The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS

Preserving the validity of the Two-Higgs Doublet Model up to the Planck scale

We examine the constraints on the two Higgs doublet model (2HDM) due to the stability of the scalar potential and absence of Landau poles at energy scales below the Planck scale. We employ the most general 2HDM that incorporates an approximately Standard Model (SM) Higgs boson with a flavor aligned Yukawa sector to eliminate potential tree-level Higgs-mediated flavor changing neutral currents. Using basis independent techniques, we exhibit robust regimes of the 2HDM parameter space with a 125 GeV SM-like Higgs boson that is stable and perturbative up to the Planck scale. Implications for the heavy scalar spectrum are exhibited.Comment: 36 pages, 4 figures, 4 tables (Version 3: typographical error in eq. (A.28) corrected

Modified Linear Projection for Large Spatial Data Sets

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial data sets. These sorts of data sets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial data sets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this paper, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.Comment: 29 pages, 5 figures, 4 table

Delays in Open String Field Theory

We study the dynamics of light-like tachyon condensation in a linear dilaton background using level-truncated open string field theory. The equations of motion are found to be delay differential equations. This observation allows us to employ well-established mathematical methods that we briefly review. At level zero, the equation of motion is of the so-called retarded type and a solution can be found very efficiently, even in the far light-cone future. At levels higher than zero however, the equations are not of the retarded type. We show that this implies the existence of exponentially growing modes in the non-perturbative vacuum, possibly rendering light-like rolling unstable. However, a brute force calculation using exponential series suggests that for the particular initial condition of the tachyon sitting in the false vacuum in the infinite light-cone past, the rolling is unaffected by the unstable modes and still converges to the non-perturbative vacuum, in agreement with the solution of Hellerman and Schnabl. Finally, we show that the growing modes introduce non-locality mixing present with future, and we are led to conjecture that in the infinite level limit, the non-locality in a light-like linear dilaton background is a discrete version of the smearing non-locality found in covariant open string field theory in flat space.Comment: 48 pages, 14 figures. v2: References added; Section 4 augmented by a discussion of the diffusion equation; discussion of growing modes in Section 4 slightly expande

Flow networks: A characterization of geophysical fluid transport

We represent transport between different regions of a fluid domain by flow networks, constructed from the discrete representation of the Perron-Frobenius or transfer operator associated to the fluid advection dynamics. The procedure is useful to analyze fluid dynamics in geophysical contexts, as illustrated by the construction of a flow network associated to the surface circulation in the Mediterranean sea. We use network-theory tools to analyze the flow network and gain insights into transport processes. In particular we quantitatively relate dispersion and mixing characteristics, classically quantified by Lyapunov exponents, to the degree of the network nodes. A family of network entropies is defined from the network adjacency matrix, and related to the statistics of stretching in the fluid, in particular to the Lyapunov exponent field. Finally we use a network community detection algorithm, Infomap, to partition the Mediterranean network into coherent regions, i.e. areas internally well mixed, but with little fluid interchange between them.Comment: 16 pages, 15 figures. v2: published versio

Neutrino mixing, interval matrices and singular values

We study the properties of singular values of mixing matrices embedded within an experimentally determined interval matrix. We argue that any physically admissible mixing matrix needs to have the property of being a contraction. This condition constrains the interval matrix, by imposing correlations on its elements and leaving behind only physical mixings that may unveil signs of new physics in terms of extra neutrino species. We propose a description of the admissible three-dimensional mixing space as a convex hull over experimentally determined unitary mixing matrices parametrized by Euler angles which allows us to select either unitary or nonunitary mixing matrices. The unitarity-breaking cases are found through singular values and we construct unitary extensions yielding a complete theory of minimal dimensionality larger than three through the theory of unitary matrix dilations. We discuss further applications to the quark sector.Comment: Misprints correcte

Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430, arXiv:0712.0849]).Comment: for Part I see arXiv:0712.084