<Emphasis Type="Italic">l</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis>,<Emphasis Type="Italic">s</Emphasis></Superscript>-Singular values and spectral radius of partially symmetric rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of l ^k,s -singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest l ^k,s -singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest l ^k,s -singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.     

On computing minimal <Emphasis Type="Italic">H</Emphasis>-eigenvalue of sign-structured tensors

Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.     

Tensor convolutions and Hankel tensors

Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n?1). Then A is called a Hankel tensor associated with a vector v ∈ ?^N+1 if A_σ = v _k for each k = 0, 1,...,N whenever σ = (i₁,..., i_m) satisfies i₁ +· · ·+i_m = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.     

Certain pairs of irreducible characters of the groups <Emphasis Type="Italic">S</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The investigation of the pairs of irreducible characters of the symmetric group S _n that have the same set of roots in one of the sets A _n and S _n ? A _n is continued. All such pairs of irreducible characters of the group S _n are found in the case when the least of the main diagonal’s lengths of the Young diagrams corresponding to these characters does not exceed 2. Some arguments are obtained for the conjecture that alternating groups A _n have no pairs of semiproportional irreducible characters.     

Further results on <Emphasis Type="Italic">B</Emphasis>-tensors with application to location of real eigenvalues

We give a further study on B-tensors and introduce doubly B-tensors that contain B-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of B-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.     

<Emphasis Type="Italic">k</Emphasis>-invariant nets over an algebraic extension of a field <Emphasis Type="Italic">k</Emphasis>

Let K be an algebraic extension of a field k, let σ = (σ _ij ) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σ _ij are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ? P ? K, up to conjugation by a diagonal matrix.     

On a conjecture of Bulman-Fleming and Gilmour

If S is a monoid, the set S×S equipped with componentwise S-action is called the diagonal act of S and is denoted by D(S). We prove the following theorem: the right S-act S ⁿ (1≠n∈?) is (principally) weakly flat if and only if \(\prod _{i=1}^{n}A_{i}\) is (principally) weakly flat where A _i , 1≤i≤n are (principally) weakly flat right S-acts, if and only if the diagonal act D(S) is (principally) weakly flat. This gives an answer to a conjecture posed by Bulman-Fleming and Gilmour (Semigroup Forum 79:298–314, 2009). Besides, we present a fair characterization of monoids S over which the diagonal act D(S) is (principally) weakly flat and finally, we impose a condition on D(S) in order to make S a left PSF monoid.     

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

In this paper, the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). Furthermore, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor \({\mathcal {A}}\) is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is positive, and \({\mathcal {A}}\) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is non-negative.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Let (M, g) be a 4-dimensional Einstein Riemannian manifold. At each point p of M, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p. In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O(4) acts as a transformation group between ST bases at T _p M and for the so-called 2-stein curvature tensors, the group Sp(1) ? SO(4) acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of SO(4) which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups SO(2), T ², Sp(1) or U(2) are obtained and the classes of curvature tensors whose transformation group into new ST bases is one of the mentioned groups are determined.     

Largest <Emphasis Type="Italic">H</Emphasis>-eigenvalue of uniform <Emphasis Type="Italic">s</Emphasis>-hypertrees

The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1 ≤ s ≤ k - 1; and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δ ^s/k ).     

Finite Groups Whose <Emphasis Type="Italic">n</Emphasis>-Maximal Subgroups Are Modular

Let G be a finite group. If M_n< M_n?1< · · · < M₁< M₀ = G with Mi a maximal subgroup of M_i?1 for all i = 1,..., n, then M_n (n > 0) is an n-maximal subgroup of G. A subgroup M of G is called modular provided that (i) 〈X,M ∩ Z〉 = 〈X,M〉 ∩ Z for all X ≤ G and Z ≤ G such that X ≤ Z, and (ii) 〈M,Y ∩ Z〉 = 〈M,Y 〉 ∩ Z for all Y ≤ G and Z ≤ G such that M ≤ Z. In this paper, we study finite groups whose n-maximal subgroups are modular.     

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.     

Rational realization of the minimum ranks of nonnegative sign pattern matrices

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,?, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in R^r?1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.     

Accurate solutions of <Emphasis Type="Italic">M</Emphasis>-matrix Sylvester equations

This paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an M-matrix Sylvester equation AX + XB = C by which we mean both A and B have positive diagonal entries and nonpositive off-diagonal entries and \({P=I_m \otimes A+B^{\rm T} \otimes I_n}\) is a nonsingular M-matrix, and C is entrywise nonnegative. It is proved that small relative perturbations to the entries of A, B, and C introduce small relative errors to the entries of the solution X. Thus the smaller entries of X do not suffer bigger relative errors than its larger entries, unlikely the existing perturbation theory for (general) Sylvester equations. We then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute X as accurately as the input data deserve.     

Weak <Emphasis Type="Italic">WT</Emphasis><Subscript>2</Subscript>-class of differential forms and weakly <Emphasis Type="Italic">A</Emphasis>-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.     

Regulating Hartshorne’s connectedness theorem

The tensor square conjecture states that for \(n \ge 10\), there is an irreducible representation V of the symmetric group \(S_n\) such that \(V \otimes V\) contains every irreducible representation of \(S_n\). Our main result is that for large enough n, there exists an irreducible representation V such that \(V^{\otimes 4}\) contains every irreducible representation. We also show that tensor squares of certain irreducible representations contain \((1-o(1))\)-fraction of irreducible representations with respect to two natural probability distributions. Our main tool is the semigroup property, which allows us to break partitions down into smaller ones.     

Last Multipliers for Riemannian Geometries,Dirichlet Forms and Markov Diffusion Semigroups

We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field Z of (0, 2)-type on a given Riemannian geometry (M, g) as a measure of how far away is Z from being divergence-free (and hence \(g^C\)-harmonic) with respect to g. The some topics are studied also for the Riemannian curvature tensor of (M, g) and finally for a general tensor field of (1, k)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carré du champ associated to the infinitesimal generator of the semigroup.     

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

The characterization of theta-distinguished representations of GL(<Emphasis Type="Italic">n</Emphasis>)

Let θ and θ’ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP84], of a metaplectic double cover of GL _n . The tensor θ ? θ’ is a (very large) representation of GL _n . We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s = 0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragredients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of θ. As a corollary, θ is shown to admit a new “metaplectic Shalika model”.