Nonexistence and partial existence of first integrals for diffeomorphisms

Existence and Nonexistence of Solutionsfor

Existence and nonexistence of positive solution for sixth-order boundary value problems

‎In this paper‎, ‎we formulate the sixth-order boundary value problem as Fredholm integral equation by finding Green's function and obtain the sufficient conditions for existence and multiplicity of positive solution for this problem‎. ‎Also nonexistence results are obtained‎. ‎An example is given to illustrate the results of paper‎.

Harmonic Homeomorphisms: Existence and Nonexistence

These notes were prepared for International Workshop on Harmonic and Quasiconformal Mappings in Chennai, August 2010. They are based on several papers written jointly with Tadeusz Iwaniec, Jani Onninen, and Ngin-Tee Koh. The source papers are available from www.arxiv.org

On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems

We consider generalized Liénard polynomial differential systems of the form ˙ x = y, ˙ y = −g(x) − f (x) y, with f (x) and g(x) two polynomials satisfying deg(g) ≤ deg(f). In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.

Existence and Nonexistence of Descriptive Patterns

In the present paper, we study the existence of descriptive patterns, i. e. patterns that cover all words in a given set through morphisms and that are optimal in terms of revealing commonalities of these words. Our main result shows that if patterns may be mapped onto words by arbitrary morphisms, then there exist infinite sets of words that do not have a descriptive pattern. This answers a qu...