The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.
Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an $L$-shaped tile, which periodically tessellates the plane. Given an initial tile $L(l,h,x,y)$, Aguil$\acute{o}$ et al. define a discrete iteration $L(p)= L(l+2p, h+2p, x+p, y+p), p=0,1,2,\ldots $, over $L$-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by $L(l,h,x,y)$, which is said to be a procreating $k$-tight tile if $L(p)( p=0,1,2,\cdots )$ are all $k$-tight tiles. They classify the set of $L$-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over $L$-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable $k$-tight $L$-shaped tile $L(l , h, x, y), 0\le |y-x|\le 2k+2$. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.
We extend the classical risk model to the case in which the premium income process, modelled as a Poisson process, is no longer a linear function. We derive an analog of the Beekman convolution formula for the ultimate ruin probability when the inter-claim times are exponentially distributed. A defective renewal equation satisfied by the ultimate ruin probability is then given. For the general inter-claim times with zero-truncated geometrically distributed claim sizes, the explicit expression for the ultimate ruin probability is derived.
In this note, we apply numerical analysis to the first Painlev\'{e} equation, find the conditions for it to have oscillating solutions and therefore solve an open problem posed by Peter A. Clarkson.
Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of $m$ components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions. Some related results will also be obtained.
Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.
In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders $2q+2$, or LRMTS$(2q+2)$, where $q=6t+5$ is a prime power. Using a computer, we find examples of such structure for $t\in T=\{0,1,2,3,4,6,7,8,9,14,16,18,20,22,24\}$. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for $LR$-design in [8], we obtain the existence for LRMTS$(v)$ and $LRDTS(v)$, where $v=12(t+1)\prod\limits_{m_i\geq0}(2\cdot7^{m_i}+1) \prod\limits_{n_i\geq0}(2\cdot13^{n_i}+1)$ and $t\in T$, which provides more infinite family for LRMTS and LRDTS of even orders.
By using fixed-point theorems, some new results for multiplicity of positive solutions for some second order m-point boundary value problems are obtained.The associated Green's function of these problems are also given.
This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.
Wenger's graph $H_m(q)$ is a $q$-regular bipartite graph of order $2q^m$ constructed by using the $m$-dimensional vector space $F_q^m$ over the finite field $F_q$. The existence of the cycles of certain even length plays an important role in the study of the accurate order of the Turan number $ex(n;C_{2m})$ in extremal graph theory. In this paper, we use the algebraic methods of linear system of equations over the finite field and the ``critical zero-sum sequences" to show that: if $m \geq 3$, then for any integer $l$ with $l\neq 5, 4 \leq l \leq 2 {\rm ch}(F_q)$ (where \ ch$(F_q)$ is the character of the finite field $F_q$) and any vertex $v$ in the Wenger's graph $H_m(q)$, there is a cycle of length $2l$ in $H_m(q)$ passing through the vertex $v$.
In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph $G$, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.
In this paper, we obtain a lower semicontinuity result with respect to the strong $L^{1}-$convergence of the integral functionals $$ F(u,\Omega)=\int_{\Omega}f\big(x,u(x),{\cal {E}} u(x)\big)dx $$ defined in the space $SBD$ of special functions with bounded deformation. Here ${\cal {E}} u$ represents the absolutely continuous part of the symmetrized distributional derivative $Eu$. The integrand $f$ satisfies the standard growth assumptions of order $p>1$ and some other conditions. Finally, by using this result,we discuss the existence of an constrained variational problem.
In this paper, we present a series of new preconditioners with parameters of strictly diagonally dominant $Z$-matrix, which contain properly two kinds of known preconditioners as special cases. Moreover, we prove the monotonicity of spectral radiuses of iterative matrices with respect to the parameters and some comparison theorems. The results obtained show that the bigger the parameter $k$ is(i.e., we select the more upper right diagonal elements to be the preconditioner), the less the spectral radius of iterative matrix is. A numerical example generated randomly is provided to illustrate the theoretical results.
This paper concerns with the number and distributions of limit cycles of a quintic subject to a seven-degree perturbation. With the aid of numeric integral computation provided by Mathematica 4.1, at least 45 limit cycles are found in the above system by applying the method of double homoclinic loops bifurcation, Hopf bifurcation and qualitative analysis. The four configurations of 45 limit cycles of the system are also shown. The results obtained are useful to the study of the weakened 16th Hilbert Problem.
This paper establishes a local limit theorem for solutions of backward stochastic differential equations with Mao's non-Lipschitz generator, which is similar to the limit theorem obtained by [3] under the Lipschitz assumption.
In this article, a Timoshenko beam with tip body and boundary damping is considered. A linearized three-level difference scheme of the Timoshenko beam equations on uniform meshes is derived by the method of reduction of order. The unique solvability, unconditional stability and convergence of the difference scheme are proved. The convergence order in maximum norm is of order two in both space and time. A numerical example is presented to demonstrate the theoretical results.