Weak-foci of high order and cyclicity

Agraïments: This work was done when H. Liang was visiting the Department of Mathematics of Universitat Autònoma de Barcelona. He is very grateful for the support and hospitality. The first author is supported by the NSF of China (No. 11201086 and No. 11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guangdong Province, China (No. Yq2013107).Agraïments: The second author is partially supported by UNAB13-4E-1604.A particular version of the 16th Hilbert's problem is to estimate the number, M(n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M(n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even n, the studied polynomial system of degree n was the one obtained by QiuYan2009 where the highest weak-focus order is n^2 n-2 for n=4,6, 18. Moreover, we provide a system which has a weak-focus with order (n-1)^2 for n 12. We show that Christopher's approach Chr2006, aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that M(6) 39, M(7) 34 and M(8) 63

Limit cycles coming from some uniform isochronous centers

Agraïments: The first author is supported by the NSF of China (No. 11201086), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No.2012LYM0087) and the Excellent Young Teachers Training Program for colleges and uni- versities of Guangdong Province, China (No. Yq2013107).This article is about the weak 16--th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers $$x= -y x^2 y (x^2 y^2)^n, y= x x y^2 (x^2 y^2)^n,$$ of degree 2n 3 and we perturb them inside the class of all polynomial differential systems of degree 2n 3. For n=0,1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n=2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center

Centers of projective vector fields of spatial quasi-homogeneous systems with weight (m,m,n)(m,m,n) and degree 22 on the sphere

Agraïments: The first author is supported by the NSF of China (No. 11201086), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2012LYM 0087) and the Excellent Young Teachers Training Program for colleges and universities of Guangdong Province, China (No. Yq2013107).In this paper we study the centers of projective vector fields Q_T of three-dimensional quasi-homogeneous differential system d/dt=Q() with the weight (m,m,n) and degree 2 on the unit sphere S^2. We seek the sufficient and necessary conditions under which Q_T has at least one center on S^2. Moreover, we provide the exact number and the positions of the centers of Q_T. First we give the complete classification of systems d/dt=Q() and then, using the induced systems of Q_T on the local charts of S^2, we determine the conditions for the existence of centers. The results of this paper provide a convenient criterion to find out all the centers of Q_T on S^2 with Q being the quasi-homogeneous polynomial vector field of weight (m,m,n) and degree 2

Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields

Agraïments: The first author is supported by the NSF of China (No. 11201086, No.11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guang-dong Province, China (No. Yq2013107).Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms. To show the power of this approach, we study the cyclicity of the holomorphic center =iz z^2 z^3 z^n under general polynomial perturbations of degree n, for n 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n=4,5, 13, we prove that the cyclicity of the holomorphic center is at least n^2 n-2. This result give the highest lower bound for M(6), M(7), M(13) among the existing results, where M(n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6) 40, H(8) 70, and H(10) 108, because until now the best result was H(6) 39, H(8) 67, and H(10) 100

Modulation of Type III Secretion System in Pseudomonas aeruginosa: Involvement of the PA4857 Gene Product

Pseudomonas aeruginosais an opportunistic pathogen that causes serious acute or chronic infections in humans.Acute infections typically involve the type Ш secretion systems (T3SS) and bacterial motility,whereas chronic infectionsare often associated with biofilm formation and the type VI secretion system (T6SS). To identifynew genes required for pathogenesis, a transposon mutagenesis library was constructed and the gene PA4857, named tspR, was found to modulateT3SS gene expression. Deletion of P. aeruginosa tspRreduced the virulence in a mouse acute lung infection model and diminished cytotoxicity. Suppression of T3SS gene expression in the tspR mutant resulted from compromised translation of the T3SS master regulator ExsA. TspR negatively regulated two small RNAs, RsmYand RsmZ, which control RsmA. Our data demonstrated that defects inT3SS expression and biofilm formation in retS mutant could be partially restored by overexpression of tspR. Taken together, our results demonstrated thatthe newly identifiedretS-tspRpathway is coordinated with the retS-gacSsystem, which regulates the genes associated with acute and chronic infections andcontrols the lifestyle choice of P. aeruginosa