ~\text {where}~ \mathcal{A} \in\mathcal{A}^I, q^I$$ This is equivalent to the following family of $T C P(\mathcal{A}, q)$'s $$w-\mathcal{A} x^{m - 1}=q, \quad x \geq 0, w \geq 0, x^ T w=0, \text { where } \mathcal{A} \in\mathcal{A}^I, q^I.$$ The family of $T C P(\mathcal{A}, q)$'s is represented as interval tensor complementarity problem and it is denoted by $\operatorname{ITCP}(\mathcal{A}^I, q^I) \cdot \sum_{x}(\mathcal{A}^I,q^I)$ is denoted as solutions set of $\operatorname{ITCP}(\mathcal{A}^I, q^I)$ and it is defined as $$\left\{x \in \mathbb{R}^{n}: q+ \mathcal{A} x^{m - 1} \geq 0, x \geq 0, x^ T (q+ \mathcal{A} x^{m-1}) =0, \mathcal{A} \in\mathcal{A}^I, q^I\right\}$$ The parametric form of $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ is represented as $T C P(\mathcal{A}(t), q(t)), t \in[0,1]$ and its solution set is defined as for some fixed $t,$ $$\sum_{x}(\mathcal{A}(t), q(t))=\left\{x \in \mathbb{R}^{n}: q(t)+\mathcal{A}(t) x^{m - 1} \geq 0, x \geq 0,x^{T} (q(t)+\mathcal{A}(t) x^{m - 1}) =0\right\}$$ Lemma 2.3. For any $\operatorname{ITCP}(\mathcal{A}^I,q^I), \sum_{x}\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $\sum_{x}\left(\overline{\mathcal{A}}, \overline{q}\right)$ are subsets of $\sum_{x}(\mathcal{A}^I,q^I)$. Theorem 2.4. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Proof. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then from Theorem \ref{pp} we get that $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$ are $P$-tensors. Since the positive convex combination of $P$-tensors is also a $P$-tensor, then $\mathcal{A}(t)=t \underline{\mathcal{A}}+(1-t) \overline{\mathcal{A}}$, $t \in[0,1]$, is a $P$-tensor. Hence $\operatorname{TCP}(\mathcal{A}(t), q(t))$ has unique solution for each $t$. Therefore, $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Theorem 2.5. The set $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not convex. Proof. Let $\left(w^{1}, x^{1}\right)^{T},\left(w^{2}, x^{2}\right)^{T} \in \sum_{(w, x)}(\mathcal{A}^I,q^I)$. Then, (2.1) $$\begin{gathered} & w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \leq \overline{q}_{i}, w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \geq \underline{q}_{i},\\ & x_{i_2}^{j} \cdots x_{i_m}^{j} \geq 0, w_{i}^{j} \geq 0, w_{i}^{j} x_{i}^{j}=0, i_1, \cdots, i_m=1, 2,\ldots,n, ~ j=1,2. \end{gathered}$$ Let $0 \leq \lambda \leq 1$. Then multiplying (2.1) by $\lambda$ for $j=1$, and multiplying (2.1) by $(1-\lambda)$ for $j=2$, and adding up these inequalities, we get $$\begin{gathered} \left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \leq \overline{q}_{i}, i=1,2, \ldots, n, \\ \left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq \underline{q}_{i}, i=1,2, \ldots, n, \\ \left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq 0,\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right) \geq 0, i_1,\cdots,i_m=1, 2,\ldots,n. \end{gathered}$$ On the other hand, $\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right)=0$ only for $t=0$ and $t=1$. Since convex combination of complementarity variables are not complementarity variables. This gives that $\left(\lambda w^{1}+(1-\lambda) w^{2}, \lambda x^{1}+(1-\lambda) x^{2}\right)^{T} \notin \sum_{(w,x)}(\mathcal{A}^I,q^I)$, and so $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not a convex set. Theorem 2.6. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then, $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ has a solution for every $q^I \in \mathbb{I R}^{n}$, that is, $\sum_{x}(\mathcal{A}^I,q^I)$ is a non-empty set. Proof. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then each $\mathcal{A} \in\mathcal{A}^I$ is an $H$-tensor, and so $T C P(\mathcal{A}, q)$ has a solution. Also, $\sum_{x}(\mathcal{A}, q) \subseteq \sum_{x}(\mathcal{A}^I,q^I)$, which yields that $\sum_{x}(\mathcal{A}^I,q^I)$ is non-empty. 3. Conclusion A methodology is developed to discuss the existence of the solution of tensor complementarity problem, where the parameters are closed intervals. We also proved that the solution set of the interval tensor complementarity problem is not necessarily convex. In this paper, we consider a general tensor complementarity problem with interval parameters, and study the conditions under which, the existence and uniqueness of the solution of the problem are guaranteed. Furthermore, we proved that the solution set of the interval tensor complementarity problem is not necessarily convex. 1. Introduction Interval analysis is a branch of numerical analysis that was born in the 1960's. It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations. The result of an interval computation is an interval, a pair of numbers, an upper and a lower bound, and this pair of numbers guarantees to enclose the exact answer. Maybe we still don’t know the truth, but at least we know how much we don’t know! [4]. How is it possible for interval analysis to guarantee that a computational result is true? The answer is very simple. Using the interval analysis we estimate at each calculation step all kinds of errors: inputs errors, rounding errors and truncation errors. One of the most famous references on IA is probably Moore’s Interval Analysis book. Throughout the paper, vectors are written as $\left\{x, y, \ldots \right\}$, matrices are shown by $ \left\{A, B,\ldots \right\}$ and tensors are written as $ \left\{ \mathcal{A}, \mathcal{B},\ldots \right\}.$ Let $[n]$, $\mathbb{R} (\mathbb{C}) $, and $\mathbb{R}^n (\mathbb{C}^n)$ denote the set $\{ 1, 2,\ldots,n\}$, the set of all real (complex) numbers, and the set of all $n$-dimensional real (complex) vectors; respectively. $x \geq 0 \ \ \ (x > 0)$ means $x_i \geq 0 \ \ \ (x_i > 0)$ for all $i \in \left[ n \right]$. Let $\mathbb{R} _{+}^n = \lbrace x \in \mathbb{R}^n \mid x \geq 0\rbrace$ be the positive cone in $\mathbb{R}^n.$ An order $m$ dimension $n$ real tensor $\mathcal{A} = (a_{i_1 i_2\cdots i_m}),$ denoted by $ \mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } ,$ consists of $n^m$ entries: \[a_{i_1 i_2 \cdots i_m} \in \mathbb{R}, \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] If $n_1=\cdots=n_m = n$, then it is said $\mathcal{A}$ is an $m$-order $n$-dimensional cubical tensor or for simplicity just $m$-order $n$-dimensional tensor. A vector is a tensor of order $1$ and a matrix is a tensor of order $2$. A tensor $\mathcal{A} = (a_{i_1 i_2 \cdots i_m}) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is called nonnegative (positive) if \[a_{i_1 i_2 \cdots i_m} \ge 0 \; (a_{i_1 i_2 \cdots i_m}>0 ), \quad \; \forall \; i_j = 1,\cdots,n ,\quad j = 1,\cdots,m.\] A tensor $ \mathcal{A}$ is said to be symmetric if its entries $ a_{i_1 i_2 \cdots i_m}$ are invariant under any permutation of $ m $ indices $ ( a_{i_1 i_2 \cdots i_m}). $ All the tensors discussed in this paper are real.\\ For any two tensors, $\mathcal{A} = (a_{i_1 \cdots i_m } ),$ and $ \mathcal{B} = (b_{i_1 \cdots i_m } ) \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ of identical orders and dimensions, their inner product is defined as $$\left\langle {\mathcal{A},\mathcal{B}} \right\rangle = \sum\limits_{i_1 \cdots i_m } {a_{i_1 \cdots i_m } b_{i_1 \cdots i_m}}. $$ Definition 1.1. If $\mathcal{A} \in \mathbb{R}^{n_1 \times \cdots \times n_m } $ is an $m$-order tensor and $ B \in \mathbb{R}^{J \times n_k }$ is a matrix, then $\mathcal{A} \times_k B $ denotes the mode-$k$ product of $\mathcal{A}$ with $B$, which is of size $ n_1 \times \cdots \times n_{k-1} \times J \times n_{k+1} \times \cdots \times n_m $, and each element of it is defined as follows \[(\mathcal{A} \times_k B)_{i_1,\ldots,i_{k-1}, j, i_{k+1},\ldots,i_m} = \sum\limits_{i_k = 1}^{n_k } a_{i_1 \cdots i_m } b_{j,i_k}.\] If we do the mode-$k$ product of $ \mathcal{A} $ and $B $ for all possible $k \in [m]$ as \[ \mathcal{A} \times_1 B \times_2 \cdots \times_m B,\] and $B $ is reduced to some row vector, say $x^T=\left( x_1,\ldots,x_n \right),$ the following notations are frequently used in this paper: \begin{align*} &\mathcal{A} x^m \equiv \mathcal{A} \times_1 x^T \times_2 \cdots \times_m x^T = \sum\limits_{i_1 \cdots i_m =1}^n {a_{i_1 \cdots i_m} x_{i_1} \cdots x_{i_m}} \in \mathbb{R}, \\ &\mathcal{A} x^{m-1 } \equiv \mathcal{A} \times_2 x^T \times_3 \cdots \times_m x^T = \sum\limits_{i_2 \cdots i_m =1}^n {a_{i, i_2 \cdots i_m} x_{i_2} \cdots x_{i_m}} \in \mathbb{R}^n. \end{align*} We call a number $\lambda \in \mathbb{C}$ an eigenvalue of $\mathcal{A}$ if it and a nonzero vector $x \in \mathbb{C}^n$ are solutions of the following homogeneous polynomial equations: (1.1) \begin{equation}\label{e3} \left( \mathcal{A} x^{m-1} \right)_i = \lambda x_i^{m - 1}, \ \ \ \forall i=1,\ldots,n, \end{equation} and call the solution $x$ an eigenvector of $\mathcal{A}$ associated with the eigenvalue $\lambda.$ If we denote $x^{[m-1]}$ as a vector in $\mathbb{C}^n$ such that its $i$th component is $x_i^{m - 1},$ then (1.1) can be simply expressed as $$\mathcal{A} x^{m-1} = \lambda x^{[m-1]}.$$ The set of all the eigenvalues of $\mathcal{A}$ is called the spectrum of $\mathcal{A}.$ The largest modulus of the elements in the spectrum of $\mathcal{A}$ is called the spectral radius of $\mathcal{A},$ denoted as $\rho(\mathcal{A}).$ Definition 1.2. Let $\mathcal{A}$ be an $m$-order and $n$-dimensional cubical tensor, then $(1)$ $\mathcal{A}$ is called an $\mathcal{Z}$-tensor if all of its non-diagonal elements are non-positive. This definition is equivalent to having $\mathcal{A}=s\mathcal{I}-\mathcal{B}$, where $s > 0$, $\mathcal{B}$ is a non-negative tensor and $\mathcal{I}=(I_{i_1 \cdots i_m })$, is the identity tensor with entries $$I_{i_1 \cdots i_m}=\left\{ \begin{gathered} 1, \ \ \ i_1=\cdots= i_m,\hfill \\ 0, \ \ \ otherwise. \hfill \\ \end{gathered} \right.$$ $(2)$ $\mathcal{A}$ is called an $\mathcal{M}$-tensor if $\mathcal{A}$ is an $\mathcal{Z}$-tensor and $\mathcal{A} = s\mathcal{I} - \mathcal{B}$, $s \geq \rho(\mathcal{B})$. If $s > \rho(B)$, then $\mathcal{A}$ is called a strong (nonsingular) $\mathcal{M}$-tensor. Definition 1.3. An $m$-order $n$-dimensional tensor $\mathcal{A}$ is said to be P-tensor, if for each nonzero $x \in \mathbb{R}^n $, there exists some index $i$ such that (1.2) \begin{equation}\label{e4} x_i \left( {\mathcal{A}x^{m - 1}} \right)_i >0. \end{equation} The tensor $\mathcal{M}(\mathcal{A} )=(m_{i_1 \cdots i_m})$ is called the comparison tensor of $\mathcal{A}$ if $$ m_{i_1 \cdots i_m}= \begin{cases}-\left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|a_{i_1 \cdots i_m}\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases} $$ Definition 1.4. A tensor $\mathcal{A}$ is called an $\mathcal{H}$-tensor, if its comparison tensor is an $\mathcal{M}$-tensor, and it is called a nonsingular $\mathcal{H}$ -tensor, if its comparison tensor is nonsingular. The tensor complementarity problem denoted by the $\operatorname{TCP}(\mathbf{q}, \mathcal{A})$, is to find a vector $\mathbf{x}$ such that: $$ \mathbf{x} \geq 0, ~\mathcal{A} \mathbf{x}^{m-1}+\mathbf{q} \geq 0,~\left\langle\mathbf{x}, \mathcal{A} \mathbf{x}^{m-1}+\mathbf{q}\right\rangle=0, $$ where $\left\langle , \right\rangle $ denotes the inner product. Theorem 1.5. [6] Tensor $\mathcal{A}$ is $P$-tensor if and only if TCP(q, $\mathcal{A}$) has a unique solution for every $q > 0.$ 2. Main Results Interval linear algebra is a mathematical field developed from classical linear algebra. The only difference is that we do not work with real numbers but we deal with the real closed intervals $x^I : = \left[ {\underline{x}, \overline{x}} \right]$, where $\underline{x} \leq \overline{x}.$ An interval tensor is a tensor which every of its elements is interval. An $m$-order $n$-dimensional cubical interval tensor is denoted by $\mathcal{A}^I : = \left[{\underline{\mathcal{A}}, \overline{\mathcal{A}} } \right]$, where ${\underline{ \mathcal{A}}}$ and ${\overline {\mathcal{A}}}$ are real tensors, and \[\mathcal{A}^I (i_1,\cdots,i_m ) = \left[ {\underline {\mathcal{A}} (i_1,\cdots,i_m), \overline {\mathcal{A}} (i_1,\cdots,i_m )} \right].\] The set of all interval tensors of size $ n_1 \times \cdots \times n_m$, is denoted by $\mathbb{I}\mathbb{R}^{n_1 \times \cdots \times n_m}. $ The parametric form of interval, interval vector and interval tensor can be expressed as follows, respectively. $${a^I=\left\{a(t) \in \mathbb{R}: a(t)=\underline{a}+t\left(\overline{a}-\underline{a}\right), \underline{a} \in \mathbb{R}, \overline{a} \in \mathbb{R}, t \in[0,1]\right\},}$$ $${x^I=\left\{x(t) \in \mathbb{R}^{n}: x(t)=\underline{x}+t\left(\overline{x}-\underline{x}\right), \underline{x} \in \mathbb{R}^{n}, \overline{x} \in \mathbb{R}^{n}, t \in[0,1]\right\},} $$ $$ {\mathcal{A}^I=\left\{\mathcal{A}(t) \in \mathbb{R}^{n \times n \cdots \times n}: \mathcal{A}(t)=\underline{\mathcal{A}}+t\left(\overline{\mathcal{A}}-\underline{\mathcal{A}}\right), \underline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, \overline{\mathcal{A}} \in \mathbb{R}^{n \times n \cdots \times n}, t \in[0,1]\right\}.}$$ Definition 2.1. For $\mathcal{A}^I=[a_{i_1 \cdots i_m}] \in \mathbb{I R}^{n \times n \cdots \times n}$, we define the comparison tensor of $\mathcal{A}^I$ and it is represented as $\mathcal{M}(\mathcal{A}^I )=(m_{i_1 \cdots i_m}) \in \mathbb{R}^{n \times n \cdots \times n}$ by setting $$ m_{i_1 \cdots i_m}= \begin{cases}-\left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m) \neq (i_1,\cdots,i_1) \\ \left|[a_{i_1 \cdots i_m}]\right|, & \text { if } (i_2,\cdots,i_m)=(i_1,\cdots,i_1)\end{cases}$$ Definition 2.2. An interval tensor $\mathcal{A}^I \in \mathbb{I R}^{n \times n \cdots \times n}$ is called an interval $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, if all $\mathcal{A} \in\mathcal{A}^I$ are $p$-tensor, $\mathcal{Z}$-tensor, $\mathcal{M}$-tensor and $\mathcal{H}$-tensor, respectively. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval tensor and $q^I \in \mathbb{I R}^{n}$ be an $n$ dimensional interval vector. Then we consider the family of $T CP(\mathcal{A}, q)$'s $$q+\mathcal{A} x^{m - 1} \geq 0, \quad x \geq 0, x^ T ( q+\mathcal{A} x^{m - 1} ) = 0, ~\text {where}~ \mathcal{A} \in\mathcal{A}^I, q^I$$ This is equivalent to the following family of $T C P(\mathcal{A}, q)$'s $$w-\mathcal{A} x^{m - 1}=q, \quad x \geq 0, w \geq 0, x^ T w=0, \text { where } \mathcal{A} \in\mathcal{A}^I, q^I.$$ The family of $T C P(\mathcal{A}, q)$'s is represented as interval tensor complementarity problem and it is denoted by $\operatorname{ITCP}(\mathcal{A}^I, q^I) \cdot \sum_{x}(\mathcal{A}^I,q^I)$ is denoted as solutions set of $\operatorname{ITCP}(\mathcal{A}^I, q^I)$ and it is defined as $$\left\{x \in \mathbb{R}^{n}: q+ \mathcal{A} x^{m - 1} \geq 0, x \geq 0, x^ T (q+ \mathcal{A} x^{m-1}) =0, \mathcal{A} \in\mathcal{A}^I, q^I\right\}$$ The parametric form of $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ is represented as $T C P(\mathcal{A}(t), q(t)), t \in[0,1]$ and its solution set is defined as for some fixed $t,$ $$\sum_{x}(\mathcal{A}(t), q(t))=\left\{x \in \mathbb{R}^{n}: q(t)+\mathcal{A}(t) x^{m - 1} \geq 0, x \geq 0,x^{T} (q(t)+\mathcal{A}(t) x^{m - 1}) =0\right\}$$ Lemma 2.3. For any $\operatorname{ITCP}(\mathcal{A}^I,q^I), \sum_{x}\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $\sum_{x}\left(\overline{\mathcal{A}}, \overline{q}\right)$ are subsets of $\sum_{x}(\mathcal{A}^I,q^I)$. Theorem 2.4. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Proof. Suppose that $T C P\left(\underline{\mathcal{A}}, \underline{q}\right)$ and $T C P\left(\overline{\mathcal{A}}, \overline{q}\right)$ have unique solutions, then from Theorem \ref{pp} we get that $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$ are $P$-tensors. Since the positive convex combination of $P$-tensors is also a $P$-tensor, then $\mathcal{A}(t)=t \underline{\mathcal{A}}+(1-t) \overline{\mathcal{A}}$, $t \in[0,1]$, is a $P$-tensor. Hence $\operatorname{TCP}(\mathcal{A}(t), q(t))$ has unique solution for each $t$. Therefore, $\sum_{x}(\mathcal{A}(t), q(t))$ is a singleton set, for every $t \in[0,1]$. Theorem 2.5. The set $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not convex. Proof. Let $\left(w^{1}, x^{1}\right)^{T},\left(w^{2}, x^{2}\right)^{T} \in \sum_{(w, x)}(\mathcal{A}^I,q^I)$. Then, (2.1) $$\begin{gathered} & w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \leq \overline{q}_{i}, w_{i}^{j}-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m} x_{i_2}^{j} \cdots x_{i_m}^{j} \geq \underline{q}_{i},\\ & x_{i_2}^{j} \cdots x_{i_m}^{j} \geq 0, w_{i}^{j} \geq 0, w_{i}^{j} x_{i}^{j}=0, i_1, \cdots, i_m=1, 2,\ldots,n, ~ j=1,2. \end{gathered}$$ Let $0 \leq \lambda \leq 1$. Then multiplying (2.1) by $\lambda$ for $j=1$, and multiplying (2.1) by $(1-\lambda)$ for $j=2$, and adding up these inequalities, we get $$\begin{gathered} \left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \overline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \leq \overline{q}_{i}, i=1,2, \ldots, n, \\ \left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)-\sum_{i_2 \cdots i_m=1}^{n} \underline{a}_{i i_2 \cdots i_m}\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq \underline{q}_{i}, i=1,2, \ldots, n, \\ \left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right) \geq 0,\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right) \geq 0, i_1,\cdots,i_m=1, 2,\ldots,n. \end{gathered}$$ On the other hand, $\left(\lambda w_{i}^{1}+(1-\lambda) w_{i}^{2}\right)\left(\lambda x_{i_2}^{1} \cdots x_{i_m}^{1}+(1-\lambda) x_{i_2}^{2} \cdots x_{i_m}^{2}\right)=0$ only for $t=0$ and $t=1$. Since convex combination of complementarity variables are not complementarity variables. This gives that $\left(\lambda w^{1}+(1-\lambda) w^{2}, \lambda x^{1}+(1-\lambda) x^{2}\right)^{T} \notin \sum_{(w,x)}(\mathcal{A}^I,q^I)$, and so $\sum_{(w, x)}(\mathcal{A}^I,q^I)$ is not a convex set. Theorem 2.6. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then, $\operatorname{ITCP}(\mathcal{A}^I,q^I)$ has a solution for every $q^I \in \mathbb{I R}^{n}$, that is, $\sum_{x}(\mathcal{A}^I,q^I)$ is a non-empty set. Proof. Let $\mathcal{A}^I $ be an $m$-order and $n$-dimensional interval $H$-tensor. Then each $\mathcal{A} \in\mathcal{A}^I$ is an $H$-tensor, and so $T C P(\mathcal{A}, q)$ has a solution. Also, $\sum_{x}(\mathcal{A}, q) \subseteq \sum_{x}(\mathcal{A}^I,q^I)$, which yields that $\sum_{x}(\mathcal{A}^I,q^I)$ is non-empty. 3. Conclusion A methodology is developed to discuss the existence of the solution of tensor complementarity problem, where the parameters are closed intervals. We also proved that the solution set of the interval tensor complementarity problem is not necessarily convex.

https://math-sci.ui.ac.ir/article_28025_55c05332c031fee405061247aa4287ec.pdf

University of Isfahan Mathematics and Society 2345-6493 9 1 2024 05 27 An investigation of kuratowski’s definition of an ordered pair An investigation of kuratowski’s definition of an ordered pair 77 89 28147 10.22108/msci.2024.137454.1571 FA Ali Molkhasi Department of Mathematics, Farhamgian University, Tehran, Iran Mahsa Ezati Department of Mathematics, Farhamgian University, Tehran, Iran Journal Article 2023 04 26 In this paper, we study the definitions of ordered pair from the point of view of Kuratowski, Wiener, Hausdorff and Morse. We first present the possible definitions of an ordered pair, and then examine the structure of the Cartesian product of sets, relations, and functions. This paper can be useful for mathematical researchers in creating new insight and better understanding of the concept of ordered pair and its related mathematical concepts, such as ordered and unordered Cartesian product, relation and function, etc. In this paper, we study the definitions of ordered pair from the point of view of Kuratowski, Wiener, Hausdorff and Morse. We first present the possible definitions of an ordered pair, and then examine the structure of the Cartesian product of sets, relations, and functions. This paper can be useful for mathematical researchers in creating new insight and better understanding of the concept of ordered pair and its related mathematical concepts, such as ordered and unordered Cartesian product, relation and function, etc.

https://math-sci.ui.ac.ir/article_28147_53cdac105bd76ae58cd33be7b4030809.pdf

University of Isfahan Mathematics and Society 2345-6493 9 1 1403 03 07 Mario Pieri's axiomatization of geometry Mario Pieri's axiomatization of geometry 91 105 28182 10.22108/msci.2024.139419.1615 FA Reza Rezavand &lrm;School of Mathematics&lrm;, &lrm;Statistics and Computer Science&lrm;, &lrm;College of Science&lrm;, &lrm;University of Tehran&lrm;, &lrm;Tehran. Iran Journal Article 1402 07 20 In this paper, we discuss the method of the Italian mathematician, Mario Pieri, for axiomatization of geometry, which is based only on two undefined terms, point and motion. We examine the postulates proposed by him, definitions of various geometrical concepts and some important theorems arising from those postulates. In this paper, we discuss the method of the Italian mathematician, Mario Pieri, for axiomatization of geometry, which is based only on two undefined terms, point and motion. We examine the postulates proposed by him, definitions of various geometrical concepts and some important theorems arising from those postulates.

https://math-sci.ui.ac.ir/article_28182_f54e28c50373fee1cc493fd1e04a75ab.pdf

University of Isfahan Mathematics and Society 2345-6493 9 1 2024 05 21 Knot theory, from past to present Knot theory, from past to present 107 136 28162 10.22108/msci.2024.140066.1631 FA Eaman Eftekhary School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5746, Tehran, Iran Journal Article 2023 12 11 In this paper, which is the first paper from a trio on important developments of low dimensional topology in the past 100 years, we review the history and major developments in knot theory. This historic account includes the initial attempts at formulating some of the main questions about knots in a mathematical language, putting the definitions and arguments in a rigorous mathematical framework, and employing tools from other fields of mathematics to extract interesting and intriguing results in knot theory. The study starts from Tait's work in nineteenth century and reviews the important steps taken before the introduction of gauge theory. In particular, we will review the prime decomposition of knots and various polynomial invariants constructed for knots and links. We finish the paper by discussing some of the important conjectures in knot theory which have surprisingly simple statement. We also review some of the recent developments around the aforementioned conjecture, including some theorems including contributions from the author. In this paper, which is the first paper from a trio on important developments of low dimensional topology in the past 100 years, we review the history and major developments in knot theory. This historic account includes the initial attempts at formulating some of the main questions about knots in a mathematical language, putting the definitions and arguments in a rigorous mathematical framework, and employing tools from other fields of mathematics to extract interesting and intriguing results in knot theory. The study starts from Tait's work in nineteenth century and reviews the important steps taken before the introduction of gauge theory. In particular, we will review the prime decomposition of knots and various polynomial invariants constructed for knots and links. We finish the paper by discussing some of the important conjectures in knot theory which have surprisingly simple statement. We also review some of the recent developments around the aforementioned conjecture, including some theorems including contributions from the author.

https://math-sci.ui.ac.ir/article_28162_154f159033a1595f9afc9139130fec1f.pdf