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Register a new userRow Cones, Perron Similarities, and Nonnegative Spectra
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In further pursuit of the diagonalizable emph{real nonnegative inverse eigenvalue problem} (RNIEP), we study the relationship between the emph{row cone} $mathcal{C}_r(S)$ and the emph{spectracone} $mathcal{C}(S)$ of a Perron similarity $S$. In the process, a new kind of matrix, emph{row Hadamard conic} (RHC), is defined and related to the D-RNIEP. Characterizations are given when $mathcal{C}_r(S) = mathcal{C}(S)$, and explicit examples are given for all possible set-theoretic relationships between the two cones. The symmetric NIEP is the special case of the D-RNIEP in which the Perron similarity $S$ is also orthogonal.
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In this paper, we generalize some conclusions from the nonnegative irreducible tensor to the nonnegative weakly irreducible tensor and give more properties of eigenvalue problems.
Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce the irreducible tubal matrix, and provide two equivalent characterizations. Then, the celebrated Perron-Frobenius theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices can be generalized to the tubal matrix setting, while some are not. One reason is the defined positivity here has a different meaning to its usual sense. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.
In this paper, we mainly focus on how to generalize some conclusions from nonnegative irreducible tensors to nonnegative weakly irreducible tensors. To do so, a basic and important lemma is proven using new tools. First, we give the definition of stochastic tensors. Then we show that every nonnegative weakly irreducible tensor with spectral radius being one is diagonally similar to a unique weakly irreducible stochastic tensor. Based on it, we prove some important lemmas, which help us to generalize the results related. Some counterexamples are provided to show that some conclusions for nonnegative irreducible tensors do not hold for nonnegative weakly irreducible tensors.
Let svec = (s_1,...,s_m) and tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(svec,tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(svec,tvec) as m,ntoinfty with s and t bounded (various authors independently, 1971-1974), and when svec,tvec are constant vectors with m/n,n/m,s/n >= c/log n for sufficiently large (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st=o(S^(2/3)). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.
We consider the problem of geometric optimization of the lowest eigenvalue for the Laplacian on a compact, simply-connected two-dimensional manifold with boundary subject to an attractive Robin boundary condition. We prove that in the sub-class of manifolds with the Gauss curvature bounded from above by a constant $K_circ ge 0$ and under the constraint of fixed perimeter, the geodesic disk of constant curvature $K_circ$ maximizes the lowest Robin eigenvalue. In the same geometric setting, it is proved that the spectral isoperimetric inequality holds for the lowest eigenvalue of the Dirichlet-to-Neumann operator. Finally, we adapt our methods to Robin Laplacians acting on unbounded three-dimensional cones to show that, under a constraint of fixed perimeter of the cross-section, the lowest Robin eigenvalue is maximized by the circular cone.
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