Laguerre Form Research Articles

Laguerre hypersurfaces with definite Laguerre tensor

x : Mn-1 ? Rn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. In this paper, we will study the Laguerre hypersurface with parallel Laguerre form and nonnegative or non-positive Laguerre tensor.

LAGUERRE CHARACTERIZATION OF SOME HYPERSURFACES

Abstract. Let x : M n−1 → R n (n ≥ 4) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated witha Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and aLaguerre second fundamental form B, which are invariants of x underLaguerre transformation group. We denote the Laguerre scalar curvatureby R and the trace-free Laguerre tensor by L˜ := L− 1n−1 tr(L)g. Inthis paper, we prove a local classification result under the assumption ofparallel Laguerre form and an inequality of the typekL˜k ≤ cR,where c= 1(n−3) √ (n−2)(n−1) is appropriate real constant, depending onthe dimension. 1. IntroductionLet x : M n−1 → R n be an umbilical free hypersurface with non-zero princi-pal curvatures. Let ξ : M → S n−1 be its unit normal. Let {e 1 ,e 2 ,...,e n−1 }be the orthonormal basis for TM with respect to dx · dx, consisting of unitprincipal vectors. Let r i = 1k i , r = r 1 +r 2 +···+r n−1 n−1 be the curvature radius andmean curvature radius of x respectively, where k

On Laguerre form and Laguerre isoparametric hypersurfaces

Let x: Mn−1 → ℝn be an umbilical free hypersurface with non-zero principal curvatures. M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi-Civita connection of its Laguerre metric.

Classification of Laguerre isoparametric hypersurfaces in R6

Let be an -dimensional hypersurface in and be the Laguerre second fundamental form of the immersion x. An eigenvalue of Laguerre second fundamental form is called a Laguerre principal curvature of x. An umbilic free hypersurface with non-zero principal curvatures and vanishing Laguerre form is called a Laguerre isoparametric hypersurface if the Laguerre principal curvatures of x are constants. In this paper, we obtain a complete classification for all oriented Laguerre isoparametric hypersurfaces in .

On Linearization by Generalized Sundman Transformations of a Class of Li�nard Type Equations and Its Generalization

We study the linearization of a class of Li ´ enard type nonlinear second-order ordinary differential equations f rom the generalized Sundman transformation viewpoint. The linearizing generalized Sundman transformation for the class of equations is constructed. The transformation is used to map the underlying class of equations into a linear second-order ordinary differential equation which is not in the Laguerre form. The general solution of this class of equations is obtained by integrating the linearized equation and applying the generalized Sundman transformation. Moreover, we apply a Riccati transformation to a general linear third-order variable coefficients ordinary differential equation to extend the underlying class of equations and we also derive the conditions of linearizability of this new class of nonlinear second-order ordinary differential equ ations using the generalized Sundman transformation method and obtain its general solution.

Hypersurfaces with parallel para‐Laguerre tensor in

Let be an ‐dimensional hypersurface with vanishing Laguerre form in , be the Laguerre second fundamental form and be the Laguerre tensor of the immersion x. We define a symmetric (0, 2) tensor which is so‐called the para‐Laguerre tensor of x, where λ is a constant. If , we say that x is of parallel para‐Laguerre tensor, where ∇ is the Levi‐Civita connection of the Laguerre metric g. An eigenvalue of the para‐Laguerre tensor is called a para‐Laguerre eigenvalue of x. The aim of this paper is to classify all oriented hypersurfaces in with parallel para‐Laguerre tensor or with three distinct constant para‐Laguerre eigenvalues one of which is simple.

On the Linearization of Second-Order Ordinary Differential Equations to the Laguerre Form via Generalized Sundman Transformations

The linearization problem for nonlinear second-order ODEs to the Laguerre form by means of generalized Sundman transformations (S-transformations) is considered, which has been investigated by Duarte et al. earlier. A characterization of these S-linearizable equations in terms of first integral and procedure for construction of linearizing S-transformations has been given recently by Muriel and Romero. Here we give a new characterization of S-linearizable equations in terms of the coefficients of ODE and one auxiliary function. This new criterion is used to obtain the general solutions for the first integral explicitly, providing a direct alternative procedure for constructing the first integrals and Sundman transformations. The effectiveness of this approach is demonstrated by applying it to find the general solution for geodesics on surfaces of revolution of constant curvature in a unified manner.

Laguerre isopararmetric hypersurfaces in ℝ4

Let x: M → ℝn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor \(\mathbb{L}\), a Laguerre form C, and a Laguerre second fundamental form \(\mathbb{B}\) which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of \(\mathbb{B}\) are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in ℝ4.

Classification of hypersurfaces with constant Laguerre eigenvalues in ℝ n

Let x: M → ℝn be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. An eigenvalue of Laguerre tensor L of x is called a Laguerre eigenvalue of x. In this paper, we classify all oriented hypersurfaces with constant Laguerre eigenvalues and vanishing Laguerre form.

APPLICATION OF THE GENERALISED SUNDMAN TRANSFORMATION TO THE LINEARISATION OF TWO SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation has not been applied to a system of equations. The motivation of this work is then to expand the application of the generalized Sundman transformation to a system of ordinary differential equations, in particular, to a system of two second-order ordinary differential equations.

The Wave Equation Together with Matheu-Hill and Laguerre Form Dynamic Boundary Conditions

The present study illustrates a series method for the solutions of one dimensional wave equation together with non-classical dynamic boundary conditions. Matheu-Hill form, a differential equation with polynomial form and Laguerre differential equation form dynamic boundary conditions were taken into consideration. Series methods were given in order for the solutions of wave equation together with these dynamic boundary conditions along with semi-infinite axis of the spatial coordinate. Wave profiles were obtained by means of wave solutions of the wave equation given by d’Alembert.

Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations

The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.

Laguerre Geometry of Surfaces in R 3

Let f : M → R3 be an oriented surface with non–degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L(f) = f (H2 – K)/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R3. And we give a classification theorem of surfaces in R3 with vanishing Laguerre form.

The S-Matrix Structure and the Inverse Scattering Transform in the J-Matrix Approach

Using the analytic properties of the S-matrix, we obtain a system of inverse scattering transform equations for nonlocal potentials with Laguerre form factors. Coulomb repulsion can be present in the system.

Propagation of Electromagnetic Pulses in an Inhomogeneous Scattering Absorbing Medium. Invariant Temporal Properties

We present analytical solutions of the Ishimaru parabolic equation for the coherence function of electromagnetic field. The equation describes the temporal properties of the pulse at the output of an inhomogeneous dissipative scattering medium. An explicit expression for the Green's function of the problem is found. It is shown that the temporal part of the Green's function has an invariant Laguerre form. Numerical results describing the shapes of the studied temporal pulses at the output of the pass interval of the medium are also given. Time scales characterizing the shapes of the Laguerre output pulses are proposed and the influence of the parameters of the medium on the properties of the resulting pulses is analyzed.

NN potentials from inverse scattering in the J-matrixapproach

An approximate inverse scattering method has beenused to construct separable potentials with the Laguerre form factors. Asan application, we invert the phase shifts of proton-proton in the1S0 and 3P2-3F2 channels and neutron-proton in the3S1-3D1 channel elastic scattering. In the latter case the deuteronwavefunction of a realistic np potential was used as input.