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Items where Division is "Faculties > Faculty of Life Sciences > Department of Botany" and Year is 2008
Group by: Creators | Item Type | No Grouping
Afreen, Bushra (2008) Identification and Characterization of a Virus Causing Carrot Red Leaf Disease. (Masters thesis). Aligarh Muslim University.
Ahmad, Naseem (2008) In Vitro Studies on Growth and Morphogenesis of Vitex negundo L. and Ruta graveolens L. (PhD thesis). Aligarh Muslim University.
Alam, Md. Masidur (2008) Effect of Various Growth Regulators on the Performance of Periwinkle 'Sadabahar' (Catharanthus roseus L.). (Masters thesis). Aligarh Muslim University.
Alka, (2008) Studies on the Induction of Variability for Quantitative Traits in Lentil (Lens culinaris Medik). (Masters thesis). Aligarh Muslim University.
Ansari, Nasiruddin (2008) Studies on the Effect of Fungi and Root-Knot Nematode on the Productivity of Chickpea. (PhD thesis). Aligarh Muslim University.
Ashraf, Mohd. Shaikhul (2008) Effect of Chromium and Nickel on Pathogenic Potential and Management of Rotylenchulus reniformis and Fusarium solani on Chickpea (Cicer arietinum L.). (PhD thesis). Aligarh Muslim University.
Bhat, Aasia Rashid (2008) Studies on the Plant Parasitic Nematodes Associated with Some Ornamental Plants. (Masters thesis). Aligarh Muslim University.
Bhat, Saima (2008) Studies on Radial Growth of some Commercially Important Plants. (PhD thesis). Aligarh Muslim University.
Goyal, Sonu (2008) Studies on the Induction of Polygenic Variability in Urdbean (vigna Mungo (L.) Hepper). (Masters thesis). Aligarh Muslim University.
Gupta, Honey (2008) Cytomorphological Studies in Cichorium intybusl (Asteraceae) Treated with Ethyl Methane Sulphonate and Methyl Methane Sulphonate. (Masters thesis). Aligarh Muslim University.
Khan, Mohd. Nasir (2008) Response of Linum usitatissimum L. to the Application of GA3, N, P, Ca and Mg. (PhD thesis). Aligarh Muslim University.
Khan, Moin Ahmad (2008) Anatomical Studies on the Activity of Vascular Cambium and Production of Vascular Tissues in Some Tropical Trees. (PhD thesis). Aligarh Muslim University.
Khan, Moin Ahmad (2008) Anatomical studies on the activity of vascular cambium and production of vasular tissues in some tropical trees. (Doctoral thesis). Aligarh Muslim University.
Laur, Smita (2008) Allelopathic Impact of Ageratum Conyzoides Linn. Towards Some Crop and Weed Plants. (PhD thesis). Aligarh Muslim University.
Mazid, Mohd (2008) Studies on the Status of Nutrients, Vitamins and Hormones in Tomato (Lycopersicon Lycopersicum) Infected with the Root-knot Nematode (Meloidogyneincognita). (Masters thesis). Aligarh Muslim University.
Mukhtar, Seemab (2008) In Vitro Morphogenetic Studies and Plant Regeneration of a Medicinal Legume, Cutoria ternatea L. (Masters thesis). Aligarh Muslim University.
Mustabeen, (2008) Assessment of Impact of Air Pollutants on Some Pulse Crops. (Masters thesis). Aligarh Muslim University.
Niyaz, Tabassum (2008) Studies on Eclipta Alba Infected with Root-Knot Nematode (Meloidogyne incognita). (PhD thesis). Aligarh Muslim University.
Perveen, Rubina (2008) Studies on the Effects of EMS Alone and in Combination with Dimethyl Sulfoxide in the Induction of Variability in Vicia faba L. (Masters thesis). Aligarh Muslim University.
Rather, Gh. Mohd (2008) Studies of Plant Community Structure in Reution to Heavy Metal Pollution of Soil at Aligarh. (Masters thesis). Aligarh Muslim University.
Rather, Mohmad Azhar (2008) Studies on Organic Soil Amendments for Management of Plant Parasitic Nematodes. (PhD thesis). Aligarh Muslim University.
Rehman, Farha (2008) Studies on the Plant Signals and Responses to Herbivory. (Masters thesis). Aligarh Muslim University.
Rizvi, Rose (2008) Evaluation of Industrial Particulate Wastes against Root Knot Nematode Meloidogyne Javanica on Eggplant (solanum melongena L.). (Masters thesis). Aligarh Muslim University.
Saba, Irm (2008) Biomanagement of Root-rot and Root-knot Disease Complex of Balsam. (Masters thesis). Aligarh Muslim University.
Shahroz, Bushra (2008) Response of Some Vegetable Crop to Waste Water. (PhD thesis). Aligarh Muslim University.
Sharma, Monika (2008) Studies on the Effect of Certain Mutagens on Cytomorphological Characters of Broad Bean (Vicia faba L.). (Masters thesis). Aligarh Muslim University.
Siddique, Iram (2008) Studies on in Vitro Regeneration Potential of Mucuna pruriens L. and Ocimum basiucum L. (PhD thesis). Aligarh Muslim University.
Singh, Kangabam Suraj (2008) Management of Root-Knot Nematode, Meloidogyne Incognita by Different Plant Products on some Leguminous Crops. (PhD thesis). Aligarh Muslim University.
Singh, Minu (2008) Influence of Triacontanol, Nitrogen and Phosphorus on the Growth, Yield and Quality of Ginger (Zingiber officinale Rosc.) and Turmeric (Curcuma longa L.). (PhD thesis). Aligarh Muslim University.
Singh, Minu (2008) Influence of Triacontanol, Nitrogen and Phosphorus on the Growth, Yield and Quality of Ginger Zingiber officinale Rosc and Turmeric Curcuma longa L. (PhD thesis). Aligarh Muslim University.
Singh, Priyanka (2008) Biocontrol of Meloidogyne incognita by Plant Growth Promoting Rhizobacteria on Tomato. (Masters thesis). Aligarh Muslim University.
Singh, Sarvajeet (2008) Coordination of Sulfur and Nitrogen in the Protection of Photosynthesis, Growth and Yield of Cadmium-Treated Mustard (Brassica juncea). (PhD thesis). Aligarh Muslim University.
Sisodia, Swapnal (2008) Allelopathic Effect of Croton Bonplandianum Baill. Towards Some Weed and Crop Plants. (PhD thesis). Aligarh Muslim University.
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Q: Getting IO exception when running Linux command in Java I'm writing a java app in windows that connect to a Linux machine through SSH.
I'm getting the following exception:
java.io.IOException: Cannot run program ssh: CreateProcess error=2, cannot find specified file.

Code:
Process p = Runtime.getRuntime().exec("ssh root@xxx.xxx.xxx.xxx ls");

While running this command in cmd everything is working, but not through java. SSH is in my Path environmental vars.

A: You need an interpreter to understand ssh command:
Process p = Runtime.getRuntime().exec(new String[]{"cmd", "ssh root@xxx.xxx.xxx.xxx ls"});

redpajama_set_name: RedPajamaStackExchange


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Post-Modern Aura
Trade Cloth – $54.95
Charles Newman (Author)
Gerald Graff (Preface by)
5 1/2 x 9
The Act of Fiction in an Age of Inflation
Charles Newman
"Inflation affects literary occupations and preoccupations quite as much as it does financial scrip." Starting from this premise, Charles Newman ventures forth on an irreverent, wide-ranging discussion of the "Post-Modern" attitude in fiction, culture, and sensibility. Newman questions the "revolutionary" claims of avant-garde novelists and literary theorists, but he is no less critical of the arguments of neoconservatives, neorealists, and advocates of "moral fiction." Newman argues that neither of these groups has confronted the unprecedented break with tradition entailed by an economics and culture of inflation. A combination of cultural critique, literary criticism, economic forecast, and historical jeremiad, The Post-Modern Aura is finally a positive statement, celebrating "The Act of Fiction" and suggesting how the forces which have been devaluing it might be overcome.
Charles Newman was educated at Yale and Oxford Universities. From 1964-75 he taught at Northwestern University where he founded and edited Tri-Quarterly. Subsequently he taught at Johns Hopkins University and Washington University in St. Louis. He has been the recipient of Fulbright, Guggenheim, Rockefeller and National Endowment grants, as well as the Zabel award for A Child's History of America. His novels include New Axis, The Promisekeeper, There Must be More to Love than Death, and White Jazz.
"Charles Newman's The Post-Modern Aura is a real eye-opener; the best thing I've read on the state of American culture in a long time, and grounded, moreover, in a tough-minded analysis of the economics of publishing--something unheard of in the annals of literary criticism." --Christopher Lasch, author of The Culture of Narcissism
"Newman's book is a hybrid, a combination of literary criticism, cultural analysis, political polemic, economic forecast, and historical lament. . . . Newman makes us feel that the future of the arts is very much involved in the health of the culture at every level, and he manages this without falling into the reductive equations that describe novels or paintings as reflections of this thing or that." --Robert Boyers, Salmagundi

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\section{Introduction}

One of the central challenges in mathematical biology is understanding mechanisms involved in development processes. Within the context of developmental biology, the emergence of large scale spatial structure, has often been theoretically explored through a common framework of deterministic partial differential equations defining reaction-diffusion systems
\cite{Murray2008,Hochberg2003,Turing37}.
While current frameworks may explain  a variety of phenomena in development, they can also suffer from over-simplification \cite{McKane2014}, with the additional caveat that finding theoretical models both consistent with mechanism-based knowledge and capable of predicting observed patterns is a highly complex task, suffering from both model and parameter fine tuning \cite{Butler2011,Xavier2013}.

Many models that describe pattern formation assume parameters are constant; however, this deterministic assumption is not suitable for certain conditions. Some systems are better suited towards a stochastic approach. When a system is coupled to external and stochastic drivers, then the parameter values can change. The stochastic driving is often represented by stochastic parameters, that is a parameter which is drawn from a certain distribution at each time step or spatial point, and referred to as {\it extrinsic noise} below.
This contrasts with most previous work on stochastic pattern formation, referred to as {\em intrinsic noise}, which assumes low copy number and it does not assume external drivers as the source of noise \cite{Schumacher2013,McKane2005,McKane2014,Biancalani2011,woolley2012effects,woolley2011influence}.

The structure of intrinsic noise is often taken to be be highly constrained and in particular uncorrelated in time, leading to white noise representations. For instance, if the noise is due to low copy number induced dynamics, Gaussian white noise forcing emerges from the chemical Langevin equation approximation  to the chemical master equation \cite{Gillespie2000}. Even with such constraints
there is a rich diversity of noise-induced phenomena, such as spatio-temporal pattern formation \cite{Schumacher2013,woolley2012effects}, stochastic oscillations \cite{McKane2005}, metastability \cite{McKane2014}, waves \cite{Biancalani2011} or enhanced oscillation amplitude \cite{Dauxois2009}.

However, when the source of the noise is {\em extrinsic}, other forms of noise are permissible. In particular, the defining special properties of white noise
may in general be relaxed and hence stochastic forcing can emerge with  non-trivial temporal correlations, often termed {\em coloured noise}. Our objective is to show how extrinsic noise influences
spatio-temporal reaction-diffusion patterns, in particular by  developing  power spectral methods to analyse the impact of coloured noise.

To proceed, we first note deterministic pattern formation reaction diffusion systems in biological applications with $n$ interacting biochemical species take the form \cite{Turing37,Murray2008}
\begin{equation}
  \frac{\partial \bm{x}}{\partial t} = \bm{D}\frac{\partial^2 \bm{x}}{\partial s^2} + \bm{f}(\bm{x})
  \label{GenericRD}
\end{equation}
in one spatial dimension, where $\bm{x} \in \mathbb{R}^n_{\geq 0}$ is a vector of $n$ chemical concentrations. The models we consider require the term $\bm{f}(\bm{x}) = (f_1(\bm{x}),\dots,f_n(\bm{x}))^T$ to be a vector of rational functions. The rational functions describe the underlying reaction network between the species
and $\bm{D}(\partial^2 \bm{x}/\partial s^2)$ with $\bm{D} = \text{diag}(D_1,\dots,D_n)$ describes the diffusion of $\bm{x}$. Let there be $M$ chemical reactions in the spatially homogeneous system whose dynamics are described by $\partial \bm{x}/\partial t = \bm{f}(\bm{x})$. Each reaction is parametrised by a reaction rate $k_i > 0$ and therefore we have a vector of reaction rates $\bm{k}=(k_1,\dots,k_M) \in \mathbb{R}^M_{> 0}$. Throughout this paper we restrict our analysis to {\em mass action kinetics} \cite{Horn1972}, rendering $\bm{f}(\bm{x})$ a vector of polynomials and we use homogeneous Neumann boundary conditions
\begin{equation}
  \frac{\partial \bm{x}}{\partial s}\biggr\rvert_{s=0,L} = 0\:,
\end{equation}
where $L$ is the domain length. Note that, although our work is only applied to systems of one spatial dimension, the theory can easily be extended to an arbitrary number of spatial dimensions.

Counter-intuitively, it has been shown that under certain conditions \cite{Murray2008} diffusion can drive an otherwise spatially uniform stable state to instability. Such unstable systems, which can form stable patterns, such as stripes or spots, are called {\em Turing Systems} \cite{Turing37,woolley2017turing}.
To avoid the complications of bistable dynamics, and later stochastic analogues such as switching between steady states, we impose the constraint that the spatially homogeneous solution has a unique stable steady state, $\bm{x}^*$, such that $\bm{f}(\bm{x}^*) = 0$ \cite{Murray2008}. To find models which have this property we use techniques from real algebraic geometry \cite{Muller2016} which provides simple tools, which ensure there exists only a single steady state in a model. Since Turing's work in 1952 many biological patterning systems have been suggested to be Turing systems \cite{Castets1990,Cartwright2002,Kondo2010}.

Fundamentally, the application of a set of partial differential equation (PDE) models for  a  biochemical reaction system assumes that the species of interest are in high enough concentration to allow continuum modelling. By contrast, when the number of particles in the biological system is low, intrinsic stochasticity of the system must be included in the model  \cite{Schumacher2013,woolley2012effects,McKane2005,McKane2014,Biancalani2011}, which typically yields studies that investigate  the impact of white noise.

However, as mentioned, stochasticity in biological systems can also arise from extrinsic noise and hence temporal variations in parameter values \cite{Picco2017}. Thus, instead  we generalise the deterministic system to include {\em stochastic} parameter values. Specifically, we focus on the effect of stochastic fluctuations in the constant term, the ``inflow'' term, of the chemical reaction network $\bm{f}(\bm{x})$. As the extrinsic noise can arise from a vast number of different mechanisms, such as varying experimental conditions \cite{Lenive2016}, it is largely free of microscopic constraints, especially the absence of temporal correlation.
However, the impact of correlated external noise has, to the authors' knowledge,  received little theoretical modelling attention. Thus we proceed to develop a framework to study external coloured noise forcing of the above deterministic system for pattern formation in biologically motivated scenarios, analysing how temporal correlation in noise, as described by colour, impacts self-organisation properties.

The paper is organised as follows.
In section \ref{Theory} we introduce required notions of chemical reaction network theory to select a class of models relevant to our framework, i.e. those whose number of steady states is unaltered by changes in parameter values, temporal or otherwise. We then introduce stochastic inflow parameters and describe their connection coloured noise. In section \ref{Computations} we highlight the impact of noise colouring on the spatio-temporal patterns formed by example of the Schnakenberg system. In particular, we discuss noise arising from stochastic subnetworks and varying experimental conditions. Our numerical results are discussed in section \ref{Conclusion} where we summarise the distinguishable differences between the various noise colours.

\section{Theoretical Results}\label{Theory}

In this section we introduce the theoretical background of this paper.
In subsection \ref{CRNT} we introduce chemical reaction networks and use results from real algebraic geometry to select models where changing a parameter does not effect the uniqueness of the steady state of the chemical reaction network. Networks that satisfy a unique steady state are required for our framework.
In subsection \ref{RandomInflows} we define stochastic inflows and describe their potential sources. In subsection \ref{PowerSpectra} we first show how the generic law of mass action reaction-diffusion system can be approximated to highlight the mathematical equivalence of external and internal noise in our framework. Next, we introduce power spectra as an analytic tool to study spatial and temporal pattern formation and, finally, we calculate a general formula for power spectra which includes the effects coloured noise.

\subsection{Chemical Reaction Network Theory}\label{CRNT}

A central aim of chemical reaction network theory (CRNT) is to describe the properties of a chemical reaction network from its reaction graph alone \cite{Feinberg1987,Feinberg1988}. One such property is the capacity for multiple steady states. Define a chemical reaction network by the multi-set $\mathcal{N} = \{\mathcal{S}, \mathcal{C}, \mathcal{R}\}$, where $\mathcal{S}, \mathcal{C}, \mathcal{R}$ are defined below. We begin by embedding the network into $n$ dimensional Euclidean space $\mathbb{R}^n$ by associating a basis vector $\bm{e}_i$ to each chemical species $X_i$ such that $X_1\to \bm{e}_1 = (1,0,0,\dots)^T$, $X_2\to \bm{e}_2 = (0,1,0,\dots)^T$ and so on. Let $\mathcal{S} = \{X_1,\dots,X_n\}$ be the set of all chemical species in the network, then a generic reaction can be expressed as
\begin{equation}
  \sum_{i=1}^n \alpha_i X_i \xrightarrow{k} \sum_{i=1}^n \beta_i X_i\; .
  \label{GenericReaction}
\end{equation}
The constants $\alpha_i$ and $\beta_i$ are {\em stoichiometric coefficients} which give information about how many molecules of $X_i$ are consumed and produced in each reaction. By letting $X_i\to \bm{e}_i$ we can formulate a {\em reaction vector} describing the consumption, or production, of a species $X_i$ in a reaction
\begin{equation}
  \bm{r} = \sum_{i=1}^n (\beta_i-\alpha_i)\bm{e}_i\; .
\end{equation}
If an entry of $\bm{r}$ is negative, then a species is consumed, whilst if an entry of $\bm{r}$ is positive, then a species is produced.

Denote the set of all reaction vectors in a network by $\mathcal{R} = \{\bm{r}_1,\dots,\bm{r}_M\}$. To complete the description of a chemical reaction network in terms of sets and their embedding into Euclidean space we introduce the notion of {\em complexes}. Complexes are linear combinations of species which appear on the left or right hand sides of reaction vectors. In equation \eqref{GenericReaction} the two complexes are $C_1 = \sum_{i=1}^n \alpha_i X_i$ and $C_2 = \sum_{i=1}^n \beta_i X_i$, where $C_1$ is the {\em reactant complex} and $C_2$ is the {\em product complex}.
Let the set of all complexes be $\mathcal{C} = \{C_1,\dots,C_l\}$. The reaction network, $\mathcal{N}$, is therefore a directed graph with vertex set $\mathcal{C}$ and a directed edge between vertices $C_i$ and $C_j$ if and only if $C_j-C_i \in \mathcal{R}$, with the same embedding, $X_i\to \bm{e}_i$.
Note that the description of the reaction network {\em does not} include any notion of rate constants $k$, however, many results in chemical reaction network theory include the reaction rates explicitly as a positive vector $\bm{k} = (k_1,\dots,k_m)^T\in\mathbb{R}^M_{>0}$.

In this paper we study the influence of noise on reaction-diffusion systems that are able to produce patterns in a well-defined parameter region. Critically, we will see that coloured noise is able to have both a constructive influence outside of this previously defined parameter region (i.e. the noise stabilises patterns where we would not expect them deterministically), as well as a destructive influence  on patterns which normally would arise. In particular, we would like to exclude the intrinsically stochastic effect of {\em stochastic switching}, which occurs for a system that has multiple steady states in some parameter region. Stochastic switching describes the phenomenon that a chemical reaction network can jump from one steady state to another when subject to finite stochastic perturbations. To avoid this scenario we use a network structure based tool described in \cite{Muller2016} which a priori excludes multistationarity.

Take a chemical reaction network $\mathcal{N}$ and embed its complexes and reactions into $\mathbb{R}^n$. Then define the matrices $m_1 = (\bm{r}_1 \bm{r}_2 \cdots \bm{r}_M)$ for $r_i\in \mathcal{R}$ and $m_2^T = (\bm{b}_1 \bm{b}_2\cdots \bm{b}_r)$ where $\{\bm{b}_i\}$ is the set of reactant complex vectors of $\mathcal{N}$. Let $diag(k_1,\dots,k_M)$ be the diagonal matrix of reaction constants. Using the law of mass action the dynamics of the species concentrations can be described by
\begin{equation}
  \frac{d\bm{x}}{dt} = \bm{f}(\bm{x}) = m_1\:diag(k_1,\dots,k_m)\: \bm{x}^{m_2}
\end{equation}
where $\bm{x}^{m_2} = (x_1^{b_{11}}x_2^{b_{12}}\cdots x_n^{b_{1n}},\dots,x_1^{b_{M1}}x_2^{b_{M2}}\cdots x_n^{b_{Mn}})^T$. Note that the reactant complexes and the reactions are structural properties of the reaction graph.
Further, let $\bm{a} \in \mathbb{R}^n$ and define its sign vector $\sigma(\bm{a}) = \{-,0,+\}^n$ by applying the sign function to each component of $\bm{a}$. Therefore, the $i^\text{th}$ component of the sign vector $\sigma(\bm{a})_i = \text{sign}(a_i) \in \{+,0,-\}$.

The link from network structure to multistationarity is outlined in \cite[Theorem 1.4]{Muller2016} and it concerns the injectivity of the map  $\bm{f}: \bm{x}\mapsto d\bm{x}/dt$. If $\bm{f}(\bm{x})$ is injective then there exists a unique vector $\bm{x}^*\in \mathbb{R}^n$ such that $\bm{f}(\bm{x}^*) = 0$. In other words the network is monostationary. The conditions for an injective $\bm{f}(\bm{x})$ hold for all parameter values $\bm{k} = (k_1,\dots,k_M)$. A corollary of \cite[Theorem 1.4]{Muller2016} is that $\bm{f}(\bm{x})$ is injective if
\begin{equation}
  \text{ker}(m_2) = \{0\}\quad\quad \text{and}\quad\quad \sigma(\text{ker}(m_1))\cap\sigma(\text{im}(m_2)) = \{0\}\: .
  \label{Inject}
\end{equation}
Note, that this condition depends on the network structure, specifically, the spaces spanned by the kernel of the reaction matrix, $m_1$, and the image of the source complex matrix, $m_2$, but not on the parameter values.

\subsection{Stochastic Inflow in Chemical Reaction Networks}\label{RandomInflows}

We will now show how chemical reaction networks (CRNs) that satisfy injectivity (preclusion of multistationarity) can be applied to parameter stochasticity, in particular, to the case of stochastic inflows.

Much effort in the CRNT literature investigates the effect of inflows into the chemical system, especially when the network is changed to a ``fully open system'' in which every species has an inflow reaction \cite{Banerjee2014,Joshi2014,Felix2016,Banaji2018}. By contrast, here we change the nature of the model by generalising the inflow reaction rates to stochastic processes, rather than simply adding inflow processes as deterministic reactions. Sometimes these inflow reactions are the called``inputs'' in engineering applications and the concentrations are the ``outputs'' and, therefore the study of chemical reaction networks with varying inflows can be rephrased into a study of the ``input-output'' relation as has been the approach in previous work \cite{Frank2013}.

Consider a chemical reaction network $\mathcal{N}$. An {\em inflow} reaction for species $X_i$ is manifested in the reaction graph as
\begin{equation*}
  \emptyset \xrightarrow{k_{\text{in}}} X_i\;.
\end{equation*}
Usually, the work in CRNT assumes the reaction rate $k_{\text{in}}$ to be constant in time (and space). However, the zero complex, $\emptyset$, symbolises a process not included in the model such as the the production of $X_i$ by another network, which we call an ``auxiliary network'', or a mechanical addition of $X_i$ to an experimental setup. Both mechanisms are often subsumed into $\emptyset$ and usually approximated as constant or ``perfect'' inflow, however, they can exhibit intricate dynamics. We model the dynamics of ``non-perfect'' inflows by a stochastic process whose origin can be two-fold.
\begin{enumerate}
  \item[(a)] {\em Stochastic sub-systems}: We assume that the inflow into the (deterministic) reaction diffusion system is provided by the output of another chemical reaction network (with a unique fixed point). When the number of particles in the {\em auxiliary network} is large the system will be in a steady state and the inflow rate $k_\text{in}$ will simply be proportional to the steady state value of a species of the auxiliary system. However, when the particle number in the auxiliary system is low, stochastic fluctuations cannot be ignored and, while still proportional to the concentration of a species in the subsystem, the actual influx parameter $k_\text{in}$ will be a stochastic process $K_\text{in}(t)$.
  \item[(b)] {\em Experimental fluctuations}: In chemical engineering it is assumed that inflow of chemicals into a reactor is a perfectly deterministic process, however, due to mechanical or other experimental imperfections the inflow into a reaction can vary stochastically. This again renders the influx parameter $k_{\text{in}}$ into a stochastic process $K_\text{in}(t)$.
\end{enumerate}

\begin{figure}
  \includegraphics[width=.5\textwidth]{Figure1.png}
  \caption{\label{Concepts} The two mechanisms contributing to stochastic inflows. The boxes on the left are usually treated as black boxes resulting in some constant inflow modelled by the parameter $k_\text{in}$. We differentiate between experimental fluctuations, symbolised by a correlation $\tau$ in (a) and auxiliary networks described by $f_\text{aux}(\bm{y})$ in case (b).}
\end{figure}

Whereas the two sources of parameter noise are conceptually different, their mathematical description is the same.
Consider a stochastic process $K(t)$ with an underlying distribution $\mathcal{D}(t)$. Therefore, at every time $t$ we have $K(t)\sim \mathcal{D}(t)$.
To be biologically relevant (i.e. to ensure all chemical concentrations are non-negative at every $t$) we require $\mathcal{D} = 0$ for all $K(t) < 0$.
To simplify the following mathematical analysis we approximate the distribution of $K(t)$ as a {\em truncated Gaussian}, since we would like to connect our analysis to the internal noise case.
Therefore, we can describe $K(t)$ as
\begin{align}
  K(0) &= k,\nonumber\\
  \Delta K &= K(t) - \mathbb{E}\left[K(t')\right] = \frac{1}{\sqrt{\Omega}} \sqrt{g(t,t')}\xi,\nonumber\\
  K(t) &= 0\qquad\text{if }\Delta K < -\mathbb{E}\left[K(t')\right],\nonumber\\
  \intertext{such that:}
  \mathbb{E}[K(t)] &= k\quad\forall t \geq 0,
\end{align}
where $\xi \sim N(0,1)$, $\Omega$ is positive constant and $g(t,t')$ is a positive function.
We assume that the standard deviation of the Gaussian perturbation to the mean parameter value is small such that $K(t)<0$ is a rare event.
Due to the negligible truncation, we approximate the moments of the truncated Gaussian as the moments of the full Gaussian such that $\langle K(t)-k\rangle = 0$ and $\langle (K(t)-k)(K(t')-k)\rangle = \Omega^{-1} g(t,t')$.
In the remainder of this paper we will show how various functions $g(t,t')$ can arise in mathematical modelling, especially when stochastic auxiliary networks are considered.

We can further connect the parameter $\sigma$ to physical quantities of the underlying chemical reaction network by again considering the sources of stochastic inflow. We assume that the origin of the stochastic inflow is stochastic auxiliary networks or experimental imperfections. Therefore, in the limit of either a perfect experiment or a deterministic auxiliary network the inflow should be deterministic and $\langle(K(t)-k)(K(t')-k)\rangle = 0$ implying $\Omega \to \infty$. For stochastic auxiliary networks the quantity $\Omega$ which parametrises the size of the stochastic fluctuations can be related to the {\em system size} \cite{Kampen2007}. Similarly to the system size expansion, we require $\Omega$ to be sufficiently large. Large $\Omega$ helps to maintain a positive solution of the stochastic partial differential equation, although, positivity cannot be guaranteed when the noise is correlated.

Next, consider a chemical reaction network $\mathcal{N}$ in which a subset of species has an inflow reaction. Denote the set of species with (stochastic) inflow by $\mathcal{S}_\text{in} \subseteq \mathcal{S}$. If there is more than one (stochastic) inflow $|\mathcal{S}_\text{in}|>1$ the stochastic process will be multi-dimensional Brownian motion. The description of the stochastic process outlined in this section still applies in this case, if we have $\bm{K}_\text{in}(t)$ as the vector of stochastic inflow such that
\begin{align}
  \mathbb{E}[\bm{K}_\text{in}(t)] &= \bm{k}_\text{in}\nonumber
  \intertext{and}
  \bm{K}_\text{in}(t)-\mathbb{E}\left[\bm{K}_\text{in}(t')\right]&= \bm{\eta}\nonumber
  \intertext{with}
  \langle\bm{\eta}\rangle &= 0\nonumber\\
  \langle\bm{\eta}\bm{\eta}^T\rangle &= BG(t,t') = \{B_{ij}G_{ij}(t,t')\}
  \label{Correlations}
  \end{align}
and $B$ representing the potential covariance between the stochastic inflows.

Since our assumptions made the inflow process additive and uncoupled to the species, the stochastic reaction network can be described by the system of  stochastic partial differential equations (SPDEs)
\begin{equation}
  \frac{\partial \bm{x}}{\partial t} = D\frac{\partial^2 \bm{x}}{\partial s^2} + \bm{f}(\bm{x}) + \frac{1}{\sqrt{\Omega}}\bm{\eta}\; ,
  \label{StochNet}
\end{equation}
where $\bm{\eta}$ is a vector of stochastic processes such that its support is $supp(\bm{\eta}) = \mathcal{S}_\text{in}$.

Note that in our limit randomising inflows leaves $\mathcal{N}$ as well as $\bm{f}(\bm{x})$ invariant, and hence, the steady state structure of the system; i.e. if we start out with an injective function $\bm{f}(\bm{x})$ then adding stochasticity will not change this injectivity.

\subsection{Power Spectra for Stochastic Inflows}\label{PowerSpectra}

To fully classify the patterns arising from the addition of stochastic inflows we calculate the power spectra of the patterns. Power spectra are an analytic tool showing which spatial and temporal frequencies are present in a pattern \cite{Adamer2017,Schumacher2013,Woolley2011a}. Peaks in power spectra give information about dominant frequencies and, hence, about oscillatory behaviour of a system in space and time.

First, we linearise equation \eqref{StochNet} about the fixed point $\bm{x} \sim \bm{x}^* + \overline{\delta \bm{x}}$. Where $\overline{\delta \bm{x}}$ represent small perturbations; these should decay in the deterministic limit as the system will converge to the steady state outside the Turing parameter regime. Inside the Turing regime the system is assumed to converge to a stable pattern rather than a stable state which is not considered in this paper. Hence, our analysis is valid only {\em outside} the Turing pattern regime. Substituting the linearisation ansatz into equation \eqref{StochNet} and keeping lowest order terms only we get
\begin{equation}
  \frac{\partial \overline{\delta \bm{x}}}{\partial t} = D\frac{\partial^2 \overline{\delta \bm{x}}}{\partial s^2} + J|_{\bm{x}=\bm{x}^*}\overline{\delta \bm{x}} + \frac{1}{\sqrt{\Omega}}\bm{\eta}\; ,
  \label{StochNetLin}
\end{equation}
where $J|_{\bm{x}=\bm{x}^*}$ is the Jacobian of $\bm{f}(\bm{x})$ evaluated at the fixed point $\bm{x}^*$, which for notational convenience we will denote as $J$.

We make one further assumption, namely the scaling of the perturbation $\overline{\delta \bm{x}}$ with the parameter $\Omega$ controlling the magnitude of the stochastic input. Let $\overline{\delta \bm{x}} = \Omega^\alpha \delta \bm{x}$. Then, if $\alpha < -1/2$ and in the limit of $\Omega \rightarrow \infty$ the leading order term is the stochastic process $\bm{\eta}$ only, similarly, if $\alpha > -1/2$, then $\bm{\eta}$ could be neglected to leading order in $\Omega$. Hence, we let $\alpha = 1/2$ and equation \eqref{StochNetLin} simplifies to
\begin{equation}
  \frac{\partial \delta \bm{x}}{\partial t} = D\frac{\partial^2 \delta \bm{x}}{\partial s^2} + J\delta \bm{x} + \bm{\eta}\; ,
  \label{StochNetLinFinal}
\end{equation}

Note, that equation \eqref{StochNetLinFinal} is mathematically equivalent to a chemical Langevin equation of compartmentalised diffusion in the limit of the compartment size, $\Delta_s$, going to zero \cite{Smith1985}. However, in our derivation the source of the noise is {\em external}. To emphasise the mathematical connection with internal noise we will in fact discretise equation \eqref{StochNetLinFinal} into a finite number of compartments, such that for an $n$ species network with $\mathcal{K}$ compartments we have
\begin{subequations}
\begin{align}
  \delta \bm{x} &= [\delta x_1,\dots,\delta x_\mathcal{K},\delta x_{\mathcal{K}+1},\dots, \delta x_{2\mathcal{K}},\dots,\delta x_{n\mathcal{K}}]^T\:,\\
  \bm{\eta} &= [\eta_1,\dots,\eta_\mathcal{K},\eta_{\mathcal{K}+1},\dots, \eta_{2\mathcal{K}},\dots,\eta_{n\mathcal{K}}]^T\:,\\
  s &\approx s_i = i\Delta_s\;\;\;\; \text{where } i = \{1,\dots,\mathcal{K}\}\:,\\
  \frac{\partial^2 \delta \bm{x}(s)}{\partial s^2} &\approx \frac{\delta \bm{x}(s_i+\Delta_s) + \delta \bm{x}(s_i-\Delta_s) - 2\delta \bm{x}(s_i)}{(\Delta_s)^2}\:,
\end{align}
\end{subequations}
and the matrices $D$ and $J$ turn into block matrices such that
\begin{equation}
  D =
  \begin{pmatrix}
  \left[D_1\right]_{\mathcal{K}\times \mathcal{K}} & 0 & 0 & \cdots\\
  0 & \left[D_2\right]_{\mathcal{K}\times \mathcal{K}} & 0 & \cdots\\
  \vdots & \vdots & \vdots & \ddots
  \end{pmatrix}.
\end{equation}
The $ij^{\text{th}}$ block of $J$ is $J_{ij}I_{\mathcal{K}\times \mathcal{K}}$ to give
\begin{equation}
  J = \begin{pmatrix}
  \left[J_{11}\right]_{\mathcal{K}\times \mathcal{K}} & \left[J_{12}\right]_{\mathcal{K}\times \mathcal{K}} & \cdots\\
  \left[J_{21}\right]_{\mathcal{K}\times \mathcal{K}} & \left[J_{22}\right]_{\mathcal{K}\times \mathcal{K}} & \cdots\\
  \left[J_{31}\right]_{\mathcal{K}\times \mathcal{K}} & \left[J_{32}\right]_{\mathcal{K}\times \mathcal{K}} & \cdots\\
  \vdots & \vdots & \ddots
\end{pmatrix}.
\end{equation}
Compartmentalisation results in a system of $n\mathcal{K}$ coupled SDEs
\begin{equation}
  d\delta \bm{x} = A\delta \bm{x}\:dt + \bm{\eta}\: dt\: ,
  \label{StochEq}
\end{equation}
with $A= D/\Delta_s^2 + J$ , $\langle\eta_i(t)\rangle = 0$ and $\langle\eta_i(t)\eta_j(t')\rangle = B_{ij}G_{ij}(t,t')$.

To calculate the power spectra of the system we introduce the discrete cosine transform \cite{Briggs1995}
\begin{equation}
  f_\kappa = \Delta_s\sum_{j=1}^\mathcal{K} \cos[\kappa\Delta_s(j-1)]f(s_j)\: .
\end{equation}
The cosine transform incorporates  the von Neumann (no flux) boundary conditions, which are commonly used for reaction-diffusion systems, however, other boundary conditions can easily be implemented \cite{Briggs1995}. Due to the boundary conditions we require $\kappa = m\pi/l$ with $m \in \{0,1,2,\dots\}$. We refer to $m$ as the {\em spatial mode}. Note that the use of $(j-1)$ instead of $j$ which is due to the fact that the compartment labelling starts at $j=1$. Hence, by applying the spatial Fourier transform we reduce the system of $n\mathcal{K}$ SDEs \eqref{StochEq} to a system of $n$ coupled SDEs
\begin{equation}
  d\delta \bm{x}_\kappa = A_\kappa\delta \bm{x}_\kappa\: dt + \bm{\eta}_\kappa\: dt\: .
  \label{StockEqSpat}
\end{equation}

Finally, applying the temporal Fourier transform
\begin{equation}
  \tilde{f}(\omega) = \int_{-\infty}^\infty f(t)e^{-i\omega t}dt\: ,
\end{equation}
we get an expression for $\delta\tilde{\bm{x}}_\kappa(\omega)$. Note that the Fourier transform always exists if we consider the system to have a fixed point \cite{Woolley2011b}.
\begin{equation}
  \delta\tilde{\bm{x}}_\kappa(\omega) = \left[-A_\kappa-i\omega\right]^{-1}\tilde{\bm{\eta}}_\kappa(\omega)\: .
\end{equation}
Therefore, the power spectra of the pattern, $P_{\delta x}(\kappa,\omega)$, can be expressed as the diagonal elements of
\begin{equation}
   \langle \delta \tilde{\bm{x}}_\kappa(\omega)\delta \tilde{\bm{x}}_\kappa^\dagger(\omega)\rangle = \Phi^{-1}\underbrace{\langle\tilde{\bm{\eta}}_\kappa(\omega)\tilde{\bm{\eta}}_\kappa^\dagger(\omega)\rangle}_{N} \left(\Phi^{-1}\right)^\dagger\: ,
\end{equation}
where we defined
\begin{equation}
  \Phi = -\left[A_\kappa + i\omega\right]\: ,
\end{equation}
and $\dagger$ denotes the hermitian conjugate of a matrix.
In order to be guaranteed real power spectra we need $N = N^\dagger$.
\bk In the case of Gaussian noise we have
\begin{equation}
  N = \mathcal{F}_\text{cos}\left(\mathcal{F}(BG(t,t'))\right),
\end{equation}
where $\mathcal{F}_\text{cos}$ denotes the cosine transform and $\mathcal{F}$ the temporal Fourier transform. Note that when all temporal correlations are equal such that $G_{ij}(t,t') = g(t,t')$ the transform factor becomes
\begin{equation}
  N = \mathcal{F}_\text{cos}\left(B\right)\mathcal{F}(g(t,t')) = B_\kappa P_\text{Correlations}(\omega)\: ,
\end{equation}
where $P_\text{Correlations}$ is the power spectrum of the correlation function $g(t,t')$ and $B_\kappa$ is the $n\times n$ matrix of covariances.
For white noise we take $g(t,t')=\delta(t-t')/dt$ to reduce \eqref{StochEq} to the It{\^o} form
\begin{equation}
  d\delta \bm{x} = A\delta \bm{x}\:dt + \bm{\eta}\: \sqrt{dt}\: ,
\end{equation}
with $\langle \eta_i(t)\eta_j(t')\rangle = B_{ij}\delta(t-t')$ and, therefore, $P_\text{Correlations} = 1$. Hence, we see that for white noise the power spectra are
\begin{equation}
  \langle \delta \tilde{\bm{x}}_\kappa(\omega)\delta \tilde{\bm{x}}_\kappa^\dagger(\omega)\rangle = \left[\Phi^{-1}\langle\tilde{\bm{\eta}}_\kappa(\omega)\tilde{\bm{\eta}}_\kappa^\dagger(\omega)\rangle \left(\Phi^{-1}\right)^\dagger\right] = \left[\Phi^{-1} B_\kappa \left(\Phi^{-1}\right)^\dagger\right] = P_\text{white}\: .
\end{equation}
For general temporal correlations $g(t,t')$ we have
\begin{equation}
  \langle \delta \tilde{\bm{x}}_\kappa(\omega)\delta \tilde{\bm{x}}_\kappa^\dagger(\omega)\rangle = \left[\Phi^{-1} B_\kappa P_\text{Correlations}\left(\Phi^{-1}\right)^\dagger\right] = P_\text{white}P_\text{Correlations}\: ,
\end{equation}
such that the spectra of the noise colour appear as a multiplicative factor modulating the white noise spectra.

\subsection{Application to the Schnakenberg System}

We illustrate our analysis via the example of the Schnakenberg kinetics. The Schnakenberg system is an $n=2$ species reaction-diffusion system which, despite its apparent simplicity, exhibits a wealth of different behaviours in the deterministic \cite{Iron2004,Maini2012,Ward2002,Flach2007} as well as stochastic \cite{Schumacher2013,Woolley2011a} regimes. The Schnakenberg system is a fully open system and, hence, both species can be subject to stochastic inflow.
The system follows the reaction scheme
\begin{equation}
  X_2 \xleftarrow{k_2}\emptyset \xrightleftharpoons[k_{-1}]{k_1} X_1\: ,\qquad\qquad\qquad   2X_1+X_2 \xrightarrow{k_3} 3X_1\:.
  \label{Schnakenberg}
\end{equation}

The dynamics of the Schnakenberg system is described by the system of PDEs
\begin{align}
  \frac{\partial x_1}{\partial t} &= D_1\frac{\partial^2 x_1}{\partial s^2} +k_1 - k_{-1} x_1 + k_3x_1^2x_2 ,\nonumber\\
  \frac{\partial x_2}{\partial t} &= D_2\frac{\partial^2 x_2}{\partial s^2} + k_2 - k_3x_1^2x_2 .
\end{align}

First we check for a potential multistationarity in the Schnakenberg system by formulating the matrices $m_1$ and $m_2$.
\begin{equation}
  m_1 = \begin{pmatrix}
  1 & -1 & 0 & 1\\
  0 & 0 & 1 & -1
\end{pmatrix}
  \quad\quad\text{and}\quad\quad m_2 = \begin{pmatrix}
  0 & 0\\
  1 & 0\\
  0 & 0\\
  2 & 1
\end{pmatrix}.
\end{equation}
It is easy to check that $\text{ker}(m_2) = \{0\}$. Further, we have
\begin{equation}
  \sigma(\text{ker}(m_1)) = \{(+,+,0,0)^T,(-,0,+,+)^T\}\;\;\text{and}\;\; \sigma(\text{im}(m_2)) = \{(0,+,0,+)^T,(0,0,0,+)^T\}\:.
\end{equation}
Hence, applying the injectivity criterion \eqref{Inject} we see that the Schnakenberg system is monostationary for all positive parameter values and, therefore, stochastic fluctuations in the inflow parameters cannot trigger stochastic switching.

Next, we consider stochastic inflows which result in $k_i \to k_i + 1/\sqrt{\Omega}\eta_i$ for $i \in \{1,2\}$ and the equations
\begin{align}
  \frac{\partial x_1}{\partial t} &= D_1\frac{\partial^2 x_1}{\partial s^2} +k_1 - k_{-1} x_1 + k_3x_1^2x_2 + \frac{1}{\sqrt{\Omega}}\eta_1,\nonumber\\
  \frac{\partial x_2}{\partial t} &= D_2\frac{\partial^2 x_2}{\partial s^2} + k_2 - k_3x_1^2x_2 + \frac{1}{\sqrt{\Omega}}\eta_2.
  \label{SchnakStoch}
\end{align}

Linearising equations \eqref{SchnakStoch} about the steady state $x^*_1 = (k_1 + k_2)k^{-1}_{-1}$, $x^*_2 = k_2k_{-1}^2k_3^{-1}(k_1+k_2)^{-2}$ and discretising space we get a system of linear stochastic differential equations similar to the ones arising from the study of internal noise \cite{Schumacher2013}
\begin{align}
d\bm{x} &= \bm{Ax} + \bm{\eta},
\label{SchnakLin}\\
\intertext{with}
A &= \begin{pmatrix}
  \bm{a} & \bm{b}\\
  \bm{c} & \bm{d}
\end{pmatrix},
\end{align}
where $\bm{a},\bm{d}$ are tridiagonal matrices with diagonal elements
\begin{align}
  a_0 &= -2D_1/\Delta_s^2 -k_{-1}+2k_3x_1^*x_2^*,\nonumber\\
  d_0 &= - 2D_2/\Delta_s^2 - k_3x_1^{*2},\\
  \intertext{and off-diagonal (sub- and super-diagonal) elements}
  a_1 &= D_1/\Delta_s^2 \equiv d_u,\nonumber\\
  d_1 &= D_2/\Delta_s^2 \equiv d_v.
\end{align}
The matrices $\bm{b}$ and $\bm{c}$ are diagonal matrices with entries
\begin{align}
  b_0 &= k_2x_1^{*2},\nonumber\\
  c_0 &= -2k_3x_1^*x_2^*.
\end{align}

Note that the vector $\bm{x}$ has $2\mathcal{K}$ components. There are $\mathcal{K}$ components for species $x_1$ and $\mathcal{K}$ components for species $x_2$.

Hence, Fourier transforming equation \eqref{SchnakLin} in space and time we can compute the power spectra,
\begin{align}
  P(k,\omega) &= \left[A_\kappa+i\omega\right]^{-1}\langle \tilde{\eta}_\kappa(\omega)\tilde{\eta}_\kappa^\dagger(\omega)\rangle\left(\left[A_\kappa+i\omega\right]^{-1}\right)^\dagger,\\
  \intertext{with}
  A_\kappa &= \begin{pmatrix}
  a_0 + 2a_1\cos{k\Delta_s} & b_0\\
  c_0 & d_0+2d_1\cos{k\Delta_s}
  \end{pmatrix}.
\end{align}

In the following section we will discuss the effect of various noise correlations on the power spectra, and hence, the patterns generated.

\section{Computational Results}\label{Computations}

In this section we highlight the computational patterns generated by a Turing system with coloured noise.
In subsection \ref{NumericalMethods} we start with a brief discussion of the numerical methods used before we discuss the differences between parameter noise as considered in this paper and internal noise in subsection \ref{WhiteNoiseSection}. We then review systems with white noise which were mainly derived in \cite{Schumacher2013} and in the following subsections we proceed to relax the assumption of uncorrelated noise to see the effect of noise colouring. This section is divided into two parts, the first one treating stochastic processes which may arise from experimental set-ups, such as exponential broadening of the noise correlation (Ornstein-Uhlenbeck noise) or excitation of the zero frequency modes. The second part will treat sources of noise due to stochastic subnetworks and noise mixtures. An overview of our main results is given in Table \ref{TabResults}

\begin{flushleft}
\begin{table}[h!!!]
\resizebox{0.9\textwidth}{!}{\begin{minipage}{\textwidth}
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Colour} & \textbf{Correlation Function} & \textbf{Effect} & \textbf{Possible Origin}& \textbf{Section}\\
\hline
White & $1$ & N/A & internal& \ref{WhiteNoiseSection}\\
Ornstein-Uhlenbeck & $1/\left(\omega^2\tau^2+1\right)$ & suppresses small $\omega$ & experimental apparatus& \ref{OrUlSection}\\
Red & $1/\omega^2$ & excites small $\omega$ & deterministic & \ref{RedSection}\\
Predator-Prey & $\left(\alpha+\beta\omega^2\right)/\left(\omega^4 + \gamma\omega^2+\delta \right)$ & induces oscillations & stochastic subsystems & \ref{OscSection}\\
\hline
\end{tabular}
\end{minipage}}
\caption{An overview of our main results on various noise colours.}
\label{TabResults}
\end{table}
\end{flushleft}

\subsection{Numerical Methods}\label{NumericalMethods}

The system of SDEs \eqref{StochEq} is simulated by an Euler-Maruyama scheme \cite{Kloeden1995} with time step $\Delta t$ such that
\begin{equation}
    \bm{x}(t+\Delta t) = \bm{x}(t) + \bm{Ax}\Delta t + \bm{\eta}\Delta t.
\end{equation}
The important step of the integration is to find the correct vector $\bm{\eta}$.

The white noise and Ornstein-Uhlenbeck noise are generated by an auxiliary noise process. In particular, at teach time step the white noise is sampled from a multivariate Gaussian distribution $$\bm{\eta}\Delta t \sim \mathcal{N}(0, \bm{B}\Delta t).$$ The Ornstein-Uhlenbeck process is a generated by the stochastic differential equation $$\frac{d\bm{\eta}}{dt} = -\frac{1}{\tau}\bm{\eta} + \frac{\sqrt{\bm{B}}}{\tau}\bm{\xi},$$ where $\bm{\xi}$ is a standard white noise vector. Therefore, at time $t$ the vector $\bm{\eta}(t)$ is added to the to the system of SDEs. The Ornstein-Uhlenbeck ``auxiliary equation'' is integrated by an Euler-Maruyama scheme as described in \cite{Milshtein1994}.

To simulate power law noise for which, in general, no auxialiary SDE exists we use inverse transforms \cite{Timmer1995}. To generate a vector $\bm{\eta}(t)$ with correlations $\langle\bm{\eta}(t)\bm{\eta}^T(t')\rangle = \bm{B}g(t-t')$, we first use the fact that $\bm{\eta} = \sqrt{\bm{B}}\bm{\xi}$ where $\bm{\xi}$ may be correlated in time but not in space, i.e. $\langle\bm{\xi}(t)\bm{\xi}^T(t')\rangle = \delta_{ij} g(t-t')$. Then, let the power spectrum of $\xi_i$ be $1/\omega^\alpha$ as described in subsection \ref{RedSection} and use the algorithm of \cite{Timmer1995} to create a time series. Multiplication of $\bm{\xi}(t)$ with $\sqrt{\bm{B}}$ gives the desired noise process, $\bm{\eta}(t)$, which can be added to the SDE system.

Auxiliary networks as in \ref{OscSection} can be simulated by either method and in this paper the auxiliary network input is generated by using the inverse transforms technique.

\subsection{White Noise}\label{WhiteNoiseSection}

First, we consider external white noise. In this special case the noise vector $\bm{\eta}$ is just a Wiener process with correlation matrix $B$. This case is mathematically identical to the case of internal noise in the weak noise limit as studied in \cite{Schumacher2013}. The main differences between internal noise and the parameter noise considered in this paper are the amplitude of the noise and the exact forms of the covariances of the stochastic processes $\bm{\eta}$.

Both approximations (the weak noise limit, as well as our ``truncated Gaussian'' approximation) assume that the stochastic effect is a perturbation to the deterministic limit. However, as derived in \cite{Schumacher2013}, the covariance matrix of the stochastic processes $\bm{\eta}$ is determined by the microscopic processes whereas for external noise, whose origin can be manifold, the covariance matrix is be arbitrary. For mathematical simplicity we choose the covariance matrix
\begin{equation}
  B_\kappa = \begin{pmatrix}
  1 & 1\\
  1 & 1
  \end{pmatrix}.
\end{equation}

Throughout the remainder of this paper we fix the parameter values to
\begin{align}
  k_1 &= k_2 = 10.0,\nonumber\\
  k_3 &= 0.01,\nonumber\\
  k_{-1} &= 20.0,\nonumber\\
  \intertext{with diffusion coefficients}
  D_1 &= 10^{-4},\nonumber\\
  D_2 &= 10^{-2},
  \label{ParamVals}
\end{align}
 which results in steady state concentrations of $x_1^* = 1$, $x_2^* = 1000$. The parameters chosen are not generic, but they represent a particularly interesting point in parameter space. The parameters are in the (stable) oscillatory regime of the Schnakenberg system and outside the Turing space. The main function of the noise will be to temporarily move the system into the Turing regime. Note that the system has large inflows (and outflows) compared to the chemical reaction parametrised by $k_3$ which allows for larger variations in the noise translating into larger pattern amplitudes. The modifications to the power spectra due to coloured noise are generic and apply all points in parameter space. However, the visibility of these modifications depends on the point in parameter space and the correlation matrix.  Similarly, the results on amplitude of the patterns are parameter-dependent. To simulate the system, we discretise the space into $40$ compartments of width $\Delta_s = 0.0025$ (which gives a total domain length of $L = 0.1$).

\begin{figure}
\centering
 \subfloat[The deterministic solution of the Turing system at the given parameter point\label{fig1:det}]{\includegraphics[width=.5\textwidth]{Pattern_u_Det.pdf}}
 \subfloat[A stochastic realisation with white noise\label{fig1:stoch}]{\includegraphics[width=.5\textwidth]{UWhiteNew.pdf}}\newline
 \subfloat[The power spectra of the analytical calculations (dashed lines) and their simulated curves averaged over 50 realisations of length $T=1000$ and subsystem size $\Omega=5000$. We attribute the differences in peak height to the finite time steps used.\label{fig1:power}]{\includegraphics[width=.7\textwidth]{WhitePowerSpec_u.pdf}}
  \caption{An overview of the deterministic and white noise behaviour of the system with parameters as in \eqref{ParamVals}. The peak in \ref{fig1:power} at zero spatial mode, $m=0$, indicates temporal oscillations which result from deterministic limit cycle oscillations.  Due to the choice of reaction kinetics the temporal oscillations of species $x_1$ and $x_2$ are in phase.}
  \label{White}
\end{figure}

Simulating equation \eqref{SchnakLin} under the influence of white noise and with the given parameter values we obtain Figure \ref{White} and its corresponding averaged power spectrum.
The power spectrum shows the temporal frequencies, $\omega$, and spatial modes, $m$, present in the pattern. The amplitude of the oscillations is about $3\%$ of the steady state value for species $x_1$ whereas for species $x_2$ it is much lower. Therefore, we assume that any patterning of $x_2$ is not generally measurable and, therefore, we focus our attention on $x_1$. The prominent temporal oscillations with no spatial dependency manifest themselves as a sharp peak in the power spectrum. This is due to the fact that the chosen parameter point is in the oscillatory regime of the Schnakenberg model. Upon close inspection slight spatial variations in the pattern of $x_1$ can be seen, but with white noise these are not very pronounced. In the following sections we show how noise colour can enhance spatial modes, create additional oscillations or even create a stable pattern.

\subsection{Ornstein-Uhlenbeck Noise}\label{OrUlSection}

Next, we investigate the effect of exponentially correlated noise also known as Ornstein-Uhlenbeck noise \cite{Hanggi1995}. This stochastic process has a finite correlation time $\tau$ which we interpret as a response time. Consider an experiment in which the inflow rate is highly sensitive to the ambient temperature. Further, suppose this temperature undergoes random fluctuations about a regulated mean value. Therefore, the correlation time $\tau$ could represent the average response time of the temperature regulator. The correlation time $\tau$ could also represent an intrinsic relaxation time of the system such as found in motility induced phase separation of bacteria \cite{Cates2015}.

We make the simplification that all the temporal correlations in equation \eqref{Correlations} originate from Ornstein-Uhlenbeck processes with the same correlation time, $\tau$, such that
\begin{equation}
  g_{ij}(t,t') = g(t,t') = \frac{1}{2\tau}e^{-\tfrac{|t-t'|}{\tau}}.
\end{equation}
Hence, it follows that
\begin{equation}
  \langle\eta_\kappa(\omega)\eta_\kappa(\omega)^\dagger\rangle = B_\kappa \frac{1}{\omega^2\tau^2 + 1}
\end{equation}
where $B_\kappa$ is the same matrix as in the white noise case and therefore $P(k,\omega)_\text{Ornstein-Uhlenbeck} = P(k,\omega)_\text{white}/(\omega^2\tau^2 +1)$.
Hence, the noise colouring will dampen temporal frequencies of $\omega \neq 0$ and, therefore, stabilise the pattern. Consequentially, in systems with Ornstein-Uhlenbeck noise we would expect any present stationary patterns that may be obscured by transient noise effects to be more prominent. We simulated the Schnakenberg system with Ornstein-Uhlenbeck noise and plotted the resulting patterns and power spectra in Figure \ref{OrUl}. The pattern of $x_2$ has the same characteristic as in the white noise case, except for a smaller amplitude. Interestingly, $x_1$ shows a very different behaviour to the white noise case. Clear spatial structures can be seen, in particular the phenomenon of polarity switching \cite{Woolley2011a,Schumacher2013}. A Turing pattern of mode $m=1$ is generated with a given polarity, namely, a minimum at $s=0$ and a maximum at $s=L$ or vice versa. Polarity switching describes the inherently stochastic phenomenon of a sudden change in polarity as can be observed in Figure \ref{OrUl}. The temporal oscillations, although still present, are reduced to a practically unobservable level. We attribute this to the fact that the exponential noise correlations dampen the oscillations present in the white noise system.

\begin{figure}
 \centering
 \subfloat{\includegraphics[width=.5\textwidth]{UOrUlNew.pdf}}
 \subfloat{\includegraphics[width=.5\textwidth]{VOrUlNew.pdf}}\newline
 \subfloat{\includegraphics[width=.47\textwidth]{OrUlSpec_u_crop.pdf}}
 \subfloat{\includegraphics[width=.47\textwidth]{OrUlSpec_v_crop.pdf}}
 \caption{The patterns and power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by an Ornstein-Uhlenbeck process and $\tau = 100$. Spatial patterns become visible in the Ornstein-Uhlenbeck case which can be attributed to its dampening effect on temporal oscillations. This can be observed as a visible excitation of the $m=1$ mode in the power spectrum of $x_1$ with the dashed line representing the analytical prediction. The well-known phenomenon of polarity switching \cite{Schumacher2013} is observed in the pattern of $x_1$. The spectra were averaged over 50 repetitions with $T=1000$ and subsystem size $\Omega = 100$.}
 \label{OrUl}
\end{figure}

\subsection{Power Law Noise}\label{RedSection}

By power law noise we mean a stochastic process whose frequency distribution follows a power law in Fourier space
\begin{equation}
  P_\text{power} = \frac{1}{\omega^\alpha}.
\end{equation}
Various types of power law noise are common in engineering and are defined by colours summarised in Table \ref{ColoursTab}. In this paper we study red noise, white noise (subsection \ref{WhiteNoiseSection}) and violet noise to illustrate the effects of a positive, zero and negative exponent, $\alpha$.
\begin{table}[h!!!]
\begin{center}
\begin{tabular}{|l|r|}
  \hline
  \textbf{Colour} & $\alpha$\\
  \hline
  Red & $\alpha = 2$\\
  Pink & $\alpha=1$\\
  White & $\alpha = 0$\\
  Blue & $\alpha=-1$\\
  Violet & $\alpha = -2$\\
  \hline
\end{tabular}
\end{center}
\caption{The various colours of power law noise. In this paper we focus on red and violet noise.}
\label{ColoursTab}
\end{table}

Red noise amplifies small temporal frequencies with $1/\omega^2 \to \infty$ as $\omega\to 0$. Therefore, the zero frequency behaviour dominates the pattern. As can be seen in Figure \ref{Pink}, due to the amplification of small $\omega$, the pattern becomes stable, even though the amplitude grows beyond the biologically viable (non-negative) bound indicating that non-linear effects need to be considered in the model.
To investigate this further, we look at the noise vector $\bm{\eta}$. For the simulation times chosen, $T=200$, the noise vector is independent of the time $t$ such that each $\eta_i$ has a constant, but random value. Therefore, the effect of red noise is a deterministic one and to get an accurate description of the system behaviour a full non-linear, deterministic compartment model with generic (random) inflow parameters needs to be analysed. The red noise case is similar to models which assume a cell-to-cell variability where a parameter is perturbed by a constant, sampled from a given distribution \cite{Lenive2016,Filippi2015}.
In contrast to conventional Turing theory in which the inflow parameter is constant throughout the domain, our analysis indicates that randomly varying inflows across the domain might actually facilitate pattern formation.

When investigating the power spectra we see that for species $x_1$ the simulations match the mathematical prediction. For $x_2$, however, we see that the peaks at the deterministic oscillation frequency have different heights. This indicates that either the oscillation amplitude is underestimated by the analytical prediction or, vice versa, the simulations exaggerate the oscillation amplitudes due to numerical limitations. Due to the agreement of prediction and simulations at all other points we disregard the different peak heights as numerical artefacts.

To illustrate the behaviour of the system on the other end of the colour spectrum we simulated violet noise which stabilised the temporal oscillations further and due to large power at large $\omega$ drove the oscillation amplitude in the linear treatment to non biologically viable values. Again, this indicates that violet noise cannot be treated in biological applications with a simple linear theory, but a full non-linear theory must be used. For violet noise the effect of the truncation of the Gaussian became too large to get an accurate prediction of the power spectra as can be seen in Figure \ref{Violet}. Further, while noise colours with positive exponent are actively studied in biology \cite{Szendro2001}, the potential emergence of blue or violet noise in biological systems is not clear.

\begin{figure}
    \centering
    \subfloat{\includegraphics[width=.48\textwidth]{UPinkNew.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{VPinkNew.pdf}}\newline
    \subfloat{\includegraphics[width=.48\textwidth]{PinkSpec_u_crop.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{PinkSpec_v_crop.pdf}}
  \caption{The patterns and power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by a red noise process. As predicted, the spectra are dominated by the behaviour the $\omega = 0$, which amounts to the stabilisation of a particular spatial mode. The species concentrations, however, go negative indicating that a full non-linear model needs to be used. The peak in the power spectrum for $x_2$ is a numerical artefact. The spectra were averaged over 50 repetitions with simulation time $T=200$. The subsystem size was $\Omega = 5000$.

  }
  \label{Pink}
\end{figure}

\begin{figure}
  \centering
    \subfloat{\includegraphics[width=.48\textwidth]{UViolet.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{VViolet.pdf}}\newline
    \subfloat{\includegraphics[width=.48\textwidth]{VioletSpec_u_crop.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{VioletSpec_v_crop.pdf}}
  \caption{The patterns and power spectra for $x_1$ (left) and $x_2$ (right) with noise generated by a violet noise process. The discrepancy in the power spectra originates from the truncation of the Gaussian process. The negative species concentrations indicate that a non-linear theory needs to be used to fully model a violet noise system. The spectra were averaged over 50 repetitions with simulation time $T=200$. The subsystem size was $\Omega = 5000$.}
  \label{Violet}
\end{figure}

\subsection{Stochastic Auxiliary Networks}\label{OscSection}

We now proceed to the second major source of random inflows, namely the dependence of a stochastic auxiliary network. An auxiliary network is a chemical reaction network which provides an input into the Turing system (see Figure \ref{AuxNetwork}). The auxiliary network is connected to the main network only via an inflow reaction and, if the auxiliary network reaches a steady state, it can be subsumed into the zero complex for practical modelling. If, however, the auxiliary network exhibits more complex dynamics such as deterministic or stochastic oscillations, then it needs to be treated as a part of the main network. We focus on auxiliary networks which are deterministically stable, but show stochastic quasi-cycles. The dynamics of such systems are modelled by using a correlated stochastic inflow parameter as outlined in section \ref{RandomInflows}.

\begin{figure}
  \centering
  \includegraphics[scale=0.5]{AuxNetwork.png}
  \caption{A schematic of an auxiliary network and its input to the main system. This is a specific example of scenario (b) of Figure \ref{Concepts}.}
  \label{AuxNetwork}
\end{figure}

As an illustrative example we use the predator-prey system described in \cite{McKane2005},
\begin{align}
  Y_1\xrightarrow{d_1}&\emptyset\xleftarrow{d_2} Y_2,\nonumber\\
  2Y_1\xleftarrow{p_1} & Y_1+Y_2\xrightarrow{p_2} Y_1.
  \label{PredPreyEq}
\end{align}
In \cite{McKane2005} it was shown that in the deterministic regime the system \eqref{PredPreyEq} has exactly one attractor for any choice of rate constants. Hence, as the inflow parameters $k_1$ and $k_2$ are proportional to the concentration of $Y_1$, they will have a constant value. However, when the copy numbers of $Y_1$ and $Y_2$ are small and stochastic fluctuations are important the predator-prey model \eqref{PredPreyEq} can exhibit so-called ``quasi-cycles'' which are stochastic analogue of limit-cycles and manifest themselves as peaks in the power spectra.

Consider this ``predator-prey noise'' in a Schnakenberg system. Suppose the inflows $k_1$ and $k_2$ in the Schnakenberg system depend on the presence of the chemical species $Y_1$, such that $k_1 \propto y_1$ and $k_2\propto y_1$, where $y_1$ denotes the concentration of $Y_1$.
Again, we assume $g_{ij}(t,t') = g(t,t')$ for mathematical simplification. In Fourier space the power spectrum of $Y_1$ is
\begin{equation}
  P_\text{Predator-Prey} = \frac{\alpha + \beta\omega^2}{\left(\omega^2+\Omega^2_0\right)+\Gamma^2\omega^2},
  \label{PredPreySpecEq}
\end{equation}
where we use the parameter values from \cite{McKane2005} $\alpha = 0.000384,\: \beta = \Gamma = 0.04,\: \Omega_0^2 = 0.016$.

Equation \eqref{PredPreySpecEq} has a peak at $\omega \sim 0.06$, so we expect peaks at a similar frequency for each non-zero spatial mode in the power spectrum of the Schnakenberg system.
The resulting patterns and the corresponding average power spectra are presented in Figure \ref{PredPrey}. The power spectra gained a second peak in the $m=0$ mode and peak in all modes $m>0$ at $\omega \sim 0.06$. Therefore, it can be seen that a deterministic parent system can inherit the dynamics of a stochastic auxiliary network and mix it with its own intrinsic dynamics.

\begin{figure}
  \centering
    \subfloat{\includegraphics[width=.48\textwidth]{UPredPreyNew.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{VPredPreyNew.pdf}}\newline
    \subfloat{\includegraphics[width=.48\textwidth]{PredPreySpec_u_crop.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{PredPreySpec_v_crop.pdf}}
  \caption{The power spectra for the species $x_1$ (left) and $x_2$ (right) with noise generated by a stochastic predator-prey network \cite{McKane2005}. The power spectra inherited a second peak from the subsystem which activates oscillatory modes on top of the deterministic oscillation frequency. The spectra were averaged over 50 repetitions and simulation time $T=200$ with $m$ indexing the spatial mode. The subsystem size was $\Omega = 100$.}
  \label{PredPrey}
\end{figure}

\subsection{Mixed Noise}

Next, we investigated the case of mixing noise processes, in particular, we consider an Ornstein-Uhlenbeck process which has a different correlation time for $x_1$ and $x_2$, $\tau_1$ and $\tau_2$ respectively. Then we let $\tau_2 \to 0$ in order to recover the white noise case for species two.

We introduce the auxiliary normal stochastic processes $\xi$ and hence, the spatial modes of the  noise $\eta_\kappa$ are described by \cite{Hanggi1995}
\begin{equation}
  \frac{d\eta_\kappa}{dt} = -\begin{pmatrix}
  \tfrac{1}{\tau_1}& 0\\
  0 & \tfrac{1}{\tau_2}
\end{pmatrix}\eta_\kappa + \begin{pmatrix}
  \tfrac{1}{\tau_1}& 0\\
  0 & \tfrac{1}{\tau_2}
  \end{pmatrix}b_\kappa\xi,
  \label{OrUlAuxEq}
\end{equation}
where $b_\kappa$ is a $2 \times 4$ matrix satisfying $b_\kappa b_\kappa^T = B_\kappa$. We Fourier transform \eqref{OrUlAuxEq} and define the matrix,
\begin{equation}
  \phi = \begin{pmatrix}
  \tfrac{1}{\tau_1}+i\omega & 0\\
  0 & \tfrac{1}{\tau_2}+i\omega
  \end{pmatrix}
\end{equation}
to give
\begin{equation}
  \eta_\kappa = \phi^{-1}\begin{pmatrix}
    \tfrac{1}{\tau_1}& 0\\
    0 & \tfrac{1}{\tau_2}
    \end{pmatrix}b\xi.
\end{equation}
Letting $\tau_2\to 0$ and computing the covariance matrix gives
\begin{equation}
  N = \langle\eta_\kappa(\omega)\eta_\kappa(\omega)^\dagger\rangle = \begin{pmatrix}
  B_{k,11}\tfrac{1}{1+\omega^2\tau_1^2} & B_{k,12}\tfrac{1}{1+i\omega\tau_1}\\
  B_{k,12}\tfrac{1}{1-i\omega\tau_1} & B_{k,22}
  \end{pmatrix},
\end{equation}
where $B_{k,ij}$ are the $ij^\text{th}$ elements of the $B_\kappa$ matrix. Note that $N$ is hermitian and therefore we expect real power spectra for the patterns of $x_1$ and $x_2$.

Simulating equation \eqref{StochNetLinFinal} with the noise process described by \eqref{OrUlAuxEq} gives rise to the mixed patterns seen in Figure \ref{Mixed}. Note, that computationally the limit $\tau_2\to 0$ is equal to setting $\tau_2$ to the time step $dt$. The patterns of both species appear similar to the pure Ornstein-Uhlenbeck patterns, however, with reduced amplitude.

\begin{figure}
 \centering
    \subfloat{\includegraphics[width=.48\textwidth]{UMixedNew.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{VMixedNew.pdf}}\newline
    \subfloat{\includegraphics[width=.48\textwidth]{MixedPowerSpec_u_crop.pdf}}
    \subfloat{\includegraphics[width=.48\textwidth]{MixedPowerSpec_v_crop.pdf}}
  \caption{The patterns generated for the species $x_1$ (left) and $x_2$ (right) with Ornstein-Uhlenbeck noise and $\tau = 100$ for $x_1$ and $x_2$ with white noise.The patterns are phenomenologically similar to the Ornstein-Uhlenbeck patterns, however, with reduced amplitude. The power spectrum of $x_2$ is a mixture of White noise and Ornstein-Uhlenbeck noise. The power spectra were averaged over 50 simulations with simulation time $T=1000$ and subsystem size $\Omega=5000$.}
  \label{Mixed}
\end{figure}

\subsection{Reduced Stochastic Inflows}

In the final subsection we instigate the case of only species $x_1$ being subjected to stochastic inflows. The species $x_2$ is assumed to only have deterministic inflow at a rate $k_2$ and $\eta_2 = 0$. Hence, the correlation matrices $N$ will be reduced to
\begin{equation}
  N = \begin{pmatrix}
  \langle\eta_{1k}\eta_{1k}^*\rangle & 0\\
  0 & 0
  \end{pmatrix}.
\end{equation}

In the noise processes studied in this paper the behaviour of the species is virtually unchanged and the patterning is robust with respect to disregarding cross-correlations.

\section{Conclusion}\label{Conclusion}

In this paper we investigated the effect of stochastic inflows on a deterministic reaction-diffusion system. We restricted the class of networks considered to monostationary systems and used results from real algebraic geometry to show how monostationarity is related to network structure. We then introduced a stochastic perturbation to an inflow reaction as a truncated Gaussian process with the expectation value at the deterministic inflow parameter. After linearising we computed the power spectra for arbitrary noise colours and showed that in simple cases the power spectra can be derived as a multiplicative factor. The remainder of the paper consisted of applying our analysis to the Schnakenberg system and highlighting the effects various noise colouring can have on a reaction-diffusion system.

We briefly restated the results from the white noise analysis, proceeding to add coloured noise and then demonstrating its effect by computing the power spectra. In particular, for the simple case when all species experience the same temporal correlations, the power spectra of the correlations appear as a multiplicative factor in the total power spectra. Hence, depending on the nature of the correlations they will suppress or excite temporal frequencies at all spatial modes.  Ornstein-Uhlenbeck noise has a Lorentzian frequency distribution and, therefore, suppresses positive frequencies. Power law noise can either completely stabilise or destabilise the pattern depending on the sign of the exponent, $\alpha$. This is due to the fact that for positive $\alpha$ temporal frequencies at $\omega = 0$ go to infinity and, therefore, all oscillations are suppressed which results in a stable pattern which resembles a pattern arising from simulations inside the Turing regime.

For the simulation times used in this paper, the noise vector was actually constant, and, therefore, deterministic methods could be used to study the stabilising effect of pink or red noise. The opposite is the case for negative $\alpha$ where the frequency contribution in the power spectrum increases as $\omega$ increases and oscillations are enhanced. However, certain aspects of the red, blue, or violet, noise cases cannot be explained by our simple analytic prediction. The biological significance and the correct analytic description of blue/violet noise could form a part of further work.

When turning to stochastic auxiliary networks we see that the reaction-diffusion system can inherit traits of the dynamics of the auxiliary network. In particular, we showed that when the auxiliary network exhibits a predator-prey dynamic with stochastic oscillations these oscillations can still be observed in the deterministic main network. We concluded the range of applications by considering mixed noise with Ornstein-Uhlenbeck input for species $x_1$ and white noise for species $x_2$. In this case, the system behaves similar to a system with pure Ornstein-Uhlenbeck dynamics and differences can only be found by a power spectral analysis. Finally, we observed in the cases considered that when only one species experiences stochastic inflow the patterns created are, except for potential special cases, similar to the ones when the inflow to both species is randomised. However, the extent to which such results may hold in generality is for further work.

Further directions could also include attempting to relate these studies to potential mechanisms of left-right symmetry breaking amplification in developmental biology, in particular the impact of induced Nodal production on one side of the embryonic node, which is hypothesised to be driven by ciliary fluid flows and also highly error-prone \cite{Blum2014}. It is generally asserted that the resulting interactions of the gene products Nodal and Lefty, which are major contenders as Turing morphogens \cite{Muller2012,Chen2002,Schier2003}, amplify this initial error-prone signal to generate robust patterning, driving downstream developmental left-right asymmetry. However, the ability of Turing systems to amplify the spatially localised, error-prone, and thus stochastic, influx of activator morphogen \cite[Figure 2]{Blum2014} to induce a robust symmetry breaking self-organisation, as well as any additional constraints required to do so, is theoretically untested. Thus examining the mechanistic basis of these postulates in this critical developmental biological process provides a fundamental application for the theoretical foundations developed here.

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Tashiyangtshi is situated on the small river valley. It is a small town but with fast growing population. The valley is popular for manufacture of bowls and containers made of turned wood, the specialty of the region.
It is Nepalese style Chorten and the model was brought from the Bowneath Chorten in Nepal. The chorten is believed to have been founded by the 13th Je Khenpo, Younten Thai, in 1782. The chorten was restored during the reign of the second king Jigme Wangchuk.


text:
UC Davis, Area 52 Establish New Business Incubator in Davis
By AJ Cheline on February 23, 2016 in Science & Technology
Incubator will provide offices, labs and workshops for UC Davis startups
Focus on medical devices, agricultural tech, robotics, energy, aerospace
Part of UC Davis DRIVE Network to support new companies
The Stuntverkoop, Davis, and Area 52, a recently formed nonprofit organization, have entered into an agreement to collaborate on a new technology business incubator for the Davis-Sacramento region. Area 52's incubator will be the second off-campus member of the university's Distributed Research Incubation and Venture Engine, or DRIVE, Network.
The shared goal of UC Davis and Area 52 is to create a workspace that provides the necessary resources for young companies looking to transform an idea into a product, especially in the areas of medical devices, agricultural technology, alternative energy, robotics and aerospace. The incubator's resources include access to advanced manufacturing technologies and tools that are essential for those areas.
Area 52 is being developed and managed by Tim Keller, a UC Davis alumnus and first-prize winner in the university's 2008 Big Bang! Business Competition. Keller founded VinPerfect Inc. in 2008 based on his invention, the SmartCap, which delivers precise oxygen control for wine bottles. Familiar with the process of turning a promising idea for a new product into a successful business, Keller is looking to create a resource to help others like him.
"Developing a tangible product is so much harder than developing a piece of software, because the costs of development are so high and because it is near impossible to find funding to create a product until you have built it," Keller said. "What we are developing is something unique that allows manufacturing companies to get to the investable stage much sooner, and for less money. Partnering with UC Davis will allow us to develop a long-term, sustainable and highly innovative manufacturing sector here in Northern California."
Area 52 will be in an existing building at 1222 Research Park Drive, Davis, within biking and walking distance of the UC Davis campus. Construction is scheduled to begin in March. The 36,000-square-foot building will have 20 offices, two wet labs, a fully equipped machine shop, composites shop, computer lab and two conference rooms, and eventually a gym and a café area for events and classes.
The building is owned by Sierra Energy, a waste gasification company also founded out of UC Davis' Big Bang! Business Competition. The renewable energy headquarters is investing more than $1 million in renovations for the facility and will occupy a portion of the building separate from the incubator space.
Mike Hart, chief executive officer of Sierra Energy, said, "We have seen an opportunity to make a real difference in the region's economy and we are taking it. People will be building new things and will have everything they need to turn those ideas into companies that can stay here in our community for the long term."
Area 52 plans to open its doors later this year, offering economical options for university and local entrepreneurs to rent office/lab space or purchase memberships to access the workshop space. Keller also expects to offer programs to students wanting to learn and develop high-tech skills.
"We're excited at the potential our collaboration with Area 52 and Sierra Energy will afford for UC Davis and regional startups to access high-quality engineering technology and medtech business incubation facilities and resources so close to campus," said Dushyant Pathak, associate vice chancellor of Technology Management and Corporate Relations at UC Davis. "Once in full swing, this facility with its advanced manufacturing and engineering tools, will complement the UC Davis-HM.CLAUSE Life Science Innovation Center which provides incubation facilities for university-affiliated startups in the biopharmaceutical, diagnostics, ag tech and related fields."
Managed by the Venture Catalyst unit within the UC Davis Office of Research, the DRIVE Network provides UC Davis-affiliated startups with access to affordable, mixed office/lab business incubation spaces in close proximity to campus. Each university-based company accepted into the DRIVE Network has access to support resources offered by Venture Catalyst. This includes a suite of services provided through the Smart Toolkit for Accelerated Research Translation, or START, Program, designed to equip UC Davis entrepreneurs with the tools they need to form and grow successful companies.
Media (s)
AJ Cheline, Office of Research, 530-752-1101, [email protected]
Andy Fell, News and Media Relations, 530-752-4533, [email protected]
Venture Catalyst
Science & Technology University
Professor Discusses Significance of Moon Landing on UC Davis Live Friday
Mukherjee Celebrates the Power of the Network
New Institute Eyes HOME in Deep Space
Guidance Issued on Closures, Power Outages
Library Provides Tips to Access Elsevier Articles
Advancing Health Worldwide
Advancing the study of medicine as a worldwide leader in human health and veterinary sciences.
Feeding a Growing Population
Connecting our research and expertise to the effort to feed a growing population under a changing climate.
Answering foundational questions about our world and how we live in it, showcasing how our research transcends science, technology, arts and humanities, and social sciences.
別れさせ屋 奈良
http://www.biceps-ua.com/
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Perched up on the top of a hill at the entrance into "the valley" Presidio Park is one of my favorite parks in San Diego. With views over Mission Bay to the Pacific Ocean, it's a favorite for sunset watchers.
As a major symbol of the city, it stands atop the hill recognized as the site where California began. It was here in 1769 that the Spanish Franciscan missionary, Father Junípero Serra, established California's first mission and presidio. The centerpiece of this park is the Serra Museum, which highlights the four different cultures that have occupied Presidio Hill from the Kumeyaay, Spanish, Mexican and early American periods.
Padre Cross erected in1913 is made up of tiles from the Presidio ruins.
Just east of the museum is Palm Canyon a half mile of large palm trees shade this canyon and it's hiking trails. Located just east, on the other side of Palm Canyon, is Inspiration Point. A popular location for weddings, it has views of Mission Valley and numerous trees that offer year-round shade making it a great spot for a picnic.
To visit Presidio Park, take Interstate 8 to the Taylor Street Exit. Head west on Taylor Street and turn left onto Presidio Drive. Follow Presidio Drive up the hill then right into the parking lot across from the Serra Museum.
Beautiful Photos! I must visit next time I am in San Diego.
Thanks Janice! San Diego is a nice place to visit and you're almost guaranteed great weather.
Beautiful…I love Presidio Park…and have captured it perfectly!!!


Speculation pits Sanders against Jim Jeffords for Senate in 2000. Jeffords' chief-of-staff suggests that Sanders won't do it because Vermont could potentially lose both men's seniority in Congress.
Forget Act 60. The real political heat these days is behind the curtain, where the merry band of Vermont's finest are pondering their moves in the upcoming millennium election of 2000. It's	really just around the corner.
The big prize on the table is the U.S. Senate seat held by Republican Jim Jeffords, and it's not only Democrats who have him in their sights.	Sources close to Vermont's independent congressman tell	Inside Track Bernie Sanders is itching to take on Jeezum Jim. There have been signs.
Yes, that was Ol' Bernardo in Rutland last Friday in what sure appeared to be campaign mode. He was speaking to students at the College of St. Joseph and Rutland High School. The Bern was getting the word out on his	latest cause — higher education: He recently introduced legislation	that would double the amount of	money put into Pell Grants to pay for college tuition. The champion of "poor people, working people and the elderly" is now	staking out new turf as the champion of higher education.
Funny, but Rutland is Jim Jeffords' home turf, and education is Jim Jeffords' bailiwick. He is, after all, the chairman of the Senate Health, Education, Labor and Pensions Committee, where national education legislation is born. Sanders' Rutland swing did not go unnoticed by the Jeffords camp. Officially, the word is that they "don't anticipate" the Bern taking Jim on in 2000.
"You look at an opportunity," says Jeezum Jim's chief of staff, Susan Boardman Russ, "and weigh it against the risks. Do you give up a secure House seat and challenge an incumbent?" she	asks.
Obviously, Susan believes the answer is hell no! "There are a lot of things about Bernie Sanders I disagree with," Russ tells Inside Track, "but he does care about Vermont. And Bernie's accumulated a lot of seniority in the House." It sure is comforting she's looking out for Bernie's political future.
In Russ' view, giving up all that House seniority to send Jeezum Jim into retirement would not be good for the state. "Even if Bernie were successful," says Russ, "it would be bad for Vermont." Senator Sanders would start at the bottom again, and so would his replacement in the House. She dubs it a "lose-lose" situation.
The Sanders' camp points to the record. In 1994, the year of the Great Republican Revolution, Jeffords got just 50.3 percent of the vote in a	race against a statewide unknown — Jan Backus.
Second, Bernie the Socialist plays well among rural conservatives, aka woodchucks, the fiercely independent rebels who turned out for Ruth Dwyer. To them, Jim's been a sellout Republican.
Third, sources say The Bern was none too pleased with Jeezum Jim's enthusiastic support for Susan Sweetser in 1996. The unwritten rule has been that, with a tiny three-member delegation, Vermont's Washington crew stayed out of the partisan fray back home. Jeffords broke that rule in the '96 race, and publicly bashed Bernie as an "ineffective" loner on Capitol Hill. Maybe it's guilt, but there's a strong feeling in the Jeffords camp that Mr. Sanders is looking for a little payback.
And take note, even with Jeffords' support, Susie Creamcheese got just 32 percent of the vote.
And fifth, in the current Impeachment Game, Jeffords is coming off like a leaf blowing in the wind. Last week when he flip-flopped on calling witnesses, State Sen. Elizabeth Ready was on the verge of putting together a press conference with a pack of Ds to whack Jim-Bob.
"I never saw a guy," says Ready, "who could ride both sides of a barbed-wire fence with such comfort."
That brings us to the other big question. If Sanders gets this one, would the Democrats sit it out?
First off, our sources say Gov. Howard Dean has taken himself out of the running. He's made it clear to both Jeffords and the Democratic National Committee he's not interested in a Senate bid in 2000. We've got him on Al Gore's list for veep. That leaves four horses in the post parade: Sens. Jan Backus, Liz Ready, Peter Shumlin and State Auditor Ed Flanagan.
What would they do if Bernie was in the mix? Chainsaw Liz says she'd probably work on Sanders' campaign. Jan of Arc says "the Democrats would have to sit down and decide if they want to run anyone." Shummy and Fast Eddie are a little less clear at this time. After all, Flanagan's already stopped by the Democratic Senatorial Campaign Committee in Washington. Shumlin's due to pay a courtesy call next week. Don't expect either pooch to roll over and thrust his paws into the air just because a Big Dog like Bernie starts barking.
And what does Mr. Sanders say about all this? "We're in the third week of a new term," says our congressman. "It's way, way too early" to think about the 2000 race."
The other side of all this is, who runs for Bernie's open seat?
Everybody! Among the pack is Burlap Mayor Peter Clavelle. Mayor Moonie is ready for statewide prime time and will rejoin the Democratic Party to do so. That is, if he beats Republitan Kurt Wright on March 2.
Chauffeur Update — Good news for Haskell Garrett. On Friday, District Court Judge Linda Levitt granted Garrett's request to be allowed to leave his Burlington home to drive Democratic Party Chairman David Curtis to the Statehouse "provided Mr. Curtis reports any violations."
Garrett was charged last month with kidnapping, stalking and aggravated domestic assault. Hey, maybe there's a way to get two birds with one stone here. While waiting around at the Statehouse to drive Mr. Curtis back to Burlap, Mr. Garrett could testify before the House and Senate Judiciary Committees. Surely they'd like to hear firsthand how tough Vermont is on domestic assault. It would be fascinating testimony.
As for other offenders on 24-hour curfew, there's still Republican Party Chairman Patrick Garahan. As far as we know, Pat doesn't have a court-assigned chauffeur ... yet.
Media Notes — WCAX-TV reporter Bridget Barry Caswell is back on the box following maternity leave. Bridget's second child is a boy. Jack Barry Caswell was named in honor of her dad, the late, great king of the Vermont airwaves, Jack Barry. And according to his pediatrician, little Jack is already warming up for talk radio. "Be sure and tell 'em Barry brought you."


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text:
Designations, Awards & Certifications
In 1937, The Ridges Sanctuary became Wisconsin's first land trust. For 80 years it has been an integral part of the rich, natural setting of the Baileys Harbor community and the Door County Peninsula. Founded to preserve the original 30 acre parcel, The Ridges has grown thoughtfully and strategically to ensure the protection of the most biologically diverse ecosystem in Wisconsin.
Cook-Albert Fuller Nature Center
Range Lights
Hike the Ridges
Accessibility at The Ridges
Nature Store
The Cook-Albert Fuller Center, Hidden Brook Boardwalk, and nature trails are open year-round. Come and explore all The Ridges has to offer. From guided hikes to tours of the Range Lights, The Ridges offers adults, families, and tour groups a wide variety of options for their visit.
Festival of Nature
The Ridges is committed to providing education programs for both children and adults. Our goal is to increase the appreciation for our rich natural landscape and to foster connections with the natural world. We provide several programs, year-round, that are great for adults, children, and families. The Ridges strives to meet its mission of education and outreach.
Blossoming Calendar
Land Acquisition Timeline
Hunting Policies
The Ridges is named for its distinctive topography - a series of 30 ridges and swales formed by the movement of Lake Michigan over the past 1100 years. Today, The Ridges owns 1,600 acres in and around Baileys Harbor. In recognition of its rich diversity, The Ridges Sanctuary has been designated as a Wisconsin State Natural Area, a National Natural Landmark by the National Park Service, a National Audubon Important Bird Area and a Wisconsin Wetland Gem.
Orchid Restoration Project
Watershed Study
Hine's Emerald Dragonfly
The Ridges and surrounding landscapes have been recognized as the most biologically diverse area in the State by the Wisconsin Department of Natural Resources. To protect The Ridges, research initiatives must provide an understanding of biodiversity and ecosystem dynamics and connections to the larger landscapes or watersheds.
Program Guide (Former Lites)
Nature of Research
Board and Member Information
The Ridges has a number of resources available to help you learn more about this amazing place.
Re-new
Call to Register
The Ridges Sanctuary
Preservation, Education, and Research
Welcome to the Cook-Albert Fuller Center!
Posted on Jun 12, 2015 by Drew Richmond Comments Off on Welcome to the Cook-Albert Fuller Center! | Posted in Homepage Rotator · Uncategorized
Photo by Len Villano
We're delighted to announce that, after years of discussion, planning and fundraising, the Cook-Albert Fuller Nature Center opened to the public on Saturday, June 13!
How does that change what you'll find when you visit The Ridges? We believe you'll experience the best of the old with a touch of the new. You'll still see all the familiar things that have made this place special for almost 78 years – extraordinary forests and wetlands, miles of scenic trails lined with rare wildflowers and nature programs for all ages. You'll still be greeted by friendly volunteers, guided
Hidden Brook Trail (Len Villano)
by naturalists and instructed by specialists on birding, botany, geology and more. We've added a few things that we believe will enhance your experience, so here's what you need to know:
We're easier to find! Our new Nature Center is located at 8166 State Highway 57 at the north end of downtown Baileys Harbor.
Both the Center and our Nature Store are open daily, 9 AM – 5 PM.
Our business office is also housed at the Center and is open Monday – Friday, 9 AM – 5 PM
Guided Sanctuary hikes now begin at the new Hidden Brook trailhead (soon to be the handicapped accessible boardwalk) located on the northeast side of the Center. Sign up at the front desk!
A $5 Day Pass includes access to Hidden Brook, the Sanctuary, the Family Discovery Trail and Logan Creek. Members and children under 18 hike free.
Nature Store Photo by Len Villano
Thank you for your support! We look forward to seeing you this season — and all year-round!
Nature Center Closed (Trails Open)
Sizing Up Winter – No School!
Ridges Guided Winter Hikes
New Moon – Night Sky Hike
Winter 2017 Ridges Lites
Fall 2016 Sanctuary
Winter 2016-2017 Lites
Natural Science Illustration at The Ridges
Homepage Photo Gallery Images
Homepage Rotator
Ridges Lites
Sanctuary Newsletter
Baileys Harbor, WI 54202
© The Ridges Sanctuary
Website by J Davis Web Design


text:
AlltheWeb Search Closes Its Doors
Rob D Young April 5, 2011
Rob D Young
/ April 5, 2011
Back in December, an image from a Yahoo in-company slideshow warned us that several major sites would be "sunset," including popular community sites like delicious and search sites like AlltheWeb and Alta Vista. For AlltheWeb the news was made official in March, when a message appeared on the site indicating that the site's final day would be April 4th. That date is now past, and anyone attempting to reach alltheweb.com will now be redirected to Yahoo search instead.
AlltheWeb opened its doors in 1999 and reached its height in popularity and size in 2003, when the total number of indexed pages rivaled Google, the leader of the time in that category. AlltheWeb was also the first group to develop instantly populating search results, a technology that has become the new standard, with the Google Instant official release and beta versions of both Bing and Yahoo using similar technology.
The closure of AlltheWeb was certainly upsetting to some of its 115,000 unique monthly visitors. However, considering the 78 million unique visitors to Yahoo each month, it should be evident that Yahoo isn't losing much as a company. The business decision for Yahoo is moving along in accordance with its revised gameplan, which focuses on pulling back its search front (in fact, the search results are even being run by former competitor Bing) and less profitable products so it can focus on community, entertainment, and other niche features.
[via Search Engine Watch]
CategorySEO
Rob has been insatiably obsessed with Google, search engine technology, and the trends of the web-based world since he began ... [Read full bio]


I think we may have access to a service manual also. If you are doing repairs and require this manual, please reply to this message and I will contact our supplier.
We haven't been able to source the service manual for the SL-2200 from our suppliers so far.
ON THE SANYO ECR-833 CASH REGISTER.
I just purchased a TOWA SX-580 cash register for personal use at home job. I would be interested in a manual to reprogram it so that i may use it for personal use.
Can you send me manaul or do i need to purchase it? I would be very interested in downloading to reprogram for my things i need.
Our suppliers advise that they do not have this model but will try to obtain it for us. It will be added to the site if we manage to procure it.
That is a very rare Sanyo register. It is unlikely that we can find a supplier for the manual. If successful, we'll add it to the site.
The SM25 is a very old model and our supplier has informed us that it is obsolete now. They do not have information for it.


Boehringer Ingelheim is to acquire the microtechnology specialist STEAG microParts GmbH from STEAG AG, Essen, Germany. STEAG microParts, established in 1990, is a worldwide leader in microsystem technology with headquarters at the hub of technology in Dortmund, Germany.
STEAG microParts' core product is the innovative Respimat® Soft MistTM Inhaler which was co-developed together with Boehringer Ingelheim. This novel medicinal inhaler, which generates a slow-moving long-lasting soft mist through purely mechanical means, allows for excellent penetration of the lungs. Respimat® Soft MistTM Inhaler offers all patients with asthma or COPD (chronic-obstructive airways disease) effective medication in precise doses.
"The acquisition of STEAG microParts fits very well into our production strategy," explained Dr Hans-Jürgen Leuchs, responsible within the Boehringer Ingelheim Board of Managing Directors for production and technology. "We see Respimat® Soft MistTM Inhaler as the inhalation device of the future; it is to be tested in clinical studies successively with different respiratory products. We thank STEAG microParts for the excellent cooperation in recent years and now look forward to having full control of the entire production chain following the acquisition. In addition, the expertise in development and production at STEAG microParts, especially in microsystem technology, will be an excellent supplement to our development activities in the field of innovative Drug Delivery".