ISSN 1439-8516 CN 11-2039/O1
In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation
Sb:=Δα/2 + b·∇,
where Δα/2 is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ Kdα-1. In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance|x- y|in short time.
Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of (G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G)=k for given integers k and n with 1 ≤ k ≤ n.
The linear 2-arboricity la2(G) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that if G is a 1-planar graph with maximum degree Δ, then la2(G) ≤ ? (Δ+1)/2? + 7. This improves a known result of Liu et al. (2019) that every 1-planar graph G has la2(G) ≤ ? (Δ+1)/2? + 14. We also observe that there exists a 7-regular 1-planar graph G such that la2(G)=6=? (Δ+1)/2? + 2, which implies that our solution is within 6 from optimal.
For random variables and random weights satisfying Marcinkiewicz-Zygmund and Rosenthal type moment inequalities, we establish complete convergence results for randomly weighted sums of the random variables. Our results generalize those of (Thanh et al. SIAM J. Control Optim., 49, 106-124 (2011), Han and Xiang J. Ineq. Appl., 2016, 313 (2016), Li et al. J. Ineq. Appl., 2017, 182 (2017), and Wang et al. Statistics, 52, 503-518 (2018).)
Let p be an odd prime number, and let E be an elliptic curve defined over a number field which has good reduction at every prime above p. Under suitable assumptions, we prove that the η-eigenspace and the η-eigenspace of mixed signed Selmer group of the elliptic curve are pseudoisomorphic. As a corollary, we show that the η-eigenspace is trivial if and only if the η-eigenspace is trivial. Our results can be thought as a reflection principle which relates an Iwasawa module in a given eigenspace with another Iwasawa module in a "reflected" eigenspace.
In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of|G|is a maximal subgroup.
In this article, we show the existence of infinitely many solutions for the fractional p-Laplacian equations of Schrödinger-Kirchhoff type equation
M([u]s,pp)(-Δ)psu + V (x)|u|p-2u=λ(Iα *|u|ps,α*|u|ps,α*-2u + βk(x)|u|q-2u, x ∈ RN,
where (-Δ)ps is the fractional p-Laplacian operator,[u]s,p is the Gagliardo p-seminorm, 0 < s < 1 < q < p < N/s, α ∈ (0, N), M and V are continuous and positive functions, and k(x) is a non-negative function in an appropriate Lebesgue space. Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma, we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.
In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are ignorable in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous transformations and continuous observation functions.
In this paper we introduce the notions of (Banach) density-equicontinuity and densitysensitivity. On the equicontinuity side, it is shown that a topological dynamical system is densityequicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measuretheoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is densitysensitive for this measure.
We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.
