Let F be a graph. A hypergraph ${\cal H}$ is Berge-F if there is a bijection $f:E(F) \rightarrow E({\cal H})$ such that e ⊂ f(e) for every e ∈ E(F). A hypergraph is Berge-F-free if it does not contain a subhypergraph isomorphic to a Berge-F hypergraph. The authors denote the maximum number of hyperedges in an n-vertex r-uniform Berge-F-free hypergraph by ex _r (n, Berge-F).

A (k, p)-fan, denoted by F _k,p, is a graph on k(p − 1) + 1 vertices consisting of k cliques with p vertices that intersect in exactly one common vertex. In this paper they determine the bounds of ex _r(n, Berge-F) when F is a (k, p)-fan for k ≥ 2, p ≥ 3 and r ≥ 3.

In this paper, the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space. They show that if the initial velocity satisfies some condition on the integral J in the “isolated mass group” (see (1.13)), then there will be finite time blow-up of regular solutions to the Euler system with J ≤ 0 (n ≥ 1) and to the Euler-Boltzmann system with J < 0 (n ≥ 1) and J = 0 (n ≥ 2), no matter how small and smooth the initial data are. It is worth mentioning that these blow-up results imply the following: The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum, with the only possible exception in the case J = 0 and n = 1 since the radiation behaves differently on this occasion.

The author shows that if a locally conformal Kähler metric is Hermitian Yang-Mills with respect to itself with Einstein constant c ≤ 0, then it is a Kahler-Einstein metric. In the case of c > 0, some identities on torsions and an inequality on the second Chern number are derived.

This paper investigates the optimal recovery of Sobolev spaces W ₁ ^r[−1, 1], r ∈ ℕ in the space L ₁[−1, 1]. They obtain the values of the sampling numbers of W ₁ ^r[−1, 1] in L ₁[−1, 1] and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms. Meanwhile, they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.

In the paper, the authors provide a new proof and derive some new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier results in the literature. And some parabolic type Liouville theorems for ancient solutions are obtained.

The author introduces the notion of a minimal resolution for BP _* BP-comodules, and gives an effective algorithm to produce minimal resolutions. This produces the data needed in the work [3] for studying motivic stable stems up to stem 90.

In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [13]). They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair (θ ₁ ,θ ₂), where θ ₁ is a congruence on the de Morgan subalgebra L°° of L and θ ₂ is a lattice congruence on the sublattice D(L) of L. They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L. Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.

In this paper, the author solves the Dirichlet problem for Hermitian-Poisson metric equation $\sqrt { - 1} {\Lambda _\omega }{G_H} = \lambda {\rm{Id}}$ and proves the existence of Hermitian-Poisson metrics on flat bundles over a class of complete Hermitian manifolds. When λ = 0, the Hermitian-Poisson metric is a Hermitian harmonic metric.

In this paper, the authors characterize Carleson measures for the weighted Bergman spaces with Békollé weights on the unit ball. They apply the Carleson embedding theorem to study the properties of Toeplitz-type operators and composition operators acting on such spaces.

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D _α ^p to D _β ^q (−1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.

In this paper, the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse, the e-core inverse and the f-dual core inverse in rings. Also, new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.

Recently, there are extensive studies on perfect state transfer (PST for short) on graphs due to their significant applications in quantum information processing and quantum computations. However, there is not any general characterization of graphs that have PST in literature. In this paper, the authors present a depiction on weighted abelian Cayley graphs having PST. They give a unified approach to describe the periodicity and the existence of PST on some specific graphs.