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The reconstruction of a matrix having a pre-defined structure from given spectral data is known as an inverse eigenvalue problem (IEP). In this paper, we consider two IEPs involving the reconstruction of matrices whose graph is a special type of tree called a centipede. We introduce a special type of centipede called dense centipede.We study two IEPs concerning the reconstruction of matrices whose graph is a dense centipede from given partial eigen data. In order to solve these IEPs, a new system of nomenclature of dense centipedes is developed and a new scheme is adopted for labelling the vertices of a dense centipede as per this nomenclature . Using this scheme of labelling, any matrix of a dense centipede can be represented in a special form which we define as a connected arrow matrix. For such a matrix, we derive the recurrence relations among the characteristic polynomials of the leading principal submatrices and use them to solve the above problems. Some numerical results are also provided to illustrate the applicability of the solutions obtained in the paper.

In this paper, we study the inverse eigenvalue problem of constructing symmetric matrices whose graph is a tree, i.e. of constructing irreducible acyclic matrices from given eigendata consisting of the smallest and largest eigenvalues of their leading principal submatrices. We solve the problem completely for the case when the given eigenvalues are distinct and the subgraph of the tree induced by { 1 , 2 , ... , j } remains connected for 1 ≤ j ≤ n. The result generalizes the results known for matrices described by some specific trees. The proofs of the main results are constructive and so provide an algorithm for computing the actual entries of the required matrix. Numerical solutions of the problem are then obtained with SCILAB by feeding the eigendata and plugging in the adjacency matrix as inputs. We also make an attempt to find the number of solutions for several special families of unlabelled trees. [ABSTRACT FROM AUTHOR]

In this paper, we consider three inverse eigenvalue problems for a special type of acyclic matrices. The acyclic matrices considered in this paper are described by a graph called a broom on n+m vertices, which is obtained by joining m pendant edges to one of the terminal vertices of a path on n vertices. The problems require the reconstruction of such a matrix from given partial eigen data. The eigen data for the first problem consists of the largest eigenvalue of each of the leading principal submatrices of the required matrix, while for the second problem it consists of an eigenvalue of each of its trailing principal submatrices. The third problem has an eigenvalue and a corresponding eigenvector of the required matrix as the eigen data. The method of solution involves the use of recurrence relations among the leading/trailing principal minors of λI - A, where A is the required matrix. We derive the necessary and sufficient conditions for the solutions of these problems. The constructive nature of the proofs also provides the algorithms for computing the required entries of the matrix. We also provide some numerical examples to show the applicability of our results. [ABSTRACT FROM AUTHOR]

Aim: Pathophysiology of Eustachian tube significantly affects the result of middle ear reconstructive surgery. The aim of this study is to ascertain the relevance of preoperative assessment of tubal function on the results of myringoplasty and to correlate the outcome of surgery with the tubal function gradings. Materials and Methods: A total of 58 cases of dry central perforation of tympanic membrane were divided into two groups. In control group A, pre-operative Eustachian tube function was not checked. In Test group B, pre-operative Eustachian tube function was checked by modified inflation-deflation test using impedance audiometer. Results: Success was assessed at 4th month postoperatively on the basis of intact drum. The results between both groups were compared using Chi-square test. There was no statistically significant difference (P< 0.05) between the control and the test group. Conclusion: The capability to reduce an applied negative pressure correlates well with the successful result of a tympanic membrane graft. However, failure to equilibrate negative pressure does not discriminate between success and failure of this operative procedure. [ABSTRACT FROM AUTHOR]