完全二部图Km,n的Kp,q-因子分解

如果完全二部图K_m,n的边集可以划分为K_m,n的K_p,q-因子, 则称K_m,n存在K_p,q-因子分解. 给出K_m,n存在K_p,q-因子分解的一个充分条件. 同时证明: 对于任意正整数k, 当p:q = k:(k + 1)时, K_m,n存在K_p,q-因子分解, 即Martin的BAC猜想成立.     

对于轮和完全图的 Ramsey 数的渐近上界

该文给出:对于偶数m≥4当n→ ∞时 r(W_m,K_n)≤l(1+o(1))C₁(m) (n/logn ) ^(2m-2)/(m-2)对于奇数m≥5当n→∞时r(W_m,K_n)≤(1+o(1))C₂(m) (n^2m/_m+1/^{_{log n)}(m+1)/(m-1)} .特别地,C₂(5)=12. 以及 c(ⁿ/_logn)^5/2≤r(K₄,K_n)≤ (1+o(1)) ^n3_/(logn)².此外,该文还讨论了轮和完全图的 Ramsey 数的一些推广.     

弱 (K1, K2) -拟正则映射的正则性

给出空间弱(K₁, K₂) -拟正则映射的定义, 并以Hodge分解及弱逆Hölder不等式为工具, 得到了其正则性结果：对任意满足 的q1, 都存在可积指数 使得对任意弱 (K₁, K₂) -拟正则映射 都有 即f为通常意义下的(K₁, K₂) -拟正则映射.     

二部多重图路因子分解的存在谱

若二部多重图λK_m,n的边集可以划分为λK_m,n 的P_v-因子,则称 λK_m,n存在P_v-因子分解.当v是偶数时, Ushio和Wang及本文的第二作者给出了λK_m,n存在P_v-因子分解的充分必要条件.同时提出了当v是奇数时λK_m,n存在P_v-因子分解的猜想.最近我们已经证明当v=4k-1时该猜想成立. 对于正整数k,文中证明λK_m,n 存在P_4k+1-因子分解的充分必要条件是: (1) 2km ≤ (2k+1)n, (2) 2kn ≤(2k+1)m, (3) m+n ≡ 0 (mod 4k+1), (4)λ (4k+1)mn/[4k(m+n)]是整数. 即证明:对于任意正整数k, 当v=4k+1时上述猜想成立，从而最终完成了该猜想成立的证明.     

定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

完全二部图的P4k-1-因子分解

如果完全二部图K_m,n的边集可以划分为K_m,n的P_v-因子,则称K_m,n存在P_v-因子分解. 当v是偶数时, Ushio和Wang 给出了K_m,n存在P_v-因子分解的充分必要条件. Ushio同时提出了当v是奇数时K_m,n存在P_v-因子分解的猜想, 但是至今为止仅知当v=3时Ushio猜想成立. 对于正整数k,本文证明K_m,n存在P_4k&#8722;1-因子分解的充分必要条件是: (1) (2k&#8722;1)m ≤2kn, (2) (2k&#8722;1)n ≤ 2 km, (3) m+n ≡ 0 (mod 4k&#8722;1), (4) (4k&#8722;1)mn/[2(2k&#8722;1)(m+n)]是整数. 即证明了对于任意正整数k, 当v=4k&#8722;1时Ushio猜想成立.     

虚二次域整环的K2群的计算

在前人工作的基础上,给出在虚二次域整环的K₂群计算中对例外位的范的界的一种较有效的估计方法,并计算了K₂Z[(1+Ö-43)/2].     

Banach空间上算子代数的K0群

对G-M型Banach空间的某些分类予以简化性合并;讨论Banach空间X上算子代数B(X)的K₀群K₀(B(X))=0的条件, 得到了改进Laustsen充分条件的一个结果,举例说明了X≈X₂不是K₀(B(X))=0的充分条件.     

完全二部图存在路因子分解的Ushio猜想的证明

如果完全二部图K_m,n的边集可以划分为K_m,n的P_v-因子, 则称K_m,n存在P_v-因子分解. 当v是偶数时, Ushio 和 Wang 给出了K_m,n存在P_v因子分解的充分必要条件. Ushio在其综述文章中提出了当v是奇数时K_m,n存在P_v-因子分解的猜想. 已经证明当v=4k-1时Ushio猜想成立. 对于正整数k, 本文证明K_m,n存在P_4k+1-因子分解的充分必要条件是: (1) 2km ≤ (2k+1)n, (2) 2kn ≤ (2k+1)m, (3) m+n ≡0 (mod 4k+1), (4) (4k+1)mn/[4k(m+n)]是整数. 即证明: 对于任何正整数k, 当v=4k+1时Ushio猜想成立，从而最终完成了Ushio猜想成立的证明.     

局部域的K2群中的一类挠元素

证明了(K₂Q_p(z_p))_I=G_p(Q_p(z_p)); 还证明若n|w(Q₅ (z₅)), 则(K₅Q₅(z₅))_n=G₅(Q5₅(z₅)), 这说明对于含有p次本原单位根的p局部域, 如果p|n, 则Browkin猜想一般不成立. 由此提出一个一般猜想. 另外, 否定了Urbanowicz的一个猜想.     

Qualitative analysis of steady states to the Sel'kov model

In this work, we are concerned with a reaction-diffusion system well known as the Sel'kov model, which has been used for the study of morphogenesis, population dynamics and autocatalytic oxidation reactions. We derive some further analytic results for the steady states to this model. In particular, we show that no nonconstant positive steady state exists if 0<p?1 and θ is large, which provides a sharp contrast to the case of p>1 and large θ, where nonconstant positive steady states can occur. Thus, these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants. The a priori estimates are fundamental to our mathematical approaches.     

Bifurcations of steady states for the Eguchi–Oki–Matsumura model of phase separation

This article is concerned with bifurcations of steady states for a model system of phase separation, which is introduced by Eguchi–Oki–Matsumura (EOM). The system consists of coupled two evolution equations and admits steady state solutions with different energies. The bifurcation phenomena of these steady states with respect to the principal parameter, which is related to the temperature, are analyzed.     

Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

A microbial continuous culture system with diffusion and diversified growth

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.     

Confined steady states of a Vlasov‐Poisson plasma in an infinitely long cylinder

We consider the two‐dimensional Vlasov‐Poisson system to model a two‐component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two‐dimensional Vlasov‐Poisson system can be derived from the full three‐dimensional model. The existence of compactly supported steady states with vanishing electric potential in a three‐dimensional setting has already been investigated in the literature. We show that these results can easily be adapted to the two‐dimensional system. However, our main result is to prove the existence of compactly supported steady states even with a nontrivial self‐consistent electric potential.     

Pattern formation in the Brusselator system

In the paper, we deal with a reaction-diffusion system well known as the Brusselator model and some improved results for the steady states of this model are presented. We first give an a priori estimates (positive upper and lower bounds) of positive steady states. Then, we obtain the non-existence and existence of positive non-constant steady states as the parameters λ, θ and b are varied, which means some certain conditions under which the pattern formation occurs or not.     

Mathematical description of the interactions of CycE/Cdk2, Cdc25A,and P27Kip1 in a core cancer subnetwork

Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27^Kip1, and Rb) at the R‐checkpoint during the G₁‐to‐S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27^Kip1 and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.     

Global dynamics and control of malicious signal transmission in wireless sensor networks

This paper studies the global dynamics of a discontinuous delayed model of malicious signal transmission in wireless sensor networks under the framework of differential inclusion. The local stability of two types of steady states are investigated for the discontinuous system by studying the corresponding characteristic equation. The sufficient conditions for the existence of two types of globally asymptotically stable steady states are obtained for the discontinuous system by using the comparison arguments method. Furthermore, the optimal control of the discontinuous system is investigated by using Pontryagin’s maximum principle. Numerical simulations of two examples are carried out to illustrate the main theoretical results. The obtained results can help us to better control and predict the spread of malicious signal transmission in wireless sensor networks.