Diophantine Integrability

How to detect the integrability of discrete systems

Several integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be des...

One Symmetry Does Not Imply Integrability

We show that Bakirov’s counterexample (which had been checked by computeralgebra methods up till order 53) to the conjecture that one nontrivial symmetry of an evolution equation implies infinitely many is indeed a counterexample. To prove this we use the symbolic method of Gel’fand-Dikii and p-adic analysis. We also formulate a conjecture to the effect that almost all equations in the family c...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

FUZZY GOULD INTEGRABILITY ON ATOMS

In this paper we study the relationships existing between total measurability in variation and Gould type fuzzy integrability (introduced and studied in [21]), giving a special interest on their behaviour on atoms and on finite unions of disjoint atoms. We also establish that any continuous real valued function defined on a compact metric space is totally measurable in the variation of a regula...

Relationships between Darboux Integrability and Limit Cycles for a Class of Able Equations

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.