Sturm-Liouville problem

We study basisness of root functions of Sturm–Liouville problems with a boundary condition depending quadratically on the spectral parameter. We determine the explicit form of the biorthogonal system. Using this we prove that the system of root functions, with arbitrary two functions removed, form a minimal system in L2, except some cases where this system is neither complete nor minimal. For the basisness in we prove that the part of the root space is quadratically close to systems of sines and cosines. We also consider these basis properties in the context of general L_p

""In this study, we investigate a Sturm-Liouville type problem with eigenparameter dependent boundary conditions and eigenparameter dependent transmission conditions. By establishing a new self-adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions.
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Resumo No estudo das Equações Diferenciais Parciais, em particular na equação do calor ho-mogênea, nos deparamos com o método de separação de variáveis para obtenção da solução formal do problema. Este método consiste na suposição da independência das variáveis em questão, e a consideração que a solução possa ser descrita como um produto de duas outras funções independentes. Neste caso, nos deparamos com uma equação diferencial ordinária no qual temos de encontrar suas soluções. Assim, neste trabalho iremos utilizar a Teoria de Sturm-Liouville para a resolução destas EDO's mais gerais, e consequentemente obter um método de encontrar a solução da equação do calor não-homogênea.

Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

We define and study the properties of Darboux-type transformations between Sturm–Liouville problems with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary conditions). Using these transformations, we obtain various direct and inverse spectral results for these problems in a unified manner, such as asymptotics of eigenvalues and norming constants, oscillation of eigenfunctions, regularized trace formulas, and inverse uniqueness and existence theorems.

""In this work we study the asymptotic properties of a new Sturm-Liouville problem with retarded argument. Contrary to previous works, differential equation includes eigen-parameter as a quadratic function. In the considered problem arise new di¢ culties.
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Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the vertex set of M through p. We take p to be an isolated umbilic point on M and describe the perestroikas of the vertex set under generic n-parameter small deformations of the surface, as well as the corresponding discriminants. One of the main consequences of our results is that, in some sense (see Theorem 2), generically the study of the discriminants of small n-deformations, with n greater than or equal to 2, simplifies to that of 2-deformations.
This work was primarily motivated by the medial representation of shapes in Computer Vision and Image Analysis, where the behaviour of vertices plays a crucial role in the qualitative changes of the skeleton or Blum medial axis of curves.
On the other hand, the study of vertices of curves has raised up a great interest in particular in Geometry and Singularity Theory, in line with several problems such as the 4-vertex Theorem, the local geometry of surfaces and Geometry of Caustics. These classical subjects have received a new impulsion due to the development of Contact Geometry, especially in the works of V. Arnold on Legendrian Collapse and Legendrian Sturm Theory showing the relation between vertices of plane curves and Legendrian singularities and Sturm-Hurwitz Theorem.