text:
\section{Introduction}

	Fractional Calculus and in particular fractional differential equations (FDEs) have received considerable attention during last few decades owing to their applicability to various branches of science and engineering. FDEs have successfully modelled phenomena related to diffusion, control theory, visco-elastic systems, signal processing, fractional-order multipoles in electromagnetism and so on
	\cite{debnath2003recent,podlubny1998fractional}.

	Numerous decomposition/ numerical methods such as Adomian decomposition method \cite{adomian2013solving,daftardar2008solvingmulti}, New iterative method (NIM) \cite{daftardar2008solving}, Homotopy perturbation method \cite{he1999homotopy} has been employed to solve FDEs. The solutions obtained by all these methods however are local in nature and it is important to explore other techniques to find exact analytical solutions of FDEs. Exact	solutions play a vital role in the proper understanding of qualitative features of the concerned phenomena and processes	in various areas of science and engineering.

	One of the analytical methods for finding exact solutions of partial differential equations (PDEs) is invariant subspace method. This method was proposed by Galaktionov and Svirshchevskii \cite{galaktionov1995invariant,galaktionov2006exact}and have been applied by many researchers thereafter \cite{ma2012refined,svirshchevskii1995lie,svirshchevskii1996invariant}. This method has been further extended for time fractional partial differential equations (FPDEs) and its utility has been illustrated in the literature \cite{gazizov2013construction,harris2013analytic,harris2014nonlinear,sahadevan2015invariant,sahadevan2016exact}. In the present paper we develop invariant subspace method for finding exact solutions of PDEs with fractional time and space derivatives. In this method fractional partial differential equations are reduced to systems of fractional differential equations in one variable which can be solved by known analytical methods.

	The organization of the paper is as follows. Section 2, deals with preliminaries and notations. In Section 3, we develop invariant subspace method for solving time and space fractional partial differential equations. In Section 4, we present illustrative examples to explain applicability of the method. It is clear from the examples that the method also works for time and space FPDEs with certain kinds of initial conditions. Finally concluding remarks are made in Section 5.

	\section{Preliminaries and Notations}
	In this section, we introduce notations, definitions and preliminaries which are used in the paper \cite{diethelm2010analysis,podlubny1998fractional}.
	\begin{definition}The fractional integral of order $\alpha > 0$ of function $f\in L^1([a,b],\mathbb{R})$ is defined as
		\begin{equation*}
		I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int\limits_{a}^{t}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}d\tau,~~t>a.
		\end{equation*}
	\end{definition}
	\begin{definition}
		Caputo fractional derivative of order $\alpha>0$ of $f\in C^n([a,b],\mathbb{R})$ is defined as
		\begin{align*}
		\dfrac{\rm d^{\alpha}f(t)}{\rm dt^\alpha}=\left\{\begin{array}{lcr}I^{n-\alpha}D^{n}f(t)=\dfrac{1}{\Gamma(n-\alpha)}\displaystyle\int\limits_{a}^{t}\dfrac{f^{(n)}(\tau)}{(t-\tau)^{\alpha-n+1}}d\tau,~\mbox{$n-1 < \alpha < n,$}\\f^{(n)}(t),\hspace{6.4cm} \mbox{$\alpha=n,~n \in \mathbb{N}.$}
		\end{array}\right.
		\end{align*}
	\end{definition}
	\iffalse{
		\noindent	Riemann-Liouville fractional integral and Caputo fractional derivative satisfy the following properties\cite{diethelm2010analysis}:
		\begin{eqnarray*}
			I^\alpha t^\beta&=&\dfrac{\Gamma(\beta+1)}{\Gamma(\beta+\alpha+1)}t^{\beta+\alpha},\hskip 1cm \alpha >0,~\beta>-1,~t>0.\\
			\dfrac{\rm d^{\alpha}t^\beta}{\rm dt^\alpha} &=&\left\{\begin{array}{lcr}
				0,\hskip 3.2cm\text{if}~ \alpha >0,~\beta\in \{0,1,2,\dots,n-1\},\vspace{.1cm}\\
				\dfrac{\Gamma(\beta+1)}{\Gamma(\beta-\alpha+1)}t^{\beta-\alpha},\hskip .6cm \mbox{if $\alpha >0,~\beta\in \mathbb{N}$ and $\beta \geq n $ or $\beta\notin \mathbb{N}$ and $\beta>n-1$}.
			\end{array}\right.\\
			I^\alpha \left(\dfrac{\rm d^{\alpha} f(t)}{\rm dt^\alpha}\right)&=&f(t)-\sum_{k=0}^{n-1}\dfrac{\rm d^{(k)}f(0)}{\rm dt^{(k)}}\dfrac{t^k}{k!},~~~~~~n-1<\alpha<n,~t>0.
		\end{eqnarray*}
	}\fi
	The $n-$th order derivative of two-parametric Mittag-Leffler function $E_{\alpha,\beta}(t)$  is given by
	\begin{equation}\label{D1}
	E_{\alpha,\beta}^{(n)}(t)=\dfrac{\rm d^{n}}{\rm dt^{n}}E_{\alpha,\beta}(t)=\displaystyle\sum\limits_{k=0}^{\infty}\dfrac{(k+n)! t^{k}}{k! \Gamma(\alpha k+\alpha n+\beta)},~~n=0,1,2,\ldots.
	\end{equation}

	Laplace transform of the Caputo derivative of order $\alpha$ is,
	\begin{equation*}\mathcal{L}\left\{\dfrac{	\rm d^\alpha f(t)}{\rm dt^\alpha};s\right\}=s^{\alpha}\tilde{f}(s)-\sum\limits_{k=0}^{n-1}s^{\alpha-k-1}f^{(k)}(0),~~n-1<\alpha\leq{n},~n\in\mathbb{N},~\mathfrak{R}(s)>0,\end{equation*}
	where~$\tilde{f}(s)=\mathcal{L}\{f(t);s\}=\int\limits_{0}^{\infty}e^{-st}f(t)dt,~~s\in\mathbb{R}.$\\

	Laplace transform of the function $\varepsilon_n(t,a;\alpha,\beta):=t^{\alpha n+\beta-1}E_{\alpha,\beta}^{(n)}(\pm at^{\alpha})$ has the form
	\begin{equation}\label{D2}
	\mathcal{L}\{t^{\alpha n+\beta-1}E_{\alpha,\beta}^{(n)}(\pm at^{\alpha});s\}=\frac{n!s^{\alpha-\beta}}{(s^{\alpha}\mp a)^{n+1}},~~\mathfrak R(s)>|a|^{\frac{1}{\alpha}},~n=0,1,\ldots. \end{equation}

	Let $I_n$ denote the n-dimensional linear space over $\mathbb{R}$ spanned by n linearly independent functions $\{\phi_i(x):~ i=0,1,\dots,n-1\}$, \textit{i.e.},
	\begin{eqnarray*}
		\hskip.4cm I_n= \mathfrak{L}\{\phi_0(x),\dots,\phi_{n-1}(x)\}=\left \{\sum_{i=0}^{n-1}k_i \phi_i(x)~ \bigg| ~k_i\in \mathbb{R}, i=0,\dots,n-1 \right \}
	\end{eqnarray*}

	A finite dimensional linear space $I_n$ is invariant with respect to a differential operator $N$ if $N[f]\in I_n, \forall f \in I_n.$

	\section{Main Results}

	Consider the following fractional partial differential equation
	\noindent \begin{eqnarray}\label{FPDE}
	\sum_{i=0}^{m}\lambda_i\dfrac{\partial ^{\alpha+i}f(x,t)}{\partial t^{\alpha+i}}=N[f(x,t)]
	=N\left(x,f,\dfrac{\partial^{\beta}f}{\partial x^{\beta}},\dots,\dfrac{\partial^{\beta+k}f}{\partial x^{\beta+k}}\right),\end{eqnarray}
	where $N[f]$ is linear/ nonlinear differential operator. $\dfrac{\partial^{\alpha+i}f}{\partial t^{\alpha+i}},i=0,1,\dots,m$ and $\dfrac{\partial^{\beta+j}f}{\partial x^{\beta+j}},~j=0,\dots,k$ are Caputo time and space derivatives respectively. $\lfloor\alpha\rfloor= r,\lfloor \beta \rfloor =s, \textrm{where}~ r,s \in \mathbb{N}$ and $\lambda_i\in \mathbb{R}.$

	\begin{theorem}\label{THM1}
		If a finite dimensional linear space $I_{n+1}=\mathfrak{L}\{\phi_j(x)~|~j=0,\dots,n\}$ is invariant under the operator N[f], then FPDE (\ref{FPDE}) has a solution of the form
		\begin{equation}\label{U}
		f(x,t)=\sum_{j=0}^{n}K_j(t)\phi_j(x),
		\end{equation}where the coefficients $\{K_j\}$ satisfy the following system of FDEs
		\begin{equation*}
		\sum_{i=0}^{m}\lambda_i\dfrac{\rm d^{\alpha+i}K_j(t)}{\rm dt^{\alpha+i}}=\psi_j(K_1(t),K_2(t),\dots,K_n(t)),~j=0,1,\dots,n.
		\end{equation*}Here $\psi_j$'s are the expansion coefficients of $N[f]$ with respect to basis $\phi_j$'s.
	\end{theorem}
	\begin{proof} Consider L.H.S of FPDE (\ref{FPDE}). Using linearity of fractional derivatives and Eq. (\ref{U}), we get
	\begin{eqnarray}\label{LHS}
	\sum_{i=0}^{m}\lambda_i\dfrac{\partial ^{\alpha+i}f(x,t)}{\partial t^{\alpha+i}}&=&\sum_{i=0}^{m}\lambda_i\frac{\partial^{\alpha+i}}{\partial t^{\alpha+i}}\sum_{j=0}^{n}K_j(t) \phi_j(x)\nonumber\\
	&=&\sum_{j=0}^{n}\left[\sum_{i=0}^{m}\lambda_i\frac{\rm d^{\alpha+i}K_j(t)}{\rm dt^{\alpha+i}}\right] \phi_j(x).
	\end{eqnarray}
	Further as $I_{n+1}$ is invariant space under the operator $N[f]$, there exist $n+1$ functions $\psi_0,\psi_1,\dots,\psi_n$ such that \begin{equation}\label{IN}
	N\left[\sum_{j=0}^{n}k_j \phi_j(x)\right]=\sum_{j=0}^{n}\psi_j(k_1,k_2,\dots,k_n)\phi_j(x),~\text{for}~ k_j\in \mathbb{R},\end{equation}
	where $\psi_j$'s are expansion coefficients of $N[f]\in I_{n+1}$ corresponding to basis functions $\phi_j$'s.\\
	In view of Eq. (\ref{U}) and Eq. (\ref{IN})
	\begin{equation}\label{RHS}
	N[f(x,t)]=N\left[\sum_{j=0}^{n}K_j(t)\phi_j(x)\right]=\sum_{j=0}^{n}\psi_j(K_1(t),\dots,K_n(t))\phi_j(x).
	\end{equation}
	Substituting Eq. (\ref{LHS}) and Eq. (\ref{RHS}) in Eq. (\ref{FPDE}), we get
	\begin{equation}\label{LHS=RHS}
	\sum_{j=0}^{n}\left[\sum_{i=0}^{m}\lambda_i\dfrac{\rm d^{\alpha+i}K_j(t)}{\rm dt^{\alpha+i}}-\psi_j(K_1(t),K_2(t),\dots,K_n(t))\right]\phi_j(x)=0.\end{equation}
	Using Eq. (\ref{LHS=RHS}) and the fact that $\phi_j$'s are linearly independent, we get the following system of FDEs
	\begin{equation*}
	\sum_{i=0}^{m}\lambda_i\dfrac{\rm d^{\alpha+i}K_j(t)}{\rm dt^{\alpha+i}}=\psi_j(K_1(t),K_2(t),\dots,K_n(t)),~j=0,1,\dots,n.
	\end{equation*}
\end{proof}
	Consider the following fractional PDE
	\begin{equation}\label{FPDE2}
	\sum_{i=1}^{m}\lambda_i\dfrac{\partial ^{i\alpha}f(x,t)}{\partial t^{i\alpha}}=N[f(x,t)]=N\left(x,f,\dfrac{\partial^{\beta}f}{\partial x^{\beta}},\dots,\dfrac{\partial^{k\beta}f}{\partial x^{k\beta}}\right),\end{equation}
	where $N[f]$ is linear/ nonlinear differential operator. $\dfrac{\partial^{i\alpha}f}{\partial t^{i\alpha}},~i=1,2,\dots,m$ and $\dfrac{\partial^{j\beta}f}{\partial x^{j\beta}}, ~j=1,2,\dots,k$ are Caputo Caputo time and space derivatives respectively. $\lfloor\alpha\rfloor= r,\lfloor \beta \rfloor =s, \textrm{where}~ r,s \in \mathbb{N}$ and $\lambda_i\in \mathbb{R}.$

	\begin{theorem}\label{THM2}
		If a finite dimensional linear space $I_n$ is invariant under the operator N[f], then FPDE (\ref{FPDE2}) has a solution of the form
		\begin{equation*}
		f(x,t)=\sum_{j=1}^{n}K_j(t)\phi_j(x),
		\end{equation*}where the coefficients $\{K_j\}$ satisfy the following system of FDEs
		\begin{equation*}
		\sum_{i=1}^{m}\lambda_i\dfrac{\rm d^{i\alpha}K_j(t)}{\rm dt^{i\alpha}}=\psi_j(K_1(t),K_2(t),\dots,K_n(t)),~j=1,2,\dots,n.
		\end{equation*}Here $\psi_j$'s are expansion coefficients of $N[f]\in I_n$ with respect to $\phi_j$'s.
	\end{theorem}
	Proof of Theorem \ref{THM2} can be given on similar lines as the proof of Theorem \ref{THM1}.

	\section{Illustrative Examples}
	In this section we solve some illustrative examples using Theorem \ref{THM1} and Theorem \ref{THM2}.
	\begin{example}
		Consider the following fractional PDE
		\begin{equation}\label{EX1}
		\dfrac{\partial ^\alpha f}{\partial t^ {\alpha}}+\dfrac{\partial ^{\alpha+1} f}{\partial t^ {\alpha+1}}+\dfrac{\partial ^{\alpha+2} f}{\partial t^ {\alpha+2}}=N[f]=\dfrac{\partial ^{\beta+1} f}{\partial x^ {\beta+1}},~~\alpha,~\beta \in (0,1].
		\end{equation}
	\end{example}
	\noindent Clearly $I_2=\mathfrak{L}\{1,x^{\beta+1}\}$ is an invariant subspace as
	\begin{equation*}
	N[k_0+k_1x^{\beta+1}]=k_1\Gamma(\beta+2)\in I_2.
	\end{equation*}In view of Theorem \ref{THM1}, Eq. (\ref{EX1}) has exact solution of the form
	\begin{equation*}\label{SOL1}
	f(x,t)=K_0(t)+K_1(t)x^{\beta+1},
	\end{equation*} where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs.
	\begin{eqnarray}
	\dfrac{\rm d ^\alpha K_0(t)}{\rm d t^ {\alpha}}+\dfrac{\rm d ^{\alpha+1} K_0(t)}{\rm d t^ {\alpha+1}}+\dfrac{\rm d ^{\alpha+2} K_0(t)}{\rm d t^ {\alpha+2}}&=&K_1(t)\Gamma(\beta+2),\label{11}\\
	\dfrac{\rm d ^\alpha K_1(t)}{\rm d t^ {\alpha}}+\dfrac{\rm d ^{\alpha+1} K_1(t)}{\rm d t^ {\alpha+1}}+\dfrac{\rm d ^{\alpha+2} K_1(t)}{\rm d t^ {\alpha+2}}&=&0.\label{12}
	\end{eqnarray}
	Using Laplace transform in Eq. (\ref{12}), we obtain
	\begin{align}
	\tilde{K_1}(s)&=\frac{K_1(0)(s^{\alpha+1}+s^{\alpha}+s^{\alpha-1})}{s^{\alpha+2}+s^{\alpha+1}+s^{\alpha}}+\frac{K_1'(0)(s^{\alpha}+s^{\alpha-1})}{s^{\alpha+2}+s^{\alpha+1}+s^{\alpha}}+\frac{K_1''(0)s^{\alpha-1}}{s^{\alpha+2}+s^{\alpha+1}+s^{\alpha}}\nonumber\\
	&=K_1(0)\left[\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k}}{(s+1)^{k+1}}+\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k-1}}{(s+1)^{k+1}}+\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k-2}}{(s+1)^{k+1}}\right]+\nonumber\\
	&K_1'(0)\left[\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k-1}}{(s+1)^{k+1}}+\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k-2}}{(s+1)^{k+1}}\right]+K_1''(0)\sum_{k=0}^{\infty}\frac{(-1)^ks^{-k-2}}{(s+1)^{k+1}}.\nonumber
	\end{align}
	Taking inverse Laplace transform and using Eq. (\ref{D2}) we get
	\begin{eqnarray}\label{13}
	K_1(t)=\sum_{i=0}^{2}K_1^{(i)}(0)\left[\sum_{l=i}^{2}\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{2k+l}E^{(k)}_{1,k+l+1}(-t)\right],
	\end{eqnarray}
	where $E^{(n)}_{\alpha, \beta}(.)$ is defined in Eq. (\ref{D1}).\\
	Substituting value of $K_1(t)$ from Eq. (\ref{13}) in Eq. (\ref{11}) and following similar procedure we deduce
	\begin{align}\label{14}
	K_0(t)=\sum_{i=0}^{2}K_0^{(i)}(0)\left[\sum_{l=i}^{2}\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{2k+l}E^{(k)}_{1,k+l+1}(-t)\right]+\Gamma(\beta+2)\nonumber\\\left[
	\sum_{i=0}^{2}K_1^{(i)}(0)\left(\sum_{l=i+2}^{4}\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{2k+\alpha+l}E^{(k)}_{1,k+\alpha+l}(-t)\right)\right].
	\end{align}
	Hence an exact solution of Eq. (\ref{EX1}) is~$f(x,t)=K_0(t)+K_1(t)x^{\beta+1}$~where $K_0(t)$ and $K_1(t)$ are given by Eq. (\ref{14}) and Eq. (\ref{13}) respectively.

	\begin{example}
		Consider the following fractional generalization of time fractional Burgers' equation.
		\begin{equation}\label{EX2}
		\dfrac{\partial^\alpha f}{\partial t^\alpha}+f\dfrac{\partial^\beta f}{\partial x^\beta}-d\dfrac{\partial^\beta }{\partial x^\beta}\left(\dfrac{\partial^\beta f}{\partial x^\beta}\right)=0,~~\alpha\in(0,1),~\alpha\neq \frac{1}{2},~\beta\in (0,1],
		\end{equation}where $d$ is diffusion constant.
	\end{example}
	\noindent Here $N[f]=-f\dfrac{\partial^\beta f}{\partial x^\beta}+d\dfrac{\partial^\beta }{\partial x^\beta}\left(\dfrac{\partial^\beta f}{\partial x^\beta}\right).~I_2=\mathfrak{L}\{1,x^\beta\}$ is an invariant subspace as $
	N[k_0+k_1x^\beta]=-k_0k_1\Gamma(\beta+1)-k_1^2\Gamma(\beta+1)x^\beta \in I_2.$\\
	Theorem \ref{THM1} implies an exact solution of the form
	\begin{equation*}\label{SOL21}
	f(x,t)=K_0(t)+K_1(t)x^\beta,
	\end{equation*} where $K_0(t)$ and $K_1(t)$ are unknown functions to be determined by solving following system of equations.
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&-K_0(t)K_1(t)\Gamma(\beta+1),\label{21}\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&-K_1(t)^2\Gamma(\beta+1).\label{22}
	\end{eqnarray}
	Assume solution of Eq. (\ref{22}) of the form $K_1(t)=\eta t^\gamma,$ where constant $\eta $ and the exponent $\gamma>-1$ can be found directly by substituting $K_1(t)=\eta t^\gamma$ in (\ref{22}), \textit{i.e.},
	\begin{equation}
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}=\eta \dfrac{\Gamma(\gamma+1)}{\Gamma(\gamma-\alpha+1)}t^{\gamma-\alpha}=-\eta ^2t^{2\gamma}\Gamma(\beta+1).
	\end{equation}
	Thus $K_1(t)=\eta t^\gamma$ is a solution if $\gamma=-\alpha$ and $\eta =\dfrac{-\Gamma(1-\alpha)}{\Gamma(1+\beta)\Gamma(1-2\alpha)},$ \textit{i.e.},
	\begin{equation*}
	K_1(t)=-\dfrac{\Gamma(1-\alpha)}{\Gamma(1+\beta)\Gamma(1-2\alpha)}t^{-\alpha},~~\alpha \neq\frac{1}{2},~\alpha \neq 1.
	\end{equation*}
	On similar lines by solving Eq. (\ref{21}) we get
	$
	K_0(t)=at^{-\alpha},
	$	where $a$ is arbitrary constant. Therefore Eq. (\ref{EX2}) has following exact solution.
	\begin{equation*}
	f(x,t)=at^{-\alpha}-\left[\frac{\Gamma(1-\alpha)t^{-\alpha}}{\Gamma(1+\beta)\Gamma(1-2\alpha)}\right]x^{\beta}.
	\end{equation*}
	In particular, for $\beta=1$ we obtain\\
	\begin{equation*}
	f(x,t)=at^{-\alpha}-\left[\frac{\Gamma(1-\alpha)}{\Gamma(1-2\alpha)}t^{-\alpha}\right]x,
	\end{equation*} which is a solution of the time fractional Burgers equation \cite{sahadevan2016exact}.\\
	A particular case of fractional Burgers equation with diffusion constant $d=0$ has been studied by Harris and Garra \cite{harris2013analytic}, in which they have considered an exact solution of the form $f(x,t)=\textrm{(constant)}+C_1(t)x^\beta.$

	\begin{example}
		Consider the time and space fractional diffusion equation
		\begin{equation}\label{EX3}
		\dfrac{\partial^{\alpha} f}{\partial t^{\alpha}}=C\dfrac{\partial^\beta f}{\partial x^\beta},~~\alpha\in (0,1],~\beta \in (1,2].
		\end{equation}
	\end{example}
	\noindent Here $N[f]=C\dfrac{\partial^\beta f}{\partial x^\beta}.$ Subspace $I_3=\mathfrak{L}\{1,x^\beta,x^{2\beta}\}$ is invariant under N[f] as
	\begin{equation*}
	N[k_0+k_1x^\beta+k_2x^{2\beta}]=Ck_1\Gamma(\beta+1)+Ck_2\frac{\Gamma(2\beta+1)}{\Gamma(\beta+1)}x^\beta\in I_3.
	\end{equation*}
	Consider an exact solution of the form	$
	f(x,t)=K_0(t)+K_1(t)x^\beta+K_2(t)x^{2\beta},
	$ where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs:
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&C~\Gamma(\beta+1)K_1(t)\label{31},\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&\frac{C~\Gamma(2\beta+1)}{\Gamma(\beta+1)}K_2(t)\label{32},\\
	\dfrac{\rm d^\alpha K_2(t)}{\rm d t^\alpha}&=&0.\label{33}
	\end{eqnarray}
	Eq. (\ref{33}) implies that $K_2(t)=a$ (say). So Eq. (\ref{32}) takes the form
	\begin{equation*}
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}=aC\dfrac{\Gamma(2\beta+1)}{\Gamma(\beta+1)},
	\end{equation*}which has the following solution:
	\begin{eqnarray*}
		K_1(t)&=&K_1(0)+\frac{aC~\Gamma(2\beta+1)}{\Gamma(\beta+1)\Gamma(\alpha+1)}t^\alpha.\end{eqnarray*}
	Similarly Eq. (\ref{31}) yields
	\begin{eqnarray*} K_0(t)&=&K_0(0)+K_1(0)\frac{C~\Gamma(\beta+1)}{\Gamma(\alpha+1)}t^\alpha +\frac{aC^2~\Gamma(2\beta +1)}{\Gamma(2\alpha+1)} t^{2\alpha}.
	\end{eqnarray*}
	Hence Eq. (\ref{EX3}) has the following solution
	\begin{eqnarray}\label{SOL32}
	f(x,t)=\left[d+\frac{bC~\Gamma(\beta+1)}{\Gamma(\alpha+1)}t^\alpha +\frac{aC^2~\Gamma(2\beta +1)}{\Gamma(2\alpha+1)} t^{2\alpha}\right]\nonumber\\
	+\left[b+\frac{aC~\Gamma(2\beta+1)}{\Gamma(\beta+1)\Gamma(\alpha+1)}t^\alpha\right]x^\beta+ax^{2\beta},
	\end{eqnarray}
	where~$a, b$ and $d$ are arbitrary constants.\\

	\noindent It can  be easily verified that $I_2=\mathfrak{L}\{1,E_\beta (x^\beta )\}$ is also an invariant subspace with respect to $N[f]$ leading to a different solution having the form
	\begin{equation*}
	f(x,t)=K_0(t)+K_1(t)E_\beta (x^\beta ),
	\end{equation*}where unknown functions $K_0(t)$ and $K_1(t)$ can be found by solving the following system of equations:
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&0\label{34},\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&CK_1(t).\label{35}
	\end{eqnarray}
	Solving the system of equations (\ref{34})-(\ref{35}), we obtain an exact solution of Eq. (\ref{EX3}) associated with $I_2=\mathfrak{L}\{1,E_\beta (x^\beta )\}$ as
	\begin{equation}\label{SOL33}
	f(x,t)=a+[bE_\alpha (Ct^\alpha )]E_\beta (x^\beta ),
	\end{equation}where $a$ and $b$ are arbitrary constants.\\

	\noindent Note that Eq. (\ref{EX3}) also admits another invariant subspace $I_2=\mathfrak{L}\{1,x^\beta\}$, which yields to the exact solution of Eq. (\ref{EX3}):
	\begin{equation}\label{SOL34}
	f(x,t)=a+\frac{bC\Gamma(\beta+1)}{\Gamma(\alpha+1)}t^\alpha+bx^{\beta},	~~a, b\in \mathbb{R}.
	\end{equation}
	Thus (\ref{SOL32}), (\ref{SOL33}) and (\ref{SOL34}) are three distinct exact solutions of Eq. (\ref{EX3}) corresponding to three distinct invariant subspaces. Note that $I_{n+1}=\mathfrak{L}\{1,x^\beta,\dots,x^{n\beta}\},$
	$~n\in \mathbb{N}$ is yet another invariant subspace for (\ref{EX3}). Thus we get infinitely many invariant subspaces for the problem under consideration, which in turn yield infinitely many solutions.\\

	\textbf{Solutions pertaining to initial conditions}\\

	\noindent 	Invariant subspace method is also useful in finding closed form solutions of fractional PDEs satisfying initial conditions. Observe that Eq. (\ref{EX3}) along with the initial condition \begin{equation*}
	f(x,0)=a_0+a_1x^\beta+a_2x^{2\beta}+\cdots+a_nx^{n\beta},~1<\beta \leq 2,~a_i \in \mathbb{R},~i=0,1,\dots n,
	\end{equation*}has closed form solutions.\\

	For example, consider the time and space fractional diffusion equation (\ref{EX3}) along with the  initial condition,
	\begin{equation}\label{IC31}
	f(x,0)=\dfrac{3x^\beta}{\Gamma(\beta+1)}.
	\end{equation}
	For solving this IVP, consider the invariant subspace $I_2=\mathfrak{L}\{1,x^\beta\}$. Using initial condition (\ref{IC31}) in Eq. (\ref{SOL34}) we obtain $a=0$ and $b=\dfrac{3}{\Gamma(\beta+1)}$.\\
	Thus we obtain the following exact solution of the IVP (\ref{EX3})-(\ref{IC31}).
	\begin{equation}\label{SOL35}
	f(x,t)=\dfrac{3Ct^\alpha}{\Gamma(\alpha+1)}+\dfrac{3x^\beta}{\Gamma(\beta+1)}.
	\end{equation}
	Note that the solution (\ref{SOL35}) coincides with the solution obtained by NIM introduced by Daftardar-Gejji \textit{et al.} \cite{daftardar2008solving}.\\
	Now consider (\ref{EX3}) together with the initial condition,
	\begin{equation}\label{IC32}
	f(x,0)=1-\dfrac{x^{2\beta}}{2}.
	\end{equation}Using the invariant subspace $I_3=\mathfrak{L}\{1,x^\beta,x^{2\beta}\}$, we get the following exact solution of the IVP (\ref{EX3}) - (\ref{IC32}).
	\begin{equation*}
	f(x,t)=\left[1-\frac{C^2~\Gamma(2\beta +1)}{2~\Gamma(2\alpha+1)} t^{2\alpha}\right]-\left[\frac{C~\Gamma(2\beta+1)}{2~\Gamma(\beta+1)\Gamma(\alpha+1)}t^\alpha\right]x^\beta-\dfrac{1}{2}x^{2\beta}.
	\end{equation*}

	\begin{example}
		Consider the following fractional generalization of telegraph equation:
		\begin{equation}\label{EX4}
		\dfrac{\partial ^{\alpha+1} f}{\partial t^ {\alpha+1}}+\dfrac{\partial ^\alpha f}{\partial t^ {\alpha}}=\dfrac{\partial ^{\beta} f}{\partial x^ {\beta}}-f,~~\alpha \in (0,1],~\beta \in (1,2].
		\end{equation}
	\end{example}
	\noindent Note that for $I_2=\mathfrak{L}\{1,E_\beta (x^\beta)\}$,
	\begin{equation*}
	N[k_0+k_1E_\beta (x^\beta)]=k_1E_\beta (x^\beta)-k_0-k_1E_\beta (x^\beta)=-k_0\in I_2.
	\end{equation*}This implies that $I_2$ is an invariant subspace, and in view of Theorem \ref{THM1}, Eq. (\ref{EX4}) has an exact solution of the form
	\begin{equation}\label{SOL41}
	f(x,t)=K_0(t)+K_1(t)E_\beta (x^\beta),
	\end{equation} where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs.
	\begin{align}
	\dfrac{\rm d ^{\alpha+1} K_0(t)}{\rm d t^ {\alpha+1}}+\dfrac{\rm d ^\alpha K_0(t)}{\rm d t^ {\alpha}}&=-K_0(t),\label{41}\\
	\dfrac{\rm d ^{\alpha+1} K_1(t)}{\rm d t^ {\alpha+1}}+\dfrac{\rm d ^\alpha K_1(t)}{\rm d t^ {\alpha}}&=0.\label{42}
	\end{align}
	Applying Laplace transform technique to Eq. (\ref{41}) we get
	\begin{eqnarray*}
		\tilde{K_0}(s)=K_0(0)\sum_{k=0}^{\infty}\frac{(-1)^ks^{-\alpha k}}{(1+s)^{k+1}}+[K_0(0)+K_0'(0)]\sum_{k=0}^{\infty}\frac{(-1)^ks^{-\alpha k-1}}{(1+s)^{k+1}}.
	\end{eqnarray*}
	Taking inverse Laplace transform we obtain
	\begin{align*}
	K_0(t)=~~&	K_0(0)\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{k(\alpha+1)}E_{1,\alpha k +1}^{(k)}(-t)+[K_0(0)+K_0'(0)]\\
	&\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{k(\alpha+1)+1}E_{1,\alpha k +2}^{(k)}(-t).
	\end{align*}
	Proceeding on similar lines, from Eq. (\ref{42}) we deduce
	\begin{equation*}
	K_1(t)=K_1(0)+K_1'(0)tE_{1,2}(-t).
	\end{equation*}
	Thus an exact solution (\ref{SOL41}) takes the following form
	\begin{eqnarray*}
		f(x,t)&=&a_1\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}t^{k(\alpha+1)}E_{1,\alpha k +1}^{(k)}(-t)+[a_1+a_2]\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}\\
		&&t^{k(\alpha+1)+1}E_{1,\alpha k +2}^{(k)}(-t)+[b_1+b_2tE_{1,2}(-t)]E_\beta(x^\beta),
	\end{eqnarray*}where $a_1, a_2, b_1$ and $b_2$ are arbitrary constants.

	\begin{example}
		Consider the following fractional IVP:
		\begin{eqnarray}
		\dfrac{\partial^\alpha f}{\partial t^\alpha}&=&\left(\dfrac{\partial^\beta f}{\partial x^\beta }\right)^2-f\left(\dfrac{\partial^\beta  f}{\partial x^\beta }\right),~~\alpha,~\beta  \in (0,1],\label{EX6}\\
		f(x,0)&=&3+\frac{5}{2}E_\beta (x^\beta )\label{IC6}.
		\end{eqnarray}
	\end{example}
	\noindent$I_2=\mathfrak{L}\{1,E_\beta (x^\beta )\}$ is invariant under $N[f]=\left(\dfrac{\partial^\beta f}{\partial x^\beta }\right)^2-f\left(\dfrac{\partial^\beta  f}{\partial x^\beta }\right)$ as
	\begin{equation*}
	N[k_0+k_1E_\beta (x^\beta )]=-k_0k_1E_\beta (x^\beta )\in I_2.
	\end{equation*}
	We consider solution of the form
	\begin{equation}\label{SOL61}
	f(x,t)=K_0(t)+K_1(t)E_\beta (x^\beta ),
	\end{equation} where $K_0(t)$ and $K_1(t)$ are unknown functions to be determined.\\
	By substituting (\ref{SOL61}) in (\ref{EX6}) and equating coefficients of different powers of $x$, we get the following system of FDEs.
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&0,\label{61}\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&-K_0(t)K_1(t).\label{62}
	\end{eqnarray}
	Solving Eq. (\ref{61}) we get $K_0(t)=K_0(0)=a$ (say), and Eq. (\ref{62}) takes the form $\dfrac{d^\alpha K_1(t)}{d t^\alpha}=-aK_1(t)$.
	Using Laplace transform technique we obtain
	$K_1(t)=K_1(0)E_\alpha(-at^\alpha).$ Thus the exact solution of (\ref{EX6}) along with the initial condition (\ref{IC6}) turns out to be
	\begin{equation*}
	f(x,t)=3+\left[\frac{5}{2}E_\alpha(-3t^\alpha)\right]E_\beta (x^\beta ).
	\end{equation*}

	\begin{example}
		Consider the following two generalization of wave equation with constant absorption term\\
		\textbf{Type (I)}
		\begin{eqnarray}\label{EX7}
		\dfrac{\partial^{2\alpha} f}{\partial t^{2\alpha}}=\dfrac{\partial^\beta }{\partial x^\beta}\left(f\dfrac{\partial^\beta f}{\partial x^\beta}\right)-1,~~\alpha,~\beta\in (0,1],\\
		f(x,0)=e+\dfrac{x^\beta}{\Gamma(\beta+1)},~f_t(x,0)=1-x^\beta.\label{IC7}
		\end{eqnarray}
	\end{example}
	\noindent Here $N[f]=\dfrac{\partial^\beta }{\partial x^\beta}\left(f\dfrac{\partial^\beta f}{\partial x^\beta}\right)-1.$ Clearly $I_2=\mathfrak{L}\{1,x^\beta\}$ is an invariant subspace of N[f] as
	\begin{equation*}
	N[k_0+k_1x^\beta]=k_1^2[\Gamma(\beta+1)]^2-1\in I_2.
	\end{equation*}
	In view of Theorem \ref{THM2}, we look for an exact solution of the form
	\begin{equation*}\label{SOL7}
	f(x,t)=K_0(t)+K_1(t)x^\beta,
	\end{equation*} where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs.
	\begin{eqnarray}
	\dfrac{\rm d^{2\alpha }K_0(t)}{\rm d t^{2\alpha }}&=&K_1^2(t)[\Gamma(\beta+1)]^2-1,\label{71}\\
	\dfrac{\rm d^{2\alpha } K_1(t)}{\rm d t^{2\alpha }}&=&0.\label{72}
	\end{eqnarray}
	Solving the system of FDEs (\ref{71})-(\ref{72}) we get the following exact solution

	\begin{align}\label{SOL71}
	f(x,t)=\left\{\begin{array}{lcr}
	\left[a_1+\left(\dfrac{b_1^2[\Gamma(\beta+1)]^2-1}{\Gamma(2\alpha+1)}\right)t^{2\alpha}\right]+b_1x^\beta,\hskip 1.5cm\mbox{if~ $0<\alpha\leq \dfrac{1}{2},$}\vspace{.4cm}\\
	\left[a_1+a_2t+\left(\dfrac{b_1^2[\Gamma(\beta+1)]^2-1}{\Gamma(2\alpha+1)}\right)t^{2\alpha}+\left(\dfrac{2b_2^2[\Gamma(\beta+1)]^2}{\Gamma(2\alpha+3)}\right)t^{2\alpha+2}\right.\\
	\left.+\dfrac{2b_1b_2[\Gamma(\beta+1)]^2}{\Gamma(2\alpha+2)}t^{2\alpha+1}\right]+\left[b_1+b_2t\right]x^\beta,\hskip 1.3cm\mbox{if~ $\dfrac{1}{2}<\alpha\leq 1$},
	\end{array}\right.
	\end{align}where $a_1,~a_2,~b_1$ and $b_2$ are arbitrary constants.\\
	Using initial conditions (\ref{IC7}) we get $a_1=e,a_2=1,b_1=\dfrac{1}{\Gamma(\beta+1)}$ and $b_2=-1.$	Hence the  exact solution of the IVP (\ref{EX7})-(\ref{IC7}) is

	\begin{eqnarray*}
		f(x,t)=\left\{\begin{array}{lcr}
			e+\dfrac{x^\beta}{\Gamma(\beta+1)},\hskip 5.7cm\mbox{if~ $0<\alpha\leq \dfrac{1}{2},$}\vspace{.4cm}\\
			\left[e+t+\dfrac{2[\Gamma(\beta+1)]^2}{\Gamma(2\alpha+3)}t^{2\alpha+2}-\dfrac{2\Gamma(\beta +1)}{\Gamma(2\alpha+2)}t^{2\alpha+1}\right]\\
			+\left[\dfrac{1}{\Gamma(\beta+1)}-t\right]x^\beta,\hskip4.5cm\mbox{if~ $\dfrac{1}{2}<\alpha\leq 1$}.
		\end{array}\right.
	\end{eqnarray*}\\
	\textbf{Type (II):}
	\noindent Consider another fractional generalization of  wave equation with constant absorption term
	\begin{equation}\label{EX72}
	\dfrac{\partial^{\alpha+1} f}{\partial t^{\alpha+1}}=N[f]=\dfrac{\partial^\beta }{\partial x^\beta}\left(f\dfrac{\partial^\beta f}{\partial x^\beta}\right)-1,~~\alpha,~\beta\in (0,1].
	\end{equation}
	From the discussion of Case (I), we know that $I_2=\mathfrak{L}\{1,x^\beta\}$ is an invariant subspace of $N[f]$. By Theorem \ref{THM1}, Eq. (\ref{EX72}) has the following solution.
	\begin{equation*}
	f(x,t)=K_0(t)+K_1(t)x^\beta,
	\end{equation*} where unknown functions $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs.
	\begin{align*}
	\dfrac{\rm d^{\alpha +1}K_0(t)}{\rm d t^{\alpha +1 }}&=K_1^2(t)[\Gamma(\beta+1)]^2-1,\\
	\dfrac{\rm d^{\alpha +1 } K_1(t)}{\rm d t^{\alpha +1 }}&=0.
	\end{align*}
	Solving these FDEs we obtain the following solution of Eq. (\ref{EX72}).
	\begin{align}\label{SOL72}
	f(x,t)=\left[a_1+a_2t+\left(\dfrac{b_1^2[\Gamma(\beta+1)]^2-1}{\Gamma(\alpha+2)}\right)t^{\alpha+1}+\left(\dfrac{2b_1b_2[\Gamma(\beta+1)]^2}{\Gamma(\alpha+3)}\right)t^{\alpha+2}\right.\nonumber\hskip-1.5cm\\
	\left.	+\left(\dfrac{2b_2^2[\Gamma(\beta+1)]^2}{\Gamma(\alpha+4)}\right)t^{\alpha+3}\right]+[b_1+b_2 t]x^\beta,
	\end{align}
	where $a_1, a_2, b_1$ and $b_2$ are arbitrary constants.\\

	\textbf{Note:} For $\alpha=1=\beta$, solutions (\ref{SOL71}) and (\ref{SOL72}) of two different fractional generalizations (\ref{EX7}) and (\ref{EX72}) respectively, reduce to
	\begin{align*}
	f(x,t)=a_1+a_2t+\left(\dfrac{b_1^2-1}{2}\right)t^{2}+\dfrac{b_1b_2}{3}t^{3}+\dfrac{b_2^2}{12}t^{4}+[b_1+b_2 t]x,
	\end{align*}
	which is a solution of the ordinary wave equation  with constant absorption term
	\begin{equation*}
	\dfrac{\partial^{2} f}{\partial t^{2}}=\dfrac{\partial^\beta }{\partial x^\beta}\left(f\dfrac{\partial^\beta f}{\partial x^\beta}\right)-1.
	\end{equation*}

	\begin{example}
		Consider the following fractional generalization of Korteweg–de Vries equation (KdV) equation:
		\begin{equation}\label{EX8}
		\dfrac{\partial^{\alpha} f}{\partial t^{\alpha}}-6f\dfrac{\partial^\beta f}{\partial x^\beta}+\dfrac{\partial ^{2\beta}}{\partial x^{2\beta}}\left(\dfrac{\partial^\beta f}{\partial x^\beta}\right)=0,~\alpha\in (0,1) \backslash\{1/2\},~\beta\in (0,1].
		\end{equation}
	\end{example}
	\noindent Here $N[f]=6f\dfrac{\partial^\beta f}{\partial x^\beta}-\dfrac{\partial ^{2\beta}}{\partial x^{2\beta}}\left(\dfrac{\partial^\beta f}{\partial x^\beta}\right).$ $I_2=\mathfrak{L}\{1,x^\beta\}$ is an invariant subspace as
	\begin{equation*}
	N[k_0+k_1x^\beta]=6k_0k_1\Gamma(\beta+1)6k_1^2\Gamma(\beta+1)x^\beta \in I_2.
	\end{equation*}
	Hence in view of Theorem \ref{THM2}, Eq. (\ref{EX8}) has solution of the form\\
	$
	f(x,t)=K_0(t)+K_1(t)x^\beta,$ where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs.
	\begin{eqnarray}
	\dfrac{\rm d^{\alpha }K_0(t)}{\rm d t^{\alpha }}&=&6K_0K_1\Gamma(\beta+1),\label{81}\\
	\dfrac{\rm d^{\alpha }K_1(t)}{\rm d t^{\alpha }}&=&6K_1^2\Gamma(\beta+1).\label{82}
	\end{eqnarray}
	Solving (\ref{81})-(\ref{82}) we get
	\begin{eqnarray*}
		K_0(t)=at^{-\alpha},
		K_1(t)=\frac{\Gamma(1-\alpha)t^{-\alpha}}{6\Gamma(1+\beta)\Gamma(1-2\alpha)}.
	\end{eqnarray*}Thus solution of Eq. (\ref{EX8}) becomes
	\begin{equation*}
	f(x,t)=at^{-\alpha}+\left[\frac{\Gamma(1-\alpha)t^{-\alpha}}{6\Gamma(1+\beta)\Gamma(1-2\alpha)}\right]x^{\beta},~a\in \mathbb{R}.
	\end{equation*}

	\begin{example}Consider the following non-linear space-time fractional dispersion equation.
		\begin{equation}\label{EX9}
		\dfrac{\partial^\alpha f}{\partial t^\alpha}=\dfrac{\partial^{\beta} }{\partial x^{\beta}}\left(	\dfrac{\partial^{\beta} }{\partial x^{\beta}}\left(\dfrac{\partial ^\beta }{\partial x^\beta}\left(\dfrac{f^2}{2}\right)\right)\right),~~\alpha,~\beta\in (0,1].
		\end{equation}
	\end{example}
	\noindent Here $I_2=\mathfrak{L}\{1,x^\beta,x^{2\beta}\}$ is an invariant subspace as
	\begin{equation*}
	N[k_0+k_1x^\beta,k_2x^{2\beta}]=\dfrac{\Gamma(4\beta+1)}{2~ \Gamma(\beta+1)}k_2^2x^{\beta}+k_1k_2\Gamma(3\beta+1)\in I_2.
	\end{equation*}
	We look for an exact solution of the form
	\begin{equation}\label{SOL9}
	f(x,t)=K_0(t)+K_1(t)x^\beta+K_2(t)x^{2\beta},
	\end{equation} where $K_0(t),~K_1(t)$ and $K_2(t)$ are unknown functions that are to be determined by solving the following system of FDEs.
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&\Gamma(3\beta+1)K_1(t)K_2(t)\label{91},\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&\dfrac{\Gamma(4\beta+1)}{2~ \Gamma(\beta+1)}K_2(t)^2,\label{92}\\
	\dfrac{\rm d^\alpha K_2(t)}{\rm d t^\alpha}&=&0.\label{93}
	\end{eqnarray}
	Eq.(\ref{93}) implies that $K_0(t)=$ constant $=c$ (say). Hence Eq. (\ref{92}) takes the form $	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}=c^2\dfrac{\Gamma(4\beta+1)}{2~ \Gamma(\beta+1)}$. Performing fractional integration on both sides, we obtain $K_1(t)=K_1(0)+\dfrac{c^2\Gamma(4\beta+1)}{2~ \Gamma(\beta+1)\Gamma(\alpha+1)}t^\alpha$. Proceeding on similar lines we get
	\begin{equation*}
	K_0(t)=K_0(0)+\dfrac{K_1(0)c\Gamma(3\beta+1)}{\Gamma(\alpha+1)}t^\alpha+\dfrac{c^3\Gamma(4\beta+1)\Gamma(3\beta+1)}{2~ \Gamma(\beta+1)\Gamma(2\alpha+1)}t^{2\alpha}.
	\end{equation*}
	Hence an exact solution of (\ref{EX9}) is
	\begin{eqnarray*}
		f(x,t)=\left[a+\dfrac{bc\Gamma(3\beta+1)}{\Gamma(\alpha+1)}t^\alpha+\dfrac{c^3\Gamma(4\beta+1)\Gamma(3\beta+1)}{2~ \Gamma(\beta+1)\Gamma(2\alpha+1)}t^{2\alpha}\right]\\
		+
		\left[b+\dfrac{c^2\Gamma(4\beta+1)}{2~ \Gamma(\beta+1)\Gamma(\alpha+1)}t^\alpha\right] x^\beta
		+cx^{2\beta},
	\end{eqnarray*}where $ a,~b$ and $c$ are arbitrary constants.\\

	Third order nonlinear time-fractional dispersion equation (corresponding to $\beta=1$) was studied in \cite{harris2014nonlinear} and the original model (corresponding to $\alpha=\beta=1$) was investigated by Galaktionov and Pohozaev \cite{galaktionov2008third}.

	\begin{example}
		Consider the following fractional version of non linear heat equation:
		\begin{equation}\label{EX10}
		\dfrac{\partial^{\alpha} f}{\partial t^{\alpha}}=N[f]=\dfrac{\partial^{\beta} }{\partial x^{\beta} }\left(f\dfrac{\partial^{\beta} f}{\partial x^{\beta}}\right),~~\alpha,~\beta\in (0,1].
		\end{equation}
	\end{example}
	\noindent$I_2=\mathfrak{L}\{1,x^{\beta}\}$ is an invariant subspace of N[f] as
	\begin{equation*}
	N[k_0+k_1x^{\beta}]=\dfrac{\partial^{\beta} }{\partial x^{\beta} }\left(k_0k_1\Gamma(\beta +1) +k_1^2\Gamma(\beta +1)x^{\beta}\right)=k_1^2[\Gamma(\beta+1)]^2 \in I_2.
	\end{equation*}
	We consider exact solution of the form
	\begin{equation}\label{SOL10}
	f(x,t)=K_0(t)+K_1(t)x^{\beta},
	\end{equation} such that
	\begin{eqnarray}
	\dfrac{\rm d^\alpha K_0(t)}{\rm d t^\alpha}&=&K_1(t)^2[\Gamma(\beta+1)]^2,\label{101}\\
	\dfrac{\rm d^\alpha K_1(t)}{\rm d t^\alpha}&=&0.\label{102}
	\end{eqnarray}
	Solving the system of FDEs given by (\ref{101})-(\ref{102}) we obtain
	\begin{align*}
	K_1(t)&=b,\\ K_0(t)&=K_0(0)+b^2\dfrac{[\Gamma(\beta+1)]^2}{\Gamma(\alpha+1)}t^\alpha.
	\end{align*}
	Substituting values of $K_0(t)$ and $K_1(t)$ in Eq. (\ref{SOL10}) we get an exact solution as
	\begin{equation*}
	f(x,t)=\left[a+b^2\dfrac{[\Gamma(\beta+1)]^2}{\Gamma(\alpha+1)}t^\alpha\right] +bx^{\beta},~a, b \in \mathbb{R}.
	\end{equation*}

	\begin{example}
		Consider the following fractional PDE:
		\begin{equation}\label{EX11}
		\dfrac{\partial^{\alpha} u}{\partial t^{\alpha}}=\dfrac{\partial^{\beta} u}{\partial x^{\beta}}+\left(\dfrac{\partial^{\beta} u}{\partial x^{\beta}}+u\right)^{-1},~~\alpha,~\beta\in (0,1].
		\end{equation}
	\end{example}
	\noindent Here $N[u]=\dfrac{\partial^{\beta} u}{\partial x^{\beta}}+\left(\dfrac{\partial^{\beta} u}{\partial x^{\beta}}+u\right)^{-1}.~I_2=\mathfrak{L}\{1,E_\beta(-x^\beta)\}$ is an invariant subspace as $
	N[k_0+k_1E_\beta(-x^\beta)]=-k_1E_\beta(-x^\beta)+\dfrac{1}{k_0}\in I_2,~k_0\neq 0.$\\
	Consider the exact solution of the form
	\begin{equation}\label{SOL11}
	f(x,t)=K_0(t)+K_1(t)E_\beta(-x^\beta),~K_0(t)\neq 0,
	\end{equation} where $K_0(t)$ and $K_1(t)$ satisfy the following system of FDEs
	\begin{eqnarray}
	\dfrac{\rm d^{\alpha} K_0(t)}{\rm d t^\alpha}&=&\frac{1}{K_0(t)},\label{111}\\
	\dfrac{\rm d^{\alpha} K_1(t)}{\rm d t^\alpha}&=&-K_1(t).\label{112}
	\end{eqnarray}
	Assuming that solution of Eq. (\ref{111}) is of the form $K_0(t)=\eta t^\gamma$ we get $K_0(t)=\pm\left(\frac{\Gamma(1-\alpha/2)}{\Gamma(1+\alpha/2)}\right)^{1/2}t^\frac{\alpha}{2}$ and using Laplace transform technique for Eq. (\ref{112}) we deduce $K_1(t)=K_1(0)E_{\alpha}(-t^\alpha)$ which leads to the following solution of Eq. (\ref{EX11}).
	\begin{equation*}
	f(x,t)= \left[\pm\left(\frac{\Gamma(1-\alpha/2)}{\Gamma(1+\alpha/2)}\right)^{1/2}t^{\frac{\alpha}{2}}\right]+[bE_{\alpha}(-t^\alpha)]E_\beta(-x^\beta),~b=K_1(0).
	\end{equation*}

	\section{Conclusions}
	Present article develops invariant subspace method for solving nonlinear time and space fractional partial differential equations. In this method FPDEs are reduced to systems of fractional differential equations which can be solved by existing analytic methods. In general fractional PDEs admit more than one invariant subspace, each of which yields an exact solution. In fact linear fractional PDEs admit infinitely many invariant subspaces (cf. (\ref{EX1}) and  (\ref{EX3})). Invariant subspace method is also used to derive closed form solutions of time and space fractional PDEs along with certain kinds of initial conditions. It is observed that invariant subspace method is very effective and rigorous tool for finding exact solutions of wide class of linear/ non linear FPDEs.

	\section*{Acknowledgements}

	Sangita Choudhary acknowledges the National Board for Higher Mathematics, India, for the award of Research Fellowship.

redpajama_set_name: RedPajamaArXiv


text:
Q: Grails 4 war deployment to Tomcat fails with no such method: reactor.core.publisher.Mono.from(Publisher)Mono/invokeStatic similar question: Grails got error when war file deployed to Heroku (Factory method 'requestMappingHandlerAdapter' threw exception)
but I'm facing it when deploying a war file generated by grails 4.0.11 into Tomcat(v8.5.39):
Caused by: org.springframework.beans.BeanInstantiationException: Failed to instantiate [org.springframework.web.servlet.mvc.method.annotation.RequestMappingHandlerAdapter]: Factory method 'requestMappingHandlerAdapter' threw exception; nested exception is java.lang.BootstrapMethodError: java.lang.IllegalAccessError: no such method: reactor.core.publisher.Mono.from(Publisher)Mono/invokeStatic
        at org.springframework.beans.factory.support.SimpleInstantiationStrategy.instantiate(SimpleInstantiationStrategy.java:185)
        at org.springframework.beans.factory.support.ConstructorResolver.instantiate(ConstructorResolver.java:622)
        ... 33 more
Caused by: java.lang.BootstrapMethodError: java.lang.IllegalAccessError: no such method: reactor.core.publisher.Mono.from(Publisher)Mono/invokeStatic
        at org.springframework.core.ReactiveAdapterRegistry$ReactorRegistrar.registerAdapters(ReactiveAdapterRegistry.java:194)
        at org.springframework.core.ReactiveAdapterRegistry.<init>(ReactiveAdapterRegistry.java:70)
        at org.springframework.core.ReactiveAdapterRegistry.getSharedInstance(ReactiveAdapterRegistry.java:179)
        at org.springframework.web.servlet.mvc.method.annotation.RequestMappingHandlerAdapter.<init>(RequestMappingHandlerAdapter.java:169)

I'm in the process of upgrading the project from grails 3.3.11.
WAR is being build with ./gradlew bootWar.
Local development env starts without problems.
dependencies in my build.gradle:
    compile 'org.springframework.boot:spring-boot-starter-logging'
    compile 'org.springframework.boot:spring-boot-autoconfigure'
    compile 'org.grails:grails-core'
    compile 'org.springframework.boot:spring-boot-starter-actuator'
    provided 'org.springframework.boot:spring-boot-starter-tomcat'
    compile 'org.grails:grails-web-boot'
    compile 'org.grails:grails-logging'
    compile 'org.grails:grails-plugin-rest'
    compile 'org.grails:grails-plugin-databinding'
    compile 'org.grails:grails-plugin-i18n'
    compile 'org.grails:grails-plugin-services'
    compile 'org.grails:grails-plugin-url-mappings'
    compile 'org.grails:grails-plugin-interceptors'
    compile 'org.grails.plugins:cache'
    compile 'org.grails.plugins:scaffolding'
    compile 'org.grails.plugins:gsp'
    compile 'org.grails:grails-web-fileupload'
    compile 'org.apache.tomcat:tomcat-jdbc'
    runtime "com.bertramlabs.plugins:asset-pipeline-grails:3.2.4"
    compile "org.grails.plugins:async"
    compile 'com.google.code.gson:gson:2.8.5'
    testCompile 'org.grails:grails-gorm-testing-support'
    testCompile 'org.grails:grails-web-testing-support'
    testRuntime 'org.seleniumhq.selenium:selenium-htmlunit-driver:2.47.1'
    testRuntime 'net.sourceforge.htmlunit:htmlunit:2.18'
    agent "org.springframework:springloaded:1.2.8.RELEASE"
    runtimeOnly "io.methvin:directory-watcher:0.9.6"
    compileOnly "io.micronaut:micronaut-inject-groovy"
    compile 'org.postgresql:postgresql:42.2.20'
    compile 'org.codehaus.groovy:groovy-dateutil:2.5.14'
    compile 'org.grails.plugins:hibernate5'
    compile 'org.hibernate:hibernate-core:5.4.18.Final'
    compile 'org.hibernate:hibernate-ehcache:5.4.18.Final'
    compile 'org.grails.plugins:spring-security-core:4.0.3'
    compile 'org.grails.plugins:spring-security-acl:4.0.0.M2'
    compile 'org.grails.plugins:postgresql-extensions:7.0.0'
    compile 'org.springframework.security.extensions:spring-security-saml2-core:1.0.3.RELEASE'
    compile 'org.grails.plugins:quartz:2.0.13'
    compile("org.quartz-scheduler:quartz:2.2.3") {
        exclude group: 'slf4j-api', module: 'c3p0'
    }
    compile 'com.google.api-ads:ads-lib:4.2.0'
    compile 'com.google.api-ads:adwords-axis:4.2.0'
    compile "com.googlecode.java-diff-utils:diffutils:1.2.1"
    compile 'net.coobird:thumbnailator:0.4.8'
    compile 'org.apache.ant:ant:1.9.4'
    compile 'commons-io:commons-io:2.5'
    compile 'com.cloudbees.thirdparty:zendesk-java-client:0.6.2'
    compile 'org.yaml:snakeyaml:1.23'
    compile 'cglib:cglib:2.2.2'
    compile 'org.apache.httpcomponents:httpclient:4.5'
    compile 'org.apache.pdfbox:pdfbox:1.8.9'
    compile 'commons-validator:commons-validator:1.6'
    compile 'io.fastjson:boon:0.33'
    compile 'com.thoughtworks.xstream:xstream:1.4.8'
    compile 'com.amazonaws:aws-java-sdk-s3:1.11.423'
    compile 'com.amazonaws:aws-java-sdk-core:1.11.423'
    compile 'com.amazonaws:aws-java-sdk-lambda:1.11.423'
    compile 'com.jcraft:jsch:0.1.53'
    compile 'com.opencsv:opencsv:5.2'
    compile project(':common')
    compile project(':query')
    compile project(':query-generator')
    compile 'org.grails.plugins:audit-logging:4.0.3'
    compile 'bouncycastle:bcprov-jdk14:138'
    compile 'org.grails.plugins:csv:1.0'
    compile 'org.grails.plugins:cookie:2.0.5'
    runtime 'org.grails.plugins:grails-console:2.1.1'
    compile 'org.grails.plugins:database-migration:3.1.0'
    compile 'org.liquibase:liquibase-core:3.10.1'
    compile 'org.grails.plugins:qrcode:0.9'
    compile 'org.grails.plugins:mail:3.0.0'
    compile 'org.grails.plugins:grails-executor:0.4'
    compile 'org.reflections:reflections:0.9.10'
    compile 'org.grails.plugins:sentry:11.7.25'
    runtime 'org.jolokia:jolokia-core'
    testRuntime 'net.sourceforge.htmlunit:htmlunit:2.18'
    testRuntime 'org.seleniumhq.selenium:selenium-support:2.53.1'
    testCompile 'org.seleniumhq.selenium:selenium-chrome-driver:2.53.1'
    compile 'org.grails.plugins:views-json:2.0.4'
    testCompile 'org.grails.plugins:greenmail:2.0.0.RC3'
    compile (group: 'com.nimbusds', name: 'nimbus-jose-jwt', version: '8.10')
        { exclude group: 'org.ow2.asm' }
    testCompile group: 'org.skyscreamer', name: 'jsonassert', version: '1.5.0'
    compile 'org.codehaus.gpars:gpars:1.1.0'
    testCompile 'org.grails:grails-test-mixins:3.3.0'
    compile 'com.syncron.amazonaws:simba-athena-jdbc-driver:2.0.2'
    testCompile 'org.mock-server:mockserver-netty:5.11.2'

A: I did a dependency scan and searched for anything reactive related.
It found reactive-streams as a dependency from micronaut.
+--- org.grails:grails-core -> 4.0.10
|    +--- org.codehaus.groovy:groovy:2.5.14
|    +--- org.slf4j:slf4j-api:1.7.30
|    +--- org.slf4j:jcl-over-slf4j:1.7.30
|    |    \--- org.slf4j:slf4j-api:1.7.30
|    +--- io.micronaut:micronaut-inject:1.3.7
|    |    +--- org.slf4j:slf4j-api:1.7.26 -> 1.7.30
|    |    +--- javax.annotation:javax.annotation-api:1.3.2
|    |    +--- javax.inject:javax.inject:1
|    |    +--- io.micronaut:micronaut-core:1.3.7
|    |    |    +--- org.slf4j:slf4j-api:1.7.26 -> 1.7.30
|    |    |    +--- org.reactivestreams:reactive-streams:1.0.3

Excluding it resolved the problem but I don't know what consequences it might cause.
configurations.all {
    exclude group: 'org.reactivestreams', module: 'reactive-streams'
}

Deployment to Tomcat as war works and the app appears to be stable so far.

redpajama_set_name: RedPajamaStackExchange


text:
Skip to main contentPoliticsImpeachmentWhite HouseCongressPollingThe TrailerFact CheckerThe Fix
Trump: Joe Lieberman is top contender for new FBI director
White House reporter
May 18, 2017 at 4:12 PM EDT
President Trump said Thursday that Joe Lieberman was his top choice to become FBI director, filling the slot left open after Trump fired James B. Comey, his previous FBI head, last week.
Trump, speaking to a group of television anchors at the White House Thursday, said that Lieberman, the former senator from Connecticut and Democratic vice presidential nominee, is his leading candidate to run the agency.
The president also told the group that he was close to a decision, an assertion he repeated later at a joint appearance with President Juan Manuel Santos of Colombia.
"We're very close to an FBI director," Trump told reporters as he posed for photos with Santos in the Oval Office, saying his choice would be announced soon.
Some senators on Capitol Hill Thursday were expressing skepticism Thursday about choosing a politician to lead the independent investigations bureau.
Lieberman, originally a Democrat and later an independent, was not always Trump's top pick, either. Previously, the president had expressed interest in Senate Majority Whip John Cornyn (R-Tex.) for the post. Cornyn took himself out of the running, saying he believed he could best serve the president by remaining in the Senate.

redpajama_set_name: RedPajamaCommonCrawl


This entry was posted on December 2, 2018 at 1:07 pm and is filed under Encouragement, Holy Spirit Power, Something to Think About.	You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


Classic mash combined with delicious tuna.
Light, tasty and served in no time.
Traditional dish with spicy ingredients.
Easy to forego meat with this dish.
This meal gives you wings.
Some say its the peak of pleasure.
A landmark of the Indonesian cuisine.
100% vegan - devilishly good.
Delicious food in less than 15 minutes!


text:
Evangelism as God Sees It
by Dr. Alva J. McClain (1888-1968)
It may surprise some readers that the English word evangelism does not occur once in the Authorized Version. And even its related term evangelist occurs but three times (Acts 21:8; Ephesians 4:11; 2 Timothy 4:5). But these few passages do not convey a fraction of the large place given in the Word of God to what has been named "evangelism." To discover this, one must examine the Greek term.
The Word for Evangelism
It is a compound word made of eu, meaning "good or well," and angelos, meaning "messenger." In the New Testament the word occurs in three forms: euangelion, which means "good news" and is uniformly translated "gospel"; euangelizo, which means "to tell good news" and is generally translated "preach" or "preach the gospel"; and euangelistes, which means "one who tells good news" and is translated "evangelist." The close relation between these Greek words, which occur 130 times in the New Testament, would be more apparent to the English reader if they had been rendered respectively "evangel," "evangelize," and "evangelist."
An examination of the New Testament passages reveals that the work which is called "evangelism" arises directly out of the Christian "evangel" or gospel. There were evangelists in the early church because there was an evangel to preach. The evangelists did not produce the evangel. The evangel produced the evangelists! Because this is so, all discussions of the work of evangelism should begin with its message.
The Message of Evangelism
What is the evangel, or gospel, of New Testament evangelism? It has several general characteristics: First, the gospel is a message of good news and therefore cannot be any mere system of law, ethics, or social program; second, the gospel is good news from God—a divine revelation, not a human philosophy; third, the gospel is good news concerning a person, the Lord Jesus Christ—what He is and what He did. More specifically, it concerns the incarnate Son of God who died for men's sins and rose again from the dead (see Romans 1:1-3 and 1 Corinthians 15:1-4); fourth, the gospel is the good news of complete salvation in Christ by grace through faith apart from all human works and righteousness (see Romans 4:3-5; 11:6; Ephesians 2:8-10). Anything else, no matter how good it may appear, is not the gospel that saves men from sin and its final doom.
The Work of Evangelism
Viewed from one standpoint, the Lord Himself is an Evangelist, for the Scriptures state that He preached beforehand the gospel to Abraham (Galatians 3:8). But the work of evangelism during the present age has been committed wholly to men. Ephesians 4:11-12, a very important passage on the subject of evangelism, tells which men are committed to the work of evangelism. "And He gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers; For the perfecting of the saints, for the work of the ministry, for the edifying (building up) of the body of Christ." The entire program of Christian evangelism is here. Note several things.
First, these names in this passage describe gifts to the church, not offices in the church. What is the difference? Offices are elective, wholly within the power of the church to confer or to withhold; and there are but two mentioned in the Bible—the eldership and the deaconate (1 Timothy 3). Gifts, on the other hand, are spiritual endowments, directly bestowed by a sovereign Christ. They may be and should be recognized by the church, but they cannot be conferred by any church. The church by election may create elders and deacons, but no church can make an evangelist or a pastor or teacher. Only God can do this.
Second, each one of the gifts mentioned in this Ephesians passage describes a distinct function in the church. The apostles were eyewitnesses to the fact of the gospel and founded the church. The apostles and prophets interpreted these facts under the guidance of the Holy Spirit, first orally and finally in the New Testament writings. The evangelists spread the good news, planting churches in new places. The pastors and teachers took over the new churches as shepherds of the flock.
Third, not all these gifts were intended to be permanent. Once the church was founded and equipped with the New Testament records, there was no further need of apostles, and the gifts associated with them passed away. But the work of enlarging the church, of pastoring and teaching it, must go on. Hence, down through the centuries and now, the remaining gifts are a permanent possession.
Fourth, a final purpose of these gifts is to build up the body of Christ through the ministry of its own membership. The pastors and teachers exist "for the perfecting of the saints." And this perfecting of the saints is "for the work of the ministry." And this work of ministering by the saints results in the building up of the body of Christ. God's purpose in giving evangelists and pastors and teachers to the church was not to build up a closed clerical order which would monopolize the ministry of soul-winning, but to exercise this ministry with and through a trained membership.
The Responsibility of Evangelism
Thus I conclude, the responsibility of evangelism rests upon every member of the church as a general duty. No one can escape it. But in a special sense this responsibility rests upon that select group of men who have in special measure the divine gift of evangelism. A study of the New Testament conveys the impression that the evangelist was an itinerant, going to new places, starting new churches. Once the church was started, the pastor was to teach the membership and work with and through them to extend the work of evangelism in that locality, while the evangelist moved on to a new place.
Various modern-day churches have departed from the program of the Word in two serious respects: First, many pastors have failed to teach their people to become a self-propagating body. Perhaps there are too many elected to the eldership who do not have the divine gift of pastoring and teaching. Or, on the other hand, perhaps too many pastors fail because they are trying to teach their members to do something which they are not doing themselves. Laziness can destroy the value of the gift. And this first blunder has led to the second: Because the average church has not become self-propagating, it has turned this task over to the evangelist, thus hindering his wider ministry of preaching in new places. And the evangelist, in turn, has too often become satisfied to confine his ministry to already existing churches, to the neglect of the unevangelized fields.
The Biblical Assumptions of Evangelism
Certain presuppositions stand behind all true New Testament evangelism. They may not be dealt with in any formal way, yet they are always present in the background. For lack of space these assumptions can be stated only briefly.
First is the reality of sin and its doom. Tell me the attitude of a church toward sin, and I will tell you its attitude toward evangelism. The cult of Eddyism (Christian Science), which denies the reality of sin, has lectures but no evangelists. And Modernism (religious liberalism), which weakens the sense of sin, creates an atmosphere which stifles evangelism.
Second is the assumption that salvation is by free grace, the work of God—not by men. Since evangelism is the heralding of the good news, it cannot live in the dead sea of legalism. It is not good news to tell men that they must save themselves. It is good news to know that the work is done. The preaching of law and commandments as a way of salvation may produce proselytes but not converts. It may change opinions, but it will not change hearts.
Third is the assumption that a response to evangelism depends on human responsibility. Man is not responsible to save himself. He cannot save himself. But man is responsible for his choices. He is not required to climb the steep ascent to heaven, but he is responsible to choose the Way of Life when it is set before him. Wherever the sense of personal responsibility is dulled, whether by a hyper-Calvinistic theology or a mechanistic psychology, true evangelism ceases.
Fourth is the assumption of the absolute lordship of Christ. Before the Lord gave His Great Commission to evangelize the nations, He reminded the disciples that "all power (authority)" had been given Him both in heaven and on earth. If this is true, then every human soul must deal at last with Christ, either for salvation or for judgment. There is no other Savior. There is no other Judge. This is what Paul had in mind when he wrote, "Knowing therefore the terror of the Lord, we persuade men" (2 Corinthians 5:11). Again, if Christ has all authority, then He has the right to command me as a Christian—my life, my talent, my substance—in the work of evangelism. This is what Paul meant when he said, "Woe is unto me, if I preach not the gospel" (1 Corinthians 9:16).
— Dr. Alva J. McClain authored numerous theological books and served as the president of Grace Theological Seminary in Winona Lake, IN.


text:
Q: Loading a hash from a file and comparing I have a file (abc.txt) which contains values such as :
ENSP00000200691_1 70.00 1.64119
ENSP00000200691_3 100.00 1.55602
ENSP00000201943_1 93.00 1.14492
ENSP00000201943_2 64.00 1.2178
ENSP00000201943_3 75.00 1.43107
ENSP00000203407_2 90.00 2.04312
ENSP00000203407_4 71.00 1.8197

I want to create a hash where values such as ENSP00000200691 will be the key (_1,_2.. are different examples of the same case). Now the comparison to make is :
When the second coloumn values is lower than its corresponding examples of the same case, and yet the third coloumn value is higher of the corresponding - it should print the lines.
For example.. the first two lines are of such a case where the second coloumn value 70<100 while yet 1.64119 > 1.55602 (this should be printed)
Rest of the other two cases are of where $2(example1) > $2 (example2) and also $3(example1) > $3 (example2).
There can be around 10 examples of the same hash(case)

A: Here is a guess at how to solve your (homework?) problem:
#!/usr/bin/perl

use strict;
use warnings;

my %info;

while (my $line=<DATA>) {
    chomp $line;
    my ($protein_id, $length, $value)=split /\s+/, $line;
    $protein_id=~s/_\d+$//;

    if (exists $info{$protein_id}) {
        if ($length<$info{$protein_id}->{length} &&
            $value>$info{$protein_id}->{value}) {
            print "$line\n";
        }
    }
    else {
        $info{$protein_id}={ length=>$length, value=>$value };
    }
}

__DATA__
ENSP00000200691_1 70.00 1.64119
ENSP00000200691_3 100.00 1.55602
ENSP00000201943_1 93.00 1.14492
ENSP00000201943_2 64.00 1.2178
ENSP00000201943_3 75.00 1.43107
ENSP00000203407_2 90.00 2.04312
ENSP00000203407_4 71.00 1.8197

The output when run is this:
ENSP00000201943_2 64.00 1.2178
ENSP00000201943_3 75.00 1.43107

You didn't quite specify whether subsequent lines should always be compared to the first protein id, or if the last one seen, or the last one printed.


text:
Q: How do I get a queued brew to start? I have a brew in the queue but I wouldn't start it just sits there and it has all ingredients needed... So how do I start it?

A: You need to move the last beer you brewed out of the Holding Area. Either move it to storage, or send all of the bottles out to market to be sold.
You can see this by going to the menu, selecting Manage and then Storage.
Your team won't start brewing until the holding area is empty.


text:
Storystreams
Looking forward: A new MLS team in Los Angeles emerges
Contributors: Alicia Rodriguez
Share All sharing options for: Looking forward: A new MLS team in Los Angeles emerges
Your source for coverage of the new MLS team in LA.
12 Total Updates Since
Oct 27, 2014, 11:02am PDT
Reflections on day one of LAFC's existence
By Alicia Rodriguez
Los Angeles Football Club was unveiled on Thursday as MLS' newest team. We reflect on the first day, discuss what's promising and perhaps concerning, and more.
LAFC's Tom Penn: "We're fully funded" on stadium
Los Angeles FC co-owners Tom Penn and Henry Nguyen discussed the stadium situation for the new MLS club on Thursday during the team's unveiling event. We share their comments on the search for a location, funding, and more.
Tom Penn speaks about LAFC academy promise, plans
New Los Angeles FC co-owner Tom Penn spoke to reporters about a variety of topics concerning the new club, including the plans and status of the development academy, and whether Chivas USA's academy would play a role with LAFC.
Hamm on owning LAFC, prospects of women's team
Los Angeles FC co-owner Mia Hamm spoke to the press on Thursday following the announcement about the team. She discussed her role in the group, the prospects of a women's pro team, and more.
Watch: Garber on creation of LAFC
MLS Commissioner Don Garber spoke about the creation of the new team in Los Angeles, LAFC, on Thursday. We share video of his remarks.
Full release of new MLS team in Los Angeles
We share the full release of the new MLS team in Los Angeles, announced on Thursday.
Los Angeles FC has 22 owners
According to its website, the new MLS club in Los Angeles has 22 owners. We list them all.
New team to be called Los Angeles Football Club
A website is live, and it looks to be for the new MLS team in LA, Los Angeles Football Club. We have the latest news, and provide a link to the new site.
WATCH: New LA MLS club unveiling event
The new era of MLS in Los Angeles begins Thursday, with the official unveiling of the new team and ownership group. We've got the stream of the press conference to watch live.
5 big questions ahead of Thursday's LA2 unveiling
With the new MLS team in LA set to be unveiled, in some capacity, on Thursday, we ask several big questions for the new club.
Twellman: New LA club to play at LA Coliseum
ESPN's Taylor Twellman reported about the temporary home of the new LA MLS club. We go over the report, and share video of the pundit talking about the story.
CUSA cease operations; LA2 to debut in 2017
Chivas USA are officially done, as the league has announced the club will cease operations immediately. We go over that and several other key news items on a dark day for the club.


The aliens have decided that for their experiments suitable cows that live on Earth. You know about the plans and now we need to stop them.
© sgames.org - free online game Save the cow ().
Play online flash game Save the cow for free. Do not forget to rate this game Save the cow and share this game with your best friends.
Description of the game Save the cow line. How to play the online game The aliens have decided that for their experiments suitable cows that live on Earth. You know about the plans and now we need to stop them. Armed with a gun, shoot at the monsters, which are planted on the ground of their ships. So try to keep as many poor animals while earning valuable points. Between each round of the game Save the cow is available to you gun shop where you can buy something new.


text:
Hudson Valley Filmed Movie Surpasses $100 Million in Ticket Sales
Jess Published: June 18, 2021
Youtube, Paramount
A movie, which filmed in several locations around the Hudson Valley, just broke a pandemic record.
Throughout the last few weeks, we've been getting "back to normal." That of course is a reference to the pre-pandemic world. When the U.S. was hit with the COVID-19 pandemic back in March of 2020 everything immediately shut down. Besides small local businesses, the entertainment industry was one of the hardest-hit industries due to the pandemic.
Production was shut down for many films and television shows and movies that were set to be released in theaters we're pushed back for months.
One of the films with, several, rescheduled dates was A Quiet Place Part II. The John Krasinski-directed horror film starring Emily Blunt was famously shot throughout New York with many scenes being filmed in Orange, Ulster, Dutchess and Putnam counties.
A Quiet Place Part II was released on May 28th, 2021, and according to Variety.com, it has overcome a pandemic milestone. Variety reports:
The Paramount Pictures film is the first of the pandemic era to surpass $100 million in U.S. ticket sales. Through the weekend, "A Quiet Place 2" has generated $108 million at the domestic box office and another $80 million internationally.
Variety explains that normally, this milestone wouldn't be such a big deal however, "the pandemic has been crushing to the movie theater business and ticket sales have still been slow to recover."
You can still see A Quiet Place Part II in theaters, but after 45 days it will also be available on the Paramount Plus streaming service.
Have you seen A Quiet Place Part II? If you did, did you recognize the Hudson Valley in any of the scenes?
Kate Pierson of the B-52's Is Selling Her Sprawling Hudson Valley 'Love Shack' Compound
Kate Pierson's Kate's Lazy Meadow in Shandaken
Celebrities Who Live in the Hudson Valley
The Most Wikipedia Searched Person From Your Hudson Valley Town
The Hudson Valley's Most Wikipedia Searched Celebs


Jacob and Megan Brandeis are a dying breed, authors of books that you can find in a brick and mortar store - something that is quickly dying out thanks to the convenience of the Store. It might have started small, but the Store has grown into a massive super company that not only conveniently supplies everything you need, but based on your data can recommend things you didn't even know you wanted or needed. When their publisher tells them that the book they have been working on for the past two years is no longer going to be published because of changes in the market they reach out in desperation to the Store to see if it can be published and after a promising start their hopes are dashed and they come up with a new plan - write a book about the Store instead.
It seems like a simple plan, even though it means moving their two children Alex and Lindsay to small town Nebraska where the Store has their massive distribution centre. It is a huge change in lifestyle and comes with some strangely lavish perks - including a house that seems too good to be true, a school with all the high tech gadgets you could wish for, and the convenience of drone deliveries from their local stores. It all seems too good to be true, and the Barndeis family quickly discovers that being part of the Store family also means something darker, with hidden surveillance and claws in the velvet glove the Store extends to the family. It soon becomes clear that Jacob is struggling to tow the company line, and standing out from the crowd at the Store is not necessarily a good thing. As his family is sucked into the 'cult' of the Store, Jacob finds himself slowly but surely on the edge - and once he tips over their may be no coming back. Surely Jacob is overreacting, surely the Store is just a convenient and savvy way to shop - or is it?
I was eagerly looking forward to The Store, mainly because Patterson writes some amazing speculative fiction about what might be just around the corner - fiction that makes you think about what is happening in a fresh light. While the Store doesn't have a name other than the Store, it does seem to be based around the idea of a company similar to Amazon, and I have to confess that it was hard not to compare the Store to Amazon as I was reading (sorry Amazon!). One of the most interesting, not to mention scary, aspects of The Store is that all the freedoms people have lost in the book are freedoms they gave away - they weren't stolen - and it makes me wonder how many people think about the information they give away every day, and the special treatment companies get each day. For younger generations it is perfectly natural to post a lot of personal information online, and it was interesting to see the divide between young and old clearly expressed in the book - but in a way that makes the older generation seem out of touch and 'dated' rather than in the right.
The Store was devoured in a single sitting and when I reached the end it was a very satisfying feeling to have reached the end - especially with the way the book ended. Patterson is an expert at making you wonder what is happening right to the end, and The Store is no exception. This book may not appeal to all of Patterson's fans because it is more speculative fiction than thriller or crime fiction, but I found it well crafted and expertly drawn out to give you maximum enjoyment and maximum time to figure out what is going on. This is the kind of book that would make an amazing movie because it is a tense psychological thriller that makes you question what you know, and makes you wonder what you would do in the same situation. If the Store really existed would you be content to let them rule your lives completely, or would you fight to keep your personal freedoms and choices?


Call for applications for one of the best Technical Studentships at CERN Switzerland. This is fully funded Technical Training Program and all the expenses will be covered by CERN.
f your university or institute requires or encourages you to acquire work experience through an internship, imagine doing this at CERN in Geneva. It's more than work experience. In fact, it's a student programme like nowhere else on Earth and an impressive addition to your CV!
If you are a student looking to complete practical training in domains related to general or civil engineering, surveying or safety, you will have the opportunity to work at the cutting edge of technology, contribute and broaden your knowledge in areas as varied as industrial and tertiary building structures, design and drafting, underground structures, roads, drainage, geotechnical engineering, safety risk management to name but a few.
An allowance of 3283 Swiss Francs per month (net of tax).
Also apply for some amazing programs of CERN Switzerland.
You have completed at least 18 months of your undergraduate studies (Bachelor or Master's) at the time of the next student selection taking place in December 2018. PhD students are not eligible for this programme.
How to apply for CERN Technical Studentships 2018?
Application deadline is not given. Apply as soon as possible.


Rachael Suhie has 4 articles published.
Rachael Suhie is a junior at the University of Connecticut and a current intern at Meryl Moss Media. She loves reading and writing as well as a hot cup of coffee! When she's not in class or working, she likes to spend time outside with her dogs.


text:
Hotels in British Columbia
Hotels in Vernon
Hotels with pool in Vernon
Holiday Inn Express Vernon
4716 - 34th Street, V1T 5Y9 Vernon, Canada
Vernon, 1.9 km from the city centre. 4716 - 34th Street See on map
Located in the heart of Okanagan Valley. - Number of rooms: 85 - Accommodation: air conditioning, TV, in-room movies (charge), telephone,...
27th Street, 4801, V1T 4Z1 Vernon, Canada
Vernon, 2.1 km from the city centre. 27th Street, 4801 See on map
This Vernon hotel is across from the Village Green Mall. The hotel offers a restaurant, a bar, and a liquor store. An independently run...
Sandman Inn & Suites Vernon
4201 32nd Street, V1T 5P3 Vernon, Canada
Vernon, 1 km from the city centre. 4201 32nd Street See on map
Featuring an indoor pool and sit-down restaurant, Sandman Inn & Suites Vernon is in Vernon. Free Wi-Fi is included. Rooms here include a...
Super 8 Vernon
32nd Street, 4204, V1T 5P4 Vernon, Canada
Vernon, 1 km from the city centre. 32nd Street, 4204 See on map
With a stay at Super 8 Vernon in Vernon, you'll be convenient to Village Green Centre and Polson Park. This motel is within close...
Vernon Lodge
32nd Street Vernon, 3914, V1T 5P1 Vernon, Canada
Vernon, 856 m from the city centre. 32nd Street Vernon, 3914 See on map
Business travelers will take pleasure in their stay at the Best Western Vernon Lodge and Conference Center which provides an atmosphere...
Prestige Hotel & Conference Centre Inn Vernon
32nd Street, 4411, V1T 9G8 Vernon, Canada
Vernon, 1.3 km from the city centre. 32nd Street, 4411 See on map
Featuring a modern restaurant and lounge, this decorative hotel is located on Highway 97 and less than 5 minutes from downtown Vernon....
The Outback Lakeside Resort
9845 Eastside Road, V1H 1Z2 Vernon, Canada
Vernon, 15.7 km from the city centre. 9845 Eastside Road See on map
These Vernon, British Columbia vacation homes are located on the shores of Okanagan Lake. Private beaches are available on the property....
Predator Ridge Resort
301 Village Centre Place Vernon, V1H1T2 Vernon, Canada
Vernon, 11.6 km from the city centre. 301 Village Centre Place Vernon See on map
Located in Vernon, Predator Ridge Resort is within the vicinity of Kalamalka Provincial Park and Allan Brooks Nature Centre. This golf...
Sparkling Hill Resort
Sparkling Place 888, V1H 2K7 Vernon, Canada
Vernon, 321 m from the city centre. Sparkling Place 888 See on map
Sparkling Hill Resort is approximately nine kilometers south of the urban centre of the city of Vernon in the North Okanagan. Located on...
Fairfield Inn & Suites Vernon
Anderson Way 5300, V1T 9V2 Vernon, Canada
Vernon, 2.5 km from the city centre. Anderson Way 5300 See on map
An indoor swimming pool with a waterslide and a hot tub are featured at this Vernon hotel. Free Wi-Fi is available in all areas. Daily...
The Castle At Swan Lake
7905 Greenhow Road, V1B 3S2 Vernon, Canada
Vernon, 9 km from the city centre. 7905 Greenhow Road See on map
When you stay at The Castle at Swan Lake by Pomeroy in Vernon, you'll be close to Historic O'Keefe Ranch. This hotel is within the...
Days Inn Vernon
26th Street, 5121, V1T 8G4 Vernon, Canada
Across from the Village Green Mall in Vernon, British Columbia, this property features a heated indoor pool and access to golf and...
Canadas Best Value Inn & Suites Vernon
4006-32nd Street, V1T 5P1 Vernon, Canada
Vernon, 822 m from the city centre. 4006-32nd Street See on map
With a stay at Canadas Best Value Inn & Suites in Vernon, you'll be convenient to Village Green Centre and Polson Park. This hotel is...
Econo Lodge Vernon
When you stay at Econo Lodge Vernon in Vernon, you'll be near the airport and convenient to Village Green Centre and Polson Park. This...
Schell Motel
2810 35th St, Vernon, Canada
Vernon, 432 m from the city centre. 2810 35th St See on map
Ubicación del establecimiento Desde Schell Motel, en Vernon, tendrás fácil acceso a Parque Polson y a Granja apícola Planet Bee. Este...
Tiki Village Motor Inn
Tiki Village Inn and Suites is known for Polynesian decor and the beautifully groomed Japanese courtyard. Enjoy the summer sun in or...
Visit Vernon
Destinia offers you a search engine designed for you to choose from among all the hotels with pools in Vernon in the market. This type of accommodation is perfect for the summer, to escape from the heat and also in winter, as it allows you to practise the most complete sport of all: swimming.
Pools are a service that more and more luxury hotels tend to provide, since they offer endless possibilities of enjoyment for guests. For that reason we offer you a large selection of hotels with pools in Vernon for you to choose the one that best suits your needs and expectations. More and more people are taking an interest in booking a hotel with a pool in Vernon and our filter of hotels with pools in Vernon may be very helpful.
As always, we make every effort to offer you the best hotels with pools in Vernon at the best price, to allow you to feel as comfortable as possible and use your money to pay for other travel expenses that are equally or more important.
Hotels in the centre of Vernon
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With experience with international management, financial consulting and fraud investigations as well as working directly with cannabis operators for three years, Simone is an asset for determining the rules and regulations for a municipal cannabis program. She insures that a City consider all economic and business impacts set forth by regulation both on a state and local level, and its impact on cannabis operators. The percentage of excise tax, local licensing costs, coupled with state and federal (280E) cannabis taxation are all important factors. If done incorrectly, this could have a negative impact on the city in the long run. She is great assistance in developing an appropriate discretionary review process, multi-phased merit based application process, as well as developing the permitting fee and tax structure and costs.


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House prices up by 10% annually in November
Vicky Shaw, PA Personal Finance Correspondent
Annual house price growth hit 10% in November, according to an index.
The increase marked an acceleration from 9.9% growth in October, Nationwide Building Society said.
Prices rose by 0.9% month on month, taking the average UK property value to £252,687.
Robert Gardner, Nationwide's chief economist, said: "House prices are now almost 15% above the level prevailing in March last year when the pandemic struck the UK.
"There have been some signs of cooling in housing market activity in recent months; for example, the number of housing transactions were down almost 30% year on year in October.
"But this was almost inevitable, given the expiry of the stamp duty holiday (in England and Northern Ireland) at the end of September, which gave buyers a strong incentive to bring forward their purchase to avoid additional tax.
"Indeed, activity has been extremely buoyant in 2021. The number of housing transactions so far this year has already exceeded the number recorded in 2020 with two months still to go and is actually tracking close to the number seen at the same stage in 2007, before the global financial crisis struck.
"Moreover, underlying activity appears to be holding up well. The number of mortgages approved for house purchases in October was still running above the 2019 monthly average.
"Early indications also suggest that labour market conditions remain robust, despite the furlough scheme finishing at the end of September.
"If this is maintained, housing market conditions may remain fairly buoyant in the coming months, especially since the market has momentum and there is scope for ongoing shifts in housing preferences, as a result of the pandemic, to continue to support activity."
But Mr Gardner said the outlook remains uncertain, adding: "It is unclear what impact the new Omicron variant will have on the wider economy.
"While consumer confidence stabilised in November, sentiment remains well below the levels seen during the summer, partly as a result of a sharp increase in the cost of living. Moreover, inflation is set to rise further, probably towards 5% in the coming quarters.
"Even if economic conditions continue to improve, rising interest rates may exert a cooling influence on the market. Indeed, house price growth has been outpacing income growth by a significant margin and, as a result, housing affordability is already less favourable than was the case before the pandemic struck."
Samuel Tombs, chief UK economist at Pantheon Macroeconomics, suggested mortgage rates could head upwards. Swap rates, which lenders use to price their loans, have increased, and profit margins on home loans are already "very tight by past standards", he said.
"Admittedly, the link between variations in mortgage rates and changes in house prices isn't stable," he said.
"Households can lengthen mortgage terms to offset the hit to monthly repayments from higher rates. In addition, the proportion of incomes that households are prepared to devote to housing might have risen permanently due to the pandemic and the long-term shift to working from home.
"Households also likely will draw on the large pot of savings they have accumulated during the pandemic to finance home purchases.
"Nonetheless, we continue to think that mortgage rates will rise enough over the coming months, to ensure that house prices stagnate in the first half of 2022 and then rise only slowly in the second half, thereby slowing the year-over-year growth rate of prices to about 2.0% by the end of next year."
Tomer Aboody, director of property lender MT Finance, said Government support including the stamp duty holiday and the furlough scheme has helped support the market.
He added: "The future is slightly uncertain with the ever-changing global picture thanks to Covid and the possibility of interest rate rises, which will slow down growth."
Guy Gittins, chief executive of estate agency Chestertons, said: "As the market witnessed a great deal of appetite from house-hunters right after the summer holidays, particularly during October, we are expecting the number of agreed sales to remain high for the rest of the year and into quarter one 2022."
Martin Beck, senior economic adviser to the EY Item Club, said: "It appears that any downward pressure on property prices from stamp duty returning to its pre-pandemic level at the start of October has been countered by other factors."
He said a recent decline in mortgage approvals "bodes well for the housing market avoiding a significant correction now that stamp duty has returned to its normal level".
He added: "The pandemic appears to have prompted a structural increase in demand for houses from people seeking more outdoor space and room to work from home – the 'race for space'.
"That demand is being facilitated by the substantial 'excess' savings accumulated by households during lockdowns – which rose to an estimated £180 billion in the autumn – and a jobs market emerging from the pandemic in a remarkably little-damaged date.
"Moreover, despite the MPC (Bank of England Monetary Policy Committee) adopting a more hawkish tone on the prospect of interest rate rises, mortgage rates fell to a record low in October.
"And uncertainty stemming from the new Omicron variant reinforces the EY Item Club's expectation that the MPC will hold off raising rates until next year. So, while house price growth is likely to slow during 2022, an outright fall seems unlikely."
Kim Kardashian just subtly shouted Pete Davidson out on Instagram
Global Premium Furniture and Furnishings Market Report 2021: Focus on Furniture, Housewares, Textiles, Lighting - ResearchAndMarkets.com
Kim Kardashian just highlighted the P in her Instagram caption and fans are convinced it's a reference to Pete Davidson.
DUBLIN, January 20, 2022--The "Global Premium Furniture and Furnishings Market - Analysis By Product (Furniture, Housewares, Textiles, Lighting, Others), Distribution Channel, End Use, By Region, By Country (2021 Edition): Market Insights and Forecast with Impact of COVID-19 (2021-2026)" report has been added to ResearchAndMarkets.com's offering.
The disgraced socialite was convicted of trafficking teenage girls for sex after a three-week trial in New York.
Hurry! The famous 'Amazon coat' is on mega-sale starting at just $88 — but only 'til midnight
Uber popular for its warmth, plethora of pockets and super cute style, this coveted coat is 67 percent off.
From divorcees to fuller classrooms, the 1921 census of England and Wales depicts an era of great change
Census data is kept under seal for a century. The 1921 records will be the last we see until 2052.
A prosecutor in Norway said Thursday that a far-right extremist who killed 77 people in 2011 still is "a very dangerous man" and therefore a poor candidate for release after 10 years in prison, as Norwegian law permits. On the final day of a three-day parole hearing, prosecutor Hulda Karlsdottir said in her closing argument that Anders Behring Breivik "has not shown any genuine remorse in court" and his behavior there is part of a "PR stunt." Breivik professed white supremacist views and flashed Nazi salutes on the hearing's opening day while claiming to have renounced violence.
Digital Trends in the BFSI Industry Amidst COVID-19 and Beyond, 2022 Survey Report - ResearchAndMarkets.com
DUBLIN, January 20, 2022--The "Digital Trends in the BFSI Industry Amidst COVID-19 and Beyond" report has been added to ResearchAndMarkets.com's offering.
At the same time, Asia-Pacific is set to record the highest CAGR by 2028
The market prospects of Asia-Pacific are attributed to the increasing clinical trial management product awareness and dense population base. COMPETITIVE INSIGHTS The market has an increasing number of market players, along with a considerable scope of innovation.New York, Jan. 20, 2022 (GLOBE NEWSWIRE) -- Reportlinker.com announces the release of the report "GLOBAL CLINICAL TRIAL MANAGEMENT MARKET FORECAST 2022-2030" - https://www.reportlinker.com/p05761999/?utm_source=GNW Bio-Optronics Inc, Med
Pickering Interfaces Joins OpenTAP Open Source, Test Automation Software Partner Program
CLACTON-ON-SEA, England, January 20, 2022--Pickering Interfaces, the leading supplier of modular signal switching and simulation solutions for electronic test and verification, is now an OpenTAP partner. Pickering joins other notable test companies in the open source, shared software workshop that enables users to contribute, collaborate and create automated test solutions.
Insights on the Healthcare API Global Market to 2027 - Featuring Accenture, Apple and Lyniate Among Others - ResearchAndMarkets.com
DUBLIN, January 20, 2022--The "Healthcare API Market Research Report by Deployment Mode, by End-user, by Region - Global Forecast to 2027 - Cumulative Impact of COVID-19" report has been added to ResearchAndMarkets.com's offering.
MOSCOW (AP) — Russia accused the West on Thursday of plotting "provocations" in Ukraine even as it blames Moscow of planning aggressive military action in the neighboring country. Russian Foreign Ministry spokeswoman Maria Zakharova alleged that Ukrainian and Western claims of an imminent Russian attack on Ukraine were a "cover for staging large-scale provocations of their own, including those of military character." "They may have extremely tragic consequences for the regional and global securi
Global Commercial Vehicle Air Deflector and Commercial Vehicle Market Report Forecast [2022-2028]
Report Forecast [2022-2028], Global Commercial Vehicle Air Deflector and Commercial Vehicle Market Size, Share, Growth, Trends, Competitive Landscape, Forecast ReportPune, India, Jan. 20, 2022 (GLOBE NEWSWIRE) -- Part I: Commercial Vehicle Air Deflector Market Research Report: The global commercial vehicle air deflector market size is set to experience dynamic growth in the upcoming years owing to the rising production of commercial vehicles across the world, finds Fortune Business Insights™ in
Global Robotic Process Automation Markets, 2021-2030: Partnerships Among Software Vendors and Service Providers & Investments In Emerging RPA Companies and Startups
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Cloud Storage Market to Reach USD 353.81 billion and Exhibit at a CAGR of 24.48% from 2021 to 2028 : The Brainy Insights
Cloud Storage Market Size by Component (Solution, Services), Deployment Type (Private, Public, Hybrid), User Type, Industry Vertical, Regions, Global Industry Analysis, Share, Growth, Trends, and Forecast 2021 to 2028Newark, NJ, Jan. 20, 2022 (GLOBE NEWSWIRE) -- As per the report published by The Brainy Insights, The global Cloud Storage market is anticipated to grow from USD 61.37 billion in 2020 to USD 353.81 billion by 2028, at a CAGR of 24.48% during the forecast period 2021-2028. The surge


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Virtual Author Talk with Fairfield Alumna Sarah Sleeper MFA'12
The Fairfield University Bookstore will host a virtual conversation with MFA alumna Sarah Sleeper about her debut novel Gaijin on Facebook Live: Thursday, April 8 at 6:30 p.m.
Join us Thursday, April 8 at 6:30 p.m. on Facebook Live @FairfieldUBookstore for lively virtual conversation between author and Fairfield MFA alumna Sarah Sleeper and Fairfield University MFA Professor Karen Osborn, as they discuss Sleeper's new novel Gaijin. The Japanese word gaijin means "unwelcome foreigner," a slur directed at non-Japanese people in Japan.
One of the novel's main characters, Lucy, is a budding journalist at Northwestern University, obsessed with an exotic new student, Owen Ota, who becomes her lover and her sensei. When he disappears without explanation, Lucy is devastated and sets out to find him. This story of the "unwelcome foreigner" is not an easy one, and Sleeper gives it the precision, sensitivity, and depth it deserves. Mark the date: April 8, and tune in to the Fairfield University Bookstore's Facebook page at 6:30 p.m. to join the event.
Gaijin is Sleeper's first novel. Her short story, "A Few Innocuous Lines," won an award from Writer's Digest. Her non-fiction essay, "On Getting Vivian," was published in The Shanghai Literary Review. Sleeper's poetry was published in A Year in Ink, San Diego Poetry Annual and Painters & Poets, and exhibited at the Bellarmine Museum. In the recent past she was an editor at New Rivers Press, and editor-in-chief of the literary journal Mason's Road. Sleeper completed her MFA in Creative Writing at Fairfield University's College of Arts and Sciences in 2012. Prior to that she had a 25-year career as a business writer and technology reporter, during which she won three journalism awards and a fellowship at the National Press Foundation.
For more information regarding upcoming bookstore events please follow the Fairfield University Bookstore on Facebook.
Sarah Sleeper MFA'12 to Discuss her Debut Novel With MFA Professor Karen Osborn
Date: Thursday, April 8, 2021
Location: Facebook Live: @FairfieldUBookstore
Join the Facebook Live Event →
Tags: Alumni Stories
April 2021 Story Archive →


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PARTNERING TOGETHER TO END
WORLD HUNGER.
KFC NASSAU ADD HOPE CAMPAIGN 2019
KFC NASSAU'S ADD HOPE CAMPAIGN RAISES NEARLY $30,000 TO SUPPORT HANDS FOR HUNGER.
With its service extending beyond the four walls of its restaurants and into the local communities it serves, KFC Nassau accelerated its annual quest to help alleviate hunger in The Bahamas with the early launch of its Add Hope Campaign. Add Hope enables the local community, KFC customers and employees to donate to those facing food insecurity within Bahamian communities including those affected by Hurricane Dorian. This year, the campaign raised almost $30,000 to support leading local hunger-relief organization, Hands for Hunger. The donation will make a tangible impact by providing food for those in need through Hands for Hunger's daily mission of tackling the hunger epidemic. For every $1 received Hands for Hunger is able to provide a nutritious meal to those who need it most.
KFC Nassau has partnered with Hands for Hunger for nearly 10 years through financial support, product donations, and employee volunteerism. The partnership between the two organizations enhances the shared passion to take action to help those in need. Through Hands for Hunger and its network of agencies and volunteers, KFC Nassau is able to reach local communities efficiently and effectively.
"We're always proud to partner with Hands for Hunger to help address and reduce the significant issue of food insecurity in our local communities. This year, the need was even greater due to the unfortunate impact of Hurricane Dorian, which led us to launch our Add Hope Campaign earlier than usual, which allowed us to make an impact in a timely way," said Eldira Backford, KFC Nassau Add Hope Campaign Ambassador. "We are grateful for the overwhelming response from the local public and our team in joining KFC and Hands for Hunger in our shared commitment to ending food insecurity one community at a time."
KFC Nassau held its Add Hope award and cheque presentation on November 20th, acknowledging the hard work showcased by its staff during such a difficult time and presenting Hands for Hunger with a cheque for $29,357 to assist in providing access to nutritious food to Bahamians in need through its partnered agencies.
KFC Nassau is part of the Yum! global group of brands and is operated by Restaurants (Bahamas) Limited throughout New Providence. More information on KFC Nassau can be found on its website, www.kfcnassau.com, or on Facebook, www.facebook.com/kfcnassau. Information on Hands for Hunger can be found at www.handsforhunger.org
Photo Caption Slide 1: Kendria Ferguson, Executive Director - Hands for Hunger, Mrs. Meadows, Lead Manager – KFC Marathon Road, Eldira Backford, Add Hope Campaign Ambassador, Willard Grant, Director of Operations, KFC Nassau
Photo Caption Slide 2: Monique Brennen, Volunteer and Community Events Manager – H4H, Keisha Ellis, Strategic Initiatives Manager - H4H, Kendria Ferguson, Executive Director - Hands for Hunger, Eldira Backford, Tamara Flowers, Disaster Relief Liaison – H4H
Our teams around the globe are hard at work building brands with purpose. World Hunger Relief (WHR) is an initiative that unites our system around a common cause. The success of the movement is centered around the way our teams take it and make it their own – from uniquely branded campaigns like KFC's Global Initiative Add Hope or China's two week campaign 'Love Connects Us All' – to raise awareness and funds to help end hunger.
We work together with supportive partners throughout the world to help end hunger.
Hands For Hunger is a humanitarian organization committed to the elimination of unnecessary hunger and the reduction of food waste through the creation of meaningful and engaging partnerships formed amongst all sectors of the Bahamian community.
Assisting 91.4 million people in around 83 countries each year, the World Food Programme (WFP) is the leading humanitarian organization saving lives and changing lives, delivering food assistance in emergencies and working with communities to improve nutrition and build resilience.


- Tablet Support: Enjoy Apple Music with an experience designed for a wider range of Android devices.
- Performance improvements for images and audio playback.
In August, Apple Music updated on Android with support for Android Auto, letting Android smartphone owners control playback of Apple Music songs directly from the infotainment center in their vehicle. Android Auto support was part of Apple Music's 2.6.0 beta on Android, which also included numerous other features already found on Apple Music on iOS: lyric searches, updated artist pages, and the new weekly playlist called "Friends Mix."
Even the peasants have Apple Music now.
It's been available on Android tablets since the Android app launched, you mean to say they are getting around to optimizing the app for tablets specifically which is new.
And while I agree I don't foresee them doing it voluntarily I do see RCS at least being somewhat of a pending competition and it wouldn't hurt to incentivize people to more exclusive features of Apple through freebies like iMessage that if offered, would eradicate WhatsApp rapidly. Also on the security portion of it, RCS isn't encrypted and that could be another push. Apple is running into shallow water with the prices of their phones putting them out of reach of even beginning to get into the ecosystem to enjoy the full benefits let alone partial ones. It's theories really but I do see just reasons for them to open the gates, even if it's watered down.
I just remembered that Steve Jobs said FaceTime would be open source when it was announced, but I believe that never happened due to a patent dispute. I don't know if I missed that being mentioned in the discussion. Apple has been getting more aggressive in the services arena, so it's not impossible.
Apple's showing more support for Android tablets than most Android devs.
Whoop-dee-doo, now all, what, 100 owners of Crapdroid tablets will enjoy the benefits of Apple Music? Oh, I forgot, they probably won't, since they're so cheap they went for the $100 dumpster-diver tablets, and are likely too cheap or unethical to actually pay for their monthly music subscription!
I'm surprised there wasn't already a tablet app.
I don't believe many apps on Android have app specifically designed for tablets. That's an Apple thing. They generally just scale up the size.
Well yes iTunes was a great source of media management but you aren't required to own an iPod to use it, granted it does have a revenue generating effect. But to the point of them not being able to split time, they seem to do it just fine with their services that run on Windows and I doubt their overly saturated developer infrastructure would be taxed by taking on something that isn't a wild goose chase for once. Apple picks up and drops new ideas and avenues for products internally on a daily basis and after working with them for over 3 years this would really not be a detriment to dedicate some resources to something long term that's got 95% of the groundwork covered as it is. There is also co-development that already exists to handle their own stuff that does exist like this already to cover both sides of things. They "don't" store any of those features, the backend of the work is through their already optimized systems that don't cost any more to operate than they do now and they already have a capacity to (and do) handle continents worth of traffic that would be generated for the handshakes of device-to-device while the majority of the workload (data) is handed off to the service providers.
iMessages and FaceTime would both consume additional resources on Apple's servers that you seem to be ignoring as well. Who is going to pay for that?
Depends on the new Amazon Apple relationship, I would say that Apple Music will eventually be on the Fire Tablets since it's shockingly coming to the Echo.
I have Apple Music on my $40 Fire HD 8. I just sideloaded the apk. The UI isn't perfect, but it's as good as most Android tablet apps.
But, if Services is going to continue to take a more leading role- I agree that some cross platform work on their paid services is definitely in order (obviously, they feel the same... hence, this article).
Not sure why you (or anyone) would think offering their free services like iMessage & FaceTime to non Apple users would be a good idea. That's ridiculous.
Encouraging people to go ahead & use other hardware, give Apple $0 & 0¢, yet enjoy multiple free services that they bear the cost of??!!! Ummm... towards what end?
Quite intelligently, it seems like they're going a very different direction- sounds like their new video service will be 100% free, but ONLY available to people with Apple hardware.
As far as free services offered beyond the Apple ecosystem being a bad idea, for who exactly? For Apple, possibly. But how would that not be beneficial to anyone else to have a universalized system? iTunes is free and I don't see anyone complaining about how unfair that is for Apple's business.
iTunes was offered free so people could buy music and load it on their iPods.
If Apple spends time and money developing software like FaceTime and iMessage for other platforms, that's time and money they can't spend on software for people who buy their products. Also, Apple would have to support FaceTime and iMessage on other platforms, which would also cost money.