In the past several years, there has been considerable progress made on a general leftde…nite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-de…nite’has its origins in di¤erential equations but Littlejohn and Wellman [25] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A gener...

This paper concerns a new type of approximant for series analysis. By incorporating linear recurrence techniques, such as are involved with second order di erential approximants, algebraic approximants can be extended to nonlinear di erential approximants. The discussion of nonlinearities is constrained by discussing approximants just one step removed from second order algebraic approximants. T...

The method of modi®ed equations is studied as a technique for the analysis of ®nite di€erence equations. The non-uniqueness of the modi®ed equation of a di€erence method is stressed and three kinds of modi®ed equations are introduced. The ®rst modi®ed or equivalent equation is the natural pseudo-di€erential operator associated to the original numerical method. Linear and nonlinear combinations ...

The aim of this paper is to derive the self-improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p(x, t)-nonstandard growth. More precisely, we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved di e...

We extend Ehrhard Regnier's di erential linear logic along the lines of Laurent's polarization. We provide a denotational semantics of this new system in the well-known relational model of linear logic, extending canonically the semantics of both di erential and polarized linear logics: this justi es our choice of cut elimination rules. Then we show this polarized di erential linear logic re ne...

We survey some constructive aspects of di erential Galois theory and indicate some analogies between ordinary Galois theory and di erential Galois theory in characteristic zero and nonzero.

this article concerned on the study of signature submanifolds for curves under lie group actions se(2), sa(2) and for surfaces under se(3). signature submanifold is a regular submanifold which its coordinate components are dierential invariants of an associated manifold under lie group action, and therefore signature submanifold is a key for solving equivalence problems.

We study a renormalization operator for families of one dimensional maps close to x 7! p + r(1 x ) 2 , 2 ( 1 2 ; 1). Such functions occur in the study of cascades of homoclinic doubling bifurcations in three dimensional di erential equations. For values of close to 1 2 , we prove the existence of a xed point of the renormalization operator, whose linearization at the xed point has two unstable ...

We study germs of singular varieties in a symplectic space. In [A1], V. Arnol’d discovered so called ‘‘ghost’’ symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of di¤erential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of ...