We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bund...

The Colin de Verdière parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parameters are calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on com...

Split-plot experiments are appropriate when some factors are more difficult and/or expensive to change than others. They require two levels of randomization resulting in a non-independent error structure. The design of such experiments has garnered much recent attention, including work on exact D-optimal split-plot designs. However, many of these procedures rely on the a priori assumption that ...

Chromatic dispersion and fiber attenuation pose a great problem in the detection of optical signals. Dispersion causes pulse broadening which limits the information carrying capacity of the fiber while attenuation limits the maximum transmission distance along the fiber. An approximate Gaussian pulse propagation model is obtained from the solution of Nonlinear Schrödinger Equation to represent ...

In this paper, we consider the problem of computing numerical solutions for stochastic differential equations (SDEs) of Itô form. A fully explicit method, the split-step forward Milstein (SSFM) method, is constructed for solving SDEs. It is proved that the SSFM method is convergent with strong order γ = 1 in the mean-square sense. The analysis of stability shows that the mean-square stability p...

The split-step Fourier method for solving numerically nonlinear Schrödinger equations (NLS) is considered as NLS with rapidly varying coefficients. This connection is exploited to justify the split-step approximation using an averaging technique. The averaging is done up to the second order and it is explained why (in this context) symmetric split-step produces a higher order scheme. The same a...