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Search: All articles in the CMB digital archive with keyword cycles

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1. CMB 2015 (vol 58 pp. 818)

Llibre, Jaume; Zhang, Xiang
 On the Limit Cycles of Linear Differential Systems with Homogeneous Nonlinearities We consider the class of polynomial differential systems of the form $\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their nonexistence and existence. Keywords:polynomial differential system, limit cycles, differential equations on the cylinderCategories:34C35, 34D30

2. CMB 2015 (vol 59 pp. 144)

Laterveer, Robert
 A Brief Note Concerning Hard Lefschetz for Chow Groups We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial. Keywords:algebraic cycles, Chow groups, finite-dimensional motivesCategories:14C15, 14C25, 14C30

3. CMB 2014 (vol 58 pp. 356)

Sebag, Julien
 Homological Planes in the Grothendieck Ring of Varieties In this note, we identify, in the Grothendieck group of complex varieties $K_0(\mathrm Var_\mathbf{C})$, the classes of $\mathbf{Q}$-homological planes. Precisely, we prove that a connected smooth affine complex algebraic surface $X$ is a $\mathbf{Q}$-homological plane if and only if $[X]=[\mathbf{A}^2_\mathbf{C}]$ in the ring $K_0(\mathrm Var_\mathbf{C})$ and $\mathrm{Pic}(X)_\mathbf{Q}:=\mathrm{Pic}(X)\otimes_\mathbf{Z}\mathbf{Q}=0$. Keywords:motivic nearby cycles, motivic Milnor fiber, nearby motivesCategories:14E05, 14R10

4. CMB 2013 (vol 57 pp. 413)

Samei, Karim
 On the Comaximal Graph of a Commutative Ring Let $R$ be a commutative ring with $1$. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined a graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra + Rb = R$. In this paper, we consider a subgraph $\Gamma_2(R)$ of $\Gamma(R)$ which consists of non-unit elements. We investigate the behavior of $\Gamma_2(R)$ and $\Gamma_2(R) \setminus \operatorname{J}(R)$, where $\operatorname{J}(R)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of $\Gamma_2(R)$ and the topological properties of $\operatorname{Max}(R)$. Diameter, girth, cycles and dominating sets are investigated and the algebraic and the topological characterizations are given for graphical properties of these graphs. Keywords:comaximal, Diameter, girth, cycles, dominating setCategory:13A99

5. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
 The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25
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