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\twocolumn[\begin{@twocolumnfalse}
\noindent\LARGE{\textbf{From geometric optics to plants: eikonal equation for buckling}}
\vspace{0.6cm}

\noindent\large{\textbf{Sergei Nechaev$^{\ast}$\textit{$^{a,b}$} and
Kirill Polovnikov,\textit{$^{c}$}}}\vspace{0.5cm}

\noindent\textit{\small{\textbf{Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX\newline
First published on the web Xth XXXXXXXXXX 200X}}}

\noindent \textbf{\small{DOI: 10.1039/b000000x}}
\vspace{0.6cm}

\noindent \normalsize{Optimal embedding in the three-dimensional space of exponentially growing
squeezed surfaces, like plants leaves, or 2D colonies of exponentially reproducing cells, is
considered in the framework of conformal approach. It is shown that the boundary profile of a
growing tissue is described by the 2D eikonal equation, which provides the geometric optic
approximation for the wave front propagating in the media with inhomogeneous refraction
coefficient. The variety of optimal surfaces embedded in 3D is controlled by spatial dependence
of the refraction coefficient which, in turn, is dictated by the local growth protocol.}
\vspace{0.5cm}
\end{@twocolumnfalse}]

\section{Introduction}

\footnotetext{\textit{$^{a}$~J.-V. Poncelet Laboratory, CNRS, UMI 2615, 119002 Moscow, Russia;
E-mail: sergei.nechaev@gmail.com}}

\footnotetext{\textit{$^{b}$~P.N. Lebedev Physical Institute, RAS, 119991 Moscow, Russia}}

\footnotetext{\textit{$^{a}$~Physics Department, M.V. Lomonosov Moscow State University, 119992
Moscow, Russia}}

Variety of shapes of 2D growing tissues emerges due to the incompatibility of local internal
(differential) growth protocol with geometric constraints imposed by embedding of these tissues
into the space. For example, buckling of a lettuce leaf can be naively explained as a conflict
between natural growth due to the periphery cells division (typically, exponential), and growth of
circumference of a planar disc with gradually increasing radius. Due to a specific biological
mechanism which inhibits growth of the cell experiencing sufficient external pressure, the division
of inner cells is insignificant, while periphery cells have less steric restrictions and
proliferate easier. Thus, the division of border cells has the major impact on the instabilities in
the tissue. Such a differential growth induces an increasing strain in a tissue near its edge and
results in two complimentary possibilities: i) in-plane tissue compression and/or redistribution of
layer cells accompanied by the in-plane circumference instability, or ii) out-of-plane tissue
buckling with the formation of saddle-like surface regions. The latter is typical for various
undulant negatively curved shapes which are ubiquitous to many mild plants growing up in air or
water where the gravity is of sufficiently small matter \cite{kelp,swinney}.

A widely used energetic approach to growing patterns exploits a continuous formulation of the
differential growth and is based on a rivalry between bending and stretching energies of elastic
membranes \cite{hebrew,lewicka,audoly, gemmer, swinney, shape-of-leaf, muller,goriely}, reflecting
the choice between options (i) and (ii) above. For bending rigidity of a thin membrane, ${\cal
B}$, one has ${\cal B} \sim h^3$, while stretching rigidity, ${\cal S}$ behaves as ${\cal S}
\sim h$, where $h$ is the membrane thickness \cite{rayleigh}. Therefore, thin enough tissues, with
$h\ll 1$, prefer to bend, i.e. to be negatively curved under relatively small critical strain.

The latter allows one to eliminate the "stretching" regime from consideration, justifying the
geometric approach for infinitesimally thin membranes \cite{1,2,marder,marder-sharon, marder-pap,
gemmer} (see also \cite{3}). Here the determination of typical profiles of buckling surfaces relies
on an appropriate choice of metric tensor of the non-Euclidean space, and is realized via the
optimal embedding of the tissue with certain metrics into the 3D Euclidean space. It should be
mentioned, that the formation of wrinkles within this approach seems to be closely related to the
description of phyllotaxis via conformal methods \cite{levitov}.

In this letter we suggest a model of a hyperbolic infinitesimally thin tissue, whose periphery
cells divide freely with exponential rate, while division of inner cells is absolutely inhibited.
Two cases of proliferations, the one-dimensional (directed) and the uniform two-dimensional, are
considered. The selection of these two growth models is caused by the intention to describe
different symmetries inherent for plants at initial stages of growth. As long as the in-plane
deformations are not beneficial, as follows from the relationship between bending and stretching
rigidities, all the redundant material of fairly elastic tissue will buckle out. In order to take
into account the finite elasticity of growing tissue, resulting from the intrinsic discrete
properties of a material, we describe the tissue as a collection of glued elementary plaquettes
connected along the hyperbolic graph, $\gamma$. The discretization implies the presence of a
characteristic scale, of order of the elementary cell (plaquette) size, below which the tissue is
locally flat.

As we rely on the absence of in-plane deformations, this graph has to be isometrically embedded
into the 3D space. The desired smooth surface profile is obtained in two steps: i) isometric
mapping of the hyperbolic graph onto the flat domain (rectangular or circular) with hyperbolic
metrics, ii) subsequent restoring of the metrics into the 3D Euclidean space above the domain. We
demonstrate that such a procedure leads to the "optimal" buckling of the tissue and is described by
the eikonal equation for the profile, $f(x,y)$, of growing sample, which by definition, is a
variant of the Hamilton-Jacobi equation.

The paper is organized as follows. We introduce necessary definitions in the Section \ref{s:1}; the
model under consideration and the details of the conformal approach are provided in the Sections
\ref{s:2}, \ref{s:3} and Appendix; the samples of various typical shapes for
two-dimensional uniform and for one-dimensional directed growth, are presented in the Section
\ref{s:4}; finally, the results of the work are summarized in the Section \ref{s:5}, where we also
speculate about possible generalizations and rise open questions.

\section{Buckling of thin tissues in cylindric and planar geometries}

\subsection{Basic facts about the eikonal equation}
\label{s:1}

To make the content of the paper as self-contained as possible, it seems instructive to provide
some important definitions used at length of the paper. The key ingredient of our consideration is
the "eikonal" equation, which is the analogue of the Hamilton-Jacobi equation in geometric
optics. As we show below, the eikonal equation provides optimal embedding of an exponentially
growing surface into the 3D Euclidean plane. Meaning of the notion "optimal" has two different
connotations in our approach:

i) On one hand, from viewpoint of the Hamilton-Jacobi theory, the eikonal equation appears in the
minimization of the action $A=\int_\gamma L dt$ with some Lagrangian $L$. According to the Fermat
principle, the time of the ray propagation in the inhomogeneous media with the space-dependent
refraction coefficient, $n(x,y)$, should be minimal.

ii) On the other hand, the eikonal equation emerges in our work in a purely geometric setting
following directly from the conformal approach.

First attempts to formulate classical mechanics problems in geometric optics terms goes back to the
works of Klein \cite{klein} in 19th century. His ideas contributed to the corpuscular theory in a
short-wavelength regime, as long as the same mechanical formalism applied to massless particles,
was consistent with the wave approach. Later, in the context of general relativity, this approach
was renewed to treat gravitational field as an optic medium \cite{rumer}.

The Fermat principle states that the time $dt$ for a ray to propagate along a curve $\gamma$
between two closely located points $M({\bf x})$ and $N({\bf x}+d{\bf x})$ in an inhomogeneous
media, should be minimal. The total time $T$ can be written in the form $T=\frac{1}{c}\int_{M}^{N}
n(\mathbf{x}(s))ds$ where $n(\mathbf{x})=\frac{c}{\textrm{v}(\mathbf{x})}$ is the refraction
coefficient at the point $\mathbf{x}=\{x^i\}$ of a $D$-dimensional space $(i=1,...,D)$, $c$ and
$\textrm{v}(\mathbf{x})$ are correspondingly the light speeds in vacuum and in the media, and
$d|\mathbf{x}|=ds$ is the spatial increment along the ray. Following the optical-mechanical
analogy, according to which the action in mechanics corresponds to eikonal in optics, one can write
down the "optic length" or eikonal, $S=cT$ in Lagrangian terms: $S=\int_{M}^{N} L(\mathbf{x},
\dot{\mathbf{x}}) ds$ with the Lagrangian $L(\mathbf{x},\dot{\mathbf{x}})= n(\mathbf{x}(s))
\sqrt{\dot{\mathbf{x}}(s)\dot{\mathbf{x}}(s)}$, where $\dot{\mathbf{x}}^2= \sum_{i=1}^D
\big(\frac{dx^i}{ds}\big)^2$. We would like to mention here, that optical properties of the media
can be also treated in terms of induced Riemann metrics in vacuum:
\begin{equation}
S=\int_{M}^{N}n(\mathbf{x}(s)) ds = \int_{M}^{N} \sqrt{\dot{\mathbf{x}} g(\mathbf{x})
\dot{\mathbf{x}}} ds
\label{eq:eikonal-metrics}
\end{equation}
where $g_{ij} = n^2(\mathbf{x})\delta_{ij}$ stands for induced metrics components in isotropic media case.
Thus, from the geometrical point, the ray trajectory can be understood as a "minimal curve" in a
certain Riemann space. This representation suggests to consider optimal ray paths as geodesics in
the space with known metrics $g$.

Stationarity of optic length, $S$, i.e. $\delta S=0$, together with the condition
$|\dot{\mathbf{x}}|=1$, defines the Euler equation:
\begin{equation}
\frac{d}{ds}\left(n(\mathbf{x})\frac{d\mathbf{x}}{ds}\right)= \nabla n(\mathbf{x})
\label{eq:euler}
\end{equation}
from which one can directly proceed to the Huygens principle by integrating \eq{eq:euler} over $s$:
$\nabla S(\mathbf{x})=n(\mathbf{x})\frac{d\mathbf{x}}{ds}$. Squaring both sides of the latter
equation we end up with the eikonal equation:
\begin{equation}
\left(\nabla S(\mathbf{x})\right)^2=n^2(\mathbf{x})
\label{eq:eikonal}
\end{equation}
The eikonal equation Eq.\eq{eq:eikonal} has the same form as the Hamilton-Jacobi equation in
mechanics for action in the $D+1$-dimensional space, which in turn can be understood as the
relativistic equation for the light, propagating in the Riemannian space.

\subsection{The model: formalization of physical ideas}
\label{s:2}

In our work the eikonal equation arises in the differential growth problem in a purely geometric
setting. Consider a tissue, represented by a colony of cells, growing in space without any
geometric constraints. The local division protocol is prescribed by nature, being particularly
recorded in genes and is accompanied by their mutations \cite{genes}. The exponential cell
division is implied, as already mentioned above. To make our viewpoint more transparent, suppose
that all cells, represented by equilateral triangles, divide independently and their
proliferation is initiated by the first "protocell". Connecting the centers of neighboring
triangles by nodes, we rise a graph $\gamma$. The number of vertices, $P_{\gamma}(k)$, in the
generation $k$, grows exponentially with $k$: $P_{\gamma}(k) \sim c^k$ ($c>1$). It is known that
exponential graphs possess hyperbolic metrics, meaning that they can be isometrically (with
fixed branch lengths and angles between adjacent branches) embedded into a hyperbolic plane. Thus,
it is clear, that the corresponding surface, pulled on the isometry of such graph in the 3D
Euclidean space, should be negatively curved.

To have a relevant image, suppose that we grow the surface by crocheting it spirally
starting from the center \cite{chroch}. Demanding two nearest neighboring circumference layers, $P(r)$ and
$P(r+\Delta r)$, to differ by a factor of $c$ (where $c=\mathrm{const}>1$), i.e., $P(r+\Delta
r)/P(r)=c$, we construct an exponentially growing (hyperbolic) surface -- see the \fig{fig:01}a
known as Amsler surface \cite{amsler}. The crocheted surface has well-posed properties on large
scales, but should be precisely described on the scale of order of the elementary cell. As we have
mentioned, the microscopic description is connected with the specific local growth protocol. The
simplest way to generate the discrete hyperbolic-like surface out of equilateral triangles,
consists in gluing 7 such triangles in each graph vertex and construct a piecewise surface, shown
in the \fig{fig:01}b. On the scale less than the elementary cell $ABC$ this surface is flat. Thus,
the size $a$ ($|AB|=|AC|=|BC|=a$) of the triangle $ABC$ stands for the rigidity parameter, playing
the role of a characteristic scale in our problem, below which no deviations from the Euclidean
metrics can be found. Later on we shall see that buckling of growing surface essentially depends
on this parameter.

\begin{figure}[h]
\centering
\includegraphics[width=8cm]{eikonal_f01}
\caption{(a) Hyperbolic surface obtained by spiral crocheting from the center; (b) Hyperbolic
piecewise surface constructed by joining 7 equilateral flat triangles (copies of the triangle
$ABC$) in each vertex. The triangle $ABC$ is lying in $z = x+iy$ plane in the 3D Euclidean space,
$|AB|=|AC|=|BC|=a$.}
\label{fig:01}
\end{figure}

We discuss buckling phenomena for two different growth symmetries shown schematically in the
\fig{fig:02}a-b: i) uniform two-dimensional division from the point-like source (\fig{fig:02}a), and ii) directed
one-dimensional growth from the linear segment (\fig{fig:02}b). In \fig{fig:02}a-b different
generations of cells are shown by the shades of gray. For convenience of perception, sizes of cells
in each new generation are decreasing in geometric progression, otherwise it would be impossible to
draw them in a 2D flat sheet of paper and the figure would be incomprehensible. In Figs.
\ref{fig:02}c,d we imitate the protocols of growth depicted above in Figs. \ref{fig:02}a,b by
embedding the exponentially growing structure in the corresponding plane domain equipped with the
hyperbolic metrics. The advantage of such embedding consists in the possibility to continue all
functions smoothly through the boundaries of elementary domains, that cover the whole plane without
gaps and intersections. Details of this construction and
its connection to the growth in the 3D Euclidean space are explained below.

\begin{figure}[h]
\centering
\includegraphics[width=8cm]{eikonal_f02}
\caption{(a) Uniform two-dimensional hyperbolic growth out of the unit domain in the plane;
(b) One-dimensional hyperbolic growth out of the linear segment; c) Tessellation of the hyperbolic
Poincar\'e disc by the images of flat Euclidean triangles; d) Tessellation of the domain in the
hyperbolic half-plane by the images of flat Euclidean triangles.}
\label{fig:02}
\end{figure}

It is known (see, for example \cite{marder}) that the optimal buckling surface is fully determined
by the metric tensor through the minimization of the discrete functional of special energetic form.
Namely, define the energy of a deformed thin membrane, having buckling profile $f(x,y)$ above
the domain, parameterized by $(x,y)$, as:
\begin{equation}
E\{f(x,y)\} \sim \sum_{i,j} \left(\Big(f_{ij}\Big)^2-\sum_{\alpha,\beta}\Delta^{\alpha}_{ij}
g_{\alpha\beta} \Delta^{\beta}_{ij} \right)^2
\label{eq:E}
\end{equation}
where $g_{\alpha\beta}$ is the induced metrics of the membrane, $f_{ij} \equiv |f(x_i, y_i) -
f(x_j, y_j)|$ is the distance between neighboring points and $\Delta_{ij}$ is the equilibrium
distance between them. The typical (optimal) shape $\bar{f}(x,y)$ is obtained by minimization of
\eq{eq:E} for any rigidity. However, the metric tensor, $g_{\alpha\beta}$ is a priori unknown
since its elements depend on specifics of the differential growth protocol, therefore some
plausible conjectures concerning its structure should be suggested. For example, in \cite{marder} a
directed growth of a tissue with one non-Euclidean metrics component, $g_{xx}(y)$, was considered.
The diagonal component $g_{xx}(y)$ was supposed to increase exponentially in the direction of the
growth, $y$, and crumpling of a leaf near its edge was finally established and analyzed.

\subsection{Conformal approach}
\label{s:3}

The preset rules of uniform exponential cells division determine the structure of the hyperbolic
graph, $\gamma$, while the infinitesimal membrane thickness allows for the isometrical embedding of
the graph $\gamma$ into the 3D space. We exploit conformal and metric relations between the surface
structure in the 3D space and the graph $\gamma$ embedded into the flat domain with the hyperbolic
metrics. The embedding procedure consists of a sequence of conformal transformations with a
constraint on area preservation of an elementary plaquette. This eventually yields the knowledge of
the Jacobian (the "coefficient of deformation"), $J(x,y)$, for the hyperbolic surface, which is
embedded into the 3D space via the orthogonal projection. Equipped by the key assumption, that a
smooth yet unknown surface $f(x,y)$ is \emph{function}, our procedure straightforwardly implies a
differential equation on the optimal surface. Note that a version of the Amsler surface cannot be
reconstructed in the same way since it is not a function above some planar domain.

To realize our construction explicitly, we first embed isometrically the graph $\gamma$: i) into
the Poincar\'e disk ($|w|<1$) for the model of uniform planar growth, and ii) into the strip of the
half-plane ($\textrm{Im}\, r > 0, -1<\textrm{Re}\, r<0$) for the model of one-dimensional growth. In the \fig{fig:03}
we have drawn the tessellation of the Poincar\'e disc and of the strip by equilateral curvilinear
triangles, which are obtained from the flat triangle $ABC$ of the hyperbolic surface (see
\fig{fig:01}b) by conformal mappings $z(w)$ and $z(r)$ discussed below. Note, that a conformal
mapping preserves the angles between adjacent branches of the graph. The graph
$\gamma$, shown in the \fig{fig:03}, connects the centers of the triangles and is
isometrically embedded into the corresponding hyperbolic domain. Besides, the areas of images
of the domain ABC are the same.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f03}
\caption{Tessellation of the hyperbolic plane by the images of the
curvilinear triangle $ABC$: (a) for Poincar\'e disc; (b) for a strip of the upper half-plane. The
graph $\gamma$ connects the centers of images of $ABC$.}
\label{fig:03}
\end{figure}

For the sake of definiteness consider the graph $\gamma$, isometrically embedded into the
hyperbolic disk, shown in the \fig{fig:03}a. Now, we would like to find the surface in the 3D
Euclidean space above the $w$-plane such that its Euclidean metrics coincides with the
non-Euclidean metrics in the disk. The Hilbert theorem \cite{hilbert} prohibits to do that for the
class of $C^2$-smooth surfaces. However, since we are interested in the isometric embedding of
piecewise surface consisting of glued triangles of fixed area, we can proceed with the standard
arguments of differential geometry \cite{diff-geom}. The metrics $ds^2$ of a 2D surface,
parameterized by $(u, v)$, is given by the coefficients
\begin{equation}
E=\mathbf{r}_u^2, \quad F=\mathbf{r}_v^2, \quad G=(\mathbf{r}_u, \mathbf{r}_v)
\end{equation}
of the first quadratic form of this surface:
\begin{equation}
ds^2=E\, du^2 + 2F\, du dv + G\, dv^2
\end{equation}
The surface area then reads $dS= \sqrt{EG-F^2}\, du dv$.

The area $S_{ABC}$ of the planar triangle $ABC$ on the plane $z = x + iy$ can be written as:
\begin{equation}
S_{ABC}=\int\limits_{\triangle ABC} dx dy = \mathrm{const}
\label{eq:03}
\end{equation}
where the integration is restricted by the boundary of the triangle. Since we aimed to conserve the
metrics, let us require that the area of the hyperbolic triangle $ABC$, after the conformal
mapping, is not changed and, therefore, it reads:
\begin{equation}
S_{ABC} = \int\limits_{\triangle ABC} |J(z,w)| du dv; \quad J(z,w)=\left|\begin{array}{cc} \disp
\partial_u x & \disp \partial_u y \medskip  \\ \disp
\partial_v x & \disp \partial_v y
\end{array}\right|
\label{eq:04}
\end{equation}
where $J(z,w)$ is the Jacobian of transition form $z$ to new coordinates, $w$. If $z(w)$ is
holomorphic function, the Cauchy-Riemann conditions allow to write
\begin{equation}
J(w) = \left|\frac{dz(w)}{dw}\right|^2\equiv |z'(w)|^2.
\end{equation}
From the other hand, we may treat the value of the Jacobian, $J(w)$, as a factor relating the
change of the surface element under transition to a new metrics, the co-called "coefficient of
deformation". As long as the metrics in the hyperbolic domain should reproduce the Euclidean metrics of the
smooth surface, $f(u, v)$, one should set $J = \sqrt {EG-F^2}$, where $E,G,F$ are the coefficients
of the first quadratic form of the surface $f$. Now, if $f(u,v)$ is function above $w$-plane, its
Jacobian adopts a simple form:
\begin{equation}
J(u,v) = \sqrt{1+(\partial_u f)^2+(\partial_v f)^2}
\label{eq:06}
\end{equation}

Making use of the polar coordinates in our
complex $w$-domain, $\{(\rho, \phi): u = \rho\cos\phi, v = \rho\sin\phi\}$, we eventually arrive
at nonlinear partial differential equation for surface profile $f(\rho,\phi)$ in the polar coordinates above $w$:
\begin{equation}
\Big(\partial_\rho f(\rho,\phi)\Big)^2+\frac{1}{\rho^2}\Big(\partial_\phi
f(\rho,\phi)\Big)^2=|z'(w)|^4-1
\label{eq:08}
\end{equation}

In the case of the hyperbolic strip domain, \fig{fig:03}b, the equation for the growth profile above the domain
can be written in local cartesian coordinates, $r = s + it$:

\begin{equation}
\Big(\partial_s f(s, t)\Big)^2+\Big(\partial_t f(s,t)\Big)^2=|z'(r)|^4-1
\label{eq:07}
\end{equation}

Note, that the inequalities $|z'(w)| > 1, |z'(r)| > 1$, following from \eq{eq:08}-\eq{eq:07}, determine
the local condition of existence of non-zero real solution and, as we discuss below, can be interpreted as the
presence of a finite scale surface rigidity.

To establish a bridge between optic and growth problems, let us mention that, say,
equation \eq{eq:08}, coincides with the two-dimensional eikonal equation \eq{eq:eikonal} for the
wavefront, $S(w)$, describing the light propagating according the Huygens principle in the
unit disk with the refraction coefficient
\begin{equation}
n(w) = \sqrt{|z'(w)|^4-1}
\label{eq:n}
\end{equation}

\begin{figure}[ht]
\centering
\includegraphics[width=6cm]{eikonal_f04}
\caption{Orthogonal projection above the Poincar\'e disc: area of the curvilinear triangle in Euclidean
space coincides with the area of the triangle in hyperbolic metrics in Poincar\'e disc.}
\label{fig:04}
\end{figure}

We can construct the conformal mappings $z(r)$ and $z(w)$ of the flat equilateral triangle $ABC$ in
the Euclidean complex plane $z=x+iy$ onto the circular triangle $ABC$ in the complex domains $r = s
+ it$ and $w=\rho(\cos\phi+i\sin\phi)$ correspondingly. The absolute value of the Gaussian
curvature is controlled by the number, $V$, of equilateral triangles glued in one vertex: the
surface is hyperbolic only for $V>6$. The surfaces with any $V>6$ have qualitatively similar
behavior, however the simplest case for analytical treatment corresponds to $V=\infty$, when the
dual graph $\gamma$ is loopless. The details of the conformal mapping of the flat triangle with
side $a$ to the triangle with angles $\{0,0,0\}$ in the unit strip $r$ are given in the Appendix.
The Jacobian $J(z(r))$ of conformal mapping $z \to r$ reads:
\begin{equation}
J(r)=|z'(r)|^2 =\frac{h^2}{a^2}|\eta(r)|^8
\label{eq:ded1}
\end{equation}
and the Jacobian of the mapping $z \to w$, is written through the function $r(w)$ that conformally
maps the triangle from the strip onto the Pincar\'e disk:
\begin{equation}
J(w)=|z'(w)|^2 =\frac{3h^2}{a^2} \frac{|\eta(r(w))|^8}{|1-w|^4}
\label{eq:ded2}
\end{equation}
where
\begin{equation}
r(w) = e^{-i\pi/3}\frac{e^{2i\pi/3}-w}{1-w}-1; \quad h= \left(\frac{16}{\pi}\right)^{1/3}
\frac{\Gamma(\frac{2}{3})}{\Gamma^2(\frac{1}{3})} \approx 0.325
\end{equation}
In both cases \eq{eq:ded1} and \eq{eq:ded2}, the function in the right-side of the equation is the
Dedekind $\eta$-function \cite{jacobi}:
\begin{equation}
\eta(w)=e^{\pi i w/12}\prod_{n=0}^{\infty}(1-e^{2\pi i n w})
\label{eq:dedeta}
\end{equation}

\section{Results and their interpretation}
\label{s:4}

The eikonal equation, \eq{eq:eikonal}, with \emph{constant} refraction index, $n$, corresponds to
optically homogeneous 2D domain, in which the light propagates along straight lines in Euclidean
metrics. From the other hand, in this case the eikonal equation yields the action surface with zero
Gaussian curvature: a conical surface above the disk, $S(\rho, \phi) \sim \rho$, for the uniform 2D
growth and a plane above the strip, $S(s, t) \sim t$, for the directed growth. Note, that at least
one family of geodesics of these surfaces consists of lines that are projected to the light
propagation paths in the underlying domain. We will show below that the geodesics of the eikonal
surface conserve this property even when the media becomes optically inhomogeneous.

For growth, the constant refraction index corresponds to an isometry of a planar growing surface
and absence of buckling. The conformal transformation, that results in the corresponding "coefficient
of deformation", $J^2(u, v) = n^2(u, v) + 1$, is uniformly compressive and the tissue remains
everywhere flat. Thus, it becomes clear, why the essential condition for buckling to appear is the
\emph{differential growth}, i.e. the spatial dependence of local rules of cells division.

We solve \eq{eq:08} and \eq{eq:07} numerically with the Jacobian, corresponding to exponentially
growing circumference, \eq{eq:ded1}-\eq{eq:ded2}, for different parameters $a$. We have chosen the
Dirichlet initial conditions along the line (for directed growth above the strip) and along the
circle of some small enough radius (for uniform 2D growth above the disk). The right-hand side of the
eikonal equations for the specific growth protocol is smooth and nearly constant up some radius and
then becomes more and more rugged. The constant plateau in vicinity of initial stages of growth is
related with the fact, that exponentially dividing cells can be organized in a Euclidean plane
up to some finite generations of growth. However, as the cells proliferate further, the isometry of
their mutual disposition becomes incompatible with the Euclidean geometry and buckling of the
tissue is observed. Note, that the Jacobian is angular-dependent, that is the artefact of chosen
triangular symmetry for the cells in our model. The existence of real solution, $\bar f(u, v)$ of
the eikonal equation is related to the sign of its right-hand side and is controlled by the parameter
$a$, while the complex solution $f(u, v) = f_R(u, v) + if_I(u, v)$ can be found for every $a$.

First, we consider the 2D growth above the Poincar\'e domain, starting our numerics from low
enough values of $a$, for which the right-hand side of the eikonal equation, \eq{eq:08}, is strictly
positive on the plateau around the source of growth. Physically that means flexible enough tissues,
since, by construction, we require $a$ to be a scale on which the triangulated tissue does not
violate flat geometry. The real solution $\bar f(u,v)$ for these parameters exists up to late
stages of growth, see \fig{fig:bluebell} left. Note,
that a conical solution at early stages of growth is related to the plateau in the Jacobain and, as
it was discussed above, corresponds to the regime when cells can find place on the surface without
violation flat geometry. From the geometric optics point of view, that corresponds to constant
refraction index and straight Fermat geodesic paths in the underlying 2D domain. We show in the
\fig{fig:bluebell} that under increasing of $a$ the initial area of conical behavior is shrinking,
since the critical generation, at which the first buckling mode appears, is lower for larger cells.
In course of growth, the surface is getting negatively curved for some angular directions,
consistent with chosen triangular symmetry. It is found reminiscent of the shape of bluebells and,
in general, many sorts of flowers.

At late stages of growth, as we approach the boundary of the Poincar\'e disk, $\rho \to 1$ at some
fixed value of $\phi$, corresponding values of the right hand side of \eq{eq:08} become negative,
leading to the complex solution of the eikonal equation. Fortunately, we may infer some useful
information from the holomorphic properties of the eikonal equation in this regime, not too close
to the boundary of the disc. Applying the Cauchy-Riemann conditions to the solution of the eikonal
equation, $f$, we have: $\partial_u f_R = \partial_v f_I$ and $\partial_v f_R = -\partial_u f_I$.
Thus, the function $\bar f$ can be analytically continued in the vicinity of points along the curve
$\Gamma$ in the $(uv)$ plane, at which the right hand side of the eikonal equation nullifies. Moreover,
using this property, one can show, that the absolute value of the complex solution in the vicinity
of $\Gamma$ smoothly transfers to the real-valued solution, as one approaches the $\Gamma$ curve:
\begin{equation}
\begin{array}{rll}
\disp \lim_{(u, v) \to \Gamma} \left(\nabla |f(u, v)|\right)^2 &=& \left(\nabla f_R(u,
v)\right)^2|_{\Gamma} \equiv \left(\nabla \bar f(u, v) \right)^2 \medskip \\
\disp |f(u, v)| &=& \sqrt{f_R^2(u, v) + f_I^2(u, v)}
\end{array}
\label{mod}
\end{equation}

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f06}
\caption{The bluebell phase, obtained by numerical solution of \eq{eq:08} for two flexible tissues:
$a = 0.07$ (first row) and $a = 0.14$ (second row). Figures on the right show appearance of buckling
instabilities at the edge with growth.}
\label{fig:bluebell}
\end{figure}

The non-existence of real solutions of the eikonal equation at late stages is a direct consequence
of the presence of finite bending scale, on which the tissue is locally flat. As it was
mentioned above and is shown in the \fig{fig:bluebell}, low values of $a$ lead to elongated conical
regime. Since $a$ stands for the scale on which the circumference length of the tissue doubles, in the $a \to 0$ limit the
real solution exists everywhere inside the disk, but it is everywhere flat (conical). Hopefully, the analytic continuation allows one
to investigate buckling for negative values of $n^2(u,v) = J^2(u, v)-1$ by taking the absolute
value of the solution, at least not far away from the zero-curve $\Gamma$. In this regime buckling
instabilities on the circumference of the bluebell arise. In the \fig{fig:s-s} we show
proliferation of buckling near the critical point. First, the evolution of buckling instabilities at the edge
can be understood as a subsequent doubling of peaks and saddles along the direction of growth. Then some hierarchy in peaks size
is seen. We note, that this hierarchical organization is a natural
result due to the theoretic-number properties of the Dedekind $\eta$-function. Though it is known, that in real plants and
flowers buckling instabilities do not proliferate profoundly, since the
division process is getting limited at late stages of growth, the formal continuation of the eikonal
equation beyond $\Gamma$ predicts a self-similar buckling profile at the circumference of growing tissues.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f07}
\caption{Development of buckling instabilities at the edge of flower for rigidity parameter $a = 0.14$.
The right figures show hierarchical organization of the flower's circumference in detail.}
\label{fig:s-s}
\end{figure}

Now we pay attention to the directed growth above the half-plane domain. Here we solve the equation
\eq{eq:07} with the Dirichlet boundary conditions, set along the line $t = 1$, and the tissue is
growing towards the boundary $t = 0$ in the upper halfplane $\textrm{Im}\, r>0$. At low stages of growth
the solution is flat until the first buckling mode appear, \fig{fig:direct_sol}. The subsequent
growth is described by taking the absolute value of the solution, since no real solution exists
anymore. As in the former case, the behavior is controlled by the value of $a$.

When the growth approaches the boundary, the edge of the tissue becomes more and more wrinkled.
Emergence of new buckling modes is the consequence of the Dedekind $\eta$-function properties: doubling
of parental peaks at the course of growth. Under the energetic approach for a leaf, very similar fractal structures
can be inferred from the interplay between
stretching and bending energies in the limit of extremely thin membranes: while the cell density
(and the corresponding strain, $\sigma$) on the periphery increases, the newly generating
wavelengths decrease, $\lambda\sim \sigma^{-1/4}$, \cite{mahadevan}.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f08}
\caption{Numerical solutions of \eq{eq:07} for the directed growth.
Figures show enhancing of buckling at the edge.}
\label{fig:direct_sol}
\end{figure}

Increasing the size $a$ of the elementary flat triangle domain, we figure out, that for some
critical value, $a_{cr}$, the starting plateau of the corresponding Jacobian crosses the zero level
and becomes negative. Our model implies no solutions for such stiff tissues. This limitation is
quite natural since we do not consider in-plane deformations of the tissue. In reality, for $a
> a_{cr}$ the tissue is so stiff, that it turns beneficial to be squeezed in-plane rather than to
buckle out. One may conjecture that $a$ is the analogue of the Young modulus, $E$, that is
known to regulate the rigidity of the tissue in the energetic approach, along with the thickness,
$h$, and the Poisson modulus, $\mu$, in their certain combination, known as bending stiffness, $D =
\frac{Eh^3}{12(1-\mu^2)}$.

It is worth mentioning that at first stages of growth, until the instabilities at the circumference have not yet appeared,
at certain angles (triangle-like cells) the surface bends similar to the Beltrami's pseudosphere, that
has a constant negative curvature at every point of the surface, compare \fig{fig:bluebell} and \fig{fig:pseudo}.
The similarity is even more striking for very low $a$, when the triangulating parameter is fairly small.
It is known that the pseudosphere locally realizes the Lobachevsky geometry and can be isometrically mapped onto \emph{the finite
part} of the half-plane or of the Poincar\'e disk, \fig{fig:pseudo}a-b. According to the Hilbert theorem, \cite{hilbert}, no full
isometric embedding of the Poincar\'e disk into the 3D space exists. Thus, in order to organize
itself in the 3D space, the plant grows by the cascades of pseudospheres, resembling peaks and saddles,
that is an alternative view on essence of buckling.

Interestingly, some flowers, such as calla lilies, initially grow psuedospherically, but then
crack at some stage of growth and start twisting around in a helix. Apparently, this is another
route of dynamic organization of non-Euclidean isometry in the Euclidean space. The Dini's surface,
\fig{fig:pseudo}c is known in differential geometry as a surface of constant negative curvature and,
in comparison with the Beltrami's pseudosphere, is infinite. The problem of sudden cracking of the lilies
seems to be purely biological, but as soon as the crack appeared, the flower may relief the stresses caused by subsequent
differential growth through twisting its petals in the Dini's fashion.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f09}
\caption{(a)-(b) Pseudosphere and correspondence of boundaries on the Poincar\'e disc; (c) Dini
surface.}
\label{fig:pseudo}
\end{figure}

Turn now to the eikonal interpretation of buckling. For the sake of simplicity,
we will proceed here in the cartesian coordinates. Seeking the solution of \eq{eq:eikonal} and
\eq{eq:07} in the implicit form $H(\mathbf{x})\equiv H(x^0,x^1,x^2) = H^0$ with $x^0= i f$,
$x^1=u$, $x^2=v$, we can rewrite \eq{eq:07} as:
\begin{equation}
g^{ij}\frac{\partial H(\mathbf{x})}{\partial x^i}\frac{\partial H(\mathbf{x})}{\partial x^j}= 0;
\quad g^{ik} = \left(\begin{matrix} n^2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right)
\label{eq:rel}
\end{equation}
Eq.\eq{eq:rel} reveals the relativistic nature of the eikonal equation \cite{landau} and describes
the propagation of light in a (2+1)D space-time in the gravitational field with induced metrics
$g$ defined by the metric tensor $g^{ik}$, where $n\equiv n(x^1,x^2)$, speed of light put $c=1$.
Having $g$, one can reconstruct geodesics that define the paths of the light propagation in our
space-time. The parameterized geodesics family, $x^{\lambda}(\tau)$, where $\lambda = 0, 1, 2$, can
be found from the equation:
\begin{equation}
\frac{d^2 x^{\lambda}}{d \tau^2} + \Gamma^{\lambda}_{ij} \frac{dx^i}{d \tau} \frac{dx^j}{d \tau}=0
\label{geo}
\end{equation}
where
\begin{equation}
\Gamma^{i}_{kl} = \frac{1}{2} g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial
g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} \right)
\end{equation}
are Christoffel symbols and $g_{ij}$ is the covariant form of the metrics ($g_{ij}g^{jk} =
\delta^k_i$). Calculating the symbols for the specific metrics \eq{eq:rel}, we end up with the set
of equations for the geodesics in a parametric form:
\begin{equation}
\left\{\begin{array}{l}
\disp u_{\tau \tau} - \frac{1}{n^3} \frac{\partial n}{\partial u} f_{\tau}^2 = 0, \medskip \\
\disp v_{\tau \tau} - \frac{1}{n^3} \frac{\partial n}{\partial v} f_{\tau}^2 = 0, \medskip \\
\disp f_{\tau \tau} - \frac{2}{n}\frac{d n}{d \tau} f_{\tau} = 0
\end{array} \right.
\label{traj}
\end{equation}
From the first two lines of \eq{traj}, one gets $\frac{u_{\tau \tau}}{v_{\tau \tau}} =
\frac{\partial n}{\partial u} \left(\frac{\partial n}{\partial v}\right)^{-1}$. Note, that the same relation
follows directly from \eq{eq:euler}, if the planar domain is parameterized by the same coordinates $\mathbf x = \mathbf x(u, v)$.
Thus, one may conclude, that the projections of the geodesics from the (2+1)D space-time onto the $(uv)$-plane coincide with
light trajectories in the flat domain with refraction coefficient $n(u,v)$.

\section{Conclusion and conjectures}
\label{s:5}

In this paper we discussed the optimal buckling profile formation of growing two-dimensional tissue
evoked by the exponential cell division from the point-like source and from the linear segment.
Such processes imply excess material generation enforcing the tissue to wrinkle as it approaches
the domain boundary. Resulting optimal hyperbolic surface is described by the eikonal equation for
the two-dimensional profile, and allows for simple geometric optics analogy. It is shown that the
surface height above the domain mimics the eikonal (action) surface of a particle moving in the 2D media with
certain refraction index, $n$, which, in turn, is linked to microscopic rules of elementary cell
division and symmetry of the plant. The projected geodesics of this "minimal" optimal surface coincide
with Fermat paths in the 2D media, which is the intrinsic feature of the eikonal equation. This result
suggests an idea to treat the growth process itself as a propagation of the wavefronts in the media
with certain metrics.

We have derived the metrics of the growing plant's surface from microscopic rules of cells division and
have shown that the solution of the eikonal equation describes buckling of tissues of different
rigidities. Our results, being purely geometric, rhyme well with a number of energetic approaches
to buckling of thin membranes, where the stiffness is controlled by the effective bending
rigidity. We show that presence of a finite scale on which the tissue remains flat, results in
negatively curved growing surfaces and the eikonal equation implies absence of real solution at
late stages of the growth. Though, an analytical continuation can be constructed and erratic
self-similar patterns along the circumference can be obtained. In reality high energetic costs for
the profound cell division after bifurcation point would prohibit infinite growth and intense buckling.

Recall, that the right-hand side of the eikonal equation mimics the squared refraction index, \eq{eq:n},
if buckling is interpreted as wavefront propagation in geometric optics. At length of our work it was pointed out,
that for the differential growth problem, negative square of refraction index leads to complex solution
for $f$. Does complex solution have any physical meaning for growth? We can provide the following speculation. The
complex solution appears for the late stages of growth when the finite bending scale of the tissue prohibits
formation of very low-wavelength buckling modes. Since in this regime the tissue
would experience in-plane deformations, one may improve the geometric model by letting
branches to accumulate the "potential energy". Thereby, the analogy between optics and differential growth
can be advanced by noting that the negative squared refraction index means absorbtion properties of the media. The
propagating wavefront of a moving particle, dissipates the energy in areas where the refraction index is complex-valued.
In the differential growth the proliferation of buckling modes may be limited by the energy losses at branches, that would
suppress buckling.

The challenging question concerns the possibility to extend our approach to the growth of
three-dimensional objects, for example, of a ball that size $R$ grows faster than $R^2$. In this
case, the redundant material can provoke the surface instabilities. We conjecture that some analogy
between the boundary growth and optic wavefronts survives in this case as well.

Authors are grateful to M. Tamm, A. Grosberg, M. Lenz, L. Mirny and L. Truskinovsly for valuable discussions
of various aspects of the work and to A. Orlov for invaluable help in numerical solution of the
eikonal equation. The work is partially supported by the IRSES DIONICOS and RFBR 16-02-00252A
grants.

\begin{appendix}

\section*{Appendix: Conformal transformation of the flat triangle to the Poincare domain}

The conformal mapping $z(w)$ of the flat equilateral triangle $ABC$ located in $z$ onto the
zero-angled triangle $ABC$ in $w$, used in the derivation of \eq{eq:ded1}, is constructed in four
sequential steps, shown in the \fig{fig:suppl01}.

\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{eikonal_f05}
\caption{Conformal mapping $z(w)$ is realized as a composition of three mappings: $z(\zeta)$
[(a)--(b)], $\zeta(r)$ [(b)--(c)], and $r(w)$ [(c)--(d)]. Finally we have $z(\zeta(r(w)))$.}
\label{fig:suppl01}
\end{figure}

First, we map the triangle $ABC$ in $z$ onto the upper half-plane $\zeta$ of auxiliary complex
plane $\zeta$ with three branching points at 0, 1 and $\infty$ -- see the \fig{fig:suppl01}a-b.
This mapping is realized by the function $z(\zeta)$:
\begin{equation}
z(\zeta)=\frac{\Gamma(\frac{2}{3})}{\Gamma^2(\frac{1}{3})} \int_0^{\zeta}
\frac{d\xi}{\xi^{2/3}(1-\xi)^{2/3}}
\label{eq:z(zeta)}
\end{equation}
with the following coincidence of branching points:
\begin{equation}
\left\{
\begin{array}{lcl}
A(z=0)& \leftrightarrow &  A(\zeta=0)\medskip \\
B(z=1) & \leftrightarrow & B(\zeta=1) \medskip \\
C(z=e^{-i\frac{\pi}{3}}) & \leftrightarrow & C(\zeta=\infty)
\end{array} \right.
\label{eq:points1}
\end{equation}

Second step consists in mapping the auxiliary upper half-plane $\Im\zeta>0$ onto the circular
triangle $ABC$ with angles $\{\alpha,\alpha,0\}$ -- the fundamental domain of the Hecke group
\cite{hecke} in $r$, where we are intersted in the specific case $\{\alpha,\alpha,0\}=\{0,0,0\}$ --
see \fig{fig:suppl01}b-c. This mapping is realized by the function $\zeta(r)$, constructed as
follows \cite{koppenfels}. Let $\zeta(r)$ be the inverse function of $r(\zeta)$ written as a
quotient
\begin{equation}
r(\zeta) = \frac{\phi_1(\zeta)}{\phi_2(\zeta)}
\label{eq:quot}
\end{equation}
where $\phi_{1,2}(\zeta)$ are the fundamental solutions of the 2nd order differential equation of
Picard-Fuchs type:
\begin{multline}
\zeta(\zeta-1) \phi''(\zeta)+\big((a+b+1)\zeta-c\big) \phi'(\zeta) \\ + ab\phi(\zeta)=0
\label{eq:fundam}
\end{multline}

Following \cite{caratheodory,koppenfels}, the function $r(\zeta)$ conformally maps the generic
circular triangle with angles $\{\alpha_0=\pi|c-1|,\alpha_1=\pi|a+b-c|, \alpha_\infty=\pi|a-b|\}$
in the upper halfplane of $w$ onto the upper halfplane of $\zeta$. Choosing $\alpha_{\infty}=0$ and
$\alpha_0=\alpha_1=\alpha$, we can express the parameters $(a,b,c)$ of the equation \eq{eq:fundam}
in terms of $\alpha$, taking into account that the triangle $ABC$ in the \fig{fig:suppl01}c is
parameterized as follows $\{\alpha_0,\alpha_1,\alpha_\infty \}=\{\alpha,\alpha,0\}$ with
$a=b=\frac{\alpha}{\pi}+\frac{1}{2}, c=\frac{\alpha}{\pi}+1$. This leads us to the following
particular form of equation \eq{eq:fundam}
\begin{multline}
\zeta(\zeta-1) \phi''(\zeta)+\Big(\frac{\alpha}{\pi}+1\Big)\big(2\zeta-1\big) \phi'(\zeta)
\\ +\Big(\frac{\alpha}{\pi} +\frac{1}{2}\Big)^2\phi(\zeta)=0
\label{eq:fundam2}
\end{multline}
where $\alpha = \frac{\pi}{m}$ and $m=3,4,...\infty$. For $\alpha=0$ Eq.\eq{eq:fundam2} takes an
especially simple form, known as Legendre hypergeometric equation \cite{golubev,hille}. The pair of
possible fundamental solutions of Legendre equation are
\begin{equation}
\begin{array}{l}
\phi_1(\zeta)=F\big(\frac{1}{2},\frac{1}{2},1,\zeta\big) \medskip \\
\phi_2(\zeta)=iF\big(\frac{1}{2},\frac{1}{2},1,1-\zeta\big)
\label{eq:hyp}
\end{array}
\end{equation}
where $F(...)$ is the hypergeometric function. From \eq{eq:quot} and \eq{eq:hyp} we get $r(\zeta) =
\frac{\phi_1(\zeta)}{\phi_2(\zeta)}$. The inverse function $\zeta(r)$ is the so-called modular
function, $k^2(r)$ (see \cite{golubev,hille,jacobi2} for details). Thus,
\begin{equation}
\zeta(r) \equiv k^2(r) = \frac{\theta_2^4(0,e^{i\pi r})}{\theta_3^4(0,e^{i\pi r})}
\label{eq:zeta(r)}
\end{equation}
where $\theta_2$ and $\theta_3$ are the elliptic Jacobi $\theta$-functions \cite{jacobi2,mum},
\begin{equation}
\begin{array}{l}
\disp \theta_2\left(\chi,e^{i\pi w}\right)=2e^{i{\pi \over 4} r}
\sum_{n=0}^{\infty} e^{i\pi r n(n+1)}\cos (2n+1)\chi \medskip \\
\disp \theta_3\left(\chi,e^{i\pi r}\right)=1+2\sum_{n=1}^{\infty} e^{i\pi r n^2}\cos 2n\chi
\end{array}
\end{equation}
and the correspondence of branching points in the mapping $\zeta(r)$ is as follows
\begin{equation}
\left\{
\begin{array}{lcl}
A(\zeta=0) & \leftrightarrow & A(r=\infty) \medskip \\
B(\zeta=1) & \leftrightarrow & B(r=0) \medskip \\
C(\zeta=\infty) & \leftrightarrow & C(r=-1)
\end{array} \right.
\label{eq:points1}
\end{equation}

Third step, realized via the function $r(w)$, consists in mapping the zero-angled triangle $ABC$ in
$r$ into the symmetric triangle $ABC$ located in the unit disc $w$ -- see \fig{fig:suppl01}c-d. The
explicit form of the function $r(w)$ is
\begin{equation}
r(w) = e^{-i\pi/3}\frac{e^{2i\pi/3}-w}{1-w}-1
\label{eq:r(w)}
\end{equation}
with the following correspondence between branching points:
\begin{equation}
\left\{
\begin{array}{lcl}
A(r=\infty) & \leftrightarrow & A(w=1) \medskip \\
B(r=0) & \leftrightarrow & B(w=e^{-2\pi i/3}) \medskip \\
C(r=-1) & \leftrightarrow & C(w=e^{2\pi i/3})
\end{array} \right.
\label{eq:points1}
\end{equation}

Collecting \eq{eq:z(zeta)}, \eq{eq:zeta(r)}, and \eq{eq:r(w)} we arrive at the following expression
for the derivative of composite function,
\begin{equation}
z'(\zeta(r(w))) = z'(\zeta)\, \zeta'(r)\, r'(w)
\end{equation}
where $'$ stands for the derivative. We have explicitly:
$$
z'(\zeta)=\frac{\Gamma(\frac{2}{3})}{\Gamma^2(\frac{1}{3})}\,
\frac{\theta_3^{16/3}(0,\zeta)}{\theta_2^{8/3}(0,\zeta)\;\theta_0^{8/3}(0,\zeta)}
$$
and
$$
\zeta'(r)|=i\pi\frac{\theta_2^4\;\theta_0^4}{\theta_3^4}; \qquad i\frac{\pi}{4}\theta_0^4=
\frac{d}{d\zeta}\ln\left(\frac{\theta_2}{\theta_3}\right)
$$
The identity
\begin{multline}
\disp \theta'_1(0,e^{i\pi\zeta})\equiv
\left.\frac{d\theta_1(\chi,e^{i\pi\zeta})}{d\chi}\right|_{\chi=0} \\ =
\pi\theta_0(\chi,e^{i\pi\zeta})\,\theta_2(\chi,e^{i\pi\zeta})\, \theta_3(\chi,e^{i\pi\zeta})
\nonumber
\end{multline}
enables us to write
\begin{equation}
\left|z'(r)\right|^2=h^2 \left|\theta'_1\left(0, e^{i\pi r}\right)\right|^{8/3}
\label{3:jacobian}
\end{equation}
where $h= \left(\frac{16}{\pi}\right)^{1/3}\frac{\Gamma(\frac{2}{3})}{\Gamma^2(\frac{1}{3})}$, and
\begin{equation}
\theta_1(\chi,e^{i\pi r})=2e^{i\frac{\pi}{4} r} \sum_{n=0}^{\infty}(-1)^n e^{i\pi
n(n+1) r}\sin (2n+1)\chi \medskip \\
\label{3:ell1}
\end{equation}
Differentiating \eq{eq:r(w)}, we get
$$
r'(w) = \frac{i\sqrt{3}}{(1-w)^2}
$$
and using this expression, we obtain the final form of the Jacobian of the composite conformal
transformation $J(z(\zeta(r(w))))$:
\begin{equation}
J(z(w)) = |z'(w)|^2 = 3h^2\frac{|\eta(r(w))|^8}{|1-w|^4}
\label{eq:composite}
\end{equation}
where
$$
\eta(r) = \big(\theta_1'(0,e^{i\pi r})\big)^{1/3}
$$
is the Dedekind $\eta$-function (see \eq{eq:ded2}), and the function $r(w)$ is defined in
\eq{eq:r(w)}.

\end{appendix}

redpajama_set_name: RedPajamaArXiv


Zych, M.; Wojnar, W.; Dudek, S.; Kaczmarczyk-Sedlak, I. Rosmarinic and Sinapic Acids May Increase the Content of Reduced Glutathione in the Lenses of Estrogen-Deficient Rats. Nutrients 2019, 11, 803.
Zych M, Wojnar W, Dudek S, Kaczmarczyk-Sedlak I. Rosmarinic and Sinapic Acids May Increase the Content of Reduced Glutathione in the Lenses of Estrogen-Deficient Rats. Nutrients. 2019; 11(4):803.
Zych, Maria; Wojnar, Weronika; Dudek, Sławomir; Kaczmarczyk-Sedlak, Ilona. 2019. "Rosmarinic and Sinapic Acids May Increase the Content of Reduced Glutathione in the Lenses of Estrogen-Deficient Rats." Nutrients 11, no. 4: 803.


Hines Park, located along the beautiful Flambeau River, offers amenities such as camping, rest rooms (during summer months), two fishing piers, a boat launch, a large playground, basketball courts, and sand volleyball courts. A large log pavilion, complete with a fireplace and enclosed serving room, is available for reservation. The campground includes nine sites all with water and electric; one site has sewer, as well. Sites 1-4 & 6-9 are $20/night; site #5 (with sewer) is $25/night.


Living a less-distracted life : admit that you're tired and reclaim rest.
I know, I know; motherhood and rest rarely go together. We are a weary bunch, aren't we, running to and fro and round in circles. Despite our exhaustion we just keep going because we have to.
Perhaps it's more pronounced at this time of year as we take a sharp inhalation and deal with the plethora of to-dos that accompany the festive season. My diary is quickly filling up with deadlines, the end of school term is fast approaching and I am completely disorganised in regards to Christmas. Yes, it's overwhelming and if I'm honest, there's just no time for early nights and afternoon kips right now.
Che is tired, too. I kept him home from school on Friday for a rest day and he lulled about reading books, watching a movie and building lego. There was a part of me that wished I could do the same thing; ignore all the necessary chores and emails and just lie in bed; rejuvenating and being kind. I know I need it but I can't see it happening anytime soon. And so I started thinking about quick rests; little things I can do that will restore and revive. I mention them in my yoga classes all the time and yet applying them to my every day takes a little more discipline. But you know what, they work, and sometimes they're more beneficial that a deep day sleep (I usually wake up groggy and cranky and that is never good!).
drink water. It's usually at around 4:30 that I experience a slump and unfortunately it coincides with dinner prep. So every evening I stand at the kitchen window and drink a big glass of water. Within ten minutes I have the energy to get through the dinner, bath, bed routine.
get outside. I've taken to lying on the trampoline and purposefully taking a few deep, nourishing breaths. I then let out loud sighs (much to the family's delight). Fresh air – always good.
lie on your bed with your feet together and your knees out wide. This position is called supta budda konasana or reclining butterfly pose. If you feel that it stretches your groin too much just place a pillow or rolled blanket under your upper thighs. In this position become aware of your breath and focus on your exhalation for a few moments – you'll notice you start to grow heavy and soft and relaxed. Then, count your breath backwards from 27. "I am breathing in 27, I am breathing out 27, I am breathing in 26…" When you get to 0 start to awaken your body by wriggling your fingers and toes. Get up slowly and carry on with your day. This is also a great practice to do with older children. First, choose an item to place on the belly – a feather, a small teddy, a paper boat – and then encourage them to watch and feel the feather/teddy/boat rise and fall with the rhythm of the breath.
close your eyes whilst breastfeeding. I understand this isn't always possible if you've got more than one child but when breastfeeding my two I made a point of lying down to feed at least once a day. I'd put my phone away, get really comfortable and close my eyes. Sometimes I would sleep, sometimes five minutes of shut-eye was enough.
Thanks for sharing this Jodi. I just wrote about how busy my weekend was and how I too need to learn how to rest and rejuvenate more in my day.
Fresh air is so important. I find that just 5 minutes outside with no distraction does so much for my day. Though I need to remember to keep up with it and not fall back into my busy, distracted life. Easier said than done unfortunately.
Water and fresh air. It seems odd that we live a life so full on that we require to be reminded to breath and drink water. Two very basic things. Thank you, Jodi!
All wonderful daily practices! I never truly realized this feeling of exhaustion existed. People always told me having a little one was so much work but I never really understood until Owynn was born. I yearn for the night when I can sleep a solid 8 hours. It is such a special experience but all my taking care of self has gone out the window. One day I know a balanced routine will come into place. But just taking a few moments each day alone is so key to preserving my energy and happiness. Thanks for the reminder!
Just reading this was relaxing!
The drinking water at a set time, is a great idea. I really need to remember that.
Great techniques Jodi thankyou! I also like the way you suggested a way to do some relaxation with children. My firstborn is three and a half and can get a bit anxious and overwhelmed so I'd love any posts about yoga or relaxation techniques for children.
Great post. Needed to hear these tips and will be putting them into practise, especially the reclining butterfly pose and the counting – I often want to relax but have no idea how to, now I do.
Exhaustion can permeate your bones, you carry it around like a burden on your back. It alters your personality and changes how you act in situations. Of to try the butterfly pose now.
Thank you for sharing these techniques. I like the five minutes of shut eye while breastfeeding. Our girl requires me to stay with her for naps…or there are none…and I so often get on my phone once she is in a deep sleep. I'm going to make it a point to close my lids from time to time and see what a difference that makes.
I AM exhausted. I hadn't the faintest idea how sore my shoulders and neck were until I slowly arched my back, titled my head slowly and took slow, deliberate breaths. Before I sleep, I will practice supta buddha konasana.
I know this is an old post but I'm really enjoying going through your space and gleaning ideas to help make this year as happy, healthy and peaceful as possible.


"Mauricio's Mexican Restaurant" is a little place tucked away in a semi-run-down shopping center on Route 66 in east Glendora. While the surroundings may not be very easy on the eyes, it's definitely worth the time to come by and order up some great Mexican fare. Mauricio always greets his guests with a hearty "Where you been?" and says goodbye with "See you tomorrow!" The food is so good here that I'm sure many people take him up on that!
Today, Jennifer and I had been riding around the local foothills on a brand new R1200 GSA. What an amazing motorcycle…the smooth, powerplant, behemoth gas tank, and superb suspension begs you to keep on riding and riding…which we might have done, if we weren't so hungry for Mauricio's!
The biggest challenge at this charming little place is deciding what to order. Jennifer decided on the beef vegetable soup (Caldo Res) and I picked the eggs and chorizo…Both meals with coffee, fresh tortillas and complimentary sweet rolls was $21 including tip.
Mauricio's Restaurant is located at 2256 E Route 66, Glendora, CA 91741. You can call for take out at 626.963.7515!
Happy riding, and stay safe!
MMMMmmmmmmmm, that looks good! We'll have to check that place out some time.
Riding around on "a" new GSA or "your" new GSA?
No, I wish it was mine….what an awesome ride!


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Is Ridley Scott still relevant to modern cinema?
Tue 30th Nov 2021 17.30 GMT
There can be no denying that Ridley Scott is one of the most important filmmakers of the modern era. A master of the big-budget blockbuster, in many ways, Scott can be regarded as something of the British answer to Steven Spielberg, just without the more childish elements to his filmography that Spielberg's filmography boasts.
The go-to auteur for a historical romp, Scott is certainly not a one-dimensional filmmaker. He has cast his net wide over his lengthy career, with movies such as The Duellists, Alien, Blade Runner, Legend, Thelma & Louise, Gladiator, Black Hawk Down, Kingdom of Heaven and American Gangster all hailed as his definitive pictures.
Not afraid of tackling a complex subject, Scott's highlighted films present a wide arrange of thematic discussions, and in his 1980s era, the director was well ahead of the curve stylistically. Famed for his depictions of urban environments ranging from Golden Age Rome to the 12th Century Holy Land and a dystopian Los Angeles, without Scott's best works, popular culture would look very different.
His legacy is something that should never be ignored, as it feels Scott is often overlooked due to the number of box office flops he has released over the past couple of decades. Whether it be the visually spectacular but rather hollow Prometheus, Exodus: Gods and Kings or the Russell Crowe starring 2010 film Robin Hood, Scott's output is not as consistent as it once was, but let's not forget that he is now into his 80s. However, Scott has shown that he is still apt at putting a visual delight together when push comes to shove, as 2015's The Martian showed.
With a career bouncing high and low with critical success and subsequent commercial failure, 2021 was meant to be the year that Ridley Scott reasserted his dominance at the box office amongst all the Marvel piffle and terrible children's movies – but thus far things have not gone to plan. This year Scott has released not one, but two blockbusters, all within the space of five weeks. On paper, and in the run-up to release, The Last Duel and House of Gucci were set to be two of the biggest films of the year, wrestling for the share festival season awards with Dune, No Time to Die and the new Venom entry.
So far, however, The Last Duel has been considered an economic flop. The film deals with the true 14th Century story of Jean de Carrouges, who challenges his friend and squire Jacques Le Gris to a duel after Carrouges' wife accuses Le Gris of rape. Split into three parts from different perspectives, The Last Duel was set to be the most refreshing of this year's blockbusters — and, in many ways, it still is.
The screenplay was written by Nicole Holofcener, Ben Affleck and Matt Damon, and based on Eric Jager's 2004 book, The Last Duel: A True Story of Trial by Combat in Medieval France. Written by a woman, and discussing the implications of rape and something of a metaphor in today's cultural climate of the Me Too movement, The Last Duel has tanked at the box office. This is owing to its 18+ rating, tough themes, and the fact that many people just aren't ready to venture out to the cinema yet due to the pandemic still being in full swing, with a third wave ominously looming around the corner.
Ridley Scott announces the arrival of new Blade Runner TV series
The film hasn't tanked critically, though, and surely this is where it matters. Scott has created a picture of worth, and it says a lot more than many of his previous outings have done. Seemingly, however, credibility doesn't matter. It turns out, perhaps unsurprisingly, that the veteran filmmaker was most displeased with how The Last Duel has performed at the box office, and he hasn't been shy about expressing it. Appearing on Marc Maron's WTF podcast last week, Scott offered some insight into why he thinks his film has bombed financially. He said: "What we've got today, (are) the audiences who were brought up on these fucking cellphones. The millennian do not ever want to be taught anything unless you're told it on a cellphone".
While perhaps speaking emotionally and out of passion for his work, Scott needs to rethink his assertions. It is Millenials, whom he seems to be confusing with Gen Y, that are currently propping up The Last Duel, noting its storytelling, even if it is slightly overcooked. Whilst Scott's point about phones certainly has some credence, it is a misfire. Scott is directing his rage at the wrong demographic. The director has every right to be angry, but it's the confluence of factors outlined prior that have culminated in The Last Duel tanking in the financial sphere.
Scott also tacitly inferred another critical point, as it appears as though he places more emphasis on money than credibility. Ridley Scott has made a film of note, that will no doubt age well, and in a couple of years, will be more widely appreciated. It's safe to say that now was not the right time for The Last Duel to be released, and, if he'd have held off a few months, it might have been a different story.
Then we come to House of Gucci, although it has more explicit appeal to younger Millenials and Gen Yers through its casting of Lady Gaga and Adam Driver, fusing a crime thriller with the fashion world, it has tanked critically in comparison to The Last Duel. By all accounts, it is a flop and not eligible for the awards that many touted it for — but where are Scott's comments on this? If House of Gucci proves to be a total bomb, questions will be levelled at Scott's future within the film industry. It's hard not to consider that his most recent work arrived at what could well be his final chance to sit atop of the mountain of Hollywood, and through ill-timed releases and misguided statements, Scott may have just proved that cinema has changed too much and that his corner of the blockbuster sphere is no longer relevant.
But for now, only time will tell. We'll see how well this has aged next year.

redpajama_set_name: RedPajamaCommonCrawl


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Located on the Costa del Sol in Andalusia, Benalmadena offers a lovely sea side location and the language school provides a small school environment, just right for those who want a Spanish school offering a quieter pace of life and plenty of sunshine!
This Mediterranean based Spanish language school appeals to adults of all ages throughout the year (and teenage students in the Easter / Summer holidays).
Every day, every student had to ask all the other students a question and all students had to answer – something as simple as what is your favourite colour and why. Flash cards were also used by the Spanish teacher as an aid to help the language students frame questions.
Many of the students were studying at the Spanish language school for 3 or 4 months and I joined them in mid stream. The first day was difficult but thereafter it was plain sailing. The other students and the teachers were very friendly and I have already been in touch with some of students. The Spanish language school in Benalmadena, was perfect.
Located in a small town, on the coast in Andalucia, Benalmadena offers a lovely sea side location and the language school provides a small school environment, just right for those who want a Spanish school offering a quieter pace of life and plenty of sunshine!
The Spanish language school in Benalmadena was established in 1976 and offers Spanish language courses, year round, in it's main college for adult students (18 yrs plus). Adults are offered general language courses (at all levels of ability), DELE preparation and Combined Courses. In addition there is a Spanish Teenage programme offered (Easter & Summer) in Benalmadena, which opened in 1990.
The Spanish School is located on the top floor of a modern building, with large, bright classrooms (12 in total), all equipped with audio-visual facilities and air conditioning. There is also a library with a wide range of reading material available, a multimedia room with free internet access, a 25-seat cinema used for conferences and seminars as well as for Spanish- version movies, a large sunshade terrace with a small cafeteria, common room, teachers' room, etc.
Students love the local area, amongst pleasant, low level villas with good local amenities and an easy stroll down to the beaches and marina.


Leaders Shefford Town & Campton were held at home for the first time on Saturday.
Their lead in the Premier Division supported by Sportsform was narrowed to five points with a 2-2 draw against AFC Oakley M&DH who were extending their unbeaten run to five games.
Jim Burnside plus an own goal from Robbie Alder marked the home scoresheet against M&DH replies via Elijah Hukin and Laurence Bentham.
Back into second are Caldecote who won 2-1 to end Sharnbrook's unbeaten home record. Liam Murray and Sam Julian were on their scoresheet whilst Niko Balde netted for the home side.
Just a point behind in third are Crawley Green Reserves who made it three wins on the bounce with a 4-2 victory at Ickwell & Old Warden.
Rob Newman, Jay Champkins, Graham Clarke and Temitaye Akerele all found the back of the net against home replies via a Danny Fox penalty and a strike from Scott Alleyne.
Also climbing the table into fourth are Wilstead, 3-0 home winners over Kempston Rovers Development thanks to a brace of goals from Colin Marshall and a single strike from Shaun Winconek.
Dropping to fifth are Wootton Blue Cross after they were beaten 3-1 at home by Cranfield United who remain unbeaten. Josh Bamford with a brace was joined on the United scoresheet by Ali Smith against a lone home reply via Kane Griffith.
Stevington who netted the afternoon's biggest victory with a 5-2 home win over Flitwick Town. Martin Danabrega led the way with a hat-trick and was joined on the scoresheet by Keiran Souter and William Pinney to bring up the nap hand against Town replies via a Ritchie Bevan penalty and Josh Reynolds.
Elsewhere following five straight defeats Queens Park Crescents shared the spoils in a 1-1 home draw with Marston Shelton Rovers. Dominic Parrella scored for the Crescents and Wes Burrows for the Rovers.
Wixams are now back on top of Division One supported by O'Neills after they claimed their fourth away win on the bounce with a 4-0 victory at Flitwick Town Reserves - who were being beaten on home soil for the first time this season.
Grant Tibbett, Gareth White, Charlie Battrams and Alex Summers all found the back of the net.
Whilst the biggest win of the day belonged to Lea Sports PSG, 6-1 winners at Sandy. Ali Mohiyaldeen, Nathan Prime and Joe Hennem all scored twice against a lone home reply from Matt Johnson.
Elsewhere The 61 FC Luton Reserves lifted themselves off the bottom of the table with their first win of the season, 2-1 at home over Biggleswade FC Reserves. Kwaku Aning and Steve Peterson netted the vital goals against one in reply via Bahador Chizari.
The biggest win of the day belonged to Shefford Town & Campton Reserves who ran out 8-2 home winners over Luton District & South Beds League side Sporting Lewsey Park. Sam King and Josh Lummis both fired home hat- tricks alongside single goals for Chris Clark and Cameron Sharp.
Also netting a home win over Luton District & South Beds League opposition were Riseley Sports with a 3-2 victory over Christians in Sports thanks to goals from James Jermyn, Mitchell Crook and Josh Poole.
Yet the hat-trick of wins over Luton District & South Beds League opposition was foiled by St Joseph's who won 2-1 at Totternhoe Reserves for whom Tom Howson netting their lone goal.
They will be joined in the hat by Cople & Bedford SA who, after being deadlocked 0-0 at AFC Kempston Town & Bedford College Reserves, won the penalty shoot-out 4-2 thanks to successful kicks from Paul Nigro, Ben Marwood, Gary Canzano and Lucas Banton.
Alas it was the end of the road for Henlow who were beaten 3-0 at United Counties League side Potton United Reserves.
Renhold United Reserves are four points clear in Division Two and unbeaten in their last six outings following their 4-3 home victory over second place Henlow Reserves. Michael France with a brace was joined on their scoresheet by Thomas O'Neil and Jonny Richens. The Henlow replies came via Russell Ayles, James Campbell and Jonny Rowe.
However the biggest win of the day belonged to Westoning who ran out 6-1 winners at Luton Leagrave AFC. The game in effect was all over inside the opening 30 minutes has they raced into a 3-0 lead with goals from Ben Hymus, Lewis Wilson and Aaron Turner. Hartley Coleman briefly gave the home side hope by reducing the scoreline to 3-1 but Westoning were to restore their three goal advantage when Wilson netted his second of the afternoon just three minutes from the break.
The home side's cause was not helped when Del Fry saw red in the 54th minute and, after Cody O'Neil made it 5-1 four minutes later, the scoring was completed with the last kick of the game which saw Wilson net from the penalty spot to claim his hat-trick.
Also hitting six goals were now third place Bedford Albion, 6-2 home victors over Marston Shelton Rovers Reserves. Paul Babbington scored four times alongside single strikes from Ali Bucknall and Pasquale Grasso against Rovers replies via Gary Russon and Lewis Steele.
Elsewhere AFC Oakley M&DH Reserves retained their 100% home record with a 2-1 win over Atletico Europa. Kade Walker and James Grandidge grabbed the M&DH goals against one in reply from Paddy Alimanji.
Whilst Elstow Abbey took their winning sequence to four games with a 2-1 home win over Caldecote Reserves, James Reynolds netting both goals against a lone reply via Nicky Thomas.
Also winning on home soil were CS Rovers, 2-0 winners over Sundon Park Rovers. Connor Darocha netted both goals.
Whilst after three straight defeats Wixams Wanderers stopped the rot with a a 1-1 home draw against Wilstead Reserves. Jordan Benham was the Wanderers marksman against a Wilstead reply from Sean Harrod.
Kempston Athletic are now just a point clear in Division Three after they suffered only their second defeat of the season, 5-0 at home against second from bottom side Stevington Reserves who were claiming their first win. Harry Austin led the way with a hat-trick and the nap hand was brought up by 55 year old Clinton Tatham who claimed a brace.
Sitting in second are Black Swan who, after three straight defeats, returned to winning ways with a 9-1 victory at Dinamo Flitwick. Danny McGovern with a hat-trick was joined on the Swan's scoresheet by Corey Richards who netted twice plus Freddie Bowman, Ashley Gardener, Joseph Grossi and Levi Yearwood. The lone home reply came from Ray Hall.
Just goal difference behind the Swan in third are Harlington who returned to winning ways on home soil with a 3-1 victory over bottom side Wootton Village. Peacre Tracey, Ryan Martin and Farrell Whitman netted the goals against a lone Village reply from Shuan Quinn.
In fourth are Clifton who lost their 100% home record when held to a 3-3 draw by Bedford Albion Reserves. Cameron Lewis with a brace plus a single goal for Ross Donaldson marked the home scoresheet against Albion replies via Josh Emmerton, Brandon Hung and Tom Wade.
For fifth place Lidlington United Sports it was a return to winning ways on home soil with a 3-0 victory over Sandy Reserves thanks to goals from Edward James, Nick Scott and Dale Crocker.
Whilst seventh place Flitwick Town A remain unbeaten on home soil this season after beating White Eagles 3-1, Dave Lyons netting twice and Jason Strong once against a lone Eagles reply from Hincu Ion.
There was also home joy for Caldecote A who won for the first time on home soil this season in beating Shefford Town & Campton A 5-2, who were losing their 100% away record. Tony Norman with a hat-trick and Jake Bruce brough up the nap hand against Town replies via Bradley Vooght and Jacob Jeavon.


Born as a bet, this brand proved to be more than winning, opening in no time 8 points of sale in central Italy. Behind the success there is a bold entrepreneur and a strategy all about transparency and simplicity.
The creation of this brand has given attention to the point of sale, the digital strategy and the printed material: all this to reach a target as wide as possible.
The aesthetics of ancient taste arrives on the tables of all Romans.


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LinkedIn is always changing—and the savviest jobseekers change right along with it
LinkedIn Masterclass
Learn how to make the most of LinkedIn as you search for jobs and build a career
Guest speaker: Omar Garriott
Date: Tuesday, Oct. 18
Time: 12 noon–1 pm (Sask. time)
Location: Online via Zoom (link to be sent out prior to event)
Register on Eventbrite
Info: alumni.artsandscience@usask.ca
PLEASE NOTE: Webinars will be recorded and made public on the College of Arts and Science YouTube channel once the webinar is over.
Join from PC, Mac, iPad, iPhone or Android device:
Passcode: 58609880
Join by telephone
Dial (for higher quality, dial a number based on your current location):
US: +1 312 626 6799 or +1 346 248 7799 or +1 646 558 8656 or +1 669 900 9128 or +1 253 215 8782 or +1 301 715 8592
Canada: (639) 638-7474 or +1 647 374 4685 or +1 647 558 0588 or +1 778 907 2071 or +1 204 272 7920 or +1 438 809 7799 or +1 587 328 1099 or +1 613 209 3054
LinkedIn is always changing—and the savviest jobseekers change right along with it.
It can be hard to know where to focus on LinkedIn. That's why we've partnered with a former LinkedIn employee to give you access to their online masterclass. You can expect to:
Participate in live roleplays
Make changes to your LinkedIn profile that will immediately start attracting recruiters
Work on techniques that will 10X increase your chances of getting interviews at any organization
Omar Garriott leads Qualtrics's education business. Previously, he was the senior product marketing manager at LinkedIn, managing content, campaigns and product development for students and universities. Prior to his role at LinkedIn, he led iPad marketing to schools (K-12 and higher ed) for Apple, and led product launches to students and schools for Adobe. Garriott has also spent time as a third-grade teacher for Teach for America, as a non-profit fundraiser and a lobbyist with national social enterprise PeerForward, as a summer consultant in FSG's education practice and in social marketing with Porter Novelli. He earned a bachelor's degree with distinction in commerce from the University of Virginia and an MBA from University of California, Berkeley, Haas School of Business. He also holds a Non-profit Management Executive Certificate from Georgetown University and sporadically tweets from @socialchanger.
Senior Art Show
A citywide exhibition for Grade 12 students to showcase their artist talent
Walter C. Murray Lecture: Origins of Meaning
Philosopher Dr. Dorit Bar-On (PhD) discusses how language might have evolved from animal communication


Wentyfalls Stn ready to accept trains when you have built it.
This kit is of a station that would look good on any train layout. I have made it look like a NSWGR station but is isn't. though it could be? The kit has a footprint of 470mm x 145 mm x 72 mm high. Like all our kits, this one is easy to construct using our locking tab system. Just paint your station any colour scheme you like and away you go, glueing it together. The see through paling fence, locomotive or track do not come with the kit. The see through paling fence is available on our web site as well "No PF 00003" Easy to follow instructions supplied. Wentyfalls Stn ready to accept trains when you have built it.


This wine red Nike Blazer SB were much sought after by sneakerheads since it has been released. The shoe based on white rubber, while upper is released on wine red color except yellow accent on swooshes. Just released as a pair of great shoes.
Copyright © 2016 - 2019 www.soccerts.com. All Rights Reserved. Processed in 0.1302018 second(s),2 Queries.


In a world awash with adaptations of young adult dystopian fiction, The Hunger Games series still towers above its brethren. The first two outings were fresh and thrilling experiences, full of colour, action and the newly minted star power of Jennifer Lawrence. She remains the chief attraction, just as her character Katniss Everdeen seems to hold the fate of Panem in her palm, but the youthful charm is wearing off. Hampered by the commercially astute and artistically questionable decision to split the finale in two, Mockingjay – Part 1 finds moments of entertainment in an otherwise incomplete whole.
Re-entry back into Panem is a slow one. Katniss, now hiding out with the rebels of District 13, is stuck down a drab bunker of green jumpsuits and looming concrete. She's the number one weapon the rebels have, and alternative President Alma Coin (Julianne Moore) and Plutarch Heavensbee (Philip Seymour Hoffman) are determined to deploy her to full effect. She's the rallying cry that will unite the oppressed against the tyranny of the Capitol. But Katniss is not for coaching. Reading from a cue card reaps dismal results. A woman of action, she needs to get in the thick of it to tap into the compassionate resolve that made her a symbol for so many.
At least this provides returning director Francis Lawrence with an excuse to remove her from the gloomy bunker. With newcomer Natalie Dormer's film crew tracking her every step, she visits hospitals, weeps over the sight of massacres and morphs into Rambo to shoot down bombers with her trusty bow. Lawrence (Francis) handles the set pieces effectively, capturing the zip that served the first two films so well. Katniss aside, there's a tense rescue mission and the spectacular destruction of key landmarks to pass the time.
The problem comes outside the action. With the story stretched so thinly, there's not enough to entertain between explosions. Key characters are reduced to little more than cameo appearances. Newly sober Haymitch (Woody Harrelson) swings by for a brief pep talk while blissfully materialistic stylist Effie (Elizabeth Banks) is shackled in her jumpsuit strait jacket. Meanwhile, the dashing young men of the franchise amble about in the background with very little opportunity to dash. Finnick (Sam Claflin) barely speaks until the end while Gale (Liam Hemsworth) moons after Katniss like a giant puppy spurned of attention.
Everyone is pushed to one side to make way for a central relationship that badly lets the film down. When she should be getting on with the job at hand, Katniss instead chooses to mope over wet blanket Peeta (Josh Hutcherson), stuck in the Capitol with President Coriolanus Snow (Donald Sutherland). Usually so assured, she becomes drearily flaky, even refusing to deliver a simple message to camera when fears over Peeta's safety overwhelm her. The previous films were allowed to keep the story small, restrained by the geographical confines of the arena. Now, when it needs to widen out, Mockingjay is painfully restricted by a love story that keeps the focus on an unconvincing romance. That thousands are dying all around seems of little concern.
Thank God for Sutherland as the sneeringly smug Snow. He delights in his opportunity to play the slick, callous leader. There's something of Alan Rickman circa Robin Hood: Prince of Thieves in his mischievous game playing, only without the childish tantrums. He's also a stark reminder that there are bigger things at stake than the endangered flame of teenage love. A chilling conversation with Katniss elevates the climax while proving that the real fun lies between Sutherland and the impossibly charismatic Lawrence (Jennifer) who manages to rise above her character's increasingly irritating traits.
Mockingjay – Part 1 isn't a bad film. It's just not a complete one. Very much the first part of an overextended finale, the strain starts to show on everyone as they struggle to fill time before the credits. In brief chunks, there's enough to entertain without ever really impressing. When the story concludes next year, let's hope a little more happens.


It's official!! John Mellencamp and Meg Ryan are engaged!!
Ryan posted the news on Instagram which you can see below. Mellencamp and Ryan were first linked in 2011, called it quits in 2014, and then reconciled in 2017.
This will be the fourth marriage for John Mellencamp. Ryan was previously married to Dennis Quaid.


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WWF's Corporate Sustainability Summit Calls for an End to Single-Use Plastics
WWF-Hong Kong's collaboration with the business sector to reduce single-use plastics, including plastic tableware and plastic packaging, was emphasised in a panel discussion at WWF's Corporate Sustainability Summit on Friday 30 November at the Hong Kong Convention and Exhibition Centre. Over 150 representatives from companies and corporations pledged to continue embracing sustainability in their business practices, including CO2 emission reductions. Companies that participated in WWF-Hong Kong's LCMP (Low Carbon Manufacturing Programme)and LOOP (Low-carbon Office Operation Programme) avoided over 171,000 tonnes of carbon emissions in the past year against a business-as-usual scenario. It would take over 7,400,000 trees one year to absorb that amount of carbon.
"Corporate partnerships play a vital role in our work to influence the course of sustainability and conservation in Hong Kong and across the region," said Peter Cornthwaite, CEO of WWF-Hong Kong. He thanked the attending companies for their support to raise awareness of key environmental issues and mobilize consumer action through various events, communications and campaigns. "Our partnerships with the business sector aim to change behaviour and drive consumer results that would not be possible otherwise, delivering direct conservation impact on key issues, in priority places, and in support of endangered species like the tiger and Chinese white dolphin."
Dr Patrick Yeung, Manager, Oceans Conservation, WWF-Hong Kong, noted it is essential to forge collaborative partnerships to implement sustainability as a core value of business planning, product development, company policy and operational practices to address issues such as single-use plastics. "To truly grapple with plastic pollution, all relevant stakeholders must first take action to change from single-use to reusable lifestyles and stop the flow of plastic waste into ecosystems. WWF supports a holistic, systems-based approach that engages with stakeholders along the entire value chain, beginning with manufacturing, through use and consumption, to disposal and waste management."
WWF-Hong Kong will continue to work together with the business sector to the reduce the most pressing threats to Hong Kong's biodiversity, and together find solutions to conservation challenges such as the protection of our oceans, issues of marine litter, over-fishing and climate change.
WWF's second annual Corporate Sustainability Summit took place on Friday 30 November 2018 at the Hong Kong Convention and Exhibition Centre.
© WWF-Hong Kong
Welcome remarks by the CEO of WWF-Hong Kong, Mr. Peter Cornthwaite.
(From left to right) Guests of panel discussion on Leading a Pledge on No Single Use Plastics: Dr Caroline Wolff, Sales Director APAC, Coatings, Adhesives & Specialties, Covestro (Hong Kong) Limited; Dr Patrick Yeung, Manager, Oceans Conservation, WWF-HK; Ms. Doris Lee, Sustainability Manager, E-pac Packaging Services Company Limited; Mr. Toiny Pang, Environmental Sustainability Manager, Marks & Spencer; Ms. Suzanne Cheung, Lead Consultant, Conservation Policy, WWF-HK, Ms. Wing Lee, Founder, WeUse; and Moderator Ms. Karen Ho, Head of Corporate and Community Sustainability, WWF-HK
Building a future in which humans live in harmony with nature.
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© All photos, graphics and images on this site remain the copyright of WWF, unless otherwise noted, and should not be downloaded without prior permission. . .


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Esik (, também escrito Issyk, Yesik e Yesyk) é uma vila na Almaty (região) do Cazaquistão, o centro administrativo do Enbekshikazakh (distrito). Localizado no rio Issyk, ao pé do Tian Shan, 53 km a leste de Almaty e 112 km sobre as montanhas de Lago Issyk Kul. População:

Localidades na região de Almaty

redpajama_set_name: RedPajamaWikipedia


Domenikos Theotokopoulos was a famous painter in the late 1500s. He was better known as El Greco–"The Greek". (And thank goodness. I'm sure art teachers everywhere are grateful.) There are many aspects of his art and personality that are hard to categorize. He didn't fit in with anything in his time. But what stood out to me when I read about him was a story about his friend finding him on a summer day sitting in a darkened room, because, as he said, he found the darkness more conducive to thought than the light of the day, which disturbed his "inner light".
I believe I know what El Greco was getting at. If you have lived any amount of time on this earth, you likely have faced (or will face) some sort of situation that makes you feel like you sit alone in darkness. Maybe El Greco had some sort of charmed life, but most of us don't have to artificially manufacture darkness. It finds us. And at times, shuts us in with pressingly close walls. In total darkness you naturally look for light. It's when we feel the dark that we truly understand how important it is to seek light. Any light. And even the smallest flame becomes enough simply because it's all you have. When El Greco sat alone in darkness he was searching. He knew there was treasure to be found and he knew where to look: In the dark. Sometimes the greatest treasures are buried in the dark.
Sometimes the greatest treasures are buried in the dark.
What we focus on in times of darkness is critical. Times of intense trials break many people. They stumble aimlessly; groping at whatever gives stability. They resort to false forms of strength: anger, hostility, bitterness. But then there are others who refuse to accept the dark, and focus instead on finding the light. They are the ones who find true strength. They are the ones who stand. They learn lessons that give purpose to the heartache and in turn, help others. They fight the darkness. And they find they most definitely can win. They are not overcome. They are overcomers. By staring directly into the flame, the darkness quiets. If you are in the dark, realize there is a choice to be made and it can be quite literally life or death.
The quote above was originally written in Greek (possibly before that it was Aramaic but leave it be for now…lol) "Overcome" can be translated as "apprehend" or "comprehend". The darkness doesn't understand the light. Light is the darkness' weakness. That's why even a small flame is enough to dispel the blackest dark. In fact, sometimes it is because of the darkness that we even notice there is a flame.
We all have the light in us. We are all made in the image of God. This does not make us gods any more than a son who bears the image of his father would be his father. It does however, make us very much like him. God is light. And in him is no darkness. Our very DNA reflects the fingerprints of our maker. I believe our spirits know this and it is why we all strive to find the "inner light".
Yes, we have a spirit. Actually, "You are a spirit. You have a body." (Thank you C.S. Lewis.) Your spirit is the essence of who you are. It is your very breath. The Hebrew word for spirit is the same as wind or breath. This is also true in Greek. When the ancient prophet Ezekiel was told to prophesy over the old dry bones and make them live again, he did. They all came together (a trick many a paleontologist might envy!) Then sinews formed on them, and flesh, and skin covered them, "but there was no breath in them". They were not alive. He was told then "prophesy to the breath. Prophesy, and say to them, Come from the four winds, O breath and breathe on these slain that they may live." And it was only at that time that the breath came into them and they lived and stood on their feet. Our spirit gives us life. Let me show you now how our spirit also gives us light.
I like to dabble in philosophical science. Does that sound super smart? If it does then that's awesome because I think I made it up. Stick with me though. Bioluminescence. It's a big word. It's actually pretty simple though: Bioluminescence is when living things emit light. The next time you see a firefly, comment to you friends about the beauty of its bioluminescence and they will be extremely impressed."Lumen" means light and do you notice the suffix "escence"? It comes from the same Latin origin as our English word essence—both words refer to the true substance, being, or spirit of something. Bioluminescent animals are uniquely made to give off light. Much the same way we are made to give off light in a spiritual sense. The essence of who you and I are –our very breath in us- the DNA that makes us God's children, is part of that light. We were made to shine. Not succumb to darkness.
The cancer cells became so full of light, they dissipate. The light literally shines in the darkness and the darkness does not overcome it. The darkness doesn't win. Cancer is never funny but picturing all those life destroying, heart shattering, family devastating cancer cells "committing suicide" is hilarious. And scientists are finding out all it takes is LIGHT? Light always fights the dark. Whether it is sickness and disease growing within or just plain "dis-ease" being pushed in from the outside. Refuse the dark. Find your bioluminescence and focus it.
But did you notice how it wasn't just the light alone that actually killed the cancer cells? They needed a photosensitizer–something to make the cells more sensitive to the light. We can find our inner light, focus on it, and get by pretty good, but we are not able to truly conquer our dis-eases without that photosensitizing agent. It is not good enough to just illuminate our darkness, especially when we have the power to conquer it completely.
Did you ever hear of the four men who wrote four different books about a guy they all knew? They mainly did it to document all the crazy things that happened when they hung out together. The first three tell the story start to finish. But the last one was written by the guy's best friend so the way he told the story was quite different. Not just facts and order. It focused more on who this man was. His essence. It is masterfully written poetic prose, and at the same time philosophical madness. In the beginning he describes his friend by saying "In him was life, and the life was the light of men. The light shines in the darkness, and the darkness has not overcome it." Then he said his friend was the "true light who gives light to everyone." He is saying his friend IS light. He truly believed his friend was God. And honestly I do too. I believe the inner light in all of us becomes a flood lamp when we tap into the original source. Think of him as the photosensitizing agent. If you never read anything else in the Bible, I'm telling you read John's book about Jesus.
The world does not make it easy. There seems to be a cosmic battle for each of us. It is a fight to live this life and continue to choose to give back good instead of evil when the world is closing in on you with darkness. But we do in fact choose. When chaos confronts (and it will all of us) we choose light or dark. Will you seek the light and find the Light of the world? If you have the Light of the world, you have everything you need to get through any darkness you face.
I do sit in the dark like you say, to meditate at times. When into to my gong meditations I put a mask over my eyes to make it completely dark. It is easiest to get into a deep meditation then, which usually sheds light on done issue I have.
Very insightful post. Keep going to the light. Xo.