Spectral properties of odd-bipartite <Emphasis Type="Italic">Z</Emphasis>-tensors and their absolute tensors

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly evenbipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Ztensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.     

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 ).     

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.     

On list incidentor (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">l</Emphasis>)-coloring

A proper incidentor coloring is called a (k, l)-coloring if the difference between the colors of the final and initial incidentors ranges between k and l. In the list variant, the extra restriction is added: the color of each incidentor must belong to the set of admissible colors of the arc. In order to make this restriction reasonable we assume that the set of admissible colors for each arc is an integer interval. The minimum length of the interval that guarantees the existence of a list incidentor (k, l)-coloring is called a list incidentor (k, l)-chromatic number. Some bounds for the list incidentor (k, l)-chromatic number are proved for multigraphs of degree 2 and 4.     

<Emphasis Type="Italic">s</Emphasis>-Vertex Pancyclic Index

A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C _k such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K _1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K ₃}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp _s (G), is the least nonnegative integer m such that L ^m (G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,

$vp_s(G)\le \left\{\begin{array}{l@{\quad}l}\qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$

And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.

     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Universal zero-one <Emphasis Type="Italic">k</Emphasis>-law

The limit probabilities of first-order properties of a random graph in the Erd?s–Rényi model G(n, n^?α), α ∈ (0, 1), are studied. For any positive integer k ≥ 4 and any rational number t/s ∈ (0, 1), an interval with right endpoint t/s is found in which the zero-one k-law holds (the zero-one k-law describes the behavior of the probabilities of first-order properties expressed by formulas of quantifier depth at most k).Moreover, it is proved that, for rational numbers t/s with numerator not exceeding 2, the logarithm of the length of this interval is of the same order of smallness (as n→∞) as that of the length of the maximal interval with right endpoint t/s in which the zero-one k-law holds.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

An accelerated augmented Lagrangian method for linearly constrained convex programming with the rate of convergence <Emphasis Type="Italic">O</Emphasis>(1/<Emphasis Type="Italic">k</Emphasis><Superscript>2</Superscript>)

In this paper, we propose and analyze an accelerated augmented Lagrangian method (denoted by AALM) for solving the linearly constrained convex programming. We show that the convergence rate of AALM is O(1/k ²) while the convergence rate of the classical augmented Lagrangian method (ALM) is O(1/k). Numerical experiments on the linearly constrained l ₁?l ₂ minimization problem are presented to demonstrate the effectiveness of AALM.     

Facets of the <Emphasis Type="Italic">r</Emphasis>-Stable (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">k</Emphasis>)-Hypersimplex

Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\). We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\). We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\)-vectors.     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

The general rigidity result for bundles of <Emphasis Type="Italic">A</Emphasis>-covelocities and <Emphasis Type="Italic">A</Emphasis>-jets

Let M be an m-dimensional manifold and A = D _k ^r /I = R⊕N _A a Weil algebra of height r. We prove that any A-covelocity T _x ^A f ∈ T _x ^A *M, x ∈ M is determined by its values over arbitrary max{width A,m} regular and under the first jet projection linearly independent elements of T _x ^A M. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result T ^A *M ? T ^r *M without coordinate computations, which improves and generalizes the partial result obtained in Tomá? (2009) from m ≥ k to all cases of m.We also introduce the space J ^A (M,N) of A-jets and prove its rigidity in the sense of its coincidence with the classical jet space J ^r (M,N).     

The <Emphasis Type="Italic">p</Emphasis>-Adic order of the <Emphasis Type="Italic">k</Emphasis>-Fibonacci and <Emphasis Type="Italic">k</Emphasis>-Lucas numbers

Let (F _k,n ) _n and (L _k,n )n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a _n+1 = ka _n + a _n?1 with initial values F _k,0 = 0, F _k,1 = 1, L _k,0 = 2 and L _k,1 = k. In this paper, we characterize the p-adic orders ν _p (F _k,n ) and ν _p (L _k,n ) for all primes p and all positive integers k.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

<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.