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.
Steady-state analysis for a class of hyperbolic systems with boundary inputs
Results of a steady-state analysis performed for a class of distributed parameter systems described by hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of the system with two state variables and two boundary inputs, the analytical expressions for the steady-state distribution of the state variables are derived, both in the exponential and in the hyperbolic form. The influence of the location of the boundary inputs on the steady-state response is demonstrated. The considerations are illustrated with a practical example of a shell and tube heat exchanger operating in paralleland countercurrent-flow modes.