Polynomial differential equations with many real ovals in the same algebraic complex solution
Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F 2 FolR(2, d) | F has a non-constant rational first integral}. Given F 2 IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of the generic level (G = c). Let O(d) = sup{O(F) | F 2 IntR(2, d)}. The main purpose of this paper is to prove that O(d) = +1 for all d _ 5.
Geometric Theory of Foliations
I - Differentiable Manifolds.- II - Foliations.- III - The Topology of the Leaves.- IV - Holonomy and the Stability Theorems.- V - Fiber Bundles and Foliations.- VI - Analytic Foliations of Codimension One.- VII - Novikov's Theorem.- VIII - Topological Aspects of the Theory of Group Actions.- Appendix - Frobenius' Theorem.- 1. Vector fields and the Lie bracket.- 2. Frobenius' theorem.- 3. Plane fields defined by differential forms.- Exercises.
Dynamic hemivariational inequalities in contact mechanics
Abstract This paper gives a review of results on the existence and uniqueness of solutions to some mechanical contact problems. The second-order evolution problems with a general normal compliance response condition and a normal damped response one in viscoelasticity are discussed. Finally, a new result for a problem which includes the slip displacement-dependent friction and the Tresca model is provided.