اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNormal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps
91
0
0.0
(
0
)
تأليف
Wei Guo Foo
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $xlongmapsto varphi(x)$, $ylongmapsto psi(x,y)$ by using Mosers method of normal forms. We first compute a basis of the Lie algebra ${frak{g}}_{{{y_{xx}=0}}}$ of fibre-preserving symmetries of $y_{xx}=0$. In the formal theory of Mosers method, this Lie algebra is used to give an explicit description of the set of normal forms $mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to $y_{xx}=0$.
قيم البحث
اقرأ أيضاً
An important result from self-similar models that describe the process of galaxy cluster formation is the simple scaling relation $Y_{rm SZE}D_{rm A}^{2}/C_{rm XSZE}Y_{rm X}= C$. In this ratio, $Y_{rm SZE}$ is the integrated Sunyaev-Zeldovich effect
flux of a cluster, its x-ray counterpart is $Y_{rm X}$, $C_{rm XSZE}$ and $C$ are constants and $D_{rm A}$ is the angular diameter distance to the cluster. In this paper, we consider the cosmic distance duality relation validity jointly with type Ia supernovae observations plus $61$ $Y_{rm SZE}D_{rm A}^{2}/C_{rm XSZE}Y_{rm X}$ measurements as reported by the Planck Collaboration to explore if this relation is constant in the redshift range considered ($z<0.5$). No one specific cosmological model is used. As basic result, although the data sets are compatible with no redshift evolution within 2$sigma$ c.l., a Bayesian analysis indicates that other $C(z)$ functions analyzed in this work cannot be discarded. It is worth to stress that the observational determination of an universal $C(z)$ function turns the $Y_{rm SZE}D_{rm A}^{2}/C_{rm XSZE}Y_{rm X}$ ratio in an useful cosmological tool to determine cosmological parameters.
This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.
We give a new CR invariant treatment of the bigraded Rumin complex and related cohomology groups via differential forms. We also prove related Hodge decomposition theorems. Among many applications, we give a sharp upper bound on the dimension of the
Kohn--Rossi groups $H^{0,q}(M^{2n+1})$, $1leq qleq n-1$, of a closed strictly pseudoconvex manifold with a contact form of nonnegative pseudohermitian Ricci curvature; we prove a sharp CR analogue of the Frolicher inequalities in terms of the second page of a natural spectral sequence; and we generalize the Lee class $mathcal{L}in H^1(M;mathscr{P})$ -- whose vanishing is necessary and sufficient for the existence of a pseudo-Einstein contact form -- to all nondegenerate orientable CR manifolds.
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. If the solution is a special function that is studied in the literature, then the result is more manageable usi
ng the properties of that function. It is straightforward to find the regular and irregular singular points of such an equation by a computer algebra system. However, one needs the corresponding indices for a full analysis of the singularity structure. It is shown that the $theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure which can be visualized through a Riemann P-symbol leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type. Whereas for ODEs the decomposition is intrinsic, for PDEs it is
necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. The harmonic map equation is examined, and in this case both the 1-form and the vector field arise naturally.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
