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Existence, multiplicity and uniqueness results for a second order m-point boundary value problem
Yuqiang Feng,Sanyang Liu 대한수학회 2004 대한수학회보 Vol.41 No.3
Let f:[0,1]times [0,infty)rightarrow [0,infty) be continuousand ain C([0,1],[0,infty)),and let xi_{i}in (0,1) with0<xi_{1} <xi_{2}<cdots<xi_{m-2}<1,a_{i}, b_{i}in [0,infty)with 0<sum_{i=1}^{m-2}a_{i}<1 andsum_{i=1}^{m-2}b_{i}<1.This paper is concerned with thefollowing m-point boundary value problem: x^{''}(t)+a(t)f(t,x(t))=0, tin (0,1),x^{'}(0)=sum_{i=1}^{m-2}b_{i}x^{'}(xi_{i}),x(1)=sum_{i=1}^{m-2}a_{i}x(xi_{i}) . The existence,multiplicity and uniqueness of positive solutions of this problemarediscussed with the help of two fixed point theorems in cones, respectively.
Multiple periodic solutions for eigenvalue problems with a p-Laplacian and non-smooth potential
Guoqing Zhang,Sanyang Liu 대한수학회 2011 대한수학회보 Vol.48 No.1
In this paper, we establish a multiple critical points theorem for a one-parameter family of non-smooth functionals. The obtained result is then exploited to prove a multiplicity result for a class of periodic eigenvalue problems driven by the p-Laplacian and with a non-smooth potential. Under suitable assumptions, we locate an open subinterval of the eigenvalue.
A successive quadratic programming algorithm for SDP relaxation of the binary quadratic programing
Xuewen Mu,Sanyang Liu,Yaling Zhang 대한수학회 2005 대한수학회보 Vol.42 No.4
In this paper, we obtain a successive quadratic pro-gramming algorithm for solving the semidenite programming (SDP)relaxation of the binary quadratic programming. Combining witha randomized method of Goemans and Williamson, it provides anecient approximation for the binary quadratic programming. Fur-thermore, its convergence result is given. At last, We report somenumerical examples to compare our method with the interior-pointmethod on Maxcut problem.
A Lightweight Detection Mechanism against Sybil Attack in Wireless Sensor Network
( Wei Shi ),( Sanyang Liu ),( Zhaohui Zhang ) 한국인터넷정보학회 2015 KSII Transactions on Internet and Information Syst Vol.9 No.9
Sybil attack is a special kind of attack which is difficult to be detected in Wireless Sensor Network (WSN). So a lightweight detection mechanism based on LEACH-RSSI-ID (LRD) is proposed in this paper. Due to the characteristic of Low-Energy Adaptive Clustering Hierarchy (LEACH) protocol, none of nodes can be the cluster head forever. Meanwhile, in order to consume less energy, both factors which are called the remaining energy of nodes and relative density of nodes are taken into account. Therefore, Sybil attack can be found by analyzing the RSSI-ID tables. Different from the previous detection methods, even though Sybil attack occurs in the initialization phase, the malicious nodes can be detected by sink node. What`s more, when each malicious node frequently changes identification, it will be detected in a short time. Through the simulations, it is revealed that the LRD mechanism can detect the Sybil attack with high detection rate and accuracy.
Some Properties of the Closure Operator of a Pi-space
Mao, Hua,Liu, Sanyang Department of Mathematics 2011 Kyungpook mathematical journal Vol.51 No.3
In this paper, we generalize the definition of a closure operator for a finite matroid to a pi-space and obtain the corresponding closure axioms. Then we discuss some properties of pi-spaces using the closure axioms and prove the non-existence for the dual of a pi-space. We also present some results on the automorphism group of a pi-space.
Multiplicity results for a class of second order superlinear difference systems
Guoqing Zhang,Sanyang Liu 대한수학회 2006 대한수학회보 Vol.43 No.4
Using Minimax principle and Linking theorem in crit-ical point theory, we prove the existence of two nontrivial solutionsfor the following second order superlinear dierence systems(P)8<:2x(k 1) + g(k;y(k)) = 0; k2 [1;T];2y(k 1) + f(k;x(k)) = 0; k2 [1;T];x(0) = y(0) = 0;x(T + 1) = y(T + 1) = 0 ;where T is a positive integer, [1,T] is the discrete intervalf1;2;:;Tg; x(k) = x(k + 1) x(k) is the forward dierence operator and42x(k) = 4 (4 x(k)).
MULTIPLE PERIODIC SOLUTIONS FOR EIGENVALUE PROBLEMS WITH A p-LAPLACIAN AND NON-SMOOTH POTENTIAL
Zhang, Guoqing,Liu, Sanyang Korean Mathematical Society 2011 대한수학회보 Vol.48 No.1
In this paper, we establish a multiple critical points theorem for a one-parameter family of non-smooth functionals. The obtained result is then exploited to prove a multiplicity result for a class of periodic eigenvalue problems driven by the p-Laplacian and with a non-smooth potential. Under suitable assumptions, we locate an open subinterval of the eigenvalue.
Jiang, Zhaolin,Liu, Sanyang Korean Mathematical Society 2003 대한수학회보 Vol.40 No.3
In this paper, a new kind of matrices, i.e., $level-{\kappa}$ II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.
Zhaolin Jiang,Sanyang Liu 대한수학회 2003 대한수학회보 Vol.40 No.3
In this paper, a new kind of matrices, i.e., level-kPi-circulant matrices is considered. Algorithms for computingminimal polynomial of this kind of matrices are presented by meansof the algorithm for the Gr"{o}bner basis of the ideal in thepolynomial ring. Two algorithms for finding the inverses of suchmatrices are also presented based on the Buchberger's algorithm.
MULTIPLICITY RESULTS FOR A CLASS OF SECOND ORDER SUPERLINEAR DIFFERENCE SYSTEMS
Zhang, Guoqing,Liu, Sanyang Korean Mathematical Society 2006 대한수학회보 Vol.43 No.4
Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems $$(P)\{{\Delta}^2x(k-1)+g(k,y(k))=0,\;k{\in}[1,\;T],\;{\Delta}^2y(k-1)+f(k,\;x(k)=0,\;k{\in}[1,\;T],\;x(0)=y(0)=0,\;x(T+1)=y(T+1)=0$$ where T is a positive integer, [1, T] is the discrete interval {1, 2,..., T}, ${\Delat}x(k)=x(k+1)-x(k)$ is the forward difference operator and ${\Delta}^2x(k)={\Delta}({\Delta}x(k))$.