Mathematical Combinatorics

ISSN 1937 - 1055 VOLUME 1, 2013 INTERNATIONAL JOURNAL OF MATHEMATICAL COMBINATORICS EDITED BY THE MADIS OF CHINESE ACADEMY OF SCIENCES AND BEIJING UNIVERSITY OF CIVIL ENGINEERING AND ARCHITECTURE March, 2013 Vol.1, 2013 ISSN 1937-1055 International Journal of Mathematical Combinatorics Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture March, 2013 Aims and Scope: The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sci- ences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces, , etc.. Smarandache geometries; · · · Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics. Generally, papers on mathematics with its applications not including in above topics are also welcome. It is also available from the below international databases: Serials Group/Editorial Department of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106, USA Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535, USA Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075 http://www.gale.com Indexing and Reviews: Mathematical Reviews(USA), Zentralblatt fur Mathematik(Germany), Referativnyi Zhurnal (Russia), Mathematika (Russia), Computing Review (USA), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by an email to [email protected] or directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: [email protected] Price: US$48.00 Shaofei Du Editorial Board (2nd) Capital Normal University, P.R.China Email: [email protected] Editor-in-Chief Baizhou He Beijing University of Civil Engineering and Linfan MAO Architecture, P.R.China Chinese Academy of Mathematics and System Email: [email protected] Science, P.R.China Xiaodong Hu and Chinese Academy of Mathematics and System Beijing University of Civil Engineering and Science, P.R.China Architecture, P.R.China Email: [email protected] Email: [email protected] Yuanqiu Huang Deputy Editor-in-Chief Hunan Normal University, P.R.China Email: [email protected] Guohua Song H.Iseri Beijing University of Civil Engineering and Mansfield University, USA Architecture, P.R.China Email: hiseri@mnsfld.edu Email: [email protected] Xueliang Li Editors Nankai University, P.R.China Email: [email protected] S.Bhattacharya Guodong Liu Deakin University Huizhou University Geelong Campus at Waurn Ponds Email: [email protected] Australia Ion Patrascu Email: [email protected] Fratii Buzesti National College Dinu Bratosin Craiova Romania Institute of Solid Mechanics of Romanian Ac- Han Ren ademy, Bucharest, Romania East China Normal University, P.R.China Junliang Cai Email: [email protected] Beijing Normal University, P.R.China Ovidiu-Ilie Sandru Email: [email protected] Politechnica University of Bucharest Yanxun Chang Romania. Beijing Jiaotong University, P.R.China Tudor Sireteanu Email: [email protected] Institute of Solid Mechanics of Romanian Ac- Jingan Cui ademy, Bucharest, Romania. Beijing University of Civil Engineering and W.B.Vasantha Kandasamy Architecture, P.R.China Indian Institute of Technology, India Email: [email protected] Email: [email protected] ii International Journal of Mathematical Combinatorics Guiying Yan Luige Vladareanu Chinese Academy of Mathematics and System Institute of Solid Mechanics of Romanian Ac- Science, P.R.China ademy, Bucharest, Romania Email: [email protected] Mingyao Xu Y. Zhang Peking University, P.R.China Department of Computer Science Email: [email protected] Georgia State University, Atlanta, USA Famous Words: Mathematics, rightly viewed, posses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture. By Bertrand Russell, an England philosopher and mathematician. International J.Math. Combin. Vol.1(2013), 01-37 Global Stability of Non-Solvable Ordinary Differential Equations With Applications Linfan MAO Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China Beijing University of Civil Engineering and Architecture, Beijing 100045, P.R.China E-mail: [email protected] Abstract: Different from the system in classical mathematics, a Smarandache system is a contradictory system in which an axiom behaves in at least two different ways within the same system, i.e., validated and invalided, or only invalided but in multiple distinct ways. Such systems exist extensively in the world, particularly, in our daily life. In this paper, we discuss such a kind of Smarandache system, i.e., non-solvable ordinary differential equation systems by a combinatorial approach, classify these systems and characterize their behaviors, particularly, the global stability, such as those of sum-stability and prod-stability of such linear and non-linear differential equations. Some applications of such systems to other sciences, such as those of globally controlling of infectious diseases, establishing dynamical equations of instable structure, particularly, the n-body problem and understanding global stability of matters with multilateral properties can be also found. Key Words: Global stability, non-solvable ordinary differential equation, general solution, G-solution, sum-stability, prod-stability, asymptotic behavior, Smarandache system, inherit graph, instable structure, dynamical equation, multilateral matter. AMS(2010): 05C15, 34A30, 34A34, 37C75, 70F10, 92B05 §1. Introduction Finding the exact solution of an equation system is a main but a difficult objective unless some special cases in classical mathematics. Contrary to this fact, what is about the non-solvable case for an equation system? In fact, such an equation system is nothing but a contradictory system, and characterized only by having no solution as a conclusion. But our world is overlap and hybrid. The number of non-solvable equations is much more than that of the solvable and such equation systems can be also applied for characterizing the behavior of things, which reflect the real appearances of things by that their complexity in our world. It should be noted that such non-solvable linear algebraic equation systems have been characterized recently by the author in the reference [7]. The main purpose of this paper is to characterize the behavior of such non-solvable ordinary differential equation systems. 1Received November 16, 2012. Accepted March 1, 2013. 2 Linfan Mao Assume m, n 1 to be integers in this paper. Let ≥ X˙ = F (X) (DES1) be an autonomous differential equation with F : Rn Rn and F (0) = 0, particularly, let → X˙ = AX (LDES1) be a linear differential equation system and x(n) + a x(n−1) + + a x = 0 (LDEn) 1 · · · n a linear differential equation of order n with a a a x (t) f (t,X) 11 12 · · · 1n 1 1 a21 a22 a2n x2(t) f2(t,X) A = · · · X = and F (t,X)= , ··· ··· ··· ··· · · · · · · an1 an2 ann xn(t) fn(t,X) · · · where all a , a , 1 i, j n are real numbers with i ij ≤ ≤ X˙ = (x ˙ , x˙ , , x˙ )T 1 2 · · · n and f (t) is a continuous function on an interval [a,b] for integers 0 i n. The following i ≤ ≤ result is well-known for the solutions of (LDES1) and (LDEn) in references. Theorem 1.1([13]) If F (X) is continuous in U(X ) : t t a, X X b (a> 0, b> 0) 0 | − 0|≤ k − 0k≤ then there exists a solution X(t) of differential equation (DES1) in the interval t t h, | − 0| ≤ where h = min a, b/M , M = max F (t,X) . { } (t,X)∈U(t0,X0) k k Theorem 1.2([13]) Let λi be the ki-fold zero of the characteristic equation det(A λI )= A λI =0 − n×n | − n×n| or the characteristic equation λn + a λn−1 + + a λ + a =0 1 · · · n−1 n with k + k + + k = n. Then the general solution of (LDES1) is 1 2 · · · s n αit ciβi(t)e , i=1 X Global Stability of Non-Solvable Ordinary Differential Equations With Applications 3 where, ci is a constant, βi(t) is an n-dimensional vector consisting of polynomials in t deter- mined as follows t11 t21 β1(t)= · · · tn1 t11t + t12 t21t + t22 β2(t)= ········· tn1t + tn2 ··························· t11 k1−1 t12 k1−2 t + t + + t1k (k1−1)! (k1−2)! · · · 1 t21 tk1−1 + t22 tk1−2 + + t (k1−1)! (k1−2)! 2k1 βk (t)= · · · 1 ································· tn1 k1−1 tn2 k1−2 t + t + + tnk (k1−1)! (k1−2)! · · · 1 t1(k1+1) t2(k1+1) βk1+1(t)= ······ t n(k1+1) t11t + t12 t21t + t22 βk1+2(t)= ········· tn1t + tn2 ··························· t1(n−ks+1) ks−1 t1(n−ks+2) ks−2 t + t + + t1n (ks−1)! (ks−2)! · · · t t 2(n−ks+1) tks−1 + 2(n−ks+2) tks−2 + + t (ks−1)! (ks−2)! 2n βn(t)= · · · ···································· tn(n−ks+1) ks−1 tn(n−ks+2) ks−2 t + t + + tnn (ks−1)! (ks−2)! · · · with each t a real number for 1 i, j n such that det([t ] ) =0, ij ≤ ≤ ij n×n 6 λ , if 1 i k ; 1 ≤ ≤ 1 λ2, ifk1 +1 i k2; αi = ≤ ≤ ; ··· ·················· λs, ifk1 + k2 + + ks−1 +1 i n.

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