The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained. Part B: Graph monomials and sums of squares In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares. Part C: Eigenvalue asymptotics of banded Toeplitz matrices This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above. Part D: Stretched Schur polynomials This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript</p>

To prove that a polynomial is nonnegative on Rn, one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semi-definite programming (SDP) computation that is much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables with k = 4 fixed those that are SOS occupy a vanishingly small fraction of those that are nonnegative on Rn, as n -> 1. With an eye toward the case of small n, we refine Blekherman'[s bounds by incorporating the underlying Newton polytope, simultaneously sharpening some of his older bounds along the way. Our refined asymptotics show that certain Newton polytopes may lead to families of polynomials where efficient SDP can still be used for most inputs.

Many papers have been presented on the study of change points detection.　Nonetheless, we would like to point out that in dealing with the time series with switching regimes, we should also take the characteristics of change periods into account.　Because many patterns of change structure in time series exhibit a certain kind of duration, those phenomena should not be treated as a mere sudden turning at a certain time. In this paper, we propose procedures about change periods detection for nonlinear time series.　One of the detecting statistical methods is an application of fuzzy classification and generalization of Inclan and Tiao’s result.　Moreover, we develop the genetic-based searching procedure, which is based on the concepts of leading genetic model.　Simulation results show that the performance of these procedures is efficient and successful.　Finally, two empirical applications about change periods detection for Taiwan monthly visitors arrival and exchange rate are demonstrated.