Let x be an indeterminate over ℂ. We investigate solutions $$\begin{array}{c}\hfill \alpha (s,x)=\sum _{n\ge 0}{\alpha }_n\left(s\right)x^n,\end{array}$$ α _n  : ℂ → ℂ, n ≥ 0, of the first cocycle equation $$\begin{array}{c}\hfill \alpha (s+t,x)=\alpha (s,x)\alpha \left(t,F(s,x)\right),s,t\in ,\left(\mathrm{Co}1\right)\end{array}$$in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x)) _{s ∈ ℂ...}

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain  ⊂ ℝ the functional is ${\mathit{I}}_{\mathit{ϵ}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int }}}_{\mathit{\Omega }}{\mathit{ϵ}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{ϵ}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where belongs to the subset of functions in ${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega }\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies ()·  = 1 where is...

Given β > 1, let T _β $\begin{array}{ccc}{\mathit{T}}_{\mathit{\beta }}\mathrm{:}\mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}& \mathrm{\to }& \mathrm{\left[}\mathrm{0}\mathit{,}\mathrm{1}\mathrm{\right[}\\ \mathit{x}& \mathrm{\to }& \mathit{\beta x}\mathrm{-}\mathrm{\lfloor }\mathit{\beta x}\mathrm{\rfloor }\mathit{.}\end{array}$ The iteration of this transformation gives rise to the greedy β-expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β. ...