Height growth of solutions and a discrete Painlev\'e equation

Consider the discrete equation $$y_{n+1}+y_{n-1}=\frac{a_n+b_ny_n+c_ny_n^2}{1-y_n^2},$$ where the right side is of degree two in $y_n$ and where the coefficients $a_n$, $b_n$ and $c_n$ are rational functions of $n$ with rational coefficients. Suppose that there is a solution such that for all sufficiently large $n$, $y_n\in\mathbb{Q}$ and the height of $y_n$ dominates the height of the coefficient functions $a_n$, $b_n$ and $c_n$. We show that if the logarithmic height of $y_n$ grows no faster than a power of $n$ then either the equation is a well known discrete Painlev\'e equation ${\rm dP}_{\!\rm II}$ or its autonomous version or $y_n$ is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.Comment: 26 page

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.Comment: 12 pages. To appear in the Journal of Mathematical Analysis and Application

Nevanlinna theory for the difference operator

Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference $f\mapsto \Delta f=f(z+c)-f(z)$. An $a$-point of a meromorphic function $f$ is said to be $c$-paired at z\in\C if $f(z)=a=f(z+c)$ for a fixed constant c\in\C. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna theory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of $f$. Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard's theorem and Nevanlinna's five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.Comment: 19 page

Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (1) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equation of the form (1), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.Comment: 34 page

Growth of meromorphic solutions of delay differential equations

Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations solved by elliptic functions

Investigation Into Informational Compatibility Of Building Information Modelling And Building Performance Analysis Software Solutions

There are significant opportunities for Building Information Modelling (BIM) to address issues related to sustainable and energy efficient building design. While the potential benefits associated with the integration of BIM and BPA (Building Performance Analysis) have been recognised, its specifications and formats remain in their early infancy and often fail to live up to the promise of seamless interoperability at various stages of design process. This paper conducts a case study to investigate the interoperability between BIM and BPA tools, and discusses the limitations to suggest development of Information Delivery Manual (IDM) aiming to propose potential solutions for typical issues facing professionals in architecture, engineering and construction (AEC) industry

The C-metric as a colliding plane wave space-time

It is explicitly shown that part of the C-metric space-time inside the black hole horizon may be interpreted as the interaction region of two colliding plane waves with aligned linear polarization, provided the rotational coordinate is replaced by a linear one. This is a one-parameter generalization of the degenerate Ferrari-Ibanez solution in which the focussing singularity is a Cauchy horizon rather than a curvature singularity.Comment: 6 pages. To appear in Classical and Quantum Gravit

Comfort signatures: How long-term studies of occupant satisfaction in office buildings reveal on-going performance

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link