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Abstract. In this paper, we show that the coefficients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. When the dimension is 2, some simplified formulas of the E-characteristic polynomial are presented. A resultant formula for the constant term of the E-characteristic polynomial is given. We prove that both the set of tensors with infinitely many eigenpairs and the set of irregular tensors have codimension 2 as subvarieties in the projective space of tensors. This makes our perturbation method workable. By using the perturbation method and exploring the difference between E-eigenvalues and eigenpair equivalence classes, we present a simple formula for the coefficient of the leading term of the E-characteristic polynomial when the dimension is 2.Key words. E-eigenvalues, tensors, E-characteristic polynomials, eigenpair equivalence class, irregularity.AMS subject classifications. 65H17. IntroductionEigenvalues of higher order tensors were introduced in 2005 [15,13] and have attracted much attention in the literature and found applications in science and engineering. The E-eigenvalues of a tensor were introduced in [15,17]. E-eigenvalues are invariant under orthonormal coordinate changes [17]. They have applications in determining positive definiteness of a multivariate system [15,19], best rank-one approximation [15,18,19,24], magnetic resonance imaging [5,20], spectral hypergraph theory [9] and symmetric embedding [21], and possess links with geometry [1,2,3,4,16]. The numbers of E-eigenvalues and E-eigenvectors have been studied in [7,14].An n-dimensional mth order hypermatrix A with entries in C is a map from {1,··· ,n} m → C. We use notation A = (a i1i2···im ) to denote such a hypermatrix. In physics [11,22], a tensor is a physical quantity. In geometry, a tensor can be regarded as a multi-linear function in a Hilbert space; see the discussion in Section 3. In such applications, in an orthonormal coordinate system, a tensor is expressed by a hypermatrix. It is important to determine parameters and properties of such a tensor, which are invariant under orthonormal coordinate changes. In Theorem 1 of Section 3 of this paper, we show that the coefficients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. For that discussion, we need to distinguish a tensor A and its hypermatrix A in a coordinate system.For some other applications, such as determining positive definiteness of a multivariate system, best rank-one approximation, spectral hypergraph theory, and symmetric embedding, there are no coordinate systems involved. People just regard a

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Abstract. R. C. Reilly calculated the variations of functions of the mean curvatures for hypersurfaces in Euclidean space. In the present paper, using Reilly's formulas, we derive some general integral formulas for hypersurfaces, which generalize the well-known Minkowski formulas, and then apply those formulas to obtain some characterizations of the hypersphere.

Mining with filling technology is not only an important part of green mining but also the ideal methods to solve the environmental issues in coal mining. The coal gangue and fly ash and furnace slag were selected as the stowing material and the SL paste as cementing material. These two kinds of material were processed into a toothpaste-like slurry without dehydration on ground and transport to underground with the help of gravity or stowing pump in order to stowing the goaf and control the movement of superincumbent strata .Industrial test at Xinsan colliery showed that such kind of paste filling material has high early strength, when filling body was used for false roof of work-face, the post-strength requirement of paste filling material can ensuring false roof stability.