The bondage number b(G) of a nonempty graph G is the smallest number of edges whose removal from G results in a graph with domination number greater than that of G. Denote P_n × P_m the Cartesian product of two paths P_n and P_m. This paper determines the exact values of b(P_n × P₂), b(P_n × P₃), and b(P_n × P₄) for n≥2.

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of RotaBaxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.

In this paper, we first give the definition of possibly non-unital function system, which is a characterization of the self-adjoint subspace of the space of continuous functions on a compact Hausdorff space with the induced order and norm structure. Similar to operator system case, we define the unitalization of a possibly non-unital function system. Then we construct two possibly non-unital operator system structures on a given possibly non-unital function system, which are the analogues of minimal and maximal operator spaces on a normed space. These two structures have many interesting relations with the minimal and maximal operator system structures on a given function system.

This paper is devoted to the study of the multiple-parameter rough Marcinkiewicz integral operators associated with certain smooth curves. It is shown that the Grafakos-Stefanov type size condition F_α (S^m-1 × S^n-1) of the kernel implies the L^p-boundedness of these Marcinkiewicz integral operators for some α>1/2 and 1+12α<p<1+2α, which is an essential improvement of certain previous results.

In this paper, we show that any complete Riemannian manifold of dimension great than 2 must be compact if it has positive complex sectional curvature and δ-pinched 2-positive curvature operator, namely, the sum of the two smallest eigenvalues of curvature operator are bounded below by δ·scal>0. If we relax the restriction of positivity of complex sectional curvature to nonnegativity, we can also show that the manifold is compact under the additional condition of positive asymptotic volume ratio.

In this paper, we study the generalized Chebyshev function related to automorphic L-functions of GLm(??), and estimate its asymptotic behavior with respect to the parameters of the original automorphic objects.

We consider the voter model with flip rates determined by {μ_e, e ∈ E_d}, where E_d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice ?d. Suppose that {μ_e, e ∈ E_d} are independent and identically distributed (i.i.d.) random variables satisfying μ_e≥1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: δ₀ and δ₁.

This paper is devoted to a study of the automorphism groups of three series of finite-dimensional special odd Hamiltonian superalgebras g over a field of prime characteristic. Our aim is to characterize the connections between the automorphism groups of g and the automorphism groups of the corresponding underlying superalgebras. Precisely speaking, we embed the former into the later. Moreover, we determine the images of the normal series of the automorphism groups and homogeneous automorphism groups of g under the embedded mapping.

For the widely orthant dependent (WOD) structure, this paper mainly investigates the precise large deviations for the partial sums of WOD and non-identically distributed random variables with dominatedly varying tails. The obtained results extend some corresponding results.

A Calabi’s theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary.

In this paper, we prove that there exists a infinite set of non-trivial local Fitting classes every element in which is decomposable as a non-trivial product of Fitting classes such that every factor in the product is neither local nor a formation. In particular, this gives a positive answer to Problem 11.25 a) in The Kourovka Notebook.

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.

In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2³. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that each 1-planar graph with maximum degree Δ is (Δ+1)-edge-choosable and (Δ+2)- total-choosable if Δ≥16, and is Δ-edge-choosable and (Δ+1)-total-choosable if Δ≥21. The second conclusion confirms the list coloring conjecture for the class of 1-planar graphs with large maximum degree.