On the flat remainder in normal forms of families of analytic planar saddles
نویسنده
چکیده
We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.
منابع مشابه
On the M-polynomial of planar chemical graphs
Let $G$ be a graph and let $m_{i,j}(G)$, $i,jge 1$, be the number of edges $uv$ of $G$ such that ${d_v(G), d_u(G)} = {i,j}$. The $M$-polynomial of $G$ is $M(G;x,y) = sum_{ile j} m_{i,j}(G)x^iy^j$. With $M(G;x,y)$ in hands, numerous degree-based topological indices of $G$ can be routinely computed. In this note a formula for the $M$-polynomial of planar (chemical) graphs which have only vertices...
متن کاملComputational studies of planar, tubular and conical forms of silicon nanostructures
Density functional theory (DFT) calculations were performed to investigate the properties of planar, tubular and conical forms of silicon nanostructures. The evaluated parameters including averaged bond lengths, binding energies, gap energies and dipole moments were then evaluated for the optimized models of study. The results indicated that the bond lengths between silicon atoms are different ...
متن کاملComputational studies of planar, tubular and conical forms of silicon nanostructures
Density functional theory (DFT) calculations were performed to investigate the properties of planar, tubular and conical forms of silicon nanostructures. The evaluated parameters including averaged bond lengths, binding energies, gap energies and dipole moments were then evaluated for the optimized models of study. The results indicated that the bond lengths between silicon atoms are different ...
متن کاملRobust normal forms for saddles of analytic vector fields
The aim of this paper is to introduce a technique for describing trajectories of systems of ordinary differential equations (ODEs) passing near saddle-fixed points. In contrast to classical linearization techniques, the methods of this paper allow for perturbations of the underlying vector fields. This robustness is vital when modelling systems containing small uncertainties, and in the develop...
متن کاملA pointfree version of remainder preservation
Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...
متن کامل